Measuring Welfare by Matching Households
across TimeDavid R. Baqaee

Ariel Burstein

Yasutaka Koike-Mori

UCLA

UCLA

UCLA∗

August 2023

Abstract
The money metric utility function is an essential tool for calculating welfare-relevant
growth and inflation. We show how to recover it from repeated cross-sectional data
without making parametric assumptions about preferences. We do this by solving
the following recursive problem. Given compensated demand, we construct money
metric utility by integration. Given money metric utility, we construct compensateddemand by matching households over time whose money metric utility value is thesame. We illustrate our method using household consumption survey data from theUnited Kingdom from 1974 to 2017 and find that real consumption calculated using
official aggregate inflation statistics overstates money metric utility in 1974 pounds
for the poorest households by around half a percent per year and understates it byaround a third of a percentage point per year for the richest households. We extend
our method to allow for missing or mismeasured prices, assuming preferences are
separable between goods with well-measured prices and the rest. We discuss how ourresults change if the prices of some service sectors are mismeasured. JEL Codes: D11,
E31.

∗

Previously circulated as “A New Method for Measuring Welfare with Income Effects using Cross-Sectional
Data” and “A Fixed Point Approach to Measuring Welfare”. We thank Andy Atkeson, John Asker, NatalieBau, Hector Chade, Dave Donaldson, Pablo Fajgelbaum, Andy Neumeyer, Jon Vogel, and Pierre-Olivier
Weill. We are also grateful to the editor and six anonymous referees for their valuable suggestions. We
acknowledge financial support from NSF grant No. 1947611 and the Alfred P. Sloan Foundation.

1

1

Introduction

Money metric utility functions are a backbone of welfare economics. They convert income
under different prices into equivalent income under base prices. For example, the money
metric for 1974 base prices is the hypothetical 1974 income a consumer needs to make her
indifferent between her budget set in 1974 and another year. Money metrics are standard
tools for quantifying economic growth and inflation because they cardinalize utility withinterpretable units. They are also essential to answer a broad range of questions like how
policies affect welfare or how social programs should be indexed.
One can calculate a money metric by deflating nominal income using a weighted average of changes in prices, where the weights are compensated (or Hicksian) budget shares(see e.g. Hausman, 1981). Since compensated budget shares are not directly observable, standard price deflators use uncompensated (or Marshallian) budget shares instead.
This shortcut leads to the correct answer if preferences are homothetic, but fails whenpreferences are nonhomothetic. This is because when preferences are nonhomothetic,
compensated and uncompensated budget shares are different, and using one in place ofthe other produces incorrect results.
In this paper, we show how to recover compensated budget shares and money metricutility without imposing homotheticity or parametric assumptions about preferences,and without estimating a demand system. To do this, consider repeated cross sectionsof households with identical preferences facing common prices. To construct the money
metric utility function, in t0 dollars, for a household with income I at time t, we must
know the compensated demand of this household for every s ∈ [t0 , t]. This is revealed ateach point in time s by the budget shares of another household with a different income
level I0 who is on the same indifference curve as the household with income I at t.1
If we can find such households, then we can calculate the money metric utility function
by integration. That is, if we know how to match households over time, we can recovermoney metric utility. Conversely, if we know money metric utility, then we can matchhouseholds through time, since households are on the same indifference curves if, andonly if, their money metric utility values coincide. The insight is that this is a fixed point
problem in terms of observables that can be solved.
Our methodology endogenously identifies the set of households for which a money
1

Even though we assume that households have common preferences that are unchanged over time (werelax this assumption in Section 3.4), our matching approach is based on revealed-preference theory andis not based on interpersonal comparisons of “well-being”. That is, we match a household with income I
under t prices with a household with income I0 under s prices if the household at t is indifferent between
these two budget sets. We do not need to postulate that two households are “equally well-off” if theirutilities are the same.

2

metric value can be calculated reliably, without out-of-sample extrapolation. That is, ourapproach does not necessarily recover the money metric for all households in the sample
because suitable matches may not exist. For example, if there is positive growth overtime, then the richest household at any point in time is on an indifference curve that noother household was on in the previous periods. This means that for such a household,we cannot calculate compensated demand in the past and hence the money metric, unless
we are prepared to extrapolate Engel curves out-of-sample.
Our method generalizes the standard practice of statistical agencies who weightchanges in prices over time using aggregate budget shares. Conventional price deflators
like the CPI or the PCE recover money metric utility under the assumptions of homothetic
and stable preferences. However, when preferences are nonhomothetic, we show that one
must use the budget shares of a unique corresponding income level in the past for each
income today instead of aggregate budget shares.2
Our paper also provides a contrast to the popular but ad hoc approach of constructing
price indices by household-income group using the budget shares of some fixed percentileof the income distribution in each period. This method lacks a theoretical foundation if
percentiles of the income distribution do not remain on the same indifference curve overtime, and the shape of the indifference curve varies as a function of income.3
Our approach differs from alternatives that calculate compensated demand basedon estimated elasticities of substitution, as it does not require the estimation of nonparametric elasticities of substitution.4 Intuitively, our method only recovers compensateddemand evaluated at observed prices, whereas the elasticities of substitution determine
how compensated demand will react to any price change, even those that have not been
observed. As a result, our procedure can measure changes in welfare for observed changes
2

Chain-weighted indices measured by statistical agencies are generally uninterpretable when preferences are nonhomothetic. However, under additional assumptions, chained indices do have meaningfulinterpretations. For example, Feenstra and Reinsdorf (2000) show that when the path of prices is linear intime, chained indices measure the cost-of-living price index for some intermediate utility level under AIDS
preferences. Caves et al. (1982) establish a similar result for Törnqvist price indices, up to a second-orderapproximation. But these are not money metrics. In this paper, we focus on the money metric utility
function.
3
National statistical agencies sometimes produce inflation statistics like this.
For example, the UK’s Office of National Statistics produces inflation indices by household
expenditure groups (see https://www.ons.gov.uk/economy/inflationandpriceindices/articles/
inflationandthecostoflivingforhouseholdgroups/october2022).
4
In this respect, our approach resembles Oulton (2012), who demonstrates how to back out compensatedbudget shares by adjusting uncompensated budget shares using a Taylor series in income. He applies thismethodology, using the Quadratic Almost Ideal Demand System of Banks et al. (1997), to estimate the costof-living index without needing to estimate price elasticities. Instead of relying on a Taylor series under aparametric functional form for demand, our approach purges income effects from substitution effects bymatching households over time who are on the same indifference curve but face different prices.

3

in prices and income but is not suited for addressing counterfactual welfare questions,such as those explored by Baqaee and Burstein (2023).
The paper is organized as follows. In Section 2, we define money metric utility andits dual, the cost-of-living index, and explain their relationship to compensated demand.
In Section 3, we demonstrate how to recover the cost-of-living index and money metricutility given cross-sectional data when all prices are fully observed over time. We present
two solution strategies, both of which exactly recover the money metric as long as the data
are continuous in both the time series and the cross section. Using an artificial example,
we show that both methods quickly converge to the truth as the number of households
and the temporal frequency of observations increase.
We also discuss how our results change when there is preference heterogeneity ineither the cross section or time series. To account for heterogeneity in preferences that
depend on observable characteristics across households, we split the sample by observed
characteristic and apply our method to each subsample separately. With unobservabletaste shocks, there are certain cases in which our methodology produces reliable results.
For example, our approach approximately recovers the true money metric as long as taste
shocks are small and uncorrelated with price changes.
In Section 4, we illustrate our method by applying it to household expenditure survey
data from the United Kingdom spanning from 1974 to 2017. We find that real consumption
calculated by deflating income with aggregate chain-weighted inflation (as measured by
official statistical agencies) overstates the money metric utility for all households below
the 60th percentile of the spending distribution in 2017 in our sample. In other words,
for expenditures below the 60th percentile, the 1974 equivalent income is less than realconsumption. The size of this gap is greatest for the poorest households, roughly 20
percentage points (0.5 percentage points per year on average), and gradually diminishes
until it reaches zero for households close to the 60th percentile.
Conversely, real consumption calculated using aggregate inflation statistics understates the money metric utility for households above the 60th percentile. For households
in the 98th percentile of our sample, who spend around £94, 000 per year, the size of thisgap is 16 percentage points over the whole sample (0.36 percentage points per year on
average).5 We are unable to compute the money metric for the richest households in 2017(98th percentile and above). The reason is that for these households, there did not exist
consumers in the past whose money metric utilities are high enough and whose observed
demand can be used in place of the compensated budget shares.
5

These results are consistent with Blundell et al. (2003), which report a relatively greater rise in the costof living for poorer households between 1975 and 1984 in the UK.

