ï»¿ WPS6026
Policy Research Working Paper 6026
Productivity and the Welfare of Nations
Susanto Basu
Luigi Pascali
Fabio Schiantarelli
Luis Serven
The World Bank
Development Research Group
Macroeconomics and Growth Team
April 2012
Policy Research Working Paper 6026
Abstract
This paper shows that the welfare of a countryâ€™s requires that total factor productivity be constructed
representative consumer can be measured using just two with prices and quantities as perceived by consumers,
variables: current and future total factor productivity not firms. Thus, factor shares need to be calculated using
and the capital stock per capita. These variables suffice after-tax wages and rental rates and they will typically
to calculate welfare changes within a country, as well sum to less than one. These results are used to calculate
as welfare differences across countries. The result welfare gaps and growth rates in a sample of developed
holds regardless of the type of production technology countries with high-quality total factor productivity and
and the degree of market competition. It applies to capital data. Under realistic scenarios, the U.K. and Spain
open economies as well, if total factor productivity is had the highest growth rates of welfare during the sample
constructed using domestic absorption, instead of gross period 1985â€“005, but the U.S. had the highest level of
domestic product, as the measure of output. It also welfare.
This paper is a product of the Macroeconomics and Growth Team, Development Research Group. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at Lserven@worldbank.org .
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Produced by the Research Support Team
Productivity and the Welfare of Nations
Susanto Basu Luigi Pascali Fabio Schiantarelli
Boston College and NBER Pompeu Fabra University Boston College and IZA
Luis Serveny
World Bank
April 4, 2012
JEL codes: D24, D90, E20, O47
Keywords: Productivity, Welfare, TFP, Solow Residual.
This paper builds on Basu, Pascali, Schiantarelli, Serven (2009). We are grateful to Mikhail Dmitriev, John
Fernald, Gita Gopinath and Chad Jones for very useful suggestions and to Jose Bosca for sharing with us his tax
data. We would also like to thank seminar participants at Bocconi University, the Federal Reserve Bank of Richmond
and Clark University for their comments.
y
Basu: Susanto.Basu@bc.edu; Pascali: Luigi.Pascali@upf.edu; Schiantarelli: Fabio.Schiantarelli@bc.edu; Serven:
Lserven@worldbank.org
1
1 Introduction
Standard models in many â€¦elds of economics posit the existence of a representative household
in either a static or a dynamic setting, and then seek to relate the welfare of that household to
observable aggregate data. A separate large literature examines the productivity residual deâ€¦ned
by Solow (1957), and interprets it as a measure of technical change or policy eÂ¤ectiveness. Yet
a third literature, often termed "development accounting," studies productivity diÂ¤erences across
countries, and interprets them as measures of technology gaps or institutional quality. To our
knowledge, no one has suggested that these three literatures are intimately related. We show that
they are. We start from the standard framework of a representative household that maximizes
intertemporal welfare over an inâ€¦nite horizon, and use it to derive methods for comparing economic
well-being over time and across countries. Our results show that under a wide range of assumptions,
welfare can be measured using just two variables, productivity and capital accumulation. We take
our framework to the data, and measure welfare change within countries and welfare diÂ¤erences
across countries.
In the simplest case of a closed economy with no distortionary taxes we show that to a â€¦rst-order
approximation the welfare change of a representative household can be fully characterized by three
objects: the expected present discounted value of total factor productivity (TFP) growth as deâ€¦ned
by Solow, the change in expectations of the same quantity, and the growth in the stock of capital
per person. The result sounds similar to one that is often proven in the context of a competitive
optimal growth model, which might lead one to ask what assumptions on technology and product
market competition are required to obtain this result. The answer is, None. The result holds
for all types of technology and market behavior, as long as consumers take prices as given and are
not constrained in the amount they can buy or sell at those prices. Thus, for example, the same
result holds whether the TFP growth is generated by exogenous technological change, as in the
Ramsey-Cass-Koopmans model; by changes in the size of the economy combined with increasing
returns to scale, as in the "semi-endogenous growth" models of Arrow (1962) and Jones (1995); or
by externalities or public policy in fully-endogenous growth models, such as Romer (1986) or Barro
(1990). As we discuss below, aggregate TFP can also change without any change in production
technology in multi-sector models with heterogeneous distortions (for example, markups that diÂ¤er
across sectors): our results show that an increase in aggregate TFP due to reallocation would be as
much of a welfare gain for the representative consumer as a change in exogenous technology with
the same magnitude and persistence.
Our â€¦ndings suggest a very diÂ¤erent interpretation of TFP from the usual one. Usually one
argues that TFP growth is interesting because it provides information on the change or diÂ¤usion of
technology, or measures improvement in institutional quality, the returns to scale in the production
function, or the markup of price over marginal cost. We show that whether all or none of these
things is true, TFP is interesting for a very diÂ¤erent reason. Using only the â€¦rst-order conditions
for optimization of the representative household, we can show that TFP is key to measuring welfare
changes within a country and welfare diÂ¤erences across countries. We interpret TFP purely from
2
the household side, producing what one might call "the household-centric Solow residual."1 Here
we follow the intuition of Basu and Fernald (2002), and supply a general proof of their basic insight
that TFP, calculated from the point of view of the consumer, is relevant for welfare.
The intuition for our result comes from noting that TFP growth is output growth minus share-
weighted input growth. The representative household receives all output, which ceteris paribus
increases its welfare. But at the same time it supplies some inputs: labor input, which reduces
leisure, and capital input, which involves deferring consumption (and perhaps losing some capital
to depreciation). The household measures the cost of the inputs supplied relative to the output
the
gained by real factor pricesâ€“ real wage and the real rental rate of capital. TFP also subtracts
inputs supplied from output gained, and uses exactly the same prices to construct the input shares.
The welfare result holds in a very general setting because relative prices measure the consumerâ€™s
marginal rate of substitution even in many situations when they do not measure the economyâ€™s
for
marginal rate of transformationâ€“ example, if there are externalities, increasing returns or im-
perfect competition. In an extension, we show that our basic insight holds even in some situations
where households are not competitive price-takers, for example, if there are multiple wages for
identical workers, leading to labor market rationing.
This intuition suggests that in cases where prices faced by households diÂ¤er from those facing
â€¦rms, it is the former that matter for welfare. We show that this intuition is correct, and here our
s
household-centric Solow residual diÂ¤ers from Solowâ€™ original measure, which uses the prices faced
by â€¦rms. Proportional taxes are an important source of price wedges in actual economies. We
show that the shares in the household-centric Solow residual need to be constructed using the factor
prices faced by households. Since marginal income tax rates and rates of value-added taxation
can be substantial, especially in rich countries, this modiâ€¦cation is quantitatively important, as we
show in empirical implementations of our results.
We then move to showing analogous results for open economies. Here we show that our previous
results need to be modiâ€¦ed substantially if we construct TFP using the standard output measure,
real GDP. To the three terms discussed above we need to add the present discounted value of
expected changes in the terms of trade, the present discounted value of expected changes in the
rate of return on foreign assets, and the growth rate of net foreign asset holdings. Intuitively,
s
both the terms of trade and the rate of return on foreign assets aÂ¤ect the consumerâ€™ ability to
obtain welfare-relevant consumption and investment for a given level of factor supply. Holdings of
net foreign assets are analogous to domestic physical capital in that both can be transformed into
consumption at a future date.
While these results connect to and extend the existing literature, as we discuss below, they are
diÂ¢ cult to take to the data. It is very hard to get good measures of changes in asset holdings by
country for a large sample of countries.2 Furthermore, measuring asset returns in a comparable way
1
The term is due to Miles Kimball.
2
The important work of Lane and Milesi-Ferretti (2001, 2007) has shed much light on this subject, but the
measurement errors that are inevitable in constructing national asset stocks lead to very noisy estimates of net asset
growth rates.
3
across countries would require us to adjust for diÂ¤erences in the risk of country portfolios, which is
a formidable undertaking. Fortunately, we are able to show that these diÂ¢ culties disappear if we
switch to using real absorption rather than GDP as the measure of output.3 In this case, exactly
the same three terms that summarize welfare in the closed economy are also suÂ¢ cient statistics in
the open economy. Thus, we can measure welfare change empirically in ways that are invariant to
the degree of openness of the economy.
These results pertain to the evolution of welfare in individual economies over time. The indexes
we obtain are not comparable across countries. Thus it is natural to ask whether our methods
shed any light on a pressing and long-standing question, the measurement of relative welfare across
countries using a method â€¦rmly grounded in economic theory. It turns out that they do. Perhaps
our most striking â€¦nding is the result that we can use data on cross-country diÂ¤erences in TFP
and capital intensity, long the staples of discussion in the development and growth literatures, to
measure diÂ¤erences in welfare across countries. More precisely, we show that productivity and
the capital stock suÂ¢ ce to calculate diÂ¤erences in welfare across countries, with both variables
computed as log level deviations from a reference country.
To understand this result, it helps to deepen the intuition oÂ¤ered above. Our analysis is based
on a dynamic application of the envelope theorem, and it shows that the welfare of a representative
agent depends to a â€¦rst order on the expected time paths of the variables that the agent takes as
exogenous. In a dynamic growth context, these variables are the prices for factors the household
supplies (labor and capital), the prices for the goods it purchases (consumption and investment),
and beginning-of-period household assets, which are predetermined state variables and equal to the
s
capital stock in a closed economy. Apart from this last term, the householdâ€™ welfare depends on
the time paths of prices, which are exogenous to the household. Thus, the TFP that is directly
relevant for household welfare is actually the dual Solow residual. We use the national income
accounts identity to transform the dual residual into the familiar primal Solow residual.4
Our cross-country welfare result comes from using the link between welfare and exogenous
s
prices implied by economic theory to ask how much an individualâ€™ welfare would diÂ¤er if he faced
the sequence of prices, not of his own country, but of some other country. We can perform the
thought experiment of having a US consumer face the expected time paths of all goods and factor
prices in, say, France, and also endow him with beginning-of-period French assets rather than
US assets. The diÂ¤erence between the resulting level of welfare and the welfare of remaining
in the US measures the gain or loss to a US consumer of being moved to France. Note that
in
our welfare comparisons are from a deâ€¦nite point of viewâ€“ this example, from the view of a
US consumer. In principle, the result could be diÂ¤erent if the USA-France comparison is made
by a French consumer, with diÂ¤erent preferences over consumption and leisure. Fortunately, our
empirical results are qualitatively unchanged and quantitatively little aÂ¤ected by the choice of the
"reference country" used for these welfare comparisons.
3
We are indebted to Mikhail Dmitriev for pointing out this result.
4
See Barro and Sala-i-Martin (2004, section 10.2).
4
The same insights that apply to the time series are relevant for the cross section: TFP needs
to be deâ€¦ned using the prices perceived by households, and if the economy is open then other
terms become relevant. Thus, tax rates, terms of trade, and foreign asset holdings also matter
for cross-country welfare comparisons. As before, we can reduce the measurement complications
enormously by using absorption rather than GDP as the deâ€¦nition of output in our household-
centric TFP measure.
These results show that we can perform interesting welfare comparisons using readily-available
national income accounts data. We illustrate our methods using data for several industrialized
countries for which high-quality data are available: Canada, France, Italy, Japan, Spain, the United
Kingdom and the United States. We show the importance of â€¦scal considerations in constructing
measures of welfare change over time. For example, if we assume that government spending
is wasteful and taxes are lump-sum, the UK has the largest welfare gain among our group of
countries over our sample period, 1985-2005, while Spain lags far behind due to its low TFP growth
rate. Indeed, the US, a much richer country, has faster welfare growth than Spain under these
assumptions. Allowing for distortionary taxation and assuming that government expenditures are
chosen optimally, Spain has the highest welfare growth among all countries, with the UK a shade
behind, and the US much further back.
However these welfare growth rates are country-speciâ€¦c indexes, and cannot be used to compare
welfare across countries. We next apply our methodology to cross country-comparisons and show
how these relative welfare levels evolve over time. In our benchmark case of optimal government
spending and distortionary taxation, the US is the welfare leader throughout our sample period.
At the start of our sample, we â€¦nd that France and the UK are closest to the US in terms of welfare,
with France having a slight advantage over the UK. By the end of the sample, France and most of
the other economies fall further behind the US in terms of welfare levels, with the two exceptions
being Spain and the UK. Spain converges towards the US level of welfare in the â€¦rst several years
of the sample, and then holds steady at a constant percent gap. The UK, by contrast, converges
towards the US at a relatively constant rate, and by 2005 is within a few percent of the US level of
per-capita welfare.
These measures have a clear interpretation because they are derived from a well-posed opti-
s
mization problem. Starting from a precise statement of the householdâ€™ optimization problem also
forces us to confront two issues in national income and welfare measurement. First, our derivation
shows that â€œconsumptionâ€?should be deâ€¦ned as any good or service that consumers value, whether
or not it is included in GDP. Similarly, "capital" should include all consumption that is foregone
now in order to raise consumption possibilities for the future. These items include, for example,
environmental quality and intangible capital. Of course, both are hard to measure and even harder
to value, since there is usually no explicit market price for either good. But our derivation is quite
clear on the principle that the environment, intangibles and other non-market goods should be
included in our measure of â€œwelfare TFP.â€? We follow conventional practice in restricting the out-
put measure for our TFP variable to market output (and the inputs to measured physical capital
5
and labor), but in so doing we, and almost everyone else, are mismeasuring real GDP and TFP.
Second, our starting point of a representative-consumer framework implies that we automatically
ignore issues of distribution that intuition says should matter for social welfare. We believe that
distributional issues are very important. However, our objective of constructing a welfare measure
from aggregate data alone implies that we cannot incorporate measures of distribution into our
framework. Thus, we maintain the representative-consumer framework, but without in any way
minimizing the importance of issues that cannot be handled within that framework.
The paper is structured as follows. The next section presents our analytical framework, and
uses it to derive results on the measurement of welfare within single economies. Section 3 applies
similar methods to derive results for welfare comparisons among countries. (Full derivations of the
results in Sections 2 and 3 are presented in an appendix.) We present a number of extensions to
our basic framework in Section 4, allowing for multiple types of goods and factors, distortionary
taxes, government expenditure, an open economy, and labor market rationing. We then take the
enhanced framework to the data, and discuss empirical results in Section 5. We discuss relations
of our work to several distinct literature in Section 6. Finally, we conclude by summarizing our
â€¦ndings and suggesting fruitful avenues for future research.
2 The Productivity Residual and Welfare
Both intuition and formal empirical work link TFP growth to increases in the standard of living, at
least as measured by GDP per capita.5 The usual justiâ€¦cation for studying the Solow productivity
residual is that, under perfect competition and constant returns to scale, it measures technological
change. However, should we care about the Solow residual in an economy with non-competitive
output markets, non-constant returns to scale, and possibly other distortions where the Solow
residual is no longer a good measure of technological progress? Here we build on the intuition of
Basu and Fernald (2002) that a slightly modiâ€¦ed form of the Solow residual is welfare relevant
even in those circumstances and derive rigorously the relationship between a modiâ€¦ed version
of the productivity residual (in growth rates or log levels) and the intertemporal utility of the
representative household. The fundamental result we obtain is that, to a â€¦rst-order approximation,
utility reâ€¡ects the present discounted values of productivity residuals (plus possibly other terms).
s
Our results are complementary to those in Solowâ€™ classic (1957) paper. Solow established
that if there was an aggregate production function then his index measured its rate of change.
s,
We now show that under a very diÂ¤erent set of assumptions, which are disjoint from Solowâ€™ the
familiar TFP index is also the correct welfare measure. The results are parallel to one another.
Solow did not need to assume anything about the consumer side of the economy to give a technical
interpretation to his index, but he had to make assumptions about technology and â€¦rm behavior.
We do not need to assume anything about the â€¦rm side (which includes technology, but also â€¦rm
behavior and industrial organization) in order to give a welfare interpretation, but we do need to
5
For a review of the literature linking TFP to GDP per worker, in both levels and growth rates, see Weil (2008).
