Lecture 2. (1) Aggregation (2) Permanent Income Hypothesis. Erick Sager. September 14, PDF Free Download

Lecture 2 (1) Aggregation (2) Permanent Income Hypothesis Erick Sager September 14, 2015 Econ 605: Adv. Topics in Macroeconomics Johns Hopkins University, Fall 2015

Erick Sager Lecture 2 (9/14/15) 1 / 37 Last Time (8/31/15) Considered data Wealth, Consumption, Income Cross-Sectional, Times-Series, Life-Cycle Inequality

Erick Sager Lecture 2 (9/14/15) 1 / 37 Last Time (8/31/15) Considered data Wealth, Consumption, Income Cross-Sectional, Times-Series, Life-Cycle Inequality Started thinking about aggregation Gorman Form c i (p, w i ) = a i (p) + b(p)w i = C(p, w) = C(p, W )

Erick Sager Lecture 2 (9/14/15) 2 / 37 Today (9/14/15) Finish: Aggregation Negishi Constantinides

Erick Sager Lecture 2 (9/14/15) 2 / 37 Today (9/14/15) Finish: Aggregation Negishi Constantinides Start: Permanent Income Hypothesis (PIH) Restrict asset space Derive results for consumption, savings, wealth Empirical evaluation of theory

Erick Sager Lecture 2 (9/14/15) 3 / 37 Aggregation Under Complete Markets Overview Pre-trade heterogeneity undone by complete markets Usually admits a representative agent Highly tractable, useful baseline Little/no role for distributional effects

Erick Sager Lecture 2 (9/14/15) 4 / 37 Aggregation Under Complete Markets Issue Consider heterogeneous agents with complete markets Suppose Gorman aggregation fails c i (p, w i ) a i (p) + b(p)w i

Erick Sager Lecture 2 (9/14/15) 4 / 37 Aggregation Under Complete Markets Issue Consider heterogeneous agents with complete markets Suppose Gorman aggregation fails c i (p, w i ) a i (p) + b(p)w i How do we write the preferences of a representative agent? Can we compute prices and allocation with heterogeneity? Negishi Approach FWT: Competitive Equilibrium allocation = Pareto Efficient allocation Allocation can be found as solution to Social Planner Problem Choose Planner weights that recover Competitive Equilibrium allocation

Erick Sager Lecture 2 (9/14/15) 5 / 37 Negishi Approach Next Slides: Characterize Competitive Equilibrium allocation Characterize Planner s allocation Show how to choose planner weights

Erick Sager Lecture 2 (9/14/15) 6 / 37 Competitive Equilibrium: Consumers N = 2 consumer types u : R + R s.t. u > 0, u < 0, u C 2, Inada Endowed with wealth {a 1 0, a 2 0} Given sequence of prices {p t } t=0 Complete Asset Markets Agent i s problem: v(p, a i 0) = max {c i t } t=0 { β t u(c i t) t=0 s.t. } p t c i t p 0 a i 0 t=0 λ i multiplier on budget constraint for agent i {1, 2}

Erick Sager Lecture 2 (9/14/15) 7 / 37 Competitive Equilibrium: Firms Representative firm: owns capital, invests Endowed with k 0 Consumers own shares in firm, a 1 0 + a 2 0 = A 0 Production function f : R + R + s.t. f > 0, f < 0, and differentiable Firm s Problem: A 0 = max {k t+1} t=0 First order conditions: { t=0 ( pt p 0 ) } (f(k t ) + (1 δ)k t k t+1 ) 1 = p ( ) t+1 f (k t+1 ) + 1 δ p t Recursive representation: ) A t = (f(k t ) + (1 δ)k t k t+1 + p t+1 A t+1 p t

Erick Sager Lecture 2 (9/14/15) 8 / 37 Competitive Equilibrium: Market Clearing p t c i t + τ>t p τ c i τ = p t a i t p t c i t + p t+1 a i t+1 = p t a i t c i t + p t+1 a i t+1 = a i t p t 2 i=1 c i t + p t+1 p t 2 a i t+1 = 2 i=1 i=1 a i t C t + p t+1 A t+1 = A t p t C t + k t+1 = f(k t ) + (1 δ)k t

Erick Sager Lecture 2 (9/14/15) 9 / 37 Negishi Planner s Problem Endowed with k 0 (µ 1, µ 2 ) planner weights for each consumer

