Sufficient Statistics for Measuring
Forward-Looking WelfareDavid R. Baqaee

Ariel Burstein

Yasutaka Koike-Mori

UCLA

UCLA

UCLA*

July 22, 2024

Abstract
We provide a method to measure welfare in money-metric terms, accounting for future expectations. Our approach relies on two key assumptions: (1) separability ofthe expenditure function between present and future, and (2) existence of households
without idiosyncratic undiversifiable risk. We infer expectations about the futurefrom observed consumption-savings choices of this subset of households. Our sufficient statistics methodology accommodates incomplete markets, lifecycle motives,
non-rational expectations, non-exponential time discounting, and arbitrary functional
forms. Application requires estimates of the intertemporal substitution elasticity, pricechanges over time, and repeated cross-sectional data on household income, balance
sheets, and expenditures. Using PSID data from the United States, we find that static
measures overstate cost-of-living increases, especially for younger and poorer households. Our estimates can be used to study the welfare consequences of dynamic
stochastic shocks that affect households along different margins and time horizons.

* We thank Andy Atkeson, Adrien Auclert, Maya Eden, Valentin Haddad, Oleg Itskhoki, Chad Jones,

Patrick Kehoe, Narayana Kocherlakota, John Fernald, Kieran Larkin, Hannes Malmberg, Ricardo Reis,
Pierre-Olivier Weill, and Ivan Werning for comments. We acknowledge financial support from NSF grantNo. 1947611 and the Alfred P. Sloan Foundation.

1

1

Introduction

Measuring the welfare of households is a central task of economics. In macroeconomics, it
is widely recognized that a comprehensive measure of welfare must be forward-looking —it must account not just for present consumption, but also expected future consumption.
The typical macroeconomic model explicitly incorporates intertemporal decision-making,fully specifies preferences, beliefs, and parameters, and performs welfare counterfactuals
taking into account both time and uncertainty (e.g. Lucas, 1987).
In contrast, conventional sufficient-statistic measures of welfare used in applied andempirical work, like real income and real consumption, eschew fully specifying every
aspect of the problem. These measures can be calculated without specifying the demandsystem or providing a model of income and prices. They are constructed by deflating
nominal expenditure or income by a price deflator, and can be micro-founded using themoney-metric utility function.1
Despite their generality, these sufficient-statistic measures apply only to static anddeterministic decision problems and can easily produce misleading estimates of changes
in intertemporal welfare.2 For example, welfare may rise even though real income or realconsumption falls.3
Therefore, when dynamic considerations are important, researchers tend to rely on
fully-specified models to measure welfare. The main difficulty for the sufficient-statistic
approach is that much of what one needs to observe in order to compute a measureof dynamic welfare is not easily observable. Static sufficient-statistic measures requireknowing prices and spending on goods and services in different periods. By contrast,
dynamic measures require knowing state-contingent spending plans, state-contingent
goods and asset prices, and probabilities of different states being realized. This intractablemeasurement problem is perhaps the central reason why forward-looking measures of real
wealth have remained firmly in the domain of economic theory rather than measurement
practice.
This state of affairs, where structural work uses forward-looking wealth-like measures
1

The money-metric was introduced by McKenzie (1957), but popularized by Samuelson (1974). For atextbook presentation, see Deaton and Muellbauer (1980).
2
Real income and real consumption use consumer price deflators and money-metric utility uses a
Konüs (1939) price deflator. These deflators are constructed for static decision problems. For example,
the Consumer Price Index manual (IMF, 2004) or the survey paper by Jorgenson (2018) on “Production andWelfare Measurement”, have little to say about how expectations about the future should influence themeasurement of welfare and cost-of-living indices.
3
Expenditures can fall even though welfare rises if, for example, there is a beneficial shock in the future
and the elasticity of intertemporal substitution is greater than one or if there is a negative shock today anda positive shock in the future and the elasticity of intertemporal substitution is less than one.

2

of welfare but applied empiricists use income-like measures, has long been a source oftension in economics. For example, Samuelson (1961) concludes his paper on dynamic
welfare measurement with this:“When we work with simple and exact models, in which no extraneous statistical difficulties of measurement could arise, the only valid measure of welfare
comes from computing wealth-like magnitudes not income magnitudes. In theabsence of perfect certainty, the futures prices needed for making the requisite
wealth-like comparisons are simply unavailable.”This paper develops a new approach to overcome some of these challenges. To doso, we first generalize the definition of money-metric utility so that it can be appliedin settings with incomplete markets. In static settings with complete markets, welfare
is measured in money-metric terms by the expansion in the budget constraint (income)required to reach a given indifference curve. When markets are incomplete, there isno single intertemporal budget constraint and the expansion required to reach a given
indifference curve depends on which budget constraint is shifted. We define the moneymetric utility of each allocation to be the equivalent one-time lump sum wealth required
to reach the indifference curve associated with that allocation.4 If markets are complete,then this definition coincides with the usual one.
We then propose a sufficient-statistics methodology to estimate this forward-looking
measure of welfare for a population with common preferences. In order to do this, we
make two key assumptions. The first assumption is that preferences are separable between
the present and the future and that the elasticity of intertemporal substitution is not equal
to one.5 The second assumption is that there is a subset of households that do not faceidiosyncratic undiversifiable risk, which we call “rentier” households.
Given these two assumptions, we can obtain forward-looking measures of welfare
without further assumptions about utility functions (e.g. CES across goods), risk preferences (e.g. expected utility), time preferences (e.g. exponential discounting), beliefs(e.g. rational expectations), and financial frictions (e.g. complete markets), or first-orderapproximations.
4

To be more precise, for an agent facing a dynamic stochastic problem in year τ, we ask: what is the
one-time lump sum payment the agent must receive in some base year (with no other income sources
thereafter) such that the agent is indifferent between their initial problem and this counterfactual problem.
This amount of wealth is the money-metric utility associated with the problem the agent faced in year τ, inbase year dollars.
5
Specifically, our notion of time separability is that the expenditure function implied by intertemporal
preferences be separable between current prices and future prices and probabilities.

3

We sketch the basic idea of our approach. First, for the rentier subsample, we backout the change in continuation value of the future relative to the present using changes in
consumption-savings choices, conditional on estimates of the elasticity of intertemporal
substitution (EIS). If the EIS is less than one, then an increase in the consumption-wealthratio suggests that inflation in the future bundle is falling relative to inflation in the present
bundle. This means that forward-looking measures of inflation that account for how
all prices, not just contemporaneous prices, change will be lower than static measuresof inflation. Accordingly, money-metric utility will be higher than what is implied by
deflating wealth by static inflation.6
Importantly, to calculate dynamic welfare we must use changes in the compensatedconsumption-wealth ratio of rentiers, which neutralizes wealth effects and responds only
to substitution effects. This is critical because consumption-savings decisions are highly
non-homothetic and respond strongly to permanent income in the data (see, e.g., Straub,2019). To back-out the compensated consumption-wealth ratio without first specifying
and estimating a dynamic model, we match rentiers in the cross-section over time, building
on ideas from Baqaee et al. (2024).
Changes in consumption-wealth ratios are not sufficient to back out welfare for nonrentiers. We recover money-metric utility for non-rentiers by relying on a generalizationof Engel’s law. Specifically, if the vector of budget shares is a one-to-one function of utility
conditional on relative prices, then two households facing the same relative prices are on
the same intertemporal indifference curve if, and only if, their budget shares are the same.
This allows us to construct money-metric values for non-rentiers by matching them withrentiers with similar static budget shares.7
Our method requires three pieces of information. First, a repeated cross-sectional
survey of static household expenditures that includes some rentier households whose
wealth is observed. Second, a time series of static price changes. Finally, knowledge of thecompensated elasticity of intertemporal substitution (which could be a constant or varyas a function of wealth and time). We also require that households’ preferences be stablefunctions of observable characteristics. That is, we rule out unobservable taste shocks,
where households with similar characteristics have different preferences.
To illustrate our method, we apply our results to the United States using the Panel6

Our sufficient statistic formulas take changes in prices of goods and services and in consumption-wealthratios as given. For answers to counterfactual questions, one would have to provide counterfactual prices
and counterfactual changes in consumption-wealth ratios, which requires a more fully-specified structuralmodel.
7
This requires that rentiers and non-rentiers are drawn from the same population (i.e. same preferences,beliefs, and prices), given observed characteristics (e.g. gender, age, and location).

4

Study of Income Dynamics (PSID) and price data from the Bureau of Labor Statistics.
We begin by selecting a subsample of rentiers. We do this by computing a proxy for thepresent value of expected future labor and transfer income for each household. We saythat a household is a rentier if the present value of their future labor and transfer income
is less than 10% of their total wealth. Taking estimates of the EIS from Best et al. (2020),
we recover an ideal cost-of-living index using changes in consumption-wealth ratios forthese households. We then extend the money-metric to cover non-rentiers by matchinghouseholds in the same period together via static budget shares as described above.
We find that conventional CPI measures overestimate the true dynamic cost-of-living
price index in our sample. Mechanically, this is because the EIS is less than one and there isan increase over time in compensated consumption-wealth ratios.8 Furthermore, we findmore heterogeneity in the dynamic cost-of-living index than in the static cost-of-living
index across both the wealth distribution and by age group. Static measures of inflation
typically find that inflation is overstated for rich households and understated for poor
households.9 In contrast, we find that dynamic inflation rates tend to be lower for poorer
households. Furthermore, dynamic non-homotheticities are much more powerful thanstatic ones in our data. However, standard errors for the dynamic price index are larger
due to small sample size.
Our methodology is useful as an input for reduced-form empirical work studying
the welfare effects of dynamic treatments with uncertain outcomes. Many policies andshocks have complex effects that affect households in ex-ante uncertain ways along many
dimensions and at different time horizons. We provide a way to study the welfare effectsof such treatments without requiring that researchers enumerate, estimate, and aggregate
all the possible ways the treatment affects the household and how the household’s beliefs
and contingent plans change in response to that treatment.
To illustrate this approach, we study the welfare consequences of job loss in our
sample. To do so, we regress our estimates of money-metric utility on job loss. We findthat involuntary job loss is associated with a roughly 20% reduction in money-metricutility for households younger than 60. For households above 60, the losses are smaller
and less statistically significant. Unlike alternative numbers from the literature, our results
do not assume that households discount the future using market rates, or that households
have perfect foresight about their future path of earnings.
8This does not imply that consumption-wealth ratios rise at the individual level since individual-level
consumption wealth ratios are not compensated. Furthermore, at the individual level, consumption wealthratios may also respond to lifecycle considerations associated with aging.
9
For example, see Jaravel and Lashkari (2024).

5

Related Literature. This paper is closely related to the literature on consumer priceindices. Although the vast majority of papers in this literature abstract from time and risk,
we use insights from this literature, originally developed for static problems, to study
dynamic ones. Feenstra (1994) inverts CES demand curves to infer the value of new
goods from changes in expenditures on existing goods.10 We extend this idea to infer thevalue of missing future prices relative to present prices using changes in consumptionwealth ratios.
Hamilton (2001) matches households that are on the same indifference curve overtime via budget shares on food.11 Atkin et al. (2024) provide a micro-foundation for thisapproach and use it to calculate welfare across the income distribution. We use a similar
idea to Hamilton (2001), but we match rentiers to non-rentiers within the same periodusing static budget shares. This means that we do not have to estimate or correct for
substitution effects, as in Atkin et al. (2024).
Our paper is also related to a literature that non-parametrically measures non-homothetic
price indices using repeated cross-sectional data, like Blundell et al. (2003), Jaravel and
Lashkari (2024), and Baqaee et al. (2024). Specifically, we generalize Baqaee et al. (2024)
to environments with intertemporal preferences and incomplete markets.
The literature on dynamic price indices is comparatively small, but can trace its origins
to the inception of index number theory. For example, Fisher and Pigou both recognizedthat the ideal cost-of-living index should incorporate information about the future, though
they did not offer a specific remedy. Alchian and Klein (1973) argue that a proper definition of the price index must be based on intertemporal consumption, and they propose
including asset prices in the CPI. Pollack (1975) studies conditions under which the intertemporal cost-of-living index can be broken up into sub-indices.
In the context of national income accounting, Hulten (1979) points out that productivity
shocks today drive capital accumulation in the future, and so the Solow (1957) residual
understates the importance of technological change. He proposes to use interest rates
to calculate a dynamic technology residual. Relatedly, Basu et al. (2022) show that,to a first-order, the welfare of a country’s infinitely-lived representative consumer canbe summarized by the net-present value of technology shocks plus the initial capitalstock. Jones and Klenow (2016) construct country-level welfare measures that account
10

This approach is frequently used to infer the value of new goods or quality change in static settings,
see, for example Broda and Weinstein (2006), Broda and Weinstein (2010), Blaum et al. (2018), Aghion et al.
(2019), Argente and Lee (2021), and Argente et al. (2023). This approach can also be used to measure thegains from trade, i.e. the value of imported goods, as in Arkolakis et al. (2012).
11
This approach, especially paired with an AIDS functional form, is frequently used to measure inflationin historical settings and settings where data quality is low. See, for example, Costa (2001), Almås (2012),Almås et al. (2018), and Nakamura et al. (2016).

6

for mortality, leisure, and risk-aversion by fully specifying preferences and calibrating
stochastic processes for the determinants of utility.
To measure dynamic measures of inflation and welfare, Reis (2005) and Aoki and
Kitahara (2010) calibrate parametric models of household preferences and beliefs, andcompute aggregate cost-of-living indices by feeding in the path of observed prices. Reis
(2005) uses additively time-separable homothetic preferences and considers only financial
wealth, whereas Aoki and Kitahara (2010) use Epstein and Zin (1989) preferences andallow for both financial and non-financial wealth. Both papers use homothetic preferences
and assume that all assets can be traded — that is, they abstract from idiosyncratic
uninsurable risk and borrowing constraints. In contrast, our method accommodates
uninsurable risk, borrowing constraints, and non-homothetic preferences.12
More recently, Fagereng et al. (2022) and Del Canto et al. (2023) use Taylor approximations to calculate how consumer welfare responds to shocks to asset prices and monetary
policy, respectively. Fagereng et al. (2022) estimate how various asset prices changed overtime in Norway and weigh these changes in asset prices by discounted net asset sales.
Del Canto et al. (2023) estimate local projections of how monetary shocks change goods
and asset prices, and then weigh these price changes by discounted budget shares ofhouseholds.
Our paper differs from these papers in some important ways. First, we do not impose
perfect foresight, complete markets, or rely on first-order approximations and certainty
equivalence. Second, we do not directly estimate or model future asset or goods prices,beliefs, discount factors, and future holdings of assets or purchases of goods. Instead, weback out the welfare impact of changes in future prices and probabilities from changes
in consumption-savings decisions. Last, our method accounts for non-homotheticities,
which only matter beyond first-order approximations. However, our approach does not
unpack which changes in future returns or prices or probabilities are responsible for thechanges in measured welfare.
Our paper is also related to generalizations of price theory to incomplete markets. A
notable example is Farhi et al. (2022), who decompose changes in demand in an incomplete
market world into income and substitution effects. Our paper is related to this task since, toconstruct a money-metric measure of welfare, we define and use a notion of compensateddemand that treats income and substitution effects differently.
12

Alvarez and Jermann (2004) measure the welfare cost of consumption fluctuations using asset prices.
Similar to this paper, their approach does not require specifying a utility function. However, their paper isdifferent to ours both in terms of the question and the methodology.