6
assume the existence of a representative consumer. Both results assume the existence of a potential
function (Hulten, 1973), and show that TFP is the rate of change of that function. Which result
is more useful depends on the application, and the trade-oÂ¤ that one is willing to make between
having a result that is very general on the consumer side but requires very precise assumptions on
technology and â€¦rm behavior, and a result that is just the opposite.
2.1 Approximating around the Steady State
More precisely, assume that the representative household maximizes intertemporal utility:
1
X 1 Nt+s
V t = Et U (Ct+s ; L Lt+s ) (1)
(1 + )s H
s=0
where Ct is the per-capita consumption at time t, Lt are per-capita hours of work, L is the per-
capita time endowment, and Nt is population. H is the number of households, assumed to be â€¦xed
and normalized to one from now on. Population grows at constant rate n and per capita variables
at a common rate g. For a well deâ€¦ned steady state in which hours of work are constant while
consumption and real wage share a common trend, we assume that the utility function has the
King, Plosser and Rebelo (1988) form:
1
U (Ct+s ; L Ls ) = Ct+s 1 (L Lt+s ) (2)
1
with 0 < < 1 or > 1 and (:) > 0.6 Denote Xt an index for per capita variables in the sense
that, along the steady state growth path, their level at time t is proportional to Xt and deâ€¦ne
Ct+s
ct+s = Xt+s . We can rewrite the utility function in a normalized form as follows:
1
X
Vt s
vt = (1 )
= Et U (ct+s ; L Lt+s ) (3)
Nt Xt s=0
(1+n)(1+g)1
where = (1+ ) is assumed to be less than one. The budget constraint (with variables
scaled by Nt Xt ) is:
(1 ) + pK
t (1 + rt )
kt + bt = kt 1 + bt 1 + pL Lt +
t t pC ct
t (4)
(1 + g) (1 + n) (1 + g) (1 + n)
Kt
New capital goods are the numeraire, kt = Xt Nt denotes capital per worker normalized by Xt ,
K
Pt Pt L PtC
Bt
bt = I
Pt Xt Nt
are normalized real bonds: pK
t = Pt
L C
I , pt = P I X , pt = P I denote, respectively, the
t t t
user cost of capital, the wage per hour of eÂ¤ective worker, and the price of consumption goods.
(1 + rt ) is the real interest rate (again in terms of new capital goods) and t = I
t
Pt Xt Nt
denotes
normalized proâ€¦ts per capita.
6
If = 1, then the utility function must be U (C1 ; ::; CG ; L L) = log(C) (L L): See King, Plosser and
Rebelo (1988).
7
Log linearizing around the non stochastic steady state, intertemporal household utility can be
written (to a â€¦rst order approximation) as:
1
X
s pK k
vt = v + Et pL LbL +
pt+s pK + bt+s
b pC cbt+s
t p
(1 + g) (1 + n) t+s
s=0
1 1 b
+ kbt
k 1 + bbt 1 (5)
b
where v is the steady state value of utility, x = log xt log x denote log deviation from the steady
state. In obtaining this result we have used the FOCs of the household maximization problem; for
time t these consist of:
C
Uct t pt =0 (6)
L
ULt + t pt =0 (7)
(1 ) + pKt+1
t + Et t+1 =0 (8)
(1 + g) (1 + n)
1
t + Et (1 + rt+1 ) t+1 =0 (9)
(1 + g) (1 + n)
plus the two trasversality conditions, one for bonds and one for capital. Equation (5) is an imme-
diate consequence of the envelope theorem and expresses normalized utility as a function of the
variables that the consumer takes as exogenous or predetermined, e.g. prices and the initial stock of
capital and bonds. Using the log linear approximation of the budget constraint around the steady
state:
(1 ) + pK b (1 + r) pK k
k bt + bbt
k b k kt 1 bbt
b 1
b
pL LLt pL LbL
pt pK
b
(1 + g) (1 + n) (1 + g) (1 + n) (1 + g) (1 + n) t
bt + pC cbt + pC cbt = 0
c p (10)
we can re-write equation (5) in terms of quantities:
1
X
b pK k bt+s 1 b
vt = v + Et s
pC cbt+s + ibt+s
c i pL L Lt+s k 1 + k kt 1 (11)
(1 + g) (1 + n)
s=0
Equation (11) says that intertemporal utility (in log deviation from the steady state) reâ€¡ects
the expected present discounted value of terms that represent the sum of the components of â€¦nal
demand (in log deviation from the steady state), weighted by their steady state contribution to
demand, minus primary inputs (in log deviation from the steady state) times their respective steady
state total remuneration. In addition, intertemporal utility depends upon the initial level of the
8
capital stock.
2.2 Connecting with the Productivity Residual
We are now close to relating utility to a modiâ€¦ed version of the productivity residual. Let us start
by obtaining a â€¦rst order approximation for the level of utility in terms of the log level productivity
residual. In doing so, we will use the fact that, to a â€¦rst order approximation, the level of normalized
value added in deviation from the steady state is given by:
b
yt = log yt log y = sc bt + sibt
c i (12)
P C CN PII
where sc PY Y
and si PY Y
are respectively the steady-state shares of consumption goods and
investment goods out of value added. Using (12), equation (11) can be re-written as:
vt v X 1 h i 1 k b
= Et s
b
yt+s b
sL Lt+s sK bt+s
k 1 + kt 1 (13)
pY y pY y
s=0
pL L pK k
where sL = pY y
and sK = pY y(1+g)(1+n)
denote the distributional shares of labor and capital
respectively. If < 1, so that vt > 0, (vt v) can be approximated by v(log vt log v) =
Vt Vt
v(log Nt log Nt ), where the subscript SS denotes the value of time varying variables along
SS
the steady state growth path. Then the equation above can be rewritten as:
1
X
v Vt Vt s 1 k Kt 1 Kt 1
log log = Et (log P Rt+s (log P Rt+s )SS )+ log log
pY y Nt Nt SS pY y Nt 1 Nt 1 SS
s=0
(14)
where:
Yt Kt 1
log P Rt+s = log sL log Lt sK log (15)
Nt Nt 1
Y
log Ntt is the sum of log consumption and log investment per capita, each weighted by its steady
Y
state expenditure shares. Productivity, log P Rt ; is deâ€¦ned log Ntt minus the log level of factor inputs
per capita, log Lt and log Kt
N
t 1
1
multiplied by their respective steady state distributional shares, sL
and sK .7
v
In order to interpret equation (14), notice that pY y
measures the percentage increase in income
necessary to generate a one percentage point increase in lifetime utility (see the Appendix for
v V Vt
details). Therefore the right hand side of equation (14), pY y
log Ntt log Nt , represents
SS
the income equivalent corresponding to the log deviation of per-capita utility from the steady
state. This quantity is an increasing function of the sequence of log deviation from the steady state
of productivity residuals, appropriately discounted and the log deviation from the steady state of
Kt 1
the initial level of the per-capita capital stock, Nt 1
.
7
See details in the appendix
9
When > 1, so that vt < 0; the interpretation remains essentially the same but the left hand
v Vt Vt
side of equation (14) now equals pY y
log Nt log Nt . This quantity is greater than zero
SS
for positive deviations of per capita utility from its value along the steady state growth path.
In order to illustrate the relationship between the change in per-capita welfare and the Solow
residual, we return to (14) and take its diÂ¤erence through time. In this case, we will rely on the
following (Divisia) deâ€¦nition of growth in value added:
log Yt = sc log(Ct Nt ) + si log It (16)
where the growth rate of each demand component is aggregated using constant steady state shares8 .
Using this deâ€¦nition of value added in growth terms and taking â€¦rst diÂ¤erences of equation (14), we
show in the Appendix that the money-value of the growth rate of per-capita welfare, as a proportion
of GDP, can be written as follows9 :
1
X
v Vt s
log = Et log P Rt+s
pY y Nt
s=0
1
X
s
+ [Et log P Rt+s Et 1 log P Rt+s ] (17)
s=0
1 k Kt 1
+ log
py y Nt 1
where log P Rt+s denotes the "modiâ€¦ed" Solow productivity residual:
Yt+s Kt+s 1
log P Rt+s = log sL log Lt+s sK log (18)
Nt+s Nt+s 1
and log Yt = sc log Nt Ct + si log It : We use the word "modiâ€¦ed," for two reasons. First, we do
not assume that the distributional shares of capital and labor add to one, as they would if there were
zero economic proâ€¦ts and no distortionary taxes. Zero proâ€¦ts are guaranteed in the benchmark case
with perfect competition and constant returns to scale, but can also arise with imperfect competition
and increasing returns to scale, as long as there is free entry, as in the standard Chamberlinian
model of imperfect competition. Second, the distributional shares are calculated at their steady
state values and, hence, are not time varying (as it is conventionally assumed). Rotemberg and
Woodford (1991) argue that in a consistent â€¦rst-order log-linearization of the production function
s
the shares of capital and labor should be taken to be constant, and Solowâ€™ (1957) use of time-
varying shares amounts to keeping some second-order terms while ignoring others.
The term Et log P Rt+s Et 1 log P Rt+s represents the revision in expectations of the log level
of the productivity residual, based on the new information received between t-1 and t. In addition,
Kt 1
log Nt 1
captures the change in the initial endowment of the capital stock per capita.
8
Here we are slightly departing from convention, as value added is usually calculated with time-varying shares.
9 V
Also in this case, when > 1, so that vt < 0; the left hand side of equation (17) equals pv y log Ntt
Y
10
Note that the revision terms in the second summation will reduce to a linear combination of
the innovations in the stochastic shocks aÂ¤ecting the economy at time t. Moreover, if we assume
that the modiâ€¦ed log level productivity residual follows a univariate autoregressive process, then
only the innovation of such process matters. In addition, all the terms in the â€¦rst summation are
simply a function of current and past values of productivity.
Summarizing, by focusing only on the representative consumer optimization problem, we have
shown that the Solow residual is a welfare relevant object. Since we have made no assumption about
the production function or about product market structure, this result is very general. It holds not
only in competitive economies with constant return to scale technologies and no externalities, but
also in distorted economies with externalities. Even though in those economies the Solow residual
does not capture technology, it matters for welfare, together with the initial endowment of capital.
To aid the quantitative interpretation of our the results, we can express them in terms of
"equivalent consumption" per capita, denoted by Ct . Ct is deâ€¦ned as the level of consumption
per capita at time t that, if it grows at the steady state rate g from t onward, with leisure set
at its steady state level, delivers the same intertemporal utility per capita as the actual stream of
consumption and leisure. More precisely, Ct satisâ€¦es:
1
X (1 + n)s
Vt
= s (Ct (1 + g)s )1 (L L)) (19)
Nt (1 + )
s=0
1
= Ct 1 (L L)
(1 ) (1 )
Taking log diÂ¤erences of equation (19) and using equation (17), one obtains:
" 1 1
#
(1 ) X X 1 k Kt 1
s s
log Ct = Et log P Rt+s + Et log P Rt+s + log (20)
sc py y Nt 1
s=0 s=0
It is easily seen that this â€¦nal expression holds for smaller or greater than one. This means that if
(1 )
we multiply the right hand side of (17) by sc , we can interpret it as the growth rate of per-capita
equivalent consumption between t and t 1:
3 Implications for Cross Country Analysis
We can use the framework developed here in order to make cross country comparisons in welfare
and to show how they are aÂ¤ected by diÂ¤erences in productivity and capital accumulation. Wel-
fare comparisons across countries have been investigated recently by Jones and Klenow (2010).
Instead of focusing on single period (or steady state) utility, as they do, we consider lifetime utility
and consider out of steady state dynamics and how they are related to capital accumulation and
productivity. This allows us to properly take into account the dynamic welfare eÂ¤ects associated
with the fact that lower consumption or leisure today may raise capital accumulation and support
11
greater consumption in the future. We do not, however, allow for cross country diÂ¤erences in life
expectancy or in inequality as in the Jones and Klenow paper. In our approach the cross-country
diÂ¤erences in welfare are summarized solely in terms of productivity and capital endowments and
not in terms of consumption and leisure.
A comparison of welfare across countries requires either assuming that their respective repre-
sentative agents possess the same utility function, or making the comparison from the perspective
of the representative agent in a reference country. We will in any case assume that the discount
factor, , is the same in each country. These assumptions suggest that it is more reasonable to
focus on a subset of countries that are relatively homogenous and this is exactly what we will do
by using core OECD countries in our empirical illustration.
Let us go back to a modiâ€¦ed version of equation (5) and assume we have taken the log-linear
expansion around the US steady state:
1
X pK;us k us
i
vt = v us
+ Et s us
[pL;us Lus (log pL;i
t+s log pL;us ) + (log pK;i
t+s log pK;us )
(1 + g) (1 + n)
s=0
us
+ (log it+s log us
) pC;us cus (log pi
t+s log pus )]
us 1 us i 1
+ k (log kt 1 log k us ) + us bus (log bi 1 log bus )
t (21)
i
If preferences are common across countries, vt , can be interpreted as the normalized lifetime utility
i
of an individual in country i. Alternatively, vt , can be thought of representing normalized lifetime
welfare of a US individual when facing the sequence of prices, initial endowments, proâ€¦ts and Xt
of country i.
In both cases, using the log-linearized version of the budget constraint in equation (10) expanded
around the US steady state, we can obtain (proceeding in a way parallel to the one used in the
previous section):10
1
X
v us i
us Y;us us (log vt log v us ) = Et s
[sus (log ci
c t+s log cus ) + sus (log ii
i t+s log ius ) (22)
p y
s=0
sus (log Li
L t+s
i
ln Lus ) sus (log kt+s
K 1 log k us )]
1 k us i
+ Y;us us (log kt 1 log k us )
p y
i Ii
i
Deâ€¦ne the productivity term obtained using US shares as P Rt+s = sus log Ct+s + sus log Nt+s
c i i
t+s
10 i
When vt represents the normalized utility of and individual from country i; it is obvious that (22) follows from
i
(10). When vt is normalized utility of the US individual when faced with the price sequence and the endowments
of country i, the result still holds. It does not follow that the US individual will choose the quantities of country i
unless the utility function is identical across countries. However,after expanding the budget constraint around the US
steady state, the algebraic sum of the terms in prices for the US individual facing country i prices and endowments
equals the algebraic sum of the terms in the quantities for the individual from country i. Hence (22) holds in this
case too.
12
i
Kt+s
sus log Li
L t+s sus log
K i
Nt+s
1
. Using this modiâ€¦ed measure of productivity, where input shares are
1
common across countries, we show in the appendix that equation (22) can be re-written in per-
capita terms as:
v us Vi Vtus
1
X h i
i
us Y;us us log ti log = Et s
log P Rt+s us
log(P Rt+s )SS (23)
p y Nt Ntus SS s=0
!
1k us Ki us
Kt 1
+ Y;us us log ti 1
log
p y Nt 1 Ntus1 SS
If we expand the US intertemporal utility around the US steady state and subtract the resulting
equation from (23), we can write:
v us Vi V us X 1 h i 1 k us Kti us
Kt 1
i 1
us Y;us us log ti log tus = Et s
log P Rt+s us
log P Rt+s + log log
p y Nt Nt pY;us y us Nti 1 Ntus1
s=0
(24)
Welfare diÂ¤erences across countries, therefore, are summarized by two components. The â€¦rst com-
ponent depends on the diÂ¤erences in the capital endowments. The second component is related
to the well-known log diÂ¤erence between TFP levels, which accounts empirically for most of the
diÂ¤erence is per capita income across countries (Hall and Jones (1999)). In the development ac-
counting literature it is interpreted as a measure of technological or institutional diÂ¤erences between
countries. This interpretation, however, is valid under restrictive assumptions on market behavior
and technology (i.e. perfect competition, constant returns to scale, etc.). We provide a diÂ¤erent
interpretation of the relevance of cross country diÂ¤erences in TFP, by showing that a "modiâ€¦ed"
version of the log-diÂ¤erence between TFP levels is essential for welfare comparison across countries,
together with diÂ¤erences in their per capita capital endowment. This last term is similar to the
capital intensity term used in the development accounting literature. Our result holds for any kind
of technology and market structure, as long as a representative consumer exists, takes prices as
given and is not constrained in the amount he can buy and sell at those prices. Notice however,
that our measure of TFP is modiâ€¦ed with respect to the traditional growth accounting measure in
three ways. First, measuring welfare diÂ¤erences requires comparing not only current log diÂ¤erences
in TFP but the present discounted value of present and future ones. Second, the distributional
and expenditure shares used to compute the log diÂ¤erences in TFP need to be calculated at
their steady-state value in the reference country. Third, as we will argue in section 4.4, domestic
absorption is used, instead of GDP, in calculating the productivity residual.
