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David R Baqaee, Ariel Burstein, Welfare and Output With Income Effects and Taste Shocks, The Quarterly Journal of Economics, Volume 138, Issue 2, May 2023, Pages 769–834, https://doi.org/10.1093/qje/qjac042
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Abstract
We present a unified treatment of how welfare responds to changes in budget sets or technologies with taste shocks and nonhomothetic preferences. We propose a welfare metric that ranks production possibility frontiers that differs from one that ranks budget sets and characterize it using a general equilibrium generalization of Hicksian demand. This extends Hulten’s theorem, the basis for constructing aggregate quantity indices, to environments with nonhomothetic and unstable preferences. We illustrate our results using both long- and short-run applications. In the long run, we show that if structural transformation is caused by income effects or changes in tastes, rather than substitution effects, then Baumol’s cost disease is twice as important for our preferred measure of welfare. In the short run, we show that standard chain-weighted deflators understate welfare-relevant inflation for current tastes. Finally, using the COVID-19 recession, we illustrate that chain-weighted real consumption and real GDP are unreliable metrics for measuring welfare or production when there are taste shocks.
I. Introduction
In this article, we study how a change in the economic environment affects welfare. For example, how does an individual’s welfare change when her budget constraint changes, or how does national welfare change when technologies change? If preferences are homothetic and there are no taste shocks, there is a well-known formula that answers these questions: the chain-weighted (Divisia) index of real consumption.1 Assuming that there are no income effects or changes in tastes is highly convenient, but also highly inconsistent with reality. In this article, we relax these assumptions and characterize changes in welfare, changes in chained-weighted consumption, and the gap between them in terms of measurable sufficient statistics.
To compare two different choice sets, indexed by t0 and t1, for a consumer with preferences ≽, we ask: “how much must the t0 endowment change to make ≽ indifferent between the two choice sets?” If the preference relations at t0 and t1 are different, due to changes in tastes, then one must take a stance on which preference relation is used for the comparison because the answer depends on this choice.2 Our baseline measure of welfare is the equivalent variation using t1 preferences, although we also characterize and report results for other welfare questions (i.e., compensating variation and t0 preferences) and explain how they differ.
We begin by studying this problem in partial equilibrium, taking prices as given, where our welfare metric compares and ranks different budget constraints. This welfare metric is appropriate for an infinitesimal agent whose choices do not alter prices. We call this a micro welfare metric. We then propose a generalization of money-metric measures that ranks production possibility frontiers (PPFs) and takes into account that prices are endogenous to collective choices. We call this a macro welfare metric. We show that comparing two PPFs using their corresponding equilibrium budget constraints, which is what standard micro money metrics do, can be misleading if preferences are nonhomothetic or unstable.
Calculating micro welfare requires integrating Hicksian (or compensated) demand curves with respect to prices. We prove that there are general equilibrium counterparts to Hicksian demand curves, which we call Hicksian sales, and show that integrating them with respect to technologies yields macro welfare. This general equilibrium integral also generalizes Hulten’s theorem to measure welfare in environments featuring taste shocks and nonhomotheticities. In this article, we focus on neoclassical economies with homogeneous agents, but a companion paper (Baqaee and Burstein 2021) generalizes our results to economies with heterogeneous agents and distortions.
We provide exact and approximate characterizations of the change in micro and macro welfare. In contrast to the standard chain-weighted consumption index formula, which weighs changes in prices or technologies using observed shares, welfare-relevant indices weigh changes in prices or technologies using Hicksian shares. We show that compared to our baseline welfare measure, this implies that chain-weighted consumption undercounts expenditure switching due to income effects or taste shocks (but not substitution effects).
To understand why chain-weighted indices undercount expenditure switching due to income effects and taste shocks, consider the following example. Over the postwar period, spending on healthcare grew relative to manufacturing. Since older and richer consumers spend more on healthcare, suppose this was caused by consumers getting older and richer. In this case, a chain-weighted consumption index does not correctly account for expenditure switching by consumers. Intuitively, when we compare the past to the present, we must use demand curves that are relevant for the older and richer consumers of today, and not demand curves that were relevant in the past. Whereas a chained deflator weighs changes in prices that happened during the 1950s using demand from the 1950s, a welfare-relevant index uses demand from today to weigh changes in prices throughout the sample. We show that the chain-weighted consumption index is higher than the welfare-relevant index if income- or taste-driven expenditure switching is positively correlated with changes in prices.
Our results for welfare and the gap between welfare and real consumption are expressed in terms of measurable sufficient statistics. In both partial and general equilibrium, we show that computing the change in welfare does not require direct knowledge of the taste shocks or income elasticities. Instead, what we must know are expenditure shares and elasticities of substitution at the final allocation. For micro welfare, these are the household’s expenditure shares and elasticities of substitution in consumption. For macro welfare, these are the input-output table and elasticities of substitution in production and consumption. Our results can be used for ex post accounting and ex ante counterfactuals.
For very simple economies with one factor, constant returns to scale, and no intermediates, the difference between welfare and chain-weighted consumption is approximately half the covariance of supply and demand shocks. We generalize this formula to more complex economies and show how the details of the production structure, like input-output linkages, complementarities in production, and decreasing returns to scale, can interact with nonhomotheticities and preference shocks to magnify the gap between welfare and chain-weighted consumption. Even if there are only idiosyncratic microeconomic shocks, the discrepancies between welfare and chain-weighted real consumption that we emphasize do not average out at the aggregate level.
We illustrate the relevance of our results for understanding long-run and short-run phenomena with three applications.
Long-run application:Baumol (1967) argues that aggregate productivity growth slows down if industries with relatively low productivity growth become larger as a share of the economy over time. To be specific, from 1947 to 2014, aggregate total factor productivity (TFP) in the United States grew by |$60\%$|. If the U.S. economy had kept its original 1947 industrial structure, then aggregate TFP would have grown by |$78\%$| instead. We show that if this transformation is caused solely by income effects and demand instability, rather than substitution effects, then our baseline measure of welfare-relevant TFP grew by only |$47\%$| instead of |$60\%$|.
Short-run application: Our second application looks at shorter horizons but uses more disaggregated (and volatile) data. We compute changes in the welfare-relevant price index and compare these to the chained index using product-level nondurable consumer goods data between 2004 and 2019. We find that the chain-weighted price index understates inflation rates if we use 2019 preferences. This is because goods that became more popular over time, due to taste shocks, have higher inflation rates. At annual frequency, the gap is around 1 percentage point, and it grows to around 4 percentage points over the whole sample. This also means that for past preferences, rather than current preferences, the welfare-relevant inflation rate is lower than the chained index because goods that became less popular experienced lower inflation.
Business cycle application: Our final application draws on the COVID-19 recession to illustrate the difference between macro and micro notions of welfare. During this recession, household expenditures switched to favor certain sectors at the same time that those sectors experienced higher inflation. We show this implies that micro welfare using mid-2020 preferences, taking changes in prices as given, fell by more than macro welfare, taking into account that changes in prices are themselves caused by demand shocks. Furthermore, unlike macro welfare, real consumption, real GDP, and aggregate TFP are unreliable metrics for measuring changes in productive capacity because they depend on irrelevant details like the order in which supply and demand shocks hit the economy.
In addition to preference stability and homotheticity, a chained index accurately measures welfare only if prices and quantities change continuously and are measured correctly. Many of the well-known reasons chained indices fail to measure welfare are due to violations of these measurement assumptions. For example, it is well known that real consumption fails to account for product creation and destruction if we do not measure the quantity of goods continuously as their price falls from or goes to their choke price (Hicks 1940; Feenstra 1994; Hausman 1997; Aghion et al. 2019); real consumption does not properly account for changes in the quality of goods (see Syverson 2017); and real consumption fails to properly account for changes in nonmarket components of welfare, like changes in the user cost of durable consumption or leisure and mortality (see Jones and Klenow 2016). In these cases, the problem is that some of the relevant prices or quantities in the consumption bundle are missing or mismeasured, and correcting the index involves imputing a value for these missing prices or quantities. Nonhomotheticities and taste shocks are different from mismeasured prices because they violate the maintained assumptions about preferences, not prices, and correcting the index requires the use of welfare-relevant (rather than observed) expenditure shares. For this reason, we abstract from these mismeasurement issues and assume that prices and quantities have been correctly measured. If prices and quantities are mismeasured or missing, then our results would apply to the quality-adjusted, corrected version of prices instead of observed prices.
I.A. Other Related Literature
Measuring changes in welfare using money metrics is standard in microeconomic theory (see Deaton and Muellbauer 1980, chap. 7). We characterize the gap between this notion of welfare and real consumption with nonhomotheticities and taste shocks. Our general equilibrium results relax the standard assumption in growth accounting that there exists a stable and homothetic final good aggregator (extending Domar 1961; Hulten 1978). This is also an important maintained assumption in the literature on disaggregated and production network models of the economy (see Carvalho and Tahbaz-Salehi 2019 and the references therein).
Our approach focusing on the money metric at final preferences can be contrasted with common practice in the literature on index numbers, which focuses on Konüs price indices for intermediate levels of utility or tastes between t0 and t1 (see Diewert 1976; Caves, Christensen, and Diewert 1982; Feenstra and Reinsdorf 2007). These papers show that under some assumptions (i.e., translog or constant elasticity of substitution, CES), commonly used indices like Törnqvist and Sato-Vartia do answer a meaningful question. The advantage of this approach is that constructing these indices requires far less information; the disadvantage is that, unlike our preferred welfare measure, these indices are not money metrics that can be used for policy or counterfactual analysis, and they do not provide specific information about the reference indifference curve being used (what budget level, price vector, and preferences it corresponds to). Furthermore, in practice, most index numbers are constructed by chaining, and the aforementioned results do not apply to chained indices. An additional contribution of our article is to characterize how welfare measures differ from chained (Divisia) indices.3 Finally, relative to this literature, we also provide a unified analysis of nonhomotheticity and taste shocks, and we define and characterize a welfare measure for comparing PPFs rather than budget sets (taking into account that prices are endogenous to choices).