Erick Sager Lecture 2 (9/14/15) 9 / 37 Negishi Planner s Problem Endowed with k 0 (µ 1, µ 2 ) planner weights for each consumer Negishi Planner s Problem: β t λ RC t v NP (µ 1, µ 2, k 0 ) = max {c 1 t,c2 t,kt+1} t=0 s.t. t=0 multiplier on resource constraint β t[ ] µ 1 u(c 1 t ) + µ 2 u(c 2 t ) c 1 t + c 2 t + k t+1 = f(k t ) + (1 δ)k t

Negishi Planner s Problem Erick Sager Lecture 2 (9/14/15) 9 / 37 Endowed with k 0 (µ 1, µ 2 ) planner weights for each consumer Negishi Planner s Problem: β t λ RC t v NP (µ 1, µ 2, k 0 ) = max {c 1 t,c2 t,kt+1} t=0 s.t. t=0 multiplier on resource constraint First order conditions for all t: β t[ ] µ 1 u(c 1 t ) + µ 2 u(c 2 t ) c 1 t + c 2 t + k t+1 = f(k t ) + (1 δ)k t Implies for all t: λ RC t µ i u (c i t) = λ RC t ( ) = βλ RC t+1 f (k t+1 ) + 1 δ u (c 1 t ) u (c 2 t ) = µ 2 µ 1

Erick Sager Lecture 2 (9/14/15) 10 / 37 Planner Weights Implies for all t: u (c 1 t ) u (c 2 t ) = µ 2 µ 1 (i) planner fully insures: constant relative MUC s (ii) consumption is allocated proportionately to planner weights Under Competitive Eq: u (c 1 t ) u (c 2 t ) = λ1 λ 2 Implement CE allocation: Implies via Euler equations: 1 = p t+1 p t µ 2 µ 1 = λ 1 λ 2 ( ) β f (k t+1 ) + 1 δ 1 = λrc ( ) t+1 λ RC β f (k t+1 ) + 1 δ t

Erick Sager Lecture 2 (9/14/15) 11 / 37 Constantinides Approach Next Slides: Generalization of Negishi Planner Decompose into allocation across agents representative consumer Example: Maliar and Maliar (2003)

Constantinides Approach Erick Sager Lecture 2 (9/14/15) 12 / 37 i = 1, 2,..., N consumers π i population weight s.t. N i=1 π i = 1 Negishi Planner s Problem: N max {c 1 t,c2 t,kt+1} t=0 s.t. t=0 β t i=1 µ i u(c i t) N π i c i t + k t+1 = f(k t ) + (1 δ)k t i=1 Constantinides Decomposition: Allocation across agents { N } N U(C t ) = max µ i u(c i t) s.t. π i c i {c i t C t t }N i=1 i=1 i=1

Erick Sager Lecture 2 (9/14/15) 13 / 37 Taking Stock Method for computing Representative Agent with heterogeneity Only assumption on preferences is strict concavity Gorman Form (homotheticity or quasilinearity) not required Requires complete markets Gorman Form makes no assumption on asset markets Next: example with closed-form solution

Erick Sager Lecture 2 (9/14/15) 14 / 37 Maliar and Maliar (2003) Unit continuum of infinitely lived households indexed by i I [0, 1] Let µ i be measure of type i agents s.t. I dµ i = 1

Erick Sager Lecture 2 (9/14/15) 14 / 37 Maliar and Maliar (2003) Unit continuum of infinitely lived households indexed by i I [0, 1] Let µ i be measure of type i agents s.t. I dµ i = 1 Endowed with one unit of time, labor (h i t) vs leisure (1 h i t) Idiosyncratic labor productivity shocks: ε i t E s.t. E[ε] = 1 Trade Arrow securities a t+1 (ε) at price p t (ε) Endowed with k 0 and a 0 Household i s problem max {c t,h t,k t+1,a t+1(ε)} t=0 E 0 t=0 [ (c β t i t ) 1 σ 1 σ + ψ (1 hi t) 1 γ ] 1 γ s.t. c i t + kt+1 i + p t (ε)a i t+1(ε)dε w t ε i th i t + (1 + r t )kt i + a i t(ε i t) E

Erick Sager Lecture 2 (9/14/15) 15 / 37 Maliar and Maliar (2003) Representative Firm CRS production technology, Y t = z t F (K t, L t ) z t is an aggregate shock Spot markets for capital and labor (unlike before) Static profits: Π t = z t F (K t, L t ) w t L t (r t + δ)k t

