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Working paper 87-6
Aggregate Fluctuations and Economic Growth:
A Case of Random-Walk Hypothesis
Ching-Sheng Mao
Federal Reserve Bank of Richmond
October 1987
This study is part of my Ph.D. dissertation submitted to the University of Rochester.
I have benefited greatly from the guidance of Robert King and Alan Stockman. I also
wish to thank Michael Dotsey, Marvin Goodfriend and Tony Kuprianov for their useful
comments. The analyses and conclusions are solely the author's and do not necessarily
reflect the views of the Federal Reserve Bank of Richmond or the Federal Reserve
System. Of course, all errors are my own.
ABSTRACT
This paper presents a model economy in which the "balanced" growth is
determined endogenously. The growth process in this economy does not
depend on exogenous specifications such as human capital accumulation or
technological progress. Rather, it is determined within the model and
governed by two economic forces: (1) the intertemporal substitution of
consumption and labor and (2) the intertemporal production opportunities.
In equilibrium, the real quantities (i.e., consumption, capital, employment
and output) will all evolve as logarithm random walks with drift. Therefore,
the time series generated by this model is not trend stationary and the
propagation of technological disturbances is permanent. This result is
consistent with the empirical findings of Nelson and Plosser (1982).
-1 -
AGGREGATE FLUCTUATIONS AND ECONOMIC GROWTH:
A CASE OF RANDOM-WALK HYPOTHESIS
1. INTRODUCTION
That the study of aggregate fluctuations can be conceptually separated from the
theory of economic growth seems to be the implicit view in the modern theory of business
cycles. Based on this view, models of economic fluctuations have been examined primarily
in a context where growth is either assumed away or triggered by exogenous considerations
such as population growth and technological progress. Consistent with this view is the
empirical strategy in which economic time series were often subject to a variety of
detrending schemes in order to remove the growth component. To a large extent, this view
may be attributed to the empirical observation that many economic aggregates like
consumption and real GNP tend to wander regularly along a steadily growing path over a
long span of time.
This empirical regularity has often been taken to suggest that the
economic cycles can be explained without reference to the growth of the economy which,
presumably, has little to do with short-run and transitory perturbations. Because of its
conceptual appeal and empirical convenience, this view has received wide acceptance in the
literature.
By contrast, the approach taken in this paper is that the nature of aggregate
fluctuations cannot be abstracted from the process of economic growth, a point emphasized
in the recent works of Romer (1985) and King and Robelo (1986). It is my contention that
business cycles can be better understood when growth dynamics are taken into account. In
a sense, these contrasting views reflect the longstanding debate among economists about
the proper ways to describe economic processes. For instance, can aggregate economic
data, say real output, be best described by a stationary stochastic process? The fact that
-2 -
many economic aggregates appear to be "explained" fairly well using high-order stationary
autoregressions does not imply that they are indeed generated by such processes. Nelson
and Plosser (1982), for example, could not reject the hypothesis that many U.S. aggregate
time series data are characterized by logarithmic random walks (more generally,
nonstationary integrated processes). Certainly, this issue cannot be settled on a purely
statistical and empirical basis.
A somewhat confusing element that has contributed to the above debate is the
stochastic theory of economic growth. In a well-known result, existing growth models
assert that per capita quantities, including output, capital and consumption, possess
stationary distributions under reasonable assumptions (see Brock and Mirman (1972) for
an example). This result implies that technological shocks are transitory in nature and real
quantities can therefore be practically described by stationary processes. However, the
stationarity of per capita quantities does not necessarily carry over to the aggregates. To
see this, one must recognize that most existing growth models assume, implicitly or
explicitly, an exogenous process for fixed factors of production such as labor. Depending
upon how these driving processes are specified, stationary as well as nonstationary time
series can be generated.
The difficulty in endogenously determining economic growth is related to the
well-known dilemma of indeterminacy of firm size when a constant returns to scale
technology is specified. That is, any factor allocation fulfilling the firm's zero profit and
marginal conditions will be equally acceptable from the firm's point of view. Therefore,
aggregate dynamics are usually generated by artificially specifying a stochastic process for
some set of scale factors. As shown in this paper, however, constant returns to scale need
not present this problem when in a general equilibrium framework consumers' labor supply
decisions can be used by firms to pin down their sizes. Based on this observation, a model
-3 -
economy can be constructed in which labor supply (or population) is endogenously
determined rather than exogenously specified.
Firms then use this information in
formulating their optimal production plan. The process of sustained growth determined
through this mechanism does not originate from the increasing returns of technology as in
Romer (1985), nor is it related to human capital technology as in King and Robelo (1986).
