Lecture 2. (1) Permanent Income Hypothesis. (2) Precautionary Savings. Erick Sager. September 21, PDF Free Download

Lecture 2 (1) Permanent Income Hypothesis (2) Precautionary Savings Erick Sager September 21, 2015 Econ 605: Adv. Topics in Macroeconomics Johns Hopkins University, Fall 2015

Erick Sager Lecture 2 (9/21/15) 1 / 36 Last Time (8/31/15) Finished thinking about aggregation Gorman Form c i (p, w i ) = a i (p) + b(p)w i = C(p, w) = C(p, W )

Last Time (9/14/15) Erick Sager Lecture 2 (9/21/15) 2 / 36 Negishi Planner s Problem: V ({µ i } N i=1, k 0 ) = max {{c i t }N t=1,kt+1} t=0 s.t. β t N t=0 i=1 ( N µ i u(c i t) ) i=1 µ ic i t + k t+1 = f(k t ) + (1 δ)k t

Last Time (9/14/15) Erick Sager Lecture 2 (9/21/15) 2 / 36 Negishi Planner s Problem: V ({µ i } N i=1, k 0 ) = max {{c i t }N t=1,kt+1} t=0 s.t. Constantinides Decomposition: β t N t=0 i=1 ( N µ i u(c i t) ) i=1 µ ic i t + k t+1 = f(k t ) + (1 δ)k t

Last Time (9/14/15) Erick Sager Lecture 2 (9/21/15) 2 / 36 Negishi Planner s Problem: V ({µ i } N i=1, k 0 ) = max {{c i t }N t=1,kt+1} t=0 s.t. Constantinides Decomposition: 1. Individual Allocation U(C t ) = max {c i t }N i=1 β t N t=0 i=1 ( N { N µ i u(c i t) s.t. i=1 µ i u(c i t) ) i=1 µ ic i t + k t+1 = f(k t ) + (1 δ)k t } N π i c i t C t i=1

Last Time (9/14/15) Erick Sager Lecture 2 (9/21/15) 2 / 36 Negishi Planner s Problem: V ({µ i } N i=1, k 0 ) = max {{c i t }N t=1,kt+1} t=0 s.t. Constantinides Decomposition: 1. Individual Allocation U(C t ) = max {c i t }N i=1 2. Aggregate Allocation max {C t,k t+1} t=0 s.t. β t N t=0 i=1 ( N { N µ i u(c i t) s.t. i=1 β t U(C t ) t=0 µ i u(c i t) ) i=1 µ ic i t + k t+1 = f(k t ) + (1 δ)k t } N π i c i t C t i=1 C t + k t+1 = f(k t ) + (1 δ)k t k 0 given

Erick Sager Lecture 2 (9/21/15) 3 / 36 Last Time (9/14/15) Maliar and Maliar (2003) Complete markets Idiosyncratic labor productivity shocks

Erick Sager Lecture 2 (9/21/15) 3 / 36 Last Time (9/14/15) Maliar and Maliar (2003) Complete markets Idiosyncratic labor productivity shocks 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 I εi th i tdµ i L t

Erick Sager Lecture 2 (9/21/15) 3 / 36 Last Time (9/14/15) Maliar and Maliar (2003) Complete markets Idiosyncratic labor productivity shocks Planner s Problem U(C t, 1 L t ) = First Order Conditions: 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 I εi th i tdµ i L t c i t = I ( ) α i 1 σ ( α i ) C t, 1 h i 1 σ dµ 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 (9/21/15) 4 / 36 Last Time (9/14/15) Maliar and Maliar (2003) I [ (c α i i t ) 1 σ 1 σ + ψ (1 hi t) 1 γ ] dµ i = (C t) 1 σ 1 γ 1 σ + Ψ (1 L t ) 1 γ t 1 γ

Erick Sager Lecture 2 (9/21/15) 4 / 36 Last Time (9/14/15) Maliar and Maliar (2003) I [ (c α i i t ) 1 σ 1 σ + ψ (1 hi t) 1 γ ] dµ i = (C t) 1 σ 1 γ 1 σ + Ψ (1 L t ) 1 γ t 1 γ Labor Wedge endogenously arises: ( ) α i 1 ( γ ε i 1 1 ( γ Ψ t ψ t) ( I I (αi ) 1 1 γ dµ γ (ε i t) 1 1 γ dµ i) i ( = ψ α i ) γ 1 γ (ε i t) 1 1 γ dµ i) I

