A Unied Approach to Aggregate Price and Welfare Measurement

Transcription

1 A Unied Approach to Aggregate Price and Welfare Measurement Stephen J. Redding Princeton David E. Weinstein Columbia May, / 81

2 Motivation Existing measures of the aggregate cost of living assume constant preference parameters for each good Rules out taste shocks for individual goods Micro data display substantial shifts in expenditure shares conditional on prices Consistent with taste shocks for individual goods We develop a unied approach to the price index and the demand system for CES preferences that allows taste shocks Rationalize observed data on prices and expenditure shares Make consistent comparisons of welfare over time Feasible to implement across a broad range of sectors Key insight: demand system inversion to express unobserved tastes in terms of observed prices and expenditure shares 2 / 81

3 CES Unied Price Index (UPI) We develop a unied approach for CES preferences that consistently estimates the cost of living and demand when demand for each good is time varying Rationalizes observed data on prices and expenditure shares as an equilibrium of the model Allows for entry and exit of goods over time Identies a unique elasticity of substitution (s) Satises money-metric utility (change in the cost of living depends on prices and expenditures) Yields consistent aggregation from micro to macro Shows that existing exact price indexes are biased in the presence of mean zero demand shocks Related to all major micro, macro, and statistical approaches to price measurement Extensions to non-homothetic CES, nested CES, mixed CES, logit, mixed logit, and translog preferences 3 / 81

4 Related Literature CES demand in economics Armington (1969), Dixit & Stiglitz (1977), Krugman (1980, 1991), Blanchard & Kiyotaki (1987), Eaton & Kortum (2002), Anderson & van Wincoop (2003), Antràs (2003), Melitz (2003), Helpman, Melitz & Yeaple (2004), Broda & Weinstein (2006, 2010), Hsieh & Klenow (2009), and Arkolakis, Costinot & Rodriguez-Clare (2012) Demand estimation McFadden (1974), Deaton & Muellbauer (1980), Anderson, de Palma & Thisse (1992), Feenstra (1994), Berry, Levinsohn & Pakes (1995), McFadden & Train (2000) New goods and welfare Feenstra (1994), Bresnahan & Gordon (1997), Hausman (1997), Broda & Weinstein (2006, 2010), Petrin (2002) Price indexes Fisher (1922), Törnqvist (1936), Diewert (1976), Sato (1976), Vartia (1976), Feenstra (1994), Balk (1999), Neary (2004) 4 / 81

5 Outline CES Unied Price Index Relation to Existing Price Indexes Estimating the Elasticity of Substitution Data and Results Conclusions 5 / 81

6 CES Demand and Price Index CES unit expenditure function over demand-adjusted prices: P t = " Â k2w t Pkt j kt 1 s # 1 1 s, Demand system equation for expenditure shares: S kt = (P kt /j kt ) 1 s Â`2Wt (P`t /j`t ) 1 s = (P kt/j kt ) 1 s P 1 s, k 2 W t t Demand evolves according to: ln j kt = ln j k + ln q kt, ln q kt F µ q, c 2 q, where ln q kt observed after a good is supplied to the market and the conventional approach is special case of c 2 q = 0 6 / 81

7 Product Turnover Share of each individual common good in total common good expenditure (W t,t 1 W t and W t,t 1 W t 1 ): S kt P kt C kt Â`2Wt,t 1 P`t C`t = (P kt /j kt ) 1 s Share of common goods in total expenditure: Â`2Wt,t 1 (P`t /j`t ) 1 s, k 2 W t,t 1 l t  k2w t,t 1 P kt C kt =  k2w (P t,t 1 kt/j kt ) 1 s  k2wt P kt C kt  k2wt (P kt /j kt ) 1 s CES unit expenditure function for common goods: P t = "  k2w t,t 1 Pkt j kt 1 s # 1 1 s. 7 / 81

