Lecture 2. (1) Permanent Income Hypothesis (2) Precautionary Savings. Erick Sager. February 6, 2018

Transcription

1 Lecture 2 (1) Permanent Income Hypothesis (2) Precautionary Savings Erick Sager February 6, 2018 Econ 606: Adv. Topics in Macroeconomics Johns Hopkins University, Spring 2018

2 Erick Sager Lecture 2 (2/6/18) 1 / 39 Housekeeping (2/6/2018) Updated syllabus Office hours set Problem set due 2/27 Referee report selection due 2/14 Presentation paper selection due 2/14 both me and Joseph

3 Erick Sager Lecture 2 (2/6/18) 2 / 39 Last Time (1/30/18) Considered data Wealth, Consumption, Income Cross-Sectional, Times-Series, Life-Cycle Inequality

4 Erick Sager Lecture 2 (2/6/18) 2 / 39 Last Time (1/30/18) Considered data Wealth, Consumption, Income Cross-Sectional, Times-Series, Life-Cycle Inequality Talked about aggregation Started with Gorman Form c i (p, w i ) = a i (p) + b(p)w i = C(p, w) = C(p, W )

5 Last Time (1/30/18) Erick Sager Lecture 2 (2/6/18) 3 / 39 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

6 Last Time (1/30/18) Erick Sager Lecture 2 (2/6/18) 3 / 39 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

7 Last Time (1/30/18) Erick Sager Lecture 2 (2/6/18) 3 / 39 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

8 Last Time (1/30/18) Erick Sager Lecture 2 (2/6/18) 3 / 39 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

9 Erick Sager Lecture 2 (2/6/18) 4 / 39 Last Time (1/30/18) Maliar and Maliar (2003) Complete markets Idiosyncratic labor productivity shocks

10 Erick Sager Lecture 2 (2/6/18) 4 / 39 Last Time (1/30/18) 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

11 Erick Sager Lecture 2 (2/6/18) 4 / 39 Last Time (1/30/18) 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

12 Erick Sager Lecture 2 (2/6/18) 5 / 39 Last Time (1/30/18) 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

13 Erick Sager Lecture 2 (2/6/18) 5 / 39 Last Time (1/30/18) 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: ( ( ψ ) I α i 1 γ γ (ε i t) 1 1 γ dµ i) Ψ t ( ) I (αi ) 1 σ σ dµ i

14 Erick Sager Lecture 2 (2/6/18) 5 / 39 Last Time (1/30/18) 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: ( ( ψ ) I α i 1 γ γ (ε i t) 1 1 γ dµ i) Ψ t ( ) I (αi ) 1 σ σ dµ 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

15 Erick Sager Lecture 2 (2/6/18) 6 / 39 Today (2/6/18) Permanent Income Hypothesis (PIH) Restrict asset space Derive results for consumption, savings, wealth Empirical evaluation of theory Excess Sensitivity Puzzle Excess Smoothness Puzzle Reconciliation of Puzzles (Campbell and Deaton (1989)) Next: Precautionary Savings

16 Permanent Income Hypothesis Erick Sager Lecture 2 (2/6/18) 6 / 39

17 Erick Sager Lecture 2 (2/6/18) 7 / 39 Next Time Asset Markets 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?

18 Erick Sager Lecture 2 (2/6/18) 7 / 39 Next Time Asset Markets 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? ( this one)

19 Erick Sager Lecture 2 (2/6/18) 8 / 39 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 π(s t ) : y i t(s t ) : probability of a particular history agent i s income following history s t at time t

20 Erick Sager Lecture 2 (2/6/18) 9 / 39 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

21 Erick Sager Lecture 2 (2/6/18) 10 / 39 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 )

22 Erick Sager Lecture 2 (2/6/18) 11 / 39 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

23 Erick Sager Lecture 2 (2/6/18) 11 / 39 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 β 1 = 1 and β 2 = 0 β 1 = 0 and β 2 = 1