Instead, it is a self-induced process governed by two economic forces: (1) the intertemporal
substitution of consumption and labor and (2) the intertemporal production opportunities.
The novelty of this model is that diminishing returns to factors of production do
not constrain the growth of the economy as they do in conventional models. The reason for
this is the absence of fixed factors in the technology. As a result, economic agents are fully
capable of transferring resources across time. This transfer permits greater propagation in
technological disturbances. In fact, with identically and independently distributed (i.i.d.)
productivity shocks and homogeneous preferences, the model predicts that the aggregate
capital stock, output, consumption, and labor employment will all evolve as logarithmic
random walks with drift. Temporary real shocks that occur at a particular point in time
will be propagated permanently into the future and shift the entire equilibrium path. Such
processes are not trend stationary and clearly do not possess stationary distributions.
This result is quite similar to that obtained by King and Robelo (1986), but the
growth mechanisms in our model do not depend on an explicit exogenous technology for
producing human capital.
Rather, the mechanisms are endogenously governed by the
consumers' willingness and ability to smooth out their consumption over time. The model
has important implications for macroeconometrics.
In particular, because of the
nonstationarity in levels, we have to search for transformation schemes in order to apply
stationary time series methods. Our results suggest the use of first differencing as a way of
achieving stationarity, a longstanding practice that has rarely been justified on the basis of
-4 -
coherent economic theory. It is noteworthy that the random walk process in our model
identifies the specific form of nonstationarity used in the test of Nelson and Plosser (1982).
The rest of the paper is organized as follows. In section 2, I construct a model
economy where population growth is endogenously determined. The model incorporates
important elements of both real business cycles and stochastic growth models. Section 3
discusses the determination of the equilibrium. A solution for the optimal decision rules
and value function is presented in section 4.
The quantity dynamics are analyzed in
section 5. The paper is concluded in section 6.
2. THE MODEL
Consider an economy populated by a large number of identical households, whose
preferences depend positively on the amount of goods consumed and negatively on the
labor participation of the family. The preferences of the stand-in representative household
are characterized by the utility function u(ct, nt), where ct and nt denote the household's
consumption and labor participation at time t, respectively. The labor participation nt in
this model measures the population or the labor force of the representative family. One
may imagine that at the beginning of each period, the head of the household decides how
many children he is willing to bear which will then engage in the economy's production
process. Under this interpretation, the production of children generates a negative service
to the household's utility because it represents the amount of labor supply.'
For
simplicity, we abstract from the problem of family formation and assume the child bearing
cost is negligible.
For analytical tractability, we confine ourselves to the case where the utility
-5 -
function u(ct, nt) is homogeneous of degree y, where y is a constant between 0 and 1. Note
that the parameter
a
can also be used to measure the intertemporal substitution elasticity
of consumption and labor, which is equal to 1/(1-y). In addition, we assume
(a) ul > 0, u2 < 0, u 1 1 <0, u2 2 < 0;
(b) The "preferred set" {(ct,nt) I u(ct,nt) > iU} is strictly convex;
(c) For a given utility level, limnt, 0 [-u 2 /u 1 ]
=
0 and limc
These assumptions are standard in economic analysis.
[-u 2 /u 1]
=
Specifically, (a) requires the
diminishing marginal utility of consumption and increasing penalty of labor supply.
Assumption (b) implies that the indifference curve is upward sloping and strictly convex
toward the axis of nt. A necessary and sufficient condition for this to hold is
2uu2 u
12
-
1
u2
2 -
u2 U1
1
> 0
Assumption (c) requires that the slope of the indifference curve be 0 as nt - 0 and infinite
as ct
-.
o . This assumption is sufficient for ruling out the corner solutions.
Now, we turn to the the production side of the economy. At the beginning of time
t, the representative household is equipped with pre-determined capital stock kt-i and
labor supply nt-1
He observes a technological shock At and produces yt units of goods
according to the following constant returns to scale technology:
=t
AtF(kt_,, nti1)
The technological shock At is assumed to be a positive i.i.d. random variable drawn from a
known probability distribution. The production technology specified above is identical to
-6 -
that in Long and Plosser (1983) and Mao (1986). Because of the assumed lag structure in
the technology, the economy behaves like a stochastic endowment economy.
The economic problem of the representative household is to achieve maximal
lifetime utility by allocating the stochastic output yt, which is given at the beginning of
each period, between consumption and savings (capital) over time, and by choosing
whatever labor supply (population) that will bring about this optimal plan. For simplicity,
we assume the capital stock is fully depreciated so that kt can be regarded as investment at
time t.
This assumption also serves to insure that any persistence of the equilibrium
quantities does not derive from the storage technology of the capital stock.