As in the within-country case, we can conduct the comparison by using the concept of equivalent
consumption. In this context, for any country i, Ct ;i , is deâ€¦ned for a constant level of leisure, â€¦xed
at the US steady state level.
13
Vti 1 1
= Ct ;i (L Lus ) (25)
Nti (1 ) (1 )
It then follows that:
" 1
#
(1 ) X i 1 k us Ki K us
log Ct ;i log Ct ;us = Et s
log P Rt+s us
log P Rt+s + log ti 1
log tus1
sus
c pY;us y us Nt 1 Nt 1
s=0
(26)
Note that along the steady state growth path, assuming that g is common across countries,
(26) can be rewritten after simple algebra as:
log(Ct ;i )SS log(Ct ;us )SS =
1 i
It us
It
= us i
sus log(Ct )SS + sus log
c i sus log(Ct )SS + sus log
c
us
i
sc Nti Ntus
SS
!SS
us i us
(1 ) (1 + g us ) (1
+ nus ) k us Kt 1 Kt 1
log log
(1 + g us ) (1 + nus ) pY;us y us Nti 1 SS
Ntus1 SS
sus
L
(log Li log Lus ) (27)
sus
c
The second and third line of equation (27) contain the diÂ¤erence in the log value of demand
components (aggregated with US expenditure weights), adjusted by the diÂ¤erence in capital de-
preciation (on the third line). The fact that the latter term represents diÂ¤erences in depreciation
is most easily seen when g and n are set to zero. This result is in the spirit of Weitzman (1976,
2003) who emphasizes the role of Net National Product (NNP) in welfare comparisons.11 However,
the term on the fourth line of (27) suggests that diÂ¤erences in per capita NNP (or a log linear
approximation to it) are not a suÂ¢ cient statistic for welfare even in the steady state and have to
be adjusted for diÂ¤erences in leisure.
4 Extensions
We now show that our method of using TFP to measure welfare can be extended to cover multiple
types of capital, labor and consumption goods, taxes, and government expenditure. These exten-
sions also modify in obvious ways the formulas for within and across countries welfare comparisons.
The â€¦rst extension modiâ€¦es our baseline results in only a trivial way, but the others all require
more substantial changes to the formulas above. These results show that the basic idea of using
TFP to measure welfare holds in a variety of economic environments, but also demonstrate the
advantage of deriving the welfare measure from an explicit dynamic model of the household. The
model shows exactly what modiâ€¦cations to the basic framework are required in each case.
11
We thank Chad Jones for drawing our attention to this point.
14
4.1 Multiple Types of Capital, Labor and Consumption Goods
The extension to the case of multiple types of labor, capital and consumption goods is immediate.
For simplicity, we could assume that each individual is endowed with the ability to provide diÂ¤erent
types of labor services, Lh;t and that the utility function can be written as:
1
U (C1;t+s ; ::; CZ;t+s ; L1;t+s ; :::; LHL ;t+s ) = C(C1;t+s ; ::; CZ;t+s )1 L L(L1;t+s ; :::; LHL ;t+s )
1
(28)
where L(:) and C(:) are homogenous functions of degree one, HL is the number of types of labor
and Z is the number of consumption goods. Denote the payment to a unit of Lh;t , PtLh :12 Similarly
consumers can accumulate diÂ¤erent types of capitals Kh;t and rent them out at PtKh . Take capital
good 1 as the numeraire. Equation (11) now becomes:
1
" Z HK HK HK
#
X X X X X pK kh
vt v = Et s
pC ci bi;t+s +
c pI ihbh;t+s b
pL Lh Lh;t+s h bh;t+s
i h i h k 1
(1 + g) (1 + n)
s=0 i=1 h=1 h=1 h=1
HK
X 1
+ kh bh;t
k 1 (29)
h=1
Redeâ€¦ne normalized real GDP in deviation from SS as:
Z
X HK
X
b
yt = sci bi;t +
c sihbh;t
i (30)
i=1 h=1
Using the two equations above, we get:
1
" HK HK
#
vt v X X X
= Et s
b
yt+s b
sLh Lh;t+s sKh bh;t+s
k 1
pY y
s=0 h=1 h=1
HK
X 1 kh
+ bh;t
k 1 (31)
pY y
h=1
pL Lh pK kh
where sLh = h
pY y
and sKh = h
(1+g)(1+n)pY y
. Proceeding exactly as in the previous section, equa-
tions (16), (19), (23) and (25) continue to characterize the relationship between utility and the
productivity residual, with the only diÂ¤erence that the latter is deâ€¦ned now as:
HK
X HL
X
Yt Kh;t 1
log P Rt = log sLh log Lh;t sKh log (32)
Nt Nh;t 1
h=1 h=1
12
We assume that the nature of the utility function is such that positive quantities of all types of labors are supplied.
15
4.2 Taxes
The derivations in Section 2.2 can be modiâ€¦ed to accommodate an environment with either distor-
tionary and/or lump-sum taxes. Since the prices in the budget constraint (11) are those faced by
the consumer, if there are taxes, prices should all be interpreted as after-tax prices. At the same
time, the variable that we have been calling â€œproâ€¦ts,â€? , can be viewed as comprising any transfer
of income that the consumer takes as exogenous. Thus, it can be interpreted to include lump-sum
taxes or rebates. Finally one should think of bt as including both government and private bonds
(assumed to be perfect substitutes, purely for ease of notation).
More precisely, in order to modify (11) to allow for taxes, let K be the tax rate on capital
t
income, R be the tax rate on revenues from bonds, L be the tax rate on labor income, C be the
t t t
ad valorem tax on consumption goods, and I be the corresponding tax on investment goods13 . We
t
assume that the revenue so raised is distributed back to individuals using lump-sum transfers. (We
consider government expenditures in the next sub-section.) Equation (4) now takes the following
form:
1
" #
X pK 1 K k pK
bt+s r 1 R bbrt+s
s
vt = v + Et pL 1 L
LbL +
pt+s + + bt+s pC 1 + C
p
cbt+s
(1 + g) (1 + n) (1 + g) (1 + n)
s=0
1
X pK K k r Rb
Et s
pL L
LbL +
t+s bK + bR pC C
cbC
(1 + g) (1 + n) t+s (1 + g) (1 + n) t+s t+s
s=0
1 1 b
+ kbt
k 1 + bbt 1 (33)
As in the benchmark case, this equation is an immediate consequence of the envelope theorem
and expresses normalized utility as a function of those variables that the household takes as exoge-
nous. However, diÂ¤erently than in the benchmark case, the exogenous variables in the householdâ€™s
maximization are not only the prices and the initial stocks of capital and bonds, but also the
tax rates on labor and capital income, consumption and investment. Using the new log-linearized
individual budget constraint:
(1 ) + pK 1 K pK 1 K k
pK K k
0 = k bt
k kbt
k 1 pK +
bt bK
(1 + g) (1 + n) (1 + g) (1 + n) (1 + g) (1 + n) t
pL 1 L b
LLt pL 1 L
LbL + pL
pt L
LbL
t
+pC 1 + C
cbt + pC 1 +
c C
cbC + pC
pt C
cbC
t bt
R R
1+r 1 bb r 1 b r Rb
+bbt
b bt 1 b
rt bR (34)
(1 + g) (1 + n) (1 + g) (1 + n) (1 + g) (1 + n) t
13
For simplicity, we are assuming no capital gains taxes and no expensing for depreciation. These could obviously
be added at the cost of extra notation.
16
Normalized utility can be re-expressed in terms of the endogenous variables14 :
1
X
vt v = Et s
[ 1+ C
pC cbt+s + 1 +
c I
ibt+s
i 1 L b
pL LLt+s
s=0
1 K pK k
bt+s 1 b
k 1] + k kt 1 (35)
(1 + g) (1 + n)
At this point, it is interesting to notice that changes in the tax rates do not appear in the right-
hand side of the equation above. The intuition for this result is that the endogenous quantities
ect
already reâ€¡ variations in the tax rates. In other words, changes in the tax burden are already
captured by the changes in consumption, investment and labor supply that they determined.
To make contact with the data, note that the national accounts at market prices deâ€¦ne nominal
expenditure using prices as perceived from the demand side. Thus, equation (12) can be written
exactly as before and still be consistent with standard national accounts data, where sc and si in
yt = sc bt + sibt are inclusive of indirect taxes (subsidies) on consumption and investment. On the
b c i
other hand, the national accounts measure factor payments as perceived by â€¦rms, before income
taxes. Hence we can write:
X 1
vt v 1 b
= Et s
[bt
y 1 L b
sL Lt+s 1 K
sK bt+s
k 1] + k kt 1 (36)
pY y
s=0
where sL and sK are the gross income shares of labor and capital respectively. Thus, the data-
consistent deâ€¦nition of the welfare residual with taxes needs to be based on a new deâ€¦nition of
log P Rt , where the shares of labor and capital returns are net of taxes. More speciâ€¦cally, equation
(15) can be re-written as:
Yt+s L K Kt+s 1
log P Rt+s = log 1 sL log Lt+s 1 sK log (37)
Nt+s Nt+s 1
b
In conclusion, after properly redeâ€¦ning yt+s and log P Rt , the results discussed in the second and
the third section of the paper can be generalized in a model with distortionary time-varying taxes
on consumption and investment goods and on the household income coming from labor, capital or
â€¦nancial assets.
4.3 Government Expenditure
With some minor modiâ€¦cation, our framework can be extended to allow for the provision of public
goods and services. We illustrate this under the assumption that government activity is â€¦nanced
with lump-sum taxes. Using the results from the previous subsection, it is straightforward to extend
the argument to the case of distortionary taxes.
Assume that government spending takes the form of public consumption valued by consumers.
We rewrite the instantaneous utility function as
14
See the Appendix for details on the derivation.
17
1
U (Ct+s ; CG;t+s ; Lt+s ) = C(Ct+s ; CG;t+s )1 (L Lt+s ) (38)
1
where CG denotes per-capita public consumption and C(:) is homogenous of degree one in its
arguments. Equation (11) now becomes:
1
X UcG cG bG;t+s
c b pK k bt+s
vt v = Et s
+ pC cbt+s + ibt+s
c i pL LLt+s k 1
(1 + g) (1 + n)
s=0
1 b
+ k kt 1 (39)
CG;t
where cG;t = Xt . The deâ€¦nition of GDP in deviation from steady state is now:
yt = sc bt + sibt + scG bG;t
b c i c (40)
P G CG UcG cG
where scG = PY Y
and P G is the public consumption deâ€¡ator. Let scG = . Then we can
write:
vt v X 1 h i 1 k
Yy
= Et s
b
yt+s b
sL Lt+s sK bt+s
k 1 + scG scG bG;t+s +
c bt
k 1 (41)
p pY y
s=0
Hence in the presence of public consumption the Solow residual needs to be adjusted up or down
depending on whether public consumption is under- or over-provided (i.e., scG > scG or scG < scG
respectively). If the government sets public consumption exactly at the utility-maximizing level for
the household, scG = scG and no correction is necessary. In turn, in the standard neoclassical case
in which public consumption is pure waste scG = 0, the welfare residual is computed on the basis
of private â€¦nal demand â€“i.e., GDP minus government purchases.
What if government purchases also yield productive services to private agents? This could be the
case if, for example, the government provides education or health services, or public infrastructure,
which may be directly valued by consumers and may also raise private-sector productivity. In such
case, the above expression remains valid, but it is important to note that the net contribution of
public expenditure to welfare would not be fully by captured by scG scG bG;t+s . To this term
c
we would need to add a measure of the productivity of public services, which in the expression is
implicitly included in the productivity residual yt+s sL Lt+s sK bt+s .
b b k
4.4 Open Economy
In a closed economy without government bt represents the net stock of domestic bonds. In deriving
our basic equation (13), however, we have not made use of the fact that net bond holdings equal
zero in equilibrium when bt denotes private bonds. Therefore (13) applies also to an open economy.
In the latter case, we interpret bt as net foreign assets, and replace GDP in (13) with domestic
18
absorption as a measure of output. However, we can still write:
vt v X 1 h i 1 k b
= Et s
bt
a b
sL Lt+s sK bt+s +
k kt 1 (42)
pa a pa a
s=0
where a denotes domestic absorption and is deâ€¦ned as:
bt = sc bt + sibt
a c i (43)
and sL , sK , sc and si are also shares out of domestic absorption. Suppose one wants to use a
standard measure of output, real GDP, deâ€¦ned as consumption, plus investment, plus net exports.
Then utility can be written as a function of a more conventionally deâ€¦ned productivity residual
and of additional components that capture terms of trade and capital gains eÂ¤ect. Moreover the
initial conditions should also include the initial value of net foreign assets. We can show this by
s
starting from the deâ€¦nition of a countryâ€™ current account:
CAt = Bt Bt 1 = it Bt 1 + PtEX EXt PtIM IMt (44)
where Bt is now the value of the net foreign assets, EXt and IMt are total exports and total
imports and PtEX and PtIM are their respective prices. In normalized form (44) becomes:
(1 + rt )
bt = bt 1 + pEX ext
t pIM imt
t (45)
(1 + g) (1 + n)
Log-linearizing we obtain:
(1 + r) b b br
bbt =
b bt 1 + rt + pEX exbEX
b pt pIM imbIM + exct
pt ex c
pIM imimt (46)
(1 + g) (1 + n) (1 + g) (1 + n)
EX
Pt PtIM
EXt IMt
where ext = I
Pt Xt Nt
; imt = I
Pt Xt Nt
; pEX =
t PtI and pIM =
t PtI . Equation (11) can now be
rewritten as:
1
" #
X (1 + r) bbt+s 1
b pK kbt+s 1
k
vt v = Et s
p cbt+s + ibt+s + bbt+s
C
c i b L b
p L Lt+s
(1 + g) (1 + n) (1 + g) (1 + n)
s=0
1 b 1 b
+ k kt 1 + bbt 1 (47)
Deâ€¦ne the normalized real GDP in deviation from the steady state as:
yt = sc bt + sibt + sx ext
b c i c c
sm imt (48)
19
where sc , si , sx and sm are respectively the shares of consumption, investment, exports and imports
out of total value added. Using the equations (46) and (48) into (47), we get:
1
X
vt v br=pY y
= Et s
b
yt+s b
sL Lt+s sK bt+s
k 1 +( rt+s + sx pEX
b bt+s sm pIM )
bt+s
pY y (1 + g) (1 + n)
s=0
1 k 1 b b
+ Y btk 1 + bt 1 (49)
p y pY y
where sL and sK are also shares out of total value added. Hence, in an open economy the standard
br=pY y
Solow residual needs to be adjusted for the returns on net foreign assets, b
(1+g)(1+n) rt , and for terms
capturing the terms of trade, sx pEX
bt sm pIM .
bt An improvement in the terms of trade has eÂ¤ects
analogous to an increase in TFP - both give the consumer higher consumption for the same input
of capital and labor (and therefore higher welfare). See Kohli (2004) for a static version of this
result.
b
The terms in rt+s also capture present and expected future capital gains and losses on net
foreign assets due either to exchange rate movements or to changes in the foreign currency prices
of the assets. Finally, the initial conditions include not only the (domestic) capital stock, but also
the net stock of foreign assets.
