Accounting for Heterogeneity

Transcription

1 Accounting for Heterogeneity David Berger Luigi Bocola Alessandro Dovis NU and NBER Stanford and NBER Penn and NBER Chicago Fed August / 48

2 Motivation Recent advances on the role of heterogeneity in macro models Uninsurable income risk as source of business cycle fluctuation Transmission mechanism of monetary and fiscal policy Several challenges for their evaluation One industry standard for model evaluation in macro is Business Cycle Accounting (Chari, Kehoe and McGrattan, 2008) Aggregate data seen as deviations from an efficient benchmark Use measured wedges for model diagnostics Our objective is to generalize BCA to the analysis of heterogeneity 2 / 48

3 What we do Prototype economy: neoclassical growth model with aggregate and idiosyncratic risk Households can trade full set of Arrow securities Individual specific wedges Use panel data to measure micro wedges Show how to use wedges for model diagnostics/discrimination Today: New Keynesian models with incomplete markets (HANK) 3 / 48

4 New Keynesian Models with Incomplete Markets Growing literature Kaplan, Moll, Violante (2017), Auclert and Rognlie (2018), Challe, Matheron, Ragot and Rubio-Ramirez (2017), Ravn and Sterk (2017), McKay and Reis (2017), Werning (2016),... Key mechanism: idiosyncratic risk aggregate demand We use our framework to evaluate this channel We perform counterfactuals in the spirit of CKM Preliminary results suggest that the mechanism is quantitatively important 4 / 48

5 Plan for today 1 Prototype economy and accounting procedure 2 One example: Huggett (1993) economy 3 Measuring the wedges 4 Quantitative experiment: wedge analysis of HANK model 5 / 48

6 Prototype economy Agents: households, firms, intermediaries, government Aggregate z t and idiosyncratic v t shocks. s t = (z t, v t ) denotes history of events. p(s t ) probability of event s t Competitive goods, labor, and financial markets Distortions introduced via individual specific wedges Labor wedge: τ l(z t, v t ) Risk sharing wedge: τ s(z t, v t ) Efficiency wedges: A(z t ), θ(v t ) Investment wedge: τ k(z t ) 6 / 48

7 Preferences and technology Households utility over consumption and leisure: t=0 s t p (s t ) β t U (c t (s t ), l t (s t )) Firms produce consumption goods using technology A(z t )F(K(z t 1 ), L e (z t )) L e (z t ) = p(v t z t )θ(v t )l(z t, v t ) v t Financial intermediaries invest in capital taxed at τ k (z t ), depreciates at rate δ Aggregate resource constraint K(z t ) = (1 δ)k(z t 1 ) + X(z t ) v t p(v t z t )c(s t ) + X(z t ) A(z t )F(K(z t 1 ), L e (z t )) 7 / 48

8 Households problem Every period receives wage W(s t ) = θ(v t )W(z t ), returns on financial assets (Arrow security a(s t )), and lump sum transfers T(s t ). They are born with wealth ω 0 and choose consumption and savings subject to c (s t ) + s t max p (s t ) β t U (c (s t ), l (s t )) s t t=0 Q(s t, s t+1 )a(s t, s t+1 )[1 τ s (s t, s t+1 )] [1 τ l (s t )]W(s t )l(s t ) + a(s t ) + T(s t ) 8 / 48

9 Firms and financial intermediaries Firms rent labor l(s t ) and capital K(z t 1 ) to maximize A(z t )F ( K(z t 1 ), L e (z t ) ) R(z t )K(z t 1 ) v t p(v t z t )W(z t, v t )l t (z t, v t ). Financial intermediaries accumulate capital K(z t ) and finance their operations by issuing (state-contingent) debt to households subject to max t=0 z t β t Q (z t ) d (z t ) d (z t ) + B (z t ) + K (z t ) [1 τ k (z t )] [R (z t ) + 1 δ] K ( z t 1) + + z t+1 Q (z t, z t+1 ) B (z t, z t+1 ) 9 / 48

