Consumption / Savings Decisions

Transcription

1 Consumption / Savings Decisions 1 The Permanent Income Hpothesis (This is mostl repeated from the first class) Household consumption decisions characterized b 1. Euler equation: u (c t ) = β p t p t+1 u (c t+1 ) = β t+1 u (c t+1 ) 2. Budget constraint with equalit: p t c t = t=0 p t t t=0 Cleanest formulation of the hpothesis is for the case of β = 1 (but principle is more general) For β = 1, Euler becomes c t+1 = c t and budget becomes so t (e t c t ) = 0 t=0 c t = 1 ( ) s s s=0 Interpretation: Now assume that: the income process is uncertain 1

2 the household acts as though it wasn t (certaint equivalent behaviour) Certaint equivalent behaviour is not optimal in general But it s optimal if preferences are quadratic - we ll see this later Consumption becomes c t = 1 ( [ ]) E t s s s=0 Notice timing Implicit: evolution of household s assets 1.1 Contrast with traditional Kenesian view Kenes (1936): The fundamental pschological law, upon which we are entitled to depend with great confidence both a priori from our knowledge of human nature and from the detailed facts of experience, is that men are disposed, as a rule and on the average, to increase their consumption as their income increases but not b as much as the increase in the income. Kenesian consumption function: Kenesian MPC: α 1 < 1 PIH MPC c t = α 0 + α 1 t Marignal propensit to consume out of wealth: c t (E t[ s=0 s s]) = c Marginal propensit to consume out of purel temporar income: t t E t [ s=1 s s ] constant) Marginal propensit to consume out of permanent income ct t = 1 (holding = 1 (assuming Ets t = 1) In general, MPC out of shocks to current income depends on income process. It could be greater than 1 if income growth is highl seriall correlated. See below. 2

3 1.2 The random walk prediction Take first difference: c t = 1 ([ ]) s E t t+s E t 1 t+s s=0 (1) Therefore Testable! E t 1 ( c t ) = 0 1. egress c t on information known at t 1 2. elax β = 1 but keep the certaint-equivalence assumption: c t = a 0 + a 1 c t 1 + ε t egress c t on c t 1 and information known at t 1. This is like the previous case but ou allow a 0 0 and a elax certaint-equivalence Test Euler equation: ( ) u (c t ) E t 1 β t = 1 u (c t 1 ) CA case with riskless interest rate ( ) ) σ ct E t (β = 1 c t 1 c σ t = c σ t 1 (β) 1 + ε t Assume values for σ egress c σ t on c σ t 1 and information known at t 1 Evidence from Hall [1978]: 3

4 Econ 210, Fall 2013 As long as income is somewhat predictable, this distinguishes PIH from Kenesian or more generall from: ct = D(L)t 1.3 Test of consumption smoothing Start from (1), estimate stochastic process for income and check. Campbell and Deaton [1989]: Income growth positivel correlated permanent income more volatile than current income. Using (1): εt = 1.78εt ct = for = 1.01 quarterl. Standard deviation of ct should be 1.78 times σε. Data: 27.3 for total; 12.4 for nondurables, less than σε = 25.2: excess smoothness! One explanation: slow adjustment to innovations in permanent income 4

5 1.4 Test of Euler equation (without assuming values for σ) Derive log-linear version of Euler equation [Hansen and Singleton, 1983]: ( ( ) ) σ ct+1 E t β t+1 = 1 c t Define ( ) σ ct+1 z t+1 t+1 c t Assume z t+1 is lognormal given time-t information: log z t+1 = log t+1 σ log c t+1 N (µ t, v t ) Define ε t+1 log z t+1 µ t and note ε t+1 N (0, v t ) Using the properties of the lognormal distribution: Now use the Euler equation: [ E t (z t+1 ) = exp µ t + v ] t 2 E t (z t+1 ) = 1 β [ exp µ t + v ] t = 1 2 β µ t + v t 2 = log β log (z t+1 ) ε t+1 + v t 2 = log β log t+1 σ log c t+1 ε t+1 + v t 2 = log β [ log β log c t+1 = σ + v ] t log t+1 ε t+1 2σ σ σ (2) log c t+1 should not be predictable with time-t information other than interest rates. Campbell and Mankiw [1990] estimate: log c t+1 = µ + λ log t+1 + θr t+1 + ε t+1 5

