Avoiding aggregation in estimating marginal utilities

Transcription

1 Avoiding aggregation in estimating marginal utilities Ethan Ligon June 24, 2014

2 Dynamics of consumer behavior In just about any model of dynamic consumer behavior we get restrictions on the evolution of marginal utilities; e.g., u (c it ) = βe t R t+1u (c it+1) for the case in which the consumer has access to asset markets. Exploiting these restrictions for estimation or testing requires us to take a stand on what the function u (c) is.

3 What is u (c)? The notation we've used makes it pretty clear that u : R R. But what is this consumer consuming? The Usual Answers (A WRONG TURN!) Consumption c is a xed share of household Total expenditures within a period; or... Just food expenditures; or... Total non-durable expenditures plus the value of ows of services from owned consumer durables, perhaps including housing. As above, but adding in expenditures on stocks of goods stored and brought into the period, while netting out expenditures on goods carried over to a subsequent period.

4 Expenditure Surveys Broad expenditure surveys generally regarded as the gold standard for measuring consumption, especially if repeated longitudinally for a panel of households. Expenditures (or consumption) are better than income, because they're a better measure of permanent income or wealth than is realized income in a particular year. The World Bank has had great success in using expenditures over time using its LSMS family of surveys. The main complaint about these is simply that there aren't enough of them, and that too seldom do they form a panel.

5 The Usual Use of Expenditure Data The usual use of expenditure data such as that from an LSMS-style survey involves constructing a comprehensive expenditure aggregate (Deaton and Zaidi, 2002). This is intended to be a measure of all current consumption expenditures (including all goods and services). Nothing wrong with this aggregate in principle; but May be dicult to back out services from assets or family labor; and There's information in the composition of expenditures that's neglected. There are too few comprehensive surveys because they are too expensive! Lanjouw and Lanjouw (2001) report that the cost of elding a single round of an LSMS survey ranges from $300,000 to $1,500,000, or about $300 per household. Just using such surveys to construct consumption aggregates is inecient.

6 When does it make sense to use an expenditure aggregate for dynamic analyses? Set aside problems of constructing a consumption aggregate. In what world does it even make sense to model consumer preferences in this way? First observation Nobody really thinks consumers are just consuming a single numeraire good, denominated in some currency units. Instead, we should think of u as an indirect utility function. Two-stage budgeting `works' Provided preferences time separable, we can think of u : R n R, and of indirect utility: v(x, p) = max c R n u(c) such that p c x.

7 Indirect utility instead of direct utility If we know (or can estimate) v, then we can swap in v/ x for u, which gives us something with greater logical consistency; e.g., v x (x v it, p t ) = βe t R t+1 x (x it+1, p t+1). For policy & targeting We often want to know which households would benet most from additional resources. This is the same as knowing (an ordering of) partialv x. But what about prices? The same people who constructed the consumption aggregate also probably constructed a price index π(p). Then to complete our justication for using a consumption aggregate, we merely require that v(x, p) v(x/π(p), 1).

8 When is using constant expenditures valid? When will v(x, p) v(x/π(p), 1) hold? What restrictions does this place on the underlying direct utility function? Using Roy's identity: c i = v/ p i v/ x = v (x/π) xπi π v (x/π) 1 π = x π i π. This tells us that demands are all linear in total expenditures x, and pass through the origin. And this is the case if and only if the utility function is homothetic.

9 We've taken a wrong turn I think we've taken a serious wrong turn; the decision to use a consumption aggregate to begin with. There's a dierent path we can take which is Theoretically consistent; Has comparatively modest data requirements; Allows us to focus on attention on goods and services that we think can be well-measured; and Completely avoids the construction of price indices.

10 A Frisch Approach What I'm going to do: I'm going to talk about how we might use disaggregated consumption data to construct something more useful for dynamic analysis. The approach I'll describe: Avoids trying to construct a complete consumption aggregate; Doesn't require homothetic utility or linear Engel curves; Yields measures of marginal utility which are what theory tells us to measure; Should allow for much less expensive data collection. What I may get to Although I'll go through the steps required to use disaggregated consumption data, I won't actually conduct any serious dynamic analyses in this talk. I will show results of a test of consumption-smoothing.