4

Whereas real consumption calculated using the aggregate inflation rate has large errors
relative to our true estimated money metric, a decile-specific chained deflator produces
smaller errors in our UK dataset. Of course, one needs to compute the true money
metric first, before knowing whether or not the ad hoc approach is a good approximation.
Furthermore, computing quantile-specific chained deflators requires more information
than our method.
In Section 5, we extend our methodology to allow for missing prices. To do this,we require the restriction that the expenditure function be separable between observed
and unobserved prices. Under this additional assumption, we show that money metricutility can be recovered if we know the compensated elasticity of substitution between
observed and unobserved goods. This generalizes the influential Feenstra (1994) approachto imputing missing prices beyond the homothetic CES case.
Specifically, we show how to back out the change in the relative price of observed andunobserved goods using changes in the compensated budget share of the observed goods.
For example, if the compensated budget share on observed goods is rising, and observed
goods are net complements with unobserved goods, then this indicates that the relative
price of unobserved goods is falling. This can then be used to calculate money metricutility. We also show that the elasticity of substitution between observed and unobserved
goods, which is required to infer missing prices, can be identified without knowledge of
those missing prices.
We provide an empirical illustration of this extension in Section 6. We assume thatsome service prices are mismeasured, estimate elasticities of substitution between these
services and other goods, and apply our methodology. We find that the price of thecompensated bundle of services has been rising faster than official data for rich but not forpoor households. This implies that the money metric is overstated for rich but not poor
households.
We conclude in Section 7. Proofs and supplementary materials are in an Online
Appendix.
Related Literature. Our paper is closely related to Blundell et al. (2003) and Jaraveland Lashkari (2024), both of which develop nonparametric approaches to measuring welfare for nonhomothetic preferences using cross-sectional household-level data. Althoughinspired by them, our approach is different and builds on Lemma 1 from Baqaee andBurstein (2023), which expresses the money metric as an integral of compensated demand
curves. We discuss the alternative approaches of Blundell et al. (2003) and Jaravel and
Lashkari (2024).
5

Blundell et al. (2003) bound the money metric by using revealed-choice arguments.
For each income level at time t, Blundell et al. (2003) construct a bundle that is strictly
better and a bundle that is strictly worse in time s , t. The price of these two bundles
then bound the true money metric value.6 Our approach has an advantage over Blundell
et al. (2003) in that it provides a point estimate, rather than only bounds, for the money
metric utility. On the other hand, in order for our methodology to recover point estimates
for the money metric utility without approximation errors, the data must be observed
continuously.7 We show in Online Appendix Section A.7 that our point estimates arealways within their bounds in real-world data.
Jaravel and Lashkari (2024) use a correction term to address nonhomotheticity inhousehold-level chain-weighted indices. Whereas our approach endogenously delineatesa set of households for whom money metric utility can be calculated, without relying on
out-of-sample extrapolation, the Jaravel and Lashkari (2024) method aims to uncover
the money metric for all households observed at any point in time. That is, unlike ourmethodology, their approach does not provide a boundary on the set of households whosemoney metric values can be reliably computed. In Online Appendix Section A.8, we apply
the Jaravel and Lashkari (2024) method to artificial examples. If the support of the crosssectional distribution of utilities changes over time, then their algorithm can diverge or
result in large errors (and these errors persist even as we increase the sample size andfrequency of observation).
In contrast to both Jaravel and Lashkari (2024) and Blundell et al. (2003), we also extend
our methodology to situations where some prices and expenditures are unobserved. Since
our method can be extended to allow for unmeasured prices, our paper is also related
to the literature that measures welfare allowing for incomplete information about prices.
Most papers with nonhomothetic preferences follow the approach of Costa (2001) and
Hamilton (2001). These papers take advantage of horizontal shifts in Engel curves to
identify money metric utility changes. The frontier in this literature is Atkin et al. (2024),
who show how to identify welfare changes assuming that preferences are quasi-separable
between the measured and unmeasured goods.
Our paper, instead, generalizes the Feenstra (1994) method beyond the homothetic CES
case. One advantage of our approach is that we do not need to make strong parametric6

We exposit and implement an amended version of their methodology in Online Appendix Section A.7,
fixing a typographical error in their algorithm for the lower bound.
7
We interpolate budget shares to turn discrete data continuous. If this interpolation is inaccurate, thenthis introduces approximation errors into our method. Such errors are not specific to our method andresult from the fact that sums do not perfectly measure integrals. For example, even when preferences are
homothetic, interpolation error affects the accuracy of standard chained-weighted price deflators.

6assumptions within the set of observed prices. This is in contrast to Atkin et al. (2024)
who need to fully model the demand system for the subset of goods with observed prices.
This advantage of our approach comes at the cost that we require a stronger form of
separability between the observed and unobserved prices than Atkin et al. (2024). We
discuss these issues in more detail in Section 5.
Our approach can also be contrasted with more parametric approaches where welfare measures are computed using a fully specified demand system (e.g. Deaton andMuellbauer 1980). Specific functional forms for nonhomothetic preferences are used tounderstand phenomena as diverse as structural transformation (e.g. Boppart 2014, Cominet al. 2021, and Fan et al. 2023), international trade patterns (e.g. Matsuyama 2000, andFajgelbaum et al. 2011), and savings behavior and inequality (e.g. Straub 2019). Our
approach provides a nonparametric way to compute welfare measures from the data
without relying on low-dimensional functional forms.

2

Money Metrics and the Cost of Living

We start by defining the objects of interest: money metric utility and the closely related
cost-of-living function. Consider a rational preference relation  defined over consumption bundles c in RN . Suppose that these preferences can be represented by a utility
function U(c) that maps consumption bundles to utility values. Given this utility function, we can define the indirect utility functionv(p, I) = max{U(c) : p · c ≤ I},cmapping a vector of prices p and expenditures I to utility values. We interchangeably refer
to I as income, but in the data, we measure I using expenditures. Define the expenditure
function to bee(p, U) = min{p · c : U(c) ≥ U}.
c

We assume that the expenditure function is continuously differentiable in all its arguments.
The expenditure and indirect utility functions are used to define money metrics andcost-of-living indices.
Definition 1 (Money Metric and Cost of Living). For a fixed reference vector of prices p̄,the money metric function maps budget sets defined by (p, I) toe(p̄, v(p, I)).

7

For a fixed reference budget set defined by (p̄, Ī), the cost-of-living index maps prices, p, toe(p, v(p̄, Ī)).
The money metric function, e(p̄, v(·)), converts the value of different budget sets (p, I)
into equivalent dollars under some baseline prices p̄. It is itself an indirect utility function because a budget set (p, I) is preferred to another budget set (p0 , I0 ) if, and only if,e(p̄, v(p, I)) > e(p̄, v(p0 , I0 )). Because the money metric ranks budget sets and assigns theman interpretable value, it is useful for measuring growth.8
The cost-of-living function, e(·, v(p̄, Ī)), converts the value of some baseline budget
constraint (p̄, Ī) into equivalent income under different sets of prices.9 Because the cost-ofliving index converts a common utility level, v(p, I), into equivalent income under different
price systems, it is useful for measuring the cost-of-living adjustment to maintain a fixed
standard of living.
In sum, the function e(p0 , v(p, I)), mapping (p0 , p, I) into a scalar, is an object of paramountinterest. The money metric is the cross section of this function that holds p0 constant andthe cost-of-living index is the cross section that holds (p, I) constant. Our aim is to recover
this object from the data.
Denote the compensated budget share for good i by bi (p, U) where p is a vector of
prices and U is a utility level. The following lemma, which is a corollary of Lemma 1
from Baqaee and Burstein (2023), and follows from Shephard’s lemma and the gradienttheorem, provides a characterization of both the cost-of-living index and the money metricusing compensated budget shares.
Lemma 1 (Money Metric and the Cost of Living). The money metric of a budget set (p, I) in
terms of p̄ prices can be expressed aslog e(p̄, v(p, I)) = log I −Z X

bi (ξ, v(p, I))d log ξi ,

(1)

C i∈N

where C is any absolutely continuous path connecting p̄ to p.10 The cost of living for a budget set
8

The equivalent and compensating variation are related to the money metric. Specifically, to measure
the change in welfare from some initial budget set (p, I) to some other budget set (p0 , I0 ), the equivalentvariation is e(p, v(p0 , I0 )) − I and the compensating variation is I0 − e(p0 , v(p, I)).
9
In index number theory, the cost-of-living index is also called the Konüs (1939) index.
R t1 Pd log ξit10
Formally, the path integral in (1) is defined by ti∈N bi (ξt , v(p, I)) dt dt where {ξt : t ∈ [t0 , t1 ]}0parameterizes the path C from p̄ and p as a function of a scalar t. The integrals in (1) and (2) are both path
independent and only depend on the end points.

8

(p̄, Ī) in terms of p prices can be expressed aslog e(p, v(p̄, Ī)) = log Ī +Z X

bi (ξ, v(p̄, Ī))d log ξi .

(2)

C i∈N

According to Lemma 1, both the money metric and the cost-of-living index can be
expressed as integrals of compensated budget shares with respect to changes in prices.
However, compensated demand curves are not directly observable, so operationalizing
this result requires having a way to identify compensated budget shares. This is what we
focus on in the next section.