6 Econ 210, Fall 2013 using lagged values of consumption or income growth or interest rates as instruments for log t+1. Excess sensitivit Campbell and Mankiw [1990]: a fraction of consumers just consume their income. Liquidit constraints? Campbell and Deaton [1989]: delaed response of consumption to innovations. Would also explain excess smoothness Statisticall: predictable changes in consumption Wh the difference with Hall [1978]? 1950:1 (large paments to WWII veterans)? Carroll [1997]: don t ignore the vt term in (2)! Buffer-stock behaviour predicts sstematic relation between vt and wealth/income ratio More generall: precautionar savings 2 Precautionar savings: two-period example No complete markets Onl riskless borrowing-saving Uncertain future income max u (0 a) + β X a s 6 Pr (s) u (a + 1 (s))

7 FOC: u ( 0 a ) + β s Pr (s) u (a + 1 (s)) = 0 (3) Define â as the solution to u ( 0 â) + βu ( â + s Pr (s) 1 (s) ) = 0 Is it the case that a = â (certaint equivalence)? Proposition 1. a > â if u ( ) is convex Proof. Assume the contrar: a â u (a + 1 (s)) u (â + 1 (s)) s Pr (s) u (a + 1 (s)) Pr (s) u (â + 1 (s)) s s ( > u â + ) Pr (s) 1 (s) s (u convex) = u ( 0 â) β u ( 0 a ) β b definition (a â) which contradicts (3) Precautionar savings if u ( ) > 0 Inevitable as c True for CA: u (c) = c1 σ 1 σ u (c) = c σ u (c) = σc σ 1 u (c) = σ (1 + σ) c σ 2 > 0 For quadratic preferences: certaint equivalence 7

8 3 Infinite-horizon consumption problem with uncertaint [Bewle, 1977, Aiagari, 1994] We talked about this as one example of a dnamic optimization problem under uncertaint. Now we ll analze it in detail ecursive representation: V (x, s) = max c,x (s ) u(c) + β s Pr(s s)v (x (s ), s ) (4) s.t. x (s ) [x c] + (s ) c x + b u (c) s λ (s) µ = 0 with equalit if c < x + b β Pr (s s) V (x (s ), s ) x λ (s) = 0 u (c) β s Pr(s s) V (x (s ), s ) x Envelope condition Euler equation: V (x, s) x = u (c(x, s)) u (c) β s Pr(s s)u (c(x (s ), s )) with equalit if c < x + b 3.1 The β = 1 case For β = 1, the Euler equation is u (c) s Pr(s s)u (c(x (s ), s )) so marginal utilit follows a supermartingale Theorem 1 (Doob). Let {Z t } be a nonnegative supermartingale. Then Z t a.s. Z, where Z is a random variable with E [Z] < + 8

9 Example 1. Suppose Z t 1 u t if t 7 Z t = otherwise Z t 1 where u t U [0, 1], and Z 0 = 8. Then Z t is a nonnegative supermartingale. It converges to some random variable Z with support in [1, 8] Proposition 2. For β = 1, the solution of the household problem has Pr [lim t c(s t ) = + ] = 1, i.e. consumption diverges to infinit almost surel Proof. B the Martingale Convergence Theorem, the marginal utilit of consumption converges almost surel to a finite number. income realizations). (What finite number could depend on the sequence of Suppose than number is greater than zero. Then consumption converges almost surel to a finite number. Since after an sequence s t the realization of t could be (s 1 ) forever, then the maximum possible constant consumption after histor s t is c(s t+n ) = (s 1) + 1 x(st ) for all n 0. But the household can improve upon this constant consumption b ratcheting up consumption b 1 ((s t) (s 1 )) forever whenever an income level other than (s 1 ), which contradicts optimalit. Therefore u (c t ) must converge to zero Therefore (assuming u (c) > 0 c), c t must diverge to (See Chamberlain and Wilson [2000] for a rigorous version of this proof and exact conditions under which it holds) Strong precautionar motive Does not depend on u (c) (but in a wa it does) 3.2 The iid-caa case in detail (without making assumptions on β and ) Suppose u(c) = exp( γc) Do not impose a borrowing limit (just a no-ponzi condition) 9

10 Allow c < 0 Suppose is iid Useful properties of CAA: u (c) = γ exp( γc) = γu(c) u(a + b) = u(a)u(b) u 1 (v) = 1 γ log( v) 1 u (c) = u ( c) Household solves V (x) = max u(c) + β Pr(s )V ( [x c] + (s )) c s Guess: V (x) = Au(λx) with λ = 1 Using guess: V (x) = max c [ u(c) + βae = max u(c) + βae c ( 1 u ( [ u )] [ [x c] + ] ( 1) (x c) + 1 )] Let α x c 1 x c = 1 x α ( 1) (x c) = ( 1) α + 1 x so ( 1 V (x) = max u α ( ) 1 = u x ) x α max α [ ( + βae u [ u( α) + βae ( 1) α + 1 x + 1 [ ( u ( 1) α + 1 )]] )] 10