Conceptually, it makes very good sense that all these extra terms come into play when taking
the GDP route to the measurement of welfare in the open economy. Measuring them empirically
poses major challenges, however. One needs reliable measures of changes in foreign asset holdings
for a large sample of countries. Asset returns would have to be measured in risk-adjusted terms to
make them comparable across countries. In addition, forecasts of future asset returns and the terms
of trade would be required as well. In contrast, all these problems disappear if the measurement
of welfare is based on real absorption rather than GDP as the measure of output, in which case
the same terms that summarize welfare in the closed economy suÂ¢ ce to measure it in the open
economy. The implication is that we can measure welfare empirically in ways that are invariant to
the degree of openness of the economy.
4.5 Multiple Wages and Labor Market Rationing
So far we have assumed that the household is a price-taker in goods and factor markets, and that it
faces no constraints other than the intertemporal budget constraint. We have exploited the insight
s
that under these conditions relative prices measure the representative consumerâ€™ marginal rate of
s
substitution between goods, even when relative prices do not measure the economyâ€™ marginal rate
of transformation. We now ask whether our conclusions need to be modiâ€¦ed in environments where
the household does not behave as a price taker. We present two examples, and then draw some
tentative conclusions about the robustness of our previous results.
Our examples focus on the labor market. It seems reasonable to assume that consumers are
price-takers in capital markets; most individuals take rates of return on assets as exogenously given.
20
The assumption is still tenable when it comes to the purchase of goods, although some transaction
prices may be subject to bargaining. The price-taking assumption seems most questionable when it
comes to the labor market, and indeed several literatures (on labor search, union wage setting, and
eÂ¢ ciency wages, to name three) begin by assuming that households are not price takers in the labor
market. Thus, we study two examples. One is in the spirit of the dual labor markets literature,
where wages are above their market-clearing level in some sectors but not in other. We do not
model why wages are higher in the primary sector, but this can be due to the presence of unions or
government mandates in formal but not in informal employment, or eÂ¢ ciency wage considerations
in some sectors but not in others. Wages in the secondary market are set competitively. The
second example is in the spirit of labor market search, and has households face a whole distribution
of wages. In both cases households would prefer to supply all their labor to the sector or â€¦rm that
pays the highest wage, but are unable to do so. In this sense, both examples feature a type of labor
market rationing. (In both cases the diÂ¤erent wages are paid to identical workers, and are not due
to diÂ¤erences in human capital characteristics.)
First, consider the case in which the household can supply labor in two labor markets. The
primary market pays a high wage PtL and the secondary market pays a lower wage PtL < PtL .
Although the worker prefers to work only in the primary sector, the desirable jobs are rationed; he
e
cannot supply more than L hours in the high-wage job in each period. The representative household
faces the following budget constraint:
PtI Kt + Bt = (1 ) PtI Kt Le L e
L) + PtK Kt PtC Nt (50)
1 + (1 + it ) Bt 1 + Nt Pt L + Nt Pt (Lt 1+ t
Assuming that the labor rationing constraint is binding, our previous derivations remain valid and
equations (16), (19), (23) and (25) continue to hold, as long as we redeâ€¦ne the modiâ€¦ed productivity
residual as:
Yt Kh;t 1
log P Rt = log sL log Lh;t sK log (51)
Nt Nh;t 1
P LL
where the distributional share of labor sL PY Y
is computed using the low wage, paid in the dual
labor market, rather than the average wage. The intuition for this result comes from the fact that
e
the marginal wage for the household is P L while Nt (P L P L )L can be considered as a lump-sum
t t t
transfer and can be treated exactly like the proâ€¦t term in the budget constraint. (Thus, we can
also allow for arbitrary variations over time in the primary wage PtL or the rationed number of
e
hours L without changing our derivations.)
This example shows that in some cases our methods need to be modiâ€¦ed if the household is no
longer a price-taker. However, in this instance the modiâ€¦cation is not too diÂ¢ cultâ€” one can simply
decrease the labor share by the ratio of the average wage to the competitive wage. Furthermore,
this example shows that imperfect competition in factor markets can introduce an additional gap
between the welfare residual and the standard Solow residual that is like a tax wedge, making
21
our modiâ€¦cations to standard TFP even more important if one wants to use TFP data to capture
welfare. As is the case with taxes, welfare rises with increases in output holding inputs constant,
even if there is no change in actual technology.
Note that we would get a qualitatively similar result if, instead of labor market rationing,
we assumed that the household has monopoly power over the supply of labor, as in many New
Keynesian DSGE models. As in the example above, we would need to construct the true labor
s
share by using the householdâ€™ marginal disutility of work, which would be less than the real wage.
In this environment, we would obtain the welfare-relevant labor share by dividing the observed
s
labor share by an assumed value for the average markup of the wage over the householdâ€™ marginal
rate of substitution between consumption and leisure.
The second example shows that there are situations where our previous results in sections 2 and
3 are exactly right and need no modiâ€¦cation, even with multiple wages and labor market rationing.
Consider a household that comprises a continuum of individuals with mass Nt . Suppose that each
b
individual can either not work, or work and supply a â€¦xed number of hours L. In this environment,
the household can make all its members better oÂ¤ by introducing lotteries that convexify their
choice sets. Suppose that the household can choose the probability qt for an individual to work.
The representative household maximizes intertemporal utility:
Nt+s h i
1
X 1 0 b 1
Vt = s qt U (Ct ; L L) + (1 qt )U (Ct ; L) (52)
(1 + ) H
s=0
0
where qt U (Ct ; L b
L) + (1 1 0
qt )U (Ct ; L) denotes expected utility prior to the lottery draw. Ct
1
and Ct denote respectively per-capita consumption of the employed and unemployed individuals,
while average per-capita consumption, Ct is given by:
0 1
Ct = qt Ct + (1 qt )Ct (53)
Assume that the individuals that work face an uncertain wage PtL , which is observed only after labor
supply decisions have been made. More speciâ€¦cally, individual wages in period t are iid draws from
a distribution with mean Et PtL . Notice that, by the law of large numbers, the household does not
face any uncertainty regarding its total wage income. Thus, the budget constraint for the household
becomes:
Lb
PtI Kt +Bt = (1 ) PtI Kt 1 +(1 + it ) Bt K
1 +qt Nt Et Pt L+Pt Kt 1 + t PtC qt Ct + (1
0 1
qt )Ct Nt
(54)
Following Rogerson and Wright (1988) and King and Rebelo (1999),15 we can rewrite the per-period
utility function as:
15
In obtaining this result we use the fact that the marginal utility of consumption of the individuals in the household
1
b
(L L)
needs to be equalized at the optimum. This implies: c0 = c1
t t (L)
:
22
1
U (Ct ; Lt ) = Ct 1 (Lt ) (55)
1
b
where Lt = qt L denotes the average number of hours worked and:
Lt b 1 Lt 1
(Lt ) = (L L) + (1 ) (L) (56)
b
L b
L
In summary, the maximization problem faced by the household is exactly the same as the one
described in section 2, even if identical workers are paid diÂ¤erent wages 16 . All the results we have
derived previously also apply in this new setting. The second example leads to a diÂ¤erent result
from the â€¦rst for two reasons. First, it is an environment with job search rather than job queuing.
Second, the number of hours supplied by each worker is â€¦xed. Under these two assumptions, the
labor supply decision is made ex ante and not ex post.
From these two examples, it is clear that dropping the assumption that all consumers face the
same price for each good or service canâ€” but need notâ€” change the precise nature of the proxies
we develop for welfare. Even in the case where the measure changed, however, our conclusion
that welfare can be summarized by a forward-looking TFP measure and capital intensity remained
robust. While the exact nature of the proxy will necessarily be model-dependent, we believe that
our basic insight applies under fairly general conditions.
5 Sources of Welfare Changes and DiÂ¤erences
In this section we discuss how our index of welfare changes over time for each country in our data
set. Expressed in terms of equivalent consumption, our calculations are based on the following
equation, based on (20), that encompasses all the cases we will consider:
" 1 1
#
(1 ) X X 1 k Kt 1
s s
log (Ct ) = Et log P Rt+s + Et log P Rt+s + log
(sC + sCG ) py y Nt 1
s=0 s=0
(57)
where productivity change is deâ€¦ned as:
It L K Kt 1
log P Rt = sC log Ct +sI log +s log CG;t 1 sL log Lt 1 sK log
N t CG Nt 1
(58)
16
King and Rebelo (1999) show that in this framework the representative agent has an inâ€¦nite Frisch labor supply
elasticity. This result follows from the assumption that all agents in the household have the same disutility of labor.
Mulligan (2001) shows that even when all labor is supplied on the extensive margin, one can obtain any desired
Frisch elasticity of labor supply for the representative agent by allowing individual agents to have diÂ¤erent disutilities
s
of labor. In a more elaborate example, we could use Mulliganâ€™ result to show that the only restrictions on the
preferences of the representative agent are those that we assume in Section 2.
23
(1+ C )P C CN (1+ I )P I I
where the shares are sc P C CN +P I I+P G CG N
; si P C CN +P I I+P G CG N
and, as explained earlier, the
value of sCG depends on the assumptions made about government consumption. If the latter is cho-
P G CG N
s
sen so as to optimize the representative householdâ€™ welfare, then sCG = scG P C CN +P I I+P G CG N
.
Alternatively, if government consumption is pure waste, sCG = 0. Recall that Ct and CG;t denote
It
private and public consumption per capita, while Nt denotes investment per capita. Equation (57),
based on domestic absorption, is appropriate also in the open economy case, and allows us to con-
sider a variety of cases with respect to taxation and government spending: 1) wasteful government
spending with lump sum taxes (in which case distortionary taxes are set to zero in the productivity
equation); 2) optimal government spending with lump sum taxes; 3) wasteful government spending
with distortionary taxes; 4) optimal government spending with distortionary taxes.
For the cross country welfare comparisons, our empirical calculations are based on the following
equation, which generalizes (26):
" 1
#
;i ;us (1 ) X i 1 k us Ki K us
ln Ct ln Ct = Et s
log P Rt+s us
log P Rt+s + log ti 1
log tus1
(sus + sCus )
C pY;us y us Nt 1 Nt 1
G s=0
(59)
where the productivity of the reference country (US in our benchmark case) is:
us us
Kt 1
It
us us
log P Rt = sus log Ct +sus log
C I
us
+s us log CG;t 1 L;us
sus log Lus 1 K;us
sus log
Ntus CG L t K
Ntus1
(60)
while the productivity for country i is calculated as:
i i
Kt 1
i It
log P Rt = sus log Ct + sus log
C
i
I
i
+ sCus log CG;t 1 L;us
sus log Li
L t 1 K;us
sus log
K
Ntis G
Nti 1
(61)
us i
Shares and tax rates used in the calculation of log P Rt and log P Rt+s are now those of the
reference country (the US in our basic set of results) and shares are deâ€¦ned as in the within case.
5.1 Data and Measurement
In order to discuss how our index of welfare changes over time within a country and how it compares
across countries, we use yearly data on consumption, investment, capital and labor services for the
years 1985-2005 and for seven countries: the US, the UK, Japan, Canada, France, Italy and Spain.
We are unable to include Germany in the sample, since data for uniâ€¦ed Germany are available only
since 1995 in EU-KLEMS. We use two diÂ¤erent data sets to compare welfare within a country and
across countries.
To analyze welfare changes over time within a country, we combine data coming from the OECD
24
Statistical Database and the EU-KLEMS dataset17 . Our index of value added is constructed from
the OECD dataset as the weighted growth of household â€¦nal consumption, gross capital formation
and government consumption (where appropriate) at constant national prices, using as weights their
respective shares of value added. According to our theory, these shares should be kept constant
at their steady state level, but in practice we use shares averaged across the twenty years in our
sample.
In constructing the growth rate of the "modiâ€¦ed" productivity residual, we subtract from the
log-changes in value added, the log changes in the capital and labor stocks used to produce it,
weighted by their average respective shares out of total compensation. Data on aggregate produc-
tion inputs are provided by EU-KLEMS. Log-changes in capital are constructed using the estimated
capital stock constructed by applying the perpetual inventory method on investment data. In our
benchmark speciâ€¦cation, log-changes in the labor stock are approximated by log-changes in the
amount of hours worked by persons engaged. Alternatively, we use a labor service index which is
computed as a translog function of types of workers engaged (classiâ€¦ed by skill, gender, age and
sex), where weights are given by the average share of each type of worker in the value of total labor
compensation. We assume that economic proâ€¦ts are zero in the steady state so that we can recover
the gross (tax unadjusted) share of capital as one minus the labor share.
In order to compare welfare across countries, we combine data coming from the Penn World Ta-
bles and the EU-KLEMS dataset. More speciâ€¦cally, our basic measure of value added is constructed
from the Penn World Tables as the weighted average of PPP converted log-private consumption,
log-gross investment and log-government consumption, using as weights their respective shares of
value added in the reference country; as in the within case, we use shares that are averaged across
the twenty years in our sample.
To construct the modiâ€¦ed log-productivity residual in each country, we subtract from this
measure of value added the amount of capital and the amount of labor used to produce it, weighted
by their respective share of compensation in the reference country, also in this case kept constant
at their average value. The stock of capital in the economy is constructed using the perpetual
inventory method on the PPP converted investment time series from the Penn World Tables. The
stock of labor is computed in two diÂ¤erent ways using the EU-KLEMS dataset. In our benchmark
speciâ€¦cation, the amount of labor services used in the economy is approximated by the hours
worked. In the alternative speciâ€¦cation, it is computed by aggregating over diÂ¤erent types of
persons engaged using a translog function, where weights are given by the shares of compensation
to each type of labor out of total compensation to labor and are kept constant at their average
value in the reference country.
Finally, to correct our welfare calculations for the presence of distortionary taxation, we use
data on average tax rates on capital and labor provided by BoscÃ¡ et al. (2005). The tax rates are
computed by combining realized tax revenues, from the OECD Revenue Statistics, with estimates
of the associated tax bases derived from the OECD National Accounts. These data update the tax
17
The EU-KLEMS data are extensively documented by Oâ€™Mahony and Timmer (2009).
25
rates constructed by Mendoza et al (1994) and introduce some methodological improvements in
their calculation, most of which are described in Carey and Tchilinguirian (2000). In essence, they
involve some adjustments to the deâ€¦nition of the various tax bases.
5.2 Within Results
Since the change in welfare over time depends on the expected present discounted value of TFP
growth, as shown by equation (57), we need to construct forecasts of future TFP. To do so, we
estimate univariate time-series models using annual data for the seven countries in our data set.
Our sample period runs from 1985 to 2005 for all countries.
We use the various aggregate TFP measures suggested by our theory (in log levels), and estimate
simple AR processes for each country. The persistence of TFP growth is a key statistic, since it
shows how the entire summation of expected productivity residuals changes as a function of the
innovation in the log level of TFP. We report the persistence of TFP using simple, reduced-
form forecasting equations for two diÂ¤erent deâ€¦nitions of TFP, which we will use as benchmarks
throughout.
The â€¦rst concept is TFP in the case where we assume that government purchases are wasteful,
and taxes are lump-sum. For this case, as discussed above, we construct output by aggregating
consumption and investment only, but using shares that sum to (1 scG ), and we do not correct
the capital and labor shares for the eÂ¤ects of distortionary taxes. In this case, the capital and labor
shares sum to one. The second case is the one where we assume government spending is optimally
chosen, but needs to be â€¦nanced with distortionary taxes. In this case, the output concept is
the share-weighted sum (in logs) of consumption, investment and government purchases, and the
capital and labor shares are corrected for both income taxes and indirect taxes (both of which
reduce the after-tax shares). Note that in both cases the output concept measures absorption
rather than GDP (unless the economy is closed, in which case the two concepts coincide). Thus,
following our discussion in Section 4.4, both concepts (and indeed all the TFP measures that we
use in this section) are appropriate for measuring welfare in both closed and open economies. In
both cases, we assume that pure economic proâ€¦ts are zero in the steady state.
