Full text of Economic Quarterly (Federal Reserve Bank of Richmond) : First Quarter 2020, Vol. 106, No. 1

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Economic Quarterly— Volume 106, Number 1— First Quarter 2020— Pages 1–17

Labor-Market Wedge under
Engel Curve Utility: Cyclical
Substitution between
Necessities and Luxuries
Yongsung Chang, Andreas Hornstein, and Marios Karabarbounis

O

ne of the leading research questions in macroeconomics concerns the identi…cation of the sources of economic ‡uctuations.1 Economists often identify these sources through accounting procedures that are based on “wedges,” that is, violations of
a model economy’s equilibrium conditions conditional on data.2 For
example, representative agent models impose tight restrictions on the
comovement of consumption, hours, and real wages. For an optimal
allocation of consumption and hours worked, the marginal rate of substitution (MRS) between leisure and consumption has to equal the
real wage. Conditional on consumption, hours worked should increase
with the real wage, but for reasonable parameterizations of the representative household’s preferences, this prediction is inconsistent with
observed movements in aggregate consumption, hours worked, and real
wages over the business cycle. On the one hand, the MRS increases
rapidly during expansions, as the marginal utility of consumption relWe would like to thank Mark Bils, Pete Klenow, John Jones, Nicolas Morales,
Mike Finnegan, and James Lee for helpful comments and suggestions. We thank
Andrew Owens for his excellent research assistance. This work was supported by
the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019S1A5A2A03043067). The views expressed in this article are those
of the authors and not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve System. E-mail: Yongsung.Chang@gmail.com; Andreas.Hornstein@rich.frb.org; Marios.Karabarbounis@rich.frb.org.
1
See, for example, Christiano, Eichenbaum, and Evans (2005) or Smets and
Wouters (2007).
2
See Hall (1997) and Chari, Kehoe, and McGrattan (2007) for expositions of wedge
accounting.

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Federal Reserve Bank of Richmond Economic Quarterly

ative to leisure quickly decreases, but on the other hand, there is no
corresponding strongly procyclical movement in real wages. This gap
between the MRS and the real wage, the so-called labor-market wedge,
when treated as an exogenous distortion is an important source of economic ‡uctuations in this class of models.3 Of course, one would prefer
to explain the wedge rather than treat it as an exogenous shock.4
Recently, Jaimovich, Rebelo, and Wong (2019) documented that
during the Great Recession, consumers reduced the quality of the goods
and services they consumed. Since part of the labor wedge is due to the
countercyclical marginal utility of consumption, procyclical variation of
quality can reduce the volatility of the labor wedge. While Jaimovich,
Rebelo, and Wong (2019) provide a general framework that includes
quantity-quality substitution, the measurement of quality is very challenging. Instead, in this paper we study the “average quality” e¤ects
stemming from composition changes in the household’s consumption
basket and nonhomothetic income-expenditure paths, that is, Engel
curves. It is straightforward to obtain information on the shape of Engel curves from cross-sectional data such as the Consumer Expenditure
Survey (CEX).
We show that accounting for the substitution between necessities
and luxuries dampens the cyclical movement of the labor-market wedge
but only by a small amount. In booms, households’ consumption of
luxuries (e.g., food away from home) tends to increase relatively more
than the consumption of necessities (e.g., food at home). This substitution along the Engel curve slows down the increase in the MRS
because the marginal utility of consumption falls more slowly as consumers move toward luxuries. For a parameterization of nonhomothetic
Engel curves consistent with the cross-sectional household expenditure
pattern across income quintiles in the CEX, we show that cyclical composition changes in the consumption basket can account for at most 16
percent of the volatility in the labor wedge measured in the aggregate
time series data.
3
Note that our (narrow) de…nition of the labor wedge represents only a part of the
broader de…nition of the labor wedge as the gap between the MRS and the marginal
product of labor. See, e.g., Bils, Klenow, and Malin (2018). Nevertheless, as Karabarbounis (2014) argues, our narrow wedge accounts for most of the volatility in the overall
wedge.
4
The existing literature o¤ers various interpretations for this wedge, including
changes in home-production technology, Benhabib, Rogerson, and Wright (1991), government spending being a part of private consumption, Christiano and Eichenbaum (1992),
various frictions in the labor market, such as wage rigidity, Galí, Gertler, and LopézSalido (2007), or search frictions, Shimer (2010), and aggregation errors, Chang and Kim
(2007).

Chang et al.: Labor-Market Wedge under Engel Curve Utility

3

This article is organized as follows. Section 1 brie‡y discusses the
measurement of the labor-market wedge and lays out a simple model
where the household’s preferences exhibit an Engel curve. In Section
2, we compute the labor wedge corrected for the Engel curve, using
data on cross-sectional household expenditure patterns across income
quintiles in the CEX. Section 3 provides a concluding remark.

1.

LABOR-MARKET WEDGE

To understand the role of the Engel curve in the measurement of the
labor-market wedge, we …rst present the standard labor wedge for
household preferences expressed with respect to an aggregate consumption good, C, and hours worked, H:
C 1 1=
1 1=
P C = W H;

U (C; H) =

H 1+1=
1 + 1=

where is the intertemporal elasticity of substitution (IES) for consumption and is the Frisch elasticity of labor supply.5 The labor
wedge is de…ned as the ratio between the MRS (between leisure and
consumption) and the real wage (W=P ):
H 1=
M UL
W
=
= M RS =
:
1=
M UC
P
C

(1)

When we denote x
^ for the cyclical component of x (de-meaned growth
rate or percentage deviation from the trend), the cyclical component
of the labor wedge can be expressed as:
1
1 ^
\
+ C^ W=P:
(2)
^= H
Figure 1 shows the cyclical component of aggregate GDP and the
labor wedge for a baseline parameterization of preferences using aggregate time series data. The measured wedge is highly volatile and
procyclical because: (i) hours worked and consumption are both procyclical, with hours being very volatile, and (ii) the real wage is neither
highly procyclical nor volatile. As shown in the table of Figure 1, (i)
hours are slightly more volatile than GDP and highly procyclical with a
0.95 elasticity with respect to GDP growth, while (ii) consumption and
the real wage exhibit similar volatility, and the real wage is only mildly
procyclical with a mere 0.19 elasticity with respect to GDP growth.
5
Since the labor-market wedge is entirely based on the intratemporal optimality
condition, we abstract from the dynamic decisions of households, e.g., savings, etc.

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Federal Reserve Bank of Richmond Economic Quarterly

As a result, the labor wedge is tightly correlated with GDP and more
than twice as volatile: a 1 percent increase in GDP is associated with
a nearly 2 percent increase in the labor wedge for our baseline parameterization, = 0:5 and = 1.
We believe our baseline parameterization is plausible since (i) there
is ample evidence for an IES in consumption that is much smaller than
one, and (ii) variations in aggregate hours re‡ect the extensive margin
as well as the intensive margin of labor-supply decisions.6 In addition,
we obtain similar results for the labor-wedge volatility for a range of
empirically plausible values of and , see columns (2) through (5) in
Table 2.
Now, suppose that the household purchases N types of consumption
goods, fc1 ; :::; cN g, at prices fp1 ; :::; pN g. The household maximizes a
utility function with intertemporal elasticities of substitution that di¤er
across goods
U (c1 ; ::; cN ; H) =
P m Cm =

N
X

i=1
N
X

1 1=

H 1+1=
1 + 1=

i
ci
i
1 1= i

pi ci = W H;

i=1

where P m and C m represent the measured aggregate price and consumption index. The FOCs are
i ci

1=

H

1=

i

=

pi ; for i = 1; :::; N

=

W;

(3)

where is the marginal utility of nominal expenditures. This speci…cation yields nonhomothetic Engel curves across goods. A good with
a small i is a necessity (e.g., food) whose marginal utility decreases
rapidly with increased consumption. A good with a large i is a luxury whose marginal utility decreases slowly. Consequently, as total
expenditures increase for …xed prices and the marginal utility of expenditures decline, consumption of luxury goods increases faster than
does consumption of necessities.
Summing over the FOCs for the consumption goods, we get the
marginal utility of expenditures
P
1 1= i
X
c~
1 1= i
iPi ci
=
= m
with
c
~
:
(4)
i ci
m
p
c
P
C
i i i
i

6

For example, Havránek (2015) in a meta analysis of 169 published articles …nds a
mean estimate of 0.5 for the IES, and Keane and Rogerson (2012) discuss the relevance
of intensive and extensive margins for estimates of the aggregate labor-supply elasticity.