10 Competitive equilibrium Given {τ l (s t ), τ k (z t ), A(z t ), θ(v t ), τ s (s t )} and transfers {T (s t )}, a competitive equilibrium is prices {Q (s t, s t+1 )}, allocations {c (s t ), l (s t ), K(z t )}, positions {a (s t, s t+1 ), B(z t, z t+1 )}, such that: 1 Households, firms and intermediaries optimize U l (s t ) U c (s t ) = [ 1 τ l ( s t, s t+1 )] A(z t )θ(v t )F L ( z t ) βp(s t+1 s t ( )U c s t+1 ) U c (s t = Q ( s t ) [ (, s t+1 1 τs s t )], s t+1 ) 1 = ) Q(z t, z t+1 )[1 τ k (z t, z t+1 )][A(z t+1 )F K (z t, z t+1 ) + (1 δ)] z t+1 2 Goods, labor and financial markets clear for every z t 3 Government s budget balances for every z t 10 / 48

11 A useful result Let {c (s t ), l (s t ), K (z t ), Y(z t )} be resource-feasible. There exists transfers {T (s t )} and wedges {τ l (s t ), τ k (z t ), A(z t ), τ s (s t ), θ(v t )} that rationalize this allocation as an equilibrium outcome of the prototype economy Implication: A detailed economy imposes restrictions on wedges Corollary: If we observe W(s t ) too then we can recover wedges using aggregate and micro data and this mapping is unique Implication: Unique mapping between our wedges and CKM wedges for certain utility functions 11 / 48

12 Relation to CKM Suppose households have utility function U (c, l) = c1 σ 1 1 σ χ l1+ψ 1 + ψ and p(ω 0 ) is degenerate. Then, the aggregate allocation from the prototype model is the outcome of the prototype economy in CKM with aggregate wedges given by ) 1 α ( A Z (z t ) = A(z t ) 1 + cov v t (θ(v t), l (z t, v t )) L (z t ) ( ) 1/(1 α) τ L (z t A(zt ) ) = 1 A Z (z t p (v t z t ) (1 τ l (s t )) ) where ϕ (s t ) c(s t )/C(z t ) and η(s t ) l(s t )/L(z t ) v t ϕ (st ) σ η(s t ) ψ 12 / 48

13 Examples In general, wedges capture frictions and/or model misspecification To build intuition, we provide a simple example of how an Huggett (1993) economy maps into our wedges The paper has other examples Sticky wages and idiosyncratic labor wedges Heterogeneous risk aversion and risk sharing wedge A simple HANK model 13 / 48

14 Huggett (1993) with tight borrowing limits Model details No capital accumulation Aggregate and idiosyncratic risk Households trade one-period bond Elastic labor supply Implications for wedges No labor wedge Risk sharing wedge due to incomplete markets 14 / 48

15 Preferences, Technology and Shocks Households have preferences subject to U(c, l) = c1 σ 1 σ χ l1+ψ 1 + ψ c (s t ) + b (s t ) W(s t )l (s t ) + b ( s t 1) R ( z t 1) b (s t ) 0 Technology for producing final good Y(z t ) = A(z t ) v t p(v t z t )e(v t )l(s t ) z t AR(1) process in logs ( ) e(vt+1 ) log = σ(z t+1) e(v t ) 2 + σ(z t )ε v,t+1 σ (.) > 0 15 / 48

16 Markets Financial markets: households can save in a nominal uncontingent bond in zero net supply. Households cannot borrow Households effectively in financial autarky and consume all their income Labor market is perfectly competitive 16 / 48

17 Allocation Households optimality and budget constraint χl(s t ) ψ = A(zt )e(v t ) c(s t ) and c(s t ) = A(z t )e(v t )l(s t ) So, the allocation is given by { [e(v c(s t t )A(z t )] 1+ψ ) = χ { [e(v l(s t t )A(z t )] 1 σ ) = χ } 1 σ+ψ } 1 σ+ψ 17 / 48

