Welfare And Output With Income Effects And Taste Shocks
Welfare and Output withIncome Effects and Taste ShocksDavid BaqaeeUCLAAriel Burstein1UCLAMarch 20221The conclusions and analysis are our own, calculated in part on data from NielsenConsumer LLC and provided through the NielsenIQ Datasets at the Kilts Center forMarketing 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 andpreparing the results reported herein.
MotivationI How does welfare respond to changes in choice sets?“How much endowment at t0 to make consumer indifferentbetween choice set in t0 and t1 ?”I If preferences stable and homothetic: can express welfare aschained (or Divisia) index. Welfare logspendingt1spendingt0 Zt1t0pi ,t ci ,t spendt d log pi ,tiI Foundation for aggregation procedures to calculate aggregatequantities and prices.I What if demand is unstable and/or non-homothetic.
What We DoI Consider preferences that are unstable and/or non-homothetic.I Characterize welfare change and chained objects in PE and GEin terms of suff. stats.I Chaining treats all expenditure-switching equally (due to prices,income, taste shocks), but welfare does not.I PE/GE distinction matters when pref. unstable or non-homoth.I Generalize Hulten’s Theorem.I Quantitative applications for long-run growth and fluctuations.1. Welfare-relevant Baumol’s cost disease.2. Firm-level shocks and industry-level outcomes.3. PE vs. GE during Covid-19 recessionPath-dependence/Index-drift of RGDP and TFP.
Selected LiteratureI Biases of real consumption/GDP:Fisher & Shell (1968), Hausman (1981), Feenstra (1994), Basu et al. (2012),Aghion et al. (2019), Syverson (2017), Jones & Klenow (2016).I Index numbers with taste shocks or non-homotheticity:Caves, Christensen, Diewert (1982), Deaton & Muellbauer (1980), Feenstra &Reinsdorf (2007), Redding & Weinstein (2020), Atkin et al. (2020).I Growth accounting and disaggregated macro:Solow (1951), Domar (1961), Hulten (1978), Long & Plosser (1983), Gabaix(2011), Acemoglu et al. (2012), Baqaee & Farhi (2019).I Structural TransformationBaumol (1961), Kongsamut, Rebelo, Xie (2001), Buera & Kaboski (2009),Herrendorf et al. (2013), Boppart (2014), Comin et al. (2020), Alder et al.(2019).
AgendaMicroeconomic ProblemMacroeconomic ProblemApplicationsExtensionsConclusion
AgendaMicroeconomic ProblemMacroeconomic ProblemApplicationsExtensionsConclusion
Set upI Preference relations { x } over vector of consumption goods c.(x is e.g. age, fads, advertising, state of nature, no choices/preferences over x)I Represent preferences by u (c ; x ) with indirect utility v (p, I; x ).I Consider change from (pt0 , It0 , xt0 ) to (pt1 , It1 , xt1 ).I Consider how welfare changes.I Consider how (chained) real consumption changes.I Compare the difference.
Micro Welfare and Real ConsumptionDefinitionChange in real consumption is log Y log I Zt1t0 ipi ,t ci ,tItd log pi ,t .
Micro Welfare and Real ConsumptionDefinitionChange in real consumption is log Y log I Zt1t0 ipi ,t ci ,tItd log pi ,t .Income needed under pt0 to make xt1 indifferent to (pt1 , It1 ).mv (pt0 , eEV It0 ; xt1 ) v (pt1 , It1 ; xt1 )
Micro Welfare and Real ConsumptionDefinitionChange in real consumption is log Y log I Zt1t0 ipi ,t ci ,tItd log pi ,t .Income needed under pt0 to make xt1 indifferent to (pt1 , It1 ).mv (pt0 , eEV It0 ; xt1 ) v (pt1 , It1 ; xt1 )I EV because it is a money-metric representation of u (CV is not).I Uses only ordinal (not cardinal) information on preferences.I t1 preferences because more relevant than t0 preferences.I For welfare, quality 6 taste.
Change in real consumption versus welfareDefine b(p, u , x ) to be budget share given p, u, and x.