A recent and related paper is Redding and Weinstein (2020), who also study welfare changes with taste shocks and nonhomotheticities. Their approach depends on untestable assumptions about cardinal properties of utility functions. On the other hand, we use a money-metric approach that relies only on ordinal preference relations and not on the way utility is cardinalized. We compare our approach to that of Redding and Weinstein (2020) in Section II.D.
Our approach to calculate ex post welfare changes requires knowing price changes. When information on prices is incomplete, if preferences are nonhomothetic, an alternative approach is to infer ex post changes in welfare by relying on changes in prices, expenditures, price elasticities, and Engel curve slopes for only a subset of goods, given assumptions on separability and stability in preferences (see Hamilton 2001; Atkin et al. 2020). In a different vein, Jaravel and Lashkari (2022) provide a procedure to measure microeconomic welfare changes without direct knowledge of elasticities of substitution in the absence of taste shocks and under some additional assumptions.4 We are interested in ex post welfare measurement with nonhomotheticities, but unlike the aforementioned papers, we are also interested in taste shocks, counterfactuals, and general equilibrium.
Our study is also related to the literature on structural transformation and Baumol’s cost disease. As explained by Buera and Kaboski (2009) and Herrendorf, Rogerson, and Valentinyi (2013), this literature advances two microfoundations for structural transformation. The first explanation is all about relative prices differences: if demand curves are not unit-price-elastic, then changes in relative prices change expenditure shares (e.g., Ngai and Pissarides 2007; Acemoglu and Guerrieri 2008; Buera, Kaboski, and Rogerson 2015). The second explanation emphasizes shifts in demand curves caused by income effects or taste shocks—households spend more of their income on some goods as they become richer (e.g., Kongsamut, Rebelo, and Xie 2001; Boppart 2014; Comin, Lashkari, and Mestieri 2021; Alder, Boppart, and Müller 2022) or older (Cravino, Levchenko, and Rojas 2019). Our results suggest that structural transformation driven by relative price changes has different welfare implications than structural transformation driven by nonhomotheticity or taste changes.
The structure of the article is as follows. In Section II, we set up the microeconomic problem and provide exact and approximate characterizations of welfare and chain-weighted real consumption. In Section III, we set up the macroeconomic general equilibrium model and provide exact and approximate characterizations of welfare and chain-weighted real output changes. Whereas in Section III we present our macro results in terms of endogenous sufficient statistics, in Section IV we solve for these endogenous sufficient statistics in terms of microeconomic primitives and consider some simple but instructive analytical examples. Our applications are in Section V. We discuss how to use our results in dynamic settings and how to treat new goods in Section VI, and we conclude in Section VII. Proofs are in the appendix and supplementary materials are in an Online Appendix.
II. Microeconomic Changes in Welfare and Consumption
In this section, we consider changes in budget constraints in partial equilibrium. We ask how consumers value these changes and compare this with chain-weighted real consumption. This section builds intuition for Section III, where we model the equilibrium determination of prices.
II.A. Environment and Definitions
Consider a set of preference relations, {≽x: x ∈ X}, over bundles of goods |$c \in \mathbb {R}^N$| where N is the number of goods. The vector c includes all relevant goods, and if ≽x is intertemporal, then c is a path of current and future consumption bundles. These preferences are indexed by some vector of parameters, denoted by x, that the consumer does not make choices about but that can affect preference rankings over bundles of goods. For example, x could include calendar time, age, exposure to fads, or state of nature.5
For every x ∈ X, we represent the preference relation ≽x by a utility function U(c; x). Because the consumer makes no choices over x, preferences over x, if they exist, are not revealed by choices. Hence, whereas U(c; x) > U(c′; x) if, and only if, c ≽xc′, a comparison of U(c; x′) and U(c; x) is not meaningful and does not encode any information, because it is affected by how U(· ; x) and U(· ; x′) are cardinalized.
Note that x should not be interpreted as quality because consumers have preferences and choices over quality but not over parameters of their own utility function (tastes). Hence, the welfare implications of quality changes are different to those of taste changes. Moreover, quality has interpretable cardinal units (e.g., GHz for the processor of a computer) whereas tastes do not. Quality characteristics must be included as a part of the description of the consumption bundle c, not in x. See Online Appendix C for a more detailed discussion.
There are two properties of preferences that are analytically convenient benchmarks throughout the rest of the analysis.
Preferences are homothetic if whenever c ∼xc′ then αc ∼x αc′ for every α > 0.
When ≽x is homothetic, we can write U(c; x) so that for every α > 0, U(αc; x) = αU(c; x).
Preferences are stable if ≽x is the same as |$\succeq _{x^{\prime }}$| for every x and x′ in X.
If preferences are stable, then the utility function U(c; x) is separable in c and x.
Consider shifts in the budget set as prices and income change from |$p_{t_0}$| and |$I_{t_0}$| to |$p_{t_1}$| and |$I_{t_1}.$| Here, t0 and t1 simply index the vector of prices and income being compared. Motivated by our applications, we refer to this index as time. This change in the budget set is accompanied by changes in preferences from |$x_{t_0}$| to |$x_{t_1}$|.
Because changes in utility do not have meaningful units, we use a money metric to measure how the consumer values different budget sets. Our baseline measure of microeconomic welfare is defined as follows.
In words, EVm is the change in income (in logs), under initial prices |$p_{t_0}$|, that a consumer with preferences |$\succeq _{x_{t_1}}$| would need to be indifferent between the budget set defined by initial prices |$(p_{t_0},e^\phi I_{t_0})$| and the new budget set defined by new prices and income |$(p_{t_1},I_{t_1})$|. The new budget set |$(p_{t_1},I_{t_1})$| is preferred to the initial one |$(p_{t_0}, I_{t_0})$| if, and only if, EVm is positive.
As Fisher and Shell (1968) point out, this is different to the following question that one may wish to answer: “how much better off is the consumer in t1 compared to the consumer in t0?” This question is unanswerable using only ordinal choice information. As discussed already, a comparison of two different utility functions, |$U(\cdot ;x_{t_0})$| and |$U(\cdot ;x_{t_1})$|, depends on how each utility function cardinalizes the underlying preference relation, which is arbitrary because utility functions are defined only up to monotone transformations. Thus, a welfare measure based only on the ordinal preference relation must hold preference parameters x constant in the comparison. For a discussion of papers that adopt a cardinal approach, see Section II.D.
Other than keeping x constant, we make two other choices in Definition 3. First, we focus on final rather than initial preferences, and second, we use equivalent rather than compensating variation. In principle, one could study initial preferences and compensating variation instead. In general, since these alternative measures ask different questions, they give different answers, unless preferences happen to be both stable and homothetic. Our methods can be also used to understand these alternative welfare measures, but to streamline the exposition we leave some of the additional details for these alternative measures in Online Appendix A.
In our baseline, we focus on final preferences |$\succeq _{x_{t_1}}$|, as opposed to initial preferences |$\succeq _{x_{t_0}}$|, because for temporal comparisons, the asymmetry of time makes current preferences more relevant than preferences in the past. As Fisher and Shell (1968) write, “every practical question which one wants the cost of living index to answer is answered with reference to current, not base-year tastes.” Of course, in some contexts one may be interested in using past preferences to value choice sets, and for counterfactuals one may want to use current rather than future preferences.6
We also focus on equivalent variation as our benchmark, rather than compensating variation. We focus on equivalent variation because unlike the compensating variation, it is a money metric (see McKenzie and Pearce 1982). Specifically, the equivalent variation is itself an index of utility that transforms the utility value of different outcomes into dollar values under a common price system (prices in t0). This is not true for compensating variation, as discussed in Section II.D.
We define real consumption based on a chain-weighted index. This is the standard procedure used by national income accountants and statistical agencies to construct aggregate quantities given data on the evolution of prices p and consumption bundles c.
Equation (2) is called a Divisia quantity index. In practice, because perfect data are not available in continuous time, statistical agencies approximate this integral via a (Riemann) sum, which they call a chained index. We abstract from the imperfections of these discrete-time approximations in this article.8 Moreover, we assume that the data on prices and quantities is perfect—completely accurate, comprehensive, and adjusted for any necessary quality changes. This is because the important and well-studied biases between measured and welfare-relevant objects associated with imperfections in the data, like the lack of quality adjustment, missing prices, or infrequent measurement, are different to the biases we study.
II.B. Characterization of Welfare
The following lemma expresses changes in welfare given changes in prices and income. It is a key result that we use throughout the paper.
The budget shares |$b_{i}(p_t,u_{t_1},x_{t_1})$| that are used to calculate welfare in equation (4) can be reinterpreted as those under prices pt of a fictional consumer with homothetic and stable preferences with expenditure function |$\tilde{e}(p,u)=e(p,u_{t_1};x_{t_1})u$|. This implies that to compute |$b_{i}(p_t,u_{t_1},x_{t_1})$| and EVm, we need to know the budget shares and elasticities of substitution at t1 but not the income elasticities or the demand shocks. This holds generally, but we illustrate it using the convenient nonhomothetic CES functional form below.
If |$b_{t_0}$| is known but |$b_{t_1}$| is not, we first have to calculate |$b_{t_1}$| before applying equation (9). Online Appendix D shows how this can be done given knowledge of taste shocks and income elasticities of demand.