For all countries, the log level of TFP is well described by either an AR(1) or AR(2) stationary
process around a linear trend. In Table (1) we report the estimation results obtained using the two
deâ€¦nitions of TFP stated above, together with the Lagrange Multiplier test for residual â€¦rst order
serial correlation (shown in the last line of each panel in the table), conâ€¦rming that we cannot reject
the null of no serial correlation for the preferred speciâ€¦cation for each country. For all countries,
the order of the estimated AR process is invariant to the TFP measure used. In all cases, we can
comfortably reject the null of a unit root in the log TFP process (after allowing for a time trend).
We use the estimated AR processes to form expectations of future levels or diÂ¤erences of TFP,
which are required to construct our welfare indexes.
We use equation (57), derived from (20), to express the average welfare change per year in each
country in terms of changes in equivalent consumption. Given the time-series processes for TFP
26
in each country, we can readily construct the â€¦rst two terms in equation (57), the present value of
expected TFP growth, and the change in expectations of that quantity. The third term, which
depends on the change in the capital stock can also be constructed using data from EU-KLEMS.
We assume that the composite discount rate, ; is common across countries and we set it equal to
0:95:18 For the expenditure and distributional shares, we use their country speciâ€¦c averages over
the sample period.
The results are in Table 2. We see that assumptions about â€¦scal policy aÂ¤ect the results in
signiâ€¦cant ways. We â€¦rst illustrate our methods by discussing the results for the US, which are
given in the last row. We then broaden our discussion to draw more general lessons from the full
set of countries.
In the â€¦rst column of Table 2, we construct the output data and the capital and labor shares
under the assumption that government expenditure is wasteful and taxes are lump-sum. In this
column, "utility-relevant output" comprises just consumption and investment, aggregated using
weights that sum to less than one.19 In this case, the average annual growth rate of welfare in the
US is equivalent to a permanent annual increase in consumption of about 2.5 percent. Recall from
Section 4.4 that this result applies whether we think of the US as an open or a closed economy.
all
The same is true for all the other results in Table 2â€“ apply to open as well as closed economies.
Now we study the case of optimal government spending, still under the assumption that taxes
are not distortionary. Thus, at the margin the consumer is indiÂ¤erent between an additional unit of
private consumption and an additional unit of government expenditures. In this case, output con-
sists of consumption, investment and government purchases, aggregated using nominal expenditure
shares that sum to one. In a closed economy this concept of TFP corresponds to the standard Solow
residual. Note that in all cases our output concepts correspond to diÂ¤erent measures of absorption;
this is why they are relevant for both closed and open economies.
Welfare growth for the US is only slightly higher when we assume that expenditures are optimal:
2.6 versus 2.5 percent for the lump-sum tax cases. (We will see that this result is not universal
within our sample of countries.) On the whole, the diÂ¤ering assumptions about the value of
government expenditure do not change the calculated US growth rate of welfare signiâ€¦cantly. Note,
however, that this result does not mean that the US consumer is indiÂ¤erent between wasteful and
optimal government spending. Steady-state welfare is surely much lower in the case where the
government wastes 20 percent of GDP. However, our results show that the diÂ¤erence in welfare in
the two cases is almost entirely a level diÂ¤erence rather than a growth rate diÂ¤erence.
We repeat our welfare calculations under the assumption that the government raises revenue
via distortionary taxes. The results are in columns 3 and 4 of Table 2. As shown above, if taxes
are distortionary we need to construct the factor shares in the Solow residual using the after-tax
wage and capital rental rate perceived by the household, implying that the shares will sum to less
18
We construct our measure of following the method of Cooley and Prescott (1995), who â€¦nd = 0:947.
19
The weight on consumption is the nominal value of consumption divided by nominal expenditures on consumption,
investment and government purchases. The weight on investment is the nominal value of investment, divided by the
same denominator.
27
than one. We construct the new shares using the tax rates described in the previous section. The
quantitative eÂ¤ect of this change is signiâ€¦cant. In both of the cases we consider (wasteful spending
and optimal spending), per capita welfare growth expressed in terms of consumption growth rates is
higher by nearly half a percentage point per year. Intuitively, if taxes are distortionary then steady-
state output is too low; thus, any increase in output, even with unchanged technology, is a welfare
improvement. It is quantitatively important to allow for the fact that taxes are distortionary and
not lump-sum. For the US, it matters more for the growth rate of welfare than whether we assume
that government spending is wasteful or optimal. We take as our benchmark the case shown in
the last column, where spending is optimally chosen (from the point of view of the household) and
taxes are distortionary. In this case, average US welfare growth is equivalent to a growth rate of
per-capita consumption of 3 percent per year.
Assumptions about â€¦scal policy naturally matter more in countries with a high rate of growth
of government purchases per capita and with a high growth rate of factor inputs. For example,
both facts are true for Spain over our sample period. The growth rate of welfare in Spain nearly
doubles from the â€¦rst column, where its 2.1 percent annual welfare growth rate is literally middling,
to the last, where its 4 percent growth rate is the highest among all the countries in our sample.
Assumptions about â€¦scal policy also matter signiâ€¦cantly for Canada and Japan, and change the
welfare growth rates of these countries by a full percentage point or more. In percentage terms,
the change is particularly dramatic for Canada. Under three of the four scenarios, the UK leads
our sample of countries in welfare growth rates; in the last case, it is basically tied with Spain.
Finally, we show the full time series of the welfare indexes for each country graphically, for
our two benchmark cases of wasteful spending with lump-sum taxes and optimal spending with
distortionary taxes. In Figures 1 and 2 we report the evolution over time of our welfare indexes
for each country, in log deviations from their values in 1985. In Figure 1, the UK is the clear
growth leader, with France and the US nearly tied in a second group, and Canada trailing badly.
In Figure 2, by contrast, there are three clear groups: the UK and Spain lead, by a considerable
margin; the US, France, and Japan comprise the middle group; Italy and Canada have the lowest
welfare growth rates. Two countries show signiâ€¦cant declines in growth rates, both starting in
the early 1990s. The â€¦rst is Japan, which in the â€¦rst few years of our sample grew in line with
the leading economies, Spain and the UK, and then slowly drifted down in growth rate to end the
sample in the middle group, with France and the US. Similarly, Italy used to grow at the pace
of the middle group, but then experienced a slowdown which, by the end of the sample, caused
it to leave the middle group and form a low-growth group with Canada. Thus, our results are
consistent with the general impression that Italy and Japan experienced considerable declines in
economic performance over the last two decades relative to the performance in the earlier postwar
period.
In Table 3, we investigate which of the two components of welfareâ€“TFP growth or capital
accumulationâ€“contributed more to the growth rate of welfare in our sample of countries. For the
purpose of this decomposition, we treat the expectation-revision term as a contribution to TFP. In
28
order to keep the table uncluttered, we drop the case where government spending is optimal and
taxes are lump sum. The â€¦rst column, in which government spending is wasteful and taxes are
treated as lump-sum, shows that four of the seven countries have achieved two-thirds or more of
their welfare gains mostly via TFP growth. The exceptions are the three countries that are known
to have had low TFP growth over our sample period: Japan, Canada, and Spain. Moving to the case
of distortionary taxes raises the TFP contribution (by reducing the factor shares), as does changing
the treatment of public spending as optimal rather than wasteful (which raises the growth rate of
output, and thus TFP). In the case of optimal spending with distortionary taxes, all countries get
a majority of their welfare growth from TFP. Only in Japan and Canada is the contribution of
TFP to welfare less than 70 percent, and in most cases it is 75 percent or more.
We check the robustness of the previous results to using a more reâ€¦ned measure of labor input.
As noted above, for the main results we use total hours worked as the measure of labor. However,
if workers are heterogeneous along dimensions that aÂ¤ect their productivity and are paid diÂ¤erent
wages as a consequence, we should use a labor input index that recognizes this fact. This amounts
to implementing our extension in Section 4.1. The results, for the case of optimal spending with
distortionary taxes, are shown in Figure 3. Qualitatively, there is little change. The UK and Spain
are still bunched at the top, followed by the US and France. However, with the labor index, Japan
drops a little further behind these four leading economies, and ends the period with a cumulated
welfare growth rate in between the US and France and the trailing economies, Italy and Canada.
Overall, the results look very similar to our baseline case.
Finally, we compare the results we have just obtained using our theory-based welfare metric to
those implied by standard proxies for welfare change. In Table 4, we present the average growth
rates of GDP and consumption per capita for our group of countries over our sample period, as well
as the average growth rate of our welfare measure under the assumption of optimal government
spending and distortionary taxes. First, note the diÂ¤erences in magnitude. Welfare usually grows
faster than do conventional measures like consumption per capita. The diÂ¤erence is typically in the
order of one full percentage point per year, which is a striking diÂ¤erence in growth rates. Second,
these diÂ¤erent measures sometimes produce quite diÂ¤erent rankings among countries. Take France,
for example. Judged by consumption growth, this country comes at the very bottom of our group
of seven countries, signiâ€¦cantly behind even Canada. In terms of welfare growth, on the other hand,
France comes second.
5.3 Cross Country Results
We now turn to measuring welfare diÂ¤erences across the countries in our sample. For each country
and time period, we calculate the welfare gap between that country and the US, as deâ€¦ned in
equation (59). Recall that this gap is the loss in welfare of a representative US consumer who is
moved permanently to country i starting at time t, expressed as the log gap between the "equivalent
consumption" of the consumer in the two cases. In this hypothetical move, the consumer loses the
per-capita capital stock of the US, but gains the equivalent capital stock of country i. From time
29
t on, the consumer faces the same product and factor prices and tax rates, and receives the same
lump-sum transfers and government expenditure beneâ€¦ts as all the other consumers in country i.
In a slight abuse of language, we often use refer to the incremental equivalent consumption as "the
welfare diÂ¤erence" or "the welfare gap."
Note that these gaps are all from the point of view of a US consumer. Hence, all the shares in
(59), even those used to construct output and TFP growth in country i, are the US shares. This
naturally raises the question whether our results would be quite diÂ¤erent if we took a diÂ¤erent
country as our baseline. We return to this issue after presenting our basic set of results.20
We present numerical results in Table 5. Since the size of the gap varies over time, we present
the gap at the beginning of our sample, at the end of our sample, and averaged over the sample
period. We present results for three cases: wasteful spending, with lump-sum and distortionary
taxes, and optimal spending with distortionary taxes. These numerical magnitudes are useful
references in the discussion that follows.
However, the results are easiest to understand in graphical form. We plot the welfare gap for
the countries and time periods in our sample in Figures 4 and 5. Note that by deâ€¦nition the
gap is zero for the US, since the US consumer neither gains nor loses by moving to the US at
any point in time. The vertical axis shows, therefore, the gain to the US consumer of moving to
any of the other countries at any point in the sample period, expressed in log points of equivalent
consumption. Figure 4 shows the results for the case of wasteful spending and lump-sum taxes.
Figure 5 shows the results for our benchmark case, where we allow for distortionary taxes and
assume that government expenditure is optimally chosen. Since both â€¦gures show qualitatively
similar results, for brevity we discuss only the benchmark case.
It is instructive to begin by focusing on the beginning and end of the sample. At the beginning
of the sample, expected lifetime welfare in both France and the UK was less than 20 percent
lower than in the US (gaps of 16 and 19 percent, respectively). This relatively small gap reâ€¡ects
both the long-run European advantage in leisure and the fact that in the mid-1980s the US was
still struggling with its productivity slowdown, while TFP in the leading European economies was
growing faster than in the US. Capital accumulation was also proceeding briskly in those countries.
By the end of the sample, the continental European economies, Canada and Japan are generally
falling behind the US, because they had not matched the pickup in TFP growth and investment
experienced in the US after 1995. Italy experiences the greatest relative "reversal of fortune,"
ending up with a welfare gap of nearly 70 percent relative to the US. The results for France are
qualitatively similar, but far less extreme. France starts with a welfare gap of 16 percent, and
slowly slips further behind, ending with a gap of 21 percent. In continental Europe, only Spain
shows convergence to the US in terms of welfare: it starts with a gap of 41 percent, and ends with
a gap of 36 percent. However, after 1995 Spain holds steady relative to the US, but does not gain
further.
20
We conjecture that if we took a second-order approximation to the welfare gap, then the shares in our computation
would be averages of the shares in the two countries, and hence bilateral comparisons would be invariant to the choice
of a reference country. We leave the investigation of this hypothesis to future research.
30
The only economy in our sample that exhibits convergence to the US throughout our sample
is the UK. Indeed, as Figure 4 shows, under the assumption of wasteful spending and lump-sum
taxes, the UK overtakes the US by the end of our sample period. Table 5 shows that in more
realistic cases where taxes are assumed to be distortionary the welfare level of the UK is always
below that of the US, but the UK shows strong convergence, slicing two-thirds oÂ¤ the welfare gap in
two decades in our benchmark case. This result is interesting, because the UK experienced much
the same lack of TFP growth in the late 1990s and early 2000s as the major continental European
economies.21 However, the UK had very rapid productivity growth from 1985 to 1995. The
other "Anglo-Saxon" country in our sample, Canada, had a welfare level about 30 percent below
that of the US in 1985, but the welfare gap had grown by an additional 50 percent by the end of
the sample. This result is due primarily to the diÂ¤erential productivity performance of the two
countries: TFP in Canada actually fell during the 1990s, and rose only slowly in the early 2000s.
Perhaps the most striking comparison is between the US and Japan. Even in 1985, when its
economic performance was the envy of much of the world, Japan was the least attractive country
in our sample to an US consumer contemplating emigration; such a consumer would give up nearly
50 percent of his consumption permanently in order to stay in the US instead of moving to Japan.
However, like the UK and Spain, Japan was closing the gap with the US until its real estate bubble
burst in 1991. The relative performance of the three countries changes dramatically from that
point: unlike the UK, which continues to catch up, and Spain, which holds steady, Japan begins
to fall behind the US, â€¦rst slowly and then more rapidly. Having closed to within 43 percent of
the US welfare level in 1991, Japan ends our sample 53 percent behind. This cautionary history
suggests that it would be interesting to see what the same calculations will show for the US in
another 10 or 15 years, after the bursting of the US real estate bubble and the associated â€¦nancial
crisis.
As we did for the within-country results, we investigate whether the cross-country welfare gaps
are driven mostly by the TFP gap or by diÂ¤erences in capital per worker. The results are in Table
6. We focus on the last column of Panel C, which gives results averaged over the full sample period
for our baseline case of optimal spending with distortionary taxes. We â€¦nd that for â€¦ve of the six
countries, TFP is responsible for the vast majority of the welfare gap relative to the US. Indeed,
for Japan TFP accounts for more than 100 percent of the gap (meaning that Japan has generally
had a higher level of capital per capita than the US). Thus we arrive at much the same conclusion
as Hall and Jones (1999), although our deâ€¦nition of TFP is quite diÂ¤erent from the one they used,
and we do not focus only on steady-state diÂ¤erences. The exception to this rule is the UK. The
average welfare gap between the US and the UK is driven about equally by TFP and by capital.
Panel B shows that by the end of the sample, the UK had surpassed the US in "welfare-relevant
TFP," and more than 100 percent of the gap was driven by the diÂ¤erence in per-capita capital
between the two economies.
We now check the robustness of the preceding results along two dimensions.
21
For discussion and a suggested explanation, see Basu, Fernald, Oulton and Srinivasan (2004).
31
First, as noted above, we wish to see whether our welfare rankings among countries is sensitive
to the choice of the baseline country. We thus redo the preceding exercises taking France as
the baseline country. France is the largest and most successful continental European economy
in our sample, and by revealed preference French households place much higher weight on leisure
than do US ones.22 We summarize the results for our baseline case of optimal spending with
distortionary taxes in Figure 6. For ease of comparison with the preceding cross-country â€¦gures,
we still normalize the US welfare level to zero throughout, even though the comparison is done
from the perspective of the French consumer and is based on French shares. Reassuringly, we see
that the qualitative results are unchanged. France and the UK start closest to the US in 1985, but
even they are well behind the US level of welfare. The UK converges towards the US welfare level
and so, from a much lower starting point, does Spain. All the other economies, including France,
fall steadily farther behind the US over time. Interestingly, from the French point of view almost
all the other countries are shifted down vis-a-vis the US relative to the rankings from the US point
of view. It appears that the representative French consumer believes that the US is further ahead
of France than does the representative US consumer!