Change in real consumption versus welfareDefine b(p, u , x ) to be budget share given p, u, and x.Lemma log Y log I Zt1 bi (pt , ut , xt )d log pi ,t .t0EVm log I Zit1t0 bi (pt , ut , xt )d log pi ,t .11iwhere b(p, u , x ) denotes budget share at (p, u , x )If preferences stable and homothetic, bi (pt , ut , xt ) bi (pt , ut1 , xt1 ).For welfare, substitution effect 6 income effect & taste shocks.derivation
Key Intuition for Consumption vs. Welfare log Y log I Zt1 bi (pt , ut , xt )d log pi ,t .t0EV m log I Zit1t0 bi (pt , ut , xt )d log pi ,t .11iI Consider change in welfare comparing 1950 to 2014.I Spend more on healthcare in 2014 due to aging and income.I Chained index uses 1950 demand to weight prices in 1950.I Welfare-relevant uses 2014 demand to weight prices in 1950.I Welfare uses xt1 since households are older and ut1 since we aregiving them income in 1950 to make them indifferent to 2014.
Example of ImplementationI Consider e.g non-homothetic CES with taste shocks where1! 1 θe(pt , ut , xt ) ωi xit pit1 θ utξiut .iI To measure welfare, integrate b(p, ut1 , xt1 )d log p:EVm log I log bit1ipit01 1 θ ! 1 θpit1.I If we know elasticity of substitution, we don’t need to knowincome elasticities or taste shocks (or even separate them).cross-sectional implementation a la Jaravel & Lashkari (2022)cardinal utility measure
Second-order Approx.Real consumption is log Y log It Zt1 bitt0 i Nd log pitdtdt .
Second-order Approx.Real consumption is log Y log It Zt1 bitt0 i NConsider Taylor expansion in t1 around t0 : log Y log I Eb ( log p) {zfirst-order}d log pitdtdt .
Second-order Approx.Real consumption is log Y log It Zt1 bitt0 i Nd log pitdtdt .Consider Taylor expansion in t1 around t0 :1 log Y log I Eb ( log p) Covb ( log b, log p),2 {zfirst-order} {zsecond-order}If spending rises for goods that got more expensive, then log Y falls.
Consumption vs. Welfare: Second-order Approx.Consider Taylor expansion in t1 around t0 : log Y log I Eb [ log p] {z}First-order,EV m log I Eb [ log p].To a first-order, welfare and real consumption are the same.
Consumption vs. Welfare: Second-order Approx.Consider Taylor expansion in t1 around t0 :1 log Y log I Eb [ log p] (1 θ0 )Varb ( log p) {z} 2{z}First-orderexpenditure-switchingdue to substitution effect,EV m log I Eb [ log p] 12(1 θ0 )Varb ( log p).
Consumption vs. Welfare: Second-order Approx.Consider Taylor expansion in t1 around t0 :1 log Y log I Eb [ log p] (1 θ0 )Varb ( log p) {z} 2{z}First-order expenditure-switchingdue to substitution effect1Covb ( log x , log p)2 {z},expenditure-switchingdue to taste shockEV m log I Eb [ log p] 12 1 Covb ( log x , log p)(1 θ0 )Varb ( log p).
Consumption vs. Welfare: Second-order Approx.Consider Taylor expansion in t1 around t0 :1 log Y log I Eb [ log p] (1 θ0 )Varb ( log p) {z} 2{z}First-order expenditure-switchingdue to substitution effect11Covb ( log x , log p) ( log I Eb [ log p]) Covb (ε, log p) ,2 {z} 2 {z}expenditure-switchingdue to taste shockEV m log I Eb [ log p] 12expenditure-switchingdue to income effect(1 θ0 )Varb ( log p) 1 Covb ( log x , log p) 1 ( log I Eb [ log p]) Covb (ε, log p).