3

Recovering the Money Metric by Matching Households

In this section, we discuss how Lemma 1 can be deployed to recover money metric utility
functions and cost-of-living indices if one has access to repeated cross-sectional data of
consumers with common and stable preferences who all face common prices at eachpoint in time but have different incomes. We start this section by introducing our main
theoretical result. We then provide two solution methods, and test them with artificial
discrete data to assess their accuracy. We end the section by discussing how our results
are affected by taste shocks and mismeasurement.

3.1

Theoretical Result

Consider an absolutely continuous path of prices pt ∈ RN as a function of time t ∈ [t0 , T].
Suppose we observe vectors of budget shares B(I, t) ∈ RN for consumers with preferences and income levels I ∈ [It , It ] for time t ∈ [t0 , T].11 Our aim is to recover the money metricutility function based on reference prices pt0 evaluated at budget set (pt , I) for t ∈ [t0 , T]
and I ∈ [It , It ]. We denote this function by u(I, t) ≡ e(pt0 , v(pt , I)).
The function u(I, t) converts the value of the budget constraint defined by prices pt andincome I into income under base prices pt0 . Once we are equipped with u(I, t), it is alsostraightforward to compute the money metric for other base prices.12 By varying baseprices, for fixed (pt , I), we can also recover the cost-of-living index.
11

We can always produce an absolutely continuous path of prices by linearly interpolating between
discrete-time observations. We can construct an associated budget share at each instant in time by linearlyinterpolating budget shares over time. See Footnote 15 for more details about interpolation.
12
Suppose we wish to obtain the money metric for some other base prices: ũ(I, t) = e(ps , v(pt , I)) for somes ∈ [t0 , T]. The solution is ũ(I, t) = I0 where I0 satisfies u(I0 , s) = u(I, t). By construction, v(pt , I) = v(ps , I0 ),hence ũ(I, t) = e(ps , v(pt , I)) = e(ps , v(ps , I0 )) = I0 .

9Denote the uncompensated budget share of good i by BM(the superscript M standsifor Marshallian). For every good i,BMi (pt , I) = Bi (I, t)
whenever t ∈ [t0 , T] and I ∈ [It , It ]. For any cardinalization of the indirect utility function
and its associated compensated demand curves, the following identity between compensated and uncompensated budget shares also holds:bi (pt , v(pt , I)) = BMi (pt , I).
Using the money metric cardinalization of indirect utility, and slightly abusing notation,
we can combine the previous two identities to obtain:13
bi (pt , u(I, t)) = Bi (I, t).
Using this identity, Lemma 1 can be rewritten as the following recursive integral equation.
Proposition 1 (Money metric as the Solution to the Integral Equation). For t ∈ [t0 , T], themoney metric u(I, t) ≡ e(pt0 , v(pt , I)) is a fixed point of the following integral equation
log u(I, t) = log I −

Z tX
t0

Bi (u−1 (u(I, t), s), s)id log pis
ds,
ds

(3)

with boundary condition u(I, t0 ) = I. Here, u−1 (·, s) is the inverse of u with respect to its firstargument (income) given its second argument (time) is equal to s. That is, u−1 (u(I, t), s) is a levelof nominal income I∗ in s such that u(I∗ , s) = u(I, t).
Since the money metric exists, the integral equation (3) necessarily has a solution.
Proposition A.1 in Online Appendix Section A.3 uses the contraction mapping theorem
to show that the solution to this integral equation is also unique.
Proposition 1 follows immediately from Lemma 1 once we recognize that in the integral
equation above, Bi (u−1 (·, s), s) : R+ → [0, 1] maps utility values to the budget share of goodi at time s. That is, it is the compensated budget share of i.
To better understand (3), observe the simplification that occurs when preferences are
homothetic. In this case, budget shares do not depend on income levels, only on time.
13

Our “abuse of notation” is that we do not index compensated budget shares by the utility cardinalization. This is to simplify notation, since we are interested in compensated budget shares only under themoney metric cardinalization.

10

Therefore, when preferences are homothetic, (3) simplifies tolog u(I, t) = log I −

Z tX
t0

i

Bi (s)

d log pis
ds,
ds

(4)

which eliminates the need to find a fixed point. This equation, called a Divisia (1926)
index, justifies the standard chain-weighting practices adopted in the national accounts
for calculating price and quantity indices.
If we can solve (3), then we can compute the compensated budget shares b(ps , ū) for autility level ū at time t under prices ps at time s by using the budget shares of a different
household on the same indifference curve at time s. That is, we “match” households withincome I∗ at time s to households with income I at time t if u(I∗ , s) = u(I, t). The budget
shares of this “matched” household, B(I∗ , s), are equal to the compensated budget sharesb(ps , ū).
Proposition 1 provides a way to recover the money metric and cost-of-living functions
without needing direct knowledge of the potentially very high-dimensional demand system BM(pt , I). Recall that the number of cross-price elasticities scales in the square ofithe number goods, and generically depends on both income and relative prices. Proposition 1 obviates the need to undertake this onerous estimation exercise by using thedemand from other households and time periods in place of a counterfactual model ofcompensated demand.
In the next section, we provide two solution methods for solving the integral equationin Proposition 1 with discrete data.

3.2

Two Solution Methods

The money metric is a fixed point of equation (3), which is a system of nonlinear equations,albeit an infinite-dimensional one. We provide two solution methods. The first is a simple
iterative procedure that converges to the desired solution as we approach the continuoustime limit. The second is a recursive solution that is equivalent to the iterative one in thecontinuous-time limit but has better properties when the data are discrete.
For both methods, suppose that we have data on a grid of points {t0 , . . . , tM } wheretn < tn+1 , with tM = T. For each t, we observe budget shares B(I, t) for any income levelI ∈ [It , It ].14
14

In our empirical application, in Section 3.4, we fit a smooth curve through micro data to obtain B(I, t)
for I ∈ [It , It ] since cross-sectional household-level data on expenditures is noisy.

11

Iterative Solution.
with n = 1:

Use the following iterative procedure for each n ∈ {1, . . . , M} starting

log u(I, tn ) ≈ log I −n−1
X

B(Im∗ , tm ) · ∆ log ptm ,

(5)

m=0

where Im∗ satisfiesu(Im∗ , tm ) = u(I, tn−1 ),(6)with the boundary condition u(I, t0 ) = I. For any m ≤ n − 1 for which we cannot find Im∗
satisfying (6), obtain a candidate Im∗ using a loglinear approximation in I. If the candidate
Im∗ is not in the observed income distribution [Itm , Itm ], then we cannot evaluate B(Im∗ , tm )without out-of-sample extrapolation, so we do not calculate u(I, tn ). As a final check,we exclude u(I, tn ) if there does not exist Im∗ ∈ [Itm , Itm ] such that u(Im∗ , tm ) = u(I, tn ) for allm ≤ n − 1.
There are two approximation errors in the iterative solution. The first is that, in(5), we are approximating an integral using a discrete Riemann sum. The second is that,in (6), we are using u(I, tn−1 ) rather than u(I, tn ) on the right-hand side (since we do notknow u(I, tn ) in step n). However, as we approach the continuous-time limit, the estimates
produced by (5) converges to the exact solution in (3). This is because the summation in(5) converges to an integral and u(I, tn−1 ) in (6) converges to u(I, tn ). Since (3) has a uniquesolution, the continuous-time limit of (5) converges to the money metric. To summarize, if
data are continuous, then the result is an exact solution to the money metric that requires
no estimation or interpolation.15
The iterative procedure that we describe is useful for building intuition. However,
one can also find a fixed point by solving the system of equations directly. This gives arecursive variation of the iterative procedure described above. The two approaches areequivalent in the continuous-time limit.
15

In practice, we use the trapezoid rule rather than the left-Riemann sum to approximate integrals. That
B(I∗ ,tm )+B(I∗,tm+1 )∗
m+1
is, we use min place of B(Im, tm ) in (5). This numerical refinement is equivalent to linearly2interpolating prices (in logs) and budget shares over time between discrete-time observations. If the truebudget shares corresponding to the linearly interpolated path of prices are not themselves linear, then thiswill introduce an interpolation error into our results. This error disappears as the price shocks between any
two consecutive periods become small.

12

Recursive Solution. Apply the iterative solution in (5) and (6) and call the resulting
money metric u0 (I, t). For each i ≥ 1, and each n ∈ {1, . . . , M}, starting with n = 1, define
log ui+1 (I, tn ) ≈ log I −n−1
X

B(Im∗ , tm ) · ∆ log ptm ,

(7)

m=0

where Im∗ satisfies
ui+1 (Im∗ , tm ) = ui (I, tn ).