Erick Sager Lecture 2 (9/21/15) 5 / 36 Last Time (9/14/15) Permanent Income Hypothesis (PIH) Restrict asset space Exogenously incomplete asset markets Cannot write asset contracts contingent on specific future states c t + 1 1 + r t a t+1 y t + a t Strict Permanent Income Hypothesis Utility: u(c) = (α/2)(c t c) 2 Time Preference: β(1 + r) = 1 No Ponzi Condition

Last Time (9/14/15) Erick Sager Lecture 2 (9/21/15) 6 / 36 PIH Characterization Consumption is random walk: E t [c t+1 ] = c t Consumption equals permanent income c t = r ( ) j 1 a t + E t [y t+j] r ) (a t + h t 1 + r 1 + r 1 + r j=0 Consumption does not depend on income variance Consumption responds to news c t = r 1 + r j=0 ( ) j 1 ( E t [y t+j] E t 1[y t+j]) 1 + r Assets offset expected income fluctuations 1 ( ) j 1 1 + r a t+1 = E t [ y t+j] 1 + r j=1

Erick Sager Lecture 2 (9/21/15) 7 / 36 Today (9/21/15) Finish: Empirical evaluation of theory Excess Sensitivity Puzzle (β 1 > 0): c t = β 0 + β 1 y t 1 + ɛ t Excess Smoothness Puzzle Reconciliation of Puzzles (Campbell and Deaton (1989))

Erick Sager Lecture 2 (9/21/15) 8 / 36 Excess Smoothness Suppose income is serially correlated and persistent

Erick Sager Lecture 2 (9/21/15) 8 / 36 Excess Smoothness Suppose income is serially correlated and persistent Strict PIH predicts: σ c > σ y

Erick Sager Lecture 2 (9/21/15) 8 / 36 Excess Smoothness Suppose income is serially correlated and persistent Strict PIH predicts: σ c > σ y In the data: σ c < σ y

Erick Sager Lecture 2 (9/21/15) 8 / 36 Excess Smoothness Suppose income is serially correlated and persistent Strict PIH predicts: σ c > σ y In the data: σ c < σ y Illustrate this puzzle using the Strict PIH model Use three different income processes:

Erick Sager Lecture 2 (9/21/15) 9 / 36 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1

Erick Sager Lecture 2 (9/21/15) 9 / 36 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics

Erick Sager Lecture 2 (9/21/15) 9 / 36 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics c t = r ( ( ) ) 1 ɛ t + γɛ t 1 + r 1 + r

Erick Sager Lecture 2 (9/21/15) 9 / 36 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics c t = r ( ( ) ) 1 ɛ t + γɛ t 1 + r 1 + r Assume r = 0.04, γ = (1 + r)/2 ( r σ c = 1 + γ ) σ ɛ = 0.04 1 + r 1 + r 1.04 (1 + 0.5)σ ɛ 0.06σ ɛ

Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t

Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics

Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics c t = ɛ t

Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics c t = ɛ t Then too volatile: σ c = σ ɛ

Erick Sager Lecture 2 (9/21/15) 11 / 36 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t

Erick Sager Lecture 2 (9/21/15) 11 / 36 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics

Erick Sager Lecture 2 (9/21/15) 11 / 36 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics c t = 1 + r 1 + r γ ɛ t

Erick Sager Lecture 2 (9/21/15) 11 / 36 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics c t = 1 + r 1 + r γ ɛ t Assume r = 0.04, γ = 0.26 σ c = 1 + r 1 + r γ σ 1.04 ɛ = 1.04 0.26 σ ɛ = 1.33σ ɛ

Erick Sager Lecture 2 (9/21/15) 12 / 36 Resolution Campbell and Deaton (1989) Suppose individuals have more information than econometrician

Erick Sager Lecture 2 (9/21/15) 12 / 36 Resolution Campbell and Deaton (1989) Suppose individuals have more information than econometrician Classic source of bias: estimating income variability with error

Erick Sager Lecture 2 (9/21/15) 13 / 36 Resolution I t is individual s information set

Erick Sager Lecture 2 (9/21/15) 13 / 36 Resolution I t is individual s information set Ω t I t is econometrician s information set

Erick Sager Lecture 2 (9/21/15) 13 / 36 Resolution I t is individual s information set Ω t I t is econometrician s information set Econometrician s prediction error is: 1 1 + r ( a i t+1 a e t+1) = j=1 ( ) j 1 ( E t [ y t+j Ω t ] E t [ y t+j I t ]) 1 + r