II.C. Comparing Welfare and Real Consumption
Combining equations (4) and (10) implies the following proposition.
This proposition clarifies the differences between welfare and real consumption. Real consumption weighs changes in prices at time t by observed budget shares at time t, taking into account expenditure switching as it happens. In contrast, welfare takes into account expenditure switching due to income effects and taste shocks from the beginning, weighing changes in prices at time t by |$b_{i}(p_t, u_{t_1}, x_{t_1})$|. Intuitively, EVm depends on budget shares evaluated at final utility (|$u_{t_1}$|) and tastes (|$x_{t_1}$|), since EVm adjusts the level of income in t0 to make consumers with t1 preferences as well off as they are in t1. For instance, if the consumer prefers the t1 to the t0 budget set, then she must be given more income in t0 to make her indifferent between t0 and t1. As we give her more income in t0, the shape of her indifference curve changes until it mirrors the one in t1. This means that the shape of the indifference curve relevant for the comparison is the one at t1.11
An immediate consequence of Proposition 1 is the well-known result that real consumption is equal to changes in equivalent variation if and only if preferences are homothetic and stable. This is because when preferences are stable and homothetic, budget shares do not depend on x or on utility u over time. That is, when preferences are homothetic and stable, |$b_{it}=b_{i}(p_t,u_{t},x_{t})=b_{i}(p_t,u_{t_1},x_{t_1})$| for every path of shocks and every t. Real consumption and welfare are also the same if the difference between the actual and Hicksian budget shares in equation (11) are orthogonal to price changes.
To gain more intuition, we provide a second-order approximation as the time period goes to zero, t1 − t0 = Δt → 0. Formally, for any variable z, we write |$\Delta \log z \approx (\frac{\partial \log z}{\partial t}) \Delta t + \frac{1}{2} (\frac{\partial \log z^2}{d^2 t}) (\Delta t)^2$|, where the remainder in the approximation is of order (Δt)3.12 Furthermore, throughout we use the shorthand |$\mathbb {E}_{z}(y)$| for ∑iziyi for any pair of vectors z and y where zi ≥ 0 and ∑izi = 1. Similarly, Covz(w, y) is shorthand for |$\mathbb {E}_{z}(w y)-\mathbb {E}_{z}(w)\mathbb {E}_{z}(y)$|.
We discuss real consumption and welfare in turn. The first two terms in equation (12) are just the change in nominal income deflated by average price changes. The last term (multiplied by |$\frac{1}{2}$|) captures how expenditure switching affects real consumption. If expenditures rise for goods whose prices are rising, then this lowers real consumption. In principle, the expenditure shares can change for three different reasons: substitution effects, income effects, and taste shocks.
Consider the expenditure-switching terms in expression (15) one by one. If goods are substitutes, θ0 > 1, then welfare is convex in prices and variance in price changes boosts welfare by allowing substitution towards goods that become relatively cheap. The second line of expression (15) captures the effect of taste shocks and income effects. If the composition of demand shifts in favor of goods that become relatively cheap, due to either taste shocks Covb(Δlog x, Δlog p) < 0 or income effects |$Cov_{b}\left(\varepsilon ,\Delta \log p\right) \left(\Delta \log I-\mathbb {E}_{b}\left(\Delta \log p\right)\right) <0$|, then real consumption increases.
Having understood expression (15), consider now expenditure-switching terms in expression (16). Welfare places the same weight on substitution effects as real consumption does. However, it places a larger weight on expenditure switching due to income effects and taste shocks. Whereas Δlog Y only takes into account expenditure switching as it occurs over time, EVm accounts for expenditure switching due to income effects and taste shocks from the start. Therefore, expenditure switching due to income and tastes are multiplied by |$\frac{1}{2}$| for Δlog Y and 1 for EVm. This implies that, for example, the increase in welfare from a price reduction in a good i with increasing demand (due to an increase in xi or a relatively high ϵi) is not fully reflected in real consumption, implying EVm > Δlog Y. Even if preferences are unstable or nonhomothetic, real consumption strays from welfare only if income effects or taste shocks covary with price changes.15
II.D. Alternative Welfare Measures
In this section, we discuss two alternative welfare measures that are widely used in the literature: changes in utility indices and compensating variation.
1. Utility Index
A common approach in the literature is to use ut as a measure of welfare and define the associated “ideal” price index |$P_t\equiv \frac{e(p_t,u_t;x_t)}{u_t}$| by analogy to the homothetic and stable case. This is particularly common with CES preferences with either taste shocks or nonhomotheticities (e.g., Ehrlich et al. 2019; Redding and Weinstein 2020; Comin, Lashkari, and Mestieri 2021; Argente, Hsieh, and Lee forthcoming). If the utility function is not changing, then ut obviously ranks choices according to the underlying preference relation, but the magnitude of the changes in ut and the associated price index Pt are not interpretable. This is because they depend on the overall level of utility elasticities ξ, which are arbitrary. Furthermore, when preferences are nonhomothetic, there is no normalization of the ξ parameters such that changes in ut are equal to EVm (see Online Appendix D.2).
This issue is even more severe when there are taste shocks. In this case, changes in ut and Pt depend on the overall change in taste shifters xt. Once again, the overall level of xt in equation (6) is not pinned down by choice behavior. Therefore, to compute ut one needs to assume a value for the overall level of xt’s period by period. This means ut no longer corresponds to the ranking of choices according to any preference relation, but is determined by the untestable assumption that determines the overall level of xt’s in each period. For example, Redding and Weinstein (2020) set the average of xt’s to be constant over time. However, the same data are consistent with the average of xt’s rising or falling in arbitrary ways through time. Each assumption results in a different answer. As Muellbauer (1975) forcefully argues: “It is not possible to set up a market experiment which would unambiguously indicate whether the consumer’s efficiency as a utility machine has changed.” As he points out, “no such problems attend an ordinal cost-of-living index” like the ones we characterize.16
2. Compensating Variation
Our baseline measure of welfare changes is equivalent variation. An alternative is the compensating variation, which is the reduction in t1 income that makes the consumer indifferent to their t0 budget set. As mentioned earlier, EVm is a money metric that ranks all choices, whereas CVm is not. To see this, consider a household in t0 choosing between |$(p_{t_1}, I_{t_1})$| and |$(p^{\prime }_{t_1}, I^{\prime }_{t_1})$|. It follows that |$u^{\prime }_{t_1} = v(p^{\prime }_{t_1}, I^{\prime }_{t_1};x) > v(p_{t_1}, I_{t_1};x) = u_{t_1}$| if, and only if, |$\frac{e(p_{t_0}, u^{\prime }_{t_1};x)}{I_{t_0}} > \frac{e(p_{t_0}, u_{t_1};x)}{I_{t_0}}$|. That is, the household prefers whichever choice has the highest EVm. Hence, EVm is itself an index of utility. On the other hand, CVm cannot be used to compare t1 and |$t^{\prime }_1$| because CVm for t1 and |$t^{\prime }_1$| are expressed in terms of income adjustments in t1 and |$t^{\prime }_1$| prices, respectively. Compensating variation can be used to compare t0 to t1 and t0 to |$t^{\prime }_1$| but it cannot be used to compare t1 to |$t^{\prime }_1$|.
III. Macroeconomic Changes in Welfare and Consumption
In the previous section we showed how to value changes in budget sets for a consumer who takes prices as given. For a whole society however, prices are endogenous to collective choices. In this section, we extend our analysis to show how to assign value to different PPFs rather than budget sets. The macro, or society-level, notion of welfare we define is not the same as micro welfare evaluated in general equilibrium.
We first introduce a closed, representative-agent, neoclassical economy with nonhomothetic and unstable preferences. We generalize our definitions of welfare, now at the macroeconomic level, and present exact and approximate expressions for real GDP and welfare in terms of endogenous sufficient statistics. In Section IV, we solve for these endogenous sufficient statistics in terms of observable primitives. Without nonhomotheticities and taste shocks, our economy is the same as the one studied by Baqaee and Farhi (2019). We extend their approximation formulas to capture real GDP and welfare when there are income effects and taste shocks.
III.A. Environment and Definitions
Let A be the N × 1 vector of technology shifters and L be the F × 1 vector of primary (exogenously given) factor endowments.20 The production possibility set (and its associated frontier) is the set of feasible consumption bundles that can be attained given A and L. Given our assumption that production functions have constant returns to scale, the PPF is linear if there is only one factor of production.
For each A, L, and x, we denote equilibrium prices (for both goods and factors) by p(A, L, x) and aggregate income, or nominal GDP, by I(A, L, x). These equilibrium prices and incomes are unique up to the choice of a numeraire.
Consider shifts in the PPF as technologies and factor endowments change from |$(A_{t_0}$|, |$L_{t_0})$| to |$(A_{t_1}$|, |$L_{t_1})$|, along with changes in preferences from |$x_{t_0}$| to |$x_{t_1}$|. We generalize our microeconomic measure of welfare in the following way.
The superscript M in EVM represents the fact that this is the macro equivalent variation, in contrast to EVm for the micro welfare measure. In words, EVM is the proportional change in initial factor endowments required to make the consumer with preferences |$\succeq _{x_{t_1}}$| indifferent between the PPF defined by |$(A_{t_0},e^\phi L_{t_0})$| and the new PPF, defined by |$(A_{t_1}, L_{t_1})$|.