7

Roadmap. Section 2 illustrates a key idea in our method using a complete-markets
example with constant relative risk aversion (CRRA) preferences: namely, that changes
in consumption-wealth ratios can be used to infer dynamic welfare changes. Section 3
generalizes the environment to allow for incomplete markets, uninsurable risk, borrowing
constraints, and non-homothetic preferences. Section 4 contains the main results of thepaper showing how to extend the basic idea in Section 2 to this more complex environment,
first for rentiers and then non-rentiers. Section 5 provides some extensions of the basic
framework such as allowing for secular changes in mortality risk and labor-leisure choice.
Section 6 constructs a measure of dynamic welfare for households in the PSID. Section 7
uses these measures to study the dynamic welfare losses from job loss. We conclude in
Section 8.

2

Complete Markets and CRRA Preferences

In this section, we consider a special case that demonstrates one of the key ideas of thispaper. We make some strong assumptions: there is only one consumption good in eachperiod, preferences are homothetic with constant relative risk aversion, and financial
markets are complete. We relax these assumptions in Section 3.
Suppose there are cohorts with planning horizon T at each calendar date τ. Intertemporal preferences are represented by
U(c, π) =

T X
X
t=0

st

β(st )π(st |τ)c(st )1−1/σ,
1 − 1/σwhere

σ , 1,(1)

where T is the agent’s planning horizon, t denotes the number of periods after the startdate, st is the state in period t, the history of realizations up to period t are denotedby st , state-contingent intertemporal discounting (which could depend on life-cycle) are
captured by β(st ), the probability of history st being realized, conditional on starting at
date τ, is π(st |τ), consumption in period t given history st is c(st ), and σ , 1 is the EIS.
Assume that financial markets are complete. Suppose at each point in time τ, there
is a cohort with preferences (1) facing the following sequence of budget constraints overtheir life:p(s0 |τ)c(s0 ) +Xp(s |τ)c(s ) +

X

t

t

ak (s0 ) = w,

(2)

k∈S1

k∈St+1ak (st ) = Rst (st−1 |τ)ast (st−1 ),8(3)

p(sT |τ)c(sT ) ≤ RsT (sT−1 |τ)asT (sT−1 ).

(4)

The first constraint ensures that consumption in the period zero plus purchases of Arrowsecurities sum to initial wealth, where S1 is the set of possible realizations of the state inperiod 1. The second constraint ensures that, at each subsequent history st , consumption
plus purchases of Arrow securities that pay off in t + 1 are equal to the returns frompreviously purchased Arrow securities. The final constraint is a no-Ponzi condition
that ensures consumption in every terminal history must be less than the returns frompreviously purchased Arrow securities. Without loss of generality, we assume that initial
wealth w includes present and discounted future income earned by households.
The value function for a household with initial wealth w in cohort τ isV(τ, w) = max {U(c, π) subject to (2), (3), and (4)}.
c,a

(5)

Our objective is to compare the choice set facing different cohorts of households, always
keeping the preference parameters, like the planning horizon T and the taste shifters β(st )fixed. For example, we compare the value function of 50 year olds in 2005 to the valuefunction of 50 year olds in 2019. This comparison has to take into account not only wealthand present prices but also future prices, probabilities, and returns. We do not compare
the welfare of a single household at different points in their life because preferences andplanning horizons change along the life-cycle.13
For each τ and w, the τ0 -money-metric utility function, u, is defined implicitly via
V(τ, w) = V(τ0 , u).
The equation above defines a scalar-valued function, which we denote by u(τ, w|τ0 ). In
words, u(τ, w|τ0 ) is the amount of initial wealth in τ0 such that the household is indifferent
between (τ, w) and (τ0 , u). To compare two different choice sets, say (τ, w) and (τ′ , w′ ), we
compare u(τ, w|τ0 ) and u(τ′ , w′ |τ0 ). The choice set defined by (τ′ , w′ ) is preferred to (τ, w)
if, and only if, u(τ′ , w′ |τ0 ) is higher than u(τ, w|τ0 ). Since both of these numbers are interms of τ0 dollars, we can calculate the rate of growth of money-metric wealth between(τ, w) and (τ′ , w′ ) as u(τ′ , w′ |τ0 )/u(τ, w|τ0 ). We can also calculate the change in the cost of13

As explained by Fisher and Shell (1968), or more recently Baqaee and Burstein (2023), welfare comparisons, based on revealed preference theory, always involve comparisons of choice sets for the same
preference relation — they do not involve intertemporal comparisons of interpersonal utility values. In thisexample, the preference relation ranks lotteries over consumption streams and hence welfare comparisonsof lotteries are meaningful. However, the planning horizon T, discount parameters β(st ), and the elasticity σ
are parameters of the preference relation. The preference relation does not rank its parameters, and hence,
they must be held constant in any welfare comparison.

9living between τ0 and τ′0 as u(τ, w|τ′0 )/u(τ, w|τ0 ). This is the change in the initial wealthneeded to reach a (τ, w)-indifference curve in τ′0 relative to τ0 . To streamline notation, we
frequently suppress τ0 and write u(τ, w) in place of u(τ, w|τ0 ), with the understanding that
τ0 is implicitly held constant.
Given that preferences are homothetic, the money-metric is also related to the consumptionequivalent number defined by Lucas (1987). To see this, define c(st |τ, w) to be the optimal
consumption in history st of cohort τ with initial wealth w — the maximizers in (5). It iseasy to verify that u(τ, w) is the scalar that solves
U(u(τ, w) × c(·|τ0 , 1)) = U(c(·|τ, w)),

(6)

where c(·|τ0 , 1) is the consumption plan of an agent in τ0 with unit wealth. In words, theu(τ, w) is the proportional increase in the consumption profile of an agent with unit wealthin τ0 required to make the agent indifferent to the optimal allocation given wealth w andstarting date τ.
Combine the Euler equation,c (st |τ, w)
β(s )π(s |τ)c (s0 |τ, w)
t

t

!− σ1

=

t
Y
l=0

R−1
sl+1

  p (st |τ)sl |τ
,p (s0 |τ)with the intertemporal budget constraint,
T X 
t
  
 X
Y
  

0
0
t
l
t
p s |τ c s |τ, w +p s |τ
R−1
s
|τ
c
s
|τ,
w
= w,
sl+1
t=1

st

l=0

to solve for consumption at each state in terms of primitives. Substituting the resulting
solutions into (6) and solving for the money-metric yields14
1
 1−σ
σ
,  PT P
1−σ Qt
t
t
t
l
σ−1
(s
β(s
)π(s
|τ)
p
|τ)
R
(s
|τ)


s
t=0
l=0 sl+1u(τ, w) = w  PT P t
 .
Q

σ
t
1−σ
t
t
t
l
σ−1 
st β(s )π(s |τ0 ) p (s |τ0 )t=0
l=0 Rsl+1 (s |τ0 )(7)This equation shows that the money-metric can be obtained by appropriately deflating
wealth. The difficulties of calculating dynamic money-metrics mentioned by Samuelsonσ PT P t t
 σ−1
t
1−1/σ

st β(s )π(s |τ)c(s |τ,1)
t=0
PP
Using c s |τ, w = c s |τ, 1 w, (6) can be expressed as u(τ, w) = w T. Fromβ(st )π(st |τ0 )c(st |τ0 ,1)1−1/σ
t=0
t
Q
s−σ




t

p(s |τ)t−1l
the Euler equation we obtain c st |τ, 1 = β(st )σ π(st |τ)σc s0 |τ, 1 , and substituting
l=0 Rsl+1 s |τ p(s0 |τ)
this into the intertemporal budget constraint, we can solve for c st |τ, 1 for all st in terms of primitives.
Substituting this into the expression for u(τ, w) above yields (7).

14

t



t

10

(1961) are apparent. Even in this highly stylized example, computing (7) requires knowledge of the entire path of discounting, probabilities, prices, and returns households expect
over their life at both τ and τ0 , none of which is easy to observe. Note that, if agents do
not have perfect foresight about the future, then data from the future (i.e. at τ + 1, τ + 2,etc.) is of no use since we only observe a single sample-path ex-post (and agent’s ex-ante
beliefs need not coincide with ex-post realizations).
Note the dramatic simplification that comes from assuming that the problem is static.
In this case, (7) simplifies to. p(s0 |τ),u(τ, w) = w
p(s0 |τ0 )which is just nominal wealth in τ deflated by consumer price inflation between τ and τ0 .
To circumvent the intractable measurement problem in (7), we take advantage of thefact that information about the future can be gleaned from households’ consumptionsavings choices. To that end, denote the consumption-wealth ratio by
p s0 |τ c s0 |τ, w
B (τ, w) = B (τ) =
,
w
P

P

where the first equality, implying independence from w, follows from homotheticity ofpreferences. We use the symbol BP because B stands for budget share out of total wealthand P indicates the present as opposed to the future. Using the Euler equation and theintertemporal budget constraint, we can show that the consumption-wealth ratio satisfies
BP (τ) =p(s0 )1−σ +p(s0 )1−σ
.
Qt
σ
t
t
1−σ
σ−1
t
l
st β(s )π(s |τ) p (s |τ)
l=0 Rsl+1 (s |τ)

PT P
t=1

(8)

Intuitively, an increase in the price of consumption in period 0, p(s0 ), raises the consumption wealth ratio if, and only if, σ < 1. Similarly, a decrease in future prices, or an increasein future rates of return, or an increase in the probability of a state with cheap low prices
being realized in the future will raise the consumption wealth ratio if, and only if, σ < 1.15
Crucially, the denominator in (8) carries information about the household’s expectations about the future. Substituting (8) into (7) yields the following proposition.
Proposition 1 (Money Metric for Special Case). Money-metric welfare for a household in
15This logic builds on the approach in Feenstra (1994), which infers changes in relative prices fromchanges in relative expenditures given knowledge of the elasticity of substitution.

11

cohort τ with wealth w, in terms of τ0 dollars, is given by:
log u(τ, w) = log w − logp(s0 |τ)Bp (τ)
1+log.
p(s0 |τ0 ) 1 − σ
Bp (τ0 )(9)

To understand Proposition 1, suppose that σ < 1 so that consumption goods in different
periods are complements. In this case, if cohort τ saves a smaller fraction of wealth thancohort τ0 , then this indicates that the price of consuming in the future is lower thanconsuming in the present for cohort τ than for cohort τ0 . This allows us to back out
the change in future prices relative present prices, comparing cohort τ and τ0 , using thechange in savings rate and the elasticity of intertemporal substitution. The bigger thedifference in savings rates between the two cohorts, the bigger is the difference between
the two cohorts in future prices relative to current prices.
To arrive at (9), we made some very strong assumptions. We assumed that there
is only one consumption good in each period, that financial markets are complete, thathousehold preferences are homothetic, and that the utility function exhibits constantrelative risk aversion. In what follows, we relax these assumptions.

3

General Environment and Measure of Welfare

In this section, we set up the general environment. We first define preferences. We thenintroduce the decision problem of households, allowing for incomplete financial markets.
Given preferences and constraints, we then generalize the notion of money-metric welfare.
Our main theoretical results are in the next section, after we set up the environment inthis section.

3.1

Time-Separable Preferences

Agents have preferences ⪰ over stochastic consumption streams represented by a utility
function U (c, π) , which need not be CRRA. The set of goods available each period is N
and we denote the consumption of good n in history st by cn (st ). Letting ST be the set of
all possible terminal histories implies that c has dimension |ST | × |N|.
Define the following transformation of the utility function:e(q, π, U) = min q · c : U (c, π) = U ,c(10)where q has the same dimensionality as c. We refer to e as the shadow intertemporal ex12

penditure function and to q as shadow prices. We use the qualifier “shadow” to emphasizethat e is a purely theoretical construct and agents need not be solving the expenditure
minimization problem defined in (10) in practice.
Throughout the paper, we impose the following time-separability condition on preferences.
Definition 1 (Time Separability). The expenditure function is time separable if the function
defined in (10) can be written as
o
 
 n
e(q, π, U) = ẽ P q(s0 ), U , F q(st )

t>0

 
, π, U , U ,(11)where ẽ, P, and F are scalar-valued functions. The function ẽ is increasing in all threearguments and homogeneous of degree one in the first two arguments. The functions P
and F are increasing and homogeneous of degree one in q, and non-decreasing in U.
We say that preferences are time separable whenever the shadow expenditure function
e is separable between present shadow prices and future probabilities and shadow prices.
This implies that spending shares across goods in the present depend on future prices
only through changes in U (i.e. wealth effects). Conversely, spending shares in the future(across dates, states, and goods) depend on present prices only through wealth effects.
Condition (11) is violated if changes in intertemporal prices, say the real interest rate, cause
spending shares in the present to change even holding intertemporal utility constant. (See
Lemma 1 in Section 4 for more details.)
In Appendix B we provide an alternative equivalent definition of time separability interms of the direct utility function (and quantities) instead of the expenditure function (andprices).16 Condition (11) is not equivalent to separability of the direct utility function unlesspreferences are homothetic. Formally, when preferences are homothetic, separability ofthe expenditure function is equivalent to requiring that the utility function be expressible
as
o
 
 n

U(c, π) = U P̃ c(s0 ) , F̃ c(st )
,π ,
t>0

where U is increasing and homogenous of degree one in its arguments, and P̃ and F̃ areincreasing and homogeneous of degree one in c.17 For example, since the preferences in
the illustrative example of the previous section, (1), are homothetic and directly separable,
they also satisfy our notion of time separability.
16

The notion of separability we use is sometimes referred to as “quasi-separability” (see, e.g., Atkin et al.,2024 who use this terminology in a static context).
17
See Theorem 4.3 in Blackorby et al. (1998).