Chang et al.: Labor-Market Wedge under Engel Curve Utility

5

Allowing for a labor wedge in equation (3) and using the marginal
utility of expenditures, the true labor wedge, , is then de…ned by the
expression
Cm
M UL C m
W
:
(5)
=
=
c~
c~
Pm
Compared to the standard measure of the labor wedge in (1) with
aggregate consumption, this wedge with multiple goods is likely to
be less cyclical because in economic booms households’ consumption
moves toward luxuries whose marginal utility decreases more slowly.
The cyclical component (growth rate) of the labor wedge is7
X
1
1 ^
\m ;
1
! i c^i W=P
(6)
^ = H
+ C^ m
H 1=

i

P

i

P

where C^ m = i ! i c^i and P^ m = i ! i p^i are Divisia quantity and price
indices of aggregate consumption. Measured quantity and price indices
of aggregate consumption are essentially constructed as Divisia indices.
Using these quantity and price measures of aggregate consumption in
expression (2), we obtain the di¤erence between the measured wedge
and true wedge
^m

^ =

X
i

2.

1

1

! i c^i

1

i

1

N
X

! i c^i =

i=1

X

1

1

! i c^i :

i

i

EMPIRICAL ANALYSIS

Engel Curves from the CEX
We use eight categories of household expenditures in the CEX: food
at home, food away from home, transportation (excluding vehicle purchases), housing, health care, apparel, entertainment, and cash contribution. In Table 1, …rst and second column, we report their expenditure
shares in 2005 and 2015. The expenditure shares of the eight categories
are quite stable over the decade, and in total (CEX8) they make up
about 75 percent of total expenditures— which is close to 89 percent of
the consumption-related expenditure (total expenditure net of those on
personal insurance and pensions, CEXNET). We exclude vehicle purchases because vehicles are durable goods, and we exclude “insurance
and pensions”because they may re‡ect the household’s savings rather
than consumption.

7

^

From the de…nition of c~ in equation (4), we get c~ =
is the expenditure share of the ith good.

P

i

1

1
i

! i c^i ; where ! i

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Federal Reserve Bank of Richmond Economic Quarterly

For each consumption category i, the Engel curve parameter, i ,
can be estimated as follows. The FOCs of the household’s utility maximization for consumption goods (3) imply that for any two goods,
ln ci =

i

ln cj

i ln(pi =pj ):

(7)

j

Let cQk
i denote the quantity of consumption for category i by the household in the kth quintile of the income distribution. Assuming that
households face the same prices, we get
!
!
Q5
p
c
pi cQ5
j
j
i
i
ln
=
ln
;
(8)
j
pi cQ1
pj cQ1
i
j
and we can infer the relative Engel curves between categories i and j,
i = j , from the cross-sectional nominal consumption ratios of the respective categories for households in the …fth and …rst income quintiles.
Based on the cross-sectional CEX of 2005 and 2015, we compute
the relative (to total expenditure) Engel curve parameters, si , third
and fourth column of Table 1,
si =

Q1
ln pi cQ5
i =pi ci

ln (P C Q5 =P C Q1 )

:

(9)

The relative Engel curve parameters for the two years di¤er somewhat,
but they do not change much over the decade, and their ranking stays
roughly constant. The last column of Table 1 displays the average
relative Engel curve parameters for the two years, which we use in our
calculation of the composition-adjusted labor wedges.
For a given aggregate intertemporal elasticity of substitution, we
calculate the levels of the corresponding Engel curve parameters as i =
si . The measured relative Engel curve parameters indicate an above
(below) average response of a category’s consumption to an increase of
income for si > 1(< 1). The average relative parameter is about 1:1,
thus the average Engel curve parameter is close to .
While the CEX contains information that we can use to calculate
the slope of household Engel curves, it does not contain information
on prices, and it is well-known that aggregate nominal expenditures
from the CEX and the more widely used NIPA Personal Consumption
Expenditures (PCE) diverge over time. For the prices of CEX consumption categories, we use the corresponding price index from the
CPI, except for “entertainment” and “cash contribution.” For the latter two categories, we use the aggregate CPI since the CPI does not
have separate price indexes for them. Aggregate nominal CEX expenditures are growing at a much slower pace than aggregate PCE in the
NIPA because the CEX systematically understates durable goods and

Chang et al.: Labor-Market Wedge under Engel Curve Utility

7

luxuries in households’ expenditures. Figure 2 shows that aggregate
PCE increased 4.6 times from 1985 to 2015, whereas aggregate CEX
expenditures (CEXNET) has increased 2.4 times. We, however, focus
on the cyclical components of consumption, and the de-meaned growth
rates of the two consumption aggregates comove fairly closely, as shown
in Figure 3. The correlation coe¢ cient for the two consumption growth
rates is 0.45, and the projection of the growth rates of aggregate PCE
on those of aggregate CEX yields an R2 of 0.80.

Cyclical Behavior of Labor-Market Wedges
We …rst show that the cyclicality of the labor-market wedge constructed
with our aggregate measure of consumption from the CEX is comparable with that of labor wedges constructed from more standard measures of aggregate consumption. We then show that the labor wedge
constructed from the disaggregated CEX categories is less cyclical than
the labor wedge from the CEX aggregate. We start with our baseline
parameterization and then show that similar results obtain for other
empirically reasonable parameterizations.
The …rst column of Table 2 displays the cyclicality of the labor
wedge for our baseline parameterization and di¤erent measures of consumption.8 The …rst three rows of Table 2 display the cyclicality of
the labor wedge based on the standard single-goods utility for three
measures of aggregate consumption: all items of PCE in the NIPA,
“PCE-All,” nondurable goods and services PCE, “PCE-NDS,” and a
Divisia-Aggregate of our eight CEX expenditure categories, “CEX8Aggregate.”The PCE-All is more cyclical than the PCE-NDS, but since
our framework applies to nondurable goods, the PCE-NDS is the appropriate aggregate consumption measure. The labor wedge cyclicality
from the CEX8-Aggregate and the PCE-NDS are of similar magnitude,
with the CEX8-Aggregate-based labor wedge slightly less cyclical.
We now use the eight CEX consumption categories and construct
a labor wedge, “CEX8-Engel,” that allows for di¤erences in income
expansion paths of consumption (fourth row of Table 2). Comparing
CEX8-Engel with CEX8-Aggregate, we can see that accounting for differences in income elasticities across commodities reduces the volatility
of the labor wedge by 9.3 percent. In other words, recognizing the di¤erences in marginal utility across commodities together with the procyclical/countercyclical nature of luxuries/necessities makes true marginal
8
Again, as in Figure 1, “cyclicality” is de…ned as the regression coe¢ cient of the
labor-market wedge growth rate on the GDP growth rate.

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Federal Reserve Bank of Richmond Economic Quarterly

utility move less than is implied by the usual aggregate consumption
measure and results in a less volatile labor wedge.
In the remaining columns of Table 2, we report the cyclicality of the
labor wedge based on alternative values of the preference parameters
and . Using a smaller intertemporal elasticity of consumption magni…es the labor-wedge cyclicality— it is even harder to justify the cyclical
behavior of consumption and hours as an optimal choice of the stand-in
household. With = 0:1, the cyclicality based on the CEX8-Aggregate
increases to 4.55— the wedge moves …ve times as much as GDP over
the business cycle. The cyclicality of the “true”wedge (CEX8-Engel) is
3.85, roughly 16 percent smaller than the standard measure. Using the
larger value, = 1, that is, log utility in consumption, accounting for
nonhomothetic Engel curves reduces the wedge cyclicality by only 6.2
percent. A larger labor-supply elasticity reduces the cyclicality of the
wedge because the marginal utility of leisure increases at a slower rate
in booms. The same reduction in the cyclicality of the marginal utility of consumption from using disaggregated Engel curves then implies
a larger percentage reduction in the labor-wedge cyclicality. Overall,
correcting the movement of the marginal utility of consumption based
on the di¤erences in the Engel curve across the eight consumption categories in the CEX decreases the cyclicality of the wedge by 6 percent
to 16 percent; see row (6) of Table 2.
We obtain an upper bound on how much one can reduce the labor
wedge through modi…cations of the marginal utility of consumption by
making the marginal utility of consumption a constant, = 1; equation (2) and row (5) of Table 2. From equation (2) it follows that this
speci…cation provides an upper bound for any speci…cation of preferences for which the implied consumption index and labor supply are
positively correlated and the real wage is essentially acyclical.9 For example, with = 1 and = 0:5, assuming a constant marginal utility of
consumption reduces the estimated cyclicality of the wedge by half relative to the benchmark case. Our treatment based on nonhomothetic
Engel curves across eight categories in the CEX materialize 18.5 percent of this potential reduction in the cyclicality of wedge. Note also
that the relative contribution of our correction of the wedge remains
at 18.5 percent regardless of ’s and ’s; see row (8) of Table 2. In
the Appendix we show that this feature is a consequence of …xing the
relative Engel curve parameters and de…ning their levels proportional
to the aggregate intertemporal elasticity of substitution.
9

In particular, it includes preference speci…cations with a quality-quantity trade-o¤
along the lines of Jaimovich et al. (2019).