(8)

For any m ≤ n − 1 for which we cannot find Im∗ satisfying (8), obtain a candidate Im∗ using
loglinear approximations in I. If the candidate Im∗ is not in the observed income distribution[Itm , Itm ], then we cannot evaluate B(Im∗ , tm ), so we do not calculate u(I, tn ). Continue until
ui+1 (I, t) = ui (I, t) for all feasible values of I and t. Then set u(I, t) = ui (I, t).
Once the recursive solution converges, it solves the fixed-point problem without any
out-of-sample extrapolation of budget shares. The difference between the iterative andrecursive solution is that we replace u(I, tn−1 ) on the right hand side of equation (6) withu(I, tn ) in (8). Proposition A.1 in Online Appendix Section A.3 shows that the continuoustime version of this recursive procedure is a contraction mapping and must necessarily
converge to the unique solution (which is the money metric).
Both procedures endogenously delineate values of (I, tn ) for which u(I, tn ) can be computed without extrapolating budget shares. We only compute money metric utility u(I, tn )if u(I, tn ) is between the upper- and lower-bound of u(·, tm ) for every m < n. This ensuresthat the income level Im∗ that solves (6) or (8) is in the support of the income distribution
where budget shares are observed.
In artificial examples with discrete data, the recursive solution has smaller errors than
the iterative solution, although, both methods work well. When we use real data fromthe UK, in Section 4, the results are almost unchanged between the iterative and recursive
methods. Since the iterative procedure is simpler and faster to compute, we only show
results for the iterative method for our empirical results.
Figure 1 illustrates the outcome of our procedure. The left panel of Figure 1a shows
the budget share on some good against nominal income for three different points in time.
The fact that the lines are downward sloping means that this good is a necessity. In thisexample, incomes grow over time, so the range of nominal income levels shifts up overtime.
In the data we observe budget shares as a function of income over time (uncompensated13Figure 1: Budget share for some good against nominal income and money metric utility
in different periods.
(a) Nonhomothetic
CompensatedUncompensated
6

8
t0
t1
t2

5

log I

log u

6
4

4
3

2
0.1

0.2

0.3

0.5
0.6
0.4
Budget Share

0.7

2
0.1

0.8

0.2

0.3

0.5
0.6
0.4
Budget share

0.7

0.8

(b) Homothetic
CompensatedUncompensated
6

8
t0
t1
t2

5

log I

log u

6
4

4
3

2
0.32

0.34

0.36
0.38
Budget share

0.4

2
0.32

0.42

0.34

0.36
0.38
Budget share

0.4

0.42

budget shares), but to construct the money metric we require budget shares as a function ofutility (compensated budget shares). The right panel of Figure 1a (color version available
online) displays the compensated budget shares for the same good. The horizontal solid
purple line in the right panel of Figure 1a shows for each period the compensated budget
share for the good evaluated at some fixed utility level ū. The change in budget shares,
holding utility constant, are pure substitution effects over time due to changes in relative
prices. As implied by Lemma 1, multiplying the compensated budget shares by log price14

changes and summing over time gives the money metric utility for the household withutility ū at time t2 .
But, we cannot directly observe the figure on the right. How do we infer compensatedbudget shares? The upward-sloping solid purple line in the left panel of Figure 1a plots,
for each period s, the income that gives the utility of ū, that is u−1 (ū, s), and the associatedbudget share for the good, Bi (u−1 (ū, s), s). In other words, we can infer compensatedbudget shares for ū by using the observed budget share along the purple line in the left
panel. Then we can construct the mapping between income and utility at each point (thepurple line) by iteratively applying the summation in (5).
To understand why Proposition 1 is unnecessary when preferences are homothetic,
Figure 1b plots the same information as Figure 1a but for homothetic preferences. Since
there are no income effects, budget shares at a point in time do not vary with householdincome or utility. That is, uncompensated and compensated budget shares coincide.
Therefore, we can construct the money metric using a price index based on uncompensatedbudget shares by good.

3.3

Example with Artificial Discrete Data

To illustrate how our method fares when faced with discrete data, rather than continuous data, we consider a simple artificial example. Suppose the expenditure function is
nonhomothetic CES
1

 1−γ

 X

1−γ 
 .
(9)
e p, U = ωi Uεi pi
i

The money metric function for t0 reference prices is
1

 1−γ
X


1−γ 
 ,
u(I, t) = ωi V εi pi,t0iwhere V is the indirect utility function and solves I = e pt , V .16 We evaluate the accuracyof our algorithm by comparing this exact expression for u(I, T) with the results of ournumerical procedure applied to artificial data generated using these preferences.
For illustration, we set γ = 0.25, ε1 = 0.2, ε2 = 1, ε3 = 1.65, which are values taken
16

See Hanoch (1975), Matsuyama (2019), and Comin et al. (2021) for more information on these
preferences. Baqaee and Burstein (2023) show that the money metric for a nonhomothetic CES has aclosed-form expression in terms of observable budget shares and the elasticity of substitution: u(I, t) =1
P
 pi,t 1−γ  1−γ
I × i Bi (I, t) pi,t0. Note that the budget shares depend on the difference in εi and not their overalllevel. This implies that εi are identified, and matter, up to an additive constant.

15

from Comin et al. (2021). We generate repeated cross-sectional data on income and budget
shares over three goods for a finite number of households facing a common price vector
over forty years. The distribution of income in the first period is lognormal (parameterized
to match the distribution of household expenditures in the 1974 UK household survey,
described in the next section). The share parameters are calibrated so that the budget
share of each good for the median household in the first period are uniform. All incomesand prices grow exponentially, at different rates, over the sample period.17
Figure 2: Maximum error as a function of the frequency of observation and sample size(b) recursive procedure

Error Size

Error Size(a) iterative procedure

0.05

0.00
10-1

50

0.00

50
10-1

100
100
Frequency

0.05

100
100

1

10 200Sample Size

Frequency

101 200

Sample Size

Notes: Throughout, we hold the path of prices and incomes constant. Our baseline calibration is annual
frequency corresponding to a value of 100 = 1 observation per year on the x-axis. If we observe the data
once every decade, then the frequency is 1/10, and if we observe the data every month, then the frequency
is 12. The left panel uses the iterative and the right panel the recursive solution method in Section 3.2.

We apply both the iterative and recursive solution methods. We use linear interpolation
to evaluate budget shares for I between two observed income levels. To assess the accuracyof our procedure, we use the infinity norm—the maximum absolute value of the log
difference between the true money metric function and our estimate in the final period.
The error is very small. For example, with 100 households and annual data, the maximum
error in the final period is 0.0077 for the iterative procedure and 4 × 10−5 for the recursive
procedure. So, the error is less than 1% of income for the iterative procedure and around
1/250th of 1% for the recursive procedure. Figure 2 shows how this error varies as wevary the number of households and the frequency of observations. As expected, the error
17

The ratio of top to bottom nominal expenditures every period is around 18. The annual (log) growthrate of nominal expenditures is 5.7%, and the annual growth rates of prices of each good are 5%, 4.25%, and3.5%, respectively.

16

converges to zero as we approach the continuous-time limit. The error also falls as thenumber of households in the sample increases.18

3.4

Tastes Shocks and Mismeasured Expenditures

In practice, data are imperfect and noisy. There are two potential sources of error in thedata: (1) expenditures by good may be mismeasured or affected by taste shocks; (2) prices
may be missing or mismeasured. In this section, we focus on mismeasured expendituresdue to measurement error and or taste shocks. We address missing or mismeasured prices
in Section 5, where we impose stronger assumptions on preferences.
If there are arbitrary unobservable shocks to preferences or measurement error, thenour methodology cannot be used reliably. However, there are certain tractable cases withshocks where we can still apply our methodology. In this section, we discuss these cases.
We begin by considering the straightforward scenario where preferences vary as a functionof observable characteristics—for instance, households with children have distinct tastes
compared to those without.19
Proposition 2 (Tastes Vary by Observed Characteristics). If there are differences in preferencesthat are functions of observable characteristics, then split the sample by characteristic and apply
Proposition 1 to each subsample separately.20
Next, we consider the more difficult case where observed expenditures depend on
unobservable taste shocks or measurement error. Suppose that observed budget sharesareB̃(I, t|κ) = B(I, t) + κ(I, t),where B(I, t) are the true expenditure shares. That is, B(I, t) are expenditure shares generated by the preferences that we wish to construct the money metric for. However, wecannot observe B(I, t) because the data feature either taste shocks or mismeasurement. The
term κ(I, t) is defined to be the difference between observed budget shares and budget
shares generated by the preferences that we are interested in.21 The scalar κ ≥ 0 controls
the magnitude of these errors.
18
In Online Appendix Section A.8, we apply the Jaravel and Lashkari (2024) algorithm to this example
and find similarly small errors. However, in the same appendix, we provide other examples where the
Jaravel and Lashkari (2024) approach yields large errors or diverges.
19This assumption is similar to that considered in Section 2.3 of Jaravel and Lashkari (2024).
20
Similarly, if we observe two groups of households that face different prices at a point in time (e.g.
households living in different locations), then we can apply our method to each sample separately.
21
See Baqaee and Burstein (2023) for a detailed analysis of how welfare should be defined when preferences are subject to taste shocks. In general, in the presence of taste shocks, B(I, t) need not correspond to
the preferences of any individual in the cross section.