13

The following is a parametric example of a utility function that satisfies Definition 1
but is non-homothetic.
Example 1 (Exponential Discounting Expected Utility). Suppose that the utility function
solves the following equation

U

σ−1
σ

=

T
X

βt Uεt

t=0

X
st

γ σ−1
 γ−1

σ
X
γ−1 



ωnt Uϵn cn (st ) γ ,π(st ) 

(12)

n

where t denotes age, β is the discount factor, ωnt are taste parameters for good n at age t, γand ρ control substitution elasticities across goods, states, and time, and εt and ϵn govern
wealth effects for consumption at age t and consumption of good n.18 These preferences
are time separable.19
We impose time separability of preferences throughout the rest of the paper. Note thattime separability is a primitive condition on preferences, not on the decision problems
agent face. That is, preferences can be time separable without agents facing the decision
problem defined by (10) in practice.

3.2

Decision Problem of Households

We assume that decision makers face a finite planning horizon, i.e. T < ∞, and index
each cohort by the start date τ.20 Consumers choose their consumption decisions andportfolio of assets to maximize utility subject to a sequence of state-contingent budget
constraints. The first period budget constraint requires that the sum of consumption andasset purchases equals initial wealth,
X
n∈N
18
19

pn (s0 |τ)cn (s0 ) +X

ak (s0 ) = w,

(13)

k∈K

Shifts in ωnt over t are life-cycle effects and not taste shocks, since intertemporal preferences are stable.
We can also consider a non-homothetic extension of Epstein and Zin (1989):  σ−1  σ−1h i σ−1σσσ= C st+ βt+1 Uεt+1 EU st+1
,
U st


t+1

EU s



ρ

 ρ−1
  ρ−1
X


ρ 
 ,
= 
π(st+1 )U st+1


st+1

where

U = U(s0 ),

γ

 γ−1
  X
γ−1 
t γ 
t
ϵn


and C s = ωnt U cn (s )  .

n

The elasticity of substitution across time, σ, can differ from that across states, ρ. If σ = ρ, we obtain (12).
20
In our baseline, we assume that mortality risk does not depend on τ. See Section 5 for a discussion of
how our results can be extended to account for secular trends in mortality risk.

14where pn (s0 |τ) and cn (s0 ) is the price and consumption of good n in the initial period given
starting point τ. There are K different asset types, which may not span the state space, andthe quantity purchased is denoted by ak (s0 ). The price of every asset is normalized to beone. If there are durable goods, say housing, then the stock of durables must be included
as an asset, ak , and the user-cost of service flows must be included as a price, pk . This is
how we treat housing in our empirical application.
At each subsequent history st , the agent faces the budget constraintXn∈Npn (s |τ)cn (s ) +ttXak (s ) =
t

k∈K

X
k∈K

Rk (st |τ)ak (st−1 ) + y(st |τ),(14)

where Rk (st |τ) is the return of asset k in history st and y(st |τ) is an exogenous payoff. We
think of y(st |τ) as the payoff from assets that cannot be bought or sold (e.g. the stock ofhuman capital that pays a wage every period). If y(st |τ) = 0 for every st , we say that thehousehold is a rentier.21
Households face borrowing constraintsXkak (st ) ≥ −X(st |τ)(15)

for some exogenous state-contingent borrowing constraint X(st |τ) ≥ 0. We impose a noPonzi condition that X(sT ) = 0 for every terminal history sT to ensure the agent cannot end
the problem in debt.
The decision problem faced by households depend on the tuple of prices, returns,

incomes, probabilities, borrowing constraints, and wealth: p, R, y, π, X, w . Prices, p,
returns, R, probabilities, π, and borrowing constraints, X, are functions of the start date τ,
but the stream of cashflows, y, and initial wealth, w, are household-specific. Hence, eachhousehold’s problem can be indexed by (τ, w, y). Define the value function associatedwith (τ, w, y) byV(τ, w, y) = max U (c, π) subject to (13), (14), and (15) .
c,a

(16)

The value function ranks decision problems according to underlying preference relation.
21

Under complete markets, every household is a rentier since all assets can be bought and sold. In Section
5, we extend the set of rentiers to include households with a risk-free cash flow y(st |τ) = y(t|τ) that do not
face ad-hoc borrowing constraints.

15

3.3

Measuring Welfare and the Cost-of-Living

We measure welfare using a generalization of the money-metric that allows for forwardlooking decisions, uncertainty, and incomplete markets.
Definition 2 (Dynamic Money Metric). Consider a reference period τ0 , with reference
prices, returns, and probabilities about the future: p(·|τ0 ), R(·|τ0 ), π(·|τ0 ) , with p(·|τ0 ) > 0
and R(·|τ0 ) > 0. The τ0 -money-metric, u, associated with a decision problem (τ, w, y) isdefined implicitly via
V(τ, w, y) = V(τ0 , u, 0).
In words, the money-metric, u(τ, w, y|τ0 ), maps the decision problem (τ, w, y) into theequivalent one-off lump-sum payment the household would need, under the baseline τ0 ,to ensure indifference. In a static deterministic world, the generalized money-metric wedefine coincides with the textbook money-metric (e.g., Deaton and Muellbauer, 1980).22
Proposition 2 (Money-metric cardinalizes utility). The money-metric utility function represents the same preference ordering as the original value function in (16).
We require that base prices and returns be non-zero, p(·|τ0 ) > 0 and R(·|τ0 ) > 0, to ensure
the value function is well-defined under baseline prices and returns.
Given the money-metric, we can also define changes in the cost-of-living for different
cohorts in the following way.
Definition 3 (Dynamic Cost-of-Living). The change in the cost-of-living between two
dates, τ0 and τ′0 , for some reference problem (τ, w, y) isu τ, w, y|τ′0.
u τ, w, y|τ0That is, to compare the cost-of-living between τ0 and τ′0 , we use the money-metric toconvert {p(·|τ), R(·|τ), y, π(·|τ), X(·|τ), w} into equivalent lump-sum payments in τ0 and τ′0
and compare the ratio of these numbers. In a static deterministic environment, the change
in the cost-of-living collapses to the traditional ideal (Konüs) price index of consumer
theory.
The objective in this paper is to infer the money-metric u by combining cross-sectional
survey data on household consumption and finances along with prices of goods andservices over time.
22

With incomplete markets, there are many different, non-equivalent, ways one could measure welfare
depending. For example, we could convert each problem into a certainty equivalent annuity value. We
focus on equivalent one-off lump-sum payments because this is the welfare measure that we can recoverfrom the data given our assumptions.

16

4Main Results

We present our main result in steps. First, we start with some preliminaries in Section
4.1. In Section 4.2 we present a method for recovering the generalized money-metric for
rentiers. In Section 4.3 we show how to recover the money-metric for non-rentiers.

4.1

Shadow Prices and Compensated Demand

In this section, we establish that for every decision problem there is a corresponding dual
shadow expenditure minimization problem where the shadow prices and shadow wealthrationalize the household’s consumption choices. This duality is useful because it allows
us to define the notion of a “compensated” elasticity of intertemporal substitution and“compensated” budget shares. These objects are important in allowing us to recover thegeneralized money-metric.
Since each decision problem is indexed by (τ, w, y), denote consumption of good n
in history st by cn (st |τ, w, y). The next proposition shows that for every decision problem (τ, w, y), there exists a set of shadow prices q∗ (τ, w, y) that rationalize the allocationsc(·|τ, w, y).
Proposition 3 (Dual Problem). There exist q∗ (τ, w, y) such that, for every st and n,c∗n (st |q∗ , π, V(τ, w, y)) = cn (st |τ, w, y),where c∗n (st |q∗ , π, V(τ, w, y)) are quantities that minimize the shadow expenditure function (10).
Moreover, we can set shadow prices for goods in the first period equal to their observed prices:q∗n (s0 |τ, w, y) = pn (s0 |τ),for every n ∈ N.
In other words, if the household faced shadow prices q∗ (τ, w, y) and minimized shadow
expenditures subject to a utility constraint, then the consumption plan the household
would choose coincides with the optimal allocation from (16).
In a static deterministic environment, cn (st |τ, w, y) collapses to uncompensated (Marshallian) demand for good n. On the other hand, the shadow quantity c∗n (st |q∗ , π, V(τ, w, y))collapses to compensated (Hicksian) demand for good n. One of the important differences
between c∗n and standard Hicksian demand is that it depends on shadow prices, rather
than actual prices, and these shadow prices can be household-specific.
17

Proposition 3 makes it possible to define a notion of compensated elasticity of intertemporal substitution for an agent facing the problem (τ, w, y). Denote spending on thepresent relative to shadow wealth byP
bP (q, π, U) =0 ∗ 0n∈N qn (s )cn (s |q, π, U)e(q, π, U).

Accordingly, the share of savings, i.e. spending on the future (superscript F), isbF (q, π, U) = 1 − bP (q, π, U).
Definition 4 (Elasticity of Intertemporal Substitution). The compensated EIS for a household facing problem (τ, w, y) is defined to beσ(τ, w, y) = 1 −X ∂ log bP (q∗ , π, V(τ, w, y)/bF (q∗ , π, V(τ, w, y)
∂ log q∗n (s0 )n∈N,where q∗ are shadow prices given in Proposition 3.
That is, the compensated EIS for a household facing (τ, w, y) is defined to be how spending
on consumption versus savings changes, for this household, if the shadow price of every
consumption good in the present rises by the same amount, holding utility constant. The
compensated EIS is a crucial statistic that we will need for our main results.

4.2

Obtaining the Money Metric for Rentiers

The next proposition limits attention to the subset of rentiers. Rentiers are households forwhom y is identically equal to 0 in every state of nature, and are denoted by (τ, w, 0).
Before stating the proposition, we introduce some notation. For each household withwealth w, cashflows y, in period τ, denote current expenditures byE(τ, w, y) =Xn∈Npn (s0 |τ)cn (s0 |τ, w, y)and the share of budget spent on good n by
Bn (τ, w, y) =pn (s0 |τ)cn (s0 |τ, w, y).
E(τ, w, y)18

Denote the consumption to wealth ratio by
BP (τ, w, y) =E(τ, w, y).
w

The following proposition characterizes the money-metric utility of rentiers as thesolution to a fixed point problem. To do this, assume that, for every state st , pricespn (st |τ), asset returns Rk (st |τ), and probabilities π(st |τ) are absolutely continuous functionsof calendar time τ.
Proposition 4 (Money Metric for Rentiers). If preferences are time separable and σ(x, w, 0) , 1
almost everywhere, then the money-metric satisfies the following fixed point problemZ τ X
P
∗


d
log
p
d
log
B
(x,
w
,
0)/dx
n
x
 dx,
Bn (x, w∗x , 0)
log u(τ, w, 0|τ0 ) = log w −−∗dx
1 − σ(x, wx , 0) 
τ0 n∈N(17)

where w∗x solves the equationu(x, w∗x , 0|τ0 ) = u(τ, w, 0|τ0 ).

(18)

for each cohort x ∈ [τ0 , τ]. The boundary condition is that u(τ0 , w, 0|τ0 ) = w.
To streamline notation, we suppress the dependence of the money-metric on the baseperiod, τ0 , where possible. Before discussing the general case, we build intuition byconsidering the special case of Proposition 4 where preferences are homothetic.
Corollary 1 (Homothetic Preferences). If preferences are homothetic, Proposition 4 implies thatthe money-metric is given bylog u(τ, w, 0) = log w −Z τd log BP (x, 0)
d log pi
1
dx +
dx .
Bn (x)dx
dx
τ0 1 − σ(x, 0)
n
{z
}
{z
} |

Z τX
τ0

|

(19)

adjustment for future

chained inflation

This is quite similar to our starting example in Proposition 1. The only differences
relative to that example are that (1) the EIS need not to be constant; (2) there may be
multiple goods per period and the price index for present consumption is a chain-weighted
average of individual goods prices.
Corollary 1 simplifies further if we assume that the EIS is constant (as in the illustrative
example in Section 2) tolog u(τ, w, 0) = log w −

Z τX
τ0


d log pilog BP (τ, 0)/BP (τ0 , 0)
Bn (x)
dx +
.
dx
1
−
σ
n
19

Compared to Corollary 1, the final term is now a simple log difference in the consumption
share of wealth, comparing households in the initial period, τ0 , to households in period τ.
We now discuss the general case with non-homothetic preferences. In this case, therelative budget shares on goods in each period, the consumption-wealth ratio, and the
EIS all depend on wealth. Proposition 4 shows that it is the compensated version ofthese objects that must be used when preferences are non-homothetic. This is ensured
by equation (18), which requires that the household is kept indifferent along the path
between (τ, w, 0) and (τ0 , u(τ, w, 0), 0). That is, w∗x is the level of wealth required at periodx ∈ [τ0 , τ] such that the consumer stays on the same intertemporal indifference curve.
The crucial fact about Proposition 4 is that this defines a fixed point problem. To arriveat u(τ, w, 0), we need to integrate compensated budget shares and compensated changes
in savings rate. On the other hand, to perform the necessary compensation, we need toknow u(τ, w, 0).
Proposition 4, which generalizes static results in Baqaee et al. (2024), is a fixed-point
problem that depends on observables and the EIS. The observables are wealth w, budget
shares Bn on goods as a function of time and wealth, changes in goods prices from period
to period d log pn /dt, and changes in expenditures relative to wealth BP as a function oftime and wealth. Given these observables, and estimates of the EIS, we can solve (17) for
the generalized money-metric.
Solution Method. To apply Proposition 4, we can follow the same procedures explained
in detail in Baqaee et al. (2024). We begin by guessing a candidate solution u0 (τ, w, 0), forexample, by deflating nominal wealth at each date using the chained inflation index andignoring the role of the future. We then use this initial guess on the right-hand side of (17)to get a new guess. We then iterate on this until convergence. This procedure will always
converge since the fixed point to (17) is, locally, a contraction mapping.
Boundaries. Proposition 4 can only be applied reliably inside a suitable boundary. This isbecause the budget shares B(τ, w, 0) and consumption-wealth ratios BP (τ, w, 0) are observed
only for some subset of wealth levels, say w ∈ [wx , wx ] for x ∈ [τ0 , τ]. This limits therange of values of w for which we can calculate the money-metric without out-of-sample
extrapolation. Intuitively, if for cohort τ and wealth w, the money-metric value u(τ, w, 0) is
not in [u(x, wx , 0), u(x, wx , 0)] for some x ∈ [τ0 , τ], then we cannot recover u(τ, w, 0) without
extrapolation. This is because there are no households in cohort x ∈ [τ0 , τ] that are on thesame indifference curve as the rentier with wealth w at time τ.
Fortunately, Proposition 4 automatically provides the boundary over which the money20

metric can be calculated without extrapolation. The initial boundary at t = τ0 is just therange in the data: [wτ0 , wτ0 ]. As we solve (17) forward, for each τ > τ0 , we can update
the boundary because the information required to compute only depends on previous
values of the money-metric (see Baqaee et al., 2024 for a discussion of this issue in a static
context).
Justifying static price indices. There are some cases of Proposition 4 where the useof a static deflator for dynamic welfare is justifiable (outside of the obvious case where
households are fully myopic). The first one is if the EIS is infinite, σ = ∞. In this case, (17)simplifies toZ τXd log pnlog u(τ, w, 0) = log w −dx.
(20)
Bn (x, w∗x , 0)dx
τ0 n∈N
That is, dynamic welfare is given by nominal wealth deflated using a deflator that relies
only on static price changes between τ0 and τ. This justifies using real wealth to measure
welfare.
The second case is when the EIS is zero and preferences are homothetic, in which case(17) can be rewritten asE(τ′ , w′ , 0)u(τ′ , w′ , 0)
= log
−
log
u(τ, w, 0)E(τ, w, 0)Z τ′ Xτn∈N