Erick Sager Lecture 2 (9/21/15) 14 / 36 Resolution Assume: ( ) yt 1 1+r a = t+1 ( ) ( ) ζ11 ζ 12 yt 1 + ζ 21 ζ 22 1 1+r a t ( u1t u 2t )

Erick Sager Lecture 2 (9/21/15) 14 / 36 Resolution Assume: ( ) yt 1 1+r a = t+1 ( ) ( ) ζ11 ζ 12 yt 1 + ζ 21 ζ 22 1 1+r a t ( u1t u 2t ) Rewritten in vector notation: x t = Ax t 1 + u t

Erick Sager Lecture 2 (9/21/15) 14 / 36 Resolution Assume: ( ) yt 1 1+r a = t+1 ( ) ( ) ζ11 ζ 12 yt 1 + ζ 21 ζ 22 1 1+r a t ( u1t u 2t ) Rewritten in vector notation: x t = Ax t 1 + u t Rewrite using e 1 = (1, 0) and e 2 = (0, 1): e 1 x t = y t e 2 x t = 1 1 + r a t+1

Erick Sager Lecture 2 (9/21/15) 15 / 36 Resolution j-step ahead forecasts: E t [x t+j ] = A j x t

Erick Sager Lecture 2 (9/21/15) 15 / 36 Resolution j-step ahead forecasts: Rewrite: E t [x t+j ] = A j x t E t [ y t+j ] = e 1 A j x t+j 1 ( ) j 1 1 + r a t+1 = e 1 A j x t 1 + r j=1

Erick Sager Lecture 2 (9/21/15) 15 / 36 Resolution j-step ahead forecasts: Rewrite: Rewrite: E t [x t+j ] = A j x t E t [ y t+j ] = e 1 A j x t+j 1 ( ) j 1 1 + r a t+1 = e 1 A j x t 1 + r e 2 x t = e 1 j=1 j=1 ( ) j 1 A j x t 1 + r

Erick Sager Lecture 2 (9/21/15) 15 / 36 Resolution j-step ahead forecasts: Rewrite: E t [x t+j ] = A j x t Rewrite: E t [ y t+j ] = e 1 A j x t+j 1 ( ) j 1 1 + r a t+1 = e 1 A j x t 1 + r e 2 x t = e 1 Implies parameter restrictions: j=1 j=1 ζ 21 = ζ 11 ( ) j 1 A j x t 1 + r ζ 22 = ζ 12 + (1 + r)

Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 1 + r a t+1 = u 1t u 2t

Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 1 + r a t+1 = u 1t u 2t Campbell and Deaton (1989) estimate the VAR x t = Ax t 1 + u t Find that ζ 11 ζ 21 Suppose ζ 21 = ζ 11 χ such that: ( ) ζ11 ζ A = 12 ζ 11 χ ζ 12 + 1 + r

Erick Sager Lecture 2 (9/21/15) 17 / 36 Resolution What does assuming excess sensitivity imply for excess smoothness?

Erick Sager Lecture 2 (9/21/15) 17 / 36 Resolution What does assuming excess sensitivity imply for excess smoothness? Some heroic algebra shows: c t = u 1t u 2t 1 χ 1+r

Erick Sager Lecture 2 (9/21/15) 17 / 36 Resolution What does assuming excess sensitivity imply for excess smoothness? Some heroic algebra shows: c t = u 1t u 2t 1 χ 1+r Magnitude of excess sensitivity determines consumption response Cases: χ 0, χ 1 + r There is no contradiction between excess sensitivity and excess smoothness; they are the same phenomenon.

Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness?

Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings

Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints

Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints Prudence in utility

Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints Prudence in utility Friedman / Buffer Stock model Add impatience to precautionary motives

Erick Sager Lecture 2 (9/21/15) 19 / 36 Precautionary Savings Borrowing Constraints Empirical Observation (Parker (1999), Souleles (1999), others)

Erick Sager Lecture 2 (9/21/15) 19 / 36 Precautionary Savings Borrowing Constraints Empirical Observation (Parker (1999), Souleles (1999), others) Consumption is sensitive to anticipated income changes Study government transfers (tax rebates, social security changes) Consumption response larger for low income/liquid wealth households

Erick Sager Lecture 2 (9/21/15) 20 / 36 Precautionary Savings Borrowing Constraints Consider Strict Permanent Income Hypothesis

Erick Sager Lecture 2 (9/21/15) 20 / 36 Precautionary Savings Borrowing Constraints Consider Strict Permanent Income Hypothesis Add No Borrowing Constraint: a t+1 0