Second, we redo the baseline results (from the point of view of the US consumer) using an
index of heterogeneous labor input. Our method demands that we construct the labor index for
each country weighting the hours of diÂ¤erent types of workers by the US shares. However, unlike
the within-country case in the previous sub-section, where we used country-speciâ€¦c shares, this
procedure yields quite diÂ¤erent results than the baseline case using hours. Figure 7 presents the
results graphically for our baseline case of optimal spending with distortionary taxes. We still â€¦nd
that the UK and France are closest to the US in welfare levels, but now we no longer â€¦nd strong
evidence of convergence for the UK; both countries steadily fall behind the US over time. Spain
shows the greatest diÂ¤erence relative to our previous results. Instead of converging or holding
steady relative to the US, Spain falls behind monotonically. This result is driven by the fact that
the Spanish growth rate of labor input is very high for categories of workers that receive a high
share of labor income in the US (particularly workers with a "middle" level of education, as deâ€¦ned
by EU-KLEMS). As a check, we computed results using the index of labor input, but from the
French point of view (i.e., using French shares). We â€¦nd very similar results for Spain, showing that
this result is likely to obtain whenever one applies disaggregated labor shares from rich countries
to growth rates of labor input for middle-income (or poor) countries. For this reason, we use total
hours as our baseline measure of labor input, in both the within- and cross-country cases.
Finally, as we did for the results on within-country welfare growth, we compare our welfare
results to those based on traditional measures, namely PPP-adjusted GDP and consumption per
capita. The results are in Table 7. Focusing on Panel B, for the â€¦nal year of our sample, we see
that the three measures sometimes give identical results. For example, the US is atop the world
rankings by all three measures, although the gap between the US and the second-ranked country is
22
As noted above, data limitations prevent us from including Germany in our sample, although it would be another
natural baseline.
32
much smaller in percentage terms for welfare (6 percent) than it is for the other two variables (18
or 19 percent). On the other hand, the diÂ¤erences can be striking. For example, Canada, which is
the second only to the US in GDP and third judged by consumption, is third from the bottom in
our welfare ranking. Canada, which leads Spain by 20 percent or more in terms of consumption
and GDP per capita, trails Spain by about 10 percent in our welfare comparison. Indeed, Spain is
last within our group of countries in terms of the conventional metrics of consumption and GDP,
but ranks fourth in welfare terms, trailing only the US, UK and France. For the other countries,
the welfare measure is not so kind. Japan trails the US by only 26 percent in GDP per capita,
but double thatâ€” 52 percentâ€” in terms of welfare. Similarly, Italy has more than 60 percent of the
per-capita GDP of the US, but only about one-third the welfare level. On the other hand, France
trails the US by 40 percent in consumption per capita, but by only half that amount in terms
of a welfare. Thus, our measure clearly provides new information on welfare diÂ¤erences among
countries.
6 Relationship to the Literature
Measuring welfare change over time and diÂ¤erences across countries using observable national in-
come accounts data has been a long-standing challenge for economists. We note here the similarities
and diÂ¤erences between our approach and ones that have been taken before. We also suggest ways
in which our results might be useful in other â€¦elds of economics, where the same question arises in
diÂ¤erent contexts.
Nordhaus and Tobin (1972) originated one approach, which is to take a snapshot of the econ-
s ow
omyâ€™ â€¡ output at a point in time and then go â€œbeyond GDP,â€? by adjusting GDP in various
ow
ways to make it a better â€¡ measure of welfare. Nordhaus and Tobin found that the largest gap
ow ow
between â€¡ output and â€¡ welfare comes from the value that consumers put on leisure. Their
result motivated us to add leisure to the period utility function in our model, which is standard
s
in business-cycle analysis but not in growth theory. Nordhaus and Tobinâ€™ approach has been fol-
lowed recently by Jones and Klenow (2010) who add other corrections, notably for life expectancy
and inequality. However, this point-in-time approach does not take into account the link between
s
todayâ€™ choices and future consumption or leisure possibilities. For example, high consumption in
the measured period might denote either permanently high welfare or low current investment. Low
investment would mean that consumption must fall in the future, so its current level would not
be a good indicator of long-term welfare. Our approach is to go beyond point-in-time measures of
welfare and compute the expected present discounted value of consumersâ€™entire sequence of period
s
utility. In so doing, we also shift the focus from consumersâ€™particular choices of one periodâ€™ con-
sumption and leisure to their intertemporal choice sets, as deâ€¦ned by their assets and the sequence
of prices they face. This approach pays oÂ¤ particularly when we measure welfare diÂ¤erences across
countries.
Our intertemporal approach echoes the methods used in the literature started by Weitzman
33
(1976) and analyzed in depth by Weitzman (2003), with notable contributions from many other
authors.23 This literature also relates the welfare of a representative agent to observables; for
example, Weitzman (1976) linked intertemporal welfare to net domestic product (NDP). Unlike
our model which allows for uncertainty about the future, this literature almost always assumes
perfect foresight.24 As we discuss later, allowing for uncertainty is important when forward-looking
rules for measurement are applied to actual data. More importantly, the results in these papers are
derived using a number of strong restrictions on the nature of technology (typically an aggregate
production function with constant returns to scale), product market competition (always assumed
to be perfect), and the allowed number of variables that are exogenous functions of time, such as
technology or terms of trade (usually none, but sometimes one or two). Most of the analysis in the
literature applies to a closed economy where growth is optimal.25 Taken together, this long list of
assumptions greatly limits the domain of applicability of the results.
By contrast, we derive all our results based only on â€¦rst-order conditions from household opti-
mization, which allows for imperfect competition in product markets of an arbitrary type and for
a vast range of production possibilities, with no assumption that they can be summarized by an
aggregate production function or a convex technology set. (This makes it easy to apply our results
to modern trade models, for example, since these models often assume imperfect competition with
substantial producer heterogeneity.) We do not need to assume that the economy follows an opti-
mal growth path. We are also able to allow for a wide range of shocks, including but not limited
to changes in technology, tax rates, terms of trade, government purchases, the size of Marshallian
spillovers, monetary policy, tariÂ¤s, and markups.26 Crucially, we do not need to specify the sources
of structural shocks to the economy. The key to the generality of our results is that we condition on
observed prices and asset stocks without needing to model why these quantities take on the values
that they do.27 Perhaps most importantly, since we do not make assumptions about the technology
or distortions prevailing in each country, we are able to compare welfare across countries that could
easily have diÂ¤erent levels of technology, taxes or product-market competition.
Our work clariâ€¦es and uniâ€¦es several results in other literatures, especially international eco-
nomics. Kohli (2004) shows in a static setting that terms-of-trade changes can improve welfare in
23
A far from exhaustive list includes Asheim (1994), Arronson and LÃ¶fgren (1995), Mino (2004), Sefton and Weale
(2006), Basu, Pascali, Schiantarelli and Serven (2009), and Hulten and Schreyer (2010). Reis (2005) analyses the
related problem of computing a dynamic measure of inâ€¡ ation for a long-lived representative consumer.
24
Arronson and LÃ¶fgren (1995) allow for stochastic population growth, and Weitzman (2003, ch. 6) considers
shocks coming from stochastic depreciation of capital.
25
Sefton and Weale (2006) and Hulten and Schreyer (2010) consider an open economy with changes in the terms
of trade and Mino (2004) analyses Marshallian spillovers to R&D (all under perfect foresight).
26
See also Sandleris and Wright (2011) for an attempt to extend the basic ideas in Basu and Fernald (2002) in
order to evaluate the welfare eÂ¤ects of â€¦nancial crises. These papers try to derive methods to measure the welfare
eÂ¤ects of a particular shock, which requires specifying an explicit counterfactual path that the economy would have
followed in the absence of the shock.
27
We do need to forecast the present value of future TFP in order to implement our results in data. It is an
open question whether specifying a complete general-equilibrium structure for the model would improve our forecasts
substantially. Most macro literatures have concluded that reduced-form forecasts beat model-based ones. We decided
that the possible gain in forecasting accuracy from specifying a general-equilibrium structure would not compensate
for the loss of generality of our results.
34
open economies even when technology is constant. Kehoe and Ruhl (2008) prove a related result
in a dynamic model with balanced trade: opening to trade may increase welfare, even if it does
not change TFP. In these models, which assume competition and constant returns, technology is
equivalent to TFP. We generalize and extend these results, and show that in a dynamic environment
with unbalanced trade welfare can also change if there are changes in the quantity of net foreign
assets or in their rates of return.28 In general, we show that there is a link between observable
aggregate data and welfare in an open economy, which is the objective of Bajona, Gibson, Kehoe
and Ruhl (2010). While we agree with the conclusion of these authors that GDP is not a suÂ¢ cient
statistic for uncovering the eÂ¤ect of trade policy on welfare, we show that one can construct such
a suÂ¢ cient statistic by considering a relatively small number of other variables. Our results also
shed light on the work of Arkolakis, Costinot and Rodriguez-Clare (2011). These authors show
that in a class of modern trade models, which includes models with imperfect competition and
micro-level productivity heterogeneity, one can construct measures of the welfare gain from trade
without reference to micro data. Our results imply that this conclusion actually holds in a much
larger class of models, although the exact functional form of the result in Arkolakis et al. (2011)
may not. Finally, since changes in net foreign asset positions and their rates of return are extremely
hard to measure, we show that one can measure welfare using data only on TFP and the capital
stock, even in an open economy, provided that TFP is calculated using absorption rather than
GDP as the output concept.
Our work provides a diÂ¤erent view of a large and burgeoning literature that investigates the
productivity diÂ¤erences across countries. As noted above, if we specialize our cross-country result to
the lump sum-optimal spending case, we obtain something closely related to the results produced
by the â€œdevelopment accountingâ€? literature. We show that in that case, (the present value of)
the log diÂ¤erences in TFP levels emphasized by the developing accounting literature need to be
supplemented with only one additional variable, namely log level gaps in capital per person, in order
to serve as a measure of welfare diÂ¤erences among countries.29 This result implies immediately
that estimates of TFP losses due to allocative ineÂ¢ ciency (e.g., Hsieh and Klenow, 2009) can be
translated to estimates of welfare losses.
Our results are also related to an earlier literature on â€œindustrial policy.â€? and to more recent
literature on the eÂ¤ect of "reallocation". Bhagwati, Ramaswami and Srinivasan (1969) and Bulow
and Summers (1986) argue welfare would be enhanced by policies to promote growth in industries
where there are rents, for example stemming from monopoly power.30 The TFP term in our basic
result, equation (20), captures this eÂ¤ect. When â€¦rms have market power, their output grows
28
The result that openness does not change TFP may be fragile in models with increasing returns. If opening to
s
trade changes factor inputs, either on impact or over time, then TFP as measured by Solowâ€™ residual will change as
well, which we show has an eÂ¤ect on welfare even holding constant the terms of trade.
29
As we show, what matters is the present discounted value of TFP diÂ¤erences. Moreover, one needs to compute
TFP using diÂ¤erent shares than the ones used by the development accounting literature, and switch to a diÂ¤erent
output concept (based on domestic absorption) in an open economy.
30
A second-best policy might involve trade restrictions to protect such industries from foreign competition. If
lump-sum taxes are available, the optimal policy is always to target the distortion directly through a tax-cum-subsidy
scheme.
35
faster than the share-weighted sum of their inputs, even when their technology is constant. Thus,
aggregate TFP rises when â€¦rms with above-average market power grow faster than average. At
â€¦rst this result sounds counterintuitive, since it implies that welfare is enhanced by directing more
capital and labor to the most distorted sectors. However, the logic is exactly the same as the usual
result that â€¦rms with the greatest monopoly power should also receive the largest unit subsidies to
increase their output.
A number of recent papers suggest that a substantial fraction of output growth within countries
comes from reallocation, broadly deâ€¦ned. However, given the diÂ¤erent deâ€¦nitions of â€œreallocationâ€?
used in the literature, it is not clear how much reallocation matters for welfare. Our work sug-
gests that the literature should focus on quantifying the increment to aggregate TFP growth from
reallocation. Furthermore, it shows that TFP is what matters, not technical eÂ¢ ciency. TFP con-
tributions can come from either higher technical eÂ¢ ciency, by exploiting increasing returns to scale,
or by allocating inputs more eÂ¢ ciently across â€¦rms. When TFP and technical eÂ¢ ciency diverge, it
is TFP that matters for welfare. We have concentrated on aggregate TFP and capital accumula-
tion without asking how individual â€¦rms and sectors contribute to these welfare-relevant variables.
Domar (1961) and Basu and Fernald (2002) show how one can decompose standard TFP based
on GDP into sectoral and â€¦rm-level contributions. However, our results above show that in an
open economy one should base TFP measures on absorption rather than GDP. There is as yet no
s
parallel to Domarâ€™ (1961) decomposition for absorption or net investment, since that would require
allocating the output of individual industries to particular components of â€¦nal expenditure. The
approach of Basu, Fernald, Fisher and Kimball (2010), based on the use of input-output tables,
may be helpful in this endeavor.
Finally, our work is closely related to the program of developing suÂ¢ cient statistics for welfare
analysis, surveyed by Chetty (2009). We have proposed such a statistic for a representative con-
sumer in a macroeconomic context. As Chetty notes, such measures can be used to evaluate the
eÂ¤ects of policies. Suppose that one wishes to evaluate the eÂ¤ect of a policy changeâ€” for example,
a change in trade policy, as in Kehoe and Ruhl (2010). The usual method is to relate the policy
change to a variety of economic indicators, such as GDP, capital accumulation, or the trade bal-
ance, and then try to relate the indicators to welfare informally. Our work suggests that one can
dispense with these â€œintermediate targets,â€?and just directly relate the welfare outcome to a change
in policy, or to some other shock.
7 Conclusions
We show that aggregate TFP, appropriately deâ€¦ned, and the capital stock can be used to construct
suÂ¢ cient statistics for the welfare of a representative consumer. To a â€¦rst order approximation,
s
the change in the consumerâ€™ welfare is measured by the expected present value of aggregate TFP
growth (and its revision) and by the change in the capital stock. We also show that in order
to create a proper welfare measure, TFP has to be calculated using prices faced by households
36
rather than prices facing â€¦rms. In modern, developed economies with high rates of income and
indirect taxation, the gap between household and â€¦rm TFP can be considerable. Finally, in an open
ect
economy, the change in welfare will also reâ€¡ present and future changes in the returns on net
foreign assets and in the terms of trade. However, these latter terms disappear if absorption rather
than GDP is used as the output concept for constructing TFP, and TFP and the initial capital
stock are again suÂ¢ cient statistics for measuring welfare in open economies. Most strikingly,
these variables also suÂ¢ ce to measure welfare level diÂ¤erences among countries, with both variables
computed as log level deviation from a reference country.
We apply these results to measuring welfare growth rates and gaps in a sample of developed
countries. For reasonable assumptions about â€¦scal policy, we â€¦nd that over our sample period the
UK and Spain are the leaders in welfare growth rates. Throughout our sample period, however, the
US is the world leader in welfare levels. The UK converges steadily towards US levels of welfare,
and is within a few percent of catching the US by the end of our sample, 2005. At the start of
our sample, several countries show evidence of convergence to the US, but by the end almost all
countries are diverging away from US welfare levels. This divergence is particularly stark for Japan
and Italy, which end the sample with less than half the per-capita welfare of the US.
