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The Origins of
Velocity Functions
Thomas M. Humphrey

L

ike any practical, policy-oriented discipline, monetary economics employs useful concepts long after their prototypes and originators are
forgotten. A case in point is the notion of a velocity function relating
money’s rate of turnover to its independent determining variables.
Most economists recognize Milton Friedman’s influential 1956 version of
the function. Written v = Y/M = v(rb , re , 1/PdP/dt, w, Y/P, u), it expresses income velocity as a function of bond interest rates, equity yields, expected
inflation, wealth, real income, and a catch-all taste-and-technology variable
that captures the impact of a myriad of influences on velocity, including degree
of monetization, spread of banking, proliferation of money substitutes, development of cash management practices, confidence in the future stability of the
economy and the like.
Many also are aware of Irving Fisher’s 1911 transactions velocity function, although few realize that it incorporates most of the same variables as
Friedman’s.1 On velocity’s interest rate determinant, Fisher writes: “Each person regulates his turnover” to avoid “waste of interest” (1963, p. 152). When
rates rise, cashholders “will avoid carrying too much” money thus prompting
a rise in velocity. On expected inflation, he says: “When . . . depreciation is
anticipated, there is a tendency among owners of money to spend it speedily
. . . the result being to raise prices by increasing the velocity of circulation”
(p. 263). And on real income: “The rich have a higher rate of turnover than
the poor. They spend money faster, not only absolutely but relatively to the
money they keep on hand. . . . We may therefore infer that, if a nation grows
richer per capita, the velocity of circulation of money will increase” (p. 167).
Finally, with respect to the catch-all variable, Fisher cites all of the following
1 Among

the few is Boris Pesek (1976, pp. 857–58) who notes that Fisher’s velocity function
contains more variables than Friedman’s.

Federal Reserve Bank of Richmond Economic Quarterly Volume 79/4 Fall 1993

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Federal Reserve Bank of Richmond Economic Quarterly

as affecting velocity: “habits as to thrift and hoarding,” “book credit,” “use
of checks,” “frequency and regularity of receipts and payments,” “density of
population,” and “extent and speed of transportation” (p. 79). A comprehensive
list indeed.
The purpose of this article, however, is not to evaluate Friedman’s and
Fisher’s velocity functions. Rather it is to correct the impression that such
functions begin with Fisher. Thus J. S. Cramer (1992), in his authoritative
article “Velocity of Circulation” in Volume 3 of the New Palgrave Dictionary
of Money and Finance, traces the concept to the equation of exchange “which
is due to Irving Fisher” (p. 757). Countless macro and money and banking
textbooks echo this view. Even Merton Miller and Charles Upton’s 1975 classic Macroeconomics: A Neoclassical Introduction categorically asserts that the
term “velocity of circulation” is “associated with Irving Fisher” (1986, p. 231).
Such statements overlook 250 years of monetary theorizing. For, as demonstrated below, the notion of a functional relationship between velocity and its
determinants dates from the middle of the seventeenth century and received
frequent restatement throughout the eighteenth and nineteenth centuries before
being bequeathed to Fisher and his successors in the twentieth. Seen in this
perspective, Fisher emerges not as the originator of velocity functions but rather
as a particularly innovative recipient of them.
Before documenting the latter assertion, however, it should be noted that
one era’s velocity determinants become another’s money-stock components.
Changes in the definition of money ensure as much. Thus modern analysts
define money to include coin, paper currency, and deposits subject to check.
By contrast, most of the pre-Fisher velocity theorists discussed below defined
money as consisting solely of gold and silver coin. They excluded bank notes
and deposits on the ground that such instruments lack the unconditional power
of specie to settle final transactions and thus are not money per se but rather
devices to accelerate money’s velocity. Consequently, they saw note and deposit
expansions and contractions as velocity shifts rather than as money-stock shifts.
Their view may seem strange to the modern reader accustomed to regarding
notes and deposits as cash, but it was entirely consistent with their metallist
conception of money.

1.

THE FIRST VELOCITY FUNCTION

Sir William Petty (1623–1687) enunciated the first velocity function, albeit in
verbal rather than algebraic form, in his A Treatise of Taxes and Contributions (1662) and Verbum Sapienti (1664). He did so in an effort to estimate
the amount of money—defined by him as consisting solely of gold coin—
necessary to support the commercial activity of a nation. This amount he saw
as depending on velocity or its inverse, the ratio of money to trade. Unlike

T. M. Humphrey: Origins of Velocity Functions

3

writers such as John Briscoe (1696), who identified the requisite money stock
with national income and so assumed a ratio of unity, Petty treated it as a
fractional magnitude.2 His pathbreaking statistical studies of the economies of
Ireland and England had convinced him that the money stock was but a small
part of national expenditure, which meant that a velocity coefficient greater
than 1 existed to adjust money to the needs of trade. Here is the origin of the
notion of velocity as the multiplier that equates the stock of money with the
flow of income.3
Petty’s statistical studies also suggested certain institutional characteristics
that determine velocity. His function embodies these characteristics in the form
of five independent variables: (1) frequency of payments, (2) size of payments,
(3) income, (4) its distribution among socioeconomic classes, and (5) banking.
Of these variables, the first enters the function with a positive sign, reflecting Petty’s belief that the more frequently income recipients are paid, i.e.,
the shorter the pay period, the less cash per unit of income they need to hold
between paydays and so the higher is velocity. Illustrating this point, Petty
claimed that workers receiving wages once per week would spend a unit of
money an average of 52 times a year whereas landlords receiving rents quarterly
would spend the same monetary unit only 4 times per year.4
Unlike the payment-interval variable, Petty’s size-of-payments variable
bears a negative sign. He believed larger payments require a greater accumulation of cash in advance relative to income than do smaller payments. To
him, many small payments at short intervals spelled a higher velocity than did
a few large payments at long intervals.5
Like Fisher, Petty saw income entering the function with a positive sign.
“The most thriving men,” he said, “keep little . . . money by them, but turn
and wind it into various commodities to their great profit” (quoted in Marshall
[1923], p. 41). Evidently he believed that scale economies in cash holding
permit the rich to hold smaller balances in relation to their incomes than do the
poor so that velocity rises with incomes. Only such economies can explain why
2 On Briscoe’s formula money stock = national income, see Heckscher (1983), p. 224,
Schumpeter (1954), pp. 314–15, and Viner (1937), p. 42.
3 On Petty’s contribution see Holtrop (1929), Roncaglia (1985), Schumpeter (1954), and Wu
(1939).
4 Petty’s frequency-of-payments analysis launched a line of research leading to modern endogenous payment period models. See Grossman and Policano (1975) and the references cited
there. In such models, rises in the cost of holding cash induce agents to shorten the pay interval
and increase velocity. Such was the case in the German hyperinflation of 1923 when employers,
to avoid the astronomical depreciation cost of holding marks to meet the wage bill, started paying
workers daily rather than weekly.
5 Recent theorizing on this point tends to support Petty. Thus Grossman and Policano (1975)
model the case where households purchase some goods more frequently, and other goods less
frequently, than they receive income. The model predicts that velocity will rise with purchasers’
holdings of the first class of goods and fall with holdings of the second.

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Federal Reserve Bank of Richmond Economic Quarterly

his “thriving men” hold so little money and spend it so fast. Certainly he did
not see money as a luxury good whose velocity varies inversely with income.
Pierre Boisguilbert (1704) enunciated the luxury-good hypothesis when, in his
Dissertation de la nature des richesses, he declared that a coin spent by the
poor has a velocity “a hundred times more” than one spent by the rich “in
whose coffers large sums of money may remain useless for months and whole
years at a time” (quoted in Hutchison [1988], p. 110).
The fourth velocity-determining variable in Petty’s function is the distribution of income across socioeconomic classes. Because workers, landlords,
and other income recipients have different pay periods, their transaction needs
for cash per unit of income and thus their velocities differ. The economywide
aggregate velocity figure, being a weighted average of the various velocities
of the income groups, obviously depends on relative shares and the fraction
of the money stock each group commands. Indeed, as mentioned below, Petty
employed such weights to estimate aggregate velocity.
As for banks, Petty saw them as speeding up velocity. “Where there are
banks,” he wrote, “less money is necessary to drive a trade” (quoted in Wu
[1939], p. 37). In his view, banks economize on the use of money—that is, gold
coin—by issuing money substitutes in the form of notes. The notes effectuate
transactions formerly mediated by gold, thus freeing the latter for other uses.
With less money required to circulate trade, the velocity of the remaining stock
increases. It is easy to understand why Petty regarded the spread of banks as a
form of technological progress. Banks saved on scarce metallic reserves, thus
enabling a given volume of transactions to be supported by a smaller gold stock
or a larger volume of transactions by a given stock. “A bank,” he wrote, “doth
almost double the effect of our coined money” (quoted in Spengler [1954], p.
415). In doing so, banks helped reduce the real resource cost of effecting the
nation’s business.
Having specified velocity’s determinants, Petty used the velocity concept,
together with the exchange identity MV = Y, to estimate the minimum amount
of money required to finance a given volume of income and trade. Assuming a national income of £40 million, he reckoned that, if money traveled a
weekly circuit from employers to workers and back, annual velocity would
be 52, thus rendering a money stock of £40/52 million sufficient to meet the
needs of trade. If, instead, the income circuit involved quarterly rent and tax
payments alone, then velocity would be 4. In this case, a money stock of £10
million would be required to accommodate trade. Finally, if money had to
traverse both circuits at once, aggregate velocity V, the average of the individual circuit velocities V1 and V2 weighted by their circuit money shares
M1 /(M1 + M2 ) and M2 /(M1 + M2 ) would be roughly 7.5 and the corresponding required money stock M would total approximately £5.5 million. That is,
V = V1 [M1 /(M1 + M2 )] + V2 [M2 /(M1 + M2 )] = 52{[40m/52]/[(40m/52) +
10m]} + 4{10m/[(40m/52) + 10m]} = 7.429 ≈ 7.5 and M = Y/V = 40m/7.5 =

T. M. Humphrey: Origins of Velocity Functions

5

£5.333 million ≈ £5.5 million.6 In still another calculation, Petty, using agricultural income as a proxy for national income, estimated money’s annual income
velocity to be 10.

2.

JOHN LOCKE’S FUNCTION

John Locke’s (1632–1704) place in the history of velocity theory is secured by
three contributions.7 He was the first to explicitly relate velocity functions to
the underlying money demand functions of cashholders, a point barely hinted
at by Petty. In his Some Considerations of the Consequences of the Lowering
of Interest and Raising the Value of Money (1691) he sought to “consider
how much money [defined by him as gold coin] it is necessary to suppose
must rest constantly in each man’s hand, as requisite to the carrying on of
trade” (quoted in Vickers [1959], p. 58). For laborers and their employers, he
estimated this amount to be a fiftieth part of wages, for brokers (i.e., merchants
and tradesmen) a twentieth part of their annual returns, and for landlords and
their tenants one-fourth of the yearly revenue of land. Elsewhere, however, he
halved these requisite amounts, presumably on the grounds that credit could
substitute for money in driving trade.
Second, while retaining Petty’s income, pay period, and distributional arguments, he introduced a new variable, the interest rate, into the velocity function.
He viewed the interest rate as measuring the opportunity cost of holding money,
a noninterest-earning asset, instead of assets yielding an explicit rate of return.
A fall in the rate, he argued, lowers the cost of holding idle balances. In so
doing it increases the quantity of such balances demanded. As a result, bankers
and other monied men are, in his words, “content to have more money lie dead
by them” when rates fall (quoted in Holtrop [1929], p. 506). The consequent
rise in the quantity of money held per unit of income lowers velocity.
Motivating Locke’s analysis of velocity’s interest rate determinant was his
strong opposition to contemporary English proposals for a legal 4 percent interest rate ceiling. As noted by Leigh (1974), he feared that the imposition of
a below-equilibrium rate would depress output and employment in two ways.
First, it would deprive the country of the money needed to drive trade. By
precipitating capital outflows financed by corresponding drains of gold, the
artificially low rate would create a shortage of money as investors moved their
funds abroad to realize higher foreign yields. Second, it would lower velocity by
reducing the cost of holding idle balances in the manner described above. Together, the velocity and money-stock reductions would constitute a contraction
6 Petty actually expressed the required money stock M as half the sum of the individual
circuit stocks M1 and M2 . That his expression is equivalent to the ratio of income to aggregate velocity M = Y/V can be seen by substituting into the latter equation his assumptions
V = (M1 V1 + M2 V2 )/(M1 + M2 ) and M1 V1 = M2 V2 = Y to obtain M = (M1 + M2 )/2.
7 On Locke, see Holtrop (1929), Leigh (1974), and Vickers (1959).

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Federal Reserve Bank of Richmond Economic Quarterly

of aggregate demand. With English prices imperfectly flexible or exogenously
given from world markets by purchasing power parity considerations, the aggregate demand contractions would cause corresponding contractions in real
activity. For this reason, Locke advocated removal of rate ceilings so that
money, velocity, spending, output, and employment could return to their equilibrium levels.
Third, Locke said that velocity could be speeded up if there were fewer
middlemen standing between producers and consumers. Here is the origin of
the notion that velocity varies inversely with the number of stages of production
separating raw materials from finished product and so increases with the degree
of vertical integration.
Like Petty, Locke regarded velocity increases as beneficial. Such increases
either reduced the quantity of money required to support a given volume of
trade or raised the volume of trade that could be supported by a given stock of
money. To this end, he recommended a shortening of pay periods. By enhancing velocity, such shortening would be “better for trade, and consequently for
everybody (for more money would be stirring and less would be necessary to
do the business)” (quoted in Hutchison [1988], p. 65). He failed, however, to
note the equivalence of velocity increases and money-stock increases in raising the price level in a closed economy. Not until 1755 was this equivalence
articulated in published form. And the first economist to do so was Richard
Cantillon (1680–1734), the foremost velocity theorist of the eighteenth century.

3.

RICHARD CANTILLON

The prize for introducing the largest number of variables into an eighteenthcentury velocity function goes to Cantillon.8 Certainly his function, as presented
in his 1755 Essai sur la nature du commerce en g´ n´ ral, was the most elaborate
e e
to be found in the literature of that era. As the premier economist of his day,
he possessed a profound understanding of the real forces shaping velocity.
And as a banker and foreign exchange specialist who amassed two fortunes
speculating on the South Sea Bubble and Mississippi System schemes, he also
had a keen appreciation of the monetary and financial forces involved. Some
of these forces—urbanization, monetization, growing financial sophistication,
advent of new credit facilities and the like—pertained to France’s emerging
transition from a predominantly agricultural economy to a mercantile and manufacturing one. Others were stressed by his predecessors, Petty and Locke,
whose metallist conception of money he also shared. All were assimilated into
Cantillon’s velocity analysis. Thus his velocity function contains the following arguments: (1) income, (2) frequency of payments, (3) size of payments,
(4) stages of production, (5) interest rates, (6) distribution among social classes,
8 On

Cantillon, see Bordo (1983) and Murphy (1986).

T. M. Humphrey: Origins of Velocity Functions

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(7) banking, (8) trade credit, (9) extent of barter, (10) urbanization (monetization), (11) hoarding, (12) uncertain expectations of the future, and (13) minimum denomination restrictions on asset purchases. Of these, the first seven he
took from Petty and Locke. The last six, however, were original with him.
According to Cantillon, urbanization, hoarding, uncertainty, and minimum
denomination restrictions all tend to reduce velocity. Trade credit and barter,
on the other hand, enlarge it. Urbanization—the growth of cities and towns—
expands the sphere of money transactions relative to barter transactions and
production for one’s own use. It does so because “all country produce is furnished by labour which may . . . be carried on with little or no actual money”
whereas “all merchandise is made in cities or market towns by the labour of
men who must be paid in actual money” (1964, p. 143). The resulting monetization of economic activity boosts the demand for cash per unit of income
so that velocity falls. Hoarding likewise slows velocity as “many miserly and
timid people bury and hoard cash for considerable periods” (p. 147). Similarly, uncertainty induces people to “keep some cash in their pockets or safes
against unforeseen emergencies and not to be run out of money” (p. 147). The
consequent rise in the precautionary demand for cash lowers velocity. Finally,
minimum denomination restrictions, which establish lower limits or floors to
the size of asset purchases, retard velocity by compelling agents to “keep out
of circulation small amounts of cash until they have enough to invest at interest
or profit” (p. 147).
Working in the opposite direction is the use of trade credit, clearing arrangements, and other substitutes for money.9 These items, by allowing businessmen
to dispense with money in financing ongoing commercial transactions and by
permitting them to cancel claims against each other so that only net balances
need be paid, “seem to economize much cash in circulation, or at least to
accelerate its movement” (p. 141). Thus it “is not without reason that it is
commonly said that commercial credit makes money less scarce.” The same is
true of barter which likewise reduces the need for cash and so raises velocity.
Taking these factors into account, Cantillon estimated income velocity to
be 9. With all determinants of money demand considered, he calculated that
a country’s money stock M should be one-third of landowners’ annual rent R.
Since he reckoned rent to constitute one-third of the value of annual produce
Y, he obtained velocity V as V = (R/M)(Y/R) = Y/M = 3×3 = 9. His estimate,
which like Petty’s used farm income as a proxy for national income, was close
to Petty’s estimate of 10.

9 Cantillon’s analysis thus implies a U-shaped pattern over time for velocity in developing
economies. At first, increasing monetization causes velocity to fall. Thereafter, increasing financial sophistication and the growth of money substitutes cause velocity to rise. Recent work in the
Cantillon tradition offers strong empirical support for this hypothesis. See the studies of Ireland
(1991) and Bordo and Jonung (1987).

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Federal Reserve Bank of Richmond Economic Quarterly

4. INTRODUCTION OF INFLATIONARY EXPECTATIONS
INTO THE FUNCTION
Cantillon, in his list of velocity determinants, had neglected to include inflationary expectations. This step was taken in the first three decades of the
nineteenth century by economists who had witnessed the debacle of the French
assignats (1794–1796). This episode, Western Europe’s first hyperinflation, left
a lasting impression. It revealed that excessive monetary expansion, by generating anticipations of future inflation, could precipitate a flight from cash and
a corresponding rise in velocity such that prices would rise even faster than
the money stock. It was an easy task to incorporate this lesson into velocity
theory.
Henry Thornton (1760–1815), in his 1802 Paper Credit of Great Britain,
was the first to do so. He said that when cashholders extrapolate observed
current falls in the purchasing power of the currency into expected future falls,
the expectations themselves will speed up velocity and quicken the currency’s
depreciation (p. 108). Using this insight, he explained how the excessive issue
of French assignats had “operated on their credit, and became a very powerful
cause of their depreciation” (p. 233).
J. B. Say (1776–1832), in his 1803 Trait´ d’´ conomie politique, likewise
e e
attributed the assignats’ “prodigious” rate of turnover to cashholders’ attempts
to rid themselves of a depreciating currency as fast as possible.10 The same
point was made by Simonde de Sismondi (1773–1842) in his 1827 Nouveaux
principes. He contrasted (1) distrust of the future stability of the real economy
with (2) distrust of the future value of the currency. The first type of distrust,
he said, tends to lower velocity whereas the second type tends to raise it.11
But the most precise account of the impact of inflationary expectations on
velocity and thus on the inflation rate itself came from Nassau Senior (1790–
1864) in his 1830 Three Lectures on the Cost of Obtaining Money. Referring
to the depreciation of the assignats stemming from the loss of confidence in
their future value, he wrote: “The prices of commodities rose in proportion, not
merely to the existing depreciation [true of course by definition], but to the wellfounded apprehension of a still further depreciation” (quoted in Eshag [1963],
p. 16). The result of such perceptions of the likely future depreciation of the
currency was exactly what one would expect: “Everybody taxed his ingenuity
to find employment for a currency of which the value evaporated from hour to
hour. It was passed on as it was received, as if it burned everyone’s hands who
touched it” (quoted in Eshag, p. 16). After Senior’s exposition, it would be
hard indeed to claim that anticipated inflation had been left out of the velocity
function.
10 See
11 See

Holtrop (1929), p. 519.
Holtrop (1929), p. 520.

T. M. Humphrey: Origins of Velocity Functions

5.

9

OTHER NINETEENTH-CENTURY CONTRIBUTIONS

The preceding hardly begins to exhaust the wealth of pre-twentieth century
writing on velocity functions. Holtrop (1929, pp. 518–20) notes that in the
nineteenth century alone, writers John Stuart Mill, Thomas Tooke, Christian
von Schlozer, Heinrich Storch, Karl Heinrich Rau, and Johann Karl Rodbertus all discussed velocity functions. By far the most important contributions,
however, came from Henry Thornton and Knut Wicksell. Towering above the
rest, their pathbreaking work constitutes the peak achievement of velocityfunction analysis prior to Irving Fisher.
We have already met Thornton, the pioneer of inflation-expectations analysis. This contribution alone would warrant his mention in any survey of velocity
theory. But he contributed much more to the theory than merely introducing
an expectations argument into the velocity function. Advancing beyond his
predecessors, he defined the relevant monetary aggregate as the total stock
of circulating media rather than its narrow specie component. Moreover he
was the first to specify how two variables, namely (1) the composition of the
payments media and (2) the state of business confidence, influenced velocity.
He had observed how these variables operated to produce the velocity swings
of the turbulent 1790s and sought to correct the tendency of his predecessors to
neglect them. In addition, as a banker and financial expert who had connections
with correspondent banks throughout the country, he was particularly alert to
the fundamental changes occurring in the English credit mechanism (Hayek
[1939], p. 38). These changes, which included rapid growth in the number of
country banks, the increasing use of checks, the establishment of the London
Clearing House, and the emergence of the Bank of England as the central bank
and lender of last resort, induced him to extend Cantillon’s analysis of the
velocity-enhancing effects of financial innovation.
His first task was to show how the composition of the payments media
enters the velocity function. He argued that the total means of payment consists of coin, banknotes, and bills of exchange. Each circulates with a speed
that varies inversely with the opportunity cost of holding it. This cost is measured as the differential between the instrument’s own rate of return and the
prevailing market rate. The lower the own rate relative to the prevailing rate,
the greater the cost of holding the instrument and the stronger the incentive to
spend it instead. Thus coin and banknotes, which yield no interest, circulate
faster than interest-bearing bills of exchange. Add to this the fact that gold
coins are hoarded more than notes and so circulate more slowly than the latter
in times of panic and it becomes apparent that different instruments possess
different velocities. It follows that aggregate velocity, the weighted average of
the component velocities, depends on the composition of the payments media.
When that composition changes, so does aggregate velocity.