The
commodity feasibility constraint is therefore
Ct +
t < Att-11
t-1) E t
With an immortal family, the optimization problem can be formally stated as follows:
max nt
tE'[,
u(ct, nt) ]
0 < /3 < 1
subject to
c + kt < AtF(kt-i nti)
(1)
yt, for all t
ct > °, kt >0, nt > °, for all t
Co + ko < yO
The initial stock yo is given
where E0 is the expectations operator conditional on information available at time 0, and /3
is the time preference discount factor.
household's expectations be rational.
The model is closed by requiring that the
-7 The model as it is stated is a typical real business cycle model in that the
intertemporal production technology plays an important role in generating quantities
dynamics. Instead of treating labor as hours worked as in previous models (e.g., Long and
Plosser (1983) and Kydland and Prescott (1982)), however, labor supply in this model is a
measure of "bodies" that corresponds to the usual employment statistics.
Since the
household can fully adjust its labor supply at any point in time, production opportunities
will not be constrained by a fixed supply of factors, such as time endowment, which will
limit the degree of intertemporal substitution in production. Because of this, the economic
agent has greater flexibility to smooth his consumption over time in response to unexpected
random shocks. This flexibility permits a greater propagation of technological disturbances
as opposed to previous business cycles models.
The model can also be regarded as one example of a stochastic growth model.
However, conventional growth theory is mainly concerned with the process of capital
accumulation and its relation to consumption smoothing.
As mentioned before, the
decision of labor supply is either indeterminate or implicitly assumed to be exogenous.
From the viewpoint of business cycles, this is not a desirable feature because the variation
of labor employment constitutes an indispensable part of economic fluctuations.
By
allowing endogenous determination of individual's labor supply, the model can display
self-sustained growth even when technological progress does not persist in the economy.
The way we approach this problem is different from King and Robelo (1986), in which
endogenous growth depends on a process of human capital accumulation which is a specific
form of technological progress.
In the tradition of the business-cycle literature, our major concern is with the
stochastic properties of the time series generated by this model. Without solving the model
explicitly, the next section proceeds to define an equilibrium for this economy and to
discuss some of its properties.
-8 -
3. EQUILIBRIUM
The central planning problem specified in the last section defines a sequence of
Pareto allocations over time. Associated with this sequence is a set of supportive "shadow
prices" that can be used to map the Pareto optima into competitive equilibria. The basic
approach here is to interpret the optimal shadow prices as the competitive prices which
consumers and producers used to solve their own optimization problems in decentralized
markets. These prices will consist of not only spot prices, such as the wage rate, but also
the interest rate that prevails in the contingent market for future consumption and output.
Given these shadow prices, agents (consumers and firms) can be induced to behave in a
way that conforms to the rules specified by Pareto optima. In other words, the optimal
decision rules of the central planning model will duplicate a competitive equilibrium
determined in markets (see Sargent (1980) and Prescott and Mehra (1983) for a formal
argument). We will not spell out details of the market arrangements here.
In view of the above remarks, we study the equilibrium using the dynamic
programming algorithm. Let V(yt, At) be the value function associated with the problem
(1), defined recursively by
(2)
V(yt, At) = jkmax t u(yt-kt, nt) +
Et [V[At+ 1 F(kt, nt), At+l]]
Note that V(.) is the maximal obtainable expected return at time t given the current
**
realization of yt and At. An equilibrium is characterized by the decision rules, kt and nt
(presumably, a function of yt and At), together with a bounded value function that solves
the above optimality equation. Associated with this solution is a transient probability
distribution for outputs that are generated by the optimal decision rules and the random
-9 -
shocks. One of our goals is to characterize such a probability distribution by extracting
information from the above optimization problem.
The equilibrium prices for this economy are determined simultaneously with real
quantities Specifically, assuming that the value function exists and is differentiable with
respect to yt, then the process {V1 (ytAt)} is a sequence of spot output prices that will
support the Pareto allocations. In our model, this supporting price is equal to the marginal
utility of consumption evaluated at the optimum.2
Given these prices, the expected real
interest rate at time t, denoted rte, is defined to be
re=
V (yt, At)
-1
Et([ Vl(yt+',At+i)]
That is, (1 + re) is the ratio of the present value prices for outputs at time t and t+1. It is
the one period ahead ex ante (prior to the occurrence of At+,) interest rate that will clear
the futures market for output.