Intuitively, EVM expresses utility changes in terms of factor endowments. That is, EVM is itself an index of utility and (A, L) ≽x (A′, L′) if, and only if, |$EV^M(A_{t_0},L_{t_0},A,L;x)\ge EV^M(A_{t_0},L_{t_0},A^{\prime },L^{\prime },x)$|. The macro equivalent variation, EVM, is a useful metric because it ranks PPFs without reference to endogenous prices.
Definition 5 shows that EVM is a generalization of EVm in the sense that the two coincide when the PPF is the same as the budget constraint. However, EVM is also a generalization of consumption equivalents commonly used to measure welfare in macroeconomics (e.g., Lucas 1987). To see this, note that EVM can also be defined in the following way.
In words, EVM also measures the proportional shift in |$\mathcal {C}_{t_0}$| necessary to make |$\succeq _{x_{t_1}}$| indifferent between |$e^{\phi }\mathcal {C}_{t_0}$| and |$\mathcal {C}_{t_1}$|. This definition is isomorphic to Definition 5. This is because scaling all factor endowments by a constant shifts out the PPF by the same constant under our assumptions.
For a graphical representation of EVm and EVM, see Figure I. Macro and micro welfare answer different questions. To see the difference, consider a situation where households age between t0 and t1 but technologies and factor endowments stay the same. Because the PPF is unchanged, the change in macro welfare is zero by construction. However, if the PPF is nonlinear, the relative price of goods changes between t0 and t1: prices rise for goods that become more desirable. In this case, EVm necessarily falls even though the PPF is unchanged. Hence, EVm does not rank PPFs for a society.
The following examples further illustrate why micro welfare can be a misleading measure of the welfare impact of technological change when preferences are nonhomothetic or unstable.
(Micro versus macro welfare and growth). Consider economies that are endowed with cows, and suppose each cow produces one unit of steak and one unit of offal. At t0 the economy is endowed with one cow. We consider what happens if we double the number of cows at t1. For any stable and homothetic preferences, macro and micro welfare doubles (EVM = EVm = log 2).
Figure II, Panel A considers a case with nonhomothetic but stable preferences. For this example, suppose that the consumer at t0 considers steak and offal to be perfect substitutes because the consumer is poor and only cares about total calories. Hence, the relative price of offal and steak is one and the consumption bundle is one unit of steak and one unit offal in t0. At t1, the consumer consumes two units of steak and two units of offal. If steak is an extreme luxury good, then at t1, the relative price of steak is much higher than offal. Macro welfare measures the distance between the original PPF and the new PPF, which is twice as big since the economy in t1 produces double the number of both steaks and offal. However, in this example, micro welfare only rises by an arbitrarily small number ϵ. That is, even though the economy is twice as good, micro welfare has increased by almost zero. Intuitively, in the initial equilibrium, steak was relatively cheap (because consumers were poor), so initial income in t0 only needs to be raised by ϵ to allow the consumer to reach the t1 indifference curve.
In other words, micro welfare measures the fact that an infinitesimal consumer with an endowment of 1 + ϵ cows could purchase a total of 2 + ϵ steaks. However, if all consumers’ endowments were raised by ϵ, and all consumers tried to sell their offal to buy steak, then the relative price of steak would quickly rise because the aggregate supply of steaks has only risen by ϵ. This means that in practice, consumers with 1 + ϵ cows are not as well off as they thought they would be. Hence, to make them indifferent, we would have to give them more endowment—doing this would change relative prices again. Macro welfare is the fixed point to this problem, where the increase in the endowment is exactly enough to make consumers indifferent taking into account the fact that relative prices are endogenous.21
Figure II, Panel B considers a case with homothetic but unstable preferences instead. Starting with the same PPF as before in t0, suppose that in t1 the number of cows increases by a factor 2 − ϵ. Due to a concurrent advertising campaign, consumer preferences shift so that in t1 steak becomes more popular than offal (the indifference curves in Figure II, Panel B become flatter). In this example, since the number of cows has increased by a factor 2 − ϵ, the macro welfare change is log (2 − ϵ). However, micro welfare declines because the consumer with t1 preferences strictly prefers the t0 budget set. Intuitively, a consumer with t1 preferences could sell the offal in t0 and purchase a total of two steaks, which is preferred to 2 − ϵ steaks she can get in t1. That is, even though the economy has almost doubled in terms of its productive capabilities, micro welfare in general equilibrium has fallen. Hence, micro welfare in general equilibrium is not a good measure of how changes in choice sets affect societal welfare.22
The previous examples illustrate that using the initial budget set to represent the initial PPF is deceptive, since the initial budget set reflects both the technologies and demand in t0. Our macroeconomic notion of welfare accounts for the endogenous changes in prices by comparing the initial and final PPFs rather than the initial and final budget sets.
When relative prices do not respond to consumers’ choices (i.e., the PPF is linear), then for a given primitive shock, macro and micro welfare are always the same. Alternatively, if preferences are homothetic and stable, then macro and micro welfare are the same, regardless of the shape of the PPF. The following proposition formalizes this:
(Macro versus Micro Welfare). Consider changes in technologies A, factor quantities L, and tastes x. Macro and micro welfare changes are equal (EVm = EVM) if preferences are stable and homothetic, or if the PPF is linear.
We provide a quantitative illustration of the difference between micro and macro welfare using the COVID-19 crisis in Section V.
III.B. Characterization of Welfare and Real GDP
To characterize macro welfare, we define a Hicksian (or compensated) economy indexed by (A, L, u, x), where (A, L) determines the PPF and (u, x) specifies an indifference curve corresponding to U(c; x) = u.
(Hicksian economy). Consider a fictitious consumer whose preferences are represented by the expenditure function |$\tilde{e}(p,\tilde{u}) \equiv e(p,u;x)\tilde{u}$|, for fixed values of u and x. In words, |$\tilde{e}(p,\tilde{u})$| maps prices p and utility values |$\tilde{u}$| into required expenditures of the fictitious consumer. The Hicksian economy, indexed by (A, L, u, x), is an economy with PPF (A, L) and expenditure function |$\tilde{e}(p,\tilde{u})$|.
In the definition, u and x are parameters that define the fictional consumer’s preferences. When the true consumer’s preferences, represented by the expenditure function e(·), are stable and homothetic, then they coincide with the fictional consumer’s preferences, represented by |$\tilde{e}(\cdot)$|. In this case, the Hicksian economy is the same as the uncompensated original economy. We refer to variables, like prices and sales, in the Hicksian economy as Hicksian variables and index them by (A, L, u, x). We refer to variables in the uncompensated original economy as observed variables and index them by t.
To characterize macro welfare and compare it to real GDP, we use Hicksian sales shares, λ(A, L, u, x), which are the sales shares in a Hicksian economy. The Hicksian sales shares are the general equilibrium counterpart to partial equilibrium Hicksian budget shares. The following proposition shows that the change in macro welfare is the integral of Hicksian sales shares with respect to changes in technologies and factor quantities.
In words, growth accounting for welfare should be based on compensated or Hicksian sales shares evaluated at current technology but final preferences and utility. We define the first N summands of equation (18) to be changes in welfare-relevant TFP and the last F summands are changes in welfare due to changes in factor inputs. As we show below, this is analogous to how measured TFP is defined using real GDP.
To apply Proposition 5, the only thing we need to know about preferences is how |$e(p,u_{t_1};x_{t_1})$| varies as a function of prices for given |$u_{t_1}$| and |$x_{t_1}$|. This uniquely pins down the fictional consumers’ preferences that generate the Hicksian shares |$\lambda (A_t,L_t,u_{t_1},x_{t_1})$|. In other words, we need to know the budget shares and elasticities of substitution at the final allocation but not income elasticities or taste shocks.23
We now consider changes in real GDP, defined using the Divisia index for final goods as |$\Delta \log Y=\int _{t_{0}}^{t_{1}}\sum _{i\in N}b_{it}d\log c_{it}$|. The following result shows that changes in real GDP can be calculated by integrating observed sales shares with respect to technology changes.
In equation (19), the first N summands are equal to measured TFP, and the last F summands are the growth in real GDP caused by changes in factor endowments. Proposition 6 is a slight generalization of Hulten (1978) to environments with unstable and nonhomothetic final demand. In general equilibrium, the sales shares play the role that budget shares played in partial equilibrium. Whereas in partial equilibrium, integrating budget shares with respect to prices yielded real consumption, in general equilibrium integrating sales shares with respect to technologies and factors yields real GDP. If the PPF is not moving, that is, technology A and endowments L are constant, then pure preference shocks have no effect on real GDP even though they can change allocations.
Propositions 5 and 6 show that the difference between macro welfare and real GDP can be understood in terms of the distinction between Hicksian and observed sales shares. If preferences are homothetic and stable, then |$\lambda (A,L,u_{t_1},x_{t_1})=\lambda _{it}$| and the change in welfare is equal to real GDP.
To get more intuition for the difference between welfare and real GDP, we use a second-order approximation of the response of real GDP and welfare to technology and preference shocks as t1 − t0 = Δt → 0). To make the formulas more compact and without loss of generality, when we write local approximations we abstract from shocks to factor endowments (Δlog L = 0).
We discuss equations (20) and (21) in turn. The first term in equation (20) is the Hulten-Domar formula. The second term in equation (20), emphasized by Baqaee and Farhi (2019), captures nonlinearities due to changes in sales shares. Intuitively, if sales shares decrease for those goods with higher productivity growth, then real GDP growth slows down.
Equation (21) shows that the gap between macro welfare and real GDP is similar to that for our micro results (the signs are flipped because a positive productivity shock reduces prices). Specifically, real GDP takes into consideration changes in sales shares along the equilibrium path. These changes in sales shares could be induced by technology shocks, but they could also be due to changes in preferences and nonhomotheticities. Equation (21) shows that welfare puts more weight on changes in sales shares that are due to demand shocks and nonhomotheticities than those due to substitution effects. That is, real GDP “undercorrects” for changes in shares caused by nonhomotheticities or changes in preferences, similar to the partial equilibrium counterparts in Proposition 2 and 3.