17

Define ũ(I, t|κ) to be the solution to the integral equation
log ũ(I, t|κ) = log I −

Z tX
t0

B̃i (ũ−1 (ũ(I, t|κ), s|κ), s|κ)id log pis
ds.
ds

(10)

Proposition 1 assumes that κ = 0. That is, ũ(I, t|0) = u(I, t).
When there is idiosyncratic (mean-zero) noise at the level of individual households,
averaging over households ensures that κ = 0 as long as the law of large numbers
holds. In such situations, we can apply Proposition 1 without concerns about taste shocks
and recover the money metric for preferences in the absence of the idiosyncratic noise.
However, if the errors do not average out, they could potentially impact the results. To
analyze the extent of this influence, we derive a first-order approximation of ũ(I, t|κ) withrespect to the error term κ. The general form of this first-order approximation can befound in Lemma A.1 in Online Appendix Section A.2. Within the main text, we highlight
two tractable and salient special cases.
Proposition 3 (Taste Shocks Uncorrelated with Price Shocks). Suppose that for all I and s ≤ t,d log pwe have Cov (I, s), ds = 0. Then, to a first-order approximation around κ ≈ 0,ũ(I, t|κ) ≈ u(I, t),where the remainder term is order κ2 .
In words, if the shocks are uncorrelated with price changes, then the money metricwe construct by solving the wrong integral equation is, to a first-order approximation,correct. This approximation assumes that κ is small, but does not require that t be close
to t0 .22
Next, we consider how taste shocks that are correlated with price changes affect our
results.
Proposition 4 (Engel Curve Slopes Uncorrelated with Price Shocks). Suppose that for all ∂B(I,s) d log p 
I and s ≤ t, the slope of Engel curves is uncorrelated with price changes Cov ∂I , ds = 0.
22This result bears a superficial resemblance to previous results, for example by Baqaee and Burstein(2023), that Divisia indices approximately measure welfare correctly when taste shocks are uncorrelated withprice changes. However, Proposition 3 is different since it characterizes the solution to an integral equationand not the Divisia index. The results about the Divisia index are based on a second-order approximationthat requires that t be close to the base year t0 . On the other hand, Proposition 3 is based on a first-orderapproximation in κ, and t can be far from t0 .

18

Then, to a first-order approximation around κ ≈ 0,
Z t!
d log pũ(I, t|κ) − u(I, t) ≈ −κ
Cov (u(I, t), s),
ds,
ds
t0
where the remainder term is order κ2 .
If the slope of Engel curves is uncorrelated with price shocks (necessarily the case whenpreferences are homothetic), then the money metric we construct is biased according to
how the taste/measurement shocks (I, t) covary with price shocks. That is, although
our methodology will have errors, the sign and magnitude of these errors can be linked
to the underlying shocks in a straightforward way. In particular, if the mismeasured
expenditures are biased upwards for goods whose relative price rose, then the constructedmoney metric will be biased downwards.
We need the requirement that the slope of Engel curves be uncorrelated with priceshocks because otherwise, as we solve the integral equation forward, errors in priorvalues of ũ(I, s|κ), for s ≤ t, contaminate the matching process in a systematic way andinduce additional biases in ũ(I, t|κ).

4

Empirical Illustration

In this section, we apply our algorithm to long-run cross-sectional household data. Our
goal is to compare welfare as measured by the money metric with real consumption. We
define real consumption consistently with how it is constructed by statistical agencies in thenational accounts: nominal expenditures deflated by a chain-weighted price index thatreflects observed (either aggregate or decile-specific) budget shares.23 When preferences
are homothetic, then real consumption for every household coincides with money metricutility.
We use the Family Expenditure Survey and Living Costs and Food Survey Derived Variables
for the UK (see Oldfield et al., 2020), which is a repeated cross section of UK household
expenditures over different subcategories of goods and services from 1974 to 2017.24 The
RtP
d log pis
The analog to real consumption in our theoretical model is log RC(I, t) = log I − t N
i=1 B̄i (t) ds ds,0where B̄i (t) is some average budget share of good i in period t. If we use the aggregate budget shares, thenthe price deflator is common for all households. Alternatively, we can group households by quantiles ofthe spending distribution and use average budget shares by quantile. We compare our results with bothaggregate and decile-specific price deflators.
24
Aggregate nominal consumption growth in our sample is lower than that in the UK national accounts.
According to the UK Office for National Statistics, this difference is due to differences in sample coverage.
While these sample coverage issues affect aggregate nominal growth rates, they do not affect our results,
23

19

UK Family Expenditure Survey was also used in Blundell et al. (2003) and Blundell et al.
(2008) to estimate Engel curves, test for deviations from revealed-preference theory, andcompute bounds for a true cost of living index.
Following the practice of the Office of National Statistics (ONS), we measure prices
using the retail price index (RPI) in the period 1974–1998 and the consumer price index(CPI) in the period 1998–2017. To concord the RPI, CPI, and household expenditure
data, we assemble 17 aggregate product categories that can be used consistently overthe entire period of analysis.25 Between 1974 and 2017 prices rose relatively less for
product categories that are disproportionately consumed by richer households, such as
leisure goods and services. Even though we consider product categories that are moreaggregated than the official data, our data tracks the official inflation figures from the ONS
fairly well.26
We pool all households in our sample and assume that they have the same stablepreference relation over the 17 categories of goods and services for which we have pricedata. To investigate the validity of this assumption, we can split the sample by observablecharacteristics (following Proposition 2). We provide examples using marital status andage in Online Appendix Section A.4. This added flexibility comes at the expense ofshrinking the boundaries over which the money metric can be computed, since householdswith different characteristics (e.g. married and unmarried households) cannot be matchedto one another through time. We do not find marked differences in the money metricfunction by age or marital status.

4.1

Mapping Data to the Model

Our procedure requires expenditures I and budget shares B (I, t) at time t across all goods.
To deal with idiosyncratic noise, we fit a smooth curve to the budget share of each good i
at time t as a function of I. We use these curves as B(I, t). More precisely, we estimate thetrue Bi (I, t) function for each good i by fitting the following curve for each t using ordinary
which are at the household level.
25
See Online Appendix Section A.5 for details about our concordance table. We also calculate our results
using more disaggregated spending categories, using only CPI data, from 2001 to 2017. Online Appendix
Figure A.3 compares these results to what we get if we instead use the more aggregated 17 spending
categories instead for the same time period. The gaps relative to the chain-weighted inflation index are
qualitatively similar but moderately larger when we use more disaggregated spending categories. Unfortunately, the more disaggregated data are not available for the full sample, so we use the more aggregated datafor our benchmark. In principle, one should apply our methodology to the most disaggregated spending
categories possible in order to minimize aggregation bias.
26
See Online Appendix Figure A.5 and Online Appendix Table A.1 for comparisons of our data withaggregate inflation and inflation by decile of expenditures as reported by the ONS.

20

least squares
2Biht = α0it + α1it log Iht + α2it log Iht + εiht ,where h is the household and t is the time period. The estimated regression line gives us
B(I, t), which we then normalize to ensure that budget shares add up to one across goods
for every income level and time period. Importantly, we only evaluate the estimated
B(I, t) in-sample to avoid out-of-sample extrapolation errors. As mentioned before thispotentially limits the set of values for which we can construct the money metric, but
ensures that our estimates are more reliable. Our results are virtually unchanged if
we estimate the Engel curves nonparametrically (i.e. using locally weighted scatterplot
smoothing, LOWESS) instead of quadratic functions (see Figure A.2 in Online Appendix
Section A.4).
Since this regression is the only source of sampling uncertainty in our exercise, we
calculate standard errors for our estimates of the money metric by bootstrapping thisregression. To do this, we redraw repeated samples with replacement. Although theEngel curves are estimated with considerable uncertainty, the standard errors for themoney metric are fairly tight. This is due to the law of large numbers, since the money
metric combines many Engel curve estimates. For this reason, and to make the figuresless cluttered, when we present our results, we do not report the bootstrapped standard
errors.
We calculate money metric utility using 1974 base prices by applying our procedure
sequentially from 1974 to 2017 to the UK data.27 Computing u(I, t) requires that for eachtime s < t, we can estimate the compensated budget share b(ps , u(I, t)). That is, for each
expenditure level I at time t, we must be able to find consumers at s < t who were on thesame indifference curve as the one delivered by I at time t.
The left panel of Figure 3 illustrates how households in 2017 are matched with households in 1974 in order to estimate b(p1974 , u(I, 2017)). For example, households in the 50th
percentile of expenditures in 2017 are matched with households in the 78th percentile
of expenditures in 1974. The dashed diagonal line is the 45-degree line and is what wewould get if we matched households by percentile of the distribution. This is how pricedeflators by spending group are typically calculated by statistical agencies (we compare
our results with such a measure below).
The right panel of Figure 3 plots the distribution of log expenditures in our data and thesolid lines show the sample of households for which we can calculate u(I, t). Our algorithm
can recover the money metric up to about the 98th percentile of households in 2017. For
27

Given the money metric at some base prices, we can easily obtain the money metric at any other baseprices in tm ∈ [t0 , T], as explained in Footnote 12.