Consumption vs. Welfare: Second-order Approx.Consider Taylor expansion in t1 around t0 :1 log Y log I Eb [ log p] (1 θ0 )Varb ( log p) {z} 2{z}First-order expenditure-switchingdue to substitution effect11Covb ( log x , log p) ( log I Eb [ log p]) Covb (ε, log p) ,2 {z} 2 {z}expenditure-switchingdue to taste shockEV m log I Eb [ log p] 12expenditure-switchingdue to income effect(1 θ0 )Varb ( log p) 1 Covb ( log x , log p) 1 ( log I Eb [ log p]) Covb (ε, log p).No bias if covariances are zero.quality change
AgendaMicroeconomic ProblemMacroeconomic ProblemApplicationsExtensionsConclusion
Set upI Perfectly competitive neoclassical economy, representative agentI F primary factors, N goods yi Ai Gi {mij }j N , {lij }j F I Macro indirect utility of representative agent isV (A, L; x ) max{u (c ; x ) : c is feasible}.I Consider changes in technologies from (At0 , Lt0 ) to (At1 , Lt1 ) andpreferences from xt0 to xt1 .I These imply some (pt0 , It0 ) and (pt1 , It1 ).
Micro Welfare Poor Measure of Technology ChangeI Suppose households age from t0 to t1 but technology unchanged.
Micro Welfare Poor Measure of Technology ChangeI Suppose households age from t0 to t1 but technology unchanged.I With DRS, relative price of healthcare higher in t1 .
Micro Welfare Poor Measure of Technology ChangeI Suppose households age from t0 to t1 but technology unchanged.I With DRS, relative price of healthcare higher in t1 .I Micro welfare falls from t0 to t1 :An individual old person likes prices in t0 compared to t1 .
Micro Welfare Poor Measure of Technology ChangeI Suppose households age from t0 to t1 but technology unchanged.I With DRS, relative price of healthcare higher in t1 .I Micro welfare falls from t0 to t1 :An individual old person likes prices in t0 compared to t1 .I But technology has not changed:A society of old people indifferent between tech in t0 and t1 .I Macro welfare takes into account that prices respond to demand.
Micro Welfare Poor Measure of Technology ChangeI Suppose households age from t0 to t1 but technology unchanged.I With DRS, relative price of healthcare higher in t1 .I Micro welfare falls from t0 to t1 :An individual old person likes prices in t0 compared to t1 .I But technology has not changed:A society of old people indifferent between tech in t0 and t1 .I Macro welfare takes into account that prices respond to demand.
Macro WelfareTo quantify welfare effect of changes in technologies, we ask:Factors needed in t0 to make xt1 indifferent to t1 technologies.MV (At0 , eEV Lt0 ; xt1 ) V (At1 , Lt1 ; xt1 ).I When preferences are stable homothetic, or PPF is linear:I macro welfare is the same as micro welfare.
Good 2Graphical Representation – Micro EV mEVmU t1Good 1
U t1EVMGood 2Graphical Representation – Macro EV MGood 1
Real GDP and WelfareI Let λi (A, u , x ) be sales shares,pi yi,GDPwith demand b(p, u , x ).PropositionChange in real GDP and welfare in response to ( x , A) isZt1 λi (At , ut , xt )d log Ai ,t , log Y t0EVMZit1 t0 λi (At , ut , xt )d log Ai ,t .1i1
Simple ExamplesI Consider non-homothetic CES consumerI One sector economy with no intermediates and one factor:EV M log Y 11Covb ( log x , log A) Covb (ε, log A) Eb [ log A] .2 2 {z}{z}gap due to taste shocksgap due to income effects
Simple ExamplesI Consider non-homothetic CES consumerI One sector economy with no intermediates and one factor:EV M log Y 11Covb ( log x , log A) Covb (ε, log A) Eb [ log A] .2 2 {z}{z}gap due to taste shocksgap due to income effectsI With roundabout (larger changes in sales shares due to taste):EVM log Y 112 (1 Ωii ) Covb ( log x , log A) Covb (ε, log A) Eb [ log A] ,where Ωii is Cobb-Douglas intermediate input share.