18 Wedges Risk-sharing wedge: 1 τ s (s t, s t+1 ) = C(zt+1 ) C(z t ) c(z t, v t ) c(z t+1, v t+1 ) = e(v t) e(v t+1 ) In frictionless benchmark, individual consumption proportional to aggregate In Huggett (1993), this is not true because of missing insurance markets Need taxes on Arrow securities to rationalize Huggett (1993) allocation Labor wedge: 1 τ l (s t ) = 0 18 / 48

19 Huggett in CKM Labor wedge: τ L,t = 1 E v t [ ( ) 1/ψ ( e(vt ) C(s t ) ] 1/ψ ψ ) A(z t ) C(z t ) Efficiency wedge: ( A Z (z t ) = A(z t ) 1 + cov v t (e(v t), l (z t, v t ) 1 α )) L (z t ) 19 / 48

20 Measuring idiosyncratic wedges Need panel on consumption expenditures, wages Important requirement: data aggregate to something sensible We use CEX ( ) and the PSID ( ) Good: Bad Has all the relevant variables Some evidence aggregates looks reasonable: Blundell, Pistaferri and Saporta-Eksten (2016); Cohen-Cole, Herkenhoff and Phillips (2016) Small samples Bi-annual frequency (PSID) 20 / 48

21 Data definitions Consumption Wages Hours Dollar spending in non-durables and services for households Labor + business income for household. Have information on total salaries, bonuses, and overtime for household plus business and investment income. Total hours worked per year for household Mapping between model and data Measure at household level and adjust for number of members. We control for household/individual characteristics that are outside the model: education, age, sex, race, state, and industry 21 / 48

22 The procedure in one slide Assume functional forms for preferences/technology U(c t, l t) = c1 σ 1 1 σ χl 1+ψ t /(1 + ψ) F(K t, L et) = K α t L 1 α et Given panel data on {c it, w it, l it }, recover {θ it, τ s,it, τ l,it } using equilibrium conditions [ ] σ Ct+1 /C t 1 τ s,it+1 = c it+1 /c it θ it = w it W t τ l,it = 1 χl 1 ν i,t c it w it 22 / 48

23 Comparison with NIPA Aggregates Consumption Income CEX NIPA PSID CEX NIPA PSID Hours CEX NIPA PSID 23 / 48

24 Histogram of Individual Wedges Risk Sharing Wedge Labor Wedge Efficiency Wedge Density Density Density Risk Sharing Wedge Labor Wedge Efficiency Wedge Density Density Density / 48

25 Cross-sectional Relationship Between Individual Wedges EW vs RSW RSW vs LW EW vs LW EW RSW RSW LW EW LW EW EW vs RSW RSW RSW RSW vs LW LW EW EW vs LW LW 25 / 48

26 Time-series of Risk Sharing and Labor Wedge Risk Sharing Wedge Labor Wedge Risk Sharing Wedge Labor Wedge / 48

27 Evaluating NK models with incomplete markets Start with standard 3 equation NK model (Woodford 2003) Rotemberg pricing No capital Add idiosyncratic income risk and incomplete markets We show: Aggregate variables solve equilibrium conditions of a representative agent economy with wedges These aggregate wedges are functions of the micro wedges in our procedure We perform counterfactuals in the spirit of CKM 27 / 48

28 The Problem of Households Households solve [ ] max β t Pr (s t c (s t ) 1 σ s 0 ) c,l,b,a j 1 σ v (l (st )) t s t subject to P (z t ) c (s t ) + j J q j (z t ) a j (s t ) + b (st ) 1 + i (s t ) W (zt ) θ (v t ) l (v t, z t ) + b ( s t 1) + j J R j (z t ) a j ( s t 1 ) and possibly debt limits 28 / 48