11 Linear Engel Curves The use of consumption aggregates in the kind of dynamic analyses we've discussed assumes, implicitly or explicitly, that Engel curves are linear, and pass through the origin: For example, Sohnesen et al. (2013) construct sub-aggregates, which works perfectly if all Engel curves in the sub-aggregate are linear and have non-negative intercepts. In this case shares of all goods are constant.

12 Non-linear Engel Curves With linear Engel curves (from the origin), there's no information to be had from the composition of consumptionthat's why it's okay to use a consumption aggregate in this case. But Engel curves aren't generally (or even typically) linear! These Engel curves cross. If we simply ask a question like Did you spend more on green or on blue last year? the answer tells us whether total expenditures are greater or less than ve.

13 Incomplete Systems of Engel Curves If Engel curves are suitably non-linear, we don't even need to observe all expenditures. Even if we don't observe red, we still learn all we need from knowing the proportion of green to blue.

14 Indirect Utility Think back to our one-period consumer problem, where total expenditures are x. The consumer solves v(x, p) = max U(c 1,..., c n ) (1) {c i } n i=1 subject to a budget constraint n p i c i x. (2) i=1 Envelope Condition The marginal value of another dollar is given by the envelope condition: V x = λ. The number λ is a measure of neediness.

15 Neediness is what we want Looking back at our example Euler equation, this neediness variable is exactly what we want: λ it = βe t R t+1λ it+1. In fact, the restrictions we're interested from theory are typically even linear in λ! (This idea recently exploited by (H. Youn Kim, 2013).)

16 A Quick, Inadequate Nod to the Literature Taking the Frischian approach to measuring marginal utilities in dynamic settings isn't my idea. Best past examples come from the labor literature. Paper Expnd. Data Rank Rsp. E Heckman (1974) Retro-Panel Heckman and MaCurdy (1980) Panel 2 Browning et al. (1985) Agg Cohort 1 No Atteld and Browning (1985) Disagg Time series 1 No Kim (1993) Disagg None 1 No Blundell et al. (1994) DisAgg Cohort 2 No Blundell (1998) Disagg None 3 No H. Youn Kim (2013) Disagg Time series 3 Yes This study Disagg Panel n Yes*

17 Modeling Non-linear Engel Curves We want a demand system that generates non-linear Engel curves. Desirable properties: Demands (log) linear in parameters; No arbitrary restrictions on the rank of the demand system, (to avoid articial restrictions on shape of Engel curves); Allow for Non-separabilityshould allow arbitrary cross-price elasticities; Flexible functional form (in the sense of Diewert); Naturally expressed in terms of λs.

18 Dierentiable Demand Systems We want a Frischian demand relation that can be estimated using the kinds of data we have available on disaggregated expenditures. Atteld and Browning (1985) use a dierentiable demand approach which yields Frischian (aggregate) demands without requiring separability. Their demands can depend on all prices, yet one need only estimate demand equations for a select set of goods. Atteld-Browning obtain a Rotterdam-like demand system in quantities; We obtain something closer to an AID specication in expenditures.

19 Derivation of the Demand System Work with the consumer's prot function, π(p, r) = max ru(c) pc, c where r has the interpretation of being the price of utility. Then: 1. The price r is equal to the quantity 1/λ; 2. The prot function is linearly homogeneous in p and r; 3. By the envelope theorem π i (p, r) = c i for all i = 1,..., n; 4. And (since we want to work with expenditures) p i π i = x i.

20 Derivation of the Demand System Take the total logarithmic derivative of x i = p i π i, yielding x i d log x i = π i p i d log p i p i n π ij p j d log p j p i π ir rd log r. j=1 Recalling that π i p i = x i Dierential Demand we get an expression for d log x i = d log p i + n j=1 π ij π i p j d log p j + π ir π i rd log r. (3)

21 Restrictions on Demands from Consumer Theory The two restrictions on demands so far is that (i) they're positive and (ii) continuously dierentiable. That's about to change: Let θ ij = π ij π i p j denote the (cross-) price elasticities of demand; Let β i = π ir π r i denote the elasticity of demand with respect to r. (Equal to minus the elasticity of demand with respect to λ). Then: 1. β i = n j=1 θ ij; (from linear homogeneity of the prot function). 2. If utility function is twice continuously dierentiable, then by Young's theorem we know that θ ij = θ ji.