21the richest households, we are unable to compute u(I, t) because there are no households
in our sample that were on the same indifference curve in the past. Nevertheless, ouralgorithm covers a significant range of households. Our sample coverage is high because
the distribution of expenditures is highly fat-tailed, which means that in 1974, there
are households who are on the same indifference curve as the richest 98th percentile ofhouseholds in 2017.
Figure 3: Results of the matching process
Matched Households in 1974 and 2017

10
9

80

Distribution of Total Expenditures over Time
Highest Utility
Lowest Utility

87log Expenditure

Percentile of matched households in 1974

100

60

40

6
5
4
3

20

2
1

0

0

20

40

60

2017 percentile

80

0

100

1975 1980 1985 1990 1995 2000 2005 2010 2015

Year

Notes: The figure on the left shows, for each expenditure percentile in 2017, the expenditure percentile in1974 of the matched household that is on the same indifference curve as the 2017 household. The dasheddiagonal line is the 45-degree line. The vertical dotted lines are the boundaries for households that can bematched. The figure on the right shows the sample distribution of (weekly) log expenditures from 1974
to 2017. The upper and lower blue boxes represent the 75th and 25th percentiles, respectively. The solidlines indicate the upper and lower bounds of the sample for whom the compensated budget share can becomputed as a function of time. The lower and upper bounds in 2017 represent the 0.2th and 98th percentile,
respectively, of the spending distribution.

4.2

Results

The solid blue line in the left panel of Figure 4 plots the expenditure function e(p1974 , v(p2017 , I))
for different values of I. This expresses different income levels in 2017 (x-axis) in termsof 1974 pounds (y-axis) — the money metric utility function with 1974 base prices. We
can also use this figure to convert different income levels in 1974 (y-axis) in terms of 2017
pounds (x-axis) — the cost-of-living function.28
That is, pick an I0 on the y-axis, and find the associated I on the x-axis. Then, since v(p1974 , I0 ) = v(p2017 , I),
it must be that e(p2017 , v(p1974 , I0 )) = e(p2017 , v(p2017 , I)) = I. In words, I is the cost-of-living adjustment needed
to keep a household with budget set (p1974 , I0 ) on the same indifference curve in 2017 as in 1974.
28

22

For comparison, the dotted red line shows the equivalent incomes in 1974 if all households faced the same effective inflation rate, as given by the chain-weighted aggregate
inflation rate. When the dotted red line is above the solid blue line, this means that realconsumption based on chain-weighted aggregate inflation is higher than equivalent income using the money metric for households in the sample. Hence, the money metric ishigher than real consumption for richer households and lower for poorer households, andthe size of the gap is largest for the poorest households. That is, the poorest households are
not as well-off as implied by using an aggregate price deflator calculated as in the official
statistics. Conversely, the gap reverses around the 60th percentile of the distribution andthen widens suggesting that the richest households are better off in 1974 pounds thanwhat is implied by official statistics. Accordingly, the histograms in the right panel of
Figure 4 show that inequality across households is larger based on money metric values
than based on real consumption.

1974 Income that gives the same utility as in 2017

Figure 4: Comparison of money metric with chain-weighted real consumption.
10,000

Money metricReal consumption

0.12
Money metric

0.10
0.08

Real consumption

0.06
1,000

0.04
0.02
0
1,000
10,000100,000

10,000

Money metric and real consumption in 2017

Income in 2017(b) Histogram (using household weights)
of money metric e(p1974 , v(p2017 , I2017 )) andreal consumption using aggregate chainweighted inflation (annualized pounds, log
scale). The distributions are truncated at theupper and lower bounds of Figure 3.

(a) Real consumption using aggregate chainweighted inflation between 1974 to 2017 (annualized pounds, log scale) and the money
metric e(p1974 , v(p2017 , I2017 )). This figure converts income in 1974 into equivalent income
in 2017 and vice versa.

The left panel of Figure 5 displays the log difference between the dotted red andsolid blue lines in Figure 4. As expected, the difference is positive for poor households,
meaning that real consumption calculated using aggregate inflation is upward biased,
and negative for rich households, meaning that real consumption is downward biased.
The size of the bias is 20 log points for the poorest households. This means that over the
44-year sample, annual inflation rates calculated as in the official statistics understate thetrue welfare-relevant inflation (i.e. the deflator implied by the money metric and cost-of23

living functions in Lemma 1) for these households by around 0.5 percentage points per
year. On the other hand, for the richest households, the official inflation rate overstates
the true inflation by 0.36 percentage points per year on average.
The right panel of Figure 5 shows the errors between the true inflation rate and chainweighted decile-specific inflation. The errors are much smaller, but not zero. We stressthat this does not guarantee that quantile-specific chained deflators always approximate
the true money metric well. We expect that in contexts where growth is more rapid,
the differences can be larger. The data requirements for constructing the money metric,
following our method, are slightly less demanding than the ones required for constructing
quantile-specific chained deflators.29
Figure 5: Log difference between chain-weighted inflation and true cost-of-living inflation
8
20

6

Bias (log points)

Bias (log points)

4
10

0

2
0
−2
−4

−10

−6
−20
10,000−8

100,000

10,000

Income in 2017

100,000

Income in 2017(a) Aggregate chain-weighted inflation andtrue cost-of-living inflation.

(b) Decile-specific chain-weighted inflationand true cost-of-living inflation.

Notes: Results are reported in log points (i.e. 100 times the log difference). The sample is from 1974 to 2017.

5

Extension with Partially Observed Prices

In this section, we extend our methodology to allow for the possibility of missing or unreliably measured price changes. This may occur because the infrastructure for collecting
comprehensive price data is absent, as in developing country contexts, or because changes
in some prices are inherently difficult to measure, for example those of services and new29

Whereas quantile-specific chained deflators require a representative sampling of the entire distributionof households, our methodology can recover the money metric for a subsample of observed households
even if that subsample does not sample incomes at the same frequency as the population, as explained in
Online Appendix Section A.5. Otherwise, the data requirements of the two methodologies are the same.

24

goods. The results in this section generalize Feenstra (1994) beyond the homothetic CES
case.
To compute welfare without data on some prices, we impose the following assumption
about preferences throughout this section.
Assumption 1 (Separability). Partition the set of goods into X and Y. Suppose that
preferences are separable in the sense that the expenditure function can be written ase(p, U) = e(eX (pX , U), eY (pY , U), U),(11)where U is utils, pX and pY are vectors of prices in X and Y, and eX and eY are nondecreasing
in and homogeneous of degree one in prices.
We assume that prices and budget shares of goods in X are observed, but prices andbudget shares in Y are unobserved. Assumption 1 does not restrict cross-price elasticities
for goods within X or Y but does restrict cross-price effects between X and Y. CES
aggregators, used by Feenstra (1994), are separable in every partition of their arguments,
so our separability assumption is much weaker. Separability can be tested using theLeontief-Sono conditions, see Blackorby et al. (1998).30 We provide a proof-of-concept
illustration for our empirical application below.
Denote the compensated budget share of X goods bybX =Xbi (p, U) = bXi∈X!
eX (pX , U),U ,eY (pY , U)

where the second equality uses Assumption 1 and the fact that e is homogenous of degree
one in prices. Hence, the budget share on X goods is pinned down, for a fixed U, by aeX (pX ,U)
single scalar, eY (pY ,U) , which we can interpret as the relative price of the X and Y bundles.
Define the compensated elasticity of substitution between X and Y goods to be
1 − σ(p, U) =bX
X ∂ log 1−bXi∈X∂ log pi.

That is, σ(p, U) captures how spending on X goods changes relative to Y goods if the priceof all X goods rises by the same amount, holding U constant.31
30

The Leontief-Sono conditions, which
are necessary and sufficient for separability, imply that, for each b∂ log b i

i, j ∈ X and k ∈ Y, we must have ∂ log pjk = 0, where bi and b j are both compensated budget shares. The same
must hold if we swap X and Y.
31
This elasticity of substitution is disciplined by the curvature of the upper nest of the expenditure25

Denote the relative uncompensated and compensated budget share on i ∈ X by
BXi (I, t) =Bi (I, t),
BX (I, t)bXi (p, U) =andbi (p, U).
bX (p, U)

The following proposition extends Proposition 1 to account for unmeasured prices.
Proposition 5 (Money metric with Missing Prices). Suppose Assumption 1 holds. For t ∈[t0 , T], the money metric u(I, t) is a fixed point of the following integral equation as long asσ(ps , u(I, t)) , 1 almost everywhere for s ∈ [t0 , t]:
Z tX

d log pis
log u(I, t) = log I −ds −bXi (ps , u(I, t))ds
t0 i∈Xd log bX (ps , u(I, t))1ds,ds
σ(ps , u(I, t)) − 1
t0
(12)

Z t

where
bXi (ps , u(I, t)) = BXi (u−1 (u(I, t), s), s),bX (ps , u(I, t)) = BX (u−1 (u(I, t), s), s).