Erick Sager Lecture 2 (9/14/15) 16 / 37 Maliar and Maliar (2003) Equilibrium household allocation {c i t, h t, k i t+1, a i t+1(ε)} t=0 for each i, firm allocation {K t, h t, L t } t=0 and prices {w t, h t, r t, p t (ε)} t=0 s.t. (i) satisfy household optimality, (ii) satisfy firm optimality (iii) satisfy market clearing conditions for capital, labor and resources: K t = ktdµ i i I L t = I I ε i th i tdµ i c i tdµ i + K t+1 = z t F (K t, L t ) + (1 δ)k t (iv) Arrow securities are in net zero supply

Maliar and Maliar (2003) Erick Sager Lecture 2 (9/14/15) 17 / 37 Planner s Problem U(C t, 1 L t ) = max {c i t,hi t } i I s.t. I α i u(c i t, h i t)dµ i I ci tdµ i C t [λ c t] I εi th i tdµ i L t [λ l t]

Maliar and Maliar (2003) Erick Sager Lecture 2 (9/14/15) 17 / 37 Planner s Problem U(C t, 1 L t ) = max {c i t,hi t } i I s.t. I α i u(c i t, h i t)dµ i I ci tdµ i C t [λ c t] I εi th i tdµ i L t [λ l t] First Order Conditions: ( α c i i t = λ c t ) 1 σ and ( ) 1 ψα h i i γ t = 1 λ l tw t ε i t Aggregate: C t = L t = I I c i tdµ i = (λ c t) 1 σ I ( α i ) 1 σ dµ i ( ) 1 ψ ε i th i tdµ i γ ( = 1 α i ) 1 γ λ l (ε i t) 1 1 γ dµ i tw t I

Maliar and Maliar (2003) First Order Conditions: ( α c i i t = λ c t ) 1 σ and ( ) 1 ψα h i i γ t = 1 λ l tw t ε i t Aggregate: C t = (λ c t) 1 σ I ( α i ) ( ) 1 1 σ ψ dµ i γ ( and L t = 1 α i ) 1 γ λ l (ε i t) 1 1 γ dµ i tw t I Erick Sager Lecture 2 (9/14/15) 18 / 37

Maliar and Maliar (2003) First Order Conditions: ( α c i i t = λ c t ) 1 σ and ( ) 1 ψα h i i γ t = 1 λ l tw t ε i t Aggregate: C t = (λ c t) 1 σ I ( α i ) ( ) 1 1 σ ψ dµ i γ ( and L t = 1 α i ) 1 γ λ l (ε i t) 1 1 γ dµ i tw t I Rewrite FOCs: c i t = I ( α i ) 1 σ ( α i ) 1 σ dµ i C t 1 h i t = I ( ) α i 1 ( γ εt) i 1 γ ( α i ) (1 L t ) 1 γ (ε i t) 1 1 γ dµ i Erick Sager Lecture 2 (9/14/15) 18 / 37

Erick Sager Lecture 2 (9/14/15) 19 / 37 Maliar and Maliar (2003) I = I [ (c α i i t ) 1 σ 1 σ + ψ (1 ] hi t )1 γ dµ i 1 γ α i I = (Ct)1 σ 1 σ + ψ I ( α i ) 1 σ σ 1 ( α i ) (C t) 1 σ 1 σ dµ i 1 σ + ψ α i (Ct)1 σ 1 σ + (1 L t) 1 γ Ψt 1 γ ( α i ) γ 1 ( ε i ) 1 γ t I (αi ) 1 γ (ε i t )1 1 γ dµ i I 1 γ ( α i ) γ 1 ( 1 γ ε i ) 1 γ t ( α i ) (1 L t) 1 γ 1 γ (ε i t) 1 γ 1 dµ i 1 γ dµi i (1 Lt)1 γ dµ 1 γ ( α i ) 1 ( γ ε i ) 1 1 γ ( t Ψ t ψ ( I ) I (αi ) γ 1 (ε i 1 1 γ dµ i ( = ψ α i ) ) 1 γ γ (ε i t) 1 γ 1 dµ i t )1 γ dµ i I

Erick Sager Lecture 2 (9/14/15) 20 / 37 Taking Stock Analytical expression for the representative consumer s preferences Nearly identical to the individual agent s preferences Labor Wedge (Chari, Kehoe and McGrattan (2006)) Ψ t depends on the distribution of idiosyncratic productivity shocks Negishi weights Given U(C, 1 L Ψ), representative agent s allocation: max E 0 {C t,l t,k t+1} t=0 t=0 [ β t (Ct ) 1 σ 1 σ + Ψ (1 L t ) 1 γ t 1 γ ] s.t. C t + K t+1 = z t F (K t, L t ) + (1 δ)k t

Permanent Income Hypothesis Erick Sager Lecture 2 (9/14/15) 20 / 37

Erick Sager Lecture 2 (9/14/15) 21 / 37 Asset Markets Overview Are complete markets a good representation of the data? Consider two extremes: Complete Markets Autarky Which does the data better support? Consider some intermediate case?