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Thornton next identified as a determinant of velocity the state of mercantile
confidence arising from general business and financial conditions. Confidence
refers to the certainty of agents’ beliefs that receipts will match expenditures,
thus obviating the need to hold emergency reserves. A high state of confidence produces a low demand for precautionary balances and a rapid velocity.
Conversely, a low state of confidence stemming from distrust and alarm produces a high demand for precautionary balances and a slow velocity. Thornton
summarizes:
A high state of confidence serves to quicken [money’s] circulation. . . . [It]
contributes to make men provide less amply against contingencies. At such
a time, they trust, that if the demand upon them for a payment, which is
now doubtful and contingent, should actually be made, they shall be able to
provide for it at the moment; and they are loth to . . . make the provision much
before the period at which it shall be wanted. When, on the contrary, a season
of distrust arises, prudence suggests, that the loss of interest arising from a
detention of notes for a few additional days should not be regarded. . . . Every
one fearing lest he should not have his notes ready when the day of payment should come, would endeavor to provide himself with them beforehand.
(1939, pp. 96–98)

Thornton concluded that no single money stock always supports the same level
of nominal activity. Since velocity fluctuates with the state of confidence, more
money is required to effect a given volume of transactions when confidence is
low than when it is high.
As for financial innovations, Thornton saw them as boosting money’s
turnover rate. He explained how the invention of the clearinghouse, with its
mutual cancellation of claims, economized on the amount of money required to
settle transactions. And he cited still other developments—correspondent banking arrangements, improved communications, and the like—that had the same
effect. Like Cantillon, he drew the conclusion that such devices economize on
the use of money and speed up velocity.
Thornton’s analysis of financial innovations influenced his contemporaries.
Classical quantity theorists, notably David Ricardo (1772–1823) and the authors
of the 1810 Bullion Report, endorsed it. But so too did anti-quantity theorists.
Thornton’s work initiated the notion that monetary contraction stimulates the
very financial innovation and compensating rise in velocity that offsets the initial monetary contraction. Indeed, nobody stated this idea better than Thornton
himself. Let such a contraction occur, he said, and the resulting “great limitation
of the number of bank notes would, therefore, lead . . . to some new modes
of economy in the use of the existing notes: the effect of which economy on
prices would be the same, in all respects, as that of the restoration of the usual
quantity of bank notes” (p. 119). Coming from a leading classical quantity
theorist, this was a startling admission indeed.

T. M. Humphrey: Origins of Velocity Functions

6.

11

KNUT WICKSELL

Thornton’s work illustrates the flourishing of velocity analysis at the century’s
beginning. Knut Wicksell’s work illustrates its vitality at the century’s end.
Thus Wicksell (1851–1926) devoted the entire 30-page Chapter 6 of his 1898
volume Interest and Prices to a discussion of the determinants of “The Velocity
of Circulation of Money.”12
He began by defining money as consisting solely of gold coin. His definition rules out notes and deposits, which he treated as credit instruments
that raise the “virtual velocity” of money. He explained that such instruments,
when used in payment, free an equivalent amount of coin to facilitate purchases
elsewhere. In so doing, they effect a virtual turnover of coin and thus raise the
velocity of money. Having made this point, he next defined velocity as the
inverse of the “average period of idleness” or “interval of rest” of coin. In so
doing, he evoked the notion of money demand as velocity’s reciprocal. Finally,
he identified at least five determinants of velocity.
The first is a transactions demand for cash to bridge the gap caused by
the lack of synchronization between receipts and expected payments. The
second consists of a precautionary demand to meet unexpected payments. Although Cantillon and Thornton had incorporated these demands into the velocity
function before Wicksell, they had not derived them from probability theory.
Wicksell, however, did so. Inspired by Francis Edgeworth’s (1888) application
of probability theory to banking, he argued that the frequency with which
cash shortfalls of various amounts are likely to occur could be described by a
probability distribution whose mean represents expected shortfalls and whose
dispersion or spread measures the risk that actual shortfalls will be larger than
expected.
For his dispersion parameter, Wicksell used a statistic called the probable
deviation. Equaling 0.6745 times the standard deviation, this statistic has the
following property. When positioned on both sides of the mean, it includes half
of the elements of the distribution. That is, half the elements lie within, and
half without, one probable deviation of the mean. It follows that cashholders
wishing to secure themselves against a 50–50 chance of an unexpected shortage
of cash will hold precautionary balances equivalent to one probable deviation.
And cashholders with still greater degrees of risk aversion will hold even more.
Wicksell explained:
Suppose that experience has shown that . . . the excess of payments over
simultaneous receipts . . . tends to oscillate from year to year about a certain
mean value, a. Let the “probable deviation” be b: this means that the odds are
even . . . in favor of the payments over the period in question lying between
12 On

24.

Wicksell’s velocity analysis, see Laidler (1991), pp. 123–29, and Uhr (1960), pp. 220–

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Federal Reserve Bank of Richmond Economic Quarterly
a + b and a − b. If the business man is satisfied with this so-called simple
margin of safety, he must have by him a cash holding of a + b. But if he
demands a greater degree of security against the possible exhaustion of his
till, his cash holding must of course be somewhat larger. With a cash holding
of as little as a + 2b, the betting on the total exhaustion of his till . . . would,
according to the laws of probability, be more than 9 to 1; with a cash holding
of a + 3b it would be more than 44 to 1; and with one of a + 5b it would be
more than 2600 to 1, i.e. the till would be exhausted only about once in three
thousand five hundred years. (1936, pp. 57–58)

From this analysis it follows that the distribution’s mean and probable deviation
parameters a and b constitute arguments of the velocity function.
Wicksell also entered into his velocity function what he called simple trade
credit between businessmen. This variable bears a positive sign since the availability of trade credit reduces the amount of cash businessmen need to hold
relative to income to finance regular recurring transactions. By far the most
important determinant, however, is “organized credit” involving the operations
of commercial banks. Banks, Wicksell argued, boost velocity by multiplying the
volume of credit instruments—notes and deposits—erected on a given money
base. Once created, the notes and deposits mediate additional exchanges. In so
doing, they raise the volume of transactions per unit of money (gold) and so
enhance velocity.
To illustrate how banks evolved to raise money’s efficiency in supporting
more transactions, Wicksell sketched the following hypothetical sequence of
events. First, the emergence of banks allows agents to dispense with money
(gold) holdings by converting them into credit instruments instead. The resulting flow of gold into banks continues until those institutions eventually hold
the entire stock of the precious metal as reserves.
At the same time, bankers discover that three considerations—(1) the regularity of chance or law of large numbers, (2) the interdependence of firms such
that payments of one set of bank customers are the receipts of another, and
(3) the practice of settling offsetting claims of different customers of the same
bank through bookkeeping transfers from one account to another rather than
through the use of money—permit them to operate with fractional reserves.
These same inducements spur banks to form clearinghouse associations. Scale
and settlement economies also provide incentives to consolidate the banking
system’s reserve holdings in a centrally located bank.
Together, these developments tend to reduce the fractional reserve ratio
to negligible proportions. The ensuing potentially unlimited expansion of the
stock of credit instruments mediates a much larger volume of trade than would
the gold itself if it were used directly in making payments. Here is the essence
of Wicksell’s doctrine that bank notes and deposits raise the “virtual” velocity
of gold reserves resting in bank vaults with an actual physical velocity of zero.

T. M. Humphrey: Origins of Velocity Functions

13

Finally, Wicksell saw velocity as a function of the difference between the
market (loan) and natural (equilibrium) rates of interest. In his famous cumulative process analysis of price-level movements, he argued that excesses of
the natural rate over the market rate produce corresponding excesses of desired
investment spending over desired saving. As a result, the demand for loanable funds to finance investment exceeds the amount of such funds voluntarily
supplied by savers. Banks supply the remainder through credit (i.e., note and
deposit) creation. The consequent rise in the volume of bank credit erected
on a given money stock constitutes a rise in the virtual velocity of that stock.
This rise in turn puts upward pressure on prices. Thus price-level movements
emanate from rate differentials—more precisely from natural rate movements
given the market rate—operating through the velocity function.
This conclusion—that velocity rises with the natural rate-market rate differential—is entirely the result of Wicksell’s definition of money to exclude notes
and deposits. Had he included those items in his definition, he would have seen
the rate differential as boosting the money stock rather than its velocity. This
point notwithstanding, he provided the most complete analysis of velocity and
its determinants since Thornton. His work is proof positive that a sophisticated
literature on the subject existed before Fisher.

7.

CONCLUSION

The preceding discussion has concentrated exclusively on major landmarks in
the evolution of velocity functions. In so doing, it has no doubt neglected other
milestones. For example, nothing was said about Alfred Marshall’s work on
velocity in the 1870s and 1880s. D. P. O’Brien (1981, pp. 58–59) notes that
Marshall (1824–1924) followed Thornton and the Bullion Report in attributing
velocity’s movements to fluctuations in the state of confidence and economic activity, to financial innovation and the growth of money substitutes, to technical
progress in production, and to changes in transportation, communications and
the like.13 Much like Wicksell, Marshall viewed bank deposits not as money
but rather as a device for economizing on its use and speeding up velocity.
Nor was anything said about Thomas Attwood’s 1817 distinction between
income velocity and transaction velocity. The distinction between the two velocity concepts is often traced to Arthur Cecil Pigou, who discussed it in his
1927 book Industrial Fluctuations. It originates, however, with Attwood, who
estimated income velocity at 4 and transaction velocity at 50 per annum.14
Nor was mention made of the pathbreaking 1895 statistical work of Pierre
des Essars. His cross-country time-series estimates of the deposit turnover rates
13 On
14 On

Marshall, see also Eshag (1963), pp. 2–18, and Whitaker (1975), pp. 172–73.
Attwood, see Marget (1938), p. 358.

14

Federal Reserve Bank of Richmond Economic Quarterly

of continental European banks for the period 1884–1894 anticipated all later
empirical work on velocity. In essence, he computed deposit velocity as the
ratio of bank debits to average balances in deposit accounts.15 Irving Fisher
(1963, pp. 63 and 87) cited his findings as evidence that population density
and anticipated inflation act to raise velocity.
Also unmentioned was E. W. Kemmerer’s 1907 attempt to verify the
Thornton-Marshall hypothesis that velocity varies directly with the state of
business confidence. Not the least of Kemmerer’s achievements was his construction, from data on business failure rates and the dollar liabilities of failed
firms, of an index of business distrust. Movements of the index, he thought,
accounted for corresponding movements in velocity.
Finally, nothing was said about early versions of the MV = Py equation
of exchange. The pre-Fisher literature boasts at least 14 such equations.16 All
contain at least one velocity variable and two contain separate velocity terms
for each component of the payments media.
Nevertheless, enough has been said to document the main contention of
the article, namely that velocity functions long predate Irving Fisher and his
recent counterparts. This is not to say, however, that older and modern versions
of the function are identical. On the contrary, modern versions tend to be stated
mathematically, often in the form of least-squares regression equations yielding
numerical estimates of the equation’s coefficients.17 By contrast, older versions
of the function tended to be expressed verbally rather than algebraically.
Still, the basic notion of a stable functional relationship between velocity
and its independent determining variables has remained unchanged since the
time of Petty. So too has the practice of specifying the function’s arguments.
Thus Petty’s successors in the eighteenth and nineteenth centuries completed
his list of velocity determinants and bequeathed it to twentieth-century writers.
Seen in this perspective, the work of Fisher, Friedman, and other modern velocity theorists constitutes the culmination of a long tradition rather than the
beginning of a new one.

15 On

Des Essars’ estimates, see Kemmerer (1907), pp. 115–16.
pre-Fisher versions of the equation of exchange, see Humphrey (1984) and the references cited there.
17 For examples, see Bordo and Jonung (1987), pp. 32–39, and Goldfeld (1973), p. 633.
16 On

T. M. Humphrey: Origins of Velocity Functions

15

REFERENCES
Attwood, T. Letter to the Right Honorable Nicholas Vansittart, on the creation
of money, and on its action upon national prosperity. Birmingham, 1817.
Boisguilbert, Pierre de. Dissertation de la nature des richesses, de l’argent et
des tributs. 1704.
Bordo, Michael D. “Some Aspects of the Monetary Economics of Richard
Cantillon,” Journal of Monetary Economics, vol. 12 (August 1983), pp.
235–58.
, and Lars Jonung. The Long-Run Behavior of the Velocity of
Circulation. Cambridge: Cambridge University Press, 1987.
Briscoe, John. A Discourse of Money. London, 1696.
Cantillon, Richard. Essai sur la nature du commerce en g´ n´ ral. 1755. Edited
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with English translation and other material by Henry Higgs, London:
Macmillan, 1931, reprinted, New York: A. M. Kelley, 1964.
Cramer, J. S. “Velocity of Circulation,” in Peter Newman, Murray Milgate, and
John Eatwell, eds., The New Palgrave Dictionary of Money and Finance,
Vol. 3. London: Macmillan, 1992.
Des Essars, Pierre. “La vitesse de la circulation de la monnaie,” Journal de la
Soci´ t´ de Statistique de Paris, vol. 36 (1895), pp. 143–51.
ee
Edgeworth, Francis Y. “The Mathematical Theory of Banking,” Journal of the
Royal Statistical Association, vol. 51 (March 1888), pp. 113–26.
Eshag, Eprime. From Marshall to Keynes; An Essay on the Monetary Theory
of the Cambridge School. Oxford: B. Blackwell, 1963.
Fisher, Irving. The Purchasing Power of Money. 1911. New and revised
edition, New York: Macmillan, 1913, reprinted, New York: A. M. Kelley,
1963.
Friedman, Milton. “The Quantity Theory of Money—A Restatement,” in
Milton Friedman, ed., Studies in the Quantity Theory of Money. Chicago:
University of Chicago Press, 1956.
Goldfeld, Stephen M. “The Demand for Money Revisited,” Brookings Papers
on Economic Activity, 3:1973, pp. 577–638.
Grossman, Herschel I., and Andrew J. Policano. “Money Balances, Commodity
Inventories, and Inflationary Expectations,” Journal of Political Economy,
vol. 83 (December 1975), pp. 1093–1112.
Hayek, Friedrich A. von. Introduction to H. Thornton, An Enquiry into the
Nature and Effects of the Paper Credit of Great Britain. 1802. Reprinted,
New York: Farrar & Rinehart, 1939.

16

Federal Reserve Bank of Richmond Economic Quarterly

Heckscher, Eli F. Mercantilism, Vol. 2. London: George Allen and Unwin,
1935. Revised 2d ed. translated by Mendel Shapiro and edited by E. F.
S¨ derlund, 1955, reprinted, New York and London: Garland Publishing,
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1983.
Holtrop, Marius W. “Theories of the Velocity of Circulation of Money in
Earlier Economic Literature,” Economic History (Supplement to the
Economic Journal), vol. 1 (January 1929), pp. 503–24.
Humphrey, Thomas M. “Algebraic Quantity Equations Before Fisher and
Pigou,” Federal Reserve Bank of Richmond Economic Review, vol. 70
(September/October 1984), pp. 13–22.
Hutchison, Terence W. Before Adam Smith: The Emergence of Political
Economy, 1662–1776. New York: B. Blackwell, 1988.
Ireland, Peter N. “Financial Evolution and the Long-Run Behavior of Velocity:
New Evidence from U.S. Regional Data,” Federal Reserve Bank of
Richmond Economic Review, vol. 77 (November/December 1991), pp.
16–26.
Kemmerer, Edwin W. Money and Credit Instruments in their Relation to
General Prices. New York: Henry Holt, 1907.
Laidler, David. The Golden Age of the Quantity Theory. New York: Philip
Allan, 1991.
Leigh, Arthur H. “John Locke and the Quantity Theory of Money,” History of
Political Economy, vol. 6 (Summer 1974), pp. 200–219.
Locke, John. Some Considerations of the Consequences of the Lowering of
Interest and Raising the Value of Money. 1691. In Vol. 5 of The Works of
John Locke, pp. 3–116. London, 1823, reprinted, Aalen (Germany), 1963.
Marget, Arthur W. The Theory of Prices, Vol. 1. New York: Prentice Hall,
1938.
Marshall, Alfred. Money, Credit and Commerce. London: Macmillan, 1923.
Miller, Merton H., and Charles W. Upton. Macroeconomics: A Neoclassical
Introduction. Chicago: University of Chicago Press, 1974 and 1986.
Murphy, Antoin E. Richard Cantillon: Entrepreneur and Economist. New
York: Oxford University Press, 1986.
O’Brien, Denis P. “A. Marshall, 1842–1924,” in Denis P. O’Brien and John
R. Presley, eds., Pioneers of Modern Economics in Britain. Totowa, N.J.:
Barnes and Noble Books, 1981.
Pesek, Boris P. “Monetary Theory in the Post-Robertson ‘Alice in Wonderland’
Era,” Journal of Economic Literature, vol. 14 (September 1976), pp. 856–
84.

T. M. Humphrey: Origins of Velocity Functions

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Petty, William. Treatise of Taxes and Contributions. 1662. In Charles H. Hull,
ed., The Economic Writings of Sir William Petty, Vol. 1. Cambridge, 1899,
reprinted, New York: A. M. Kelley, 1964.
. Verbum Sapienti. 1664. In Charles H. Hull, ed., The Economic
Writings of Sir William Petty, Vol. 1. Cambridge, 1899, reprinted, New
York: A. M. Kelley, 1964.
Pigou, Arthur C. Industrial Fluctuations. London: Macmillan, 1927.
Rodbertus, Johann K. Die Preussische Geldkrisis. 1845.
Roncaglia, Alessandro. Petty: The Origins of Political Economy. Armonk: M.
E. Sharpe, Inc., 1985.
Say, Jean-Baptiste. Trait´ d’´ conomie politique. Paris: Degerville, 1803.
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Schumpeter, Joseph A. History of Economic Analysis. New York: Oxford
University Press, 1954.
Senior, Nassau W. Three Lectures on the Cost of Obtaining Money. London:
John Murray, 1830.
Sismondi, Simonde de. Nouveaux principes d’´ conomie politique, 2d ed. Paris:
e
Delauny, 1827.
Spengler, Joseph J. “Richard Cantillon: First of the Moderns,” Journal of
Political Economy, vol. 62 (August and October 1954), pp. 281–95 and
406–24.
Thornton, Henry. An Enquiry into the Nature and Effects of the Paper Credit
of Great Britain. 1802. Edited with an introduction by F. A. von Hayek,
New York: Farrar & Rinehart, Inc., 1939.
Uhr, Carl G. Economic Doctrines of Knut Wicksell. Berkeley: University of
California Press, 1960.
Vickers, Douglas. Studies in the Theory of Money, 1690–1776. Philadelphia:
Chilton, 1959.
Viner, Jacob. Studies in the Theory of International Trade. New York: Harper
and Bros., 1937.
Whitaker, John K., ed. The Early Economic Writings of Alfred Marshall, 1867–
1890, Vol. I. New York: Free Press, 1975.
Wicksell, Knut. Interest and Prices. 1898. Translated by R. F. Kahn, with an
introduction by Bertil Ohlin, London: Macmillan, 1936.
Wu, Chi-Yuen. An Outline of International Price Theories. London: George
Routledge & Sons, 1939.

Public Investment and
Economic Growth
William E. Cullison

In the years following World War II, the papers of any major city . . . told daily
of the shortages and shortcomings in the elementary municipal and metropolitan services. The schools were old and overcrowded. The police force was
under strength and underpaid. The parks and playgrounds were insufficient.
Streets and empty lots were filthy, and the sanitation staff was underequipped
and in need of men. . . . Internal transportation was overcrowded, unhealthful,
and dirty. . . . The discussion of this public poverty competed, on the whole
successfully, with stories of ever-increasing opulence in privately produced
goods.