Bn (x, 0)

d log pndx.
dx

That is, the growth in money-metric wealth from (τ, w) to (τ′ , w′ ) is just the growth innominal expenditures deflated by a chained inflation between τ and τ′ . This is consistent
with Reis (2005), who argues that when the EIS is zero and agents are rentiers, then static
real consumption growth coincides with dynamic welfare growth. (This result breaks
down however, even when the EIS is zero, if households are non-rentiers or if preferences
are non-homothetic.)
The final case we consider is when preferences are non-homothetic and the EIS is zero.
In this case, (17) can be rewritten aslog E(τ0 , u(τ, w, 0), 0) = log E(τ, w, 0) −Z τX
τ0 n∈NBn (x, w∗x , 0)

d log pndx.
dxThis implies that nominal expenditures deflated by static inflation (calculated using compensated budget shares) is equal to the expenditures of the compensated household inτ0 . If expenditures are increasing in wealth, then this is also a money-metric since it is amonotone transformation of utility. Hence, the empirical approach taken by Baqaee et al.

21

(2024), which ignores time, is justifiable if the EIS is zero and all households are rentiers.
To better understand Proposition 4, we sketch the proof. (The formal proof is in
Appendix A). Readers that are not interested in the proof can safely skip the rest of thissection.
Proof sketch for Proposition 4. First, we establish that the dual shadow prices, definedby Proposition 3, associated with (τ, w, 0) can alternately be written to depend on τ andV(τ, w, 0) instead of w directly. That is, for each history st , for rentiers we can write (in an
abuse of notation)q∗ (st |τ, w, 0) = q∗ st |τ, V(τ, w, 0) .
(21)
This is intuitive since w and V(τ, w, 0) are monotone. Therefore, we can think of q∗ as a
“Hicksian” or compensated shadow price because it depends on utility rather than wealth.
One of the reasons we focus on rentiers is that (21) need not hold for non-rentiers. For
non-rentiers, the shadow prices do not just depend on wealth and calendar time, they also
depend on expected cashflows and borrowing constraints.
Next, using the Hicksian shadow prices, we show that the money-metric, u(τ, w, 0) canbe expressed using the shadow expenditure function asu(τ, w, 0) = e(q∗ (·|τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0)).
That is, for rentiers, the money-metric value coincides with the shadow expendituresthat a household would need to be given to reach the utility level u(τ, w, 0) when facing
Hicksian shadow prices. We can manipulate this expression to get
log u(τ, w, 0) = log w − loge(q∗ (·|τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0)).
e(q∗ (·|τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0))That is, the money-metric is nominal wealth at date τ deflated using an appropriate
price index that holds utility fixed and tracks changes in Hicksian shadow prices andprobabilities.23
Next, we re-express the last term in the equation above, using the fundamental theorem
23

In a static, deterministic environment, this price deflator collapses to an ideal price index, also known
as a Konüs (1939) price index.

22

of calculus, as
Z τ0 X
T X∂ log e(q∗ (st |x, u(τ, w, 0)) , π(st |x), u(τ, w, 0)) d log q∗ (st |x, u(τ, w, 0))·∗ (st |x, u(τ, w, 0))
dx
∂
log
q
τ
t
t=0 s
!
∂ log e(q∗ (st |x, u(τ, w, 0)) , π(st |x), u(τ, w, 0)) dπ(st |x)
+
·
dx.
(22)
dx
∂π(st |x)log u(τ, w, 0) = log w +The integral, which is equal to the change in the ideal price index, consists of two setsof terms. The first set of integrands, on the top line, track how the expenditure function
responds to changes in shadow prices in all possible times and states as calendar time,indexed by x, moves from the base year τ0 to τ. In a static deterministic environment,
this collapses to how the expenditure function responds to changes in static prices. The
second set of integrands, on the second line, track how the expenditure function responds
to changes in probabilities in all possible future dates and states as calendar time, indexedby x, moves from the base year τ0 to τ. These terms have no counterparts in the standard
static deterministic framework.
These summands in the integral are very high-dimensional, potentially infinite-dimensional,
summing over all possible dates and states. Equation (22) elucidates the enormous complexity of forward-looking measures of inflation as compared to the traditional static
objects. The forward-looking measure depends on how all possible future shadow prices
and probabilities change as time moves forward. This complexity is compounded by thefact that we must weigh changes in all of these unobservable shadow prices and probabilities by the elasticities of the shadow expenditure function with respect to shadow prices
and probabilities respectively.
Fortunately, we can cut through much of this complexity as long as preferences aretime separable. Denote the compensated budget share on good n to bebn (q∗ , π, U) = P

q∗n (s0 )c∗n (q∗ , π, U)
.
∗
0 ∗
∗
n′ ∈N qn′ (s )cn′ (q , π, U)

When preferences are time separable, the complicated integrand in (22) can be rewrittenasd log bP (q∗ , π, V(τ, w, 0)) X−+
bn (q∗ (s0 ), π, V(τ, w, 0))d log q∗n (s0 ),1 − σ(τ, w, 0)n∈Nleaving the details of this derivation in the appendix. Intuitively, changes in shadow
prices and probabilities change the shadow expenditure function to the extent that theymove the compensated future price aggregator F q(st ) t>0 , π, U relative to the present23
aggregator P q(s0 ), U . This compensated relative price, in turn, changes consumption
relative to savings rates if the EIS is not equal to one. Hence, we can infer changes in theprices and probabilities relevant for the future by observing changes in saving behavior,as long as we know the EIS. Plugging this equation back into (22) and manipulating leads
to Proposition 4. With the addition of (18), we can treat (22) as a fixed point problem.

4.3

Non-Rentiers

A challenge for the applicability of Proposition 4 is that, in practice, many households inthe sample are non-rentiers (i.e. y , 0). Fortunately, we can exploit non-homotheticity
of preferences to extend u(τ, w, 0) to non-rentiers. To do so, we first make the following
observation.
Lemma 1 (Compensated Budget Shares). If preferences are time separable, then the budget
share of each good in the initial period, τ, can be expressed as a function of only present prices andoverall utility:
Bn (τ, w, y) = bn (p(s0 |τ), V(τ, w, y)).
We refer to bn as the compensated budget share of n.
Importantly, Lemma 1 implies that the budget share of each good in the present
Bn (τ, w, y) does not directly depend on wealth w and the stream of payoffs y.
The next proposition makes it possible to extend u(τ, w, 0) to cover non-rentier households. Recall that u(τ, w, y|τ0 ) denotes the money-metric value for the problem faced byan agent at calendar time τ, with wealth w, and payoff streams y in τ0 dollars.
Proposition 5 (Money Metric is a Function of Budget Shares and Time). Suppose that thevector-valued function b p, V is a one-to-one function of V. Then, there exists a function m
mapping budget shares and time into money-metric utility values in that period:u(τ, w, y|τ) = m(B(τ, w, y), τ),for every τ, w, and y.

The compensated budget shares b p, V are a one-to-one function of V if no two distinct
values of V result in the same vector of budget shares. Notably, this rules out homothetic
preferences, since once we fix time τ, then the budget shares are constant for every valueof V. In words, Proposition 5 implies that, if budget shares are one-to-one with V, thenholding time τ fixed, there exists a function m(B, τ) mapping vectors of budget shares B
at date τ into the equivalent lump sum wealth at date τ (i.e. u(τ, w, y|τ)).
24

Hence, if we know the function m, and we observe budget shares B(τ, w, y) at timeτ, then we can deduce the money-metric utility u(τ, w, y|τ) for a household facing the
problem (τ, w, y). Given u(τ, w, y|τ) we can then use Proposition 4 to convert this tomoney-metric utility for some other base date u(τ, w, y|τ0 ).
How do we learn the shape of the function m? We use the identity that u(τ, w, 0|τ) = w
for rentiers. Given this, we can learn the shape of m by solving the following least-squares
problemarg min ∥w − m̂(B(τ, w, 0), τ)∥,m̂∈M

where M is a set of functions that contains m. In words, we fit a flexible function that
relates budget shares to wealth for rentiers. We then use this fitted relationship to imputeu(τ, w, y|τ) for non-rentiers given their static budget shares. Intuitively, we infer themoney-metric wealth for non-rentiers using their static budget shares. Proposition 6
formalizes this idea.
Proposition 6 (Money Metric for Non-Rentiers). LetonB(τ) = B(τ, w, 0) : w ∈ [wτ , wτ ] ,where [wτ , wτ ] is the support of the wealth distribution of the rentiers at date τ. Let m|B(τ) be thefunction m restricted to the domain B(τ). We have thatm(·, τ)|B(τ) ∈ argmin
m̂∈M

Z wτ
wτ

(w − m̂(B(τ, w, 0), τ))2 dw.

A special case of Proposition 5 and Proposition 6 is the case where the budget share ofa specific good, usually food, is known to be strictly monotone in utility.
Corollary 2 (Engel’s Law). Suppose that there exists a good i ∈ N whose budget share,bi (p(s0 |τ), V(τ, w, y)), is strictly monotone in V. Thenu(τ, w, y) = u(τ, w∗ , 0),if, and only if,

Bi (τ, w, y) = Bi (τ, w∗ , 0).

In this simple case, if the compensated budget share of i is monotone in utility, then we
can deduce that two households (τ, w, y) and (τ, w∗ , 0) have the same utility if, and only if,
their budget shares on good i coincide.
Propositions 4, 5, and 6 can be combined to recover u(τ, w, y) for every u(τ, w, y) inside
the boundary where we can solve (17) (without extrapolation).

25

5

Extensions

In this section, we discuss some extensions to our basic framework. These extensions
are: relaxing common prices and probabilities, allowing rentiers to earn risk-free labor
income, allowing for changes in mortality, and allowing leisure.
Relaxing common prices and probabilities. We assume that, conditional on observables(like age, gender, location) households have the same preference relation within and acrosscohorts.24 We also assume that static prices only vary as a function of observables (e.g.
time or location). Similarly, cohorts of rentiers at each point in time must hold common
beliefs about future prices and rates of return. Beliefs can change over time but, withina period, they can only vary for rentiers as a function of observable characteristics (e.g.
age).
However, non-rentiers’ future state variables (i.e. beliefs, prices, returns, borrowing
constraints, cash flow) need not be the same as those of the rentiers, nor do they need to be
the same for all non-rentiers. For example, rentiers may have access to different assets or
hold different beliefs about the returns on those assets than non-rentiers. To understandwhy, recall that by Lemma 1, spending shares on goods in the present only depend on
static prices and utility. Therefore, as long as this function is one-to-one, two households
facing the same prices at a point in time choose the same budget shares across goods inthe present if, and only if, they are on the same indifference curve.
Finally, we do not require that households’ beliefs about the future be objective. All
that matters is that π(·|τ) is the lottery that households in cohort τ believe they face — thismay or may not be the result of a rational expectations equilibrium.
Risk-free cash flows. Rentier households are defined to be those with zero exogenous
cash flows: y(st |τ) = 0 for every st . However, we can extend the set of rentiers toinclude households with non-zero, time-varying, but risk-free exogenous cash flows,
24

To see what can go wrong, suppose that the preference relation of cohorts is changing over time. Then,changes in consumption-wealth ratios or budget shares may be due to changes in preference parameters(e.g. discount factor changes), and not due to changes in welfare-relevant variables like prices, returns,
or probabilities. If preference shocks are independent and identically distributed across households, thenaveraging across households with the same observables eliminates them by the law of large numbers.
For this reason, in our empirical applications, we average over households with similar characteristics.
However, if the preference shocks are systematic, then this can cause biases. Note that preference stability
is a typical maintained assumption in the literature on static welfare measures. To deal with preferences
instability, e.g. taste shocks over time, we would need to specify a model of demand to purge out changes
in choices that are driven by taste shocks as opposed to income and substitution effects, as discussed in
Baqaee and Burstein (2023).

26

y(st |τ) = y(t|τ), as well. Examples could include public sector employees, members ofteachers’ unions, pensioners on defined benefits, and tenured professors. To treat thesehouseholds as rentiers, we assume that they do not face ad-hoc borrowing constraints andthat they can access bonds with maturities {1, . . . , T}.
Specifically, the first period budget constraint, previously (13), is now
X
n∈N

0

0

pn (s |τ)cn (s |τ) +

X
k∈K

0

ak (s |τ) +

T
X
m=1

Bm (s0 |τ) = w + y(s0 |τ),(23)

where Bm (s0 |τ) is the quantity of bond of maturity m ∈ {1, . . . , T} purchased in period τ,with return R(m|τ) at date τ + m. At each subsequent history st , the budget set is thenXn∈N

pn (s |τ)cn (s |τ) +
t

t

X
k∈K

ak (s |τ) =
t

X
k∈K

Rk (st |τ)ak (st−1 |τ) + Bm (s0 |τ)R(t|τ) + y(t|τ).