Erick Sager Lecture 2 (9/14/15) 22 / 37 Asset Markets Preliminaries s t : S t : state of the economy at t set of possible states s.t. s t S t s t = {x 0,..., s t } S t : history of states up to t pi(s t ) : y i t(s t ) : probability of a particular history agent i s income following history s t at time t

Erick Sager Lecture 2 (9/14/15) 23 / 37 Asset Markets Autarky No possibility for intertemporal substitution of resources No access to asset markets No access to storage technology Then: c i t(s t ) = yt(s i t ) No insurance against income shocks No risk sharing

Erick Sager Lecture 2 (9/14/15) 24 / 37 Asset Markets Complete Markets Access to Arrow securities a i t+1(s t+1, s t ) with price q t+1 (s t+1, s t ) Sequential budget constraint (as before): c i t(s t ) + q t (s t+1, s t )a i t+1(s t+1, s t ) yt(s i t ) + a i t(s t ) s t+1 S t+1 Impose a no Ponzi condition: lim q t (s t+1, s t )a i t+1(s t+1, s t ) 0 t s t+1 S t+1 Constant relative MUCs & only aggregate risk: u (c i t(s t )) u (c j t(s t )) = αj α i = c i t(s t ) = (αi ) 1 σ (α j ) 1 σ j I C t (s t )

Erick Sager Lecture 2 (9/14/15) 25 / 37 Asset Markets Empirical Evaluation Individual consumption growth vs aggregate consumption growth: log ( c i t/c i t 1) = β log (Ct /C t 1 ) Include income growth (c.f. Mace (1991)) log ( c i t/c i t 1) = β1 log (C t /C t 1 ) + β 2 log ( y i t/y i t 1) + ε i t

Erick Sager Lecture 2 (9/14/15) 26 / 37 Asset Markets Restrictions Want alternative that supports partial risk sharing Incomplete Markets: cannot write contracts contingent on all histories s t

Erick Sager Lecture 2 (9/14/15) 26 / 37 Asset Markets Restrictions Want alternative that supports partial risk sharing Incomplete Markets: cannot write contracts contingent on all histories s t Exogenously Incomplete Markets: cannot write contracts on any future contingencies Then: c i t(s t ) + q t (s t )a i t+1(s t ) y i t(s t ) + a i t(s t ) Going forward: suppress (s t ) notation c i t + q t a i t+1 y i t + a i t

Erick Sager Lecture 2 (9/14/15) 27 / 37 Asset Markets Next Steps Write down model of consumer behavior Suppose access to exogenously incomplete markets Make additional assumptions: Permanent Income Hypothesis What are the implications for consumption, savings, income, wealth?

Incomplete Markets Erick Sager Lecture 2 (9/14/15) 28 / 37 Canonical Consumption Savings Problem: v i (a i 0, y i 0) = max E 0 {c i t,ai t+1 } t=0 t=0 β t u(c i t) s.t. c i t + 1 1 + r t a i t+1 y i t + a i t a i t+1 a i t+1 c i t 0

Incomplete Markets Erick Sager Lecture 2 (9/14/15) 28 / 37 Canonical Consumption Savings Problem: Recursive Form: v i (a i 0, y i 0) = max E 0 {c i t,ai t+1 } t=0 t=0 β t u(c i t) s.t. c i t + 1 1 + r t a i t+1 y i t + a i t a i t+1 a i t+1 c i t 0 v(a t, y t ) = max c t,a t+1 u(c t ) + βe t [v(a t+1, y t+1 )] s.t. c t + 1 1 + r t a t+1 y t + a t a t+1 a t+1 c t 0

Erick Sager Lecture 2 (9/14/15) 29 / 37 Permanent Income Hypothesis Restrictions on Canonical Problem Quadratic utility specification: u(c) = α 2 (c t c) 2 c is a bliss point of maximium utility α is a utility parameter One-period returns are certain and pinned down by the discount rate: β(1 + r) = 1 Borrowing constraints are replaced by the No Ponzi Condition for all t 0: [ ( ) j 1 E t lim a j t+j] 0 1 + r