J. K. Galbraith (1958), p. 253

A

fter a lively debate in the late 1950s and early 1960s about the merits
of John Kenneth Galbraith’s theory of social balance (The Affluent
Society), the economics profession dismissed (or forgot) Galbraith’s
admonitions about the perils of neglecting the public infrastructure. David
Aschauer, however, rekindled a great deal of interest in the efficiency of
public capital spending by showing that additional spending by governments
for nondefense capital goods apparently had a very large positive effect on
private productivity and, hence, output.
Although economists were not surprised that public infrastructure spending could promote private output growth, the magnitude of the effect found
by Aschauer was startling to most. Aschauer estimated that additional public
capital spending would increase the output of private firms by more than 1 1/2
times as much as would an equivalent dollar increase in the firms’ own capital
stock.
A Congressional Budget Office (CBO) study of the effects of public infrastructure spending concluded that Aschauer’s results merited some skepticism
because “the statistical results are not robust [and] there is a lack of corroborating evidence” (CBO 1991, p. 25). The CBO observed that other empirical
The author would like to thank Peter Ireland, Christopher Otrok, and the Federal Reserve
Bank of Richmond editorial committee for helpful comments and suggestions. The views
expressed are those of the author and do not necessarily represent those of the Federal
Reserve Bank of Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 79/4 Fall 1993

19

20

Federal Reserve Bank of Richmond Economic Quarterly

research, including cost-benefit studies, found private output to be more responsive to investments in private capital than to investments in public capital.
There were a number of other studies in response to Aschauer.1 Some of the
studies found the effects of public investment on economic growth to be smaller
than Aschauer found them to be.
Alicia Munnell, formerly of the Federal Reserve Bank of Boston, tried
a different statistical approach to measuring the productivity of government
spending. Although Munnell (1990), like Aschauer, used a production function
approach to evaluate the effects of government infrastructure spending, she
approached the problem by estimating her production functions from crosssectional state-by-state data.
Munnell (1990) used estimates of gross state product and of private inputs of capital to develop estimates of public capital stocks for 48 states over
the 1970–86 time period. She then used the state-by-state data to estimate
the production functions, concluding that “the evidence seems overwhelming
that public capital has a positive impact on private output, investment, and
employment” (p. 94).
Munnell’s (1990) estimates of the relative effects of public investment were
smaller than those made by Aschauer. Hulten (1990), commenting on Munnell,
observed that her findings of smaller relative effects were consistent with other
studies that analyzed state data but that her findings differed sharply from the
results of studies that were based upon time series.
The CBO (1991), in summarizing the results of cost-benefit studies, noted
that there has been little support for the view that across-the-board increases in
public capital programs have remarkable effects on economic output. Rather,
they concluded that “cost-benefit analysis paints a fairly consistent picture of
high returns to maintaining the existing stock of physical infrastructure and to
expanding capacity in congested urban highways and runway traffic and air
traffic control at major airports” (p. 40).
Indeed, it seems clear that a sensible approach to spending for government
infrastructure would not include across-the-board increases in government investment spending as a means of stimulating economic growth. Rather, any
project should stand on its own merits and be able to withstand a cost-benefit
analysis. Given this caveat, however, there is interest in what sorts of public
investment spending would tend to have the most impact on economic growth.
If the most important sectors can be isolated, project proposals within those
categories can be given priority in setting governmental budgetary goals.
The types of government infrastructure spending evaluated by Aschauer
and Munnell and commented upon by the Congressional Budget Office fall

1 The

many evaluations of Aschauer’s results include Aaron (1990), Hulten (May 1990),
Hulten and Schwab (1991), Jorgenson (1991), Rubin (1991), and Tatom (1991).

W. E. Cullison: Public Investment and Economic Growth

21

into the category of physical capital investment, but government also invests in
its people. This latter type of investment produces human capital if it improves
the job skills (potential and actual productivity) of its citizens.
This article examines the effects on economic growth of government investment in both physical (nonhuman) and human capital, paying particular
attention to the relative social returns of investments in human capital. As noted
earlier, Aschauer, Munnell, and others use the aggregate production function approach to evaluating the effects of government spending. While such a method
works well for evaluating the effects of government spending for physical
capital, it is not clear that it is equally appropriate for human capital.
Investments in human capital may affect aggregate production possibilities
in ways that are far more complicated than investments in physical capital. In
the case of physical capital, it seems reasonable to assume that the government
stock of physical capital enters an aggregate production function in a manner
that is symmetric to, or at least quite similar to, private capital. It is far more
difficult to isolate, a priori, the role played by government spending for human
capital in an aggregate production function.
Fortunately, other statistical techniques are available to evaluate the effects of government spending on economic growth (see Cullison [1993]). The
methodologies used for this article are Granger-causality tests and simulations
from a vector autoregressive (VAR) model. These techniques also have the
advantage of requiring data only on investment flows rather than on stocks of
capital. The data on investment flows are readily available in disaggregated
form, thus facilitating the article’s research plan of evaluating the effects of
government spending by functional component.
The Granger-causality tests are used to determine what types of government
investment spending are correlated with economic growth. The VAR model is
a modified version of the model that Ireland and Otrok (1992) used to test the
effects on economic growth of reducing the federal debt by cutting defense
spending 20 percent over six years. The attractive feature of the VAR model
is that it is atheoretical, imposing no structure on the data. As a result, it is
not necessary to know exactly how government-provided human capital enters
into the aggregate production function.

1.

DATA ON GOVERNMENT SPENDING BY FUNCTION

The Department of Commerce publishes annual data on total government expenditures by function. The functions include the following: (1) expenditures
for central executive, legislative, and judicial activities; (2) international affairs;
(3) space; (4) national defense; (5) civilian safety; (6) education; (7) health and
hospitals; (8) income support, social security, and welfare; (9) veterans benefits and services; (10) housing and community activities; (11) recreational and
cultural activities; (12) energy; (13) agriculture; (14) natural resources; (15)

22

Federal Reserve Bank of Richmond Economic Quarterly

transportation; (16) postal service; (17) economic development, regulation, and
services; (18) labor training and services; (19) commercial activities; (20) net
interest paid; and (21) other.
When government investment is defined broadly, including both human
and nonhuman capital, some items in most of the 21 categories denoted above
probably would be classified as investment. Examples discussed below include
government expenditures for space, national defense, civilian safety, education,
health and hospitals, income support, veterans benefits, housing, agriculture,
transportation, economic development, labor training, and commercial activities.
Government spending for space and national defense are likely to result in
innovations useful for private production. In addition, much spending for space
and national defense is contracted from private business. Government spending
for civilian safety (police protection) provides an environment in which the
private economy can operate efficiently. Government spending for education
enhances human capital directly. One must at least be able to read, write, and
cipher to hold even menial jobs in the current job market. Higher education is
necessary to hold better jobs.
Government spending for health and hospitals also enhances human capital
by curing maladies and injuries that can impair the productivity of individuals
in the labor force. Income support programs such as aid to families can help
to keep families together so that the children can become productive members of the labor force. Veterans benefits can help veterans reenter society as
productive members by improving their physical and mental abilities. Housing
expenditures, by providing housing for those who otherwise might not be able
to afford it, can also enhance human capital by providing better-quality workers
as well as providing the homeless an entry into the labor force (by providing
them with an address).
Government spending for agriculture has for decades provided for basic
agricultural research through the land grant college system and other arms
of the Department of Agriculture. The fruits of such research are distributed
throughout the country by the county agricultural extension system. Government spending for transportation enhances the productivity of the private economy by providing roads and other methods of getting products from producers
to purchasers. Economic development programs can bring modern technology
to less developed areas of the United States, thus putting formerly underutilized resources to work. Labor training programs can enhance human capital by
improving the job skills of recipients of the program. Government commercial
activities increase GDP in and of themselves and provide job experience to the
work force.
Given that there are so many conceivable ways in which government spending can affect the private economy, this article will start by evaluating all
21 categories mentioned above to determine which actually had empirically

W. E. Cullison: Public Investment and Economic Growth

23

observable effects. Intuitively, it would seem that education, space, national
defense, civilian safety, transportation, agriculture and labor training would
have the more pronounced effects on the growth of the private economy. As a
preliminary procedure, a simplified version of the so-called Granger-causality
test is used to determine those categories of government spending that seem
most likely to have promoted economic growth.

2.

GRANGER-CAUSALITY TESTS

A Granger-causality test examines whether the variable to be tested adds
explanatory power to an existing relationship between one (or more) other
variable(s) and its (their) lags. For example, if Zt is a dependent variable and
Zt−1 is the variable lagged one period, then Zt = f (Zt−1 , vt ) would represent a
statistical relation between the two, when vt is some unknown source of variation in the functional relation between them. For the Granger test, a known
variable would be put into the functional relation of Zt and Zt−1 with various
lags and leads to determine whether it helped to reduce vt .
The Granger-causality tests and the VAR simulations reported in this article
are consistent in using only one lagged value of the relevant variables. The tests
are restricted to one lagged value because the short span of the available annual
data necessitates economizing on degrees of freedom—the shortage of degrees
of freedom being especially acute for the VAR analysis.
Table 1 shows the results of a Granger-causality test run on each of the
various classes of government expenditures. The equation used for the test is
∆ ln(Yt ) = a + b1 · ∆ ln(Yt−1 ) + b2 · ∆ ln(Xt−1 ),

(1)

where Y is private gross domestic product, X is the government spending variable to be tested, and a, b1 , and b2 are parameters to be estimated.2 The
notations “∆” and “ln” represent, respectively, one-year first differences and
natural logarithms, and the “t” subscripts are time indexes (in years). All variables are calculated in real (1987) dollars.
As the table shows, when X = ALL GOVERNMENT (total government
spending), the t-statistic for the coefficient b2 is 0.24, which is not statistically
significant. However, education spending and spending for labor training, both
of which enhance human capital, are statistically significant at the 5 percent
level. Spending for income support, agriculture, civilian safety and net interest
(negatively signed) are significant at the 15 percent level.3
2 All

estimations in this article use ordinary least squares (OLS).
spending was not statistically significant, according to the Granger-causality
tests. This result was somewhat surprising because the Finn analysis in this issue of the Economic
Quarterly found highway capital to have a significant, if imprecise, effect on productivity. Finn’s
analysis, however, deals with the stock of highway capital, while this article deals with the flow of
3 Transportation

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Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Granger-Causality Test Results, Government Purchases,
1955 to 1992
Equation: ∆ ln(Yt ) = a + b1 · ∆ ln(Yt−1 ) + b2 · ∆ ln(Xt−1 ),
where

Y = private gross domestic product in 1987 dollars,
X = government spending variable, measured in 1987 dollars, and
∆ = an operator designating the year-to-year first difference.
X Equals
ALL GOVERNMENT
AGRICULTURE
CIVILIAN SAFETY
COMMERCIAL ACTIVITY
ECON. DEVELOPMENT
EDUCATION
ENERGY
EXECUTIVE, LEGISLATIVE &
JUDICIAL
HEALTH & HOSPITAL
HOUSING
INCOME SUPPORT
INTERNATIONAL AFFAIRS
LABOR TRAINING
NATIONAL DEFENSE
NATURAL RESOURCES
NET INTEREST PAID
POSTAL SERVICE
RECREATION & CULTURE
SPACE*
TRANSPORTATION
VETERANS’ BENEFITS
OTHER

b2

“t” Value

Corrected R2

0.037
0.03
0.295
0.002
0.028
0.269
−0.011

0.24
1.87
1.64
1.25
0.85
2.33†
−0.35

0.00
0.05
0.03
0.00
0.00
0.10
0.00

0.03
−0.05
0.007
0.151
0.004
0.080
−0.039
0.018
−0.12
0.003
−0.007
0.020
0.080
0.094
0.082

0.26
−0.54
0.179
1.71
0.10
2.74†
−0.60
0.36
−1.69
0.28
−0.121
1.12
0.745
0.95
1.28

0.00
0.00
0.00
0.04
0.00
0.14
0.00
0.00
0.04
0.00
0.00
0.06
0.00
0.00
0.00

* The effects of space spending are estimated over the 1961–92 period because space spending
was zero in 1955–60.
† Statistically significant at the 5 percent level.

government transportation spending. The Finn article also uses quite different statistical methodology. In addition, the transportation spending category used in this article includes expenditures
for air, rail, water, and transit as well as highways. In deference to Finn’s results, however,
transportation spending was also examined with the VAR model, explained below. While the Fstatistic for transportation with one lag indicated that transportation had a significant effect on real
private GDP, the 95 percent confidence interval for the impulse-response function was practically
symmetrical around zero, indicating no clear direction of the resulting change in the level of real
private GDP.

W. E. Cullison: Public Investment and Economic Growth

25

Surprisingly, neither government spending for national defense nor space
spending had statistically significant effects on the growth of the real private
economy. In the case of space, the results may have been influenced by the
shorter span of available data (1961–92).4
The results of the tests shown in Table 1 lead to the conclusion that the
types of government spending most likely to have a statistically significant
effect on economic growth are education and labor training. Thus, the analysis
implies that the most efficient way to increase economic growth by increasing
government spending would be to channel expenditures to well-thought-out
education or labor training projects without ignoring projects in agriculture,
civilian safety, and income support and policies designed to reduce government interest payments. The analysis, however, gives little information about
the relative effectiveness of the different types of government spending. For
that, it is necessary to move to the simulations from the VAR model mentioned
earlier.

3.

SIMULATIONS FROM A VAR MODEL

The Ireland-Otrok VAR model can be modified to test the effects of various types of government spending on economic growth. Since the analysis in
Section 2, above, provides evidence that government expenditures for education and labor training have statistically significant impacts on private economic
growth, the analysis that follows will examine those variables. In addition and
for completeness, the economic effects of spending on agriculture, civilian
safety, and income support will also be considered.5
The following VAR model is estimated over the 1953–91 time period.
k

Bs · Xt−s + ut ,

Xt =

(2)

s=1

where
Xt = [RDEFt , GSFt , RDEBTt , M2t , Yt ].

(3)

RDEF is the growth rate of real defense spending, RDEBT is the growth rate
of real government debt, GSF is defined as the growth rates of the various
types of real government spending, Y is the growth rate of real private gross
domestic product, and M2 is the growth rate of money.
4 When space spending is combined with other government spending data and the resulting
sums are evaluated for Granger-causality, the addition of space spending usually improves the
statistical results. That cannot be said of defense spending, the addition of which usually lowers
the statistical significance of the resulting aggregate.
5 Since the Ireland-Otrok model includes a federal debt variable, net interest paid will not
be evaluated separately.

26

Federal Reserve Bank of Richmond Economic Quarterly

Empirical Results from the Model
Table 2 reports some results of estimating the system of equations with one-,
two-, and three-year lags. F-statistics were computed to evaluate the effects on
real private GDP growth of spending on education, labor training, agriculture,
income support, and civilian safety with one-, two-, and three-year lags. The
table shows agriculture not to have been a statistically significant factor at any
of the three lag lengths. The other four types of spending showed statistical
significance at 5.5 percent or less. For the subsequent analysis/forecasts from
the VAR, the lag length k = 1 was chosen to conserve degrees of freedom.
The estimates for the parameters of the model with one lag were used
to develop impulse-response functions outlining the effects on real economic
growth of cuts in defense spending and the federal debt and increases in the
government spending categories noted above. The cuts in defense spending and

Table 2 F-Statistics for Government Spending and
Real Private GDP, 1952 to 1991

Lags

Variable

F-Statistics for
Combined Lags

Significance
Levels

Degrees of
Freedom

(F-statistics calculated as a part of VAR system)
1
1
1
1
1
1

EDUCATION
LABOR TRAINING
AGRICULTURE
CIVILIAN SAFETY
INCOME SUPPORT
ED + L TRAIN + C SAF

20.86
12.69
0.26
3.98
6.61
27.00

0.00007*
0.001*
0.613
0.055*
0.015*
0.00001*

32
32
32
32
32
32

2
2
2
2
2
2

EDUCATION
LABOR TRAINING
AGRICULTURE
CIVILIAN SAFETY
INCOME SUPPORT
ED + L TRAIN + C SAF

7.76
2.46
0.94
0.42
1.69
7.88

0.002*
0.105
0.404
0.661
0.205
0.002*

26
26
26
26
26
26

3
3
3
3
3
3

EDUCATION
LABOR TRAINING
AGRICULTURE
CIVILIAN SAFETY
INCOME SUPPORT
ED + L TRAIN + C SAF

7.07
1.43
0.85
0.21
2.12
6.76

0.002*
0.263
0.484
0.891
0.130
0.002*

20
20
20
20
20
20

* Six percent or smaller probability that the variable’s effect on GDP growth was due to chance.
Note: All variables are in 1987 dollars and measured as changes in natural logarithms.

W. E. Cullison: Public Investment and Economic Growth

27

the federal debt are reported because the next step in the analysis will be to
perform a policy experiment similar to that done in the Ireland-Otrok study
(1992) in which both defense spending and the debt were reduced.6
Figures 1-A through 1-G depict impulse-response functions that show what
might happen to the level of real private GDP if there were a one-time onestandard-deviation shock to the growth rate of a particular type of government
spending. It is customary in the literature for the researcher to apply a shock
of the magnitude of one standard deviation of the variation in the series to
be tested. Limiting the shock to one standard deviation ensures that it will be
within the purview of the data from which the model is estimated.
Figure 1-A shows the effect of a one-time $7.95 billion (one standard
deviation of the growth rate) cut in defense spending, while Figure 1-B shows
the effect of a one-time $26.8 billion (one standard deviation of the growth rate)
reduction in the federal debt. The dotted lines represent 95 percent confidence
limits for the impulse-response predictions. Since the areas between the dotted
lines in each figure include 0.0, the results do not conclusively show even the
direction of the effect on real private GDP of cutting defense or the debt.
Figures 1-C through 1-G show the responses of real private GDP to onestandard-deviation shocks to spending for agriculture, civilian safety, education,
labor training, and income support. As might be expected, the magnitudes of
the one-standard-deviation shocks vary considerably. One standard deviation
for education spending, for example, is $3.1 billion in 1987 dollars, while one
standard deviation for labor training is only $0.6 billion. The magnitudes of
the standard deviations of the various series are reported in Table 3.
As the impulse-response figures show, shocks to spending for education
and labor training might be expected to result in a cumulative increase in the
level of real private GDP, an expectation predicted with 95 percent confidence.
The impulse-response analysis for income support payments, on the other hand,
not only showed the 95 percent confidence band to be practically symmetrical
around zero, but the prediction itself to be for no change in the level of real
GDP. Income support payments, therefore, were dropped from consideration
as possible sources of economic growth, while education and labor training
expenditures were considered likely sources worthy of further examination.
A Policy Experiment with the Model
In 1991, the Bush Administration presented a proposal entitled “The Future
Years Defense Program” (popularly known as the “1991 plan”) that called for a
20 percent reduction in real defense spending between 1992 and 1997. Ireland
6 For the purpose of generating the impulse-response functions, the ordering of the variables
assumes that policy decisions that change defense spending and the distribution of its proceeds
are made before contemporaneous values of money and output are observed.

28

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Response of Growth Rate of Private GDP
A. Real Defense Spending

E. Education Spending

10.0

10.0

7.5

7.5

Percent

12.5

Percent

12.5

5.0
2.5

5.0
2.5

0

0

-2.5

-2.5

-5.0

-5.0
1

5

10

15

1

5

Years

10

15

Years

B. Debt

F. Labor Training Spending

10.0

10.0

7.5

7.5

Percent

12.5

Percent

12.5

5.0
2.5

5.0
2.5

0

0

-2.5

-2.5

-5.0

1

5

10

15

-5.0

1

5

Years

10

15

Years

C. Agriculture Spending

G. Income Support Spending

10.0

10.0

7.5

7.5

Percent

12.5

Percent

12.5

5.0
2.5

5.0
2.5

0

0

-2.5

-2.5
-5.0

-5.0
1

5

10

15

1

5

10

Years

D. Civilian Safety Spending

15

Years

H. Education, Labor Training,
and Civilian Safety Spending
10.0

7.5

7.5

Percent

12.5

10.0

Percent

12.5

5.0
2.5

5.0
2.5

0

0

-2.5

-2.5

-5.0

+

-5.0
1

5

10
Years

15

1

5

10
Years

15

W. E. Cullison: Public Investment and Economic Growth

29

Table 3 Standard Deviations of the Growth Rates
of Selected Data Series, 1952 to 1991
Data Series
Standard Deviation
(converted into billions of 1987 dollars)
Agriculture
Civilian Safety
Education
Federal Debt
Income Support
Labor Training
National Defense

$ 2.8
0.6
3.1
26.7
9.5
0.6
8.0

and Otrok (1992) evaluated the 1991 plan with their VAR model. They found,
using data from 1931 to 1991, that implementation of the 1991 plan with the
proceeds going to federal debt reduction would be likely to reduce private GNP
in the short run but increase it slightly after 13 or more years.
As a complement to the Ireland-Otrok study, the policy experiment reported
here will also evaluate the 1991 plan. The new simulations, however, will assume that only a portion of the proceeds of the defense cuts are used for federal
spending reductions. The remainder will be used to raise government spending
on a specified function. An implicit assumption in the policy experiment is
that any new spending programs would be as cost-benefit effective as has been
average government spending for each function tested over the past 40 years.
Six simulations were made assuming the defense cutbacks of the 1991
plan, but with differing uses of the proceeds. The goal of the 1991 plan, recall,
was to cut defense spending by 20 percent between 1992 and 1997. In 1987
dollars, this meant cutbacks of $17 billion in 1992, $21.7 billion in 1993, $10.2
billion in 1994, $9.0 billion in 1995, $6.5 billion in 1996, and $7.0 billion in
1997.
The simulations distributed the proceeds of the defense cutbacks either (1)
all to federal debt reduction as in Figure 2-A or (2) a portion to a one-standarddeviation increase in one of the four types of government spending with the
remainder going to federal debt reduction (Figures 2-B, 2-C, 2-D, and 2-E).
The simulation with all of the proceeds of the defense cutbacks going to debt
reduction shows the resulting level of real private GDP to be a persistent 1.5
percent below what it would have been with no change in defense spending.7
7 This result differs from the result found by Ireland and Otrok using 1931–91 data and
their slightly different model. However, when their model was reestimated over the 1955–91 and
1947–91 time periods, the results were quite similar to the results found here.