24 Erick Sager Lecture 2 (2/6/18) 11 / 39 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 β 1 = 1 and β 2 = 0 β 1 = 0 and β 2 = 1 Data shows something in between

25 Erick Sager Lecture 2 (2/6/18) 12 / 39 Asset Markets Restrictions Want alternative that supports partial risk sharing Incomplete Markets: cannot write contracts contingent on all histories s t

26 Erick Sager Lecture 2 (2/6/18) 12 / 39 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

27 Erick Sager Lecture 2 (2/6/18) 12 / 39 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

28 Erick Sager Lecture 2 (2/6/18) 13 / 39 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?

29 Incomplete Markets Erick Sager Lecture 2 (2/6/18) 14 / 39 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 t a i t+1 y i t + a i t a i t+1 a i t+1 c i t 0

30 Incomplete Markets Erick Sager Lecture 2 (2/6/18) 14 / 39 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 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 t a t+1 y t + a t a t+1 a t+1 c t 0

31 Erick Sager Lecture 2 (2/6/18) 15 / 39 Permanent Income Hypothesis Restrictions on Canonical Problem - Review - 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 Borrowing constraints are replaced by the No Ponzi Condition for all t 0: [ ( ) j 1 E t lim a j t+j] 0

32 Erick Sager Lecture 2 (2/6/18) 16 / 39 PIH Characterization Quadratic utility: u (c) = d dc [ α 2 (c t c) 2] = α(c t c) Euler equation: u (c) β()e t [u (c t+1 )] Quadratic utility, β() = 1 and the No Ponzi condition, imply: α c αc t = E t [α c αc t+1 ] c t = E t [c t+1 ] Consumption is a random walk or martingale By the Law of Iterated Expectations: c t = E t [c t+j ] j 0

33 Erick Sager Lecture 2 (2/6/18) 17 / 39 PIH Characterization Define human wealth at time t as: h t j=0 ( 1 1+r ) j Et [y t+j ] Define financial wealth as a t and total wealth as a t + h t Permanent income is the annuity value of total wealth: r 1+r (a t + h t ). Iterate forward on the budget constraint and divide by () j for each iteration j: ( ) 1 c t = y t + a t a t+1 ( ) ( ) ( ) ( ) c t+1 = y t+1 + a t+1 a t+2 ( ) j ( ) j ( ) j ( ) j c t = y t+j + a t+j a t+1+j

34 Erick Sager Lecture 2 (2/6/18) 18 / 39 Summing gives: j=0 ( ) j 1 c t+j = a t + j=0 ( ) j ( ) j 1 1 y t+j lim a t+j j Taking expectations and applying the No Ponzi Condition yields: j=0 ( ) j 1 E t [c t+j] = a t + j=0 ( ) j 1 E t [y t+j] But because consumption is a random walk, E t [c t+j ] = c t for all j 0: c t = r ( ) j 1 a t + E t [y t+j] j=0 r ) (a t + h t Therefore, consumption equals permanent income

35 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption:

36 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t

37 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income:

38 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r a t + j=0 ( ) j 1 E t [y t+j] r ) (a t + h t

39 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r a t + j=0 ( ) j 1 E t [y t+j] r ) (a t + h t Certainty Equivalence

40 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r a t + j=0 ( ) j 1 E t [y t+j] r ) (a t + h t Certainty Equivalence Consumption does not depend on income variance Any stochastic process for income with same expected value (e.g., = h t ) yields same consumption The result of quadratic preferences

41 Erick Sager Lecture 2 (2/6/18) 19 / 39 PIH Characterization Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r a t + j=0 ( ) j 1 E t [y t+j] r ) (a t + h t Certainty Equivalence Consumption does not depend on income variance Any stochastic process for income with same expected value (e.g., = h t ) yields same consumption The result of quadratic preferences Next: Consumption and Wealth Dynamics

42 Erick Sager Lecture 2 (2/6/18) 20 / 39 PIH Characterization c t = c t c t 1 c t = = c t E t 1 c t = r ( ) (a t + h t ) E t 1 [(a t + h t )] = r r a t E t 1 [a t ] + j=0 j=0 ( ) j 1 ( E t [y t+j] E t 1[y t+j]) ( ) j 1 ( ) E t [y t+j] E t 1[E t [y t+j]] The change in consumption between times t 1 and t is proportional to new information agents receive about their discounted expected income.