37
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A Appendix: Derivations
A.1 Making the problem stationary
The representative household maximizes intertemporal utility:
1
X 1 Nt+s
Vt = s U (Ct+s ; L Lt+s ) (A.1)
(1 + ) H
s=0
where Ct+s is the capita consumption of good i at time t+s, Lt+s are hours of work per capita, L
is the time endowment, and Nt+s population. H is the number of households, assumed to be â€¦xed
and normalized to one from now on. Population grows at constant rate n and per capita variables
at a common rate g. Xt is and index for per capita variables in the sense that their level at time
t is proportional to Xt :Consider the laws of motion for Nt and for Xt :
Nt = N0 (1 + n)t (A.2)
Xt = X0 (1 + g)t (A.3)
and normalize H = 1.
We can rewrite the utility function as:
1
X (1 + n)s
Vt = Nt U (Ct+s ; L Lt+s ) (A.4)
(1 + )s
s=0
For a well deâ€¦ned steady state in which hours of work are constant we assume that the utility
function has the King Plosser and Rebelo form (1988):
1
U (Ct+s ; L Ls ) = Ct+s 1 (L Lt+s ) (A.5)
1
We assume that (L Lt+s ) is an increasing and concave function of leisure, and assumed to be
Ct+s
positive. Deâ€¦ne ct+s = Xt+s . We can rewrite the utility function in the following form:
1 (1 )
U (ct+s ; L Lt+s ) = Xt+s ct+s 1 (L Lt+s )
1
or
(1 ) 1
U (ct+s ; L Ls ) = (1 + g)s(1 )
Xt ct+s 1 (L Lt+s )
1
Inserting this into Vt , we get:
1
X
(1 ) s
Vt = Nt X t U (ct+s ; L Lt+s ) (A.6)
s=0
42
(1+n)(1+g)1
where: = (1+ ) .
A.2 Budget constraint
Start from the usual budget constraint:
PtI Kt + Bt = (1 ) PtI Kt 1 + (1 + it ) Bt 1 + PtL Lt Nt + PtK Kt 1 + t PtC Ct Nt (A.7)
Divide both sides by PtI Xt Nt to get:
Kt Bt Kt 1 Xt 1 Nt 1 Bt 1 PtI 1 Xt 1 Nt 1
+ = (1 ) + (1 + it )
Xt Nt PtI Xt Nt Xt 1 Nt 1 Xt Nt PtI 1 Xt 1 Nt 1 Pt
I Xt Nt
PtL Lt Nt P K Kt 1 Xt 1 Nt 1 t PtC Ct Nt
+ + tI +
PtI Xt Nt Pt X t 1 N t 1 Xt Nt PtI Xt Nt PtI Xt Nt
K
Pt PtL C
Pt
Kt Bt (1+it )
Deâ€¦ne: kt = Xt Nt , bt = I
Pt Xt Nt
, pK =
t PtI , pL =
t I
Pt Xt
, pC =
t PtI ; (1 + rt ) = (1+ t ) , t = I
t
Pt Xt Nt
.
The budget constraint can be rewritten as:
(1 ) + pK
t (1 + rt )
kt + bt = kt 1 + bt 1 + pL Lt +
t t pC ct
t (A.8)
(1 + g) (1 + n) (1 + g) (1 + n)
A.3 Optimality conditions
Vt
The representative household maximizes normalized intertemporal utility vt = (1 ) . The La-
Nt Xt
grangean for this problem is:
1
X
s
t = Et fU (ct+s ; L Lt+s )
s=0
(1 ) + pKt+s (1 + rt+s )
+ t+s ( kt+s bt+s + kt+s 1 + bt+s 1 + pL Lt+s +
t+s t+s pC ct+s )g
t+s
(1 + g) (1 + n) (1 + g) (1 + n)
The FOCs are:
C
Uct t pt =0 (A.9)
L
ULt + t pt =0 (A.10)
(1 ) + pKt+1
t + Et t+1 =0 (A.11)
(1 + g) (1 + n)
43
1
t + Et (1 + rt+1 ) t+1 =0 (A.12)
(1 + g) (1 + n)
A.4 Approximation around SS
b
Deâ€¦ne with x = log xt log x the log deviation from the steady state of a variable (x is the steady
state value of xt ). Taking a â€¦rst order approximation in logs of the Lagrangean (which equals the
value function along the optimal path), one obtains:
X1
vt v = Et [ s
c b
(Uc cbt+s + UL LLt+s
s=0
b
L
+ p LLi;t+s pC cbt+s
c kbt+s
k bbt+s )
b
1
X K
(1 )+p (1 + r)
+ s+1
( kbt+s +
k bbt+s )
b
(1 + g) (1 + n) (1 + g) (1 + n)
s=0
1
X
sb (1 ) + pK (1 + r)
+ t+s ( k b+ k+ b
(1 + g) (1 + n) (1 + g) (1 + n)
s=0
L
+p L + pC c)]
1
X
s pK k rb
+ ( pL LbL +
pt+s pK
b pC cbt+s + bt+s +
t p b
rt+s )
(1 + g) (1 + n) t+s (1 + g) (1 + n)
s=0
(1 ) + pK b (1 + r)
+ k kt 1 + bbt
b 1 (A.13)
(1 + g) (1 + n) (1 + g) (1 + n)
Using the â€¦rst order conditions, the â€¦rst four lines equal zero and, therefore, we get:
1
X
s pK k rb
vt = v + Et pL LbL +
pt+s pK
b pC cbt+s + bt+s +
p b
rt+s )
(1 + g) (1 + n) t+s (1 + g) (1 + n)
s=0
(1 ) + pK b (1 + r)
+ k kt 1 + bbt
b 1 (A.14)
(1 + g) (1 + n) (1 + g) (1 + n)
Now log linearize the budget constraint:
(1 ) + pK b (1 + r) pK k
k bt + bbt
k b k kt 1 bbt
b 1
b
pL LLt pL LbL
pt pK
b
(1 + g) (1 + n) (1 + g) (1 + n) (1 + g) (1 + n) t
rb
b
rt bt + pC cbt + pC cbt = 0
c p (A.15)
(1 + g) (1 + n)
Using this result and the steady state version of the FOC for capital in (A.14) gives us:
44
1
X (1 ) + pK b (1 + r)
vt = v + Et s
[pC cbt+s + kbt+s
c k k kt+s 1 + bbt+s
b bbt+s
b 1
(1 + g) (1 + n) (1 + g) (1 + n)
s=0
1 b 1+r
b
pL LLt+s ] + k kt 1 + bbt
b 1 (A.16)
(1 + g) (1 + n)
Rearranging the terms, we get:
1
X (1 ) + pK b
vt = v + Et s
pC cbt+s + kbt+s
c k k kt+s 1
b
pL LLt+s
(1 + g) (1 + n)
s=0
1
X
1 (1 + r)
+ kbt
k 1 + s
bbt+s
b bbt+s
b (A.17)
(1 + g) (1 + n)
s=0
Using the FOC and the trasversality condition for bonds, the equation above becomes:
1
X (1 ) + pK b
vt = v + Et s
pC cbt+s + kbt+s
c k k kt+s 1
b
pL LLt+s
(1 + g) (1 + n)
s=0
1
+ kbt
k 1 (A.18)
Kt
Notice that the law of motion of capital: Kt = (1 )Kt 1 + It , can be rewritten as: Xt Nt =
Kt 1 Xt 1 Nt 1 It
(1 ) Xt 1 Nt 1 Xt Nt + Xt Nt which after some algebra becomes:
(1 )
kt = kt 1 + it (A.19)
(1 + g) (1 + n)
DiÂ¤erentiating it around the steady state, we get:
(1 )
kbt =
k kbt
k 1 + ibt
i (A.20)
(1 + g) (1 + n)
Inserting this equation into equation (A.17) we get:
1
X
b pK k bt+s 1 b
vt = v + Et s
pC cbt+s + ibt+s
c i pL L Lt+s k 1 + k kt 1 (A.21)
(1 + g) (1 + n)
s=0
This is equation (11) in the main text.
45
A.5 Connecting the level of productivity to the level of welfare
Deâ€¦ne value added (for normalized variables in deviation from steady state) as:
P C CN P II
b
yt = log yt log y = bit + Y bt = sc bt + sibt
c i c i (A.22)
PY Y P Y
K
pK k Pt Kt 1
Inserting this equation into (A.21), and noticing that pY y(1+g)(1+n)
is the SS value of sK;t Y
Pt Yt
we get:
1
X h i 1 b
vt = v + Et s
p Y y yt
b b
sL Lt+s sK bt+s
k 1 + k kt 1 (A.23)
s=0
Using the deâ€¦nition of the normalized variable, this can be rewritten as:
1
X
Y s Yt+s Kt+s 1
vt = v + p y Et (log log y) sL (log Lt+s log L) sK (log log k)
Nt+s Xt+s Nt+s 1 Xt+s 1
s=0
1 b
+ k kt 1 (A.24)
P C CN PII
where logYt = PY Y
log(Ct Nt ) + PY Y
log It :The previous equation can be rewritten as:
X 1
vt v s Yt+s Kt+s 1 1 k Kt 1
= Et log sL log Lt+s sK log + log f (t) (A.25)
pY y Nt+s Nt+s 1 p Yy Nt 1
s=0
where:
1
f (t) = log y sL log L sK log k + [g(1 sK ) + n(1 sL sK )]
1 (1 )
1 sK
+ [(1 sK ) log Xt ] + g
1 1
1 k
+ Y (log k + log Xt 1 ) (A.26)
p y
Y
Deâ€¦ne aggregate productivity (in log level) as: log P Rt = log Ntt sL log Lt sK log Kt
N
t 1
1
. Notice
that we are using a deâ€¦nition with constant shares. Then:
1
X
s 1 k Kt 1
f (t) = Et (log P Rt+s )SS + log (A.27)
pY y Nt 1 SS
s=0
Yt Kt 1
where we have used the fact that log (P Rt+s )SS = log Nt sL log L sK log Nt 1
equals
SS SS
log (Xt (1 + g)s y) sL log L sK log Xt (1 + g)s 1
k : Inserting the equation above into (A.25),
we obtain:
46
X 1
vt v s 1 k Kt 1 Kt 1
Yy
= Et [log P Rt+s (log P Rt+s )SS ] + log log (A.28)
p pY y Nt 1 Nt 1 SS
s=0
vt v
Assume vt > 0 ( < 1). Since v ' log vt log v, this result can be also re-written, to a â€¦rst
order approximation, as:
1
X
v s 1 k Kt 1 Kt 1
(log vt log v) = Et [log P Rt+s (log P Rt+s )SS ] + log log
pY y p Yy Nt 1 Nt 1 SS
s=0
(A.29)
which can be also expressed in terms of log-deviation from the steady state of intertemporal per-
capita utility as:
1
X
v Vt Vt s 1 k Kt 1 Kt 1
Yy
(log log ) = Et log [log P Rt+s (log P Rt+s )SS ]+ Yy
log log
p Nt Nt SS p Nt 1 Nt 1 SS
s=0
(A.30)
Vt
where log Nt denotes the log value of per-capita intertemporal utility along the balanced-
SS
growth path.
If > 1 so that vt < 0 (recall we are assuming (L L) > 0), the R.H.S. of (A.30) should equal
v Vt Vt
instead pY y
(log Nt log Nt ); which is positive for positive deviations from the steady
SS
state.
v
In order to interpret py y , notice that by applying the envelope theorem on the household
maximization problem, we get:
@vt
= py
t (A.31)
@yt
Moreover, using the deâ€¦nition of vt and yt ; together with the fact that Nt and Xt are deterministic
functions of time, it follows that:
Vt 1
@vt @N 1 1 @Vt 1 1
t Xt Nt Xt
= = = 1 @yt
@yt @yt @yt Nt Xt @Vt
1 1 1 1 1 @Vt
= 1 Yt = 1 1 @Yt
= (A.32)
Nt Xt @N X
t t
Nt Xt Nt Xt @Vt
Xt @Yt
@Vt
Taking the result above together with equation (A.31), we obtain that:
1 @Vt
py =
t (A.33)
Xt @Yt
which, together with the deâ€¦nitions of vt and yt implies that:
47
Vt
vt 1
Nt Xt Vt @Yt
y = 1 @Vt Yt
=
t pt y t Yt @Vt
Xt @Yt Nt Xt
v
In other words, py y equals the reciprocal of the elasticity of utility with respect to income evaluated
along the balanced growth path. Notice that we can re-express this concept in per capita terms
Vt Vt Y
Vt @Yt Nt @Yt Nt
@ Nt v
since: Yt @Vt = Yt @Vt = Yt V
t
: Therefore, py y can be also interpreted as the percentage increase
Nt Nt
@ Nt
t
in per-capita income necessary to generate a one percentage point increase in lifetime per-capita
utility.
Alternatively, welfare can expressed in terms of "equivalent consumption" per capita, denoted
by Ct . Ct is deâ€¦ned as the level of consumption per capita at time t that, if it grows at the steady
state rate g from t onward, with leisure set at its steady state level, delivers the same intertemporal
utility per capita as the actual stream of consumption and leisure, More precisely, Ct satisâ€¦es:
1
X (1 + n)s
Vt
= (Ct (1 + g)s )1 (L L)) (A.34)
Nt (1 + )s
s=0
1
= Ct 1 (L L)
(1 ) (1 )
Per-capita utility on the steady state growth path can be written as:
1
X (1 + n)s
Vt
= ((Ct )SS (1 + g)s )1 (L L)) (A.35)
Nt SS (1 + )s
s=0
1 1
= (Ct )SS (L L)
(1 ) (1 )
The subscript SS denotes the steady state values of time varying variables. By taking the log-
diÂ¤erence between equation (A.34) and (A.35) (for < 1), we get:
Vt Vt
(log log ) = (1 ) (log Ct log(Ct )SS ) (A.36)
Nt Nt SS
v cpc
Using the deâ€¦nition of v and the F.O.C. for consumption, it follows that = (1 )(1 ). This,
together with equation (A.29) implies that:
1
X
(1 ) s
log Ct = log (Ct )SS + Et [log P Rt+s (log P Rt+s )SS ]
sC
s=0
(1 )1 k Kt 1 Kt 1
+ Yy
log log (A.37)
sC p Nt 1 Nt 1 SS
This holds for smaller or greater than one.
48
A.6 Connecting the aggregate Solow residual with the change in welfare
Take the diÂ¤erence between the expected level of intertemporal utility vt deâ€¦ned in (A.21) and
vt 1:
1
X
s pK k
vt = Et pC c log ct+s + i log it+s pL L log Lt+s log kt+s 1
(1 + g) (1 + n)
s=0
1
X
s pK k
Et 1 pC c log ct+s 1 + i log it+s 1 pL L log Lt+s 1 log kt+s 2
(1 + g) (1 + n)
s=0
1
+ k log kt 1 (A.38)
The right hand side, after adding and subtracting, for each variable xt+s , Et xt+s , can be written
as:
1
X
s pK k
vt = Et [pC c log ct+s + i log it pL L log Lt+s log kt+s 1]
(1 + g) (1 + n)
s=0
1
X
s
+ [pC c (Et log ct+s Et 1 log ct+s ) + i (Et log it+s Et 1 log it+s )
s=0
pK k
pL LEt (log Lt+s Et 1 log Lt+s ) (Et log kt+s 1 Et 1 log kt+s 1 )]
(1 + g) (1 + n)
1
+ k log kt 1 (A.39)
Deâ€¦ne value added growth (at constant shares) as:31
pC c i
log yt = log ct+s + log it (A.40)
pY y pY y
Using the fact that nominal value added Pt Yt = PtC Ct Nt + PtI It , it is also true that:
P C CN P II
log Yt = log(Ct Nt ) + log It (A.41)
PY Y PY Y
Now, insert this into equation (A.39) and factor out pY y to obtain:
31
Here we are departing slightly from convention, as value added is usually calculated with time varying shares.