Erick Sager Lecture 2 (9/21/15) 20 / 36 Precautionary Savings Borrowing Constraints Consider Strict Permanent Income Hypothesis Add No Borrowing Constraint: a t+1 0 If borrowing constraint binds: c t = y t + a t 1 1 + r a t+1 }{{} =0

Erick Sager Lecture 2 (9/21/15) 21 / 36 Precautionary Savings Borrowing Constraints Recall optimal savings dynamics: 1 1 + r a t+1 = j=1 ( ) j 1 E t [ y t+j] 1 + r

Erick Sager Lecture 2 (9/21/15) 21 / 36 Precautionary Savings Borrowing Constraints Recall optimal savings dynamics: 1 1 + r a t+1 = j=1 ( ) j 1 E t [ y t+j] 1 + r Suppose: y t = y t 1 + ɛ t, ɛ N (0, σ 2 ɛ )

Erick Sager Lecture 2 (9/21/15) 22 / 36 Precautionary Savings Borrowing Constraints Suppose: y t = ɛ t s.t. ɛ N (0, σ 2 ɛ )

Erick Sager Lecture 2 (9/21/15) 22 / 36 Precautionary Savings Borrowing Constraints Suppose: y t = ɛ t s.t. ɛ N (0, σɛ 2 ) Then: y t+j = ɛ t+j ɛ t 1+j

Erick Sager Lecture 2 (9/21/15) 22 / 36 Precautionary Savings Borrowing Constraints Suppose: y t = ɛ t s.t. ɛ N (0, σ 2 ɛ ) Then: y t+j = ɛ t+j ɛ t 1+j Optimal savings dynamics: 1 1 + r a t+1 = j=1 ( ) j 1 E t [ y t+j] = ɛ t 1 + r

Erick Sager Lecture 2 (9/21/15) 23 / 36 Precautionary Savings Borrowing Constraints Suppose: y t = ɛ t s.t. ɛ N (0, σ 2 ɛ )

Erick Sager Lecture 2 (9/21/15) 23 / 36 Precautionary Savings Borrowing Constraints Suppose: y t = ɛ t s.t. ɛ N (0, σ 2 ɛ ) Optimal consumption: c t = r 1+r (a t + y t ) if a t+1 > 0 y t + a t if a t+1 = 0 = min { y t + a t, } r 1 + r (a t + y t ) Agent is constrained when: r 1 + r (a t + y t ) > y t + a t y t < a t Future borrowing constraints affect current consumption: c t = E t [min{y t+1 + a t+1, E t+1 [c t+2 ]}]

Erick Sager Lecture 2 (9/21/15) 24 / 36 Precautionary Savings Prudence Another mechanism: Prudence (Kimball, 1990)

Erick Sager Lecture 2 (9/21/15) 24 / 36 Precautionary Savings Prudence Another mechanism: Prudence (Kimball, 1990) Preferences for smooth consumption profile Self-insurance against low income realizations Income variability increases Precautionary Motive

Erick Sager Lecture 2 (9/21/15) 25 / 36 Precautionary Savings Prudence Consider: u : R + R, u (c) > 0, u (c) < 0

Erick Sager Lecture 2 (9/21/15) 25 / 36 Precautionary Savings Prudence Consider: u : R + R, u (c) > 0, u (c) < 0 Arrow-Pratt measure of of absolute risk aversion: A(c) u (c) u (c)

Erick Sager Lecture 2 (9/21/15) 25 / 36 Precautionary Savings Prudence Consider: u : R + R, u (c) > 0, u (c) < 0 Arrow-Pratt measure of of absolute risk aversion: A(c) u (c) u (c) Decreasing absolute risk aversion (DARA) implies: A (c) = u (c) u (c) + ( u ) 2 (c) u < 0 (c) Condition on the third derivative of the utility function: u (c) > u (c) 2 u (c) > 0

Erick Sager Lecture 2 (9/21/15) 26 / 36 Precautionary Savings Prudence Prudence implies: Uncertainty increases Consumption today decreases Savings for tomorrow increases

Erick Sager Lecture 2 (9/21/15) 26 / 36 Precautionary Savings Prudence Prudence implies: Uncertainty increases Consumption today decreases Savings for tomorrow increases Consider two period example (Leland, 1968) Board work

Erick Sager Lecture 2 (9/21/15) 27 / 36 Permanent Income Hypothesis Next Time Continue discussion of Prudence Patience: β versus (1 + r) Characterization of Buffer Stock model Marginal Propensity to Consume

Lecture 2. (1) Permanent Income Hypothesis. (2) Precautionary Savings. Erick Sager. September 21, 2015