30

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Forecasts of Difference in Output Between Base Case
and 1991 Plan
A. Debt

D. Education

2.5
2.0

2.5
2.0

1.5
1.5

Percent

Percent

1.0
0.5
0

-0.5

1.0
0.5
0

-1.0
-0.5

-1.5
-2.0
1992

96

2000

4
Year

8

12

-1.0
1992

96

B. Agriculture

4
Year

8

12

E. Labor Training

2.5
2.0

10.0
8.0

1.5

6.0

Percent

1.0

Percent

2000

0.5
0

4.0
2.0

-0.5
-1.0

0

-1.5
-2.0
1992

96

2000

4
Year

8

12

-2.0
1992

96

2000

4
Year

8

12

F. Education, Labor Training,
and Civilian Safety

C. Civilian Safety
3.0

4.0

2.5

3.0

2.0

Percent

Percent

2.0
1.0
0

1.5
1.0
0.5
0

-1.0
-2.0
1992

-0.5
96

2000

4
Year

8

12

-1.0
1992

96

2000

4
Year

8

12

+

As Figures 2-C, 2-D, and 2-E show, this outlook changes considerably
when a portion of the proceeds of the defense cuts are used to increase spending on civilian safety, education, or labor training. These simulations put the
level of real private GDP persistently above what it otherwise would have been
even though most of the proceeds of the defense cuts are still used to reduce

W. E. Cullison: Public Investment and Economic Growth

31

the federal debt. For example, the simulation channeling $3.1 billion per year
of the defense cuts (25 percent of the total defense reduction) to education
raises the level of real GDP 1.5 percent.
Surprisingly, the simulation with $0.6 billion per year of the defense cuts
going to labor training has the level of real private GDP rising a whopping 9
percent above what it otherwise would have been. The magnitude of this result
is not credible. It probably indicates that the model has been affected by some
kind of spurious correlation with respect to labor training, which is a very small
part (0.5 percent) of government spending.
The predicted effects of civilian safety spending also seem suspiciously
large. The simulation has $0.6 billion per year in additional spending for civilian safety raising the level of private GDP almost 3 percent higher than it
otherwise would have been. Given the error structure of the impulse-response
function depicted in Figure 1-D, however, the forecast errors on the civilian
safety simulation would undoubtedly be relatively large, were they available.8
The policy experiment was run with a variable that combined government
spending for civilian safety, education, and labor training (Figure 2-F). The
simulation using this variable, which was significant for the F-test reported in
Table 2, and which had an impulse-response function (Figure 1-H) that was
significantly greater than zero, predicts that the level of real private GDP will
be persistently 1.8 percent larger with the policy experiment than without it.
The increase in real GDP comes about as a result of $3.47 billion per year
apportioned among civilian safety, labor training, and education during the
years of the defense cuts. Over the six-year period, this experiment results
in a cumulative $20.8 billion increase in civilian safety, education, and labor
training and a $50.6 billion reduction in the federal debt.

4.

CONCLUSIONS AND POLICY IMPLICATIONS

First some caveats. The analysis in this article uses past data to simulate future
events. Although that approach is the only one available for empirical studies,
it is always subject to question. One should have good reason to believe that
past trends will continue if one is to put much credence in simulations of the
type reported in this article. Moreover, while one can find certain correlations
between past events and guess that one event may cause another, it is virtually
impossible for an economist to prove that one economic occurrence in the real
world caused another. Thus, the results of this study cannot be considered to
8 The

effects of spending on labor training and civilian safety were examined further to find
whether or not they were likely to have been the result of reverse causation. Reverse Grangercausality tests were run to determine whether GDP determined labor training or civilian safety
spending. Lagged GDP did not have a statistically significant effect on either.

32

Federal Reserve Bank of Richmond Economic Quarterly

be conclusive.
The results of the study, however, imply that government spending on
education and labor training (and perhaps also civilian safety) have statistically
significant, and numerically significant, effects on future economic growth. It
is noteworthy that spending for education, civilian safety, and labor training
directly affect human capital rather than physical capital. The VAR simulations
with education, labor training, and civilian safety spending show effects so
strong, in fact, that policies to reduce defense spending 20 percent and apportion the proceeds between debt reduction and one or all of those three spending
types were estimated to result in higher levels of real private GDP than would
have resulted with no reductions in defense spending.
As noted above, however, the results reported here are based upon correlations of past events and the correlations may or may not continue in the
future. Thus, programs to increase government spending for, say, education or
labor training should not be undertaken willy-nilly, justified by the promotion
of economic growth. Rather, any such program should stand up to a cost-benefit
analysis and prove itself worthy on its own merits.

REFERENCES
Aaron, Henry J. “Discussion,” in Alicia H. Munnell, ed., Is There a Shortfall
in Public Capital Investment? Boston: Federal Reserve Bank of Boston,
1990, pp. 51–63.
Aschauer, David A. “Why Is Infrastructure Important?” in Alicia H. Munnell,
ed., Is There a Shortfall in Public Capital Investment? Boston: Federal
Reserve Bank of Boston, 1990, pp. 21–48.
. “Is Government Spending Stimulative?” Contemporary Policy
Issues, vol. 8 (October 1990), pp. 30–46.
. “Public Investment and Productivity Growth in the Group of
Seven,” Federal Reserve Bank of Chicago Economic Perspectives, vol. 13
(September/October 1989), pp. 17–25.
. “Does Public Capital Crowd Out Private Capital?” Journal of
Monetary Economics, vol. 24 (September 1989), pp. 171–88.
. “Is Public Expenditure Productive?” Journal of Monetary
Economics, vol. 23 (March 1989), pp. 177–200.
. “Tax Rates, Deficits, and Intertemporal Efficiency,” Public
Finance Quarterly, vol. 16 (July 1988), pp. 374–84.
. “Government Spending and the ‘Falling Rate of Profit,’ ” Federal
Reserve Bank of Chicago Economic Perspectives, vol. 12 (May/June
1988), pp. 11–17.

W. E. Cullison: Public Investment and Economic Growth

33

Cullison, William E. “Saving Measures as Economic Growth Indicators,”
Contemporary Policy Issues, vol. XI (January 1993), pp. 1–8.
Hulten, Charles R. “Discussion,” in Alicia H. Munnell, ed., Is There a Shortfall
in Public Capital Investment? Boston: Federal Reserve Bank of Boston,
1990, pp. 104–7.
. “Infrastructure: Productivity, Growth, and Competitiveness.”
Hearing statement before the House Committee on Banking, Finance, and
Urban Affairs, Subcommittee on Policy Research and Insurance, May 8,
1990.
, and Robert M. Schwab. “Is There Too Little Public Capital?”
Paper presented at the American Enterprise Institute, Washington, D.C.,
February 1991.
Galbraith, John Kenneth. The Affluent Society. Boston: The Houghton Mifflin
Company, 1958.
Ireland, Peter, and Christopher Otrok. “Forecasting the Effects of Reduced
Defense Spending,” Federal Reserve Bank of Richmond Economic Review,
vol. 78 (November/December 1992), pp. 3–11.
Jorgenson, Dale W. “Fragile Statistical Foundations: The Macroeconomics
of Public Infrastructure Investment.” Paper presented at the American
Enterprise Institute, Washington, D.C., February 1991.
Munnell, Alicia H. “How Does Public Infrastructure Affect Regional Economic
Performance,” in Alicia H. Munnell, ed., Is There a Shortfall in Public
Capital Investment? Boston: Federal Reserve Bank of Boston, 1990, pp.
69–103.
Rubin, Laura S. “Productivity and the Public Capital Stock: Another Look,”
Economic Activity Section Working Paper Series, no. 118. Washington:
Board of Governors of the Federal Reserve System, 1991.
Tatom, John. “Public Capital and Private Sector Performance,” Federal Reserve
Bank of St. Louis Review, vol. 73 (May/June 1991), pp. 3–15.
U.S. Congress, Congressional Budget Office. How Federal Spending for Infrastructure and Other Public Investments Affects the Economy. Washington:
U.S. Government Printing Office, July 1991.

Unit Labor Costs
and the Price Level
Yash P. Mehra

A

popular theoretical model of the inflation process is the expectationsaugmented Phillips-curve model. According to this model, prices are
set as markup over productivity-adjusted labor costs, the latter being
determined by expected inflation and the degree of demand pressure.1 It is assumed further that expected inflation depends upon past inflation. This model
thus implies that productivity-adjusted wages and prices are causally related
with feedbacks running in both directions.
In this article, I investigate empirically the causal relationship between
prices and productivity-adjusted wages (measured by unit labor costs) using
cointegration and Granger-causation techniques.2 In my recent paper, Mehra
(1991), I used similar techniques3 to show that inflation and growth in unit

1 This

version has been closely associated with the work of Gordon (1982, 1985, 1988)
and differs from the original Phillips-curve model. The latter was formulated as a wage equation
relating wage inflation to the unemployment gap.
2 Let X , X , and X be three time series. Assume that the levels of these time series are
1t
2t
3t
nonstationary but first differences are not. Then these series are said to be cointegrated if there
exists a vector of constants (α1 , α2 , α3 ) such that Zt = α1 X1t + α2 X2t + α3 X3t is stationary.
The intuition behind this definition is that even if each time series is nonstationary, there might
exist linear combinations of such time series that are stationary. In that case, multiple time series
are said to be cointegrated and share some common stochastic trends. Moreover, if series are
cointegrated, then some series must adjust in the short run so as to maintain equilibrium among
multiple series. That implies the presence of short-run feedbacks (and hence Granger-causality)
among these series.
3 The statistical inference in most of the empirical work prior to Mehra (1991) has often been
conducted under the assumption that wage and price series contain deterministic trends. Recent
evidence has called this assumption into question and has shown that the trend components of
several of these time series also contain stochastic components (Nelson and Plosser 1982). A
misspecification of trend components can lead to incorrect tests of hypotheses. Mehra (1991)
therefore employed recent techniques to investigate trends in wage and price series and used the
analysis to determine the nature of causal structure between prices and unit labor costs.

Federal Reserve Bank of Richmond Economic Quarterly Volume 79/4 Fall 1993

35

36

Federal Reserve Bank of Richmond Economic Quarterly

labor costs are correlated in the long run and that the presence of this correlation appears to be due to Granger-causality running from inflation to growth in
unit labor costs, not the other way around. The results presented there indicate
that the “price markup” hypothesis is inconsistent with the data and that growth
in unit labor costs does not help predict the future inflation rate.
This article examines the robustness of the conclusions in Mehra (1991)
to changes in the measure of the price level, the sample period, and unit rootcointegration test procedures used there. In particular, the price series used in
Mehra (1991) is the fixed-weight GNP deflator that covered the sample period
1959Q1 to 1989Q3; the test for cointegration used is the two-step procedure
given originally in Engle and Granger (1987); and the stationarity of data is examined using Dickey-Fuller unit root tests. This article considers an additional
price measure, the consumer price index, which covers consumption goods and
services bought by urban consumers. In contrast, the implicit GNP deflator,
the other price measure used here, covers prices of consumption, investment,
government services, and net exports. Since the consumer price index is also
a widely watched measure of inflation pressures in the economy, the article
examines whether the causal relationships found between the general price
level and unit labor costs carry over to consumer prices.
In my earlier empirical work (1991), I used Dickey-Fuller unit root tests to
determine whether the relevant series contain stochastic or deterministic trends.
Recently, some authors including Dejong et al. (1992) have shown that DickeyFuller tests have low power in distinguishing between these two alternatives.
These studies suggest that economists should supplement unit root tests by tests
of trend stationarity. Thus, a series now is considered having a unit root if two
conditions are met: (1) the series has a unit root by Dickey-Fuller tests and
(2) it is not trend stationary by tests of trend stationarity. Furthermore, the test
for cointegration recently proposed by Johansen and Juselius (1990) overcomes
several pitfalls associated with the Engle-Granger test for cointegration.4 This
article employs these additional, refined cointegration-stationarity tests to determine the stationarity of data and to study the nature of the causal structure
between the general price level and unit labor costs.
4 The

Engle-Granger test for cointegration is implemented by regressing one series on the
other remaining series and then testing whether the residuals from that regression are stationary or
not. If the residuals are stationary, then the multiple time series are said to be cointegrated. This
test has several shortcomings: (1) the test results are sensitive to the particular series chosen as
the dependent variable; (2) the test cannot tell whether the number of cointegrating relationships
is one or more than one; and (3) tests of hypotheses in the cointegrating vectors cannot be carried
out because estimated coefficients have unknown nonstandard distributions. In contrast, the test
proposed in Johansen and Juselius (1990) does not have any of the aforementioned problems.
Their test procedure enables one to test directly for the number of cointegrating vectors and
provides at the same time the maximum likelihood estimates of the cointegrating vectors. Tests
of hypotheses in such estimated cointegrating vectors can be easily carried out. Lastly, the test
results are not sensitive to the particular normalization chosen.

Y. P. Mehra: Unit Labor Costs and the Price Level

37

The empirical evidence reported here indicates that wage and price series
contain stochastic, not deterministic, trends and that long-run movements in
prices are correlated with long-run movements in unit labor costs. That is, the
wage and price series used here are cointegrated as discussed in Engle and
Granger (1987). This result holds whether the particular price series used is the
implicit GDP deflator or the consumer price index.
Tests of Granger-causality presented here indicate that short-run movements in prices and unit labor costs are also correlated, with Granger-causality
running one way from prices to unit labor costs when the price series used is
the implicit GDP deflator. Test results with the consumer price index, however,
are consistent with the presence of bidirectional feedbacks between prices and
unit labor costs.
The empirical work here supports and extends the results in Mehra (1991).
Though the cointegration test procedures, the sample periods, and the general
price-level series used in these studies differ, both studies indicate that the
“price markup” hypothesis is inconsistent with the data when the price series
used measures the general price level. The additional results here, however, indicate such is not the case when the price series used is less broadly measured
by the consumer price index. Thus, movements in unit labor costs help predict
movements in consumer prices, but not in the general price level.
The plan of this article is as follows. Section 1 presents a Phillips-curve
model of the inflation process and discusses its implications for the relationship
between wages and prices. It also discusses how tests for cointegration and
Granger-causality can be used to examine such wage-price dynamics. Section 2
presents the empirical results, and Section 3 contains concluding observations.

1.

THE MODEL AND THE METHOD

The Phillips-Curve Model
The view that systematic movements in wages and prices are related derives
from the expectations-augmented Phillips-curve model of the inflation process.
Consider the price and wage equations that typically underlie such Phillipscurve models described in Gordon (1982, 1985) and Stockton and Glassman
(1987):
∆Pt = h0 + h1 ∆(w − q)t + h2 χt + h3 Spt

(1)

∆(w − q)t = k0 + k1 ∆Pe + k2 χt + k3 Swt
t

(2)

n

∆Pe
t

λj ∆Pt−j ,

=
j=1

(3)

38

Federal Reserve Bank of Richmond Economic Quarterly

where all variables are in natural logarithms and where Pt is the price level,
wt is the wage rate, qt is labor productivity, χt is a demand-pressure variable,
Pe is the expected price level, Spt represents supply shocks affecting the price
t
equation, Swt represents supply shocks affecting the wage equation, and ∆ is
the first-difference operator. Equation (1) describes the price markup behavior. Prices are marked up over productivity-adjusted labor costs (w − q) and
are influenced by cyclical demand (χ) and the exogenous relative price shocks
(Sp). This equation implies that productivity-adjusted wages determine the price
level, given demand pressure. Equation (2) is the wage equation. Wages are
assumed to be a function of cyclical demand (χ) and expected price level, the
latter modeled as a lag on past prices as in equation (3). The wage equation,
together with equation (3), implies that wages depend upon past prices, ceteris
paribus.5
The price and wage behavior described above suggests that long-run movements in wages and prices must be related. In fact, some formulations of (1)
and (2) predict that these two variables would grow at similar rates in the long
run.6 Furthermore, if one allows for short-run dynamics in such behavior, the
analysis presented above would also suggest that past changes in wages and
prices should contain useful information for predicting future changes in those
same variables, ceteris paribus. These implications can be examined easily
using tests for cointegration and Granger-causality between wage and price
series.
Tests for Cointegration and Granger-Causality
If wage and price series have stochastic trends that move together, then the
two time series should be cointegrated as discussed in Granger (1986). Thus,
the long-run comovement of wages and prices is examined using the test
for cointegration given in Johansen and Juselius (1990). The test procedure,
5 The

price and wage equations used here should be viewed as the reduced form equations.
Price behavior as characterized in equation (1) is based on a markup model of pricing by firms.
Nordhaus (1972) shows such pricing could be derived from optimizing behavior in which the
technology is characterized by a Cobb-Douglas production function. Gordon (1985), on the other
hand, derives a wage equation like (2) from an explicit model of labor demand and supply in
which the wage rate adjusts in response to any change in the size of the gap between the two.
6 For example, as indicated in footnote 5, the markup model of pricing behavior characterized
in equation (1) is consistent with optimizing behavior in which the technology is characterized by
a Cobb-Douglas production function. Given the additional assumptions of constant returns and
the constant relative price of capital, the production environment implies a long-term coefficient
of unity on unit labor costs in the price equation (1), h1 = 1. That result indicates that prices and
wages would grow at the same rate in the long run. Alternatively, the natural rate hypothesis, if
valid in the long run, would indicate that the sum of the coefficients on past prices in (2) should
n

λj = 1. That result also would indicate wages and prices grow at similar rates in

be one, k1
j=1

the long run.

Y. P. Mehra: Unit Labor Costs and the Price Level

39

denoted hereafter as the JJ procedure, consists of estimating a VAR model that
includes differences as well as levels of nonstationary time series. The matrix
of coefficients that appear on levels of these time series contains information
about the long-run properties of the model.
To explain the model, let Xt be a vector of time series on prices and wages.
The VAR model is
Xt = Π1 Xt−1 + Π2 Xt−2 + Πk Xt−k + t ,

(4)

where Πi , i = 1, . . . k, is a matrix of coefficients that appear on Xt−i . Under
the hypothesis that the series in Xt are difference stationary, it is convenient to
transform (4) in a way that it contains both levels and first differences of the
time series in Xt . That transformation is shown in (5).
∆Xt = Γ1 ∆Xt−1 + · · · + Γk−1 ∆Xt−k−1 + ΠXt−k + t ,

(5)

where Γi , i = 1, . . . k − 1, and Π are matrices of coefficients that appear on first
differences and levels of the time series in Xt . The component ΠXt−k in (5)
gives different linear combinations of levels of the time series in Xt . Thus, the
matrix Π contains information about the long-run properties of the model. When
the matrix’s rank7 is zero, equation (5) reduces to a VAR in first differences.
In that case, no series in Xt can be expressed as a linear combination of other
remaining series. That result indicates that there does not exist any long-run
relationship between the series in the VAR. On the other hand, if the rank of
Π is one, then there exists only one linearly independent combination of series
in Xt . That result indicates that there exists a unique, long-run (cointegrating)
relationship between the series. When the rank is greater than one, then there
is more than one cointegrating relationship among the elements of Xt .
Two test statistics can be used to evaluate the number of the cointegrating
relationships. The trace test examines the rank of Π matrix and the hypothesis
that rank (Π) ≤ r is tested, where r represents the number of cointegrating
vectors. The maximum eigenvalue test tests the null that the number of cointegrating vectors is r given the alternative of r + 1 vectors. The critical values
of these test statistics have been reported in Johansen and Juselius (1990).
Granger (1988) points out that if two series are cointegrated, then there
must be Granger-causation in at least one direction. Assume that the JJ test
procedure indicates that wage and price series are cointegrated and that the
estimated cointegrating relationship is
Pt = δ(w − q)t + U1t , δ > 0,
where U1 is the random disturbance term. Equation (5) then implies that there
exists an error-correction representation of price and wage series of the form
7 The

matrix.

rank of a matrix is the number of linearly independent columns (or rows) in that

40

Federal Reserve Bank of Richmond Economic Quarterly

k

k

∆Pt = a0 +

a2s ∆(w − q)t−s

a1s ∆Pt−s +
s=1

s=1

+λ1 [Pt−1 − δ(w − q)t−1 ] +
k

∆(w − q)t = b0 +

(6.1)

1t
k

b1s ∆(w − q)t−s +
s=1

b2s ∆Pt−s
s=1

+ λ2 [Pt−1 − δ(w − q)t−1 ] +

2t ,

(6.2)

where all variables are as defined before and where one of λ1 , λ2 = 0.8
Equation (6) indicates that whenever the price level Pt−1 deviates from the
long-run value δ(w − q)t−1 , then either prices or wages or both adjust so as to
keep these two series together in the long run. Lagged levels of the variables
now enter the VAR via the error-correction term Pt−1 − δ(w − q)t−1 . Test of the
hypothesis that wages do not Granger-cause prices is that all a2s = 0 and/or
λ1 = 0. Hence, the presence of Granger-causality is also examined by testing
whether one or both of λ1 , λ2 = 0.
Estimation and Tests of Hypotheses in Cointegrating Vectors
Suppose that the JJ test procedure indicates that price and wage series are cointegrated. In order to examine the nature of long-term correlations between price
and wage series, the cointegrating wage and price regressions are estimated
using the dynamic OLS procedure described in Stock and Watson (1993).9
The dynamic versions of these regressions are
k

Pt = a0 + a1 (w − q)t +

a2s ∆(w − q)t−s + U1t

(7.1)

s=−k
k

(w − q)t = b0 + b1 Pt +

b2s ∆Pt−s + U2t ,

(7.2)

s=−k

where all variables are as defined before and where U1 and U2 are random
disturbance terms. Equation (7) includes, in addition, past, current and future
values of first differences of the right-hand variables that appear in the cointegrating regression. Since the random disturbance terms, U1 and U2 , may be
8 If λ = λ = 0, then the matrix Π in (5) has a zero rank, indicating the absence of any
1
2
long-run relationship between wage and price series.
9 The JJ test procedure also provides maximum likelihood estimates of the cointegrating
price and wage regressions. These estimates, though superior asymptotically, do not behave well
in small samples. In contrast, Stock and Watson’s (1993) dynamic OLS behaves well in small
samples.