Chang et al.: Labor-Market Wedge under Engel Curve Utility
3.

9

CONCLUDING REMARK

Estimated DSGE models have been widely used to study economic
‡uctuations. One popular way to identify the sources of ‡uctuation in
these DSGE models is to measure shocks as “wedges”in model-implied
relationships among key aggregate time series, e.g., an optimality condition or a resource constraint. According to this method, the labormarket wedge— the gap between the real wage and the MRS between
consumption and leisure— often emerges as an important source of aggregate ‡uctuations.
In this article, we have studied the extent to which procyclical
changes in the “average quality”of aggregate consumption can account
for the volatility of the labor wedge when Engel curves are nonhomothetic. Using information on changes in consumption patterns from
the CEX, we have found that the impact of these composition e¤ects
on the labor wedge is of limited quantitative importance. They can
account for at most 6 percent to 16 percent of the labor-wedge volatility. We have also derived an upper bound on how much more general
approaches that allow for unobserved quantity-quality substitution in
consumption, such as Jaimovich et al. (2019), can reduce volatility of
the measured labor wedge. These more general speci…cations of preferences can reduce the cyclicality of the labor wedge by at most 80
percent. The particular preferences we consider, nonhomothetic Engel
curves disciplined by the cross-sectional Engel curves over eight expenditure categories in the CEX, can account for only one-…fth of that
maximal reduction.

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Federal Reserve Bank of Richmond Economic Quarterly

REFERENCES

Benhabib, Jess, Richard Rogerson, and Randall Wright. 1991.
“Homework in Macroeconomics: Household Production and
Aggregate Fluctuations.” Journal of Political Economy 99
(December): 1166–87.
Bils, Mark, Peter J. Klenow, and Benjamin A. Malin. 2018.
“Resurrecting the Role of the Product Market Wedge in
Recessions.” American Economic Review 108 (April): 1118–46.
Chang, Yongsung, and Sun-Bin Kim. 2007. “Heterogeneity and
Aggregation: Implications for Labor-Market Fluctuations.”
American Economic Review 97 (December): 1939–56.
Chari, V.V., Patrick J. Kehoe, and Ellen R. McGrattan. 2007.
“Business Cycle Accounting.” Econometrica 75 (May): 781–836.
Christiano, Lawrence J., and Martin Eichenbaum. 1992. “Current
Real-Business-Cycle Theories and Aggregate Labor-Market
Fluctuations.” American Economic Review 82 (June): 430–50.
Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans.
2005. “Nominal Rigidities and the Dynamic E¤ect of a Shock to
Monetary Policy.” Journal of Political Economy 113 (February):
1–45.
Galí, Jordi, Mark Gertler, and J. David Lopéz-Salido. 2007.
“Markups, Gaps, and the Welfare Costs of Business Fluctuations.”
Review of Economics and Statistics 89 (February): 44–59.
Hall, Robert E. 1997. “Macroeconomic Fluctuations and the
Allocation of Time.” Journal of Labor Economics 15 (January)
Part 2: s223–s250.
Havránek, Tomá¼
s. 2015. “Measuring Intertemporal Substitution: The
Importance of Method Choices and Selective Reporting.” Journal
of the European Economic Association 13 (December), 1180–204
Jaimovich, Nir, Sergio Rebelo, and Arlene Wong. 2019. “Trading
Down and the Business Cycle.” Journal of Monetary Economics
102 (April): 96–121.
Karabarbounis, Loukas. 2014. “The Labor Wedge: MRS vs. MPN.”
Review of Economic Dynamics 17 (April): 206–23.

Chang et al.: Labor-Market Wedge under Engel Curve Utility

11

Keane, Michael, and Richard Rogerson. 2012. “Micro and Macro
Labor Supply Elasticities: A Reassessment of Conventional
Wisdom.” Journal of Economic Literature 50 (June), 464–76.
Shimer, Robert. 2010. Labor Markets and Business Cycles. Princeton,
N.J.: Princeton University Press.
Smets, Frank, and Rafael Wouters. 2007. “Shocks and Frictions in
U.S. Business Cycles: A Bayesian DSGE Approach.” American
Economic Review 97 (June): 586–606.

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Federal Reserve Bank of Richmond Economic Quarterly

APPENDIX

We can rewrite the equations for the growth rates in the measured,
true, and limiting labor wedge with = 1, as follows

^m = ^1 +

1 ^m
C ;

= ^1 +

1^
C ;

^

^1 =

1^
H

\m ;
W=P

P
where C^ = si 1 ! i c^i .
In Table 2, we list the regression coe¢ cients of the growth rate in
the three labor wedges on GDP growth in rows (3), (4), and (5). Across
columns the aggregate IES and labor supply elasticity change, but the
relative IES across categories, si , remain …xed. This means that the
\m , C^ m , and C^ , are all independent
^ W=P
right-hand side variables, H,
of and .
The regressions asymptotically re‡ect the linear projections of the
labor wedges on output
(3) :

E [^m j^
y] =

E[^1 j^
y] +

1

(4) :

E [^ j^
y] =

E[^1 j^
y] +

1

(5) :

E [^1 j^
y] =

E[C^ m j^
y] =
E[C^ j^
y] =

1

+

1

+

1

1

m

1

y^

Therefore the ratios in rows (6), (7), and (8) are given by
(6) :
(7) :
(8) :

(4)
(3)
(5)
(3)

1=
1=
(6)
=
(7)

m

(
1

)=
=

m

+
(1= ) m
1
+ m=

m
m

As you can see, the relative improvements are independent of :

y^;
y^;

Chang et al.: Labor-Market Wedge under Engel Curve Utility

13

Figure 1 Cyclical Behavior of the Labor-Market Wedge

SD (%)
Cyclicality

GDP
2.06
1.00

H
2.34
0.95

C
1.31
0.56

W/P
1.50
0.19

Wedge ( )
4.55
1.88

Notes: Aggregate consumption (C) and its price are based on personal consumption expenditure (PCE) data for nondurables and services from the NIPA. Aggregate hours (H) and nominal wages (W ) are total hours and wages from the
BLS’s Labor Productivity and Cost index (LPC) for nonfarm business sectors
(https://www.bls.gov/lpc/). We use annual data and calculate their growth rates
as 100 times …rst di¤erences in logs. The labor-market wedge is computed for
=0:5 and =1. SD denotes the standard deviation, and “Cyclicality” denotes
the regression coe¢ cient on GDP growth.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Nominal Consumption Expenditures

Note: Nominal expenditures of personal consumption expenditure of all categories
(PCE-All) and those of CEX net of pension and insurance (CEXNET).

Chang et al.: Labor-Market Wedge under Engel Curve Utility

15

Figure 3 Cyclical Components of Consumption

Note: Real consumption growth of PCE nondurables and services (PCE-NDS),
CEXNET, and CEX8.

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Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Relative Engel Curves
[-1.5ex] Category
Food at Home
Food away from Home
Transportation
Housing
Health Care
Apparel
Entertainment
Cash Contribution
Sum of 8 Categories (CEX8)
Others
Sum of All Above (CEXNET)
Personal Insurance and Pension
All Items

Share (%)

Relative Engel (

i

)

2005

2015

2005

2015

Avg.