43 Erick Sager Lecture 2 (2/6/18) 21 / 39 PIH Characterization Similarly for wealth dynamics a t+1 = a t+1 a t = ra t + ()(y t c t ) Substitute expression for consumption and iterate 1 a t+1 = j=1 ( ) j 1 E t [ y t+j] Change in savings proportional to discounted expected income change Agents use savings to offset expected income fluctuations

44 Erick Sager Lecture 2 (2/6/18) 21 / 39 PIH Characterization Similarly for wealth dynamics a t+1 = a t+1 a t = ra t + ()(y t c t ) Substitute expression for consumption and iterate 1 a t+1 = j=1 ( ) j 1 E t [ y t+j] Change in savings proportional to discounted expected income change Agents use savings to offset expected income fluctuations Suppose y t+j = (1 + g) j 1 for g < r

45 Erick Sager Lecture 2 (2/6/18) 21 / 39 PIH Characterization Similarly for wealth dynamics a t+1 = a t+1 a t = ra t + ()(y t c t ) Substitute expression for consumption and iterate 1 a t+1 = j=1 ( ) j 1 E t [ y t+j] Change in savings proportional to discounted expected income change Agents use savings to offset expected income fluctuations Suppose y t+j = (1 + g) j 1 for g < r a t+1 = j=1 ( ) j g = r g Income growth increases the rate at which consumer decreases savings

46 PIH Characterization - Review - Erick Sager Lecture 2 (2/6/18) 22 / 39 Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r ( ) j 1 a t + E t [y t+j] r ) (a t + h t j=0

47 PIH Characterization - Review - Erick Sager Lecture 2 (2/6/18) 22 / 39 Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r ( ) j 1 a t + E t [y t+j] r ) (a t + h t j=0 Certainty Equivalence Consumption does not depend on income variance Consumption and Wealth Dynamics

48 PIH Characterization - Review - Erick Sager Lecture 2 (2/6/18) 22 / 39 Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r ( ) j 1 a t + E t [y t+j] r ) (a t + h t j=0 Certainty Equivalence Consumption does not depend on income variance Consumption and Wealth Dynamics Response to news c t = r ( ) j 1 ( E t [y t+j] E t 1[y t+j]) j=0

49 PIH Characterization - Review - Erick Sager Lecture 2 (2/6/18) 22 / 39 Optimal Consumption: Consumption is random walk, E t [c t+1 ] = c t Permanent Income: Consumption equals permanent income c t = r ( ) j 1 a t + E t [y t+j] r ) (a t + h t j=0 Certainty Equivalence Consumption does not depend on income variance Consumption and Wealth Dynamics Response to news c t = r ( ) j 1 ( E t [y t+j] E t 1[y t+j]) j=0 Offsets expected income fluctuations 1 ( ) j 1 a t+1 = E t [ y t+j] j=1

50 Erick Sager Lecture 2 (2/6/18) 23 / 39 Permanent Income Hypothesis Next Steps What are the implications for consumption, savings, income, wealth? Empirically evaluate Excess sensitivity and Excess smoothness Reconciliation of puzzles

51 Erick Sager Lecture 2 (2/6/18) 24 / 39 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

52 Erick Sager Lecture 2 (2/6/18) 25 / 39 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 = y t 1 (9.7) (3.20) Excess sensitivity of current consumption to lagged income

53 Erick Sager Lecture 2 (2/6/18) 26 / 39 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 c τ + c τ 1 Annual income growth: y A t = y τ y τ + y τ 1

