What does consumer heterogeneity mean for measuring changes in the cost of living?

Transcription

1 What does consumer heterogeneity mean for measuring changes in the cost of living? Robert S. Martin Office of Prices and Living Conditions FCSM Conference 3/9/ / 25

2 Disclaimer The views expressed herein are those of the author and not necessarily those of the Bureau of Labor Statistics or the U.S. Department of Labor. 2 / 25

3 Contents Introduction Theory Data Results Conclusion 3 / 25

4 Motivation: COGI vs. COLI The Consumer Price Index (CPI-U) and Chained Consumer Price Index (C-CPI-U) aim to measure changes in the cost of living Cost of Living Index (COLI) Inflation is measured as the change in expenditures required to maintain a fixed standard of well-being Cost of Goods Index (COGI) Inflation is measured as the change in expenditure required to continue consuming a fixed basket of goods 4 / 25

5 Motivation: Nonhomotheticity Need (at least implicit) model of consumer preferences Tractability v. generality Homotheticity: assumption that substitution patterns do not vary with income Heterogeneity in expenditures well-documented Is homotheticity a decent approximation for aggregate indexes? National Research Council (2002): literature tends to show low variation in inflation rates across groups in long run Short run or more recent differences Much existing research uses individual or group-specific category weights 5 / 25

6 6 / 25 Hottman and Monarch (2018) relax homotheticity in economic model and describe differences in import price inflation for different income groups Potentially very important for consumer cost of living measurement I replicate HM s method of estimating nonhomothetic consumer demand using household scanner data Preview: Some indications against homotheticity but model may not be suited for my scanner data

7 Theory: Consumer preferences Constant Elasticity of Substitution (CES) utility over sectors s (closely related goods, i.e. Coffee ) V ht = [ s S ϕ σ 1 σ hst Q σ 1 σ hst ] σ σ 1, (1) Within sectors, generalized-ces over varieties v (i.e. Folgers 24.2 oz Colombian - Ground ) similar to Stone-Geary Q hst = v Gs ϕ σ s 1 σ s vt (q hvt α v ) σ s 1 σ s σ s σ s 1, (2) Like Stone-Geary, α v loosely interpreted as subsistence quantity of good v, but allowed to be negative 7 / 25

8 Theory: Consumer preferences Constant Elasticity of Substitution (CES) utility over sectors s (closely related goods, i.e. Coffee ) V ht = [ s S ϕ σ 1 σ hst Q σ 1 σ hst ] σ σ 1, (1) Within sectors, generalized-ces over varieties v (i.e. Folgers 24.2 oz Colombian - Ground ) similar to Stone-Geary Q hst = v Gs ϕ σ s 1 σ s vt (q hvt α v ) σ s 1 σ s σ s σ s 1, (2) Like Stone-Geary, α v loosely interpreted as subsistence quantity of good v, but allowed to be negative 7 / 25

9 8 / 25 Model accommodates entering and exiting varieties (non-continuers) Regularity conditions for individual demand include α v 0 for non-continuers Using aggregate data for a price index requires α v = 0 for non-continuers Demand curves, budget shares, and cost function (P t ) follow from maximizing Eq. s (2) and (1) subject to budget constraint (which depends on prices p vt )

10 9 / 25 Aggregate COLI from period t to t + i in HM model: P t+i P t = ( s S ϕσ 1 st+i P1 σ st+i ( s S ϕσ 1 st Pst 1 σ ) 1 σ 1 ) 1 σ 1 ( Yk ) s S v G s (α v n t )p vt s S v G + s (α v n t+i )p vt+i (3) Y k Sectoral price index P st depends on σ s and variety-level demand shifters ϕ vt Compare to Laspeyres Or Törnqvist where w v = 1 2 ( P t+i P t = P t+i P t r S r S = s S Y k v G s p vt+i q vt (4) v G s p vt q vt v G s ( pvt+i p vt ) wv (5) ) p vt+i q vt+i p r S v Gs p vt+i q vt+i + vtq vt r S v Gs pvtqvt