7.1
5.7
10.3
32.6
5.7
4.1
5.1
3.5
73.8
15.0
88.8
11.2
100.0

7.2
5.4
9.9
32.9
7.8
3.3
5.1
3.2
75.1
13.6
88.7
11.3
100.0

0.78
1.30
1.28
1.10
0.84
1.23
1.45
1.64
–
–
–
2.82
1.00

0.68
1.15
1.18
1.01
0.95
1.12
1.13
1.29
–
–
–
2.5
1.00

0.73
1.23
1.24
1.06
0.90
1.18
1.29
1.47
–
–
–
2.66
1.00

Notes: The data are based on the annual overall expenditure shares and mean
expenditures of the …rst and …fth income quintiles (before taxes) from the Consumer Expenditure Surveys of 2005 and 2015. “Transportation” excludes vehicle
purchases. “Others” are other miscellaneous categories and “Cash Contribution”
is cash donation.

Chang et al.: Labor-Market Wedge under Engel Curve Utility

17

Table 2 Cyclicality of Labor Wedges
[-1.5ex]

Consumption Measure
for Marginal Utility

= 0:5
=1

= 0:1
=1

(1)

PCE-All

2.15

7.71

1.46

1.68

3.10

(2)

PCE-NDS

1.88

6.35

1.32

1.40

2.83

(3)

CEX8-Aggregate

1.52

4.55

1.14

1.05

2.47

(4)

CEX8-Engel

1.38

3.85

1.07

0.90

2.33

(5)

Constant M UC

0.76

0.76

0.76

0.29

1.71

(6)

(4)
(3)
(5)
(3)
(6)
(7)

1

-9.2%

-15.4%

-6.2%

-13.4%

-5.7%

1

-50%

-83%

-33%

-73%

-31%

18.5%

18.5%

18.5%

18.5%

18.5%

(7)
(8)

=1
=1

= 0:5
=2

= 0:5
= 0:5

Notes: Rows (1) through (5) display the regression coe¢ cient of labor-market
wedge growth rates on GDP growth rates for di¤erent measures of consumption in
the construction of marginal utility of consumption (M UC ). Rows (1) and (2) use
personal consumption expenditures (PCE) from the NIPA, all categories or nondurable goods and services only. Rows (3) and (4) use the eight categories in the
CEX, where CEX8-Aggregate uses the Divisia-Aggregate and CEX8-Engel uses
the CEX8-Components together with the relative Engel curve parameters from
the last column of Table 1. Row (5) considers the limit for
large, when M UC
is a constant and independent of the measure of consumption.

Economic Quarterly— Volume 106, Number 1— First Quarter 2020— Pages 19–40

Technology Di usion: The
Case of Internet Banking
Richard Sullivan and Zhu Wang

N

ew ideas, embodied in product and technology innovations,
are fundamental driving forces for long-run growth. However,
it often takes many years for an innovation to become widely
adopted by the population, a process termed “di¤usion.”Moreover, the
speed of di¤usion is rarely constant. Rather, we typically observe di¤usion curves that depict cumulative adoption over time to be S -shaped.
To better understand the di¤usion process, an extensive literature has
been developed that seeks to explain how, why, and at what rate new
ideas and technologies spread.
A large body of literature emphasizes the role played by communication of information (Rogers, 2003). One of the most popular theories focuses on contagion, or the so-called “word-of-mouth” e¤ect, in
which agents adopt innovations when they come in contact with others
who have already adopted; in other words, innovations spread like epidemics. Two alternative but related theories are social in‡uence and
social learning, which attribute contagion to social forces such as conformity motive or belief updating. A common theme of these theories
is that the di¤usion process is driven by internal feedback e¤ects from
prior to future adopters (see, e.g. Young 2009 for an overview of the
“internal di¤usion” models). These models are particularly appealing
for empirical uses because the internal feedback e¤ect can be formalized as a di¤erential equation that generates logistic di¤usion curves
(e.g., Griliches 1957, Mans…eld 1961, Bass 1969, 2004).
In contrast, a competing view in the literature emphasizes agents’
heterogeneity in terms of adoption costs and bene…ts (e.g., David 1969,
We thank Abigail Burns, Arantxa Jarque, Elliot Tobin, John Weinberg, and Russell
Wong for helpful comments. The views expressed herein are solely those of the
authors and do not necessarily re‡ect the views of the Federal Reserve Bank of
Richmond or the Federal Reserve System.

20

Federal Reserve Bank of Richmond Economic Quarterly

Jovanovic and MacDonald 1994, Stoneman 2002). According to this
view, di¤usion lags are not necessarily explained by incomplete information. Rather, agents may have complete information and make adoption decisions based on their heterogeneous willingness to pay for the
innovation. As a result, di¤usion is driven mainly by external factors,
such as price reduction or quality improvement, and di¤usion curves
can be S -shaped if the adoption thresholds of agents follow a positively
skewed distribution.
In this paper, we incorporate and extend the ideas from the literature to study the di¤usion of a recent technological innovation, internet
banking. We consider that bank size follows a log-logistic distribution
due to cost heterogeneity. Internet banking technology requires a …xed
cost for adoption but reduces marginal cost of operation. As a result, when it is initially introduced, large banks enjoy advantages for
adoption because of their size. Over time, due to external changes (e.g.,
demand shift, technological progress, and/or deregulation), the innovation gradually di¤uses into smaller banks. This approach is consistent
with the external di¤ usion view and predicts the timing of adoption by
bank size that is consistent with the data. Moreover, this approach is
able to generate a logistic di¤usion curve that resembles those derived
from the internal di¤ usion models. We test the theoretical hypothesis with an empirical study of internet banking di¤usion among banks
across …fty U.S. states. Using an instrument-variable approach, we
identify a positive e¤ect of average bank size on internet banking diffusion. The empirical …ndings also allow us to examine technological,
economic, and institutional factors governing the di¤usion process, and
explain the variation in di¤usion rates across geographic regions.
As mentioned above, our study is directly related to the literature on technology di¤usion. In the banking context, several recent
studies have looked at the internet and related technology adoption in
the banking industry. For example, Hernández-Murillo et al. (2010)
study a panel of commercial banks for 2003–2006 and show that banks
adopt online banking earlier in markets where their competitors have
already done so. DeYoung et al. (2007) study a sample of U.S. banks
in the late 1990s. They …nd that branching intensity and online banking are complementary and online banking adoption positively a¤ects
the bank’s future performance. Courchane et al. (2002) develop and
estimate a model for early adoption of internet banking. They …nd
that relative bank size and demographic information predictive of future demand positively in‡uence internet banking adoption. Furst et
al. (2001) estimate a logit model for internet banking adoption in a
sample of national banks. They …nd that larger banks and banks that
are younger and better-performing are more likely to adopt internet

Sullivan & Wang: Technology Di usion: Internet Banking

21

banking. However, unlike our paper, these studies focus more on individual banks’adoption decisions rather than the aggregate pattern of
di¤usion and bank-size distribution.1
The paper is organized as follows. Section 1 introduces industry
background regarding the banking sector and internet banking di¤usion. Section 2 describes the framework of our empirical study. Section
3 discusses our …ndings on internet banking di¤usion among banks
across …fty U.S. states. Section 4 concludes.

1.