29 Heterogeneity and the Euler Equation Aggregate consumption C and returns R j must satisfy the Euler equation: 1 = max v t where β Pr ( z t+1 z t) R j ( ( ( ) z t+1) C z t+1 C (z t ) z t+1 φ(v t, z t, z t+1 ) = v t+1 ) σ φ (v t, z t, z t+1) π(v t+1 z t+1, v t )[1 τ s,t+1 (s t, s t+1 )] Remark: This result builds on Chien, Cole and Lustig (2011) and Werning (2016) and holds for a large class of economies 29 / 48

30 Heterogeneity and the Phillips curve Aggregate Π, C, Y, L must satisfy the Phillips curve: π ( z t) = 1 κ (µ 1) Y ( [ ] z t) µ χy (zt ) ψ C (z t ) σ ω (z t ) A(z t) A Z(z t ) 1 + Q(z t+1 z t ) π (z t+1) ψ s where π (1 + Π(z t ))Π(z t ), ω (z t ) = and ϕ (s t ) is consumption share [ [ E v t ϕ (z t, v t ) σ/ψ θ(v t ) 1/ψ]] ψ, 30 / 48

31 Equilibrium Suppose that C, Y, L, Π, i are part of an equilibrium. Then they satisfy: π ( z t) 1 = κ (µ 1) Y ( [ z t) µ χy (zt ) ψ C (z t ) σ A(z t) i (z t ) = max {β Pr ( z t+1 z t) v t 1 + Π (z t+1 ) z t+1 ] ω (z t ) A Z(z t ) 1 ψ ( ( ) C z t+1 C (z t ) + Q(z t+1 z t ) π (z t+1) s ) σ ( φ v t, s t+1)} i ( z t) = max { ī + ρ 2 ( π ( z t ) π ) + ρ 3 ( log C ( z t ) log C ), 0 } Y ( z t) =C ( z t) + κ 2 π ( z t) 2 L ( z t) = Y (zt ) A Z (z t ) Key: Heterogeneity only matters for aggregates through φ, ω, and A Z 31 / 48

32 Discussion and Caveats In this class of models, heterogeneity only matters for aggregates through φ, ω, and A Z We can use our wedges to conduct counterfactuals to assess the importance of heterogeneity Caveat: this is system of equations is not block recursive Wedges are not typically not exogenous There are special cases in which they are Models with tight borrowing limits So, our counterfactuals are in the same spirit of CKM 32 / 48

33 Counterfactuals Parametrize the 3 equations model following Gust et. al (2012). Construct time-series for φ, ω, and A Z and estimate an AR(1) Conduct counterfactuals in the spirit of CKM Feed model with sequence of φ from the data Construct counterfactual series assuming φ = φ throughout the sample Difference between between "actual" and counterfactual nets out effect of idiosyncratic risk and market incompleteness 33 / 48

34 Constructing φ(v t, z t, z t+1 ) Here is our idea for constructing φ(v t, z t, z t+1 ) Proxy for v t by constructing groups of individuals sharing similar characteristics (E.g. income and assets) Within each group J, compute 1 (1 τ s,it+1 ) N J i J If N J is large, this statistic approximates φ(v t, z t, z t+1 ) We partition on last periods income 34 / 48

35 Constructing φ(v t, z t, z t+1 ) Time-series for Phi (CEX) Y < Y_med Y > Y_med 35 / 48

36 Constructing φ(v t, z t, z t+1 ) Time-series for Phi (PSID) Y < Y_med Y > Y_med 36 / 48

37 Effect of heterogeneity / 48

38 Next steps Evaluate HANK vs TANK Debortoli and Gali (2018) provide first order decomposition of φ(v t, z t, z t+1 ) into 3 components Gap in C growth between two agents Change in the share of both agents Dispersion within agent types They argue that only first component matters; Violante suggested this result is special (not true in their HANK model) We can measure these components in the data 38 / 48

39 Conclusion Novel framework to evaluate macro models with heterogeneous agents We used panel data from CEX and PSID to measure micro wedges We showed how to use micro wedges to evaluate business cycle implications of NK models with incomplete markets We are working on more applications 39 / 48