Simple Example with Nonlinear PPFI One sector with Cobb-Douglas decreasing returns to scale:EV M logY 1 Covb ( log x , log A)2 1 (θ0 1)(1 γ).Sales shares respond more to taste shocks if complements andDRS.
Simple Example with Nonlinear PPFI One sector with Cobb-Douglas decreasing returns to scale:EV M logY 1 Covb ( log x , log A)2 1 (θ0 1)(1 γ).Sales shares respond more to taste shocks if complements andDRS.I With only taste shocks, Micro and Macro welfare are different.EV m 1(1 γ)2 (1 (θ0 1) (1 γ))Varb ( log x ) 6 0 EV M .
Implementation: notationI LetΩij pj mijpi yi,be input-output matrix.I LetΨ (I Ω) 1 I Ω Ω2 .be Leontief inverse.I Denote Hicksian sales shareλ ev (A) λ (A, ut1 , xt1 ).
ImplementationI Consider non-homothetic CES consumer CES producers.I Observed sales:d λit λjt (θj 1)CovΩ(j ,:),t d log pt , Ψ(:,i ),tj {0} N F {zsubstitution effect }! CovΩ(0,:),t d log xt , Ψ(:,i ),t CovΩ(0,:),t (εt , Ψ(:,i ),t ) {z} {ztaste shocks .λk d log Aktkt Nincome effect}
ImplementationI Consider non-homothetic CES consumer CES producers.I Observed sales:d λit λjt (θj 1)CovΩ(j ,:),t d log pt , Ψ(:,i ),tj {0} N F {zsubstitution effect }! CovΩ(0,:),t d log xt , Ψ(:,i ),t CovΩ(0,:),t (εt , Ψ(:,i ),t ) {z} {ztaste shocks income effectI Welfare-relevant sales (starting at t1 and going back to t0 )d λitev j {0} N F λjtev (θj 1)CovΩev d log ptev , Ψev(:,i ),t .(j ,:),tI Real GDP uses λi , welfare uses λiev .prices.λk d log Aktkt N}
AgendaMicroeconomic ProblemMacroeconomic ProblemApplicationsExtensionsConclusion
Implications for Long-Run GrowthAs economies grow, low-productivity-growth sectors expand.(Baumol’s cost disease)CorollaryIf structural transformation is due only to:I substitution effects, then welfare-TFP uses chained sales shares.I income effects or taste shocks, then welfare-TFP uses terminalsales shares.
Growth in welfare TFP between t and 2014 in the US61 private-sector industries, gross-output TFP as in Jorgenson et al. (2005) and Carvalho and Gabaix (2013)0.60Welfare with only substitution effectWelfare with only income 1990200020102020I Settling dispute about income v. substitution effects hasimportant welfare implications.
Growth in welfare TFP between t and 2014 in the US61 private-sector industries, gross-output TFP as in Jorgenson et al. (2005) and Carvalho and Gabaix (2013)0.60Welfare with only substitution effectWelfare with only income effectInitial 1990200020102020I Settling dispute about income v. substitution effects hasimportant welfare implications.
Quantitative ExperimentI θ0 elasticity of sub. between primary/manufacturing/services.I θ1 elasticity of sub. within primary/manufacturing/services.I θ2 elasticity of sub. between materials.I θ4 elasticity of sub. between VA-materials.Table: US from 1948 to 2015.(θ0 , θ1 , θ2 , θ3 )Welfare TFPMeasured TFPConstant-share %
US welfare between t and 2019 using consumption data1.81.6Initial SharesWelfare with only substitution effectWelfare with only income 200020102020
Implications for Short-Run GrowthI Biases can also matter at more disaggregated level.I The more we disaggregate, the more shares vary.(increasing scope for bias).I Within-industry firm-level shocks can cause industry-levelprice/quantity to be mismeasured.I Use Nielsen UPC-level data 2004-2019 and CES with tasteshocks and elasticity of substitution 4.5 to quantify.