Now, assuming that the value function V(yt, At) exists, then the solutions of the
optimality equation (2) must satisfy the following first order conditions:
(3)
u 1 (y-kt, nt) = / Et[Vl(yt+1 , At+,) At+1]F1(kt, nt)
(4)
-u 2 (yt-kt, nt) = / Et[Vl(yt+ 1, At+i) At+]F 2 (kt, nt)
where yt+l is the optimal output at time t+1 contingent on the realization of At+i
Equations (3) and (4) require that the marginal cost and benefit arising from the trade-off
of consumption and labor over time be equal in equilibrium. In other words, a small
-10-
deviation from the optimal plan should leave the life time utility intact. As usual, this
optimal plan should also fulfill the following transversality condition:
lim BtE 0 [Vj(y*, At) yt] = 0
The transversality condition says that the expected present value of output cannot be
valued at infinity, which implies, as in growth models, that the capital stock and labor
supply will not grow "too fast" in equilibrium. We assume this condition is fulfilled for the
rest of the paper.
The difference equations (3) and (4) impose certain restrictions on the equilibrium
paths of the capital stock and the labor supply. These restrictions, although not explicitly
specified, reflect two basic choices facing individuals: (i) the substitution between goods
and (the negative of) labor at a given point of time and (ii) the allocation of output and
consumption over time. To see how the equilibrium is determined by these forces, divide
(4) by (3) to get
(5)
-u 2 (yt-k~t , nt)/ul(yt-kt, nt)
F2 (kt, nt)/Fl(kt, nt)
Equation (5) is the efficiency condition that governs the allocation of goods and labor at a
given point of time.
It requires that the rates of contemporaneous substitution in
preferences and production be equal in equilibrium. That is, the isoquant must be tangent
to the indifference curve at each point in time. As shown in figure 1, this condition alone
does not determine the equilibrium because there are a large number of candidate
allocations on the tangent loci. To locate the equilibrium, one must take into account the
allocation of goods over time.
-
11
-
The individual's intertemporal decision is illustrated in figure 2, in which the
horizontal and vertical axes measure the output at time t and t+1, respectively. For the
sake of exposition, let us assume for the moment that the labor supply is fixed at some
given level, say, n t
Then, according to the production technology, there is a
transformation possibility between current and future consumption, conditional on nt and
the realization of the random shock At+1. Since At+1 is unknown at time t, the expected
frontier E(At+i)F(kt, nt) is employed as the relevant transformation scheme. The slope of
this transformation curve (measured in absolute value) is the expected marginal product of
capital and measures the feasible substitution of consumption over time.
Given this
production opportunity, the individual allocates a portion of current output into saving to
maximize utility. Clearly, the optimum will be achieved at the point where the expected
rate of return on saving (i.e., the marginal product of capital) equals the rate of
intertemporal substitution in consumption. That is, the indifference curve will be tangent
to the production frontier in equilibrium. This is the standard efficient condition that
governs the allocation of goods over time. With labor supply being fixed at n1 , the point
E1 is the best outcome that can be obtained.
Now, suppose the individual is allowed to adjust his labor supply. Then, for each
level of labor supply we have a different efficient allocation, which is represented by a point
on the tangent loci AB. Moving along this tangent loci reflects the fact that the individual
is now capable of altering the transformation possibilities through the choice of labor
supply. To pin down the equilibrium, recall that the capital stock and labor supply must
also satisfy equation (5), the contemporaneous efficiency condition.
The allocations
fulfilling this condition are represented by a curve CD which corresponds to the tangent
loci shown in the figure 1. Clearly, the equilibrium will occur at the point E where AB and
CD intersect. This is the point where both intertemporal and contemporaneous efficiency
conditions are fulfilled at the same time.
-12-
The foregoing analysis illustrates the importance of the labor supply as an
endogenous device in coordinating the use of resources. This feature is either absent or
limited in existing models. For example, consider the classical growth model where the
labor supply is inelastic and exogenously given by the size of population. Intuitively, the
equilibrium of this economy corresponds to a particular point on the tangent loci AB, say
E1 , figure 2. Since the economic agent can only move along a given production frontier, his
ability to transfer resources over time is limited by the productivity of capital, which is
diminishing by assumption.
It is clear that, unless there is some kind of persistent
technological progress or the population is growing to such an extent that it will offset the
decreasing marginal product of capital, the economy cannot experience sustained growth.3
This is the underlying reason why stationarity will prevail in the classical model. Now,
with endogenous labor supply, the individual is no longer constrained to a fixed production
schedule. In fact, by changing his labor supply decision, not only can he alter the rate of
return on investment, but he can also manipulate intertemporal production opportunities.
Notice that his ability to do so is not limited by feasible constraints such as a fixed time
endowment.
The extra degree of freedom leaves the household additional room for
smoothing consumption and income over time.
Therefore, one would expect that the
equilibrium in this economy would display a stronger tendency for persistence. In fact, as
will be shown later, the equilibrium quantities of this model possess an extreme form of
smoothing property -
logarithmic random walks with drift.