8 CES Price Index: Change in Cost of Living F t 1,t P 1 t lt s 1 = P t 1 l t 1 Exact CES price index: F t 1,t = lt l t 1 1 s 1 " Âk2Wt,t 1 (P kt/j kt ) 1 s  k2wt,t 1 (P kt 1 /j kt 1 ) 1 s k2w t,t 1 Pkt /j kt P kt 1 /j kt 1 " S wkt = kt Skt # " # 1 S `t S `t ln Skt ln Skt 1 /  1 ln S `2W `t ln S `t, t,t 1 1 Demand system inversion: j kt = j kt j t = P kt P t Skt S t w kt, 1 s 1 1 jkt, N t,t 1  ln = 0. j k2w t,t 1 kt 1 # 1 1 s 8 / 81

9 The Unied Price Index Proposition The unied price index (UPI) which is exact for the CES preference structure in the presence of changes in the set of goods, demand shocks for individual goods, and discrete changes in prices and expenditure shares is given by F U t 1,t = lt l t 1 1 s 1 {z } Variety Adjustment " P t S 1 # s 1 t P t 1 S t. 1 {z } Common-Goods UPI Note: lim s! FU t 1,t = F Jevons translog t 1,t For s <, variety and heterogeneity terms 9 / 81

10 Outline CES Unied Price Index Relation to Existing Price Indexes Estimating the Elasticity of Substitution Data and Results Conclusions 10 / 81

11 Consumer Valuation Bias Sato-Vartia exact CES price index is biased in the presence of demand shocks ln F CG t 1,t =  wkt ln Pkt 1 P kt 1 {z } k2w t,t ln F SV t 1,t apple wkt Skt Skt 1 = ln Skt ln Skt 1 / "  k2w t,t  S `2W t,t 1 wkt ln jkt 1 j kt 1 `t S `t 1 ln S `t ln S `t 1 Mechanical positive correlation between w kt and j kt/j kt 1 : wkt (j kt /j kt 1 ) (j kt /j kt 1 ) wkt > 0. Consumers substitute towards goods whose price (P kt ) falls relative to demand (j kt ) more # 11 / 81

12 Equivalences Three expressions for the change in the cost of living: F F t 1,t = P 1 " # 1 t lt s 1 = P t 1 l t 1 Â Skt Pkt /j 1 s 1 s kt 1 P k2w t,t 1 kt 1 /j kt 1 F B t,t 1 = P t 1 P t = lt 1 l t 1 " s 1 Â k2w t,t F U t 1,t= P 1 t lt s 1 = P t 1 l t 1 Skt P t P t 1 Pkt 1 /j kt 1 P kt /j kt S t S t 1! 1 s 1 1 s # 1 1 s 3 5 where ln j t j t 1 = 1 N t,t 1 Â k2w t,t jkt ln = 0 1 j kt 1 deriv statpindex econpindex 12 / 81

13 The Big Picture Quadratic Mean of Order r = 2(1 ) r 6= 2(1 ) Sato Vartia CES 6= 1 Cobb- Douglas 6= 0 6= 1 t/ t 1 6= 1 PFW Fisher Törnqvist Feenstra CES Jevons Key : Elasticity of Substitution PFW: Purchase Frequency Weighting 'k,t/'k,t 1 = 1: No Demand Shifts t/ t 1 = 1: No Change in Variety 'k,t/'k,t 1 6= 1 6= 0 t/ t 1 6= 1 'k,t/'k,t 1 6= 1 Unified Price Index Aggregation 6= 1 'k,t/'k,t 1 6= 1 6= 0 t/ t 1 6= 1 Logit/Fréchet Laspeyres PFW Carli PFW Paasche PFW PFW Dutot 13 / 81

14 Extensions Non-homothetic CES (indirectly additive) nonhomothetic Nested CES Mixed CES Logit and mixed logit logit Translog translog Demand system inversion requires assumption of connected substitutes to be satised (Berry, Gandhi and Haile 2013) 14 / 81

15 Outline CES Unied Price Index Relation to Existing Price Indexes Estimating the Elasticity of Substitution Data and Results Conclusions 15 / 81

16 Reverse-Weighting Estimator CES expressions for change in the cost of living imply: " Q F t 1,t  S Pkt kt 1 P k2w t,t 1 kt 1 " 1 Q B t,t 1  S Pkt kt P k2w t,t 1 kt 1 # s 1 s = 4 P t P t 1 # 1 2 (1 s) 1 s = 4 P t P t 1 S t S t 1! 1 3 s 1 5 S t S t 1! 1 3 s 1 5 where (Q F t 1,t, QB t,t 1 ) are aggregate demand shifters aggdemand " # 1 s 1 s 1 Q F t 1,t =  Skt jkt 1, j k2w t,t 1 kt " # 1 s 1 s 1 Q B t,t 1 =  Skt jkt 1. j k2w t,t 1 kt 1 16 / 81