Y. P. Mehra: Unit Labor Costs and the Price Level

41

serially correlated, standard test statistics corrected for the presence of serial
correlation are used to test hypotheses in (7). Thus, wages are not significantly
correlated with the price level in the long run if the hypothesis a1 = 0 or
b1 = 0 is not rejected.
Testing for Unit Roots and Mean Stationarity
The cointegration test requires that the time series in Xt be integrated of order
one.10 That is, the data should be stationary in their first differences but not in
levels. To determine the order of integration, I use the test procedure suggested
by Dickey and Fuller (1979). In particular, the unit root tests are performed by
estimating the Augmented Dickey-Fuller regression of the form
k

yt = a0 + ρyt−1 +

a2s ∆yt−s + t ,

(8)

s=1

where yt is the pertinent variable; the random disturbance term; and k the
number of lagged first differences of yt necessary to make t serially uncorrelated. If ρ = 1, yt has a unit root. The null hypothesis ρ = 1 is tested using
the t-statistic. The lag length (k) used in tests is chosen using the procedure11
given in Hall (1990), as advocated by Campbell and Perron (1991).
The unit root tests in (8) test the null hypothesis of unit root against the
alternative that yt is mean stationary (the alternative is trend stationary if a
linear trend is included in [8]). Recently, some authors including DeJong et al.
(1992) have presented evidence that the Dickey-Fuller tests have low power in
distinguishing between the null and the alternative. These studies suggest that
in trying to decide whether the time series data are stationary or integrated, it
would also be useful to perform tests of the null hypothesis of mean stationarity
(or trend stationarity). Thus, tests of mean stationarity are performed using the
procedure advocated by Kwiatkowski, Phillips, Schmidt, and Shin (1992). The
test, hereafter denoted as the KPSS test, is implemented by calculating the test
statistic
1
nu = 2
ˆ
T

T

S2 /σ 2 (k),
t ˆ
t=1

t

where St =

ei , t = 1, 2, . . . T; et is the residual from the regression of yt
i=1

on an intercept; St is the partial sum of the residuals e; σ(k) is a consistent
ˆ
10 The

series is said to be integrated of order one if it is stationary in first differences.
procedure is to start with some upper bound on k, say k max, chosen a priori (eight
quarters here). Estimate the regression (8) with k set at k max. If the last included lag is significant
(using the standard normal asymptotic distribution), select k = k max. If not, reduce the order of
the estimated autoregression by one until the coefficient on the last included lag (on ∆y in [8])
is significant. If none is significant, select k = 0.
11 The

42

Federal Reserve Bank of Richmond Economic Quarterly

estimate of the long-run variance12 of y; and T is the sample size. The statistic
nu has a nonstandard distribution and its critical values have been provided by
ˆ
Kwiatkowski et al. (1992). The null hypothesis of mean stationarity is rejected
if nu is large. Thus, a time series yt is considered unit root nonstationary if
ˆ
the null hypothesis that yt has a unit root is not rejected by the Augmented
Dickey-Fuller test and the null hypothesis that it is mean stationary is rejected
by the KPSS test.

2.

EMPIRICAL RESULTS

This section presents empirical results. In particular, I examine the long- and
short-term interactions between wages and prices in a trivariable system consisting of the price level, productivity-adjusted wage, and a demand pressure
variable. The price level is measured either by the log of the implicit GDP
deflator (ln P) or by the log of the consumer price index (ln CPI); productivityadjusted wage by the log of the index of unit labor costs of the nonfarm business
sector (ln ULC); and demand pressure variable by the log of real over potential
GDP (denoted as GAP). Unit labor costs are measured as compensation per
hour divided by output per hour. Since supply shocks could have important
short-run effects on wages and prices, tests of Granger-causality are conducted
including some of these in the trivariable system. The supply shocks considered here include relative prices of energy and imports. Dummy variables for
the period of President Nixon’s wage and price controls and for the period
immediately following the wage and price controls are also included.13 The
data used are quarterly and cover the sample period 1955Q1 to 1992Q4.
Test Results for Unit Roots and Mean Stationarity
In order to determine first whether linear trend is present in the data, Table 1
presents t-statistics on constant and time variables from regressions of the form
12 The residual e is from the regression y = a + bTime + e . The variance of y is the
t
t
t
t
variance of the residuals from this regression and is estimated using the Newey and West’s (1987)
method as
T
T
T
2
1
e2 +
b(s, k)
et et−s ,
σ(k) =
ˆ
t
T
T
t=1

s=1

t=s+1

S
where T is the sample size; the weighing function b(s, k) = 1 + 1+k ; and k is the lag truncation parameter. The lag parameter was set at k = 8. For another simple description of the test
procedure, see Ireland (1993).
13 The relative price of energy is the ratio of the producer price index for fuels, petroleum,
and related products to the producer price index for all commodities, and the relative price of
imports is the ratio of the implicit deflator for imports to the implicit GNP deflator. The dummy
variable for the period of price controls is 1 for 1971Q3 to 1974Q1 and 0 otherwise. The dummy
variable for the period immediately following price controls is 1 for 1974Q2 to 1974Q4 and 0
otherwise. The data on prices, unit labor costs, and real GDP are from the Citibase data bank and
that on potential GDP from the Board of Governors of the Federal Reserve System.

Y. P. Mehra: Unit Labor Costs and the Price Level

43

k

∆Xt = a + bTimet +

Cs ∆Xt−s + Ut ,
s=1

where Xt is the pertinent variable; Ut a random disturbance term; and k the
number of lagged first differences of Xt needed to make Ut serially uncorrelated. If the t-statistic on the constant is large, then Xt has linear trend. In
addition, if the t-statistic on the time variable is large, then Xt has quadratic
trend. As can be seen, the t-statistics presented in Table 1 are not large for
ln P, ln CPI, ln ULC, and GAP, indicating that linear or quadratic trends are
not present in any of these time series. Hence, linear trend is not included in
tests of unit roots where the alternative hypothesis now is that of mean, not
trend, stationarity.

Table 1 Tests for Trends, Unit Roots, and Mean Stationarity

Series X

Constant Trend k
ln P
ln CPI
ln ULC
GAP
∆ ln P
∆ ln CPI
∆ ln ULC

Panel B

Panel A
t-statistics for a
Regression of ∆X on:

1.5
1.2
1.5
−0.4

.1
.1
.3
.2

3
8
3
8

Panel C
Tests for Mean
Stationarity

Tests for Unit Roots

ρ
ˆ
.99
1.00
.99
.93
.88
.88
.72

tp
ˆ
−0.3
−0.5
0.0
−2.8*
−2.3
−2.5
−3.5**

k

Confidence
Interval
for ρ

3
8
3
1
2
8
2

(1.00,
(1.00,
(0.99,
(0.83,
(0.88,
(0.85,
(0.77,

1.03)
1.03)
1.03)
1.00)
1.00)
1.01)
0.96)

nu
ˆ
1.28**
1.27**
1.19**
.23
.42
.50
.35

* Significant at the 10 percent level.
** Significant at the 5 percent level.
Notes: P is the implicit GNP deflator; CPI is the consumer price index; ULC is the unit labor
cost; and GAP is the logarithm of real GDP to potential GDP. ln is the natural logarithm and
∆ the first-difference operator. The sample period studied is 1955Q1–1992Q4. The t-statistics in
k

Panel A above are from regressions of the form ∆Xt = a0 + a1 TREND +

as ∆Xt−s , where X
s=1

is the pertinent series. ρ and t-statistics (tp ) for ρ = 1 in Panel B above are from the Augmented
ˆ
k

Dickey-Fuller regression of the form Xt = a0 + ρXt−1 +

as ∆Xt−s . The 5 and 10 percent
s=1

critical values for tp are −2.9 and −2.6. The number of lagged first differences (k) included in
ˆ
these regressions are chosen using the procedure given in Hall (1990), with maximum lags set
at eight quarters. The confidence interval for ρ is constructed using the procedure given in Stock
(1991).
The test statistic nu in Panel C above is the statistic that tests the null hypothesis that
ˆ
the pertinent series is mean stationary. The 5 and 10 percent critical values for nu given in
ˆ
Kwiatkowski et al. (1992) are .463 and .574.

44

Federal Reserve Bank of Richmond Economic Quarterly

Tests for unit roots and mean stationarity are also presented in Table 1.
As can be seen, the t-statistic (tp ) that tests the null hypothesis that a pertinent
ˆ
time series has a unit root is small for ln P, ln CPI, and ln ULC, but large for
GAP. On the other hand, the statistic nu that tests the null hypothesis that a
ˆ
pertinent time series is mean stationary is large for ln P, ln CPI, and ln ULC,
but small for GAP. These results indicate that the time series ln P, ln CPI, and
ln ULC have a unit root by the ADF test and are not mean stationary by the
KPSS test. The GAP variable, on the other hand, does not have a unit root
by the ADF test and is mean stationary by the KPSS test. Thus, the wage and
price series used here are nonstationary in levels, whereas the demand pressure
variable GAP is stationary in levels.
As indicated before, the series has a unit root if ρ = 1. Table 1 contains
estimates of ρ and their 95 percent confidence intervals.14 As can be seen, the
estimated intervals contain the value ρ = 1 and are very tight for ln P, ln CPI,
and ln ULC. In contrast, the estimated interval for ρ is fairly wide for the GAP
series (.83 to 1.0 for GAP vs. .99 to 1.03 for others). These results further
corroborate the evidence above that ln P, ln ULC, and ln CPI each have a unit
root whereas the GAP series does not.
The unit root and mean stationary tests using first differences of ln P, ln
CPI, and ln ULC are also presented in Table 1. The results here are mixed.
The inflation series, ∆ ln P and ∆ ln CPI, have a unit root by the ADF test but
are mean stationary by the KPSS test. The 95 percent confidence interval for ρ
is (.88, 1.0) for ∆ ln P and (.85, 1.0) for ∆ ln CPI. These confidence intervals
are quite wide, indicating that ρ could as well be below unity (say, ρ = .8) and
thus the inflation series could as well be stationary. The wage growth series
∆ ln ULC, on the other hand, does not have a unit root by the ADF test and is
mean stationary by the KPSS test. These results indicate that the wage growth
series is mean stationary. The empirical work presented hereafter also treats
the inflation series as mean stationary.15
Cointegration Test Results
The results presented in the previous section indicate that the price and wage
series used here have stochastic, not deterministic, trends. I now examine

14 The confidence interval for ρ is constructed using the procedure given in Stock (1991).
The 95 percent confidence interval provides the range which may contain the true value of ρ with
some probability (.95).
15 These results differ somewhat from those given in Mehra (1991). The unit root tests given
in Mehra (1991) were performed including a linear trend and indicated that gap, inflation (∆ ln P),
and growth in unit labor costs (∆ ln ULC) series are nonstationary. The additional test results—
such as those on linear trend, mean stationarity, and the confidence intervals for ρ—presented
here, however, indicate that inflation, the output gap, and growth in unit labor costs could as well
be stationary.

Y. P. Mehra: Unit Labor Costs and the Price Level

45

Table 2 Cointegration Test Results

System

ka

Trace Test

Maximum
Eigenvalue
Test

(ln P, ln ULC)

3

17.2*

17.2*

(ln CPI, ln ULC)

5

20.2**

19.8**

a

The lag length k was selected using the likelihood ratio test procedure described in footnote 16
of the text.
* Significant at the 10 percent level.
** Significant at the 5 percent level.
Notes: Trace and maximum eigenvalue tests are tests of the null hypothesis that there is no
cointegrating vector in the system. The 5 percent and 10 percent critical values are 17.8 and 15.6
for the Trace statistic and 14.6 and 12.8 for the maximum eigenvalue statistic. Critical values are
from Johansen and Juselius (1990).

whether there exists a long-run equilibrium relationship between ln P and ln
ULC or between ln CPI and ln ULC, using the test of cointegration.
Table 2 presents cointegration test results using the JJ procedure.16 As
can be seen, trace and maximum eigenvalue test statistics that test the null
that there is no cointegrating vector are large and significant, indicating that
the wage and price series are cointegrated. The cointegrating price and wage
regressions estimated using the dynamic OLS procedure are reported in Table
3. χ2 is the Chi-square statistic that tests the null hypothesis that the coefficient
1
on ln ULC in the price regression is zero. Similarly, χ2 tests the null that the
2
coefficient on ln P (or on ln CPI) is zero in the wage regression. As can be
seen, χ2 and χ2 take large values and are significant, indicating that prices and
1
2
wages are significantly correlated in the long run.17 Furthermore, the estimated
coefficients that appear on price and wage variables in these cointegrating regressions are positive and not far from unity. This indicates that wage and price
series may grow at similar rates in the long run.

16 The lag length parameter (k) for the VAR model was chosen using the likelihood ratio
test described in Sims (1980). In particular, the VAR model initially was estimated with k set
equal to a maximum of eight quarters. This unrestricted model was then tested against a restricted
model where k is reduced by one, using the likelihood ratio test. The lag length finally selected
in performing the JJ procedure is the one when the restricted model is rejected.
17 The relevant statistics have a Chi-square, not an F, distribution because standard errors
have been corrected for the presence of moving average serial correlation. The order of the moving
average correction was determined by examining the autocorrelation of the residuals at various
lags.

46

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Cointegrating Vectors; Dynamic OLS
Price Regressions

Wage Regressions

lnPt = −.29 + 1.03 ln ULCt
χ2 = 41.3
1

ln ULCt = .31 + .96 lnPt
χ2 = 83.9
2

ln CPIt = −.23 + 1.05 ln ULCt
χ2 = 111.1
1

ln ULCt = .23 + .94 ln CPIt
χ2 = 196.8
2

Notes: All regressions are estimated by the dynamic OLS procedure given in Stock and Watson
(1993), using eight leads and lags of first differences of the relevant right-hand side explanatory
variables. χ2 is the Chi-square statistic that tests the null hypothesis that ln ULC is not significant,
1
whereas χ2 tests the null that ln P or ln CPI is not significant. Both statistics are distributed Chi2
square with one degree of freedom. The standard errors in these regressions were corrected for
the presence of moving average serial correlation.

Granger-Causality Test Results
The presence and nature of short-term interactions between wage and price
series are investigated by estimating regressions of the form
k1

k2

∆ ln Pt = a0 +

a1s ∆ ln Pt−s +
s=1

a2s ∆ ln ULCt−s
s=1

k3

ˆ
a3s GAPt−s + λ1 Upt−1 +

+

(9)

1t

s=1
k1

∆ ln ULCt = b0 +

k2

b1s ∆ ln ULCt−s +
s=1

b2s ∆ ln Pt−s
s=1

k3

ˆ
b3s GAPt−s + λ2 Uwt−1 +

+

2t ,

(10)

s=1

where P is the price level measured either by the implicit GDP deflator or by the
ˆ
consumer price index; Up the residual from the cointegrating price regression;
ˆ the residual from the cointegrating wage regression;18 and k1, k2, and
Uw
k3 the lag lengths on various variables needed to make random disturbances
( 1 , 2 ) serially uncorrelated. Wages do not Granger-cause prices if all a2s = 0
and/or λ1 = 0, and prices do not Granger-cause wages if all b2s = 0 and/or
λ2 = 0.
18 The

appearance of error-correction terms in (9) and (10) follows directly from equation (5).
If wage and price series are cointegrated, then the term ΠXt−k in equation (5) captures coefficients
that appear on the linear combination of wage and price variables. That is also demonstrated in
equations (6.1) and (6.2).

Y. P. Mehra: Unit Labor Costs and the Price Level

47

Table 4 Error-Correction Coefficients and F Statistics for
Granger-Causality; Implicit GDP Deflator

Sample
Period
1956Q1–
1992Q4

1956Q1–
1979Q3

Statistics from Price Regressions
Lag Lengths
λ1
(k1, k2, k3)
(t-value) F1
d.f.

Statistics from Wage Regressions
λ2
χ2
(t-value)
F2
d.f.
w

.02(1.6)
.03(1.9)
−.01(0.4)
−.01(0.2)
−.00(0.1)

(4,0,0)
(8,0,0)
(4,4,4)
(8,8,8)
(7,0,0)a
(4,4,1)a

.03(1.5)
.04(1.7)
.04(0.9)
.05(0.9)
.03(1.6)

0.3
1.6
1.2

0.1
0.8

4,116
8,104
8,111

4,63
8,51

−.14(3.9)
−.17(4.5)
.05(1.1)
.06(1.2)

7.1**
1.9*

4,124
8,112

−.02(0.5)

22.4**

4,131

−.17(3.9)
−.18(4.1)
.06(1.0)
.14(1.3)

4.8**
0.8

4,71
8,59

.03(0.5)

(4,0,0)
(8,0,0)
(4,4,4)
(8,8,8)
(7,8,2)a
(0,4,1)a

7.3**

4,74

5.8

7.0

a

Lag lengths chosen using the procedure given in Hall (1990).
* Significant at the 10 percent level.
** Significant at the 5 percent level.
Notes: The price regressions are of the form
k1

∆ ln Pt = a0 +

k2

a1s ∆ ln Pt−s +
s=1

k3

a3s GAPt−s + λ1 U1t−1

a2s ∆ ln ULCt−s +
s=1

s=1

and wage regressions are of the form
k1

∆ ln ULCt = b0 +

k2

b1s ∆ ln ULCt−s +
s=1

k3

b3s GAPt−s + λ2 U2t−1 .

b2s ∆ ln Pt−s +
s=1

s=1

U1 is the residual from the cointegrating price regression and U2 from the cointegrating wage
regression, both reported in Table 3. F1 tests all a2s = 0, F2 tests all b2s = 0, and d.f. is the
degree-of-freedom parameter for the F statistic given in the relevant row. The price regressions
also included eight past values of the relative prices of energy and imports and dummies for
President Nixon’s price controls. The wage regressions included eight past values of the relative
price of imports and price control dummies. χ2 is the Lagrange multiplier test for the hypothesis
w
that eight lags of the relative price of energy do not enter the wage regression (the 5 percent
critical value is 15.5).

Tables 4 and 5 report estimates of λ1 and λ2 (with t-statistics in parentheses)
from regressions of the form (9) and (10). In Table 4 the price series used is
the implicit GDP deflator and in Table 5 it is the consumer price index. The
regressions are estimated using some arbitrarily chosen lag lengths (k1, k2, k3)
as well as those chosen on the basis of the procedure given in Hall (1990). In
addition, the results are presented for the subperiod 1956Q1 to 1979Q3. The
price regression (9) estimated here also included eight past values of relative

48

Federal Reserve Bank of Richmond Economic Quarterly

Table 5 Error-Correction Coefficients and F Statistics for
Granger-Causality; Consumer Price Index

Sample
Period

Statistics from Price Regressions
Lag Lengths
λ1
(k1, k2, k3)
(t-value)
F1
d.f.

1956Q1–
1992Q4

1956Q1–
1979Q3

(4,0,0)
(8,0,0)
(4,4,4)
(8,8,8)
(8,0,2)a
(0,1,1)a

−.05(2.2)
−.05(2.2)
−.05(2.1)
−.02(0.6)
−.05(2.7)

(4,0,0)
(8,0,0)
(4,4,4)
(8,8,8)
(8,4,1)a
(0,1,1)a

−.05(1.4)
−.04(1.4)
−.10(2.4)
.01(0.2)
.11(2.9)

0.70
1.58

1.4
1.5
2.2*

4,116
8,104
8,104

4,63
8,51
4,62

Statistics from Wage Regressions
λ2
χ2
(t-value)
F2
d.f.
w
−.10(1.8)
−.12(2.3)
−.04(0.8)
−.04(0.8)

4.4**
0.5

4,132
8,120

−.00(0.4)

68.7**

1,142

−.01(0.4)
−.24(2.5)
−.08(1.0)
−.15(1.1)

4.9**
1.1

4,63
8,51

−.05(0.8)

14.5**

1,73

18.1

Notes: See notes in Table 4. The regressions are estimated using the consumer price index.

prices of energy and imports, and “on” and “off” dummies for President Nixon’s
price controls. The wage regression (10) included, in addition, eight past values
of the relative price of imports and price control dummies. Coefficients for the
relative price of energy were not significant in such regressions.19
If we focus on results for the general price level presented in Table 4, it is
clear that λ1 is generally not statistically significant whereas λ2 is significant
(see t-statistics in Table 4). Moreover, other lags of ∆ ln ULC when included in
price regressions are not statistically significant, whereas other lags of ∆ ln P
when included in wage regressions are statistically significant (compare F1
and F2 statistics in Table 4). These results are consistent with the presence of
Granger-causality, not from wages to prices, but from prices to wages.
The results using consumer prices are somewhat different from those using
the general price level. As can be seen from Table 5, λ1 is generally statistically significant, even though other lags of ∆ ln ULC when included in price
regressions are not (see t- and F1 statistics in Table 5). On the other hand, λ2
19 Whether

or not some of these supply shocks enter price and wage regressions was first
tested using the Lagrange-multiplier (LM) test for omitted variables (Engle 1984). An LM test
for k omitted variables is constructed by regressing the equation’s residuals on both the original
regressors and on the set of omitted variables. If the omitted variables do not belong in the
equation, then multiplying the R2 statistic from this regression by the number of observations
will produce a statistic asymptotically distributed χ2 with k degrees of freedom.