The next section explores the homogeneous structure of the underlying economy
and provides a solution to the optimization problem. The method we use is motivated by
Boyd (1985) and Mao (1986).
13-
-
4. SOLUTION
Solving the optimization problem (1) with a homogeneous structure relies on a
simple observation. As a specific example, consider the standard textbook problem where
consumers maximize utility over a linear feasible set.
If the utility function is
homogeneous (or, more generally, homothetic), then the equilibrium will be invariant up to
a scalar shift of the budget line. In other words, the income path is a straight line starting
from the origin. The underlying reason for this, of course, is that the scalar transformation
of the budget set does not alter the preference ordering, and the equilibrium can therefore
be mapped to another under the same transformation.4 This simple fact allows us to derive
some useful properties of the value function.
To begin with, let us rewrite the optimality equation (2) as follows:
V(yt, At)
(2')
=
max u(ct, nt) + Et[
s.t.
c + ks
=
E
fistu(cs, ns)]
A F(k5 -1 ns 1 )
y5I
S > t±
Ct+ kt =t
Note that the optimization problem above amounts to the problem (1) from the vantage
point of time t. Let us assume that the sequence {cV, k5 , nS}t solves (2') with yt as the
initial stock. Then the maximum utility can be evaluated along this optimal time path as
follows:
V(yt, At) = u(yt-kt, nt) + Et[
E
PStu(y-k* ns)*]
Now, suppose the individual is endowed with a different initial income given by
-14-
ryt, r > 0. For our purposes, it does not matter whether the variation of Yt is due to an
unexpected shock at time t or a change in the production decision made at time t-1.
Given this new initial stock, what is the new equilibrium? Consider the sequence
{rkV,
rnsit which is a scalar transformation of the old equilibrium. We claim that this sequence
is optimal with ryt as the initial income. First, the path {rk*, rnS}t' lies within (in fact,
on the boundary of) the feasible set because the production technology exhibits constant
returns to scale. Secondly, since the period's utility function is homogeneous of degree y,
we have
*
*
=-T7 (yt-k*, n*) + r7 E
= T7
*
St*
-
u(ryt-'rkt, Irnt) + E[t
1
*
sE 1ptu(Tys- rks, rns)
E
3s tu(y*-k*, n)]
V(yt, At)
*
Clearly, if*~~~~
the sequence
{k*, n*1' solves the problem for yt and At, then the transformation
{-ks, rn* }' also solves the problem for ryt and At. 5 This implies that the value function
obeys the following relationship:
V(iryt, At) = U(i-yt-rkt, i-nt) + Et[
=
S
Tu(-ys-rks, rns)]
r7 V(yt, At)
Thus the value function is homogeneous of degree 7 in Yt for a given realization of At.
Hypothetically, the value function takes the following form:
(6)
V(yt, At) = A(At)y7I, 0 < Y< 1
-
15-
where A(At) is an unknown function, presumably nondecreasing in At. Notice that the
value function is strictly concave in the state variable yt. Economically, this means that
The relative risk aversion
the househould is risk averse toward output fluctuations.
coefficient of Arrow and Pratt is 1-7, the reciprocal of which is just the intertemporal
substitution elasticity of consumption. The slope of the value function evaluated at yt is
the equilibrium spot price of the current output.
The homogeneity of the value function is a building block in our solution
procedure. Specifically, it allows simplification of the Euler equations. Using (6), the
Euler equation (3) can be rewritten as follows:
ul(yt-k*, nt) = 7y
E[A7±A(At+ )]F(k*, nt)71 F1 (kt, nt)
***
where use has been made of yt+1 = At+iF(kt, nt).
Also, we have replaced the
conditional expectation operator by an unconditional one because the process {At} is
serially uncorrelated. By assumption, ul(.,.) and F(.,.) are homogeneous of degree 7-1 and
1, respectively. We can divide both sides by (nt)7- 1 to obtain
(7)
u1 [(yt-kt)/nt, 1] = y# E [A7 1 A(At+ )]F(kt/nt, 1)7 -Fl(kt/nt, 1)
This equation involves only the consumption-labor ratio and capital-labor ratio. Now,
because the rate of contemporaneous substitution in preferences and production can also be
expressed in terms of a ratio, the tangency condition (5) can be written as
g[(yt-kt)/nt] = f(kt/nt)
where f = F2/F1 and g = -u2 /u 1 . This equation defines a one-to-one correspondence
-16-
between the equilibrium capital-labor ratio and the consumption-labor ratio.
determination of either one will automatically determine the other.