INDUSTRY BACKGROUND

In our study, internet banking is de…ned as a bank providing a website
that allows customers to execute transactions on their accounts. In the
United States, the history of internet banking can be traced back to
1995, when Wells Fargo …rst allowed its customers to access account
balances online.2 Since then, banks have steadily increased their online presence. Figure 1 plots the di¤usion of internet banking among
in-state banks from 2003 through 2007, before the start of the Great
Recession.3 In-state banks refer to commercial banks focusing on operating in a single state, which accounted for more than 90 percent
of the U.S. banking population during this period.4 The …gure shows
that 51.8 percent of in-state banks had adopted internet banking by
2003, and the ratio continued to rise to 81.5 percent in 2007. A similar
di¤usion pattern can be found if we instead consider all U.S. commercial banks. By 2003, 53 percent of all commercial banks had adopted
transactional websites, and the ratio rose to 82 percent in 2007.
However, the di¤usion pattern varies signi…cantly across bank-size
groups and geographic regions. First, looking across size groups, large
banks appear to have an advantage over smaller ones in adopting the
1
Note that adoption and di¤usion are two related but di¤erent terms used in the
literature. Adoption typically refers to an individual process of adopting an innovation,
while di¤usion is a group phenomenon that refers to how an innovation spreads.
2
Internet-only banks account for a very small fraction of the U.S. banking population (less than 0.5 percent even during the dot-com boom years). In this paper, we
focus on the internet banking adoption among traditional brick-and-mortar banks. See
Wang (2007) for an analysis of internet-only banks.
3
Data source: call reports. Since 2003, depository institutions have been required
to report whether their websites allow customers to execute transactions on their accounts. Our sample ends in 2007 because adoption had become almost universal by
then and we also want to avoid the disruption of the Great Recession.
4
More speci…cally, a bank is classi…ed as an in-state bank if all its deposits are
in the state of the bank’s headquarters. As will become clear, focusing on this group
of banks allows us to avoid the complications of interstate banking when measuring
internet banking di¤usion and bank size distribution by state. In 2003, there were 7,712
commercial banks in the United States, among which 7,183 were in-state banks (i.e., 93
percent).

22

Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Internet Banking Di usion

innovation. As shown in Figure 2, 90.5 percent of in-state banks with
deposits over $300 million reported that they had a transactional website in 2003, compared with only 10.5 percent of in-state banks with
deposits under $25 million. The variation is also striking across geographic regions. Figure 3 compares internet banking di¤usion among
in-state banks across U.S. states in 2003. The northeast and the west
regions had the highest adoption rates (i.e., 65 percent to 85 percent in
each state), while the central regions of the country had the lowest (i.e.,
25 percent to 45 percent in each state). These observations raise important questions regarding technology di¤usion: Why do large banks
tend to be early adopters of the internet innovation? What determines
the di¤erent di¤usion rates across bank groups and geographic regions?
These observations and questions motivate our study. Conceptually, the bene…ts of internet banking can be viewed as twofold. First,
it brings convenience to bank customers, allowing them to use services
from banks at a distance and avoid hassles like traveling to ATMs
or branches. Second, it generates substantial cost savings to banks.
Most banking websites provide balance-transfer and bill-payments services, and some also process applications for deposits, loans, and credit

Sullivan & Wang: Technology Di usion: Internet Banking

23

Figure 2 Internet Banking Adoption by Bank Size Group
(Deposits in Millions)

cards.5 This allows banks to conduct standardized, low-value-added
transactions through the online channel while focusing their resources
on more specialized, high-value-added transactions (e.g., business lending, personal trust services, investment banking) through branches. In
fact, the ratio of bank employees (and bank tellers) to deposits has
been declining since the late 1990s.6 This is consistent with continuous
progress in information technology, including the increasing adoption
of internet banking.
5
For instance, a survey conducted by the Federal Reserve Bank of Kansas City
shows that in the tenth Federal Reserve District, more than 70 percent of commercial
bank websites provided balance-transfer and bill-payment services, and less than 20 percent allowed for online application for deposits, loans, or credit cards in 2006.
6
From 1997 through 2007, the number of bank employees per million-dollar deposits
fell from 0.44 to 0.24, and the number of bank tellers per million-dollar deposits fell
from 0.14 to 0.09. (Data sources: Commercial bank employees and tellers are from the
Bureau of Labor Statistics, and commercial bank deposits are from the Federal Deposit
Insurance Corporation.)

24

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Internet Banking Adoption by State (2003)

2.

RESEARCH FRAMEWORK

In this section, we describe the framework for our empirical study. The
framework is built upon the theoretical model proposed by Sullivan
and Wang (2017), which characterizes the relationship between banksize distribution and internet banking di¤usion.

Theoretical hypothesis
According to the theory of Sullivan and Wang (2017), the banking industry is composed of a continuum of banks that produce homogenous
banking services and take prices as given. Banks are heterogeneous in
productivity and their size follows a log-logistic distribution. As shown
in Figure 4, the log-logistic distribution …ts the banking industry data
very well.
When internet banking technology is introduced, banks of di¤erent
size need to make adoption decisions. Because the technology requires
a …xed cost of adoption but reduces the marginal cost of banking operation, this pins down a threshold size of adoption. As a result, large
banks have an advantage adopting the new technology. Over time,
as the deep model parameters (e.g., consumer willingness to pay for
banking services, average bank productivity, cost savings due to internet banking adoption, and adoption costs of internet banking) change,
the technology di¤uses into smaller banks. Particularly, given that the

Sullivan & Wang: Technology Di usion: Internet Banking

25

Figure 4 Bank-Size Distribution (In-State Banks 1990)

bank-size distribution follows a log-logistic distribution, as long as those
deep model parameters have approximately linear time trends, the diffusion path of internet banking would follow a logistic curve, a path
well documented in the technology di¤usion literature (e.g., Griliches
1957, Mans…eld 1961, Bass 1969, 2004).
Figure 5 illustrates the industry dynamic path. Before internet
banking is introduced, the banking industry stays at a log-logistic size
distribution, drawn with a dotted line. After internet banking becomes available, in the long run, the banking industry converges to a
post-innovation, long-run size distribution (which again is log-logistic),
drawn with a solid line. In between, the bank-size distribution is on a
transitional path, drawn with a dashed line. At a given time t during
the transition, a bank can always compare two options: adopting internet banking or not. Under each option, the size of the bank is denoted
as ya;t or yn;t . There is a size threshold yn;t at time t, which splits the
pre-innovation size distribution. For banks with size yn;t
yn;t , the
size distribution resembles the post-innovation, long-run distribution
1

1

in the range ya;t 2 [ t 1 yn;t ; 1), so t 1 yn;t is the minimum size of
adopters (Note that, as explained in Sullivan and Wang 2017, > 1 is

26

Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Illustration of the Industry Dynamics

the cost-saving parameter associated with adopting the innovation and
1 > > 0 is the cost elasticity in banks’production function). Meanwhile, for banks with size yn;t < yn;t ; the size distribution resembles
the pre-innovation one, so yn;t is the maximum size of non-adopters.
1

Over time, yn;t and t 1 yn;t fall due to external changes (e.g., demand
shift, technological progress, and/or banking deregulation). As a result, internet banking di¤uses into smaller banks, and the bank-size
distribution gradually converges to the post-innovation, long-run distribution.

Empirical speci cation
The focus of this article is to test the theoretical hypothesis with an
empirical study on internet banking di¤usion. The sample that we
consider includes all in-state banks in each of the …fty U.S. states from
2003 through 2007. Focusing on in-state banks allows us to avoid the

Sullivan & Wang: Technology Di usion: Internet Banking

27

complications of interstate banking when measuring internet banking
di¤usion and bank-size distribution at the state level.7
The theory in Sullivan and Wang (2017) shows that the di¤usion of
internet banking is characterized by two jointly determined endogenous
variables: aggregate internet banking adoption rate and average bank
size. According to the theory, using state-level data (where each state
is indexed by j and each year is indexed by t), the aggregate adoption of internet banking, adjusted by the Gini coe¢ cient of bank-size
distribution, can be speci…ed as
gj;t ln(

X
Fj;t
) = a0 + a1 ln(E(y)j;t ) +
ai ln(Xi;j;t ) + "j;t ;
1 Fj;t

(1)

i

F is the aggregate adoption rate of internet banking.
g is the Gini coe¢ cient of bank-size distribution.
E(y) is the average bank size.
X denotes other explanatory variables.
" is an i.i.d. random error.
To estimate the equation, we need to collect empirical variables.
Moreover, given the endogeneity of average bank size E(y), we need
to use instrument variables to correctly identify the e¤ect of average
bank size on internet banking di¤usion. The instrument variables are
supposed to only a¤ect internet banking di¤usion through average bank
size.
Below is a list of the empirical variables used in our estimation.
(See Tables 4 and 5 in the Appendix for the data sources and summary
statistics.) For most of these variables, we take the log transformation
and pre…x the variables with “ln” in the notation.
The dependent variable (a measure of internet banking di¤usion):
lnTRANODDS_GINI – Log odds ratio for the internet banking
adoption rate adjusted by the Gini coe¢ cient, constructed using two
variables: TRANS – Adoption rate for transactional websites, and
GINI –Gini coe¢ cient for bank deposits.
An endogenous explanatory variable (a measure of average bank
size):
7

While our empirical study does not directly consider interstate banks, we include
the out-of-state bank presence in the in-state banking market as a regressor to control
for the demand for the services of in-state banks.