IV. Structural Macro Results and Analytic Examples
The results in Section III are reduced form in the sense that they take changes in observed and compensated sales shares as given. In this section, we solve for changes in these endogenous objects in terms of microeconomic sufficient statistics. For clarity, we restrict attention to nested-CES economies. The general case is in Online Appendix I, and the intuition is very similar. This section also contains some worked-out analytical examples.
1. Nested-CES Economies
2. Input-Output Matrix
We stack the expenditure shares of the household, all producers, and all factors into the (1 + N + F) × (1 + N + F) input-output matrix Ω. The first row corresponds to the household. To highlight the special role played by the household, we index it by 0, which means that the first row of Ω is equal to the household’s budget shares introduced above (Ω0 = b′, with bi = 0 for i ∉ N). The next N rows correspond to the expenditure shares of each producer on every other producer and factor. The last F rows correspond to the expenditure shares of the primary factors (which are all zeros, since primary factors do not require any inputs).
3. Leontief Inverse Matrix
IV.A. General Characterization for Nested-CES Economies
According to Propositions 5 and 6, changes in real GDP and welfare can be computed by integrating observed and Hicksian sales shares with respect to technology shocks. The following proposition pins down sales shares as a function of primitives. This proposition can then be used in combination with Propositions 5 and 6 to calculate exact changes in real GDP and welfare. For readability, we again assume away shocks to factor endowments.
To compute changes in real GDP using Proposition 6, we need to know λit = λ(At, Lt, ut, xt) for t ∈ [t0, t1]. These are solutions to the differential equations (22) and (23) given some boundary condition that pins down Ψ at some point in time t ∈ [t0, t1].
On the other hand, to compute macro welfare, we need Hicksian sales shares |$\lambda (A_t, L_t, u_{t_1}, x_{t_1})$| as a function of t. These are solutions to the same differential equations, except that the terms involving taste shocks and income effects in equation (23) are set to zero. In this case, the boundary condition is that the Leontief inverse at t1 is equal to the observed Leontief inverse |$\Psi _{t_1}$| at t1. Therefore, if |$\Psi _{t_1}$| is observed, we can calculate Hicksian sales shares between t0 and t1 by starting equation (23) at t1 and going backward to t0. This process does not require knowledge of either the income elasticities ϵ nor the taste shocks |$\Delta \log \mathtt {x}$|.
Each term in the differential equations in Proposition 8 has a clear interpretation. Equation (22) captures the fact that the price of each good dlog pi is determined by its (direct and indirect) exposure to the prices of inputs j and factors f (captured by Ψijt and Ψift at time t).
On the other hand, the equation for dlog Ψilt shows that changes in the Leontief inverse are determined by substitutions by j, if j is an intermediary between i and l, as well as income and substitution effects if i is the household (i = 0). Finally, the Hicksian version of this equation is identical except that it does not adjust expenditures due to income effects and taste shocks along the transition.
(Micro Welfare). Proposition 8 can also be used to compute changes in microeconomic welfare EVm in a general equilibrium model. To do this, we compute |$p(A_{t_0}, L_{t_0}, u_{t_0}, x_{t_0})$| and |$p(A_{t_1}, L_{t_1}, u_{t_1}, x_{t_1})$| using Proposition 8 and then plug these price changes into equation (9). Unlike macroeconomic welfare, microeconomic welfare changes generically require knowledge of both income elasticities and taste shocks because they are needed to calculate the initial and or final prices.
To build more intuition about the difference between real GDP and macro welfare, consider economies with only a single factor of production. In this case the system is simplified since we know that the economy’s single primary factor always has a sales share equal to one. This allows for a simple characterization of both welfare and real GDP up to a second-order approximation.
Proposition 9 is a general equilibrium counterpart to Proposition 3. We first discuss real GDP. The first term in equation (24) is the Hulten-Domar term. The other terms are second-order terms resulting from the fact that sales shares change in response to shocks. The first term captures nonlinearities due to the fact that sales shares respond to changes in relative prices caused by technology shocks (these effects are emphasized by Baqaee and Farhi 2019). The terms on the second line of equation (24), which are the ones we focus on in this article, capture changes in sales shares due to changes in tastes or nonhomotheticities.
Equation (25) shows that while real GDP correctly accounts for substitution due to supply shocks, to measure welfare it needs to be corrected for expenditure switching due to changes in final demand caused by taste shocks or income effects. Whereas in partial equilibrium, the gap between welfare and real GDP is proportional to the covariance of price and demand shocks (see Proposition 3), equation (25) shows that in general equilibrium, the relevant statistic is the covariance of demand shocks with a network-adjusted notion of supply shocks.
IV.B. Analytical Examples
We now work through some simple examples to illustrate the forces that drive a gap between real GDP and welfare.
We work through some simple examples with multiple factors of production to illustrate how nonlinear PPFs affect the previous results.
(Macro versus Micro Welfare Change). Finally, we demonstrate the difference between macro and micro welfare changes using the previous example. The economy in the previous example has multiple factors and unstable preferences. Therefore, macro and micro notions of welfare are different because the PPF is no longer linear.
Online Appendix E contains additional examples showing how input-output connections can amplify or mitigate the gap between macro welfare and real GDP.
V. Applications
In this section, we consider three applications of our results. The first application is about long-run growth, quantifying the difference between welfare-relevant and measured aggregate productivity growth in the presence of income effects and demand instability. The second application is about short-run fluctuations and shows that correlated product-level supply and demand shocks within industries drive a gap between real consumption and welfare even in the short run. Our final application is a business cycle event study, where we use the COVID-19 recession to demonstrate the difference between macroeconomic and microeconomic welfare using a quantitative model. We also use this model to show that demand instability can make real GDP an unreliable metric for changes in production.
V.A. Application 1: Long-Run Growth and Structural Transformation
As economies grow, sectors with low productivity growth tend to expand compared to sectors with faster productivity growth. This means that over time, aggregate productivity growth is increasingly determined by those sectors whose productivity growth is slowest. This phenomenon is often called Baumol’s cost disease.
1. Two Polar Extremes
Computing the Hicksian sales shares to obtain Δlog TFPev requires an explicit structural model of the economy. However, there are two polar cases in which Δlog TFPev can be calculated without specifying the detailed model. The first extreme is when demand is stable and homothetic, so that changes in sales shares are due only to relative price changes (substitution effects). The second extreme is when there are no substitution effects in sales shares (as in a Cobb-Douglas economy), and changes in sales shares are only due to income effects or demand instability. If structural transformation is driven by a combination of substitution effects and nonhomotheticities or demand instability, then the change in welfare TFP will be somewhere in between these two cases, as discussed in Online Appendix F. The following corollary of Proposition 5 summarizes the change in welfare TFP in these polar cases.
In the first case, because preferences are homothetic and stable, |$\lambda (A_{t}, L_{t}, u_{t_1}, x_{t_1})=\lambda _{it}$|, so welfare TFP is equal to TFP in the data. In the second case, since there are no substitution effects in production or demand, compensated sales shares do not respond to productivity changes, so |$\lambda (A_{t}, L_{t}, u_{t_1}, x_{t_1})=\lambda _{it_1}$|.
To quantify Corollary 1, we use US-KLEMS data on sales shares and TFP growth for 61 private-sector industries between 1947 and 2014. We calculate changes in industry-level gross-output TFP following the methodology of Jorgenson, Ho, and Stiroh (2005) and Carvalho and Gabaix (2013).26Figure III plots EVM comparing 1947 to subsequent years under alternative assumptions about substitution and income elasticities. For comparisons that are close to 1947, the change in welfare is not very sensitive to our assumptions about elasticities. This is because at higher frequencies, the shocks are small and industry sales shares are reasonably stable. However, the assumptions about substitution and income elasticities play a larger role over longer horizons. Comparing 1947 to 2014, the constant-initial-sales-share term grows by around 58 log points (or |$78\%$|), whereas the chain-linked change in aggregate TFP grew by around 47 log points (|$60\%$|). Hence, Baumol’s cost disease caused aggregate TFP to fall by 10 log points and reduced aggregate productivity growth by around 23%.
If we assume that structural transformation is due solely to income effects and taste shocks, then by Corollary 1 the growth in welfare-relevant TFP from 1947 to 2014 was 37 log points (|$46\%$|) instead of the measured 47 log points (|$60\%$|)—that is, a 23% additional reduction in the growth rate.
Intuitively, welfare-based productivity increases less than TFP because, relative to 1947, preferences in 2014 favor low productivity growth sectors such as services (due to either income effects or demand instability). These sectors were cheaper compared to manufacturing in 1947 than in 2014. This means that at 1947 prices, when services were relatively cheap, households require less income growth to be indifferent between their budget constraint in 1947 and the one in 2014.27
In Online Appendix F, we provide some quantitative illustrations away from the two polar extremes discussed above. In this appendix, we compute welfare changes for different values of elasticities of substitution in consumption and production using Proposition 8. We show that welfare-relevant TFP is closer to measured TFP if the elasticity of substitution across industries (in consumption or production) is significantly lower than one.
V.B. Application 2: Inflation with Product-Level Taste Shocks
In the previous application, we considered a long-run industry-level application. With industry-level data, the gaps between welfare and chain-weighted indices are usually modest over the short run because industry-level sales shares are relatively stable at high frequency. However, this does not mean that these issues are absent in short-run data.