The
Assume that the
function g is invertible, then the equilibrium consumption-labor ratio can be expressed in
terms of the capital-labor ratio:
(8)
(yt-kt)/nt = g f(kt/nt)
Substituting (8) into the left side of (7), we have
(9)
ul[glf(kI/n*),1l = y PE [A'±A(At 1 )] F(kt/nt*,l)7
1
Fl(kt/nt,)
Equation (9) summarizes the theoretical restrictions that must be placed on the
equilibrium capital-labor ratio.6
Given the expectations of the random shocks and the
unknown function A(At+i), the only possible solution for the equilibrium capital-labor
ratio is a constant.7 Thus, let
*
(10)
*
kt/nt = a
where a is a constant dictated by the parameters of preferences, technology, and most
importantly, the stochastic properties of the random shocks. From (8), the equilibrium
consumption-labor ratio is also a constant:
(yt-k)/n = g 1 f(a)
Rearranging and using (10), we obtain
-17-
(11)
*
~~a
_]y
kt= L a + g-1 f(a)
t
t
O<q5<1
By definition,
(12)
*
*
r~~~~
g11 f(a)
ct = yrkt = [ a ± 1 f(a)
]
t
(1-0) Yt
Yt
a Yt
Also,
(13)
nt = kt/a =
Ea + g-1f(a)
Substituting (11)-(13) into the optimality equation (2'), one can determine the unknown
function A(At+i), which is a constant and equals 8
A=
u[g 1f(a),1]/[a g
1
f(a)]J
F(a, 1)
,
1 -
where
-1
IE
a + g 1f(a)
E(At+71) A7
The constant A is a meaningful quantity if and only if [I E(A7
1
)pO7] < 1. This is also a
necessary and sufficient condition for the transversality condition to hold. Later we will
show that the quantity A dictates the growth rate of this economy.
Equations (11)-(13)
are the equilibrium decision rules for investment,
consumption and labor supply. The linearity of the decision rules implies that unexpected
productivity shocks at each point of time will be distributed proportionately between
consumption and savings (i.e., capital stock).
The labor supply will also change
proportionally in order to satisfy the efficiency conditions.
Therefore, the equilibrium
-
18-
quantities will move together in response to random disturbances.
In principle, such
changes are governed by the substitution and wealth effects, but because of the general
equilibrium nature of the model prices and wealth (or permanent income) cannot be held
exogenous to uncover these effects. Some general observations, however, can be made.
The relationship between employment and output fluctuations is particularly
interesting. Consider the situation of higher than expected current income. First, the
individual tends to reduce his labor supply because he is wealthier. Secondly, because both
current and future consumption are normal goods, the individual has an incentive to spread
some portion of current income into the future.
This is done by increasing the
accumulation of capital stock. Given the initial labor supply, the future marginal products
of capital and labor will then decrease and increase, respectively. Of course, this cannot be
an equilibrium because it creates potential profits that can be exploited by increasing the
labor supply. As long as the exploitable gains outweigh the wealth effect, the labor supply
will continue to rise. In other words, the individual is always willing to substitute labor
into periods where the net expected returns are higher.
This mechanism differs from
traditional growth models, in which the smoothing of consumption relies entirely on the
adjustment of capital stock.
It is not surprising that the incentive for smoothing consumption over time also
tends to smooth the equilibrium price ratios, such as the expected real interest rate. In
fact, in the case of random walk consumption, the equilibrium rate of interest (i.e., r4e)
remains constant over time. It should be mentioned, however, that the ex post interest
rate, which has nothing to do with the savings decision, will fluctuate over time in response
to random shocks.