If we know the shape of the function σ(ps , u) for s ∈ [t0 , T], Proposition 5 can be used toobtain the money metric utility function using similar procedures to the ones in Section 3.2.
Proposition 5 is a consequence of Proposition 1. To derive it, we use the rate of change in
d log bX (ps ,u(I,t)), to infer the compensated-budgetthe compensated budget share of X goods,
dsP
d log pshare-weighted rate of change in prices for the unobserved goods i∈Y bi (ps , u(I, t)) ds is
given the elasticity of substitution σ(ps , u(I, t)). We require that σ(ps , u(I, t)) , 1 almost
everywhere in order to do this, since when σ(ps , u(I, t)) = 1, the compensated share of bX
does not respond to the relative price of X and Y goods.
Compared to Proposition 1, the fixed point in Proposition 5 has some additional terms.
First, the compensated elasticity of substitution σ(ps , u(I, t)) on the right-hand side dependson u(I, t), and since u(I, t) depends on the compensated elasticity of substitution, there isa fixed point in this term. Second, the rate of change in the budget share of X goods,
d log bX (ps ,u(I,t)), are compensated. To compute these changes, we must use the money metricdsutility function, u(I, t), to match households on the same indifference curve through timeand use changes in the budget shares of matched households over time. Hence, there isalso a fixed point in this term.
To better understand Proposition 5, it helps to consider the homothetic special case.
Example 1 (Homothetic preferences). Suppose that preferences are homothetic. In thisfunction σ(p, U) = 1 − (1−b1X )bX∂2 log e(∂ log eX )

2

.

26

case, Proposition 5 simplifies to
Z tX

d log pis
log u(I, t) = log I −ds −BXi (ps )ds
t0 i∈XZ td log BX
1ds.
ds σ(ps ) − 1
t0

(13)

When preferences are homothetic, there is no longer a fixed point problem since budget
shares and elasticities of substitution do not depend on utility. If we also assume that theupper-nest expenditure function is CES, then σ(ps ) is a constant and we get
log u(I, t) = log I −Z tX
t0 i∈X

BXi (ps )log BX (t) − log BX (t0 )d log pis
ds −
.
ds
σ−1

(14)

Equation (14) is a version of the popular Feenstra (1994) formula.32 This formula iscommonly used in the macroeconomics and trade literatures for adjusting price indices toaccount for missing price changes (typically those of new goods). Relative to this CES case,
Proposition 5 allows the elasticity of substitution to vary as a function of prices, allows fornonhomotheticities, and does not impose parametric assumptions on preferences among
the X goods and among the Y goods.
Relative to the homothetic special case in (13), the additional complication in (12) isthat changes in the budget share of X and the elasticity of substitution must both becompensated. To see the issue, restate (12) using uncompensated budget shares as
Z tX

d log pis
log u(I, t) = log I −BXi (Is∗ , s)ds −dst0 i∈Xd log BX (Is∗ , s)1ds,ds
σ(ps , u(I, t)) − 1
t0

Z t

(15)

where Is∗ is implicitly defined by u(Is∗ , s) = u(I, t).
With more structure on the demand system, this expression can be further simplified.
For example, suppose that the expenditure function in (11) can be written as 11−γ(U) 1−γ(U)
1−γ(U)
ξX X XξY Y Ye(p, U) = ωX U e (p , U)
+ ωY U e (p , U)(16)

for any level of utility U. Let V(p, I) be the indirect utility function associated with (16).
In this example, the elasticity of substitution varies as a function of utility but not as afunction of relative prices, as in Fally (2022). With this restriction, the second integral in
32

The only (relatively inconsequential) difference between (14) and Feenstra (1994) is the assumptionthat eX and eY also be homothetic CES aggregators.

27equation (15) can be evaluated explicitly, and the expression simplifies tolog u(I, t) = log I −Z tX

BXi (Is∗ , s)t0 i∈Xlog BX (I, t) − log BX (It∗0 , t0 )d log pisds −,ds
σ(u(I, t)) − 1

where σ(u(I, t)) = γ(V(pt , I)). Of course, if the elasticity of substitution σ is also constant as
a function of utility, then the denominator becomes just σ.
We now show that the compensated elasticity of substitution σ, which is the unknown
term required to apply Proposition 5, can be expressed nonparametrically in terms ofelasticities that are estimable using only data on prices in X. This is an important result as
it demonstrates that, in general, recovering σ(ps , u) does not require data on unobserved
prices. Denote by X (I, s) the uncompensated elasticity of the budget share of X withrespect to the price of the X bundle. That is, let X (I, s) be the scalar that satisfies thefollowing equation for each level of income I at each time s:
X ∂ log BX (I, s)k∈X∂ log pkd log pk = X (I, s)X

BXk (I, s)d log pk .

k∈XProposition 6 shows that the compensated elasticity of substitution between X and Y canbe deduced given knowledge of X (I, s) and income elasticities.
Proposition 6 (Identifying Substitution Elasticity of X and Y). Suppose Assumption 1 holds.
∂ log B (I,t)
Let ηi (I, t) − 1 = ∂ logi I be the income elasticity of demand for each i ∈ X at time t. Then, wehavePX (Is∗ , s) + BX (Is∗ , s) i∈X (ηi (Is∗ , s) − 1)BXi (Is∗ , s),σ(ps , u(I, t)) = 1 −1 − BX (Is∗ , s)where Is∗ is defined by u(Is∗ , s) = u(I, t).
Proposition 6 shows that if we know income elasticities for all goods in X and canestimate the uncompensated elasticity of BX with respect to prices in X, X , then we canrecover the relevant elasticity of substitution and apply Proposition 5. Estimating theincome elasticities, ηi for i ∈ X, is relatively straightforward since we simply need to fit
a curve that relates the budget share of i to income in each period. Estimating the priceelasticity X is more challenging, but we only require a single elasticity per income groupand period. That is, the number of elasticities that needs to be estimated does not dependon the number of goods.
With more structure on the demand system, then even less information is required.
We provide one example below.
Example 2 (Generalized nonhomothetic CES). Consider the case where the expenditure28

function takes the form in (16). According to Proposition 6, the function σ(·) is determinedby the following expressionPX (I, t0 ) + BX (I, t0 ) i∈X ηi (I, t0 ) − 1 BXi (I, t0 ),σ(I) = 1 −(1 − BX (I, t0 ))(17)

Since σ is not a function of relative prices, Proposition 6 needs to be applied only in theinitial period, t0 , to recover the shape of the σ function.33 If σ also does not vary withutility, as in the example in Section 3.3, then equation (17) can still be used but only needs
to be applied for one income group.
Relation to previous literature. When price data are unavailable or unreliable, a large
strand of the literature relies on Feenstra (1994), which our method generalizes. A different strand, building on Hamilton (2001) and Costa (2001), estimates changes in welfare
by inverting Engel curves. This procedure requires that relative budget shares be strictly
monotone in income (i.e. homothetic preferences are ruled out). Atkin et al. (2024) provide
a recent micro-founded treatment of this idea. To apply their method, one needs to estimate a compensated demand subsystem for the set of goods where prices are measured,a task that can suffer from a curse of dimensionality if the number of goods with observed
prices is large. In their applications, they either rely on first-order approximations or use
a CES subsystem to keep the estimation challenges manageable.
In contrast, we make stronger assumptions about preferences (separability rather thanquasi-separability). In exchange, we do not require that budget shares be strictly monotone in income. More importantly, without making further assumptions, our approach
only requires a single uncompensated price elasticity as a function of income in eachperiod (rather than a compensated system). Given estimates of this elasticity, we can
nonparametrically and nonlinearly back out the elasticity of substitution between themeasured and unmeasured goods and use this to nonlinearly solve for welfare changes.