11 9 / 25 Aggregate COLI from period t to t + i in HM model: P t+i P t = ( s S ϕσ 1 st+i P1 σ st+i ( s S ϕσ 1 st Pst 1 σ ) 1 σ 1 ) 1 σ 1 ( Yk ) s S v G s (α v n t )p vt s S v G + s (α v n t+i )p vt+i (3) Y k Sectoral price index P st depends on σ s and variety-level demand shifters ϕ vt Compare to Laspeyres Or Törnqvist where w v = 1 2 ( P t+i P t = P t+i P t r S r S = s S Y k v G s p vt+i q vt (4) v G s p vt q vt v G s ( pvt+i p vt ) wv (5) ) p vt+i q vt+i p r S v Gs p vt+i q vt+i + vtq vt r S v Gs pvtqvt

12 Tradeoffs Törnqvist index Flexible (see Diewert 1976) No parameters to estimate COLI interpretation requires homotheticity HM index Assumes specific model Need to estimate σ, σ s s, α v s, ϕ vt s, ϕ hst s Explicitly allows nonhomotheticity 10 / 25

13 Theory: Completing the model Market demand with n t consumers in period t ( ) pvt σs ϕ σs 1 vt q vt = (α v n t ) + Y j G s p 1 σs jt ϕ σs 1 t (α j n t )p jt, jt j G s (6) Market structure for firms/varieties is monopolistic competition Price = Markup Marginal Cost p vt = ε vt ε vt 1 δ vt(1 + ω s )q ωs vt (7) ( ) where ε vt = qvt αv n t q vt σ s is the elasticity of demand perceived by the firm and is the firm s markup εvt ε vt 1 11 / 25

14 Theory: Estimation Carried out sector-by-sector using price (p vt ), sales (p vt q vt ) data for each variety v and time period t To reduce dimensionality, HM model α v as: α v = 1 ] [β j n smin (q vt q vt > 0), for v G j 1 t s, j {C, E} (8) where Gs C = set of continuing varieties, and Gs E = set of non-continuing varieties Two parameters to estimate: βs C and βs E Model regularity requires βs C < 1 and βs E 0. Aggregation requires βs E = 0 12 / 25

15 First stage of HM s procedure extends the approach of Feenstra (1994) to estimate the σ s, ω s, β C s, and β E s Simultaneity bias For market demand equation (6) and firm pricing (supply) equation (7), multiply each by p vt, take logs, and difference both over time and with respect to a reference variety k within the same sector s k,t ln (p vt q vt α v n t p vt ) = (1 σ s ) k,t ln p vt + ν vt (9) k,t ln p vt = ω s k,t ln (p vt q vt )+ 1 ( ) εvt 1 + ω s 1 + ω s k,t ln +κ vt ε vt 1 (10) ν vt = (1 σ s ) [ t ln ϕ kt t ln ϕ vt ] and κ vt = 1 1+ω s [ t ln δ vt t ln δ kt ] are the unobserved errors 13 / 25

16 Identifying assumption: for each v, double differenced supply and demand errors ν vt and κ vt are orthogonal moment equation for each v: G(β s ) = E [u vt (β s )] = 0, (11) where β s = ( σ s, β C s, β E s, ω s ), and uvt = ν vt κ vt. Need at least four varieties in a sector to identify β s Generalized Methods of Moments (GMM): Stack sample analogs of varieties into vector G (β s ), then with weighting matrix W ˆβ s = argmin β s { G (β s ) WG (β s ) } (12) 14 / 25