Whereas industry sales shares are stable at high frequency, firm or product-level sales shares are highly volatile even over the very short run. If firms’ or products’ supply and demand shocks are correlated, then measured industry-level output is biased relative to what is relevant for welfare.28 We provide an empirical illustration of the magnitude of the biases caused by taste shocks in product-level data using the Nielsen Consumer Panel database. In the body of the article, we only briefly describe the data set, and refer readers to Online Appendix H for more details. The Nielsen Consumer Panel tracks the purchasing behavior of about 40,000 to 60,000 panelists every year from 2004 to 2019 for a wide variety of nondurable consumer goods (food, nonfood groceries, general merchandise, etc.). A product in the data is defined by its unique Universal Product Code (UPC), and each product is assigned to a module. Our balanced sample covers roughly 820 modules. Panelists in the sample are assigned weights, allowing purchases by the panel to be projected to a nationally representative sample.
We model national demand for UPCs in a given module using a homothetic but unstable CES functional form. We set t1 = 2019 and then for each year t0 < 2019, we calculate welfare-relevant deflators for t0 and t1 preferences module by module for continuing products using equation (9). The price of each UPC in each year is calculated as the ratio of national expenditures on that UPC over units sold over the whole year. For each t0, we include only UPCs purchased in each quarter of each year between t0 and t1. In other words, we abstract from product entry and exit by focusing on the continuing-goods price index (see Section VI for how to deal with product entry-exit when preferences are unstable). To combine module-level inflation rates into a single number, we assume preferences in both t0 and t1 across modules are Cobb-Douglas. For the same set of UPCs, we also compute the change in inflation as measured by a chained Törnqvist index (a discrete-time approximation to the Divisia index).
Figure IV displays the welfare-relevant and chained inflation rates for each t0 assuming that the elasticity of substitution across UPCs in the same module is 4.5.29 Starting in 2018, inflation for 2019 preferences is around 1 percentage point higher than the chained index. Intuitively, this is because changes in prices and changes in demand shocks between t0 and t1 are positively correlated. That is, goods that are more popular in 2019 had relatively higher inflation rates. Following the logic of Proposition 3, this means that the chained index understates inflation for final preferences. The gap increases as we go farther back in time because the shocks are persistent and cumulate. Over the whole sample, the gap widens to 4.3 percentage points.
On the other hand, for initial preferences, the chained index overstates the inflation rate. This is because goods that are relatively more popular in t0 compared with t1 (i.e., goods that became less popular over time) experience lower inflation rates. Since goods that become more popular are also the goods that have higher inflation rates, it is natural that inflation with final tastes is higher than inflation with initial tastes. The two series are different because they answer different questions.30
We report robustness with respect to the elasticity of substitution parameter in Online Appendix H. In this appendix, we show that the size of the bias gets smaller as θ gets closer to one. This is because in the data changes in prices and changes in expenditure shares are approximately uncorrelated. When demand is Cobb-Douglas, changes in expenditure shares are driven only by taste shocks, and so taste shocks are roughly uncorrelated with price changes. Hence, following the logic of Proposition 3, the bias is smaller in the Cobb-Douglas case and larger if the calibrated elasticity is greater than 4.5.
V.C. Application 3: The COVID-19 Recession
Our final application examines how real GDP, microeconomic welfare, and macroeconomic welfare were affected during the COVID-19 recession. The COVID-19 recession is an interesting case study because sectoral expenditure shares changed substantially during this time, these changes were not explainable by changes in observed prices alone, and the movements in demand curves were correlated with movements in supply curves. These are exactly the conditions under which micro welfare, macro welfare, and real GDP can diverge.31
In this application, we do not attempt to measure the welfare costs of COVID-19 itself. This is because households do not make choices over whether they live in a world with COVID-19. Therefore, their preferences about COVID-19 itself are not revealed by their choices. Instead, we ask a more modest question: how does the household value changes in prices (micro welfare) and changes in production (macro welfare), holding fixed the presence of COVID-19.
To study this episode, we use a modified version of the quantitative model introduced in Section IV. Since this is a short-run application, we assume that factor markets are segmented by industry, so that labor and capital in each industry is inelastically supplied. We calibrate share parameters to match the 71 industry U.S. input-output table in 2018 (we exclude government sectors) from the Bureau of Economic Analysis and consider a range of elasticities of substitution. Following Baqaee and Farhi (2022), we model the COVID-19 recession as a combination of negative sectoral employment shocks and sectoral taste shifters. We hit the economy with a vector of primitive supply and demand shocks. The reductions in sectoral employment are calibrated to match peak-to-trough reductions in hours worked by sector from January 2020 to May 2020. The primitive demand shifters in household demand are calibrated to match the observed peak-to-trough change in personal consumption expenditures by sector from January 2020 to May 2020 (conditional on the supply shocks and the elasticities of substitution).32
We consider three different calibrations informed by empirical estimates from Atalay (2017) and Boehm, Flaaen, and Pandalai-Nayar (2019): high complementarities, medium complementarities, and no complementarities (Cobb-Douglas). The high complementarity scenario sets the elasticity of substitution across consumption goods to be 0.7, the one across intermediates to be 0.01, across value added and materials to be 0.3, and the one between labor and capital to be 0.2. The medium complementarities case sets the elasticity of substitution across consumption goods to be 0.95, the one across intermediates to be 0.01, across value added and materials to be 0.5, and the one between labor and capital to be 0.5. The Cobb-Douglas calibration sets all elasticities of substitution equal to unity.
Table I displays welfare changes between January 2020 and May 2020 in the calibrated model. We report separately micro and macro welfare based on pre-COVID (t0 = Q1-2018) and COVID (t1 = Q2-2020) preferences. Recall that micro and macro welfare are not equal in this economy because the PPF is nonlinear.
Elasticities . | High compl. . | Medium compl. . | Cobb-Douglas . |
---|---|---|---|
Micro pre-COVID preferences | −11.7% | −9.1% | −8.7% |
Micro COVID preferences | −13.2% | −12.3% | −10.9% |
Macro pre-COVID preferences | −16.2% | −12.5% | −10.8% |
Macro COVID preferences | −10.1% | −9.4% | −9.0% |
Chained real GDP | −12.1% | −10.6% | −9.8% |
Elasticities . | High compl. . | Medium compl. . | Cobb-Douglas . |
---|---|---|---|
Micro pre-COVID preferences | −11.7% | −9.1% | −8.7% |
Micro COVID preferences | −13.2% | −12.3% | −10.9% |
Macro pre-COVID preferences | −16.2% | −12.5% | −10.8% |
Macro COVID preferences | −10.1% | −9.4% | −9.0% |
Chained real GDP | −12.1% | −10.6% | −9.8% |
Note: Chained real GDP is computed assuming supply and demand shocks arrive simultaneously.
Elasticities . | High compl. . | Medium compl. . | Cobb-Douglas . |
---|---|---|---|
Micro pre-COVID preferences | −11.7% | −9.1% | −8.7% |
Micro COVID preferences | −13.2% | −12.3% | −10.9% |
Macro pre-COVID preferences | −16.2% | −12.5% | −10.8% |
Macro COVID preferences | −10.1% | −9.4% | −9.0% |
Chained real GDP | −12.1% | −10.6% | −9.8% |
Elasticities . | High compl. . | Medium compl. . | Cobb-Douglas . |
---|---|---|---|
Micro pre-COVID preferences | −11.7% | −9.1% | −8.7% |
Micro COVID preferences | −13.2% | −12.3% | −10.9% |
Macro pre-COVID preferences | −16.2% | −12.5% | −10.8% |
Macro COVID preferences | −10.1% | −9.4% | −9.0% |
Chained real GDP | −12.1% | −10.6% | −9.8% |
Note: Chained real GDP is computed assuming supply and demand shocks arrive simultaneously.
Table I shows that micro welfare falls by more than macro welfare under COVID preferences. This is because, as shown in Example 5, when demand rises for goods in the presence of decreasing returns to scale, micro welfare falls more than macro welfare. Intuitively, if household demand for, say, toilet paper rises, then this raises the price of toilet paper and reduces micro welfare compared to macro welfare. That is, a single consumer with COVID preferences, who values toilet paper and takes prices as given, is made worse off by the fact that demand for toilet paper rose. However, society as a whole does not take prices as given. Therefore, when comparing the t0 PPF to the t1 PPF, the fact that toilet paper was cheaper in t0 due to lower demand is irrelevant.
This pattern is reversed for pre-COVID preferences. Macro welfare at pre-COVID preferences fell by a lot because supply contracted in sectors that were popular pre-COVID (e.g., airline travel). Micro welfare losses at pre-COVID preferences are smaller because demand also fell in those sectors (e.g., airline travel) and this reduction in demand reduced their equilibrium prices. Hence, the consumer with pre-COVID preferences, who still likes airline travel, is not as worse off as the society that still likes airline travel.
This illustrates that micro and macro welfare answer different questions, and the answers to these questions can be quantitatively very different. Furthermore, comparing columns of Table I shows that the magnitude of these differences depend on the details of the production structure like the extent of complementarities in production. As we raise the elasticities of substitution in production closer to unity (Cobb-Douglas), the differences between macro and micro notions become less dramatic. This is because the PPF becomes less curved.
In Table I, we also compute real GDP assuming supply and demand shocks arrive simultaneously and linearly over time. Under this assumption, real GDP is somewhere in between the welfare measures at t0 and t1 preferences for both macro and micro. However, this is only because of our assumption that the supply and demand shocks arrive simultaneously and at the same speed. If we change the path of supply and demand shocks, real GDP changes value (even though the initial and final allocation are not changing). For example, if the supply shocks arrive before the demand shocks, then real GDP equals macro welfare changes at t0 preferences. On the other hand, if demand shocks arrive before the supply shocks, then real GDP equals macro welfare changes at t1 preferences.