Y. P. Mehra: Unit Labor Costs and the Price Level

49

or other lags of ∆ ln CPI when included in wage regressions are statistically
significant (see t- and F2 statistics in Table 5). These results are consistent with
the presence of Granger-causality between prices and wages with feedbacks in
both directions.

3. CONCLUDING OBSERVATIONS
A central proposition in the expectations-augmented Phillips-curve model of
the inflation process is that prices are marked up over productivity-adjusted
labor costs. If that proposition is correct, then long-run movements in prices
and labor costs must be correlated. Moreover, we should find that short-run
movements in labor costs help predict short-run movements in the price level.
The evidence reported here indicates that these implications are consistent with
the data when prices are narrowly measured by the consumer price index but
not when they are broadly defined by the implicit GDP deflator. For the latter
measure, short-run movements in labor costs have no predictive content for
the future price level. The general price level and unit labor costs are still
correlated in the long run. But the presence of this correlation appears to be
due to Granger-causality running from the general price level to labor costs,
not the other way around. Huh and Trehan (1992) report similar results using
business sector price and wage data.
The finding that consumer prices and unit labor costs are Granger-causal
with feedbacks in both directions differs from the one in Barth and Bennett
(1975), which found Granger-causality running one way from consumer prices
to wages. The empirical work in Barth and Bennett, however, does not test for
the presence of Granger-causality occurring via the error-correction term. If we
were to ignore this channel, other test results presented here are also consistent
with the presence of Granger-causality running one way from consumer prices
to wages (compare F1 and F2 statistics in Table 5).

REFERENCES
Barth, James R., and James T. Bennett. “Cost-Push Versus Demand-Pull
Inflation: Some Empirical Evidence,” Journal of Money, Credit, and
Banking, vol. 7 (August 1975), pp. 391–97.
Campbell, J. Y., and P. Perron. “Pitfalls and Opportunities: What Macroeconomists Should Know About Unit Roots,” in O. J. Blanchard and
S. Fischer, eds., NBER Macroeconomics Annual, 1991. Cambridge, Mass.:
MIT Press, 1991, pp. 141–200.

50

Federal Reserve Bank of Richmond Economic Quarterly

DeJong, David N., John C. Nankervis, N. E. Savin, and Charles H. Whiteman.
“Integration Versus Trend Stationarity in Time Series,” Econometrica, vol.
60 (March 1992), pp. 423–33.
Dickey, D. A., and W. A. Fuller. “Distribution of the Estimators for Autoregressive Time Series with a Unit Root,” Journal of the American
Statistical Association, vol. 74 (June 1979), pp. 427–31.
Engle, Robert F. “Wald, Likelihood Ratio and Lagrange Multiplier Tests
in Econometrics,” in Zvi Griliches and Michael D. Intriligator, eds.,
Handbook of Econometrics, Vol. 2. New York: Elsevier, 1984, pp.
775–826.
, and C. W. Granger. “Cointegration and Error-Correction: Representation, Estimation and Testing,” Econometrica, vol. 55 (March 1987),
pp. 251–76.
Fuller, W. A. Introduction to Statistical Time Series. New York: Wiley, 1976.
Gordon, Robert J. “The Role of Wages in the Inflation Process,” American
Economic Review, vol. 78 (May 1988, Papers and Proceedings), pp. 276–
83.
. “Understanding Inflation in the 1980s,” Brookings Papers on
Economic Activity, 1:1985, pp. 263–99.
. “Price Inertia and Policy Ineffectiveness in the United States,
1890–1980,” Journal of Political Economy, vol. 90 (December 1982), pp.
1087–1117.
Granger, C. W. J. “Some Recent Developments in a Concept of Causality,”
Journal of Econometrics, vol. 39 (September/October 1988), pp. 199–211.
. “Developments in the Study of Cointegrated Economic Variables,”
Oxford Bulletin of Economics and Statistics, vol. 48 (August 1986), pp.
213–28.
Hall, A. “Testing for a Unit Root in Time Series with Pretest Data Based
Model Selection.” Manuscript. North Carolina State University, 1990.
Huh, Chan Guk, and Bharat Trehan. “Modelling the Time Series Behavior of
the Aggregate Wage Rate,” Working Paper 92–04. San Francisco: Federal
Reserve Bank of San Francisco, 1992.
Ireland, Peter. “Price Stability Under Long-Run Monetary Targeting,” Federal
Reserve Bank of Richmond Economic Quarterly, vol. 79 (Winter 1993),
pp. 25–45.
Johansen, Soren, and Katarina Juselius. “Maximum Likelihood Estimation
and Inference on Cointegration—With Applications to the Demand for
Money,” Oxford Bulletin of Economics and Statistics, vol. 52 (May 1990),
pp. 169–210.

Y. P. Mehra: Unit Labor Costs and the Price Level

51

Kwiatkowski, Denis, Peter C. B. Phillips, Peter Schmidt, and Yongcheol Shin.
“Testing the Null Hypothesis of Stationarity Against the Alternative of a
Unit Root: How Sure Are We that Economic Time Series Have a Unit
Root?” Journal of Econometrics, vol. 54 (October-December 1992), pp.
159–78.
Mehra, Yash P. “Wage Growth and the Inflation Process: An Empirical Note,”
American Economic Review, vol. 81 (September 1991), pp. 931–37.
Nelson, Charles R., and Charles I. Plosser. “Trends and Random Walks in
Macroeconomic Time Series: Some Evidence and Implications,” Journal
of Monetary Economics, vol. 10 (September 1982), pp. 139–62.
Newey, Whitney K., and Kenneth D. West. “A Simple, Positive Semi-Definite,
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,”
Econometrica, vol. 55 (May 1987), pp. 703–8.
Nordhaus, William D. “Recent Developments in Price Dynamics,” in Otto
Eckstein, ed., The Econometrics of Price Determination. Washington:
Board of Governors of the Federal Reserve System, 1972.
Sims, Christopher A. “Macroeconomics and Reality,” Econometrica, vol. 48
(January 1980), pp. 1–49.
Stock, James H. “Confidence Intervals for the Largest Autoregressive Root in
U.S. Macroeconomic Time Series,” Journal of Monetary Economics, vol.
28 (December 1991), pp. 435–59.
, and Mark W. Watson. “A Simple Estimator of Cointegrating
Vectors in Higher Order Integrated Systems,” Econometrica, vol. 61 (July
1993), pp. 783–820.
Stockton, David J., and James E. Glassman. “An Evaluation of the Forecast
Performance of Alternative Models of Inflation,” Review of Economics
and Statistics, vol. 69 (February 1987), pp. 108–17.

Is All Government
Capital Productive?
Mary Finn

T

he early 1970s witnessed dramatic change in per-capita output and
labor-productivity growth rates in the United States. These growth rates
averaged 2.2 percent and 2.0 percent, respectively, for the 1950–1969
period compared to 1.3 percent and 0.8 percent for the 1970–1989 period.
Aschauer (1989) advances the idea that an important explanatory factor in
this productivity slowdown is the government’s stock of capital. Estimating a
production function that relates private sector output to private sector labor and
capital and to total government capital for the aggregate U.S. economy (1949–
1985), Aschauer finds the output elasticity of government capital to be 0.39.1
That is, for every 1 percent change in government capital, output responds by
0.39 percent. This productivity coefficient, coupled with the sharp fall in the
average growth rate of government capital from 4.1 percent for 1950–1970
to 1.6 percent for 1971–1985, constitutes the evidence underlying Aschauer’s
view.2
The Aschauer (1989) study is innovative and important. His evidence suggests that government capital plays a significant role in economic growth. His
findings are, however, surprising and somewhat unconvincing. The evidence
is surprising because the output elasticity of government capital is relatively
high and because government capital contains many different types of stocks
(e.g., museums, hospitals, airports, prisons, seawalls, and wildlife preservation
The author, an economist at the Federal Reserve Bank of Richmond, thanks William Cullison,
Tom Humphrey, Peter Ireland, Stacey Schreft, and Dawn Spinozza for helpful suggestions
and comments and Karen Myers for processing the article. Also, the author gratefully
acknowledges use of the GMM package written by Lars Hansen, John Heaton, and Masao
Ogaki. The analysis and conclusions are those of the author and do not necessarily reflect
the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
1 See Aschauer (1989), Table 1, equation (1.1). Also, more exactly, Aschauer (1989) estimates a generalized, constant-returns-to-scale, Cobb-Douglas production function. The estimation
method is least-squares and applies to the functional relationship in level form. The measure of
total government capital is the net, consolidated-government, nonmilitary, nonresidential stock.
2 These numbers are from Aschauer (1989), Table 7.

Federal Reserve Bank of Richmond Economic Quarterly Volume 79/4 Fall 1993

53

54

Federal Reserve Bank of Richmond Economic Quarterly

facilities), some of which are highly unlikely to make a direct productive contribution to output.3 Aschauer’s evidence is unconvincing not only because it
fails to distinguish the growth rate of the productive component of government
capital from the growth rate of total government capital, but also because the
output elasticity of government capital may be inflated from reverse-causation
bias.4 That is, the productivity-coefficient estimate may be capturing the effect of output on government investment spending and hence on government
capital instead of the effect of government capital on output. Output could
affect government investment spending because government investment decisions possibly depend on output performance—higher output can lead to more
tax revenue to finance such investment.
Aschauer’s (1989) work raises many questions. What is unique to government capital that could be so productive? Which components of government
capital play a role in production? What is the nature of the production channel
through which they exercise this role? Do these channels differ across components? What are the magnitudes of the associated productivity coefficients,
controlling for possible reverse causation? How do these magnitudes explain
output and labor-productivity growth rates in the post-World War II United
States? Finally, are the real returns to investing in productive government capital components high?
This article addresses these questions. The answers provide guidance for
government investment policies by elucidating how components of government
capital influence output production and by quantifying their effects on economic
growth. Lucas (1987) underscores the importance of these questions by showing
that changes in economic growth as small as 1 percentage point can have huge
social welfare effects.
The article proceeds as follows. Section 1 describes the components of total
government capital and considers their possible production roles. The resulting analysis suggests that only government-owned, privately operated capital
(GOPO), government enterprise capital (ENTP), and government highway capital (HGWY) directly contribute to private production. GOPO’s and ENTP’s
contribution to private production stems from the measurement of private sector
output. One possible way that GOPO and ENTP enter the production function
is through the same channel as most private sector capital; i.e., GOPO and
ENTP perfectly substitute for private sector capital.5 To capture this effect,
the present study retains standard production function theory but changes the

3 The figure 0.39 is large relative to 0.30, the output elasticity of private capital found in
other studies (see Lucas [1990]).
4 Aschauer’s use of the total stock, rather than the productive component, of government
capital also may affect the estimate in unknown ways.
5 “Most private sector capital” here means total private (business) sector capital less its
transportation-vehicle component.

M. Finn: Is All Government Capital Productive?

55

standard measure of capital by adding private capital, GOPO, and ENTP together. HGWY affects production because of transportation services. In particular, HGWY and the transportation-vehicle component of private (business)
sector capital together yield services that facilitate the transportation of both
final and intermediate goods and, in turn, help produce final delivered output.
Capturing this idea, Section 2 extends standard production function theory so
as to model transportation services and distinguish them from those of the
other factors of production—labor and the augmented stock of private capital
mentioned above.
Section 2 also embeds the production theory in a general equilibrium model
of the economy that allows derivation of mathematical statements of private
firms’ investment and capital-utilization decision rules. These rules are useful
in the estimation exercise, undertaken here, by bringing more information to
bear on the parameter values of the production function. Section 3 outlines
the data and method for estimating the production function and firms’ decision
rules. For this method, Generalized-Method-of-Moments, the possible reversecausation phenomenon does not distort the coefficient estimates.
The highlights of the empirical findings of Section 4 pertain to the productivity coefficient of highway capital. The point estimate is 0.16; it is statistically
significant but imprecise. For example, a 95 percent confidence interval around
this point estimate implies that the true productivity coefficient could be as
much as 0.32 or as little as 0.001. Using the point estimate 0.16, highway capital reduced output growth by 0.1 percent during the 1970–1989 period. These
results support, but strongly moderate, Aschauer’s (1989) claim that government capital is an important explanatory factor in the productivity slowdown.
Section 5 concludes with thoughts on the policy implications of the empirical
findings.

1.

COMPONENTS OF GOVERNMENT CAPITAL

This section describes the components of total government capital, listing the
main types of capital goods in each component and summarizing some quantitative features. The key quantitative features include the components’ average
shares in total government capital (see Table 1) and the characteristics of the
component shares’ trends. The sample period is 1950–1989. The appendix
provides further detail on all data underlying this discussion and indicates data
sources and measurement caveats. Also, this section considers the possible
production role for each component.
Highway Capital (HGWY)
HGWY includes highways, streets, bridges, tunnels, overpasses, viaducts, and
association lighting and erosion control structures. It is the largest component
of government capital, with an average share of 0.36. From 1965 to 1989, this
share exhibits a downward trend.

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Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Component Average Shares in Total Government Capital
Component

Share

Highway Capital

0.361

Government Enterprise Capital

0.248

Educational and Hospital Capital

0.190

Fire and Natural Resource Stocks

0.081

Equipment Capital

0.048

Administrative, Judicial, Police, and Research and Development Stocks

0.040

Government-Owned Privately Operated Capital

0.032

Total

1.000

Notes: (1) The entries are average annual shares over the period 1950–1989.
(2) See the appendix for data definitions, caveats, and the reason that equipment capital
is a separate component in this table.

HGWY influences output production in the private sector through the provision of transportation services. HGWY and the stock of private (business)
sector transportation vehicles are necessary stocks for the transportation of
both intermediate and final goods. The flow of transportation services from
these stocks directly contributes to the production of final delivered goods.
The ratio of HGWY to private sector output is small but not insignificant;
its average value is 0.18.6 This ratio trends down from 1975 to 1989.
Government Enterprise Capital (ENTP)
Government enterprises include various credit and insurance corporations (e.g.,
Commodity Credit Corporation, FDIC, and FSLIC); the U.S. Post Office;
gas and electric utilities; water and sewerage utilities; public transit agencies;
airport and maritime terminal operators; and miscellaneous service-producing
agencies (e.g., agencies that administer lotteries, parking, highway tolls, and
housing and urban renewal). Their capital consists of office buildings, electrical
transmission facilities, gas structures, parking structures, sewer systems, water
supply facilities, public transit stations (bus, streetcar, subway, and rail), railroad
structures, airport facilities, maritime buildings, harbors, amusement structures,

6 Compare

(see Table 5).

it to the average value of the private capital to private output ratio of about one

M. Finn: Is All Government Capital Productive?

57

and associated equipment.7,8 This component is the second-largest component
of government capital; on average, its share in government capital is 0.25. For
the period 1968–1989, this share shows an upward trend.
The measure of private sector production in the national income accounts
includes the output of government enterprises (see Department of Commerce
[1988]).9 The underlying national income accounting rationale is that the output
of enterprises is very similar to the output of private firms (one can, for example, compare electricity or postal services across the two). This measurement,
combined with the fact that government enterprises use their capital stocks
to produce their output, implies that enterprise capital directly contributes to
the production of private sector output. Given that outputs are similar across
enterprises and private firms, presumably so too are the associated production
techniques/methods, suggesting that the production functions of enterprises and
private firms should be treated the same. It follows that the aggregate production
function relates the private sector product measure to the sum of private and
enterprise capital. That is, private and enterprise capital are perfect substitutes
for one another in the production process.

Government-Owned, Privately Operated Capital (GOPO)
GOPO includes research and development facilities, atomic energy facilities,
nuclear weapon factories, arsenals, shipyards, and associated equipment. It is
the smallest component, with an average share of only 0.03. This share trends
downward from 1955 to 1989. During the two world wars, GOPO was quantitatively significant (see Braun and McGrattan [1993]).
GOPO directly enters the production process of private sector output. Since
GOPO contains capital goods similar to privately owned capital goods, one
possible way of capturing its productive contribution is to treat it as a perfect
substitute for private capital in the production function.
The ratio of ENTP and GOPO to private sector output is small but not negligible, with an average value of 0.14. This ratio does not exhibit a noticeable
trend over the sample period.

7 The term harbors refers to harbors, piers, canals, docks, and dredging and drainage
equipment.
8 Harbors and some airport facilities (primarily those on national parks and Indian reservations) included in this component are not owned (or operated) by enterprises. They are owned
and operated by state and local government and by federal government, respectively. These are
included here because they cooperate with, in the case of harbors, or are very similar to, in the
case of federal airports, some of the enterprise capital stocks. Since both stocks are small, their
inclusion/exclusion is not of quantitative importance. See the appendix for more detail.
9 It does not include the output of the rest of government. The rest of government is referred
to as “general government” in Department of Commerce (1988).

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Federal Reserve Bank of Richmond Economic Quarterly

Educational and Hospital Capital (EDHS)
This component consists of primary, secondary, and university-level educational
buildings, associated buildings (laboratories, libraries, student unions, and dormitories), and equipment; stocks that serve an educational purpose (e.g., public
libraries, museums, art galleries, observatories, archives, and botanical and
zoological gardens); and health care and institutional facilities (e.g., hospitals, clinics, and infirmaries). Its average share in government capital is 0.19.
This share evolves like an inverted “v,” trending upwards to peak in 1975 and
trending downwards thereafter.
Without doubt, EDHS influences output production by promoting the
knowledge and well-being of labor input. But, if the measure of labor input accounts for both labor input’s quantity, in terms of the number of manhours, and
labor input’s quality, in terms of, for example, each worker’s educational level
and age, then it is difficult to see why EDHS should have a separate productive
effect. This study uses a labor input measure that incorporates many quality
adjustments; therefore, EDHS is not included as a direct factor of production
in the quantitative analysis.
Administrative, Judicial, Police, and Research and
Development Stocks (ADMN)
Office buildings, customs houses, courthouses, prisons, police buildings, research and development facilities, and associated equipment comprise this
category. Its share, averaging only 0.04, sharply trends up from 1963 to 1989.
Could ADMN affect production? It could since it is linked to the setting
of rules and regulations governing the conduct of business and to research and
development that affects technology. For rules, regulations, and technology
determine the amount of output that can be produced from any given quantity
of inputs (see Hansen and Prescott [1993]). But linkages such as these are
subtle and indirect.
Fire and Natural Resource Stocks (NATR)
NATR consists of structures on government land that are intended for water, land and animal protection (e.g., reservoirs, irrigation facilities, seawalls,
erosion control systems, fish hatcheries, and wildlife preservation facilities),
housing for forest rangers and national park employees, fire buildings, and
associated equipment.10 As a share of government capital, it averages 0.08 and
shows a downward trend during 1953–1989.
10 This component includes housing for forest rangers and national park employees even
though the total government capital stock under review is classified as nonresidential by the
Department of Commerce.

M. Finn: Is All Government Capital Productive?

59

Some of the capital goods in NATR contribute to the output production
process. Specifically, stocks such as fire buildings, fire equipment, reservoirs,
and seawalls mitigate or prevent the destruction of other capital stocks that are
directly employed in production, for example, private capital stocks. But, the
productive role of these government stocks, while valuable, is merely supportive. Therefore, they do not qualify for the present analysis as a direct factor of
production.
The foregoing considerations suggest that only HGWY, ENTP, and GOPO
directly contribute to private production. Accordingly, they are the only capital
components entering the quantitative part of this study’s investigation. The
channels of their contribution, suggested above, require an extension of standard production function theory to allow an explicit role for HGWY and an
expanded measure of private capital to include ENTP and GOPO. The upshot
of this for the quantitative questions motivating the study is that they will
pertain to HGWY only. That is, those quantitative questions will not apply to
ENTP and GOPO since their effects cannot be separated from that of private
capital.11,12

2.

