Lecture 2. (1) Permanent Income Hypothesis. (2) Precautionary Savings. Erick Sager. September 21, 2015
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1 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
2 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 )
3 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
4 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
5 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
6 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
7 Erick Sager Lecture 2 (9/21/15) 3 / 36 Last Time (9/14/15) Maliar and Maliar (2003) Complete markets Idiosyncratic labor productivity shocks
8 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
9 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
10 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
11 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
12 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 Labor Wedge: ψ(1 h t ) γ = (1 τ t )w t c σ 1 t = τ t = 1 ψ(1 h t) γ (1 α)y t /L t c σ t
13 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 r t a t+1 y t + a t
14 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 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
15 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 r a t+1 = E t [ y t+j] 1 + r j=1
16 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))
17 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)) Start: Buffer Stock Savings Model Borrowing Constraints Prudence Patience
18 Erick Sager Lecture 2 (9/21/15) 8 / 36 Excess Smoothness Suppose income is serially correlated and persistent
19 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
20 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
21 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
22 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:
23 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: y t = ɛ t + γɛ t 1
24 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: y t = ɛ t + γɛ t 1 y t = y t 1 + ɛ t
25 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: y t = ɛ t + γɛ t 1 y t = y t 1 + ɛ t y t = γ y t 1 + ɛ t
26 Erick Sager Lecture 2 (9/21/15) 9 / 36 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1
27 Erick Sager Lecture 2 (9/21/15) 9 / 36 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics
28 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
29 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 = r 1 + r 1.04 ( )σ ɛ 0.06σ ɛ
30 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 = r 1 + r 1.04 ( )σ ɛ 0.06σ ɛ Then too smooth: σ c /σ ɛ nearly zero
31 Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t
32 Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics
33 Erick Sager Lecture 2 (9/21/15) 10 / 36 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics c t = ɛ t
34 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
35 Erick Sager Lecture 2 (9/21/15) 11 / 36 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t
36 Erick Sager Lecture 2 (9/21/15) 11 / 36 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics
37 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
38 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 = σ ɛ = 1.33σ ɛ
39 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 = σ ɛ = 1.33σ ɛ Then too volatile: σ c > σ ɛ
40 Erick Sager Lecture 2 (9/21/15) 12 / 36 Resolution Campbell and Deaton (1989) Suppose individuals have more information than econometrician
41 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
42 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 Strategy: Use information on savings as predictor of income growth
43 Erick Sager Lecture 2 (9/21/15) 13 / 36 Resolution I t is individual s information set
44 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
45 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: 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
46 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: 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 Strategy: Use information on savings as predictor of income growth
47 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 r a t ( u1t u 2t )
48 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 r a t ( u1t u 2t ) Rewritten in vector notation: x t = Ax t 1 + u t
49 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 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 = r a t+1
50 Erick Sager Lecture 2 (9/21/15) 15 / 36 Resolution j-step ahead forecasts: E t [x t+j ] = A j x t
51 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 r a t+1 = e 1 A j x t 1 + r j=1
52 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 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
53 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 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)
54 Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t r a t+1 = u 1t u 2t
55 Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t r a t+1 = u 1t u 2t Campbell and Deaton (1989) estimate the VAR x t = Ax t 1 + u t
56 Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 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
57 Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 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 r
58 Erick Sager Lecture 2 (9/21/15) 16 / 36 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 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 r Implies excess sensitivity: c t = χ y t 1 + (u 1t u 2t )
59 Erick Sager Lecture 2 (9/21/15) 17 / 36 Resolution What does assuming excess sensitivity imply for excess smoothness?
60 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
61 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
62 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
63 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.
64 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. If past savings predicts income, then savings affects permanent income
65 Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness?
66 Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings
67 Erick Sager Lecture 2 (9/21/15) 18 / 36 Next Steps Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints
68 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
69 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
70 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
71 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 Marginal Propensity to Consume out of income shocks
72 Erick Sager Lecture 2 (9/21/15) 19 / 36 Precautionary Savings Borrowing Constraints Empirical Observation (Parker (1999), Souleles (1999), others)
73 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
74 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)
75 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
76 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 Suggestive of borrowing constraints impeding consumption smoothing
77 Erick Sager Lecture 2 (9/21/15) 20 / 36 Precautionary Savings Borrowing Constraints Consider Strict Permanent Income Hypothesis
78 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
79 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 r a t+1 }{{} =0
80 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 r a t+1 }{{} =0 Optimal consumption is: E t 1 [c t ] if a t+1 > 0 c t = y t + a t if a t+1 = 0
81 Erick Sager Lecture 2 (9/21/15) 21 / 36 Precautionary Savings Borrowing Constraints Recall optimal savings dynamics: r a t+1 = j=1 ( ) j 1 E t [ y t+j] 1 + r
82 Erick Sager Lecture 2 (9/21/15) 21 / 36 Precautionary Savings Borrowing Constraints Recall optimal savings dynamics: 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 )
83 Erick Sager Lecture 2 (9/21/15) 21 / 36 Precautionary Savings Borrowing Constraints Recall optimal savings dynamics: 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 ) Then a t+1 = 0
84 Erick Sager Lecture 2 (9/21/15) 21 / 36 Precautionary Savings Borrowing Constraints Recall optimal savings dynamics: 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 ) Then a t+1 = 0 Borrowing constraint always binds: c t = y t + a t t
85 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 )
86 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
87 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: r a t+1 = j=1 ( ) j 1 E t [ y t+j] = ɛ t 1 + r
88 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: r a t+1 = j=1 ( ) j 1 E t [ y t+j] = ɛ t 1 + r Savings is a random walk
89 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: r a t+1 = j=1 ( ) j 1 E t [ y t+j] = ɛ t 1 + r Savings is a random walk Borrowing constraint binds with probability one as t
90 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 )
91 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 )
92 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
93 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 ]}]
94 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: What happens if σ ɛ increases? c t = E t [min{y t+1 + a t+1, E t+1 [c t+2 ]}]
95 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: What happens if σ ɛ increases? c t = E t [min{y t+1 + a t+1, E t+1 [c t+2 ]}]
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