UPC-level inflation rate for continuing varietiesSato-VartiaChainedWelfare-relevantWelfare - Chained1/2 Cov( log p, log x)0.040.200.020.0020052010201520202005(a) Inflation rates20102015(b) BiasI Welfare-relevant inflation is higher than chained.Price and demand shocks are positively correlated.I Gap widens as we extend horizon due to persistence.I Covariance formula works well.2020
Implications for Business CyclesI Consider application to Covid-19 recession: mixture of supplyand demand shifters.I 66 sector model with input-output linkages and complementaritiesin production, as in Baqaee-Farhi (2021).I Labor market segmented by sector.I Supply shocks: labor supplied to match reductions in employmentfrom Q1 to Q2 of 2020.I Demand shocks: taste shifters to match changes in PCEexpenditures from Q1 to Q2 of 2020.
Macro vs. Micro welfareWelfare Feb to May, 2020 using structural modelElasticitiesMedium compl.Cobb-DouglasMicro Covid preferences-12.3%-10.9%Macro Covid preferences-9.4%-9.0%Chained real consumption-10.6%-9.8%I Micro welfare losses larger than RGDP because prices rise for goods withincreasing demand.I Macro losses smaller than micro since part of price rises were due to changesin demand.I Chaining under-measures micro losses, and over-measures macro losses
RGDP as Measure of ProductionI What are implications for comparing RGDP to model predictions?I Consider two different time-paths of shocks:I Supply shocks first, then demand shifters.I Demand shifters first, then supply shocks. RGDP Q1 to Q2, 2020 using structural model.ElasticitiesSupply then demandDemand then supplyMedium compl.Cobb-Douglas-12.5%-9.4%-10.8%-9.0%I RGDP falls by more if supply shocks first, because supply anddemand shocks positively correlated.I If recovery not mirror image of downturn, RGDP off by up to 6%.
AgendaMicroeconomic ProblemMacroeconomic ProblemApplicationsExtensionsConclusion
Extension to Heterogeneous AgentsI Homogeneous agent assumption simplifies exposition, butBaqaee & Burstein (2021) generalizes:“What is the minimum change in endowments in t0 so that it ispossible to make every consumer indifferent between t0 and t1 ?”I With this definition, all macro and micro results readily generalize.
Extensions to Extensive Margin AdjustmentsThe change in equivalent variation at t1 tastes is given byEV m logIt1It0 logPt1Pt0,(1)where Pt1Pt0bc t1 P c θ0 1t1Ptc01 1 btc1xt1 bt0 1 Ptn θ0 1 θ0 11Ptn 0. I btc1 is the share on continuing goods in t1 ,I Ptc1 /Ptc0 and Ptn1 /Ptn0 is change in CES index for continuing andnewI btx0t1 is share on exiting goods under t1 tastes and pt0 prices.(2)
Steady-state comparisons in dynamic modelI Intertemporal preferencesUt β s t u(Cs ),s t ωi0 xiticisξ! θ0θ 10 1Cs i I Goods: yis Ais Gi {mijs }j N , H (lis , kis ) I Investment: Is AIs I {mIjs }j N , H (lIs , kIs ) .I Capital accumulation Ks 1 (1 δ ) (Ks Is )PropositionConsider two dynamic economies, denoted t0 and t1 , that are insteady-state. The change in macro welfare is given by i pit1 cit1EV log i pit0 cit0M log biti1pit0pit1 1 θ0 ! 1 θ1 0.
Other ExtensionsI Distorted economies.I Beyond CES.
ConclusionI Toolbox for welfare computation. How to modify Divisia/Hulten fortaste shocks and non homoth.I Chaining doesn’t correctly account for expenditure-switchingcaused by taste shocks or income effects.I In these cases, distinct macro and micro notions of welfare.I Characterize both and show gap with chained quantity significantwhen demand shifters covary with supply shifters.
Changes in Pricesd log pit Ψijt d log Ajt {z Ψift d log λftf Fj {0} N Fupstream TFP changes} {z}upstream factor price changesandd log pitev j {0} N FbackevevΨevijt d log Ajt Ψift d log λft .f F.