49
1
X
vt = pY yEt s
[ log yt sL log Lt+s sK log kt+s 1] (A.42)
s=0
1
X
s
+ [(Et log yt+s Et 1 log yt+s )
s=0
sL Et (log Lt+s Et 1 log Lt+s ) sK (Et log kt+s 1 Et 1 log kt+s 1 )]
1
+ k log kt 1
Using the fact that:
Yt
log yt = log g
Nt
Kt
log kt = log g
Nt
and dividing both terms by pY y we can rewrite equation (A.42) as:
1
X
vt s
Yy
= Et log P Rt+s
p
s=0
1
X
s
+ [Et log P Rt+s Et 1 log P Rt+s ]
s=0
1 k Kt 1
+ log f1 (A.43)
pY y Nt 1
where Et log P Rt+s Et 1 log P Rt+s represents the revision in expectations of the level of the
productivity residual (normalized by population and Xt ) based on the new information received
between t-1 and t and:
1 1 k
f1 = g(1 sK ) + g (A.44)
(1 ) pY y
vt
Since v ' ln vt (for vt > 0 ( > 1)), equation (A.43) can be re-written, to a â€¦rst order
approximation, as:
50
v v Vt
log vt = log (1 )g
pY y pY y Nt
X1
s
= Et log P Rt+s
s=0
1
X
s
+ [Et log P Rt+s Et 1 log P Rt+s ]
s=0
1 k Kt 1
+ log f1 (A.45)
pY y Nt 1
v cpc
Using the fact that = (1 )(1 ), this result can be also expressed in terms of log-change of per
capita intertemporal utility as following:
1
X
v Vt s
log = Et log P Rt+s
pY y Nt
s=0
1
X
s
+ [Et log P Rt+s Et 1 log P Rt+s ]
s=0
1k Kt 1
+ log (A.46)
pY y Nt 1
sC
+ (1 )g f1
1
v Vt
For > 1 so that vt < 0 the R.H.S. of (A.46) should equal, instead, pY y
ln Nt : Most
importantly, using equations (A.11) and (A.19) evaluated in steady state, one can easily show that
( (1sC ) (1
1
) (1 sK ) 1 k
pY y
)g = 0, so that the last line in the equation above equals zero. This
yields equation (17) in the main text.
Alternatively, we can measure changes in welfare in terms of changes in equivalent per-capita
consumption. Taking the log-change over time of equation (A.34), we have:
1 v Vt
log Ct = ln (A.47)
sC pY y Nt
v cpc
where, again, we used the fact that: = (1 )(1 ). Using (A.46), we obtain:
1
X
1 s
log Ct = Et log P Rt+s
sC
s=0
1
X
1 s
+ [Et log P Rt+s Et 1 log P Rt+s ]
sC
s=0
1 1 k Kt 1
+ log (A.48)
sC pY y Nt 1
51
This is equation (20) in the main text.
A.7 Cross-country comparison
Start from equation (22) in the text:
1
X
v us i
us Y;us us (ln vt ln v us ) = Et s
[sus (ln ci
c t+s ln cus ) + sus (ln ii
i t+s ln ius ) (A.49)
p y
s=0
sus (ln Li
L t+s
i
ln Lus ) sus (ln kt+s
K 1 ln k us )]
1 k us i
+ Y;us us (ln kt 1 ln k us )
p y
This can be re-written as:
1
X i
v us Vi Vtus It+s I us
us Y;us us ln ti ln = Et s
[sus (ln Ct+s
i
c
us
ln Ct+s ) + sus (ln
i ln )
p y Nt Ntus SS
SS i
Nt+s us
Nt+s SS
s=0
!
Ki us
Kt+s
sus
L (ln Li
t+s ln L )us
sus
K log t+s
i
1
log us
1
]
Nt+s 1 Nt+s 1 SS
!
k us
1 Ki us
Kt 1
+ Y;us us log ti 1
log
p y Nt 1 Ntus1 SS
sus
c 1 sus
K 1 i us
+ + + ln Xt+s ln Xt+s (A.50)
(1 ) (1 )
v cpc
where we have use the fact that = (1 )(1 ). Using equations (A.11) and (A.19) evaluated
sus 1 1 k
in US steady state: c
(1 ) (1 ) (1 sK ) pY y
= 0. This implies that the last line in the
i
equation above equals zero. Deâ€¦ne the productivity term obtained using US shares as P Rt+s =
Ii i
Kt+s 1
sus ln Ct+s + sus ln Nt+s
c
i
i t+s
sus ln Li
L t+s sus ln
K Nt+s=1 . Using this modiâ€¦ed measure of productivity,
where input shares are common across countries, the equation above can be re-written as:
v us Vti Vtus
1
X h i
s i us
us Y;us us ln ln = Et log P Rt+s log(P Rt+s )SS (A.51)
p y Nti Ntus SS s=0
!
k us
1 Ki us
Kt 1
+ Y;us us log ti 1
log
p y Nt 1 Ntus1 SS
This is equation (23) in the main text.
52
A.8 Distortionary taxes
We now allow for distortionary taxes on capital, labor and â€¦nancial assets, and for indirect taxes
on consumption and investment (at rates K , K , R, C , I respectively). The numeraire is now
t t t t t
1+ I PtI .
t
The household budget constraint is now:
(1 ) + pK 1
t
K
t 1 + rt 1 R
t
kt + bt = kt 1+ bt 1 + pL 1
t
L
t Lt + t pC 1 +
t
C
t ct
(1 + g) (1 + n) (1 + g) (1 + n)
(A.52)
Log-linearizing it one obtains:
(1 ) + pK 1 K pK 1 K k
pK K k
0 = k bt
k kbt
k 1 pK +
bt bK
(1 + g) (1 + n) (1 + g) (1 + n) (1 + g) (1 + n) t
pL 1 L b
LLt pL 1 L
LbL + pL
pt L
LbL
t
+pC 1 + C
cbt + pC 1 +
c C
cbC + pC
pt C
cbC
t bt
R R
1+r 1 bb r 1 b r Rb
+bbt
b bt 1 b
rt bR (A.53)
(1 + g) (1 + n) (1 + g) (1 + n) (1 + g) (1 + n) t
Log-linearizing the maximization problem as before, we get:
1
" #
X pK 1 K k pK
bt+s r 1 R bbrt+s
s
vt = v + Et pL 1 L
LbL +
pt+s + + bt+s pC 1 + C
p
cbt+s
(1 + g) (1 + n) (1 + g) (1 + n)
s=0
1
X pK K k r Rb
Et s
pL L
LbL +
t+s bK +
t+s bR pC C
cbC
(1 + g) (1 + n) (1 + g) (1 + n) t+s t+s
s=0
(1 ) + pK 1 K 1+r 1 R
+ kbt
k 1 + bbt
b 1 (A.54)
(1 + g) (1 + n) (1 + g) (1 + n)
Using the log-linearized budget constraint in the equation above, we obtain:
1
X (1 ) + pK 1 K
vt v = Et s
[ 1+ C
pC cbt+s + k bt+s
c k kbt+s
k 1 1 L b
pL LLt+s
(1 + g) (1 + n)
s=0
(1 + r 1 R )
+bbt+s
b bbt+s 1 ] +
b
(1 + g) (1 + n)
(1 ) + pK 1 K 1+r 1 R
+ kbt 1 +
k bbt
b 1 (A.55)
(1 + g) (1 + n) (1 + g) (1 + n)
Rearranging the terms, we get:
53
1
" #
X ) + pK 1 (1 K
vt = v + Et 1+ s
p cbt+s + k bt+s
c k C C
kbt+s
k 1 1 L bL
p LLt+s
(1 + g) (1 + n)
s=0
1
" #
1 b X (1 + r 1 R )
+ k kt 1 + s
bbt+s
b bbt+s
b (A.56)
(1 + g) (1 + n)
s=0
Using the FOC for bonds evaluated in steady state, the last term equals zero. Finally, using
the fact that:
(1 )
kbt+s
k kbt+s
k 1 = ibt+s
i (A.57)
(1 + g) (1 + n)
we obtain:
1
" #
X pK 1 K
vt = v + Et s
p C
1+ C
cbt+s + ibt+s
c i p L
1 L b
LLt+s kbt+s
k 1
(1 + g) (1 + n)
s=0
1 b
+ k kt 1 (A.58)
which is equation (35) in the paper.
54
Table 1: Log Productivity
Dependent variable: logP Rt
CASE 1: Wasteful Government. Lump-Sum Taxes
Canada France Italy Japan Spain UK USA
log PR(t-1) 0.459 0.967 0.835 1.424 1.174 0.962 0.798
(0.195) (0.210) (0.153) (0.186) (0.199) (0.200) (0.135)
log PR(t-2) -0.287 -0.582 -0.393 -0.449
(0.204) (0.187) (0.195) (0.179)
LM1(Prob>chi2) 0.191 0.130 0.444 0.960 0.338 0.450 0.124
CASE 2: Optimal Government. Distortionary Taxes
Canada France Italy Japan Spain UK USA
log PR(t-1) 0.689 1.107 0.874 1.468 1.388 1.104 0.801
(0.167) (0.202) (0.119) (0.184) (0.172) (0.196) (0.135)
log PR(t-2) -0.393 -0.591 -0.623 -0.491
(0.197) (0.186) (0.172) (0.182)
LM1(Prob>chi2) 0.157 0.274 0.166 0.717 0.309 0.820 0.084
Notes: Time period: 1985-2005.
55
Table 2: Annual Average Log Change in Per-Capita Equivalent Consumption
Wasteful Spending Optimal Spending Wasteful Spending Optimal Spending
Lump-Sum Taxes Lump-Sum Taxes Distortionary Taxes Distortionary Taxes
Canada 0.013 0.014 0.021 0.023
France 0.026 0.031 0.026 0.031
Italy 0.018 0.020 0.021 0.023
Japan 0.018 0.025 0.023 0.030
Spain 0.021 0.030 0.030 0.040
UK 0.032 0.036 0.036 0.039
USA 0.025 0.026 0.029 0.030
Notes: Time period: 1985-2005.
Table 3: Components of the Annual Log-Change in Per-Capita Equivalent Consumption
Wasteful Spending Wasteful Spending Optimal Spending
Lump-Sum Taxes Distortionary Taxes Distortionary Taxes
Fraction due to: Fraction due to: Fraction due to:
TFP Capital TFP Capital TFP Capital
Canada 0.445 0.555 0.658 0.342 0.690 0.310
France 0.830 0.170 0.827 0.173 0.857 0.143
Italy 0.659 0.341 0.707 0.293 0.724 0.276
Japan 0.429 0.571 0.559 0.441 0.661 0.339
Spain 0.512 0.488 0.663 0.337 0.747 0.253
UK 0.816 0.184 0.833 0.167 0.848 0.152
USA 0.830 0.170 0.852 0.148 0.858 0.142
Notes: Time period: 1985-2005. TFP includes both expected present value and expectation revision.
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Table 4: Annual Average Log-Change in Per-capita Consumption, GDP and Equivalent Consump-
tion
Consumption GDP Equivalent Consumption
Opt Gov, Dist Tax
Canada 0.016 0.016 0.023
France 0.016 0.016 0.031
Italy 0.016 0.017 0.023
Japan 0.019 0.018 0.030
Spain 0.027 0.027 0.040
UK 0.024 0.030 0.039
USA 0.020 0.022 0.030
Notes: Time period: 1985-2005.
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Table 5: Welfare Gap Relative to USA: 1985, 2005 and Average
Wasteful Spending Wasteful Spending Optimal Spending
Lump-Sum Taxes Distortionary Taxes Distortionary Taxes
PANEL A: 1985
Canada -0.256 -0.294 -0.295
France -0.069 -0.176 -0.165
Italy -0.368 -0.420 -0.437
Japan -0.511 -0.488 -0.471
Spain -0.327 -0.396 -0.414
UK -0.096 -0.182 -0.189
USA 0.000 0.000 0.000
PANEL B: 2005
Canada -0.407 -0.455 -0.451
France -0.078 -0.240 -0.213
Italy -0.569 -0.641 -0.664
Japan -0.540 -0.582 -0.526
Spain -0.396 -0.405 -0.362
UK 0.034 -0.068 -0.059
USA 0.000 0.000 0.000
PANEL C: average 1985-2005
Canada -0.328 -0.372 -0.370
France -0.065 -0.201 -0.181
Italy -0.445 -0.507 -0.525
Japan -0.498 -0.505 -0.468
Spain -0.348 -0.389 -0.376
UK -0.026 -0.120 -0.120
USA 0.000 0.000 0.000
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Table 6: Components of Welfare Gap Relative to USA: 1985, 2005 and Average
Wasteful Spending Wasteful Spending Optimal Spending
Lump-Sum Taxes Distortionary Taxes Distortionary Taxes
Fraction due to: Fraction due to: Fraction due to:
TFP Capital TFP Capital TFP Capital
PANEL A: 1985
Canada 0.936 0.064 0.945 0.055 0.945 0.055
France 0.968 0.032 0.988 0.012 0.987 0.013
Italy 1.007 -0.007 1.006 -0.006 1.006 -0.006
Japan 1.091 -0.091 1.096 -0.096 1.099 -0.099
Spain 0.855 0.145 0.880 0.120 0.885 0.115
UK 0.379 0.621 0.672 0.328 0.683 0.317
PANEL B: 2005
Canada 0.900 0.100 0.911 0.089 0.910 0.090
France 0.421 0.579 0.811 0.189 0.788 0.212
Italy 0.972 0.028 0.975 0.025 0.976 0.024
Japan 1.065 -0.065 1.060 -0.060 1.067 -0.067
Spain 0.898 0.102 0.900 0.100 0.888 0.112
UK - - -0.079 1.079 -0.233 1.233
PANEL C: average 1985-2005
Canada 0.915 0.085 0.925 0.075 0.925 0.075
France 0.666 0.334 0.891 0.109 0.879 0.121
Italy 0.999 0.001 0.999 0.001 0.999 0.001
Japan 1.115 -0.115 1.113 -0.113 1.122 -0.122
Spain 0.887 0.113 0.899 0.101 0.895 0.105
UK -1.513 2.513 0.448 0.552 0.443 0.557
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Table 7: Per-Capita GDP, Consumption and Equivalent Consumption relative to USA: 1985, 2005
and Average
Consumption GDP Equivalent Consumption
Opt Gov, Dist Tax
PANEL A: 1985
Canada -0.179 -0.106 -0.295
France -0.285 -0.250 -0.165
Italy -0.378 -0.287 -0.437
Japan -0.434 -0.211 -0.471
Spain -0.624 -0.570 -0.414
UK -0.306 -0.304 -0.189
USA 0.000 0.000 0.000
PANEL B: 2005
Canada -0.324 -0.177 -0.451
France -0.401 -0.317 -0.213
Italy -0.501 -0.370 -0.664
Japan -0.456 -0.261 -0.526
Spain -0.527 -0.419 -0.362
UK -0.190 -0.219 -0.059
USA 0.000 0.000 0.000
PANEL C: average 1985-2005
Canada -0.248 -0.163 -0.373
France -0.338 -0.265 -0.181
Italy -0.396 -0.278 -0.529
Japan -0.391 -0.176 -0.468
Spain -0.537 -0.447 -0.375
UK -0.220 -0.251 -0.116
USA 0.000 0.000 0.000
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Figure 1: Within-country welfare comparisons: log equivalent permanent consumption
Figure 2: Within-country welfare comparisons: log equivalent permanent consumption
Figure 3: Within-country welfare comparisons: log equivalent permanent consumption (computed
using the EU-KLEMS labor service index)
61
Figure 4: Cross-country welfare comparisons: log equivalent permanent consumption gap (vis-a-vis
the U.S.)
Figure 5: Cross-country welfare comparisons: log equivalent permanent consumption gap (vis-a-vis
the U.S.)
62
Figure 6: Cross-country welfare comparisons: log equivalent permanent consumption gap (vis-a-vis
the U.S.). French preferences.
Figure 7: Cross-country welfare comparisons: log equivalent permanent consumption gap (vis-a-vis
the U.S.) computed using the EU-KLEMS labor service index.
63