(24)

Here, for simplicity of notation, we assume these bonds are only available to purchase attime τ (allowing access to these bonds after τ does not alter the results).
Under these assumptions, it is straightforward to show that these households’ problem
is isomorphic to that of a rentier (with exogenous cash flows equal to zero) but witheffective wealthTXy(t|τ).
w(s0 |τ) +R(t|τ)
t=0
That is, households with risk-free exogenous cash flows and no ad-hoc borrowing constraints are isomorphic to rentiers whose wealth is augmented by the present discounted
value of exogenous cash flows.
Changes in mortality. While the baseline model accommodates changes in mortality
risk as a function of age, it assumes that mortality risk is fixed as a function of calendar
time. Allowing for secular changes in mortality risk can substantively alter the results.
This is because changes in welfare caused by changes in mortality risk are not necessarilyreflected in changes of the consumption-wealth ratio.
To extend the results to allow for exogenous changes in mortality risk, consider theutility functiono    σ−1
nσ−1σU σ = λP P̃ c s0 , U+ λP λF βF̃ c(st )t>0, π, U

 σ−1
σ

σ−1

σ−1

+ [1 − λP + λP (1 − λF )β]c̄ σ U σ +ε ,where P̃ and F̃ are present and future aggregators that are homogeneous of degree one in27

quantities (for a given U), parameters λP and λF are the probabilities of surviving in thepresent and the future, β is a discount factor, and σ is the constant EIS. Upon death, thehousehold receives a consumption-equivalent payoff of c̄ — the lower is c̄, the higher is thevalue of statistical life.25 The parameter ε determines wealth effects for the value of life.
All of these parameters may vary as a function of observable characteristics (principally,age).
The shadow intertemporal expenditure function associated with these preferences is

 1
 t
0
1−σ
σ σ
1−σ 1−σ
P(p(s
),
U)
+
λ
β
F(
q(s
)
,
π,
U)
σ

t>0
F
e q, π, λ, U = λP1−σ
U,
σ
h
i 1−σ
σ−1
ε
1 − [1 − λP + λP (1 − λF )β]c̄ σ U
where P and F are aggregators that are homogeneous of degree one in prices. For thisproblem, the lotteries the household faces are parameterized by the stochastic process
over prices and returns, π, as well as mortality risk, λ. While the expenditure function istime separable in shadow prices, q, and probabilities over market outcomes, π, it is nottime separable in mortality risk λ.
As long λ is constant as a function of calendar time, our results can be used. However, if λ changes over time, then time-separability is violated and our results cannot be
used. Survival probabilities are not isomorphic to the other probabilities in the model
because upon death, households do not make choices subject to a budget constraint.
This causes the shadow intertemporal expenditure function to become non-separable insurvival probabilities.
To extend our results to this setting, we again index decision problems by calendar
date, τ, which now also indexes survival probabilities, λP and λF . Following Schelling(1968), denote the value of statistical life for cohort τ to be the marginal willingness to pay
for increasing survival probabilities in the present and the future relative to initial wealthw:d log e p, π, λ, V(τ, w)
d log e p, π, λ, V(τ, w),
andΦF (τ, w) =,ΦP (τ, w) =d log λ0d log λ1
where the derivatives of the shadow intertemporal expenditure function are taken at the
shadow prices that rationalize consumer choices (i.e. as in Proposition 3). The marginal
25
Some papers effectively assume that c̄(σ−1)/σ = 0, but this is arbitrary. For example, it implies that thevalue of life switches from positive to negative as σ goes from above one to below one. Because c̄(σ−1)/σdetermines the household’s willingness to pay to reduce the probability of death, its value should instead
be disciplined by the value of statistical life (e.g. as in Jones and Klenow, 2016). Our formula in equation(25) directly uses information about the value of statistical life, instead of calibrating c̄.

28

willingness to pay to reduce the probability of death is an important input into costbenefit analysis.26 If preferences are non-homothetic, then these elasticities depend on
initial wealth.27
The extension of Proposition 4 to this environment is that the money-metric u(τ, w, 0)
solves the following fixed point problem:Z τ X
P
∗ 

d
log
p
d
log
B
(x,
w
)
1
n
x 
 dx

Bn (x, w∗x )−log u (τ, w, 0) = log w −


dx
1
−
σ
dx
τ0 n∈N!
Z τd log λP (x)d log λF (x)∗∗−ΦP x, wx
+ ΦF x, wx
dx
dx
dx
τ0
Z τ
 d log λF (x)σ 
1 − BP (x, w∗x )dx,+
dx
τ0 1 − σ(25)

where u(x, w∗x , 0) = u(τ, w, 0) for each x ∈ [τ0 , τ]. The first line of (25) is identical to the one in
Proposition 4. The remaining two lines adjust for changes in mortality risk. As expected,
if mortality risk does not vary as a function of calendar date, d log λF /dx = d log λP /dx = 0,then (25) simplifies to the expression in Proposition 4.
The intuition for the new terms is the following. The second line of (25) adds the valueof increased survival to the money-metric. This is very similar to how price changes mustaccounted for: we integrate the demand curve for the value of life with respect to changes
in mortality risk. The subtlety is that, as with prices, we must integrate compensateddemand curves rather than Marshallian demand curves. This implies that there is a fixed
point term in the second line unless preferences are homothetic. The final line of (25)accounts for the fact that, holding all else fixed, changes in the future survival probabilitychanges the consumption wealth ratio. Since the second line is fully accounting for thewelfare changes caused by changes in the future survival probability, the third line purges
P∗
1 d log B (x,wx )those changes caused by changes in future survival probability.
from 1−σdx
σ−1
A simplification of (25) results if the utility payoff of death is zero: c̄ σ = 0. This isbecause, in this case, the marginal value of increasing survival probability in the future
exactly offsets the adjustment in the consumption wealth ratio:ΦF (τ, w) =26

σ[1 − Bp (τ, w)] .
1−σ

See Ashenfelter (2006) for a discussion of how such statistics can be estimated.
If the aggregators P and F are independent of U and ε = 0, then the value of life relative to wealth does
not vary as a function of wealth.
27

29This means that (25) simplifies toZ τ X
P
∗


(x)
d
log
p
d
log
B
(x,
w
)
d
log
λ

1
n
P
x
 dx,
Bn (x, w∗x )−+ ΦP x, w∗xlog u (τ, w) = log w −


dx
1−σ
dx
dx
τ0 n∈Nso that changes in the consumption wealth ratio appropriately account for changes in
σ−1
welfare caused by changes in future survival probabilities. Although assuming c̄ σ = 0
is common in the literature and simplifies our results, there is no empirically compelling
reason to treat this as a benchmark.
To recap, our results can be extended to account for changing mortality provided withadditional information on the value of statistical life as a function of wealth and age. In ourempirical application, we abstract from these issues and treat mortality risk as constantover time. We think this is reasonable because the time series dimension of our sample isrelatively short (15 years). We emphasize that this does not imply that we take mortality
risk to be constant as a function of age. Variation in mortality rates due to changes in age
are fully accounted for by our baseline result in Proposition 4. Rather, the assumption we
make in our empirical results is that survival probabilities are constant as a function oftime conditional on age.
Leisure. Our baseline framework treats hours worked as exogenous. In other words,
we require that, conditional on observables like age, leisure choices do not change as afunction of calendar time. For rentiers, this can be justified if their leisure choices are ata corner and equal to the time endowment or if their labor productivity is assumed to
be zero.28 If rentiers’ leisure choices are not changing as function of calendar time, then
Proposition 4 can be applied without modification.
We can allow non-rentiers to make labor-leisure choices and still use Proposition 5 toinfer their welfare, if relative static budget shares only depend on utility and static pricesof goods and services. This assumption rules out non-separabilities between consumption
choices and leisure (e.g. we rule out complementarities between leisure and recreational
travel).
For example, consider the utility functiono    σ−1nσ−1σ+ F̃ c(st )U σ = P̃ c s0 , U

t>0

28

, π, U

 σ−1
σ

o
n
+ H̃ l(st )t≥0, π, U ,The key assumption is that the number of hours rentiers work does not vary as a function of calendar
time. So, if rentiers do earn labor income, as in the extension with risk-free cashflows, then to ensure thathours do not vary as a function of calendar time, we require that the number of hours they work is fixed
conditional on age (e.g. a nine to five job prior to retirement).

30

where P̃ and F̃ are aggregators over consumption of goods and services for the present andthe future, and H̃ is an aggregator over leisure choices. These preferences have the usefulfeature that Lemma 1 is still valid and static relative budget shares depend only on static
relative prices and U. This means that Proposition 5 can be used without modification.

6

Illustrative Application to US Data

In this section, we apply our method to estimate the money-metric using data from the
US. In the next section, we use our estimates of the money-metric to study how welfare
responds to job loss.

6.1

Data

We require data on households and on prices. For data on households, we use the Panel
Study of Income Dynamics (PSID) spanning the years 2005 to 2019. For the price data, weuse consumption price data from the Bureau of Labor Statistics (BLS). We describe eachdataset in turn.
The household data must be accurate, but it does not need to be representative of theunderlying population in terms of sampling frequency. Therefore, we can use the raw
data from the PSID without sampling weights. The PSID contains repeated cross-sectional
data on household expenditures by category, household-level balance sheets (assets andliabilities), household incomes, and demographic information. To account for how age
affects preferences (including planning horizons), we condition our results on decade oflife.29
The PSID includes household expenditure surveys, broken down into seven categories. A major omission is the user cost of owner-occupied housing. To remedy this,we impute equivalent owner-occupied housing costs by matching home owners in eachperiod to renters with similar observable characteristics and spending behavior. That is,
we predict rental expenditures using household characteristics and spending behavior
using a regression estimated on renters in the same period. This procedure is theoreticallyjustified by Proposition 5. In the final year of our sample, 2019, the PSID asked homeowners to report the rental value of their property if they were to rent it out. We use theanswers to this question, in 2019, to validate our imputation procedure. When we regress
29

Ideally, with enough observations, we could treat each age separately. Similarly, with more data, we
could split the sample along other observed characteristics that may influence preferences, like gender,
household size, location, etc.

31

surveyed housing costs on our imputed measure of housing costs (both relative to current
expenditures), we find a coefficient of 1.03 and an R2 value is 0.59. See Appendix C for
details.30
We combine the price data from the BLS with the expenditure survey from the PSID
via a correspondence between PSID spending categories and categories of goods in theConsumer Price Index (CPI). For more details about how specific variables are constructed,
see Appendix C.

6.2

Constructing Wealth and Classifying Rentiers

To apply Proposition 4, we need to observe a sample of rentier households. To classify
rentiers, we first estimate a proxy measure of total wealth for all households in the sample.
Our proxy for total wealth is the sum of financial wealth (net asset value including homeequity and defined contribution pension savings) and the present discounted value ofpredicted labor and transfer income. If the head of the household is unemployed andlooking for a job, then we exclude this household from the sample of rentiers.
To calculate the present discounted value of labor and transfer income, we predicteach household’s expected lifetime income profile based on observed characteristics, anddiscount the resulting flows using a real discount rate of 4% following Catherine et al.
(2022). The construction of net assets and capitalized income is detailed in Appendix C.
Figure 1 plots total consumption expenditures relative to proxy wealth in our data.
On average, households spend around 3% of their total proxy wealth in the present. The
aggregate consumption to wealth ratio rose during the great recession, as wealth shrankrelative to consumption, but fell during the stock market boom in the mid 2010s. Overthe whole sample, the aggregate consumption-wealth ratio fell.
If the EIS is less than one, then a decrease in the compensated consumption-wealthratio implies that the dynamic cost-of-living index is rising more quickly than traditional
inflation. If we assume that all households are identical (i.e. no lifecycle considerations), markets are complete, and preferences are homothetic, then the compensatedconsumption-wealth ratio is equal to the aggregate consumption-wealth ratio depicted in
Figure 1. This would suggest that inflation according to the dynamic cost-of-living index
should be higher than inflation according to static indices. However, in the next section,
we show that once we account for non-homotheticity, incomplete markets, and life-cycle
dynamics, dynamic cost-of-living inflation is lower than static inflation for the majority
30

We abstract from other durable consumer goods, for which the user cost would have to be estimatedin a similar way.

32

Figure 1: Aggregate consumption wealth ratio over time

aggregate consumption wealth ratio

0.036

0.035

0.034

0.033

0.032

0.031

2006

2008

2010

2012
2014
Year

2016

2018

2020

Notes: Aggregate consumption is the sum of consumption expenditures for all households in our sample.
Aggregate wealth is the sum of proxy wealth for all households in our sample.

of households.
Since Proposition 4 is only valid for rentiers, we must separate households into rentiersand non-rentiers. We say that a household is a rentier if net financial assets constitute morethan 90% of proxy total wealth. We also exclude from the rentier set those households
whose net assets are in the top and bottom 2.5%.31 Since there are few rentiers younger
than 40 years old in our sample, we do not report estimates for households below 40.

6.3

Money Metric Wealth for Rentiers

Figure 2 shows a scatterplot of log consumption wealth ratios against log wealth forrentiers in the first and last year of the sample. Both panels of Figure 2 show a strong
decreasing relationship between consumption-wealth ratios and wealth. The relationship is approximately loglinear. This is consistent with the finding in Straub (2019)that the consumption-wealth ratio strongly declines in permanent income. Whereas
static non-homotheticity is a cornerstone of the literature on consumption, dynamic nonhomotheticity, like the one depicted in Figure 2, is relatively understudied. However, as
our estimates of money-metric wealth show, in our dataset, dynamic non-homotheticity
is an order of magnitude more powerful than static non-homotheticity.
31

Our results are similar if we further restrict the set of potential rentiers to those whose financial assets
constitute more than 95% of proxy total wealth.