THE MODEL ECONOMY

The model economy presented here provides a mathematical framework to
address the quantitative questions raised above. The model specifies economic
agents’ objectives and constraints, including the production function, the market
structure, and the stochastic exogenous processes. From it firms’ investment and
capital-utilization decision rules can be derived, which are useful for estimating
the parameters of the production function.
Consider an economy with a large number of identical firms and households
and a government. Since all firms and all households are identical, one can
focus on the behavior of any one representative firm and any one representative

11 Aschauer (1989) undertakes one estimation differing from that described in the beginning
of the article by breaking the total government capital stock into separate components. These components are more comprehensive than those considered in this article. All of them are included
in his estimation. See Aschauer (1989), Table 6, for further details.
12 Other studies estimating production functions involving government capital for the United
States include Munnell (1990a, 1990b), Garcia-Mila and McGuire (1992), Hulten and Schwab
(1991), Holtz-Eakin (1992), Tatom (1992), Fernald (1992), and Lynde and Richmond (1993).
In most cases the stock of government capital is measured by a total (or comprehensive)
capital stock. Garcia-Mila and McGuire (1992) and Fernald (1992) use the highway component
of government capital. Their production function specifications differ from that of this article. See
footnote 17 for more detail. Also, Garcia-Mila and McGuire (1992) and Fernald (1992) use state
and industrial-level data, respectively, in contrast to the aggregate economy-level data employed
here.

60

Federal Reserve Bank of Richmond Economic Quarterly

household.13 Therefore, most quantity variables will be expressed in per-capita
terms. The goods and factor markets in which firms, households, and the government interact are competitive, with all agents viewing prices as beyond their
control; i.e., the agents are price takers. There are three stochastic, exogenous
variables in the economy: technology, energy prices, and government investment spending. In the following discussion, unless otherwise indicated, most
variables are current-period variables; variables with a prime ( ) attached are
next-period variables. The notation is explained in Table 2.
The representative firm maximizes profit:
Π = y − wl − rv (vuv ) − rk (kuk ) − p(ev + ek ).

(1)

Profit is the difference between the revenue from the sale of output and the cost
of labor, capital services, and energy. Output is the numeraire, so its price is
normalized at one. All factor prices are relative prices. Notice that the utilization
rate of a given amount of any one capital stock determines the flow of capital
services. Interpret a utilization rate as the number of hours worked per period
and/or the intensity of work per hour of the capital stock. The firm’s choice
variables are y, l, v, k, uv , uk , ev , and ek and are subject to the following technical
constraints. The production function
y = (zl)θ1 (kuk )θ2 sθ3 ,

0 < θi < 1 (i = 1, 2, 3)
θ 1 + θ2 + θ3 = 1

(2)

states that output positively depends on technology, labor, services from capital
(k), and transportation services.14 Transportation services directly contribute to
output production by facilitating the transportation of both final and intermediate goods associated with the production process.15 Transportation services are
an increasing function of the services from vehicles and the effective highway
stock:
g
(3)
s = (vuv )¯ ψ , ψ > 0.
The effective highway stock is defined as the aggregate highway stock adjusted
for, or divided by, its aggregate usage:
g=
¯

G
,
(VUv )

(4)

13 In this model economy, no distinction is drawn between a private firm and a government
enterprise. An implicit assumption, therefore, is that enterprises are profit maximizers.
14 Placing the exponent parameter, θ , on z is for algebraic convenience only. With this
1
specification, the steady-state growth rate of the economy is given by the growth rate of z rather
than that of z/θ1 (see King, Plosser, and Rebelo [1988]).
15 Although intermediate goods underlie this production process, they do not enter the production function (2), since implicitly it is derived by averaging across all firms’ production
functions for all goods (final and intermediate). Intermediate-good output of one firm cancels with
the intermediate-good input of another. Alternatively expressed, (2) is the value-added production
function for the economy.

M. Finn: Is All Government Capital Productive?

61

Table 2 Notation
Π
y
w
l
v

=
=
=
=
=

k

=

rv (rk )
uv (uk )
p
ev (ek )
z
s
θi (i = 1, 2, 3)
g
¯
G

=
=
=
=
=
=
=
=
=

V

=

Uv
ωi (i = 1, 2, 3, 4)
g

=
=
=

Et

=

β
c
γ
t
log
τ
n
iv (ik )
x
δv , δk , δg
ig
z
¯

=
=
=
=
=
=
=
=
=
=
=
=
=
=

E

per-capita profit
per-capita output
wage rate for labor
per-capita labor hours
per-capita stock of private transportation vehicles, in place at the
beginning of the period
per-capita stock of other private capital, in place at the beginning
of the period
rental rate for capital services from v(k)
utilization rate of v(k)
exogenous energy price
per-capita energy use required for the utilization of v(k)
exogenous technology
per-capita transportation services
parameters
per-capita effective stock of government highway capital
aggregate stock of government highway capital, in place at the
beginning of the period
aggregate stock of private transportation vehicles, in place at the
beginning of the period
economy-wide average value of uv
parameters
per-capita stock of government highway capital, in place at the
beginning of the period
expectations operator conditioned on information available in
time period t
subjective discount factor
per-capita consumption
a parameter
time period t
natural logarithm
the tax rate on income from capital services due to v and k
per-capita lump-sum transfer payment from government
per-capita private gross investment in v (k )
per-capita lump sum tax paid to government
depreciation rates of v, k, and g
per-capita exogenous government gross investment in g
mean gross growth rate of z
innovation/disturbance term
unconditional expectations operator

where VUv measures the aggregate usage. So, equations (3) and (4) capture
the notion that there is congestion of aggregate highway capital. That is, the
higher the total use of vehicles in the economy (VUv ), the lower the contribution of aggregate highways (G) to each firm’s transportation services (s). For
this reason, g is referred to as the effective highway stock: it is the highway
¯

62

Federal Reserve Bank of Richmond Economic Quarterly

stock effectively contributing to transportation services. Since g depends on
¯
the aggregate stocks, G and V, and the economy-wide average utilization rate,
Uv , each of which is beyond the representative firm’s control, it follows that
g is exogenous to the firm. Barro and Sala-i-Martin (1992) and Glomm and
¯
Ravikumar (1992) model congestion of public goods in a similar fashion to
that modeled here.16
The remaining technical constraints are the energy relationships:
ev
= ω1 (uv )ω2 , ωi > 0 (i = 1, 2)
(5)
v
and
ek
= ω3 (uk )ω4 , ωi > 0 (i = 3, 4).
(6)
k
These equations specify that energy is essential for the utilization of capital, k
and v, with an increase in utilization increasing energy use per unit of capital at
an increasing rate. These specifications follow those in Finn (1993). Their presence here serves the purpose of forming cost margins for utilization decisions
and of explicitly according a role to energy in the production process. Finn
(1993) shows that the latter is important when addressing questions involving
productivity.
Equations (2), (3), (5), and (6) together show that output exhibits constant returns to scale in l, k, v, ek , and ev . Therefore, this production structure is
consistent with the assumed competitive market structure.
The assumption of identical firms implies that in equilibrium the economy’s
per-capita amount of V is the same as each firm’s choice variable v. Furthermore, Uv coincides with uv . Noting these equilibrium results and dividing G
and V on the right-hand side of (4) by the population size leads to:
g
g=
¯
.
(7)
(vuv )
Substitute (7) into (3), and the result into (2) to obtain:
s = (vuv )(1−ψ) gψ

(8)

y = (zl)θ1 (kuk )θ2 (vuv )θ3(1−ψ) gθ3ψ .

(9)

and

16 In Barro and Sala-i-Martin (1992), a total government spending flow entering the production function is subject to congestion. The government spending flow is divided by the economy’s
aggregate private capital stock. In Glomm and Ravikumar (1992), an aggregate stock of government capital entering the production function is congested. The government capital stock is divided
by a Cobb-Douglas function of the economy’s aggregate amounts of private labor and capital. See
Glomm and Ravikumar (1992) for references to other, earlier studies advancing the congestion
idea.

M. Finn: Is All Government Capital Productive?

63

Equation (8) shows that s satisfies constant returns to scale in v and g, while
equation (9) specifies y as a constant-returns-to-scale function of l, k, v, and
g. To this list one may add ek and ev , by noting (5) and (6). These constantreturns-to-scale features are important. They imply that the production function
is consistent with steady-state or balanced growth. Therefore, in the absence
of temporary innovations to the exogenous variables, output, all three capital
stocks, and energy use will grow at the constant rate of technology growth. (See
the technical appendix of King, Plosser, and Rebelo [1988] for an explanation.)
This result is important because steady-state growth is a characteristic of many
developed market economies (see King, Plosser, and Rebelo [1988]).17
The exponents in (9) are the output elasticities or productivity coefficients
of the corresponding factors of production. Of particular interest is θ3 ψ, the
productivity coefficient of highway capital. Equation (9) differs from standard
production functions by distinguishing vuv from kuk and by including g. These
differences stem from the objective of explicitly accounting for the role of
highway capital in production.
The representative household maximizes its expected, discounted lifetime
utility:
∞

β t [log ct + γ log(1 − lt )], 0 < β < 1, γ > 0.
(10)
E
t=0
0
Here, utility in any one period positively depends on consumption and leisure.
The time endowment in each period is normalized at one. As part of its
maximizing behavior, the household engages in market activities that involve
purchasing consumption and investment goods from and selling labor and capital services to the firm. The household pays taxes to and receives transfer
payments from the government. Therefore, its choice variables in any one
period are c, iv , ik , and l and are subject to the following budget and technical
constraints. Total income must equal total spending:
wl + (1 − τ )[rv vuv + rk kuk ] + n = c + iv + ik + x.

(11)

The sum of wage and after-tax capital income and transfer payments constitutes total income. Total spending is the sum of consumption, investment,
and lump-sum taxes. The reason for the presence of taxes and transfers is
17 The production function specifications of Garcia-Mila and McGuire (1992) and Fernald
(1992) differ from equation (9). Garcia-Mila and McGuire specify a production function of private
structures, private equipment, labor, the highway stock, and government educational spending.
Fernald’s specification comes closer to that of the current study. He specifies a production function
in which, like here, highway capital and private business sector vehicle capital affect production
through transportation services. His specification assumes increasing returns to scale to labor,
private capital, and highway capital. By contrast, here the specification assumes constant returns
to scale to all of the inputs, making the production theory consistent with the balanced-growth
facts.

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Federal Reserve Bank of Richmond Economic Quarterly

explained below. Also, for any one type of capital good, its one-period future
value depends on the current undepreciated quantity of and investment in that
capital good:
v = (1 − δv )v + iv

(12)

k = (1 − δk )k + ik .

and

(13)

Ensuring internal consistency of the model economy requires a description
of the behavior of government. This behavior is kept as simple as possible
given the existence of highway capital and distortional taxation. More exactly,
highway capital evolves over time according to a technical constraint analogous
to those of the household capital stocks:
g = (1 − δg )g + ig .

(14)

Government investment spending is exogenous and must be balanced each
period by lump-sum tax revenue:
ig = x.

(15)

Government revenue from distortional capital-income taxation is rebated each
period through lump-sum transfers:
n = τ [rv vuv + rk kuk ].

(16)

The above description of government behavior is the simplest possible one
given the internal consistency requirement, since the government’s budget is
balanced each period and government investment spending is both exogenous
and independent of distortional-taxation effects on the household. (See Aschauer and Greenwood [1985], Baxter and King [1993], and Dotsey and Mao
[1993] for analyses of more realistic fiscal policy.) But, the description is adequate for addressing the quantitative questions of this study. In particular, an explanation of government investment, which could involve the reverse-causation
phenomenon, would have no effect on the production function or firms’ investment and capital-utilization decision rules that are to be used in the econometric
investigation below.18 However, because distortional capital-income taxation
does influence the decision rules and is quantitatively important for the behavior
of private capital, such taxation is included in this model economy (see Greenwood and Huffman [1991] and Finn [1993] for some supporting evidence).
Regarding the stochastic, exogenous shock structure, i.e., the z, p, and
ig processes, only that pertaining to z needs detailed description here. The z
process is a logarithmic random walk with drift:
¯
log z = log z + log z + .
18 The

causation.

(17)

econometric investigation will take account of the real world possibility of reverse

M. Finn: Is All Government Capital Productive?

65

Assume that the innovation, , is identically and independently distributed
through time with zero mean. Suppose the p process is stationary. The ig
process is assumed to have a trend component, due to z, and a stationarity
component. These assumptions ensure that there is only one source of growth
in the economy, technology growth. Furthermore, ig will grow at that rate along
the steady-state growth path.
The competitive equilibrium of the economy obtains when the representative firm and household solve their maximization problems and the government
satisfies its constraints. Of the equations implicitly defining this competitive
equilibrium, only the production function, (9), the technology process, (17),
and the following four equations are used in the estimation exercise. The intertemporal efficiency conditions:
1 = βEt

ct

yt+1
+ 1 − δk − pt+1 ω1 (ukt+1 )ω2
kt+1

(18)

ct
yt+1
(1 − τ )θ3
+ 1 − δv − pt+1 ω3 (uvt+1 )ω4
ct+1
vt+1

(19)

ct+1

(1 − τ )θ2

and
1 = βEt

govern the firm’s investment decisions in kt+1 and vt+1 , respectively. The firm
sets the marginal cost of investing an additional unit at time t equal to the time
t expected discounted marginal benefit of the return from that investment at
time t + 1. For example, in equation (18), the marginal cost is the foregone
marginal utility of consumption at time t, or 1/ct . The marginal benefit is the
product, at time t + 1, of the marginal utility of consumption, or 1/ct+1 , and
a term including the after-tax marginal product of k less its depreciation and
marginal energy costs. So, the marginal benefit is:
(1/ct+1 )

(1 − τ )θ2

yt+1
+ 1 − δk − pt+1 ω1 (ukt+1 )ω2
kt+1

.

The intratemporal efficiency conditions
ω1 ω2 (ukt )ω2 pt = (1 − τ )θ2 yt /kt

(20)

ω3 ω4 (uvt )ω4 pt = (1 − τ )θ3 yt /vt

(21)

and

determine the firm’s capital-utilization decisions, ukt and uvt , respectively. They
equate, at time t, the marginal benefits and costs of increasing utilization rates.
In equation (20), for example, the marginal benefit is the after-tax marginal
product of ukt , or (1 − τ )θ2 yt /ukt . The marginal cost is the marginal energy cost
of ukt , or ω1 ω2 (ukt )ω2−1 pt kt .

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Federal Reserve Bank of Richmond Economic Quarterly

Notice that any corresponding pair of intertemporal and intratemporal
efficiency conditions includes the productivity coefficient of the relevant capital
stock (e.g., equations [18] and [20] each include θ2 , the productivity coefficient
of kt ). This is the reason for including these equations in the estimation exercise.
That is, these equations bring more information to bear on the values of the
parameters of the production function.19
Also, the estimation exercise includes one of the model’s balanced-growth
restrictions, mentioned earlier:
E log (yt+1 /yt ) = log z,
¯

(22)

which states that the mean growth rate of output coincides with that of technology.

3.

THE ESTIMATION METHOD AND DATA MEASURES

The estimation method is Generalized-Method-of-Moments (GMM) due to
Hansen (1982) and Hansen and Singleton (1982). Those studies explain the
method and show how the GMM estimator is consistent and asymptotically
normal. Ogaki (1992, 1993) also explains GMM and provides practical guidance on its implementation.
Here, GMM is used to estimate the parameters of equations (9) and (18)
through (22). There are two reasons it is particularly appropriate for this task.
First, GMM is an instrumental-variables procedure and so avoids the possible
reverse-causation bias noted at the beginning of the article.20 Second, it is
applicable to equations that are nonlinear in both parameters and variables.
In the remainder of this section, some key features and requirements of
the estimation method are outlined with reference to the estimation equations
of this study. Also, the data are briefly described.
The application of GMM requires that each equation include only stationary
variables. Equations (18) through (22) already satisfy this requirement. Their
variables are growth rates of consumption and output, output-capital ratios,
utilization rates, and the relative price of energy, all of which are stationary.

19 This approach is similar in spirit to using the factor profit-share equations, derived from
a profit function, in estimating production function parameters. Lynde and Richmond (1993) take
this related approach.
20 Aschauer (1989) obtains least-squares estimates of the production function expressed in a
form that is not a cointegrating relationship in the sense of Engle and Granger (1987). Because
of Aschauer’s method, the possible two-way interaction between output and government capital,
described at the beginning of the article, may affect his estimates. See Engle and Granger (1987)
for further discussion of these econometric concepts.

M. Finn: Is All Government Capital Productive?

67

By taking first differences of the logarithm of equation (9), it may be transformed into a stationary form, in which all of its variables are growth rates:
ˆ
z
l
ˆ
yt+1 = θ1 [ˆt+1 + ˆt+1 ] + θ2 [kt+1 + ukt+1 ]
ˆ
+ θ3 (1 − ψ)[ˆ t+1 + uvt+1 ] + θ3 ψ[ˆ t+1 ],
v
ˆ
g

(9 )

where ˆ denotes the percentage rate of change. Using equation (17) to eliminate
the unobservable zt+1 from equation (9 ) and noting the constant-returns-toˆ
scale restriction θ1 = 1 − θ2 − θ3 gives the estimation form of the production
function
ˆ
l
ˆ
yt+1 = (1 − θ2 − θ3 )ˆt+1 + θ2 (kt+1 + ukt+1 )
ˆ
+ θ3 (1 − ψ)(ˆ t+1 + uvt+1 ) + θ3 ψ gt+1
v
ˆ
ˆ
+ (1 − θ2 − θ3 ) log z + ¯t+1 ,
¯

(9 )

where ¯t+1 = (1 − θ2 − θ3 ) t+1 .
Next, each equation is used to generate or specify, as the case may be, a
disturbance term. For equations (18) and (19), the disturbances are one-period
expectational/forecast errors. The disturbances for equations (20) and (21) take
the form of combinations of any omitted variables. In the case of equation (22),
the deviation of output growth from its mean is the disturbance. For equation
(9 ), ¯t+1 is the disturbance term.
GMM requires that the instrumental variables for any one equation belong to an information set of variables that are independent of the equation
disturbance. Also, the instrumental variables must be stationary. There is no
requirement that the instruments be econometrically exogenous. That is, candidate instruments, appropriately dated and transformed to satisfy the information
set and stationarity requirements, include endogenous variables such as output
and consumption growth. Also, instruments may include a constant term.
The instrumental variables and the corresponding disturbance terms are
used to create a set of orthogonality conditions.21 These conditions form the
basis of GMM’s criterion function, denoted by J here for convenience. The
GMM estimator of the vector of parameters, b, is the parameter vector that
minimizes the criterion function.
ˆ
Recalling equation (9 ), if reverse causation is present in the data, gt+1
will be correlated with yt+1 and hence with ¯t+1 . But, this possibility will not
ˆ
invalidate the orthogonality conditions used in the GMM procedure. It follows
that possible reverse causation will not distort or bias the GMM estimates of
the parameters in equation (9 ).
21 No

instrumental variables are chosen for equation (22) because it is already in the form
of an orthogonality condition.

68

Federal Reserve Bank of Richmond Economic Quarterly

When the number of orthogonality conditions equals (exceeds) the number
of parameters, the estimation system is exact (overidentified). In the case of
an overidentified system, Hansen (1982) shows that the minimized value of J,
multiplied by sample size, is asymptotically distributed as a Chi-square whose
degrees of freedom equal the number of overidentifying restrictions. This Chisquare, therefore, provides a measure of the model’s fit.
Ogaki (1992) shows that one way of testing coefficient restrictions (such
as constant returns to scale) is based on the test statistic:
T[J(br ) − J(bu )],

(23)

where br (bu ) is the GMM estimator imposing (relaxing) the coefficient restrictions and T is the sample size. This test statistic is asymptotically distributed
as a Chi-square whose degrees of freedom equal the number of coefficient
restrictions. Ogaki (1992) also describes various methods for correcting for
serial correlation when using GMM.22
Some of the parameters entering into the estimation equations are not estimated in this article. They are β, τ , δk , and δv . Since they do not appear in the
production function, these parameters are not central to the current exercise.
Also, the existing literature provides guidance to their values, in the cases of
β and τ , or to a simple method of obtaining them, in the cases of δk and δv .
Therefore, β and τ are set equal to the values used in many other studies (see
Finn [1993] for references), while δk and δv are set equal to the U.S. sample
average depreciation rates implied by equations (12) and (13) (see Greenwood
and Hercowitz [1991] for an example). The resultant values are the following:
β = 0.96, τ = 0.35, δk = 0.08, and δv = 0.17.
The data are annual, real, per-capita data for the United States during the
period 1950–1989. Full details of these data, their sources, and caveats with
respect to the capital-stock measures are presented in the appendix. Since there
are no exact empirical counterparts for uk and uv , the total-industry total-privatecapital utilization rate, denoted by u, proxies for both. Once estimates of the
production function parameters are obtained, they are used in equation (9) to
solve for a data measure of technology, z. This series is then used for one
purpose in Section 4.

22 The present article undertakes a diagnostic test for first-order serial correlation. It is a
t-test and is described as follows. First, obtain the equation residuals, for each equation, using the
GMM point-coefficient estimates reported in Table 3. Second, conduct an Ordinary-Least-Squares
regression of the equation residuals on their one-period lagged values and a constant. The t-test
pertains to the regressor’s coefficient in this regression equation.

M. Finn: Is All Government Capital Productive?

4.