17 Proposition Reverse-Weighting Estimator For small changes in demand ((j kt /j kt 1 )! 1), money-metric utility is satised (Q F p t 1,t! 1/Q B p t,t 1! 1), and the reverse-weighting estimator consistently estimates the elasticity of substitution (ŝ RW! p s). proof Proposition Assuming that demand shocks are uncorrelated with price shocks for each good and independently and identically distributed across goods: (j kt /j kt 1 ) i.i.d. 1, c 2 j. As the number of common goods becomes large (N t,t 1! ), the reverse-weighting estimator consistently estimates the elasticity of substitution (ŝ RW! p s). proof 17 / 81

18 Generalized-Reverse-Weighting Estimator Correlated and orthogonal components of demand shocks " g = g (s) = r c q = 1 s 1 + c # ps c p s 1 c 2, p S kt (s) = (P kt /P kt 1 ) g(s 1) S kt Â`2Wt,t 1 (P`t /P`t 1 ) g(s 1) S `t Proposition Assume that demand shocks for each good (q kt /q kt 1 2 (0, )) can be partitioned into a component that is correlated with price shocks and an orthogonal component ((e kt /e kt 1 ) i.i.d (1, ye) 2 for e kt /e kt 1 2 (0, )), and are independently and identically distributed across goods. As the number of common goods becomes large (N t,t 1! ), the Generalized-Reverse-Weighting (GRW) estimator consistently estimates the elasticity of substitution (ŝ GRW p! s D ). 18 / 81

19 Outline Unied Price Index Relation to Existing Price Indexes Estimating the Elasticity of Substitution Data and Results Conclusions 19 / 81

20 Nielsen Homescan Data Approximately 55,000 households scan in every purchase of a good with a barcode Observe price paid (including coupons) and total quantity purchased in common physical units (e.g. volume, weight, area, etc.) by UPC Around 670,000 dierent Universal Product Codes (barcodes) sold in each quarter, aggregated into 87 product groups Largest four are carbonated beverages, pet food, paper products, bread We aggregate to the national level for Q4 ( ) using nationally representative household weights from Nielsen to measure average price per UPC and total quantity sold. 20 / 81

21 Estimated Elasticities Percentile Reverse Generalized Feenstra (1994) Lower Upper Weighting (RW) Reverse Bound Bound Weighting (GRW) Min th th th th th Max Estimates also precise: standard errors typically < 0.5 more 21 / 81

22 Is it Reasonable to Assume j kt = j k? Distribution of s gt and s gt 1 T Â t s gt Elasticity Deviation from Time Mean Mean Median SD p10 p25 p50 p75 p90 Sato-Vartia Reverse-Weighting Generalized Reverse-Weighting / 81

23 Is No Product Turnover Reasonable? Density λ t / λ t-1 23 / 81

24 Proportional Change in Aggregate Cost of Living CCG-RW CUPI-RW CCG-GRW CUPI-GRW Feenstra-CES-RW Laspeyres Feenstra-CES-GRW Laspeyres Paasche Paasche Comparison superlative price indexes superlative Robustness to alternative elasticities elasticities New goods and consumer valuation bias around same size 24 / 81

25 Conclusions We develop a unied approach for CES preferences that consistently estimates welfare and demand when demand for each good is time varying Rationalizes observed data on prices and expenditure shares as an equilibrium of the model Allows for entry and exit of goods over time Identies a unique elasticity of substitution (s) Satises money-metric utility (change in the cost of living depends on prices and expenditures) Yields consistent aggregation from micro to macro Shows that existing exact price indexes are biased in the presence of mean zero demand shocks Related to all major micro, macro, and statistical approaches to price measurement Extensions to non-homothetic CES, nested CES, mixed CES, logit, mixed logit, and translog preferences Consumer valuation bias is around as large as variety bias 25 / 81

26 Thank You 26 / 81