33

Figure 2: Log consumption-wealth against log wealth for rentiers
0.0

0.0
−1.0

−1.0

−2.0

−2.0

−3.0

−3.0

−4.0

−4.0

−5.0

−5.0

−6.0
−7.0

8

10

12

14

−6.0

16

(a) 2005

8

10

12

14

16

(b) 2019

To compute money-metric values for rentiers using Proposition 4, we need to evaluate
consumption-wealth ratios and static budget shares as a function of date and wealthfor each age group. To do this, we non-parametrically regress the consumption-wealthratios on log wealth, age group, and year using kernel regression (npregress in STATA).
Similarly, for each good i ∈ N, we regress the budget share on a quadratic function of age,log wealth, and year using LASSO. In both cases, the smoothing parameters are chosen
by cross-validation.32
We apply Proposition 4 to our estimated cross-sectional curves to recover moneymetric utility as a function of date, wealth, and age group.33 For illustration, we use theinitial year, t0 = 2005, as the base year, so that money-metric wealth values map nominal
wealth in each year t into equivalent wealth in 2005. For our benchmark results, we set
the EIS, σ, equal to 0.1, the benchmark estimate from Best et al. (2020). Best et al. (2020)
estimate that the EIS is relatively homogeneous in the cross-section of households, withpoint estimates that are uniformly between 0.05 and 0.15 across different quartiles of age
and income.
Unfortunately, Best et al. (2020) do not estimate Hicksian elasticities but Marshallian
32

We use npregress since it offers a disciplined way to trade off non-parametric fit as a function of age,
wealth, and time against overfitting. For the static budget shares, we use LASSO instead of npregress, since
LASSO is computationally cheaper than npregress, and makes bootstrapping much less time consuming.
Although we do not report it, our point estimates of the money-metric are very similar for all age groups
and wealth levels if we use npregress rather than LASSO to fit the static budget shares.
33
To recover the money-metric, we need to solve the integral equation in Proposition 4. To do so, we use
the “recursive” methodology described in Baqaee et al. (2024).

34

ones. Luckily, theoretically, Slutsky’s equation implies that, if consumption is a normal
good, then the Hicksian intertemporal elasticity should be smaller in magnitude than the
Marshallian one. Since the elasticity is bounded below by zero, we experiment with lower
values of σ in the appendix and find that our results are not sensitive to using values of σ
lower than 0.1. Specifically, our results are similar if we set σ = 0.2 or σ = 0.05. Of course,
the results are highly sensitive to the assumption that σ is not close to one.34
The left column of Figure 3 plots the money-metric, for each age group, as a functionof 2019 wealth for 2005 base prices. For comparison, we also plot a naive calculationthat deflates nominal wealth in 2019 by the official CPI inflation between 2005 and 2019.
The confidence bands are calculated by bootstrap. Since there are fewer young rentiers,
sampling uncertainty is higher for younger age groups. In all cases, the dynamic moneymetric wealth is higher than wealth deflated by CPI.
The right column of Figure 3 plots the dynamic annualized inflation rate (log difference
between nominal wealth in 2019 dollars and money-metric wealth in 2005 dollars). CPI
inflation over this period was 2%, but the dynamic inflation rate is always below 2%.
Furthermore, the dynamic inflation rates vary as a function of wealth and age highlighting
the importance of non-homotheticity and lifecycle considerations.
To understand the pattern in Figure 3, we decompose the dynamic inflation rate, thedeflator in equation (17), into a static and a forward-looking part. Specifically, for ahousehold with wealth w in 2019, the change in the ideal cost-of-living index between
2005 and 2019 is:w=
logu(2019, w, 0|2005)

Z 2019 X
2005|

Bn (x, w∗x , 0)

n∈N

{z

!
d log pn
BP (2019, w, 0)
1
dx +
log P
,
dx
σ−1
B (2005, w∗2005 , 0)
{z
}
} |

static chained inflation

future relative to static inflation

where w∗x ensures that we are using compensated consumption-wealth ratios and budget
shares. The first summand is a static measure of inflation. The second summand is related
to expected future inflation relative to present inflation. If the second term is positive,then the rate at which the price of the future bundle changes is higher than the rate at
which the price of the present bundle changes.
As an example, this decomposition is shown in Figure 4 for the 60 − 69 year oldgroup. The static inflation term is not exactly the same as aggregate CPI because it weighs
34

Even though in this paper we do not estimate the compensated EIS, we note that in principle it can be
estimated using changes in present prices, without knowledge of unobserved future prices and beliefs (fora related discussion, see Proposition 6 in Baqaee et al. (2024).

35

Figure 3: Conversion of 2019 dollars to 2005 dollars

$1.2M

4
money metric wealthwealth deflated by static inflation

Annualized Inflation (p.p.)

$1.6M
$1.4M

Wealth in 2005

$1M
$0.8M
$0.6M

$0.4M

$0.3M

$1M

2

0

−2

−4
$0.3M

$3.5M

$1M

$3.5M

Wealth in 2019

Wealth in 2019(a) money-metric 40 − 49 year olds(b) annualized inflation 40 − 49 year olds
4

$10M

Annualized Inflation (p.p.)

Wealth in 2005

money metric wealthwealth deflated by static inflation

$1M

$0.3M

$1M

2

0

−2

−4
$0.3M

$3.5M

$1M

$3.5M

Wealth in 2019

Wealth in 2019(c) money-metric 50 − 59 year olds(d) annualized inflation 50 − 59 year olds
32Annualized Inflation (p.p.)

Wealth in 2005

money metric wealthwealth deflated by static inflation

$1M

1
0
−1
−2

$0.3M

$1M

−3
$0.3M

$3.5M

$1M

$3.5M

Wealth in 2019

Wealth in 2019(e) money-metric 60 − 69 year olds(f) annualized inflation 60 − 69 year olds

36

4

$10M

Annualized Inflation (p.p.)

Wealth in 2005

money metric wealthwealth deflated by static inflation

$1M

2

0

−2

−4
$0.1M
$0.3M

$1M

$0.3M

$3.5M

(g) money-metric 70 − 79 year olds

4

Annualized Inflation (p.p.)

money metric wealthwealth deflated by static inflation

Wealth in 2005

$3.5M(h) annualized inflation 70 − 79 year olds

$10M

$1M

$0.1M
$0.3M

$1M
Wealth in 2019

Wealth in 2019

$1M

2

0

−2
−4
−6
$0.3M

$3.5M

$1M

$3.5M

Wealth in 2019

Wealth in 2019(i) money-metric 80 − 89 year olds(j) annualized inflation 80 − 89 year olds

Notes: Shaded area depicts 90% confidence intervals from bootstrap.

Figure 4: Decomposing inflation for 60-69 Figure 5: Changes in log consumption
wealth ratios for 60 − 69 year olds
year olds
0.32

Annualized Inflation (p.p.)

2.0

compensateduncompensated

0.30
0.28

1.0

0.26
0.24

0.0

0.22
−1.0

−2.0
$0.3M

0.20

Dynamic Inflation
Static Inflation
Future Inflation
$1M

0.18

$3.5M

0.16
$0.3M

Wealth in 2019

37

$1M

$3.5M

changes in static prices using compensated budget shares rather than aggregate budget
shares. Nevertheless, the static component is very close to aggregate CPI inflation ataround 2% per year for all wealth levels. The very slight downward slope reflects thenon-homotheticity of static preferences, and static inflation is slightly higher for poorer
households, consistent with other studies of the US, like Jaravel and Lashkari (2024),
that show that the static cost-of-living index has tended to rise more quickly for poorer
households. Nevertheless, the slope of the static inflation line is mild compared to theslope of the dynamic inflation measure.35
The component corresponding to future inflation in Figure 4 is not zero, which meansthat future inflation is not equal to static inflation. For all wealth levels, future prices are
expected to rise by less than static prices, explaining why the dynamic all-encompassing
cost-of-living index is below the static inflation line in Figure 4. Moreover, the future
inflation term exhibits more dependence on wealth than the static term.
The future component of the dynamic inflation measure is proportional to the compensated change in the log consumption-wealth ratio. Figure 5 plots both the compensatedand uncompensated log change in the consumption-wealth ratio between 2005 and 2019as a function of nominal wealth for 60 − 69 year olds. Since compensated consumptionwealth ratios rose for all wealth levels, the dynamic inflation rate is lower than the staticone.
The uncompensated change in the consumption wealth ratio is more positive than thecompensated one. This is because there is a strong wealth effect whereby the consumption
wealth ratio declines as households become richer. For a given nominal level of wealth,households in 2005 are on a higher indifference curve than households with that same
nominal level of wealth in 2019 because of positive inflation. Therefore, the wealth effectmeans that such households would have higher consumption wealth ratios in 2019 thanin 2005, even if relative prices do not change. The changes in compensated consumption
wealth ratios, which are purged of wealth effects, are lower and reflect only substitution
effects. This figure underscores the importance of accounting for wealth effects whenusing consumption wealth ratios to infer changes in relative prices.36
35

There may be several reasons why the contribution of static non-homotheticity is so mild in our exercise.
First, this sample is limited to rentiers — this means that we are looking at a relatively rich set of households
compared to studies focusing on static inflation, which typically include very poor households in the sample.
Second, we construct a price index as a function of wealth, rather than as a function of current expenditures,
as is done in static studies of the cost-of-living. Third, our sample period of fourteen years is reasonably
short compared to previous studies, which compute changes over 50 years or longer. Finally, we have only
seven spending categories, and static non-homotheticities may be stronger at more disaggregated levels.
36
Note that changes in the compensated and uncompensated consumption-wealth ratios are different tochanges in the consumption-wealth ratio at the individual level. The reason is that the compensated ratio
holds utility constant and the uncompensated ratio holds nominal wealth constant, while at the individual

38

Our methodology does not identify which future prices or beliefs are responsible forthe patterns in Figure 3. However, the differences in the dynamic measure of inflationneed not be caused by differential exposures to future goods prices alone. Even if allhouseholds are symmetrically exposed to future goods prices, the future component ofinflation can differ across households because of differences in expected returns of assets(see Fagereng et al., 2022). For example, if poor and rich rentiers or young and old rentiers
are differentially reliant on, say, returns to real estate, equities, or bonds to finance theirconsumption, then changes in returns will differentially affect dynamic inflation rates forthese households.

6.4

Non-Rentiers

We now turn our attention to the remaining households — the non-rentiers. Proposition
4 does not apply to these households. To recover the money-metric for these households,
we rely on Propositions 5 and 6 instead. For each date τ we fit log money-metric values
to static budget shares and age group indicators for the rentiers:
log wh,τ = α′τ Xh,τ + errorh,τ ,(26)

where X are budget shares and age groups. We then use the predicted values from thisregression to impute money-metric wealth for the non-rentiers conditional on their budget
shares, age, and date.37
This is analogous to Hamilton (2001), and more recently Atkin et al. (2024), who use
relative budget shares within a subset of goods, in their case food, to infer changes in
welfare in a static context. Unlike Atkin et al. (2024), who compare relative budget sharesacross time (adjusted for substitution effects) to infer changes in money-metric income
over time, we compare relative budget shares within each period across rentier and nonrentier households. Since rentiers and non-rentiers face the same relative prices at eachpoint in time, we do not have to correct relative budget shares for substitution effects andcan deduce money-metric wealth for non-rentiers from the rentiers.
Figure 6 displays the distribution of money-metric wealth, in 2019, for all age groups.
Figure 7 plots nominal log money-metric wealth against nominal capitalized (proxy)level both are changing over time. In particular, this implies that the aggregate consumption to wealth ratio
displayed in Figure 1 is not informative about changes in either the compensated or uncompensated ratio.
37
We also consider an additional robustness check where we exclude from the set of rentiers outliers in
regression (26). That is, if a potential rentier’s predicted and measured total wealth differ significantly, as
measured by a Cook’s distance value greater than one, then we exclude these households from the set of
rentiers. The results are very similar.

39

Figure 6: Distribution of money-metric wealth in 2019
1400
1200

number

1000
800
600
400
200

*

*

0

$0.1M

$1M

$10M

Notes: Bars with asterisk are households whose estimated money-metric wealth in $2019 dollars is outsideof the wealth range of rentiers in that year.

wealth for rentiers and non-rentiers. If financial markets are complete and absent measurement error, these two figures should be 45 degree lines.38
Figure 7: Capitalized (proxy) wealth against money-metric wealth (in logs) for all years(a) Rentiers

7(b) Non-rentiers

Treatment Effect on Welfare

Many policies and shocks affect households along many margins simultaneously. For
example, job training programs, changes in tax policy, changes in monetary policy, or job
38

See Appendix C for more details.

40

loss all plausibly have dynamic effects on many different relevant variables for households.
For example, Del Canto et al. (2023) show that monetary policy shocks affect households
through goods price inflation, labor market outcomes, changes in equity prices, house
prices, bond prices, and so on.
To understand the welfare effect of such shocks, researchers can estimate the dynamic
effects of the shock on each of the different relevant variables and then use changes
in those variables, weighted by predicted pre-shock household behavior, to calculate
the welfare effect. This is the approach taken by Del Canto et al. (2023). Other thanrequiring the researcher to enumerate, measure, and estimate all the relevant variables
through which the shock affects households, the resulting welfare estimates are first-orderapproximations around perfect-foresight allocations.
Our methodology provides a complementary approach. Instead of enumerating andestimating all potentially relevant margins, we back out the component of welfare that
depends on expectations about the future from observed changes in consumption-savings
behavior. We illustrate this by studying the welfare effects of job loss using the PSID. We
regress log money-metric wealth for households on a dummy variable for job loss for thehead of the household. Our measure of job loss is equal to one if the head of householdloses her job and reports that she is searching for a new job in that period. To control for
confounds and selection, we include year fixed effects, demographic controls, and lagged
log money-metric wealth.
Table 1 reports the results. The outcome in the first two columns is log nominal
money-metric wealth, log u(τ, w, y|τ). Column (1) shows that job loss is associated with
an approximately 20 log point reduction in nominal money-metric nominal wealth. Thiseffect is statistically significant and economically large. Column (2) shows that the effects
are much weaker for older workers (above 60 years old). The identifying variation in these
regressions comes from differences in static budget shares between households who losttheir job and those who did not, given other controls. The spending patterns of households
that lose their job differ from those that did not lose their job in a way that, had they beenrentiers, their total wealth would be 20% lower. For older household heads, differencesin wealth associated to job loss are smaller compared to younger households. The impact
of job loss on wealth is much smaller than the effect on measures of contemporaneous
household income (which is around 85% lower for households that lost their job).
Columns (5) and (6) repeat these regressions but use log money-metric wealth in 2019
dollars, log u(τ, w, y|2019), as the dependent variable instead of log nominal money-metric,log u(τ, w, y|τ). Since we do not estimate the 2019 money-metric for the full sample, thelast two columns are for the restricted subset of households where we can infer the money41

Table 1: Log money-metric wealth and job loss

log nominal money metric

Job Loss(1)-0.197(0.031)

Job loss × 1(age ≥ 60)(2)(3)-0.218 -0.099(0.034) (0.031)

log money metric2019 dollars
(4)
(5)
(6)
-0.108 -0.098-0.107(0.038) (0.031) (0.038)

0.180(0.083)0.050(0.080)0.045(0.074)Lagged LHS

Yes

Yes

Yes

Yes

Yes

Yes

Controls
Observations

Yes
48,357

Yes
48,357

Yes
33,987

Yes
33,987

Yes
33,987

Yes
33,987Full Sample

Overlapping Support

Notes: Standard errors are bootstrapped. Controls are year fixed effects, age group, marital status of headof household, industry, and education level.