54 Empirical Evaluation Time Aggregation c A t y A t = c τ c τ + c τ 1 = y τ 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 (2/6/18) 27 / 39

55 Empirical Evaluation Time Aggregation c A t y A t = c τ c τ + c τ 1 = y τ 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 = y t 1 (6.83) (2.18) Erick Sager Lecture 2 (2/6/18) 27 / 39

56 Erick Sager Lecture 2 (2/6/18) 28 / 39 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 y t Resolution if λ 1/2 of aggregate income consumed by Hand-to-Mouth

57 Erick Sager Lecture 2 (2/6/18) 29 / 39 Permanent Income Hypothesis Next Steps: Excess smoothness puzzle If income is persistent, then PIH predicts: consumption variability > income variability Data: consumption variability < income variability Two resolutions of the Excess Sensitivity and Excess Smoothness puzzles Inefficient inference / Bias (Campbell and Deaton (1989)) Precautionary savings motives

58 Erick Sager Lecture 2 (2/6/18) 30 / 39 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1

59 Erick Sager Lecture 2 (2/6/18) 30 / 39 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics

60 Erick Sager Lecture 2 (2/6/18) 30 / 39 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics c t = r ( ( ) ) 1 ɛ t + γɛ t

61 Erick Sager Lecture 2 (2/6/18) 30 / 39 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics c t = r ( ( ) ) 1 ɛ t + γɛ t Assume r = 0.04, γ = ()/2 ( r σ c = 1 + γ ) σ ɛ = ( )σ ɛ 0.06σ ɛ

62 Erick Sager Lecture 2 (2/6/18) 30 / 39 Excess Smoothness Process 1: y t = ɛ t + γɛ t 1 Optimal consumption dynamics c t = r ( ( ) ) 1 ɛ t + γɛ t Assume r = 0.04, γ = ()/2 ( r σ c = 1 + γ ) σ ɛ = ( )σ ɛ 0.06σ ɛ Then too smooth: σ c /σ ɛ nearly zero

63 Erick Sager Lecture 2 (2/6/18) 31 / 39 Excess Smoothness Process 2: y t = ɛ t

64 Erick Sager Lecture 2 (2/6/18) 31 / 39 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics

65 Erick Sager Lecture 2 (2/6/18) 31 / 39 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics c t = ɛ t

66 Erick Sager Lecture 2 (2/6/18) 31 / 39 Excess Smoothness Process 2: y t = ɛ t Optimal consumption dynamics c t = ɛ t Then too volatile: σ c = σ ɛ

67 Erick Sager Lecture 2 (2/6/18) 32 / 39 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t

68 Erick Sager Lecture 2 (2/6/18) 32 / 39 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics

69 Erick Sager Lecture 2 (2/6/18) 32 / 39 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics c t = γ ɛ t

70 Erick Sager Lecture 2 (2/6/18) 32 / 39 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics c t = γ ɛ t Assume r = 0.04, γ = 0.26 σ c = γ σ 1.04 ɛ = σ ɛ = 1.33σ ɛ

71 Erick Sager Lecture 2 (2/6/18) 32 / 39 Excess Smoothness Process 3: y t = γ y t 1 + ɛ t Optimal consumption dynamics c t = γ ɛ t Assume r = 0.04, γ = 0.26 σ c = γ σ 1.04 ɛ = σ ɛ = 1.33σ ɛ Then too volatile: σ c > σ ɛ If income changes persistent (γ < 1), consumption variability larger than income variability Data shows the opposite!