-19-
5. DYNAMICS
The dynamic properties of the system can be studied in terms of the reduced forms
of the decision rules. For convenience, we will omit the superscript "*" hereafter. Using
the production technology and the decision rule for labor supply (13), equilibrium output
can be derived as follows:
yt=A~F~a,1)n~
tA1
r F(a, 1)1
t[ a + g 1 f(a) ]
t 1
t
tt
Similarly, from (11)-(13), the equilibrium capital stock, consumption and employment are
kt
F(a, 1)
ytt= At I a + g-1 f(a)
ct =
-
nt
kt = At p (yt~l
n =kt/
kt/a
-
1
ant-1
kti) = At
r
FAa,1)
At [ a
g-1 f(a)
=AtI
kt-1
ct_,
1
nt-1
At p nt-1
Taking natural logarithms, we have
(14)
log yt = log A + log yt_l + log At
(15)
log kt = log y + log kt-1 + log At
(16)
log ct = log 4 + log ct_1 + log At
(17)
log nt = log a + log nt-i + log At
Equations (14)-(17) are the reduced forms of the dynamic system. Since the random shock
log At is i.i.d., the equilibrium quantities, in logs, will all evolve as random walks with a
constant drift log IO.9
-20 -
One of the most important implications of random walks is that temporary real
disturbances will be propagated permanently into the future and shift the equilibrium time
path once and for all. The time series generated by such processes will be unpredictably
fluctuating along a constant mean path, a feature that appears to agree with a casual
observation of the real economy and is consistent with the findings of Nelson and Plosser
(1982) that many U.S. economic aggregates, including real GNP, consumption, capital, and
employment may be generated by autoregressive processes with an unit root.t 0
The drifted random walks (14)-(17) imply that the economy is characterized by
balanced growth in which all endogenous variables grow at the same rate. In fact, with an
i.i.d. random shock, the equilibrium growth rate is a white noise. Rewrite (14) as follows:
Alog yt
=
log p + log At
where A is the first difference operator. Clearly, the growth rate of this economy, Alog yt,
consists of two components: (1) the expected or "natural" growth rate, log jA, governed by
the deep parameters of preferences and technology and the probability distribution of the
random shock, and (2) the stochastic part, log At, driven by exogenous disturbances. The
equilibrium growth rate, therefore, is determined endogenously by economic agents, which
is different from conventional growth models where balanced growth is exogenously given
by population growth. Further, the endogenous growth is not related to an exogenous
technology for producing human capital, which is essential for the models of King and
Robelo (1986) and Hercowitz and Sampson (1986)."1
The random walk as a data generating process contrasts with other real business
cycles models.
In Long and Plosser (1983), for example, the output process can be
characterized as a stationary autoregression of first order, for which the propagation of
-21 -
shocks decays with the share of physical capital in output (see King and Plosser (1986)).
Since the share is empirically small (about one third in U.S.), the internal mechanisms of
the Long and Plosser model
-
consumption smoothing and capital accumulation -
do
not produce quantitatively important serial correlation in economic time series. In this
regard, the random walk with a constant drift may be interpreted as an extreme form of
serial correlation in the levels of economic aggregates. However, it is questionable to
measure serial correlation for time series generated by nonstationary processes because they
do not possess a stable autocorrelation function. In our case, however, stationarity can be
achieved simply by first differencing the time series and considering the growth rate, which
is a white noise.
For the last century, capitalistic economies have been growing at a somewhat
constant pace. In the case of the U.S. economy, for example, the annual growth rates of
real GNP have been 3 percent on average, which is consistent with the prediction of our
model. However, the growth of many aggregate time series (notably, output, employment,
and consumption) also displays a strong cyclical pattern over the course of economic
fluctuations, which our model fails to capture. The reason for this is due to the nature of
technological shocks assumed in the model. It is not difficult to see that, if random shocks
follow a Markov process, then equilibrium quantities may be characterized as an integrated
process, in which case the variation in growth rates will be serially correlated.
The
economic reason for this is because a particular realization of technological shocks that are
serially correlated will alter the curvature of the future production schedules, and hence the
intertemporal efficiency condition. Despite its ad hoc nature, this device have been widely
used in real business cycles models to enhance persistence in economic time series (e.g.,
Kydland and Prescott (1982)). The above discussion suggests that the presence of utility
impulses may also introduce additional persistence into the system through its effect on the
intertemporal substitution of consumption. This factor is often ignored in the literature.
Whether or not it is quantitatively important is left for further research.
-22 -
6. CONCLUSION
The main objective of this paper has been to provide a theoretical ground on which
economic time series should be interpreted. We start by asking whether the observed
macroeconomic data can be modeled as a nonstationary process, a hypothesis that has
important implications for the study of business cycles. To empirical economists, this view
has been accepted implicitly for a long time, but rarely justified on a theoretical basis.
Part of the problems may be attributed to the difficulty of identifying different sources of
the nonstationarity embedded in the economic data.
This paper provides an explicit
example in which the economic system itself is capable of generating nonstationary
aggregate data even in the absence of nonstationary shocks.
In this economy, the
nonstationarity cannot be removed by detrending the time series as it is usually done in
empirical studies. Instead, first differencing is a proper way of achieving stationarity.
The model analyzed in this paper may be used as a benchmark to study the
implications of fiscal policy. For example, the model may incorporate the government as
the third agent who claims a certain amount of resources (i.e., lump-sum taxes) to meet its
expenditure needs. In this case, the variation of production shocks may be interpreted as a
change in the expenditure or tax policy. Since shocks have permanent effects in our model,
we expect that a change in the tax or expenditure policy will shift the time path of the
equilibrium quantities. In the long run, however, this should not affect the "natural"
growth rate at the steady state unless the government also changes the stochastic
properties of its policy such as the mean and variance of its expenditure.