28

Federal Reserve Bank of Richmond Economic Quarterly

lnDEPOSITS –Log average bank size, constructed by the variable
DEPOSITS –Average bank deposits.8
We then consider two groups of explanatory variables in X and a
set of instrument variables I, listed as follows.
Variables in X that a¤ ect both internet banking di¤ usion and bank
size:
METRO –Ratio of banks in metropolitan areas to all banks.
LOANSPEC –Specialization of lending to consumers.9
OFF_DEP –Bank o¢ ces per value of deposits.
RMEDFAMINC –Real median family income in 1967 dollars.
POPDEN –Population density.
AGE –Average age of banks.
HHINET –Household internet access rate.
WAGERATIO –Ratio of computer analyst wage to teller wage.
BHC –Ratio of banks in bank holding companies to total banks.
DEPINT –Ratio of deposits in out-of-state banks to total deposits.
REGION and YEAR –Dummies.10
Variables in X that only a¤ ect internet banking di¤ usion:11
IMITATE –Years since the …rst bank in the state adopted a transactional website.
COMRATE –Adoption rate of high-speed internet among commercial …rms in 2003, calculated as an average of urban …rms’ and rural
…rms’ internet adoption using METRO to weight urban and rural location. Essentially, COMRATE measures in-state banks’ exposure to
other commercial …rms’internet adoption in each state.
Instrument variables in I that only a¤ ect average bank size:
DEPOSITS90 –Average bank deposits in 1990.
INTRAREG – A dummy variable for whether the state had intrastate branching restrictions after 1995.
Some variables in X a¤ect both internet banking di¤usion and average bank size. Take HHINET for example: if more households have access to the internet, local banks may get more cost savings from adopting internet banking. However, internet access also allows households
to reach nonlocal banking services (e.g., out-of-state banks), which may
then lower demand and consumer willingness to pay for local banking
services. AGE is another example: established banks typically achieve
8

Note that the empirical results would have been similar if we had used bank assets
as an alternative measure of bank size.
9
De…ned by consumer loans plus 1-4 family mortgages divided by total loans.
10
Regional dummies refer to eight geographic areas de…ned by the Bureau of Economic Analysis.
11
Sullivan and Wang (2017) show that these variables can serve as instruments to
estimate the e¤ects of internet banking adoption on average bank size.

Sullivan & Wang: Technology Di usion: Internet Banking

29

higher productivity, so they may enjoy a large size. However, established banks may also face a higher internet banking adoption cost
compared to young banks given that they have to adapt internet banking to their legacy computer systems.
We also consider two variables that only directly a¤ect internet
banking di¤usion but not average bank size: the number of years since
the …rst bank in the state adopted a transactional website (IMITATE)
and internet adoption rate among commercial …rms in the state (COMRATE). The former variable, IMITATE, is from the Online Banking
Report, a publication keeping track of the development of internet banking. The data suggest that the …rst wave of internet banking was largely
driven by exogenous factors (such as entrepreneurs’risk-taking experiments) rather than cost-bene…t calculations assumed in our model. In
fact, the correlation between a state’s …rst internet banking adoption
(measured by IMITATE in 2003) and the average bank size in 1990 is
-0.001. To some extent, this variable may capture the contagion e¤ect
suggested by the internal di¤ usion models, but we could also think that
a higher value of IMITATE may reduce internet banking adoption costs
by providing more local expertise on bank-speci…c website design and
performance. The latter variable, COMRATE, is constructed based on
the information provided by Forman et al. (2003). The e¤ect of COMRATE might be ambiguous in theory. On the one hand, a higher value
of COMRATE may help internet banking di¤usion through an imitation e¤ect. On the other hand, it may delay internet banking di¤usion
by competing away resources and pushing up local costs of internet
installation and operation. Therefore, we will rely on our empirical
estimation to evaluate the overall e¤ect of COMRATE.
The two instrument variables we include in I are intuitive: a dummy
variable for whether the state had intrastate branching restrictions after
1995 (INTRAREG) and average bank deposits in 1990 (DEPOSITS90).
The former value is from Kroszner and Strahan (1999), and the latter
is from the call reports. Both variables are expected to a¤ect internet banking di¤usion only through their e¤ects on average bank size:
INTRAREG may negatively a¤ect the average bank size by imposing high regulation costs; DEPOSITS90 may be positively correlated
with current average bank size through the persistence of underlying
productivity variables.

3.

EMPIRICAL FINDINGS

Our following discussions focus on the estimation results based on a
2SLS (two-stage least squares) model. In the …rst stage, we regress the
average bank size (lnDEPOSITS) on all the exogenous variables listed

30

Federal Reserve Bank of Richmond Economic Quarterly

in groups X and I. In the second stage, we then use the …tted value
of (lnDEPOSITS) instead of the actual value to estimate equation (1).
Both the …rst-stage and the second-stage results are reported in Tables
1 and 2. For comparison, we also include the OLS result.

Model validation
The 2SLS results suggest that the instrument variables we use are
valid. In the …rst-stage average bank-size regression, the coe¢ cients on
INTRAREG and lnDEPOSITS90 have the expected signs and lnDEPOSITS90 is statistically signi…cant. The relevance of the instruments
is also con…rmed by the F-test. As a rule of thumb, the F-statistic of
a joint test where all excluded instruments are signi…cant should be
bigger than ten in case of a single endogenous regressor. As shown
in Table 1, this is satis…ed in our regression. Moreover, because we
have two instruments for each endogenous variable, we can perform
the overidenti…cation test. This test checks whether both instruments
are exogenous assuming that at least one of the instruments is exogenous. As shown in Table 1, the 2 statistics show that we cannot reject
the null hypothesis that our instruments are exogenous.
We also test whether the 2SLS estimates are statistically di¤erent
from the OLS estimates. The is done by rerunning second-stage regressions where the residuals from the …rst-stage regressions are included
(Wooldridge 2010, Chapter 5).12 This test is robust to heteroscedasticity given that the robust variance estimator is used. The results show
that the coe¢ cient of the …rst-stage residual is statistically signi…cant,
which con…rms that instrumenting matters for the estimation.

Economic

ndings

We now turn to the economic …ndings based on the second-stage estimation results shown in Tables 1 and 2. The model …ts the data well,
with an R2 of 0.75. Most signs of estimated coe¢ cients, and all of
those that are statistically signi…cant, are consistent with the theoretical predictions. The …ndings are summarized as follows.
The coe¢ cient on the …tted value of lnDEPOSITS is positive and
statistically signi…cant. The …nding supports our theoretical hypothesis that average bank size has a positive causal e¤ect on internet banking di¤usion. Quantitatively, considering a Gini coe¢ cient equal to
12

An alternative is to run the Hausman test, but the Hausman test is only valid
under homoscedasticity and involves the cumbersome generalized inversion of a nonsingular matrix.