69

EMPIRICAL FINDINGS

This section discusses the empirical findings reported in Tables 3–7.
At the outset note that there is evidence of first-order autocorrelation in
the residuals of equations (18) through (21), providing evidence of missing
dynamic elements from these equations.23,24 This autocorrelation is taken into
account in the estimation and selection of the instruments.25
Consider the GMM estimation results in Table 3 for equations (18) through
(22) and (9 ). These results obtain for the particular choice of instruments indicated there. The productivity coefficients, θ2 and θ3 , are statistically significant
at the 5 percent level and have small standard errors. Their values seem plausible in view of the U.S. average share of total private capital in output, 0.30,
found in many studies (e.g., Lucas [1990]) and the relative magnitudes of k
and v.26 The coefficients of the energy relationships, the ωi (i = 1, 2, 3, 4),
are generally insignificant and very imprecise, presumably because of the
omitted dynamics from equations (20) and (21). But, it is important to note that
ωi (i = 1, 2, 3, 4) do not, it turns out, interfere much with the estimation of θ2
and θ3 . That is, the estimates of θ2 and θ3 are essentially determined by equations (18) and (19), with little influence stemming from the marginal energy-cost
terms.27 This point is important because θ2 and θ3 , not the ωi (i = 1, 2, 3, 4),
enter the production function estimation equation, (9 ). Furthermore, given the
plausibility of the θ2 and θ3 estimates, it leads to the judgment here that the
omitted dynamics from equations (18) and (19) are not that serious, at least
for the purpose of this study. The mean annual rate of output growth, log z,
¯
is statistically significant, precise, and reasonable. The productivity coefficient
of the capital stock v, θ3 (1 − ψ), is insignificant and imprecisely determined.
Highway capital’s productivity coefficient, θ3 ψ, is 0.16. It is significant but
imprecise. Highlighting the latter, note that a 95 percent confidence interval

23 Autocorrelation

was detected using the diagnostic test described in footnote 22.
finding may be related to the finding in Canova, Finn, and Pagan (1993) that the
dynamic restrictions imposed by many real business cycle models are empirically rejected.
25 The autocorrelation correction is the modified-Durbin method described in Ogaki (1992).
First-order serial correlation in the residuals of equations (18) through (21) implies that the instruments for equations (18) through (21) must be lagged two periods relative to the dates of the
variables appearing in those equations. The instruments for equation (9 ) must be lagged only
one period relative to the variables appearing in the equation, unless they are capital-stock growth
rates. The latter need not (but can) be lagged because the dating of the stocks is such that they
already are lagged one period relative to, say, output. (This makes the dating of the empirical
stocks conform with that of the model’s stocks.)
26 The average values of k/y and v/y are 1.06 and 0.05, respectively.
27 The evidence supporting this assertion is that similar estimation results for θ and θ obtain
2
3
when the capital-utilization energy-cost margins are entirely ignored, i.e., when equations (20)
and (21) and the terms involving utilization in equations (18) and (19) are dropped. The results for
this experiment (for the same choice of instruments as in Table 3) are: θ2 = 0.242(0.003), θ3 =
0.018(0.001), ψ = 8.533(3.602), log z = 0.015(0.003), and χ2 = 0.039(0.844).
¯
1
24 This

70

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 GMM Estimation Results
θ2
θ3
ψ
log z
¯

=
=
=
=

0.267
0.020
7.963
0.015

(0.015)
(0.002)
(3.590)
(0.004)

ω1 = 0.121 (0.063)
ω3 = 0.179 (0.108)

θ3 ψ = 0.158 (0.077)
Instruments:
I(18) = {constant}
I(19) = {constant}

ω2 = 10.769 (6.773)
ω4 = 9.745 (9.300)

θ3 (1 − ψ) = −0.138 (0.075)

χ2 = 0.045 (0.832)
1
I(20) = {constant, ut−2 }
I(21) = {constant, ut−2 }

I(9 ) = {constant, yt−1 }
ˆ
I(22) = {constant}

Notes: (1) Coefficient standard errors are in parentheses.
(2) χ2 denotes the Chi-squared statistic with one degree of freedom. Its probability
1
value is in parentheses.
(3) I(x) denotes the instrument set for equation x.
(4) Equations (18) through (21) were estimated subject to correction for first-order
serial correlation.

for θ3 ψ implies that the true value of θ3 ψ could be as high as 0.32 or as
low as 0.001. The Chi-square measure of fit, χ2 , indicates that the model’s
1
overidentifying restrictions are not rejected at a high level of confidence.
These findings, especially regarding the productivity coefficients, are robust
to a wide range of instrument sets.28 Also, tests of the constant-returns-to-scale
restrictions, i.e., θ1 = 1 − θ2 − θ3 , and the constant-returns-to-scale restriction from the transportation-services equation, (8), are neither individually nor
jointly rejected at high levels of confidence.
In short, the model specification finds a good deal of empirical support.
The key finding is that highway capital is significantly productive, with a productivity coefficient of 0.16. However, the estimate is imprecise, which must
be borne in mind when assessing the implications for growth and real returns
to government investment.
Highway capital growth has implications for output and labor-productivity
growth, working through its productivity coefficient. These implications are
summarized in Tables 4 and 5. The contribution of highway capital growth gt
ˆ
to output growth yt is measured by θ3 ψ gt . Regarding labor-productivity growth,
ˆ
ˆ
28 In

checking robustness, the instrument set for any one equation always included a constant
and possibly the appropriately lagged variables appearing in that equation. The total number of
instruments was kept small, following Tauchen’s (1986) advocacy of a small number of instruments for small samples. Only occasionally, when using some two-period lagged variables as
instruments for equations (18) through (21) or lagged capital-stock growth rates as instruments
for equation (9 ), the estimation algorithm failed to converge or sensitivity of the estimates was
detected. This result stemmed from the absence of strong correlation of those instruments with
the equation variables.

M. Finn: Is All Government Capital Productive?

71

Table 4 Output Growth Accounting

yt
ˆ
1950–1969
1970–1989
1950–1989

gt
ˆ

0.022
0.030
0.013 −0.005
0.018
0.013

contribution of gt (to yt ) evaluated at
ˆ
ˆ
point estimate upper estimate lower estimate
(θ3 ψ = 0.158)
(θ3 ψ = 0.315)
(θ3 ψ = 0.001)
0.005
−0.001
0.002

0.009
−0.002
0.004

0.00004
−0.000008
0.00002

ˆ
ˆ
contribution of gt (to yt ) relative to yt evaluated at
ˆ
point estimate upper estimate lower estimate
(θ3 ψ = 0.158)
(θ3 ψ = 0.315)
(θ3 ψ = 0.001)
1950–1969
1970–1989
1950–1989

0.218
−0.064
0.115

0.434
−0.127
0.229

0.002
−0.001
0.001

Notes: (1) The entries are average annual growth rates of the indicated variables over the time
period shown. These entries have been rounded.
(2) Upper (lower) estimate of θ3 ψ is the estimate at the upper (lower) bound of the 95
percent confidence region for θ3 ψ.

Table 5 Labor Productivity Growth Accounting
contribution of (ˆ t − ˆt ) (to [ˆ t − ˆt ]) evaluated at
g
l
y
l
point estimate upper estimate lower estimate
(ˆ t − ˆt ) (ˆ t − ˆt ) (θ3 ψ = 0.158)
y
l
g
l
(θ3 ψ = 0.315)
(θ3 ψ = 0.001)
1950–1969
1970–1989
1950–1989

0.020
0.008
0.014

0.028
−0.010
0.010

0.005
−0.002
0.002

0.009
−0.003
0.003

0.00004
−0.00001
0.00001

l
y
l
contribution of (ˆ t − ˆt ) (to [ˆ t − ˆt ])
g
l
relative to (ˆ t − ˆt ) evaluated at
y
point estimate upper estimate lower estimate
(θ3 ψ = 0.158)
(θ3 ψ = 0.315)
(θ3 ψ = 0.001)
1950–1969
1970–1989
1950–1989

0.222
−0.203
0.105

0.443
−0.405
0.209

0.002
−0.002
0.001

Notes: (1) The entries are average annual growth rates of the indicated variables over the time
period shown. These entries have been rounded.
(2) Upper (lower) estimate of θ3 ψ is the estimate at the upper (lower) bound of the 95
percent confidence region for θ3 ψ.

72

Federal Reserve Bank of Richmond Economic Quarterly

yt − ˆt , the contribution of growth in the highway capital-to-labor ratio, (ˆ t − ˆt ),
ˆ l
g l
g l
is measured by θ3 ψ(ˆ t − ˆt ).
First, look at the output growth accounting. Using the point estimate θ3 ψ =
0.16, the contribution of gt is always small but important. During the 1950–1969
ˆ
period, it contributes 0.5 percent to the output growth rate of 2.2 percent, representing 22 percent of that rate. In the productivity slowdown period, 1970–1989,
gt has reduced the output growth rate of 1.3 percent by 0.1 percent, amounting
ˆ
to 6 percent of that output growth rate. Second, examine the labor-productivity
g l
growth accounting. At the point estimate θ3 ψ = 0.16, the contribution of (ˆ t −ˆt )
is again always small but not negligible. In the period 1950–1969, the contribution is 0.5 percent to the labor-productivity growth rate of 2.0 percent, which
is 22 percent of labor-productivity growth. During the productivity slowdown,
l
(ˆ t − ˆt ) has reduced the labor-productivity growth rate of 0.8 percent by 0.2
g
percent, amounting to 20 percent of labor-productivity growth.
This accounting picture changes quite substantively when the upper and
lower bound estimates, 0.32 and 0.001, of θ3 ψ are used. The contributions of
ˆ
l
gt (or gt − ˆt ) become much more important or negligible, as the case may be.
ˆ
What is the real return to government investment in highway capital? How
does it compare with the real returns to private investment in the private capital
stocks, k and v? Table 6 summarizes the answers to these questions, pertaining
to average annual real returns over the 1950–1989 period.
The real return to government investment is measured by the marginal
product of g: θ3 ψy/g. The private marginal products of k and v, θ2 y/k and
θ3 y/v, give the real returns to private investment in those stocks.
Using the point productivity-coefficient estimates, the real returns from
investments in k and v are 25 percent and 41 percent, respectively. These
returns may seem high, but of course are consistent with the corresponding
point-coefficient estimates and output-capital ratios. If they were compared to
other returns (e.g., Treasury bill returns), it would be important to measure their
net returns (net of taxes, depreciation, and marginal energy costs) and to note
any differences in risk characteristics.
At the point estimate θ3 ψ = 0.16, the real return to government investment
of 87 percent is considerably higher than the above private real returns. The
upper and lower bound estimates of θ3 ψ imply that the true real returns could
be 174 percent or 0.8 percent, respectively.
Recall that three components of government capital, EDHS, ADMN, and
NATR, do not enter the quantitative analysis. Section 1 suggests that influences
on the production process could stem from EDHS if labor were inaccurately
measured, and from ADMN because of its association with rules and regulations as well as with research and development. If these effects exist, then the
technology measure, z, will embody them. In addition, if they are of quantitative
importance, then a systematic correlation between the growth rates of technology and each of EDHS and ADMN will be evident. On the other hand, Section 1

M. Finn: Is All Government Capital Productive?

73

Table 6 Average and Marginal Products
(yt /kt )
0.942

marginal product of kt evaluated at
point estimate
(θ2 = 0.267)

upper estimate
(θ2 = 0.297)

lower estimate
(θ2 = 0.236)

0.251

0.280

0.223

(yt /vt )
20.360

marginal product of vt evaluated at
point estimate
(θ3 = 0.020)

upper estimate
(θ3 = 0.030)

lower estimate
(θ3 = 0.010)

0.405

0.607

0.203

( yt /gt )
5.507

marginal product of gt evaluated at
point estimate
(θ3 ψ = 0.158)

upper estimate
(θ3 ψ = 0.315)

lower estimate
(θ3 ψ = 0.001)

0.872

1.736

0.008

Notes: (1) The entries are annual averages of the indicated variables over the period 1950–
1989.
(2) Upper (lower) estimates refer to the estimates of the relevant parameter at the upper
(lower) bound of its 95 percent confidence region.

suggests that NATR, in and of itself, does not influence the production process,
which implies that the technology measure, z, does not incorporate productivity
effects stemming from NATR. Therefore, if there is no reason that the growth
rates of NATR and z should be systematically linked together, then a significant
correlation between the two will not be detected.
It is interesting, therefore, to compute these correlations. Table 7 reports the
results. The only significant correlation is that involving NATR.29,30 One possible interpretation of this correlation is that changes in z, by causing changes
in output, affect changes in government investment in NATR. Or perhaps both
z and NATR are jointly responding to movements in some other variable such
as the weather. While it would be interesting to explore these interpretations
further, note that the correlation between the growth rates of z and NATR is
only marginally significant.
29 Significance is judged at the 5 percent level. The critical value for the one-sided t-test
statistic, at the 5 percent significance level and with 40 degrees of freedom, is 1.68.
30 Another government capital component, EQIP, was also omitted from the quantitative
analysis. This variable is defined and explained in the appendix. The correlation between the
growth rates of EQIP and technology is 0.357, with a t-statistic of 2.321. But, it is difficult to
interpret this correlation since EQIP is a component that should be split across the EDHS, ADMN,
and NATR categories.

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Table 7 Correlations Between Growth Rates of Technology and
Omitted Government Capital Stocks
Fire and Natural Resource Capital

0.288

(1.826)

Educational and Hospital Capital

0.248

(1.557)

Administrative, Judicial, Police, and
Research and Development Capital

0.150

(0.924)

Notes: The entries (not in parentheses) are correlations between the growth rates of technology
and the indicated capital stock over the period 1950–1989. The numbers to the right of these
entries (in parentheses) are corresponding t-statistics.

5.

CONCLUSION

The key empirical finding is that highway capital is significantly productive.
The point estimate of its productivity coefficient is 0.16, meaning that for every
1 percent change in highway capital, output responds by 0.16 percent. But, there
is much uncertainty surrounding this estimate. To highlight this uncertainty,
consider that the true productivity coefficient could be as high as 0.32 or as
low as 0.001. Further work achieving more precise estimation of the productive
effect of highway capital would be worthwhile.
Using the productivity-coefficient estimate, 0.16, the implications for output growth accounting are as follows. During the 1950–1969 period, highway
capital growth contributes 0.5 percent to the output growth of 2.2 percent, representing 22 percent of the output growth. In the productivity slowdown period,
1970–1989, highway capital growth has reduced the output growth of 1.3 percent by 0.1 percent, amounting to 6 percent of that output growth. These effects
are small but significant. They imply that government investment in highway
capital matters for output growth. However, the uncertainty surrounding the
productivity-coefficient estimate of 0.16 must qualify this assessment of the
magnitude of the contribution of highway capital growth to output growth. That
contribution could be much larger or smaller than the numbers just mentioned
suggest.
Over the period 1950–1989 the real return to government investment in
highway capital, when evaluated at the productivity coefficient 0.16, averages
87 percent per year. While, again, there is much uncertainty about this estimate,
suppose for discussion purposes that it is reliable. The real return, 87 percent, is
high. Compare it to, say, the real return to private investment in private capital
that averages 25 percent per year over the same period. Does this imply that
government investment in highway capital should be increased up to the point
that ensures equality across the two returns? It is difficult to answer such a
question about the optimal level of government investment. Much will depend
on the financing of government investment. Suppose, for example, increases

M. Finn: Is All Government Capital Productive?

75

in government investment are financed by increases in the tax rates on labor
and/or private capital income. Increases in these tax rates will work to reduce
labor and private capital, thereby leading to output losses. On the other hand,
the increase in government investment, by increasing government capital, will
cause output to increase. The optimal level of government investment is that
level which carefully balances these opposing output effects (see Glomm and
Ravikumar [1992] for an analysis of these issues in a deterministic endogenous
growth model). It is not clear that the optimal level occurs exactly at the point
of equality between the real returns to private and government capital. Further
complications arise if uncertainty is factored into the analysis. In the presence
of uncertainty, real returns to investing in different assets are generally not
equated, even in an expected sense, reflecting the differential roles that different
assets play in hedging consumption risk (see Finn [1990]). Further exploration
of optimal government investment that addresses the considerations just raised
is an important task for future research.

APPENDIX: DATA SOURCES, DEFINITIONS,
AND CAVEATS
The data sources are the following: (1) Citibase, (2) National Income and
Wealth Division, Bureau of Economic Analysis, U.S. Department of Commerce, denoted by DC, (3) Federal Reserve Bulletin, denoted by FRB, and (4)
Dale W. Jorgenson, Harvard University, denoted by DWJ. Unless otherwise
indicated, the source is Citibase.
Population (thousands of persons): civilian non-institutional population aged
16 and over.
Output (billions of 1987 dollars): gross domestic product less gross government
product.
Labor Hours (real index): aggregate domestic private quality-adjusted labor
hours index, where the quality adjustment is based on a cross-classification by
age, sex, education, class of worker, and occupation. The index is described in
Jorgenson, Gallop, and Fraumeni (1987), Chap. 8. Source: DWJ.
Utilization Rate (real index): manufacturing sector utilization rate (1950–1953)
and total industrial sector utilization rate for the remainder of the sample.
Source: FRB.
Aggregate Price Deflator (1987=100): gross domestic product deflator.

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Energy Price Index (1987=100): producer price index for fuels and related
products (covering petroleum, natural gas, coal, and electricity).
Relative Price of Energy (1987=1): ratio of the energy price index to the
aggregate price deflator.
Consumption (billions of 1987 dollars): personal consumer expenditures on
nondurable goods plus services.
Private Transportation Vehicle Capital and Investment (billions of 1987
dollars): The capital is the net, end-of-period stock of transportation vehicles
(automobiles, trucks, trailers, and buses) owned by the private (business) sector and government enterprises. The latter is proxied by taking one-tenth of
enterprise equipment capital (see the discussion on caveats below, part [e], for
an explanation). The investment is the corresponding gross investment. Source:
DC.
Private Capital and Investment, excluding that pertaining to transportation
vehicles (billions of 1987 dollars): The capital is a net, end-of-period stock consisting of nonresidential, fixed capital owned by the private (business) sector
and government enterprises and that owned by general government but privately operated, plus federal government airport facilities, plus state and local
government harbors. The terms general government and harbors are explained
in footnotes 9 and 7, respectively. The latter two components of capital are
proxied by three-quarters of the stocks in the DC “federal other structures”
and “state and local conservation and development” categories (see the discussion on caveats, part [b], for more information). The investment series is the
corresponding gross investment. Source: DC.
Total Government Capital (billions of 1987 dollars): government (federal,
state, and local), net, end-of-period, fixed, nonresidential, nonmilitary capital.
Source: DC.

Government Capital Components
In what follows, the mnemonics correspond to the paragraph titles in the text
(pp. 55–58), except for EQIP, which denotes equipment capital. The components are defined with reference to the DC categories, the titles of which are
in italics. Unless otherwise indicated, those categories are the sum of federal,
state, and local government categories. Source: DC.
HGWY: highways and streets.
ENTP: government enterprises plus federal government airport facilities plus
state and local government harbors. The latter two components of capital are
approximated as described above.

M. Finn: Is All Government Capital Productive?

77

GOPO: government-owned and privately operated.
EDHS: educational and hospital buildings.
NATR: federal conservation and development plus one-quarter of state and
local conservation and development (i.e., the residual after measuring ENTP
as described above).
ADMN: The structures component of total government capital less the structures components of ENTP and GOPO less HGWY, EDHS, and NATR (these
last three components are entirely composed of structures).
EQIP: The equipment component of total government capital less the equipment components of ENTP and GOPO. Note that HGWY, EDHS, NATR, and
ADMN do not have equipment components.

Some Caveats and Comments on the Capital-Stock Data
(a) The government enterprises category includes but does not isolate toll highways. Highways and streets also includes but does not isolate these. So there
is unavoidable double-counting of toll highways. Because toll highways are a
small part of government enterprises and highways and streets, this mismeasurement is probably not significant.
(b) Measures of federal government airport facilities and state and local government harbors are not published. Given the DC description of federal other
structures and state and local conservation and development, it seems reasonable to get approximate measures of these variables by taking three-quarters of
federal other structures and state and local conservation and development, respectively. Federal other structures are small relative to both total government
capital and government enterprises, averaging 0.4 percent and 1.7 percent,
respectively, over the sample period. Also, state and local conservation and
development is small in relation to total government capital and government
enterprises, averaging 1.2 percent and 5.1 percent, respectively, over the sample
period. Therefore, any mismeasurement arising from the use of the indicated
approximations in this article is not likely to be quantitatively significant.
(c) The fire capital-stock series described in the text on p. 58 is not separately
available, nor is there useful information for forming an approximate measure.
It is only because of the discussion in the text as opposed to the estimation
task of the article, that the fire capital stock should be included in NATR and
excluded from ADMN and EQIP, both of which, recall, are derived as residual
series. But, this mismeasurement is probably immaterial since the fire capital
stock is surely small.

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Federal Reserve Bank of Richmond Economic Quarterly

(d) The only measures of equipment capital published by DC, relevant here, are
the equipment components of total government capital, government enterprises
and government-owned and privately operated. These components are used to
form the EQIP series. No information is available for allocating EQIP among
the EDHS, NATR, and ADMN components, which would have been interesting
for the purpose of the discussion in the text.
(e) A measure of government enterprise vehicle capital is not available. Given
the existence of the series on the equipment component of government enterprises and the fact that some enterprises undertake much transportation (public
transit enterprises and the U.S. Post Office), it is desirable to get some proxy
for enterprise vehicles. A proxy of one-tenth of the government enterprises
equipment component seems reasonable in view of the list of government enterprises. Any mismeasurement arising from the use of this proxy in this study
is not likely to be important since the equipment component of government
enterprises is a small stock, averaging 0.8 percent of total government capital
and 3.3 percent of government enterprises over the sample period.

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