40 A detailed economy: NK model with incomplete markets Application motivated by several recent papers. Basic story Idiosincratic income risk Households desired saving Real interest rates cannot adjust b/c of sticky prices and constrained monetary policy Consumption needs to fall Fall in consumption accommodated by fall in income We capture this story with simple model. We then Compute implied wedges Show that story implies a particular pattern for the wedges 40 / 48

41 Preferences, Technology and Shocks Households have preferences U(c, l) = log(c) χ l1+ψ 1 + ψ Final good produced using CES aggregator [ 1 Y(z t ) = y 1/µ (z t )di Intermediate goods produced using linear technology y i (z t ) = exp{z t }n i (z t ) n i (z t )di = π(v t z t )e(v t )l(z t, v t ) v t 0 ] µ z t AR(1) process in logs. ( ) e(vt+1 ) log = σ(z t+1) e(v t ) 2 + σ(z t )ε v,t+1 σ (.) > 0 41 / 48

42 Markets Financial markets: households can save in a nominal uncontingent bond in zero net supply. Households cannot borrow Households effectively in financial autarky, consume their income Monopolistic competitive intermediate good firms, Rotemberg adjustment costs Wages clear the labor market 42 / 48

43 Households problem and optimality subject to max c,l,b t=0 s t β t π (s t ) U (c (s t ), l (s t )) P (z t ) c (s t ) + b (st ) 1 + i (z t ) W (z t ) e (v t ) l (v t, z t ) + b ( s t 1) + e(v t )T (z t ) b (s t ) 0 (Assume government taxes firms profits and transfers to households) l (z t, v t ) ψ = (1/χ) W (zt ) e (v t ) c(z t, v t ) i (z t ) ( β π (s ( t+1 s t ) c z t+1, v t+1) ) Π (z t+1 ) c (z t, v t ) s t+1 43 / 48

44 Aggregation From households problem we get l(z t, v t ) = l(z t ), and The Euler equation becomes 1 + i(z t ) 1 = max βe z t v t 1 + Π (z t+1 ) c(z t, v t ) = e (v t ) C(z t ) ( C ( z t+1 ) C (z t ) ) 1 exp{σ(z t+1)} Key mechanism: high σ(z t+1 ) increases precautionary motives. If real rates do not adjust, consumption growth must fall Shock propagates to output because of sticky prices 44 / 48

45 Wedges Risk-sharing wedge Efficiency wedges: 1 τ s (s t, s t+1 ) = C(zt+1 ) C(z t ) c(z t, v t ) c(z t+1, v t+1 ) = e(v t) e(v t+1 ) A(z t ) = exp{z t }/ (z t ) θ(v t ) = e(v t ) Labor wedge 1 τ l (s t ) = χ W(zt )/P(z t ) A(z t ) 45 / 48

46 Restrictions on wedges Model restricts wedges 1 Correlation between risk sharing wedge and efficiency wedge is 1 Households hand-to-mouth 2 Correlation between a particular moment of the risk sharing distribution and labor wedge Basic story: φ(v t, z t, z t+1 ) = π ( v t+1 z t+1 ), v t [1 τs (s t, s t+1 )] v t+1 = exp{σ(z t+1 )} σ(z t+1 ) max v t if real interest rate cannot adjust to satisfy ( ( ) C z t+1 1 = max βe z t (1 + r) v t C (z t ) z t+1 π(z t+1 z t )φ(v t, z t, z t+1 )] higher desire to save ) 1 exp{σ(z t+1)} current consumption must fall to deter household to work real wage must fall positive labor wedge 46 / 48

47 Constructing φ(v t, z t, z t+1 ) Time-series for Phi (CEX) Y < Y_med Y > Y_med 47 / 48

48 Constructing φ(v t, z t, z t+1 ) Time-series for Phi (PSID) Y < Y_med Y > Y_med 48 / 48