6

Empirical Illustration with Partially Observed Prices

As an illustration, we apply Proposition 5 to the UK data that we used in Section 4. Sinceservice prices are difficult to measure, as a test case, we partition the consumption bundle
In writing (17), we assume that X (I, t0 ) and ηi (I, t0 ) are known at t0 . This is without loss of generality
since Proposition 5 can be applied with time running forward t > t0 and backward t < t0 . Furthermore,
once we apply Proposition 5 to obtain the money metric with t0 reference prices, we can easily obtain themoney metric at ts ∈ [t0 , T] base prices, as described in Section 3.1.
33

29

into a subset of luxury services and the rest. That is, we assume that prices for leisuregoods & services and catering are not reliably observed (see Table A.2 in Online Appendix
Section A.5 for a description of these categories). These are the Y goods, which in ourdata account for roughly 30% percent of spending. We assume that prices for all othercategories of spending are measured accurately. These other categories are the X goods.
We impose Assumption 1 throughout this section.34
Table 1: Elasticity of budget share of X with respect to price index of X
OLS
(1)0.144**(0.069)Pi∈X BXi (h, t)∆ log pit

Pi∈X BXi (h, t)∆ log pit × 1(h ≥ median)F-statQuantile FE
Year FE
Obs

Y
Y
41,000IV(2)0.073***(0.019)403,945YN
41,000OLS(3)0.146**(0.069)IV(4)0.061***(0.021)

0.005(0.007)0.025(0.039)YY
41,000177,760
YN
41,000Notes: Columns (2) and (4) use the log difference in world oil prices as an instrument. All lags are twoyear differences (results are similar for annual and triennial differences). The sample years are 1974–2017.
Standard errors are clustered at the household quantile level (we have 1, 000 quantiles). Two and three stars
indicate statistical significance at the 5% and 1% level respectively.

To apply Proposition 5, we must estimate the compensated elasticity of substitution
between X and Y. To do this, we group households into a thousand groups by quantilesof the spending distribution. We run the following regression∆ log BX (h, t) = X

X

BXi (h, t)∆ log pit + controls + error,i∈X

where t is time, h is the quantile of the spending distribution, and X measures theuncompensated elasticity of the budget share of X goods with respect to the price of X
goods. To check for heterogeneity, we allow this elasticity to depend on whether quantile
h is above or below the median.
34

As a proof-of-concept, we provide a test for separability between X and Y goods in Online Appendix
Section A.6. To do this, we estimate whether the relative compensated budget shares of i, j ∈ Y respond tochanges in the price of k ∈ X.

30

We estimate this regression by OLS. Given that we include year fixed effects, identification comes from variation across households in the price change of the X bundle.
We also instrument the right-hand side variable using world oil prices (in which case wecannot include year fixed effects). The identification strategy requires that oil price shocks
exogenously move the price of goods versus services. We view our exercises as a proof ofconcept rather than a full-fledged elasticity estimation.
The results of this regression are reported in Table 1. The first two columns assume
that X does not vary as a function of expenditures, and the last two columns allow
for the possibility that X varies as a function of expenditures. Since the second row is
insignificant with small coefficients, we assume X does not vary by quantile. We also
assume that X does not vary as a function of time (we check for subsample stability byrerunning the regression on the first and second half of the time period).
The OLS and IV point estimates are X = 0.14 and X = 0.07, though with overlapping
confidence intervals. For concreteness, we take X = 0.14 and apply Proposition 6 torecover an estimate of the compensated elasticity of substitution σ for each value of I
and in each time period.35 The results, for the 25th, 50th, and 75th percentile of theexpenditures distribution are plotted in Figure 6. The estimated elasticity is below one, so
X and Y are complements, and increasing in income. Richer households are more willingto substitute between X and Y goods than poorer households.
Figure 7 uses these estimates of the compensated elasticity and computes the money
metric. The resulting money metric is plotted against the money metric from Section 4
when we assumed that all prices are perfectly observed. For low-income households,
the two money metrics are quite similar and both are below real consumption (computed
using an aggregate chain-weighted price deflator assuming that all prices are observed).
However, for households with high incomes, the money metric calculated using Proposition 5 is lower than the one calculated using Proposition 1. The fact that the dottedblue line is lower than the dashed yellow line for rich households suggests that, for thesehouseholds, prices in Y have risen more than the official price data suggest.
Figure 8 shows the percent difference between the money metric with observed prices
and the money metric with unobserved prices for different deciles of expenditures as wellas the breakdown of the difference into two terms. The first is the difference between
35

Figure A.4 in the Online Appendix displays results using the IV point estimates instead. Results are
qualitatively similar, but the adjustment to the money metric values for rich households is larger than in
Figure 7 because the implied elasticity of substitution σ is closer to one for richer households. Figure A.4
also shows that the results are very similar to our baseline if we calibrate the compensated elasticity of
substitution between X and Y goods to be constant in both the time series and the cross-section and equal
to 0.5.

31

Figure 7: Money metric e(p1974 , v(p2017 , I))
and real consumption.
1974 Income that gives the same utility as in 2017

Figure 6: Compensated elasticity of substitution between X and Y.
0.5

The value of σ

0.4

0.3

0.2
25th percentile
50th percentile
75th percentile

0.1

10,000Real consumption
Money metric (unobserved prices)
Money metric (observed price)

1,000

10,000

1970

1980

1990

2000

2010

2020

100,000

Income in 2017

Figure 8: Log difference in estimated money metrics under observed and unobserved
prices by decile of the expenditures distribution.
0.08%∆ inflation for X ∪ Y and X goods
Adj. due to ∆ log bX
0.06

0.04

0.02

0.00

10% 20% 30% 40% 50% 60% 70% 80% 90%

overall inflation and inflation for X goods implied by the two methods:
R 2017 PN
1974

R 2017 P
d log pisd log pis
b
(p
,
u)
ds
−
i
s
i∈X bXi (ps , u) ds ds
i=1
ds
1974
.
R 2017 PN
d log pis
b
(p
,
u)
ds
i
s
i=1
ds
1974

These are the dark blue bar graphs in Figure 8. The remainder is the adjustment due tochanges in the budget share of X goods, similar to the Feenstra (1994) adjustment:
d log bX (ps ,u)1dsds
1974 σ(ps ,u)−1.
R 2017 PN
d log pis
b
(p
,
u)
ds
i=1 i sds
1974

R 2017(18)

These are the light yellow bar graphs in Figure 8. This decomposition shows that inflation32

among X goods has tended to be higher than among all goods by roughly the sameamount (around 1 percentage point) for all deciles. However, the change in compensatedexpenditures on X goods has been very different. Compensated expenditures on X goods
have been falling much more quickly for rich households than poor.
Figure 9: Compensated changes in the budget share BX for different I percentiles in2017.

Figure 10: Compensated and uncompensated changes in the budget share BX forthe median household in 2017.

0.85

0.8

0.8

0.75

0.75

0.7

1970

0.7

90th percentile
50th percentile
10th percentile
1980

Compensated BX (unobserved prices)
Compensated BX (observed prices)
Uncompensated BX

0.65

1990

2000

2010

2020

1970

1980

1990

2000

2010

2020

To better understand this, we investigate how compensated expenditures on X goods
have changed over time. Figure 9 shows the compensated budget share on X goods forhouseholds at three different points in the distribution: the 10th, 50th, and 90th percentiles
in 2017. For poor households, there was almost no change on expenditures on X goods.
This explains why the adjustment term in (18) is small for these households. For the
median household, there was a modest decrease in the share of spending on X goods.
Since X and Y are complements, this indicates that the relative price of Y goods rose
relative to X goods for these households. Finally, for the richest households, there was afairly dramatic reduction in their spending on X goods from around 74% to around 63%.
This suggests that for these households, the relative price of Y goods rose fairly rapidly
compared to X goods. This explains why the adjustment term, (18), for these households
is large and negative. Furthermore, since the elasticity of substitution for rich households
is closer to one, the implied difference in the relative price of X and Y goods is larger.
This explains why the money metric according to Proposition 5 (the dotted blue line in
Figure 7) has a flatter slope than the money metric calculated according to Proposition 1(the dashed yellow line in Figure 7).
These difference in compensated expenditures are not mirrored in uncompensated expenditures. Figure 10 compares the compensated and uncompensated changes in expen33

ditures for the median household. Whereas, for the median household, the compensatedexpenditures on X goods declined somewhat over time, the uncompensated expenditures
on X goods increased very strongly. Intuitively, a household in 1974 with nominal expenditures equal to the median of the expenditure distribution in 2017 is actually fairly rich.
Such a household spends relatively less on goods (X) and relatively more on services (Y).
As we roll time forward, such a household is effectively becoming poorer, due to inflation,and this causes the expenditures on the X goods to rise due to income effects. That is, theincome effect overwhelms the substitution effect.

7

Conclusion

In this paper, we propose a method to construct money metric representations of utility—
an essential input to measuring welfare-relevant growth—using repeated cross-sectional
data. Our method does not require any estimation when the data on prices is comprehensive, aside from interpolation of how budget shares vary with income and time. If thedata on prices is incomplete, the method can still be used under a separability assumption
on preferences and knowledge of one uncompensated elasticity of substitution.
Whether prices are fully or partially observed, the unifying idea in both cases is thatmoney metric utility can be calculated using observed demand of matched households inthe cross-sectional distribution over time. Doing so involves solving a simple fixed point
equation in terms of observable variables.
Despite its advantages, our approach does not allow for preferences to vary in arbitrary and unobserved ways in the cross section or the time series, and requires that all
consumers face common prices. Furthermore, we have abstracted from intertemporalchoice. Relaxing these assumptions is an interesting avenue for future work.

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