Sullivan & Wang: Technology Di usion: Internet Banking

31

Table 1 Estimation of Internet Banking Adoption
(Dependent Varable: lnTRANODDS GINI)

First Stage

2SLS
Second Stage

lnDEPOSITS
lnIMITATE
lnCOMRATE
INTRAREG
lnDEPOSITS90
lnMETRO
lnLOANSPEC
lnRMEDFAMINC
lnPOPDEN
lnAGE
lnHHINET
lnBHC
lnWGRATIO
lnDEPINT
lnOFF_DEP
Constant
Adjusted R2
N
Weak Instrument
Test: F(2,201)y
Exogeneity of
Regressors-Wald
Test
Overidenti…cation
Test: Chi2(1)

0.3933
(0.2848)
-4.9335
(1.0055)***
-0.1001
(0.0764)
0.4572
(0.0694)***
0.7520
(0.2166)***
0.3773
(0.2138)*
0.2582
(0.5425)
0.0994
(0.0681)
0.2163
(0.1581)
1.0941
(0.6718)
1.9964
(0.4520)***
-0.5468
(0.3983)
-0.1557
(0.0477)***
-0.3453
(0.1175)***
-1.2171
(2.3079)
0.78
227

OLS

0.5716
(0.0848)***
0.1135
(0.1754)
-0.9002
(0.9023)

0.2467
(0.0436)***
0.1915
(0.1530)
-2.0247
(0.7779)***

0.1060
(0.1636)
-0.0837
(0.1441)
-0.5276
(0.3653)
-0.1059
(0.0426)**
-0.3449
(0.1063)***
1.6906
(0.3598)***
0.0764
(0.2211)
0.0033
(0.2575)
0.0949
(0.0327)***
0.3009
(0.0851)***
-8.2948
(1.3169)***
0.75
227

0.3926
(0.1280)***
0.0511
(0.1205)
-0.4229
(0.3247)
-0.0844
(0.0324)***
-0.3696
(0.0928)***
1.7774
(0.3507)***
0.5697
(0.1616)***
-0.1073
(0.2257)
0.0487
(0.0281)*
0.1244
(0.0629)**
-5.8434
(1.1062)***
0.82
227

Reduced
Form

0.3384
(0.1506)**
-3.7200
(0.7026)***
-0.0574
(0.0493)
0.2613
(0.0463)***
0.5357
(0.1231)***
0.1319
(0.1191)
-0.3799
(0.3451)
-0.0490
(0.0329)
-0.2213
(0.0872)**
2.3160
(0.3779)***
1.2176
(0.1804)***
-0.3093
(0.2177)
0.0059
(0.0342)
0.1035
(0.0762)
-8.9911
(1.3336)***
0.83
227

18.45
-4.52***
0.00

0.57 (the average value in 2003), the results imply that holding everything else constant, a 10 percent increase in average bank size would

32

Federal Reserve Bank of Richmond Economic Quarterly

Table 2 Estimation of Internet Banking Adoption (cont'd)

First Stage
d2004
d2005
d2006
d2007
Southeast
Far West
Rocky Mtn
Southwest
NE
Mideast
Great Lakes

2SLS
Second Stage

-0.0636
(0.0975)
-0.0383
(0.1251)
-0.1251
(0.1502)
-0.1317
(0.1764)
0.2575
(0.1378)*
0.9697
(0.1666)***
0.3365
(0.1515)**
0.3933
(0.1335)***
0.3811
(0.2509)
-0.3424
(0.2099)
-0.3125
(0.1332)**

0.1431
(0.0578)**
0.2627
(0.0779)***
0.4232
(0.0911)***
0.5446
(0.1061)***
-0.0623
(0.1010)
-0.4340
(0.1534)***
-0.2374
(0.0877)***
-0.0688
(0.0898)
-0.2810
(0.1406)**
0.0647
(0.1527)
0.1196
(0.0871)

OLS

Reduced
Form

0.1362
(0.0482)***
0.2750
(0.0658)***
0.4246
(0.0820)***
0.5507
(0.0980)***
0.0847
(0.0850)
-0.0500
(0.1094)
-0.1454
(0.0712)**
0.0829
(0.0796)
0.0842
(0.1112)
0.2995
(0.1223)**
0.1716
(0.0731)**

0.1068
(0.0477)**
0.2408
(0.0666)***
0.3517
(0.0883)***
0.4693
(0.1030)***
0.0849
(0.0866)
0.1203
(0.0907)
-0.0450
(0.0790)
0.1561
(0.0942)*
-0.0632
(0.1314)
-0.1308
(0.1582)
-0.0590
(0.0700)

Notes: Equations are estimated using two-stage least-squares for the time period
2003 through 2007. Robust standard errors are in parentheses. Estimated coe¢ cients for other variables in the model equations are in Table 1. * p < 0.1; **
p < 0.05; *** p < 0.01

increase the adoption odds ratio by about 10 percent. To put things
into perspective, we consider a case where the internet adoption rate
is 56.4 percent and the average bank deposits are $311 million, which
are mean values of the 2003 data. Therefore, based on the 2003 data,
a one-standard-deviation increase of average bank deposits from the
mean would increase the internet banking adoption rate from 56.4 percent to 77.1 percent.13 The …nding is in sharp contrast with the OLS
regression result. Without addressing the endogeneity of regressors, the
OLS results underestimate the impact of average bank size on internet
banking di¤usion by more than a half.
13

This is calculated by solving F , where 0:57
[ln(311 + 496) ln(311)]:

[ln( 1 FF )

ln( 1 0:564
)] = 0:5716
0:564

Sullivan & Wang: Technology Di usion: Internet Banking

33

We also …nd that population density (lnPOPDEN) has a signi…cant
e¤ect on internet banking di¤usion. The e¤ect is negative, suggesting
a higher demand for internet banking in locations with higher cost of
travel to bank branches. The average bank age in a state (lnAGE)
shows a negative e¤ect, which implies that as the average age of a
state’s banks increases, the adoption rate falls. This results is consistent
with previous …ndings that de novo banks were more likely to adopt
internet banking than incumbent banks (Furst et al. 2001). New banks
may …nd it cheaper to install internet banking technology in a package
with other computer facilities compared with older banks that must
add internet banking to legacy computer systems. Household access
to the internet (lnHHINET) is also statistically signi…cant, and greater
household access to the internet is associated with a higher adoption
of internet banking. Competition from out-of-state banks (lnDEPINT)
has a positive coe¢ cient, suggesting that more deposits in out-of-state
banks push more in-state banks to adopt internet banking (possibly
in order to compete for business). We also …nd that bank o¢ ces per
value of deposits (lnOFF_DEP) is statistically signi…cant. The positive
coe¢ cient implies that banks with more o¢ ces may try to explore the
synergy between branch banking and internet banking.14
Finally, all the year dummies are statistically signi…cant. This suggests that after controlling for the other explanatory variables, there is
a positive year trend for internet banking di¤usion. In contrast, most
regional dummies are not signi…cant or have a negative sign, in comparison with the excluded Plains states, which have the lowest internet
banking adoption. This suggests that the observed cross-region di¤erences of internet banking adoption rates are mainly driven by the other
explanatory variables in our model rather than the remaining regional
…xed e¤ects. We will discuss this further below.

Regional variations
Our empirical study identi…es a positive e¤ect of average bank size on
internet banking di¤usion. As explained by the theory, this is because
large (more e¢ cient) banks enjoy scale economies of adoption. Moreover, our empirical study can help explain the variation in internet
banking di¤usion across geographic regions. Particularly, why do the
northeast and the west regions have the highest adoption rates, while
the central regions have the lowest (see Figure 3)?
14
This …nding is consistent with optimization of branch network size that encompasses both branch-based and non-branch-based activities (Hirtle, 2007).

34

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Mean Values of Selected Variables by Region (Far
West, Plains, and New England 2003)*
Variables

TRANS
GINI
DEPOSITS90
IMITATE
HHINET
METRO
BHC
COMRATE
AGE

E¤ect on IB

Far West

Plains

New England

+
+
+
+
+

0.71
0.59
217.9
5.80
0.61
0.95
0.66
0.90
25.6

0.43
0.60
37.5
6.71
0.55
0.51
0.87
0.90
81.6

0.67
0.50
289.9
6.40
0.60
0.79
0.62
0.88
68.1

*Far West includes AK, CA, HI, NV, OR, and WA; Plains includes
IA, KS, MN, MO, NE, ND, and SD; New England includes CT, MA,
ME, NH, RI, and VT.

In order to answer the question, we run a reduced-from regression in
which the dependent variable (lnTRANODDS_GINI) is regressed on
all the exogenous variables listed in groups X and I. In doing so, we
bypass the endogenous variable of average bank size, and measure the
overall e¤ect of each exogenous variable on internet banking di¤usion
(i.e., by taking into account their direct e¤ects on internet banking diffusion and indirect e¤ects through average bank size). The results are
also reported in Tables 1 and 2. In Table 3, we present regional averages of variables that are found to signi…cantly a¤ect internet banking
di¤usion in the reduced-form regression. Far West, Plains, and New
England are used to represent the west, central and northeast regions
respectively. As shown, the Plains region had a similar Gini coe¢ cient
of bank size in 2003 as the Far West and New England, but the internet banking adoption rate was much lower. Compared with the other
two regions, we …nd that the Plains region has smaller initial bank
size, lower household internet access, fewer banks in metro markets,
and older bank vintages. Based on the coe¢ cients (marginal e¤ects)
that we uncovered from the reduced-form regression, we conclude that
these are the factors that have contributed to slow di¤usion of internet
banking in the Plains region. On the other hand, our …ndings reject
several alternative hypotheses that may have seemed appealing, including imitation of early adopters, internet adoption of commercial …rms,
and bank holding company membership. In fact, some of those would
have been the Plains region’s advantages for adoption.