Hence, if the supply and demand shocks do not disappear in the same way they arrived, measured real GDP after the recovery can be higher or lower than it was before the crisis, even if the economy returns exactly to its pre-COVID allocation. That is, if in the downturn demand shocks arrive before supply shocks (so real GDP falls by roughly 10% in the high-complementarities case, according to Table I) and, in the recovery, demand shocks disappear before the supply shocks (so real GDP rises by roughly 16%), then real GDP is as much as |$6\%$| higher when comparing preshock real GDP to postrecovery real GDP. This is despite the fact that every price and quantity is the same when comparing the preshock allocation to the postrecovery allocation. Hence, during episodes where final demand is unstable, chained real GDP and consumption are unreliable guides for measuring output or welfare, even if we chain in continuous time.33
VI. Extensions
In this section, we briefly summarize how our theoretical results can be used in dynamic settings and extended to account for new goods.
VI.A. Dynamic Economies
At an abstract level, our results can be applied to dynamic economies by using the Arrow-Debreu formalism of indexing goods by period of time and state of nature (as in Basu et al. 2022). In a dynamic economy the utility function is intertemporal and capital accumulation must be treated as an intertemporal intermediate good, as advocated by Barro (2021). In Online Appendix G, we specialize our results further to show how Proposition 5 can be used to make welfare comparisons in a dynamic multisector model with production of consumption goods and investment goods, with unstable and nonhomothetic preferences. We show that steady-state to steady-state macro welfare is equal to the change in nominal consumption expenditures deflated by the exact-algebra CES price index associated with the t1 indifference curve, exactly as for the partial equilibrium microeconomic welfare in expression (9), even though the dynamic general equilibrium economy has infinitely many factors.
VI.B. Extensive Margin
In our discussion, we do not explicitly deal with the new-goods problem (i.e., goods discontinuously appearing or disappearing). The classic treatment assumes that consumers start and stop consuming goods because the price falls from or goes to infinity. Using stable CES preferences, Feenstra (1994) provides a simple-to-implement formula to adjust price indices for entry and exit of goods. However, when preferences are unstable, it is possible that consumers start or stop consuming a good not because of a change in its availability but because of a change in their tastes. In this section, using the commonly used CES specification, we show that the welfare implications of this can be profound.
Consider a consumer whose preferences are homothetic CES with taste shifters x and elasticity of substitution θ0 > 1. Good i is unavailable if its price is infinite, pi = ∞, and available otherwise. Demand for good i may be zero either because the good is unavailable (pi = ∞) or because the consumer does not value the good (xi = 0).
Split the set of goods that consumers value at t1 into three sets: (i) |$\mathcal {C}$|: continuing goods consumed in both periods, |$b_{it_0}>0$| and |$b_{it_1}>0$|; (ii) |$\mathcal {N}$|: newly consumed goods that either were unavailable at t0 (|$p_{it_0} = \infty$| and |$p_{it_1} < \infty$|) or that were available |$(p_{it_0} <\infty$|) but are valued at t1 and not at t0 (|$x_{it_0} = 0$| and |$x_{it_1} > 0$|); (iii) |$\mathcal {X}$|: exiting goods that become unavailable at t1 (|$p_{it_0} < \infty$| and |$p_{it_1} = \infty$|).
The following proposition derives the change in welfare.
Applying equation (29) requires three pieces of information:
The t1 share of continuing goods, |$b^{c}_{t_1}$|, and changes in the t1 price index for continuing goods |$\frac{P^{c}_{t_1}}{P^{c}_{t_0}}$|.
The price index for newly consumed goods |$\frac{P^{n}_{t_1}}{P^{n}_{t_0}}$|, which combines newly available goods, for which |$\frac{p_{i_{t_1}}}{p_{i_{t_0}}}=0$|, and goods that were available in both periods but consumers did not have tastes for at t0, for which |$\frac{p_{i_{t_1}}}{p_{i_{t_0}}} \ne 0$|.
The counterfactual share of exiting goods at t0 prices but t1 preferences |$b^{{x}_{t_1}}_{t_0}$|. It is reasonable to think that |$b^{x_{t_1}}_{t_0} \in [0,b^{{x}_{t_0}}_{t_0}].$| The lower bound takes the position that exiting goods are no longer valuable to the consumer with t1 tastes, and the upper bound takes the position that exiting goods are not relatively more valuable to the consumer with t1 tastes than the consumer with t0 tastes (i.e., demand curves for those goods that exited did not shift out).
By making different assumptions about the bounds, we can derive three noteworthy special cases of expression (29).
1. No Extensive Margin
2. Feenstra (1994) with Taste Shocks
3. Entry/Exit Only Due to Taste Shocks
VII. Conclusion
In this article, we provide a toolkit for studying how welfare changes in response to changes in budget sets or production possibility sets allowing for preference instability and nonhomotheticity. In contrast to the case of stable and homothetic preferences, there is no model-free statistic like chain-weighted real consumption that simultaneously provides answers to different welfare questions (i.e., equivalent or compensating variation, initial or final preferences, partial or general equilibrium). When preferences are nonhomothetic and unstable, there are different welfare questions one may ask, and they have different answers. We characterize these measures and show the difference between them and chain-weighted real consumption.
Although our motivation and applications have focused on shocks across time, an interesting avenue to explore is welfare comparisons across locations (see Deaton 2003; Argente, Hsieh, and Lee forthcoming). Whereas in a temporal context the preferences of today are typically more relevant than the preferences of yesterday, in a spatial context both locations’ preferences are equally interesting. The distinction between macroeconomic and microeconomic welfare is also important in a spatial context. Comparing budget constraints in one location to another may be misleading as a way to compare the technologies of two economies. This is because even if PPFs in both locations are exactly the same, the relative price of goods households value more in one location will be lower in the other location.35
Appendix A: Proofs
We do not present the proofs in the same order as the propositions in the article because the proofs of earlier propositions rely on notation and logic from later propositions.
If the path of prices is continuously differentiable, we can combine Lemma 1 with the definition of real consumption.
Data Availability
The data underlying this article are available in the Harvard Dataverse, https://doi.org/10.7910/DVN/G3IW0R (Baqaee and Burstein 2022).
Footnotes
We thank Conor Foley, Yasutaka Koike-Mori, and Sihwan Yang for outstanding research assistance. We thank Fernando Alvarez, Andy Atkeson, David Atkin, Natalie Bau, Paco Buera, Javier Cravino, Joe Kaboski, Greg Kaplan, Stephen Redding, Esteban Rossi-Hansberg, Pierre Sarte, David Weinstein, and Jon Vogel for helpful comments. We are grateful to Emmanuel Farhi and Seamus Hogan, both of whom passed away tragically before this article was written, for their insights and earlier conversations on these topics. This article received support from NSF grant no. 1947611. The conclusions and analysis are our own, calculated in part on data from Nielsen Consumer LLC and provided through the NielsenIQ Datasets at the Kilts Center for Marketing Data Center at the University of Chicago Booth School of Business. NielsenIQ is not responsible for, had no role in, and was not involved in analyzing and preparing the results reported herein.
Chain-weighted indices weigh changes in prices or quantities by good-specific weights that are updated every period. The continuous time analog is called a Divisia index. Chained-weighted indices are the standard way to measure real economic activity (real GDP, productivity, real consumption, etc.), and their use is justified under homotheticity and stability of preferences (see United Nations 2009, chap. 15; IMF 2004, chaps. 15 and 17).
For background on how to account for taste shocks in welfare measures, see Fisher and Shell (1968) and Samuelson and Swamy (1974).
Under the assumption that the path of prices is linear, Feenstra and Reinsdorf (2000) show the equivalence between Divisia and a Konüs price index for an intermediate utility level under AIDS preferences. In practice, price paths tend to be nonlinear (for evidence using scanner-level data, see Ivancic, Diewert, and Fox 2011). Therefore, in contrast to Törnqvist and Sato-Vartia, chained indices cannot generically be interpreted as welfare measures corresponding to any well-defined preferences. This is because, as we discuss in Section V, Divisia (or chained) indices are path-dependent, so they can violate basic properties like assigning a higher value to a strictly larger choice set. Oulton (2008) discusses how Konüs price indices resolve the path-dependency problem of Divisia indices.
Baqaee, Burstein, and Koike-Mori (2022) show how Lemma 1 in this article can also be used to accomplish this task under similar assumptions.
For static comparisons aging can be represented using x. This is not the case in a dynamic life cycle model, where c is the entire path of consumption over the life cycle. In this case, preferences over consumption paths are stable even if age affects intratemporal choices (e.g., healthcare versus food when you are young or old). Therefore, standard welfare results with stable preferences apply for ranking price paths over the life cycle.
For optimal policy questions in societies with heterogeneous and/or changing preferences, one must specify a social welfare function that makes the necessary interpersonal utility comparisons. However, this article is about assigning value to choices given some preferences, not about how different preferences should be weighed against each other.
For any variable z, we use the shorthand dz to denote its change over infinitesimal time intervals, |$dz \equiv \frac{dz}{dt} dt$|, so that |$\Delta z=\int _{t_{0}}^{t_{1}}dz$|. The last term on the right-hand side of equation (2) suppresses dependence on t in the integral. We sometimes use this convention to simplify notation.
In discrete time, one can approximate this Riemann integral in different ways. For example, we can use left-Riemann sums (chained Laspeyres), right-Riemann sums (chained Paasche), or midpoint Riemann sums (chained Törnqvist or Fisher). In continuous time, all of these procedures are equivalent and yield the same answer.