With Quality ChangeChange in real consumption1 log Y log I Eb [ log p̃] (1 θ0 )Varb (d log p̃)21122 (1 θ0 )Covb (d log q , d log p̃) Covb (d log x , d log p̃),Change in welfareEV m log I Eb [ log p̃ log q ] 12(1 θ0 )Varb ( log p̃)1 (1 θ0 )Varb ( log q ) (1 θ0 )Covb ( log p̃, log q ) Covb ( log x , log p) ,2Hence,11EV m log Y Eb [ log q ] (θ0 1)Varb ( log q ) (1 θ0 )Covb ( log p̃, log q ) {z } 2 {z} 2 {z}average quality dispersion in qualityCovb ( log x , log p̃) Covb ( log x , log q ) .2 {z} {z}covariance of taste and pricebackcovariance of price and quality1covariance of taste and quality
Micro WelfareLemmaEV m log I Zt1t0EV m log1e (pt0 , v (pt0 , It0 ; xt1 ); xt1 )Zt1t0 log I Z i log bi (p, ut , xt )1iIt1 e (pt0 , ut1 ; xt1 ) It0 e (pt1 , ut1 ; xt1 ) log e(p, v (pt1 , It1 ; xt1 ); xt1 )d log pi log pit1t01ie (pt0 , v (pt1 , It1 ; xt1 ); xt1 ) log I back bi (pt , ut , xt )d log pi ,t .1d log pidtdt ,
Cross-Sectional ImplementationI Jaravel & Lashkari (2022) setting.I Cross-section facing common prices with same stable .I Observe p(t ) and budget shares B (I , t ) at t [t0 , T ].Corollary of Lemma 1EV m (p(t0 ), I (t0 ), p(t ), I ) log u (I , t ) solves following integral equationlog u (I , t ) logII (t0 ) Ztt0 Bi (u 1 (u(I , t ), s), s)iwith boundary condition u (I , t0 ) I .backd log pi (s)dsds
Second-order ApproximationI Real GDP: log Y λi log Ai i12 λi log Ai . i{z}expenditure switchingI Welfare:EV M λ 0 log A 12 λi log Aii λi0 λi0 u log x log A log Ai .2 i log x log A log u1I Changes due to income effects and taste shocks areundercounted.
Good 2Graphical RepresentationU t1U t0Good 1I Objective is to compare value of initial and final technology.
Good 2Graphical Representation – Micro EV mEVmU t1Good 1
U t1EVMGood 2Graphical Representation – Macro EV MGood 1
U t1EVMGood 2Graphical Representation – Macro EV MGood 1I EV M EV m if PPF is linear or preferences homothetic stable.
Cardinal Utility MeasureI Consider e.g non-homothetic CES with taste shocks where1! 1 θe(pt , ut , xt ) 1 θ ξi ωi xit pitutut P̃t ut .i( u , P̃ ) not good (quantity, price) measure if non-homo./unstablesince it is cardinal.Back
Key Intuition for Consumption vs. Welfare logY logI Z t 1 t0 å i bi(pt;ut;xt)dlogpi;t: EVm logI Z t 1 t0 å i bi(pt;ut 1 xt 1)dlogpi;t: I Consider change in welfare comparing 1950 to 2014. I Spend more on healthcare in 2014 due to aging and income. I Chained index uses 1950 demand to weight prices in 1950. I Welfare-relevant uses 2014 demand to weight prices in 1950. I Welfare uses xt
Income Tax Act 2007 2007 No 97 BC 6 Income tax liability of filing taxpayer 106 BC 7 Income tax liability of person with schedular income 106 BC 8 Satisfaction of income tax liability 108 Subpart BD—Income, deductions, and timing BD 1 Income, exempt income, excluded income, non- 108 residents' foreign-sourced income, and assessable income
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AREA MEDIAN INCOME (AMI): The median family income in the metropolitan or nonmetropolitan area EXTREMELY LOW-INCOME (ELI): Households with income at or below the Poverty Guideline or 30% of AMI, whichever is higher VERY LOW-INCOME (VLI): Households with income between ELI and 50% of AMI LOW-INCOME (LI): Households with incomes between 51% and .
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