42

metric in 2019 dollars without extrapolation. We do not estimate log u(τ, w, y|2019) for
the full sample because, as discussed above, Proposition 4 can only be applied inside asuitable boundary (without extrapolation). The point estimates for this restricted subset
are about half as large. The reason is that the sample of households for which we cancalculate the money-metric wealth in 2019 dollars is, on the whole, much older. Since older
households are less affected by job loss, when we restrict the sample, we find smaller point
estimates. To show this, columns (3) and (4) repeat the same regression as in columns (1)and (2) but restrict the sample to be the same as the one in columns (5) and (6). We findthat the point estimates are almost exactly the same. The fact that the point estimates in(3) and (4) are similar to those in (5) and (6) suggests that there is no correlation between
job loss and welfare-relevant inflation rates. Hence, the point estimates for nominal andreal losses are similar. If job loss is uncorrelated with welfare-relevant inflation rates forthe whole sample, then we can extrapolate the point estimates in (1) and (2) to apply tomoney-metric wealth in 2019 dollars for the whole sample as well.
An alternative approach in the literature for estimating the dynamic consequencesof job loss is due to Davis and Von Wachter (2011). They estimate the present-valueof earnings losses after mass-layoff events to be around 12% of counterfactual earnings(using a 5% discount rate). Our point estimates are larger than the present-value earnings
losses estimated by Davis and Von Wachter (2011). The fact that our point estimates are
somewhat different to theirs is not surprising, because even with the same data it is only
under very strong assumptions that our estimates should coincide with their present-value
calculation.
For example, our calculation does not assume complete markets and if the household’s
marginal utility is high in states where earnings are low, as is probably realistic, then aconstant discount factor understates the welfare losses of job loss. Furthermore, Davis and
Von Wachter (2011) estimate ex-post earnings losses whereas we estimate ex-ante welfare
losses, and households’ ex-ante beliefs about the consequences of job loss may be more
pessimistic than the outcomes Davis and Von Wachter (2011) estimate. Additionally, we
do not assume exponential discounting — if households are present-biased, then welfare
losses from job loss are amplified since households care more about the near-term, whenearnings are low. Finally, they focus on mass lay-off events whereas we consider any
job loss. It is plausible that the welfare losses associated with unconditional job loss aredifferent to those caused by mass lay-offs. Despite all these differences, the estimatesfrom Davis and Von Wachter (2011) provide some quantitative context about how themagnitude of our estimates compare to the standard empirical approach.

43

8Conclusion

We provide a methodology for measuring welfare and cost-of-living for households accounting for dynamics, uncertainty, market incompleteness, borrowing constraints, andnon-homotheticities. Our methodology requires repeated household consumption, income, and wealth surveys, as well as prices. The key assumptions we make are that
preferences are intertemporally separable and, given observable characteristics, all households have common preferences and face the same prices and beliefs in each period. To
calculate money-metrics and cost-of-living, we also require that some subset of households face negligible idiosyncratic undiversifiable income risk (e.g. have negligible riskylabor income). Our approach provides a way to non-parametrically measure dynamic
welfare without fully specifying a structural model. This makes it useful for reducedform analysis of the welfare effect of shocks that have dynamic stochastic effects on many
variables.

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A

Proofs

Proof of Proposition 2. Since ∂V/∂w > 0 as long as p(τ0 ) , 0 and R)(τ0 ) > 0, u is monotone
increasing in V.
■
Proof of Proposition 3. The existence of q∗ follows from the separating hyperplane theorem,
since the constraint set and indifference curves are both convex (the constraint set is anintersection of convex sets). Furthermore, since the solution is a convex optimization
problem, the Karush-Kuhn-Tucker conditions must be satisfied. The Lagrangian forhouseholds is:XX
ak (s0 |τ) − w]
L(p, R, y, π, w) = U(c, π) − λ(s0 |τ)[pn (s0 |τ)cn (s0 |τ) +
n∈N

k∈K


X
X
X


+
λ(st |τ) pn (st |τ)cn (st |τ) +ak (st |τ) −Rk (st |τ)ak (st−1 |τ) + y(st |τ)
n∈N
k∈K
k∈K
st


X
X

−
µ(st |τ) ak (st |τ) − X(st |τ)
kstX= U(c, π) + λ(s0 |τ)w +λ(st |τ)y(st |τ)

X

st

47

−
−
−

sT
X
st =s0
sT
X
0

s =s
X
t

st

λ(st |τ)X

λ(st |τ)Xµ(s |τ)
t

n∈N

k∈K
X
k

pn (st |τ)cn (s0 |τ)ak (st |τ) +ak (s |τ) +
t

X
t

s
X
st

λ(st |τ)Xk∈K

Rk (st |τ)ak (st−1 |τ)µ(s |τ)X(st |τ)t

The first order conditions for asset holdings arehi X− λ(st |τ) + µ(st |τ) =λ(st+1 |τ)Rk (st+1 |τ)
st+1

Substituting this back in, we get that the Lagrangian is equal to0L(p, R, y, π, w) = U(c, π) + λ(s |τ)w +Xstλ(s |τ)y(s |τ) −
t

t

sT
X
st =s0λ(s |τ)
t

X
n∈N

0

pn (s |τ)cn (s |τ) +tXstµ(st |τ)X(st |τ).

Define the indirect utility function to be v that satisfies this equation:e(q, π, v) = W.
From standard duality, we know that we can also writev(q, π, W) = max{U(c, π) : q · c = W}.
c

Call the maximizers above c∗∗ (q, π, W). The Lagrangian for intertemporal indirect utility
function isL∗∗ (q, π, W) = U ({c, π}) − µ q · c − W .
Setλ(st |τ)pn (st |τ)qn (s ) =0λ(s |τ)
t

and
W =w+

X λ(st |τ)stλ(s0 |τ)y(s |τ) +t

X µ(st |τ)stλ(s0 |τ)X(st |τ)HenceX λ(st |τ)X µ(st |τ)X X λ(st |τ)


t
t
t
t
 .
L∗∗ (q, π, W) = U ({c, π})+µ w +
y(s
|τ)
+
X(s
|τ)
−
p
(s
|τ)c
(s
|τ)
n
n

λ(s0 |τ)λ(s0 |τ)λ(s0 |τ)
st

st

48

st

n∈N

These problems have the same solution because the Lagrangian is the same. Hencec∗∗ (q, π, W) = c(st |τ, w, y),where qn (st ) = λ(st |τ)pn (st |τ) and W = λ(s0 |τ)w +standard duality arguments, we also know thatPtt
st λ(s |τ)y(s |τ) +Ptt
st µ(s |τ)X(s |τ).

Byc∗∗ (q, π, W) = c∗ (q, π, v(q, π, W)) = c∗ (q, π, V(q, π, W)).
■
Proof of Proposition 4. For the proof, we define the following function:bn (q(s0 ), U) =cn (s0 )qn (s0 ).
e(q, π, U)bP (q, π, U)

We proceed in steps, using a series of lemmas.
Lemma 2. If preferences are time separable, then the following holds
X cn (s0 )qn (s0 )=∂ log e(q, π, U),∂ log PbF (q, π, U) ≡ 1 − bP (q, π, U) =∂ log e(q, π, U),∂ log FbP (q, π, U) ≡

and

n∈N

e(q, π, U)∂ log Pcn (s0 )qn (s0 )=.
∂ log qn (s0 ) e(q, π, U)bP (q, π, U)

Proof. By the envelope theorem,∂ log e(q, π, U) cn (s0 )qn (s0 )=,e(q, π, U)∂ log qn (s0 )and for t > 0∂ log e(q, π, U) cn (st )qn (st )=.
e(q, π, U)∂ log qn (st )Then, we know thatb (q, π, U) =P

X cn (s0 )qn (s0 )n∈Ne(q, π, U)=X ∂ log e(q, π, U)n∈N∂ log qn (s0 )49∂ log e(q, π, U) X ∂ log P∂ log e(q, π, U)==∂ log P
∂ log qn (s0 )∂ log Pn∈NandbF (q, π, U) =X X cn (st )qn (st )st |t>0 n∈N==e(q, π, U)

X X ∂ log e(q, π, U)
st |t>0 n∈N∂ log qn (st )∂ log e(q, π, U)∂ log e(q, π, U) X X ∂ log F=0∂ log F∂ log qn (s )∂ log Ftn∈Ns |t>0where the last steps use homogeneity of degree 1 in q of P and F.
Next, we show that∂ log P= bn (q(s0 ), U).
∂ log qn (s0 )To do this, use the following equality,∂ log e(q, π, U) ∂ log e(q, π, U) ∂ log P=∂ log qn (s0 )∂ log P
∂ log qn (s0 )cn (s0 )qn (s0 )=e(q, π, U)∂ log P
= bP (q, π, U).
∂ log qn (s0 )Rearranging yields∂ log Pcn (s0 )qn (s0 )=.
∂ log qn (s0 ) e(q, π, U)bP (q, π, U)■
Lemma 3. When preferences are time separable, the elasticity of intertemporal substitutionσ∗ (q, π, U) = 1 −is given by

X ∂ log bp (q, π, U)/bF (q, π, U)∂ log qn (s0 )

n∈N

.

2
∂2 log e/ ∂ log P
σ (q, π, U) = 1 − F.
b (q, π, U)bP (q, π, U)∗

Proof. We start withX ∂ log e(q, π, U) ∂ log P 
∂ log bP (q, π, U)
1
∂

 ,
= P


b (q, π, U) ∂ log qn (s0 ) ∂ log qn (s0 )∂ log P
∂ log qk (s0 ) 
k∈N

50



X ∂ log e(q, π, U)

1
∂

 ,
b
(q,
U)
= P
k


b (q, π, U) ∂ log qn (s0 ) ∂ log Pk∈NX ∂ log e ∂bk (q, U) 
X
∂
log
e(q,
π,
U)
1
∂

 ,
= P
bk (q, U) +b (q, π, U) 
∂ log qn (s0 )∂ log P
∂ log P ∂ log qn (s0 ) 
k∈N
k∈N


P
X ∂2 log e(q, π, U)

∂
log
e
∂b
(q,
U)
1
k
k∈N

 ,
= P
b
(q,
U)b
(q,
U)
+
n
k


2
0
b (q, π, U)(∂ log P)∂ log P ∂ log qn (s ) 
k∈N


X
 ∂2 log e(q, π, U)

1

 ,
= P
b
(q,
U)
b
(q,
U)
n
k

b (q, π, U)  (∂ log P)2
k∈N

=

2

∂ log e(q, π, U)
1
bn (q, U).
bP (q, π, U) (∂ log P)2

Summing over all n ∈ N yields
X ∂ log bP (q, π, U)n∂ log qn (s0 )=∂2 log e(q, π, U) X
1bn (q, U),bP (q, π, U) (∂ log P)2n=∂2 log e(q, π, U)
1.
bP (q, π, U) (∂ log P)2Since bP + bF = 1,we have that∂ log bF (q, π, U)
bP (q, π, U) ∂ log bP (q, π, U)=− F.
b (q, π, U) ∂ log qn (s0 )∂ log qn (s0 )By definition,
1 − σ(q, π, U) =X ∂ log bP (q, π, U)n∂ log qn (s0 )−X ∂ log bF (q, π, U)n∂ log qn (s0 ),∂2 log e(q, π, U) bP (q, π, U) ∂ log bP (q, π, U)
1
= P
+ F
,
b (q, π, U) (∂ log P)2b (q, π, U) ∂ log qn (s0 )∂2 log e(q, π, U)
1.
= F
b (q, π, U)bP (q, π, U) (∂ log P)2■
Lemma 4. When preferences are time separable, the following equation holds:∂ log e(q, π, U)∂ log e(q, π, U)d log bP (q, π, U) X
d log q +dπ = −+bn (q(s0 ), U)d log qn (s0 )1 − σ∗ (q, π, U)∂ log q∂πn∈N51

Proof. From Lemma 2, we know thatX
∂ log e(q, π, U)∂ log e(q, π, U)d log q +dπ = bP (q, π, U)
bn (q, U)d log q(s0 )∂ log q∂πn∈NX X ∂ log F∂ log F

t
t 
F
 .
d
log
q
(s
)
+
dπ(s
)
+b (q, π, U)
n


t)
t)
∂
log
q
(s
∂π(s
n
t
n∈N
s |t>0

Next, from homogeneity of degree one, we know that∂ log e(q, π, U) ∂ log e(q, π, U)+= 1.
∂ log P
∂ log F
Differentiating this identity with respect to P and F yields the following equation∂2 log e(q, π, U)∂2 log e(q, π, U) ∂2 log e(q, π, U)
=−
=
2
2 .
∂ log P∂ log F∂ log P
∂ log F
Hence, fixing utility, the total derivative of bP (q, π, U) with respect to q and π is
∂2 log e(q, π, U) X ∂ log P
d log qn (s0 )0)(∂ log P)2
∂
log
q
(s
n
n∈N


2
X
X ∂ log F∂ log e∂ log F
t
t


+
d
log
q
(s
)
+
dπ(s
)
n

∂ log F∂ log P t  n∈N ∂ log qn (st )∂π(st )
s |t>0P
bn (q, U)d log qn (s0 )−∂2 log e(q, π, U) 
n∈N

 P
=


 .
P

∂ log F∂ log Ftt(∂ log P)2
d log q (s ) +dπ(s ) tbP d log bP (q, π, U) =s |t>0nn∈N ∂ log qn (st )∂π(st )

(27)

From Lemma 2 and Lemma 3, we can rewrite this as
d log bP (q, π, U)

P
=
(1
−
b
(q,
π,
U))
 P

(1 − σ∗ (q, π, U))tPs |t>0

P





 ,
∂ log F∂ log Fttd log q (s ) +dπ(s ) 0n∈N bn (p, U)d log qn (s )−n∈N ∂ log qn (st )n∂π(st )

Rearranging this givesPbP (q, π, U) n∈N bn (p, U)d log qn (s0 ) − bF (q, π, U))×d log bP (q, π, U) X=−+
bn (q(s0 ), U)d log qn (s0 ).
PP∗ (q, π, U)∂ log F∂ log F
t
t
1
−
σ
n∈N
st |t>0n∈N ∂ log qn (st ) d log qn (s ) + ∂π(st ) dπ(s )52Plug this back into (27) to get the desired result.