Erick Sager Lecture 2 (9/14/15) 30 / 37 PIH Characterization To do at white board: Optimal Consumption

Erick Sager Lecture 2 (9/14/15) 30 / 37 PIH Characterization To do at white board: Optimal Consumption Consumption is random walk

Erick Sager Lecture 2 (9/14/15) 30 / 37 PIH Characterization To do at white board: Optimal Consumption Consumption is random walk Permanent Income

Erick Sager Lecture 2 (9/14/15) 30 / 37 PIH Characterization To do at white board: Optimal Consumption Consumption is random walk Permanent Income Consumption equals permanent income

Erick Sager Lecture 2 (9/14/15) 30 / 37 PIH Characterization To do at white board: Optimal Consumption Consumption is random walk Permanent Income Consumption equals permanent income Certainty Equivalence Consumption does not depend on income variance

Erick Sager Lecture 2 (9/14/15) 31 / 37 Permanent Income Hypothesis Next Steps What are the implications for consumption, savings, income, wealth? Empirically evaluate Excess sensitivity and Excess smoothness Reconciliation of puzzles

Erick Sager Lecture 2 (9/14/15) 32 / 37 Empirical Evaluation Excess Sensitivity Is consumption a random walk? c t = γ 0 + γ 1 c t 1 + γ 2 z t 1 z t 1 is any variable known at t 1 (suppose income or stock market returns) PIH implies γ 1 = 1, γ 2 = 0 Hall (1978): γ 1 1, γ 2 > 0 and significant

Erick Sager Lecture 2 (9/14/15) 33 / 37 Empirical Evaluation Excess Sensitivity Consumption growth a random walk? c t = +µ 0 + µ 1 z t 1 Suppose z t 1 is income growth y t PIH implies µ 0 = 0 and µ 2 = 0 Flavin (1981): c t = 11.39 + 0.121 y t 1 (9.7) (3.20) Excess sensitivity of current consumption to lagged income

Erick Sager Lecture 2 (9/14/15) 34 / 37 Empirical Evaluation Time Aggregation Suppose consumption data were collected annually t represents a year τ represents six months: τ + (τ + 1) is a year Annual consumption growth: c A t = (c τ + c τ+1 ) (c τ 1 + c τ 2 ) = c τ + c τ+1 c τ 2 = c τ + c τ+1 c τ 2 + ( c τ+1 + c τ + c τ 1 + c τ 2 c τ+1 ) }{{} =0 = c τ+1 + 2 c τ + c τ 1 Annual income growth: y A t = y τ+1 + 2 y τ + y τ 1

Empirical Evaluation Time Aggregation c A t y A t = c τ+1 + 2 c τ + c τ 1 = y τ+1 + 2 y τ + y τ 1 τ 1 is the second half of t 1 c A t and y A t depend on τ 1 Therefore measure a portion of lagged income/consumption Erick Sager Lecture 2 (9/14/15) 35 / 37

Empirical Evaluation Time Aggregation c A t y A t = c τ+1 + 2 c τ + c τ 1 = y τ+1 + 2 y τ + y τ 1 τ 1 is the second half of t 1 c A t and yt A depend on τ 1 Therefore measure a portion of lagged income/consumption Instrument y t 1 with y t 2 : Excess sensitivity still present! c t = 10.63 + 0.174 y t 1 (6.83) (2.18) Erick Sager Lecture 2 (9/14/15) 35 / 37

Erick Sager Lecture 2 (9/14/15) 36 / 37 Empirical Evaluation Predicatable Income Change What if a fraction λ of consumers are hand-to-mouth? Hand-to-Mouth: consume all income each period c t = λ y t + (1 λ)ε t If past income is a good predictor of future income: Excess sensitivity might be due to large fraction λ Campbell and Mankiw (1989): c t = µ + 0.506 y t Resolution if λ 1/2 of aggregate income consumed by Hand-to-Mouth

Erick Sager Lecture 2 (9/14/15) 37 / 37 Permanent Income Hypothesis Next Time Excess smoothness puzzle Two resolutions of the Excess Sensitivity and Excess Smoothness puzzles Inefficient inference / Bias (Campbell and Deaton (1989)) Precautionary savings motives

Lecture 2. (1) Aggregation (2) Permanent Income Hypothesis. Erick Sager. September 14, 2015