-23 -
n
t
kt
I
I
F (kt,
I
n
I
t
t~~~~~~~~~
nt )
t
U(ct'
nt)
Tangent Loci
C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
ti
It
1I
n
Figure 1 -
*
t
Contemporaneous Efficiency Condition
n
t
- 24 -
Yt+1
Tangent Loci (intertemporal)
*
E(X)F(k ,nI)
t
t
E(A)F(kt,nt
*
*
U(ct, nt)+PE[V(.,
ct
Figure 2 -
kt
Intertemporal Efficiency Condition
.)]
-25 -
FOOTNOTES
1. This formulation is different from the dynastic utility function used by Barro and
Becker (1986), in which the production of children generates a positive service to the
household's utility because parents care about the welfare of their offsprings.
2. This relationship can be argued intuitively. Let us suppose that the output increases
marginally at time t due to unexpected random shocks. This extra income can be either
consumed at time t or used as production input for future consumption. But either way
should leave the total returns intact at optimum, and poses no differences to the economic
agent. Therefore, the value of such extra income, which is the price of the current output,
should be equal to the marginal utility of consumption. Note that this quantity should be
adjusted accordingly if capital is not fully depreciated.
3. In this sense, the classical model does not really explain the phenomenon of growth
because it relies on exogenous mechanisms.
4.
Recently, Boyd (1986) has shown that this principle can be applied to dynamic and
stochastic settings as well. In fact, according to Boyd, the transformation need not be
linear as long as it preserves the preference ordering. Technically speaking, any such
transformation is "isomorphic" and preserves the geometrical structure of the equilibrium.
Using this property, a complicated stochastic model might be converted to a easy-to-solve
form. See Boyd (1986) for details.
5. Another way of looking at this is to check the Euler conditions. Since V1 (.,.) equals
marginal utility of consumption in equilibrium, equation (3) can be written as follows:
***
*
*
*
*
*
u 1 (yt-kt, nt) = IEt [u[At+iF(kt, nt)-kt+i, nt+1] At+, Fl(kt, nt)
Since both u and F are homogeneous, this equation still holds under the scale
transformation {rys, rk5 }l. The same argument applies to equation (4) as well.
-26 -
6. Note that equation (9) can be viewed as an algebraic equation rather than a functional
equation. In the latter case, the solution is in general more difficult to obtain because
expectations of the future endogenous variable have to be taken into consideration. In the
present case, however, only the current endogenous variables are involved.
7. Like Mao (1986), one can imagine that the equilibrium capital-labor ratio as a function
of yt and At, or more generally, a function of the current and past random shocks {At,
At-1...}. Then, equation (9) can be rearranged as follows:
H(At, At
1
'....)
E
7[At+A(At~i)]
where H(.) is the implicit equilibrium function implied by the equilibrium capital labor
ratio. Since the right hand side is constant over time, H(.) must be a constant function.
That is, the implicit coefficients of the random shocks must be identically equal to zero,
which implies a constant capital-labor ratio.
8. The quantity A(At) can be derived in a different way. Using the homogeneity of the
utility and production function, the optimality equation (2) becomes
A(At)yt
u(yt-kt, nt) +±
-
E[It +±A(tt+1)]F(kt nd)
[ul(yt-kt, nt)(yt-kt) + U2 (yt-kt, nt)nt] +
FJ 3E[A t±A(At+l)][7Fl(kt, n*)k* + 7F2 (k*, n*)nt]
From the Euler equations (3) and (4), the above reduces to
A(At)yt7 -
[ul(yt-kt, nt)yt]
E1
*
- u[1[g-1
~a,
J[-
1Y+g-*f(a)
1(* 7-lyt
u 1 gfa,1
1]
-27 -
Therefore,
A(At)E
[
u[g-lf(a), 1]
It can be shown that this expression is equivalent to that in the text.
9.
The wealth (or permanent income), measured in log, will also evolve as a drifted
random walk. Define the wealth wt as the expected present value of current and future
income, discounted by the equilibrium real rate of interest, i.e.,
wt=Et [ sEt [1
I+re
Ys
where ret is constant in equilibrium. Use (14) and complete the summation, we obtain
t
[ 1
-
/ E(At..71 )
]J Yt
Therefore, the wealth is also linear in yt, which implies that it will also evolve as a drifted
random walk. Note that, in this case, the current income and permanent income amount
to the same thing.
10. It should be mentioned that the test conducted by Nelson and Plosser (1982) has a low
power against the alternative processes.
11.
A common feature of the models of King and Robelo (1986) and Hercowitz and
Sampson (1986) is that both impose some kind of fixed factor dynamics. Although the
growth in these models is endogenously generated, it can be shown that it may not be
sustainable in the absence of technical progress. In fact, the data generating process in this
case will be stationary. This is most apparent in the model of Hercowitz and Sampson.
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-
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