■Lemma 5. The shadow prices q∗ (τ, w, 0) can be written as a function of τ and V(τ, w, 0). That is,
we can writeq∗ (τ, w, 0) = q∗ (τ, V(τ, w, 0)) .
Furthermore,V(τ, w, 0) = v(q∗ (τ, V(τ, w, 0)) , π(·|τ), w)
for every τ and w.
Proof. The first part follows from the fact that the value function V(τ, w, 0) is monotone in
w. Hence, we can substitute the inverse of V(τ, w, 0) with respect to w into q∗ (τ, w, 0) to getq∗ (τ, V(τ, w, 0)) = q∗ (τ, V −1 (V(τ, w, 0)), 0).
For the second part, we know from Proposition 3, thatc(τ, w, 0) = c∗ (q∗ (τ, V(τ, w, 0)), π(·|τ), V(τ, w, 0)).
Hence
V(τ, w, 0) = U(c(τ, w, 0), π(·|τ))= U(c∗ (q∗ (τ, V(τ, w, 0)), π(·|τ), V(τ, w, 0)), π(·|τ)= v(q∗ (τ, V(τ, w, 0)) , π(·|τ), w).

■
Lemma 6. The following holdse(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0)) = w.
Proof. From the proof of Proposition 3, we know thatXX
e(q∗ τ, u(τ, w, y) , π(·|τ), u(τ, w, y)) = w +λ(st |τ)y(st |τ) +µ(st |τ)X(st |τ),ststwhere λ(st |τ) are lagrange multipliers on state-contingent budget constraints and µ(st |τ)are lagrange multipliers on borrowing constraints. Since y(st |τ) = 0, we know thatXe(q∗ τ, u(τ, w, y) , π(·|τ), u(τ, w, y)) = w +µ(st |τ)X(st |τ).
st53

We prove the desired result by showing that µ(st ) ≡ 0. To do this, we use backward
P′
induction. Suppose that for some t, we know that, for every t′ > t, we have k ak (st |τ) ≥ 0.
′
That is, the borrowing constraint is slack for every st following st . For the sake of deriving
a contradiction, suppose that µ(st |τ) , 0. Then
X
n∈N

pn (s

t+1

|τ)cn (s

t+1

|τ) +

X
k

ak (s

t+1

|τ) =

X
k∈K

Rk (s |τ)ak (s
t

t−1



T
|τ) < − min Rk (s |τ) X(sT−1 |τ) < 0.
kThis implies thatXkak (st+1 |τ) < 0,which is a contradiction. Hence, we know thatXkak (st+1 |τ) ≥ 0.

This implies that µ(st |τ) = 0. We finish by observing that we know that for every sT , theno-Ponzi scheme condition implies thatXkak (sT |τ) ≥ 0.

This is the first step of the backward induction.
■
With these preliminaries out of the way, we are ready to prove Proposition 4. We startwith the definition of the money-metric. That is, u(τ, w, 0) solves the following equation:V(τ, w, 0) = V(τ0 , u(τ, w, 0), 0).
From Lemma 5, we knowv(q∗ (τ, V(τ, w, 0)) , π(·|τ), w) = V(τ, w, 0) = V(τ0 , u(τ, w, 0), 0) = v(q∗ (τ0 , V(τ, w, 0), 0) , π(·|τ0 ), u(τ, w, 0)).
Hence, u(τ, w, 0) solvesv(q∗ (τ, V(τ, w, 0)) , π(·|τ), w) = v(q∗ (τ0 , V(τ, w, 0), 0) , π(·|τ0 ), u(τ, w, 0)).
Without loss of generality, by Proposition 2, cardinalize the value function using themoney-metric (since the value function is only defined up to monotone transformations).

54

Thereforev(q∗ (τ, u(τ, w, 0)) , π(·|τ), w) = v(q∗ (τ0 , u(τ, w, 0), 0) , π(·|τ0 ), u(τ, w, 0)).
Using the shadow expenditure function, we can writeu(τ, w, 0) =e(q∗ (τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0)),e(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0)),e(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0))e(q∗ (τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0)),=e(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0))e(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0))e(q∗ (τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0))=w,e(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0))=e(q∗ (τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0))

where the last line uses Lemma 6. Logging both sides givese(q∗ (τ0 , u(τ, w, 0)) , π(·|τ0 ), u(τ, w, 0)),
log u(τ, w, 0) = log w + loge(q∗ (τ, u(τ, w, 0)) , π(·|τ), u(τ, w, 0))Z τ0∂ log e(q∗ (x, u(τ, w, 0)) , π(·|x), u(τ, w, 0)) d log q∗= log w +dx
∂ log q∗τ!
∗∂ log e(q (x, u(τ, w, 0)) , π(·|x), u(τ, w, 0)) d log π(·|x)
dx,+dx∂ log π(·|x)
where the second equality uses the fundamental theorem of calculus for line integrals.
Using Lemma 4, we can rewrite the last line asZ τ Xd log pn
log u(τ, w, 0) = log w −bn (p(·|x), u(τ, w, 0))dx
τ0 n∈N!
d log bP (q∗ (x, u(τ, w, 0)) , π(·|x), u(τ, w, 0)) 1
dx,
+ ∗ ∗
σ (q (τ, u(τ, w, 0)) , π(·|x), u(τ, w, 0)) − 1 dx
Z τ Xd log pnd log BP (x, w∗x , 0) 1∗ dx.
Bn (x, wx , 0)
= log w −+dx
σ(x, w∗x , 0) − 1dx
τ0 n∈N
where for the last step, we replaced compensated budget share with uncompensatedbudget share.
■
Proof of Lemma 1. Need to show that
Bn (τ, w, y) = bn (p(s0 |τ), V(τ, w, y)).
55

By Proposition 3, we know that
Bn (τ, w, y) = P

q∗n (s0 )c∗n (s0 |q∗ , π, V(τ, w, y))pn (s0 |τ)cn (s0 |τ, w, y)P=≡ bn (q∗ , π, V(τ, w, y)).
0 |τ)c (s0 |τ, w, y)0 |τ)c∗ (s0 |q∗ , π, V(τ, w, y))
p
(s
q
(s
m
m
m
m
m∈N
m∈N

Next, we know, from Shephard’s lemma that for each n ∈ N
q∗n (s0 )c∗n (q∗ , π, V(τ, w, y)) ∂ log e(q∗ , π, V(τ, w, y))=e(q∗ , π, V(τ, w, y))∂ log q∗n (s0 )



 
∂ log e P q∗ (s0 ), V(τ, w, y) , F q∗ (st ) t>0 , π, V(τ, w, y) , V(τ, w, y)=∂ log P
∂ log P(q∗ (s0 ), V(τ, w, y)).
∂ log q∗n (s0 )Hence, we have thatq∗ (s0 )c∗n (s0 |q∗ , π, V(τ, w, y))
P n
=P
∗
0 ∗
0
m∈N qm (s |τ)cm (s |q , π, V(τ, w, y))∂ log P(q∗ (s0 ),V(τ,w,y))∂ log q∗n (s0 )∂ log P(q∗ (s0 ),V(τ,w,y))m∈N∂ log q∗m (s0 ),which is only a function of q∗ (s0 ) = p(s0 |τ) and V(τ, w, y) as needed.

■

Proof of Proposition 5. From Lemma 1, we know that
B(τ, w, y) = bn (p(s0 |τ), V(τ, w, y)).
By definition of u(τ, w, y|τ), it follows that
B(τ, w, y) = bn (p(s0 |τ), V(τ, u(τ, w, y|τ), 0)).
Since b is an injective function, we can write
0V(τ, u(τ, w, y|τ), 0)) = b−1
n p(s |τ), B(τ, w, y) .
Since V is monotone in wealth, we can write

 

0
p(s
|τ),
B(τ,
w,
y)
, 0 = m(B(τ, w, y), τ).
u(τ, w, y|τ) = V −1 τ, b−1n■56

Proof of Corollary 2. Lemma 1 shows that
Bi (τ, w, y) = bi (p(s0 |τ), V(τ, w, y)).
Hence, if bi is monotone in V, then
Bi (τ, w, y) = bi (p(s0 |τ), V(τ, w, y)) = bi (p(s0 |τ), V(τ, w∗ , 0)) = Bi (τ, w∗ , 0)if, and only if,V(τ, w, y) = V(τ, w∗ , 0).
■

B

Alternative Definition of Time Separability

In this appendix, we provide an alternative equivalent definition of time separability
that does not explicitly use the shadow intertemporal expenditure function. To do so,
consider some preference relation ⪰ with a utility function representation U(c, π). Define
the distance function to beD(c, π, U) = max{λ : U(c/λ, π) = U}.
λ∈R

The distance function is standard in duality theory and, by construction, is homogeneousof degree one in quantities and decreasing in U (as long as c are goods). See Blackorby
et al. (1998) or Cornes (1992) for textbook treatments.
The distance function carries all the same information as the utility function. In fact,
the utility function U(c, π) can be recovered from the distance function by solving for U
in the following identity
D(c, π, U) = 1.
The solution gives us U(c, π). This “implicit” representation of the utility function iscommon in the macroeconomics literature though it is rarely referred to as a distancefunction. For example, Kimball (1995) preferences and non-homothetic CES (e.g. Cominet al., 2021) preferences are usually defined via a distance function.
Theorem 4.1 in Blackorby et al. (1998) implies that separability in the expenditure
function, Definition 1, is equivalent to separability in the distance function. Specifically,
Definition 1 is equivalent to requiring that there exists a utility function representation of57

⪰ satisfying

D(c, π, U) = D̃(P̃(c(s0 ), U), F̃({c(st )}t>0 , π, U), U) = 1,

where P̃ and F̃ are homogeneous of degree 1 in c and D̃ is homogeneous of degree 1 inits first two arguments (all holding U constant) and D is decreasing in U. An example ofthis is provided in (12) where D̃, P̃, and F̃ are nested non-homothetic CES aggregators.
Specifically,σ
1  σ−1σ−1 σ−1P̃ σ + F̃ σ
D̃ =
,
U
where
γ
 γ−1

X
γ−1 



ωn0 Uεn cn (s0 ) γ  ,P̃ = Uϵ0 
n

and


γ σ−1  σ
 γ−1

σ−1
T
σ 
X
X
X
γ−1 



t εt
t 
εn
t γ 




F̃ = 
βUπ(s ) 
ωnt U cn (s ) 
 .


t=1

C

st

n

Data Construction

We use two different datasets. One is a household-level survey (PSID) and the other isdata on prices of different categories of goods (CPI). The PSID is a longitudinal survey,
interviewing households annually until 1997 and biennially thereafter. Each sample
includes about 7,000-9,000 households. We use seven spending categories and merge
them with CPI categories. We describe how we construct the variables needed for our
methodology below.
Net Assets:
The wealth module of the PSID tracks the value of components of household balance
sheets (business equity, stocks, mutual funds, bonds, automobiles, pensions, cash, etc.).
Home equity data are recorded as the value of a household’s home minus its mortgage
obligations. The PSID aggregates these variables, imputes missing values, and reports
the comprehensive variables WEALTH1 and WEALTH2. WEALTH1 represents wealthexcluding home equity, while WEALTH2 is the sum of WEALTH1 and home equity. As
Cooper et al. (2019) note, these measures exclude the value of defined-contribution (DC)account. We define net assets as WEALTH2 plus the value of DC account (recorded58

separately in the PSID) to incorporate as much of the household’s assets as possible.39
Capitalized wealth proxy:
We construct a proxy for total wealth by adding the capitalized value of labor income
and transfers to net assets. Define household income as labor income plus the variables
recorded as social security income and other welfare income. First, we estimate the agespecific income profile for each period τ using cross-sectional data. To do this, we regressa quadratic of the age of the head of household on log income controlling for household
characteristics (marital status, state of residence, race of household head, gender, andoccupation) and year fixed effects. We then use this regression to predict each household’s
income profile as their age increases. We inflate these predictions of the household’s
income in the future by an estimate of expected nominal per capita GDP growth. The
expected growth in nominal GDP comes from the Congressional Budget Office’s real-time(contemporaneous) forecast of nominal GDP growth and the population growth rate uses
realized population growth rates for the United States, assuming a constant growth after
2019. We discount these nominal income flows back to the present using a nominal rateof 6%, consisting of a 4% real rate, following Catherine et al. (2022), and a 2% expected
inflation rate. We assume that income flows are zero beyond age 90.
Owner-occupied housing:
For renters, we use the housing expenditures variable in the PSID (which includes utilities).
For owner-occupied housing, we impute housing costs by matching homeowners torenters using static budget shares in each period. This procedure should yield accurate
estimates as long as preferences are time separable.
Specifically, for each year, we run the following regression for renters:
housingh,τ =Xαi,τ spendingi,h,τ + β1,τ ageh,τ + β2,τ age2h,τ + stateFEh,τ + ϵh,τ ,i,housingwhere the left-hand side variable is expenditures on housing (including utilities), and covariates are households’ spending on non-housing categories, age, and state fixed effects.
We then use this regression to impute (predict) rental expenditures for homeowners basedon their age, spending on non-housing categories, and state of residence.
39

Cooper et al. (2019) report that adding DC account information to WEALTH2 generally matches thetotal assets reported in the Survey of Consumer Finances (SCF). If no value was provided and the value wasgiven in bins, the median household value between the bins was used for imputation.

59

In 2019, a new question was added to the PSID survey which asks the following:
If someone were to rent this (apartment/mobile home/home) today, how much do
you think it would rent for per month, unfurnished and without utilities?
We use the responses to this question to validate our procedure. A regression of the survey
values (including utilities) on our imputed values, both relative to current expenditures,
has a coefficient of 1.03 with an R2 value of 0.59. This suggests that our imputation
performs well.
Budget shares:
We align the seven categories of the PSID (food, housing, transportation, education, health,clothing, and recreation) with the CPI.40 As mentioned above, for homeowners, we imputehousing costs. The relative budget share is defined as the spending on each category
divided by total spending. We compute the consumption-wealth ratio of households by
dividing total spending in each year by wealth.

40

The corresponding codes for CPI are CPIFABSL, CPIHOSSL, CPITRNSL, CPIEDSL, CPIMEDSL, CPIAPPSL, and CPIRECSL, respectively. Education includes child care. Recreation includes Trips & vacations
and Recreation & entertainment in PSID.

60