The ratio of expenditure functions, holding fixed utility, is called a Konüs (1939) price index. Equation (5) shows that EVm requires deflating nominal income by the t1 utility and preferences Konüs price index. This is in contrast to common practice in the index number theory literature, say Diewert (1976), that uses an intermediate level of utility or preferences between |$u_{t_0}$| and |$u_{t_1}$|. We discuss how our results relate to this alternative approach in Online Appendix B. By definition, EVm depends only on initial and final prices and income, given t1 preferences. Therefore, the integral in equation (4) must be path independent and yields the same answer under every continuously differentiable path of prices that go from |$p_{t_0}$| to |$p_{t_1}.$|
In practice, estimating the elasticity of substitution θ0 must take into account the possibility of demand shocks and income effects. For example, Auer et al. (2022) estimate compensated price elasticities and apply Lemma 1 to measure the heterogeneous welfare effects of changes in foreign prices in the presence of demand nonhomotheticities.
When there are no taste shocks, real consumption, defined by equation (2), is a multigood version of the change in consumer surplus, which is the area under the Marshallian demand curve. Similarly, by equation (4), welfare is the area under a Hicksian demand curve. Hence, in a partial equilibrium context with stable preferences, the gap between real consumption and welfare is also the gap between consumer surplus and welfare, studied by Hausman (1981) and McKenzie and Pearce (1982), among others. This equivalence does not hold when preferences are unstable because in this case Marshallian consumer surplus is not the same as chained real consumption.
For local approximations throughout the article, we assume that the exogenous parameters are smooth functions of t and that the expenditure function is a smooth function of preference parameters x.
See Online Appendix D for a more detailed derivation. We assume here that all goods have positive expenditure shares. We discuss new goods in Section VI.
In terms of primitives of the utility function, |$\varepsilon _{i}=1+\left(1-\theta _0\right)(\frac{\xi _{i}}{\mathbb {E}_{b}\left(\xi \right)}-1)$| and |$d\log \mathtt {x}_{i} = d\log x_{i}-(\frac{\xi _{i}}{\mathbb {E}_{b} (\xi)})\mathbb {E}_b (d\log x)$|. As expected, only relative ξ’s and relative x’s are identified, but not the overall levels |$\mathbb {E}_b (\xi)$| and |$\mathbb {E}_b (d\log x)$|, since these are not pinned down by the underlying preference relation.
In Section II.B, we pointed out that starting at |$b_{t_1}$| computing welfare does not require knowledge of income elasticities or taste shocks if we know the elasticities of substitution. However, the approximation in expression (16) depends on income elasticities and taste shocks. The reason is because this approximation is around initial budget shares |$b_{t_0}$|. If we start with budget shares at t1 and take an approximation as t0 moves back in time, we get
See also Balk (1989) and Martin (2020) for related discussion of the cardinal and ordinal approaches.
Hence, calculating CVm requires knowledge of initial budget shares and elasticities of substitution, whereas equivalent variation EVm requires knowledge of final budget shares and elasticities of substitution. This means that EVm at final preferences is more convenient for ex post comparisons and CVm at initial preferences is more convenient for counterfactuals.
In Baqaee and Burstein (2021) we show that our results can be generalized to economies with heterogeneous agents and distortions. To measure the change in welfare from t0 to t1, we ask: “what is the minimum amount endowments in t0 must change so that it is possible to make every consumer indifferent between t0 and t1?” We show that the results in this article generalize to economies without representative agents if we define social welfare in this way, and this definition collapses to the definition we use here when there is a representative agent. We capture distortions by including wedges as part of the primitives of the economy.
Note that unlike utility, physical output of each individual good has interpretable units and can be measured directly, without the need to use a money metric.
Allowing for endogenous labor-leisure choice requires including the time endowment in L and leisure in the consumption bundle c.
Surprisingly, even though macro welfare is the solution to this seemingly complicated fixed point problem, our analytical results will show that it is in fact simpler to characterize than micro welfare in general equilibrium.
Although EVm and EVM can have different signs when preferences are unstable, they have the same sign when preferences are stable. This is because ut is equal in the micro and macro problem (by the first welfare theorem), and EVm > 0 and EVM > 0 if, and only if, |$u_{t_1}>u_{t_0}$|.
Following the observation made in Section II.D, for compensating variation at initial preferences, we need to know elasticities of substitution at the initial allocation instead of the final one.
For all of these expressions, the summations are evaluated over all goods and factors, so that i and j ∈ {0} + N + F, |$Cov_{\Omega ^{(j,:),t}}(\cdot)$| is the covariance using the jth row of Ω at time t as the probability weights, and Ψ(:, i), t is the ith column of the Leontief inverse at time t.
Technically, this is an approximation, since we define aggregate TFP in continuous time but the data are measured in discrete time (at annual frequency). However, this approximation error, resulting from the fact that the Riemann sum is not exactly equal to the integral, is likely to be negligible in practice. At our level of disaggregation, long-run TFP growth is very similar if we weight sectors using sales shares at time t or time t and t + 1 averages.
For each industry, the change in TFP is itself a chain-weighted index calculated as output growth minus share-weighted input growth. Inputs are industry-level measures of materials, labor, and capital services.
This intuition is flipped for compensating variation at 1947 preferences. Preferences in 1947 favor manufacturing over services. Therefore, at 2014 prices, when manufacturing is relatively cheap, households require a larger reduction in 2014 income to make them indifferent to 1947. More generally, if structural transformation is purely due to income effects or preference instability, then welfare-based productivity growth using CV at initial preferences is given by initial sales-share weighted productivity growth, |$\sum ^{t_1}_{t=t_0}\sum _{i\in N}\lambda _{it_0} \Delta \log A_{it}$| (which corresponds to the Initial Shares line in Figure III), so for this exercise, the Baumol adjustment is not welfare relevant. The fact that EV at final preferences and CV at initial preferences are different stems from the fact that they answer different questions, so EV uses demand from 2014 whereas CV uses demand from 1947.
In Online Appendix G, we formally show that the biases in industry-level data are not diversified away as we aggregate, even if all products are infinitesimal in their industry and all industries are infinitesimal in the aggregate economy. Furthermore, we provide conditions under which the within-industry biases are, to a second order, linearly separable from the across-industry biases. That is, the overall bias is the sum of the cross-industry bias (that we studied in the previous section) plus additional biases driven by within-industry covariance of supply and demand shocks.
Estimating the elasticity of substitution is beyond the scope of this article, therefore for our empirical illustration we draw on estimates from the literature. An elasticity of 4.5 is at the lower range of estimates reported by Redding and Weinstein (2020) and Jaravel (2019). In Online Appendix H, we report results for higher and lower elasticities. We find that the size of the bias is increasing in the elasticity of substitution. In this sense, the results in Figure IV are relatively conservative.
If we were to average initial and final tastes for each t0, then we would get a series that is fairly close to the chained index in this example. Since the chained index is, to a second order, equal to the average of the series with initial and final tastes, this suggests that the second-order approximation is relatively accurate for this example. In Online Appendix H, we also report results using monthly, rather than annual, data. In that case, the chained measure is not as close to the average of initial and final tastes because the second-order approximation is less accurate. The “taste-adjusted inflation rate” in Redding and Weinstein (2020) is lower than the chain-weighted inflation. This may appear related to our finding that inflation for initial tastes is lower than chained inflation. However, for initial tastes, our welfare-relevant inflation is simply exact hat-algebra with initial expenditure shares, discarding any information on subsequent taste shocks. This is different to Redding and Weinstein (2020) whose measure does account for changing tastes but models an increase in tastes as equivalent to a reduction in price and pins down the overall level of taste shifters by assuming that their average stays constant through time. For a better understanding of why our approach is fundamentally different to that of Redding and Weinstein (2020), see Online Appendix C.
Cavallo (2020) argues that, during this episode, the fact that price indices were not being chained at high enough frequency led to “biases” in official measures of inflation. However, as we have argued, chaining is only theoretically valid if expenditure switching is caused by substitution effects and not if expenditure switching is caused by shocks to demand.
Changes in labor by sector and personal consumption expenditures, used to calibrate supply and demand shocks, are taken from Baqaee and Farhi (2022). For related analysis of COVID-19-induced supply shocks, see Bonadio et al. (2021) and Barrot, Grassi, and Sauvagnat (2021). For related analysis of COVID-19-induced demand shocks, see Cakmakli et al. (2020).
This is related to a problem known as “chain drift” bias in national accounting. Chain drift occurs when a chained index registers an overall change between t0 and t1 even though all prices and quantities in t0 and t1 are identical. This is a specific manifestation of path dependence of chained indices (see Hulten 1973) and, by the gradient theorem for line integrals, it must be driven by either demand instability, income effects, or approximation errors due to discreteness. Chain drift bias can appear when movements in prices and quantities are oscillatory, where changes that take place over some periods are reversed in subsequent periods. Welfare changes do not exhibit chain drift since, by definition, they depend only on t0 and t1 variables.
For example, the model in Arkolakis (2016) predicts that changes in tastes induced by advertising will be correlated with changes in physical productivity, whereby more productive firms will expend more resources on advertising. In this case prices of newly consumed goods increase on average less than continuing goods.
This implies that consumers from one location prefer the budget set in the other location, which resembles the Gerschenkron (1951) effect that the relative GDP of a country is higher when evaluated at another country’s prices (the grass is greener in the other side). However, while the Gerschenkron effect is a spatial version of the discrepancy between Laspeyres and Paasche indices, our result is driven by the endogeneity of relative prices to demand forces and not by substitution bias.