Sullivan & Wang: Technology Di usion: Internet Banking

35

We also rule out several other factors that are only found to signi…cantly a¤ect internet banking di¤usion in the second stage of our 2SLS
regression, such as deposits held in out-of-state banks, population density, and bank o¢ ces per value of deposits. This is because those factors
have opposite e¤ects on the average bank size (see Sullivan and Wang,
2017). Therefore, their overall e¤ects on internet banking di¤usion become insigni…cant in the reduced-form regression where the interaction
e¤ects between internet banking di¤usion and average bank size are
taken into account.

Internal versus external di usion
Our empirical analysis sheds light on the debate regarding internal and
external di¤usion models. The classic internal di¤usion models (e.g.,
Griliches 1957, Mans…eld 1961, Bass 1969, 2004) typically assume that
the hazard rate of adoption increases with cumulative adoption due to
contagion or the “word-of-mouth” e¤ect:
dFt =dt
= vFt ;
1 Ft
where Ft is the fraction of potential adopters who have adopted the
innovation at time t, and v is a constant contagion parameter. Solving
this …rst-order di¤erential equation yields the logistic function
Ft =

1
1 + ( F10

1)e

vt

;

which implies that one could use our state-level internet banking di¤usion data to estimate a simple log-linear equation:
ln(
F

Fj;t
) = aj + vj t;
1 Fj;t

(2)

where aj = ln( 1 j;0
Fj;0 ):
Comparing with the regression model (1) used above, model (2)
suggests that the di¤usion process in a state j can now be explained by
two state-speci…c parameters: the initial condition aj and the contagion
rate vj . Such a model predicts an S -shaped logistic di¤usion curve,
which could serve as a convenient tool for data …tting or forecasting.
However, it is di¢ cult to explore deeper economic questions beyond
that, for example, why the contagion rate, or the “word-of-mouth”
e¤ect, di¤ers across regions, and why large banks rather than small
banks tend to be the early adopters.
In contrast, the external-di¤usion approach we take in this paper
provides a better micro-founded explanation. By modeling explicitly

36

Federal Reserve Bank of Richmond Economic Quarterly

the size heterogeneity of banks, we keep the appealing feature of S shaped logistic di¤usion curves but connect them to more meaningful
economic factors. Our empirical …ndings, besides providing good …tting
of the data, o¤er several additional insights:

First, employing instrument variables in the estimation con…rms
the causal e¤ect of …rm-size distribution on technology di¤usion,
which justi…es the external-di¤usion approach we take.

Second, the variation in di¤usion rates across regions can be
well explained by underlying technological, economic, and institutional factors. We …nd that, after controlling for those variables in the regressions, regional dummies are left with little
explanatory power.

Finally, technology di¤usion and …rm-size distribution are jointly
determined, so they should not be treated exogenously to each
other. As our results show, without addressing the endogeneity
problem, the OLS regression results can be signi…cantly biased.

4.

CONCLUSION

Taking internet banking as an example, we study di¤usion of costsaving technological innovations. When such an innovation is initially
introduced, large …rms enjoy adoption advantages. Over time, due to
external changes (e.g., demand shift, technological progress, and/or
deregulation), the innovation gradually di¤uses into smaller …rms. As
a result, the …rm-size distribution shifts, and the technology di¤usion
follows an S-shaped logistic curve.
We test the theoretical hypothesis with an empirical study of internet banking di¤usion among banks across …fty U.S. states. Using an
instrument variable approach, we identify a positive e¤ect of average
bank size on internet banking di¤usion. The empirical …ndings also
allow us to examine technological, economic, and institutional factors
governing the di¤usion process and explain the variation in di¤usion
rates across geographic regions.

Sullivan & Wang: Technology Di usion: Internet Banking

37

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Courchane, Marsha, David Nickerson, and Richard J. Sullivan. 2002.
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Kroszner, Randall S., and Philip E. Strahan. 1999. “What Drives
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Federal Reserve Bank of Richmond Economic Quarterly
Banking Branching Restrictions.”Quarterly Journal of Economics
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Mans…eld, Edwin. 1961. “Technical Change and the Rate of
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Exploration in Technology Di¤usion and Impact,”Federal Reserve
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2017.
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Sullivan & Wang: Technology Di usion: Internet Banking

39

APPENDIX
Table 4 Summary Statistics

Notes: Sample population includes the …fty states in the U.S. and the District
of Columbia. The sample size varies from year to year because the transactional
website adoption rate reached 100 percent for some observations and TRANODDS
cannot be calculated. The actual sample size in 2003, 2005, and 2007 is 47, 46,
and 43, respectively. See Table 1 for variable de…nitions and sources. *In millions.
**In thousands

40

Federal Reserve Bank of Richmond Economic Quarterly

Table 5 Empirical Variable De nitions and Sources
Variable

De…nition

Source

TRANS
TRANODDS

Call reports
Call reports

OFF_DEP

Adoption rate for transactional websites
Odds ratio for adoption of transactional
websites
Gini coe¢ cient for bank deposits
Average bank deposits
Ratio of banks in metropolitan areas to
all banks
Specialization of lending to consumers
(consumer loans plus 1-4 family mortgages
divided by total loans)
Bank o¢ ces per value of deposits

RMEDFAMINC
POPDEN

Median family income (in 1967 dollars)
Population density

IMITATE

Years since the …rst bank in the state adopted
a transactional website
Average age of banks
Household access rate for internet

GINI
DEPOSITS
METRO
LOANSPEC

AGE
HHINET
WGRATIO
INTRAREG
BHC
DEPINT
COMRATE
DEPOSITS90

Ratio of computer analyst wage to teller wage
Indicator variable for whether the state had
branching restrictions after 1995
Ratio of banks in bank holding companies
to total banks
Ratio of deposits in out-of-state banks
to total deposits
Adoption rate of high-speed internet among
commercial …rms
Average bank deposits in 1990

Call reports
Call reports
Call reports
Call reports
Call reports; FDIC
Summary of Deposits
U.S. Census Bureau
Statistical Abstract
of the United States
Online Banking
Report
Call reports
Statistical Abstract
of the United States
BLS
Kroszner and Strahan
(1999)
Call reports
FDIC
Summary of Deposits
Forman et al., 2003
Call reports

Regional Dummy Variables from the BLS are as follows:
(SE): AL, AR, FL, GA, KY, LA, MS, NC, SC, TN, VA, WV.
(FARWEST): AK, CA, HI, NV, OR, WA. (ROCKYMTN): CO, ID, MT, UT, WY.
(PLAINS): IA, KS, MN, MO, NE ,ND, SD. (SW): AZ, NM, OK, TX.
(GRTLAKE): IL, IN, MI, OH, WI. (MIDEAST): DC, DE, MD, NJ, NY, PA.
(NWENGLND): CT, MA, ME, NH, RI, VT.
Notes: Data are for individual states. Variables for banks are unweighted averages
for those located in individual states. Selected banks are full-service, retail commercial banks. Data for adoption of high-speed internet among commercial …rms
is for 2003. COMRATE is an average of urban …rms’ and rural …rms’ internet
adoption, using METRO to weight urban and rural location. BEA Regions are a
set of geographic areas that are aggregations of the states. The regional classi…cations, which were developed in the mid-1950s, are based on the homogeneity of
the states in terms of economic characteristics, such as the industrial composition
of the labor force, and in terms of demographic, social, and cultural characteristics. For a brief description of the regional classi…cation of states used by BEA,
see U.S. Census Bureau, Geographic Areas Reference Manual, Washington, D.C.,
U.S. Government Printing O¢ ce, November 1994, pp. 6–19.