22 Additional Identifying Restrictions So far our demand equations give us an exact description of how expenditures will change in response to innitesimal changes in prices. Two Additional Assumptions 1. Elasticities {θ ij } and {β i } are constant; 2. Our demand equation with constant parameters gives a good approximation of how demand changes with respect to larger changes in prices.

23 Estimating Equations Allowing also for household characteristics z t to serve as demand shifters, discrete-time specication of demands is n log x it = log p it θ ij log p jt +β i δ z i t β i log λ t + ξ it, j=1 where ξ it is an approximation error. (4)

24 Estimating Demands How we go about estimating the demand system depends on the nature of the data we have. I've devised methods to cover the case of: Cross-sectional data with prices Cross-sectional data without prices Panel data (with or without prices).

25 Estimation Using a Panel Suppose household panel expenditure data, with households all facing the same prices (which the econometrician does not need to observe). Then: Our Structural estimating equation Is just the sum of ve terms: log x j it = ( log p it n θ ij log p kt )Prices k=1 +β i δ i z j t Observed HH Characteristics β i log λ j tneediness + ξ j Approximation error it +β i ɛ j.unobserved HH Characteristics it

26 Reduced form estimating equation where y j it = a it + b i ( z j t z t ) + c i w j t + e j it, (5) y j = log x j it it [ a it = log p it ] n θ ij log p kt β i log λ t + β i ɛ it + ξ it k=1 b i = β i δ i e j = β it i( ɛ j ɛ it it) + ( ξ j ξ it it) c i w j t = β i ( log λ j t log λ t ).

27 Estimation y j it = a it + b i ( z j t z t ) + c i w j t + e j it, 1. Use least squares to estimate reduced form parameters (a it, b i ) (with a it good-time eects). 2. The residuals are c i w j t + e j. The rst term of this sum is what it we're interested in. 3. Arrange the c i w j t + e j as an n NT matrix Y : it The rst term captures eect of λ on demands.

28 Backing Out λ 1. The matrix [c i w j t ] is the outer product of two vectors, so it must be rank one. 2. The second matrix depends on changes in l < n household characteristics, and has rank of at most l. 3. Use singular value decomposition to obtain estimates of these two matrices. 4. Then we have c i w j t β 1 β 2 =. [ β i log λ 1 1 log λ 1 log λ 2 1 log λ 1... log λ j t. β n which allows us to identify log λ j t log λ t normalization. up to a

29 Selecting Particular Goods One of the attractive features of Frisch demand systems is that it's very simple to estimate incomplete demand systems. So: We don't have to know about expenditures for all goods; Instead, we can simply choose a few goods. The cost of using a smaller set of goods is simply that one ignores a possible additional useful source of information on neediness.

30 How to Select Goods Cost of ignoring a good i is small if Elasticity β i is small variation in expenditures simply isn't closely related to variation in neediness. Fit is poor for good i If one of our estimated equations is a poor t to data (measurement error, important unobservable household characteristics, approximation error, parameters not constant) then we don't learn much about λ from that equation. Two simple criteria for including a good High ˆβ i Low R 2 We only get estimates up to a scale parameter, but this still allows us to know how useful a good is for inferring log λ. If the t is poor, variation in expenditures may depend more on the error term in the regression than

31 Aggregate and Mean Consumption Shares in Uganda Table : Shares of Aggregate Expenditures in Uganda, (2005 and 2010) for all goods with shares greater than 3%. Agg. Shares Mean Shares Expenditure Item Imputed rent of owned house Matoke (Bunch) Sweet potatoes (Fresh) Maize (our) Medicines etc Water Food (restaurant) Beef Sugar Beans (dry)