17 Writing Eq. (11) in terms of observables and re-arranging, we have for each variety v: + [ ( ) ] 2 E k,t ln p vt = ω ] s E [ k,t ln (p vt q vt ) k,t ln p vt 1 + ω s 1 [ ] σ s 1 E k,t ln p vt k,t ln (p vt q vt α v n t p vt ) ω [ s (1 + ω s )(σ s 1) E k,t ln (p vt q vt ) k,t ln (p vt q vt α v n t p vt ) + 1 E 1 + ω s 1 + (1 + ω s )(σ s 1) E [ k,t ln p vt k,t ln ( εvt ε vt 1 )] [ k,t ln (p vt q vt α v n t p vt ) k,t ln ( εvt ] )] ε vt 1 (13) 15 / 25

18 Writing Eq. (11) in terms of observables and re-arranging, we have for each variety v: + [ ( ) ] 2 E k,t ln p vt = ω ] s E [ k,t ln (p vt q vt ) k,t ln p vt 1 + ω s 1 [ ] σ s 1 E k,t ln p vt k,t ln (p vt q vt α v n t p vt ) ω [ s (1 + ω s )(σ s 1) E k,t ln (p vt q vt ) k,t ln (p vt q vt α v n t p vt ) + 1 E 1 + ω s 1 + (1 + ω s )(σ s 1) E [ k,t ln p vt k,t ln ( εvt ε vt 1 )] [ k,t ln (p vt q vt α v n t p vt ) k,t ln ( εvt ] )] ε vt 1 (13) 15 / 25

19 Replication: Data HM use supplier trade data from Linked-Longitudinal Firm Trade Transaction Database (LFTTD) based on customs information I use the Homescan database from Nielsen covering 2008Q1 to 2010Q4 60,000 households in the sample per year who scan barcodes to record transactions Nielsen provides frequency weights for projections to U.S. population Prices, quantities, bar codes (UPC), brand information, some characteristics Demographic info for households 16 / 25

20 17 / 25 Coverage is mainly packaged food and general merchandise Only about 13% of U.S. consumer expenditures (BLS Consumer Expenditure Survey) Roughly 700,000 UPC s per year in 122 product groups (sectors)

21 Replication: Method I estimate the model both unconstrained and with βs C < 1 and βs E 0 imposed HM implement other constraints/penalties in estimation which I have not yet replicated 18 / 25

22 Results With different data, I do not expect to replicate HM s results exactly E.g. there should be more substitution between varieties at UPC level than between varieties at HS10-firm level 19 / 25

23 Table: Summaries of σ s, ω s estimates HM Replications with Homescan (1) (2) (3) σ s 10% Median % ω s 10% Median % Sectors Constraints Yes No βs C < 1, βs E 0 20 / 25

24 Table: Summaries of β C s, β E s estimates HM Replications with Homescan (1) (2) (3) βs C 10% 9.96E Median % βs E 10% -5.97E E-15 Median -2.55E % -1.08E Sectors Constraints Yes No βs C < 1, βs E 0 21 / 25

25 Table: Summaries of sales-weighted average elasticities and markups HM Replications with Homescan (1) (2) (3) P.E.D. (ε vt ) 10% Median % Markups ( εvt ε ) vt 1 10% Median % Sectors Constraints Yes No βs C < 1, βs E 0 22 / 25

26 Discussion Homescan results indeed imply greater substitution between UPCs within same product group The estimated P.E.D. from the Homescan data are still somewhat high relative to other work (i.e. Broda and Weinstein 2010) βs E 0 unless otherwise constrained I also find evidence against homothetic CES, but βs E 0 doesn t fit generalized-ces Alternative classification of continuing and non-continuing varieties? 23 / 25

27 Conclusion More work is needed to address variety-level substitution heterogeneity in COLI framework HM also find cross-sector nonhomotheticity may play a larger role in import price index heterogeneity Perhaps worth revisiting group or household-specific expenditure weights for the CPI Broader issue: what to do when the model is fighting the data in terms of theoretical constraints 24 / 25

28 CONTACT INFORMATION Robert S. Martin Division of Price and Index Number Research Bureau of Labor Statistics 2 Massachusetts Ave, NE Washington, DC Martin.Robert@bls.gov 25 / 25