72 Erick Sager Lecture 2 (2/6/18) 33 / 39 Resolution Campbell and Deaton (1989) Suppose individuals have more information than econometrician

73 Erick Sager Lecture 2 (2/6/18) 33 / 39 Resolution Campbell and Deaton (1989) Suppose individuals have more information than econometrician Classic source of bias: estimating income variability with error

74 Erick Sager Lecture 2 (2/6/18) 33 / 39 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

75 Erick Sager Lecture 2 (2/6/18) 34 / 39 Resolution I t is individual s information set

76 Erick Sager Lecture 2 (2/6/18) 34 / 39 Resolution I t is individual s information set Ω t I t is econometrician s information set

77 Erick Sager Lecture 2 (2/6/18) 34 / 39 Resolution I t is individual s information set Ω t I t is econometrician s information set Econometrician s prediction error is: 1 ( 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 ])

78 Erick Sager Lecture 2 (2/6/18) 34 / 39 Resolution I t is individual s information set Ω t I t is econometrician s information set Econometrician s prediction error is: 1 ( 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 ]) Strategy: Use information on savings as predictor of income growth

79 Erick Sager Lecture 2 (2/6/18) 35 / 39 Resolution Assume: ( ) yt 1 1+r a = t+1 ( ) ( ) ζ11 ζ 12 yt 1 + ζ 21 ζ r a t ( u1t u 2t )

80 Erick Sager Lecture 2 (2/6/18) 35 / 39 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

81 Erick Sager Lecture 2 (2/6/18) 35 / 39 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 = 1 a t+1

82 Erick Sager Lecture 2 (2/6/18) 36 / 39 Resolution j-step ahead forecasts: E t [x t+j ] = A j x t

83 Erick Sager Lecture 2 (2/6/18) 36 / 39 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 a t+1 = e 1 A j x t j=1

84 Erick Sager Lecture 2 (2/6/18) 36 / 39 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 a t+1 = e 1 A j x t e 2 x t = e 1 j=1 j=1 ( ) j 1 A j x t

85 Erick Sager Lecture 2 (2/6/18) 36 / 39 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 a t+1 = e 1 A j x t e 2 x t = e 1 Implies parameter restrictions: j=1 j=1 ζ 21 = ζ 11 ( ) j 1 A j x t ζ 22 = ζ 12 + ()

86 Erick Sager Lecture 2 (2/6/18) 37 / 39 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 a t+1 = u 1t u 2t

87 Erick Sager Lecture 2 (2/6/18) 37 / 39 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 a t+1 = u 1t u 2t Campbell and Deaton (1989) estimate the VAR x t = Ax t 1 + u t

88 Erick Sager Lecture 2 (2/6/18) 37 / 39 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 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

89 Erick Sager Lecture 2 (2/6/18) 37 / 39 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 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 +

90 Erick Sager Lecture 2 (2/6/18) 37 / 39 Resolution Restrictions imply consumption is a random walk: c t = y t + a t 1 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 + Implies excess sensitivity: c t = χ y t 1 + (u 1t u 2t )

91 Erick Sager Lecture 2 (2/6/18) 38 / 39 Resolution What does assuming excess sensitivity imply for excess smoothness?

92 Erick Sager Lecture 2 (2/6/18) 38 / 39 Resolution What does assuming excess sensitivity imply for excess smoothness? Some heroic algebra shows: c t = u 1t u 2t 1 χ 1+r

93 Erick Sager Lecture 2 (2/6/18) 38 / 39 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

94 Erick Sager Lecture 2 (2/6/18) 38 / 39 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

95 Erick Sager Lecture 2 (2/6/18) 38 / 39 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 There is no contradiction between excess sensitivity and excess smoothness; they are the same phenomenon.

96 Erick Sager Lecture 2 (2/6/18) 38 / 39 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 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

97 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time Other mechanisms that generate excess sensitivity and smoothness?

98 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings

99 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints

100 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints Prudence in utility

101 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints Prudence in utility Friedman / Buffer Stock model

102 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time Other mechanisms that generate excess sensitivity and smoothness? Precautionary Savings Borrowing constraints Prudence in utility Friedman / Buffer Stock model Add impatience to precautionary motives

103 Erick Sager Lecture 2 (2/6/18) 39 / 39 Next Time 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