32 Aggregate vs. Mean Consumption Shares in Uganda All Consumption Items

33 Aggregate vs. Mean Consumption Shares in Uganda All Food

34 Aggregate vs. Mean Consumption Shares in Uganda Slightly Aggregated Food

35 Estimates from Uganda (Slightly Aggregated Foods) Food category HHSize Rural φβ i R 2 % Zeros Other foods Restaurant meals Infant formula Tobacco Coee Peas Sorghum Maize Ground nut Fresh milk Tea Sugar Fruits Oils Meat

36 Cross-sectional distribution of log λ We can back out estimates of log λ (up to a scale parameter): These are the objects we're after!

37 Changes in Expenditures vs. Changes in λ

38 Correlations between dierent welfare measures Agg. All All (0.8) S.A. Exp. Food Food Food Agg. Exp All All Food All Food (0.8) All Food (0.5) S.A. Food S.A. Food (0.8) Pearson correlation coecients are below the diagonal; Spearman correlation coecients above. Diagonals are the proportion of total residual variation accounted for by variation in neediness.

39 Full Insurance? We can use our computed log λs to test the full insurance hypothesis of (Townsend, 1994) and others. With estimates of the marginal utilities already in hand, this boils down to the extremely simple regression log λ j t = α + β log y j t, where y j t is a measure of household income. Test is H 0 : β = 0.

40 Full Insurance Test (λ)

41 Full Insurance Test (S.A. food expenditures)

42 Final Thoughts The main point: Looking at disaggregated non-durable expenditures may be an inexpensive way to infer household marginal utilities. Focus should be on goods with non-linear Engel curves that are easy to measure. A secondary point: Marginal utilities are usually what we want in dynamic analyses and for policy purposes anyway. Estimating these directly from a small number of consumption goods seems much better than collecting data on all goods and service expenditures, adding these up, normalizing by some price index, and then assuming that Engel curves are linear.

43 The Usefulness of the Composition of Bundles Instead of collecting data on 61 dierent food expenditures, collect data on fewer, but focus on eliminating zeros: 21 food items Do as well (Correlation of 0.98) 11 food items Do almost as well (Correlation of 0.94). [fresh cassava, fresh sweet potatoes, bread, rice, fresh milk, tea, beef, sugar, cooking oil, onions, and tomatoes] Zeros the biggest problem for Uganda Cause problems for two reasons: 1. Violates assumptions of the dierential demand system. (Fix possible?) 2. Measurement error; perhaps related to the fatigue or inattention of enumerators or respondents. Doesn't eect λ estimates much (Certainly less than usual poverty measures.)

44 References Atteld, C. L. and M. J. Browning (1985). A dierential demand system, rational expectations and the life cycle hypothesis. Econometrica 53(1), Blundell, R. (1998). Consumer demand and intertemporal allocations: Engel, Slutsky, and Frisch. Econometric Society Monographs 31, Blundell, R., M. Browning, and C. Meghir (1994). Consumer demand and the life-cycle allocation of household expenditures. Review of Economic Studies 61, Browning, M., A. Deaton, and M. Irish (1985). A protable approach to labor supply and commodity demands over the life cycle. Econometrica 53, Deaton, A. and S. Zaidi (2002). Guidelines for Constructing Consumption Aggregates for Welfare Analysis. World Bank Publications. H. Youn Kim, Keith R. McLaren, K. K. G. W. (2013). Empirical demand systems incorporating intertemporal consumption

45 The Variable Elasticity of Substitution Demand System Let a household's preferences over n dierent consumption goods depend on a momentary utility function n (c i + φ i ) 1 γ i 1 U(c 1,..., c n ) = α i. (6) 1 γ i i=1 The parameters {γ i } govern the curvature of the n sub-utility functions; The parameters {α i } govern the weight of the n sub-utilities in total momentary utility; and The parameters {φ i } `translate' the commodity space in such a way to make it simple to accomodate subsistence levels for some goods, or more generally to control the marginal utility of consumption near zero for any of the goods. Novelty comes from the fact that the γ i are allowed to dier Frisch Demands in the VES System