11 Therefore [ [ ( A = max u( α) + βae u ( 1) α + 1 )]] α which verifies the guess because A is a constant. (5) Solve b taking FOCs: ( u ( α) + βa ( 1) E [u ( u( α) = βa ( 1) E [ u ( 1) α + 1 ( 1) α + 1 )] )] = 0 = 0 (6) ecall (5): [ ( A = u( α) βae u ( 1) α + 1 )] and using (6): A = u( α) u( α) 1 = 1 u( α) 11

12 eplace back in the FOC (6) [ ( u( α) = βu( α)e u ( 1) α + 1 )] [ ( 1 = βe u ( 1) α + 1 )] [ ( )] 1 = βu (( 1) α) E u [ ( )] 1 u ( ( 1) α) = βe u ( 1) α = u 1 (βe ( 1 [ u )]) ( α = u 1 βe [ u ( 1 )]) 1 = 1 log [ βe [ u ( 1 )]] γ 1 = 1 log (β) + log [ E [ u ( 1 )]] γ 1 = 1 ( [ ( log (β) γ 1 u 1 E u 1 )]) 1 (7) The consumption function is therefore c = 1 x α = 1 x 1 γ log (β) 1 + u 1 Special case of β = 1 and no uncertaint about : c = 1 x + ( [ ( E u 1 )]) 1 The term: adjusts for β 1 1 log (β) γ 1 The (constant) term: adjusts for the precautionar savings effect. u ( 1 E [ u ( 1 )]) 1 12

13 u concave means [ ( 1 E u [ ( 1 u (E 1 u )] )]) ( < u 1 ( 1 < u = 1 E ) E ( 1 u )) E Drift: consumption will tend to increase iff 1 γ log (β) 1 + u 1 ( [ ( E u 1 )]) 1 c < 1 < E x + E It could drift off to infinit even with β < 1, if there is sufficient risk and risk aversion Evolution of wealth: x t+1 = (x t c) + t+1 ( = x t 1 ) x t + α + t+1 = x t + t+1 + α so wealth is a randow walk with drift. The drift is given b (7) Implication: the variance of the wealth distribution in the population diverges to infinit! 3.3 β < 1 and decreasing absolute risk aversion ecall that the coefficient of absolute risk aversion s defined as γ(c) = u (c) u (c) Now assume that β < 1 and lim γ(c) = 0 c + Focus on the iid case for simplicit (but the result is more general) Proposition 3. Assume β < 1 and lim c + γ(c) = 0. Let c(x) be the solution to program (4) and let x max(x) = [x c(x)] + (s n ) and x min(x) = [x c(x)] + (s 1 ). There exists a value x such that x max(x) x for all x x. 13

14 Proof. Below is part of the proof. There are three things missing from it (left as an exercise) 1. Showing that the borrowing constraint does not bind for sufficientl high x 2. Showing that c (x) is increasing 3. Showing that lim x c (x) = For sufficientl high x the borrowing constraint will not bind, so the Euler equation is u (c(x)) = β s Pr(s )u (c ( [x c(x)] + (s ))) s Pr(s )u (c ( [x c(x)] + (s ))) = β u (c (x u (c (x max(x))) (8) max(x))) Note that since c (x) is increasing, the following inequalities hold: 1 s Pr(s )u (c ( [x c(x)] + (s ))) u (c (x max(x))) The fact that lim c + γ(c) = 0 implies that eplacing (9) in (8) and taking limits: u (c (x min(x))) u (c (x max(x))) = 1 u (c (x max(x))) u (c (x max(x))) 1 c(x max(x)) ˆ c(x min (x)) u (c) u (c) dc s Pr(s )u (c ( [x c(x)] + (s ))) lim x u (c (x max(x))) c(x max ˆ (x)) c(x min (x)) u (c)dc = 1 (9) lim x [u (c(x)) βu (c max(x))] = 0 Since β < 1, this implies that for high enough x, consumption is expected to fall for an realization of s. Given that c (x) is increasing, the result follows. Consumption does not grow to infinit There is an upper bound on wealth There is an invariant distribution of wealth for an given individual 14

15 If we think of the population as a collection of independent individuals, there is a stead state wealth distribution c x + b 45 c(x) b - b + (s 1 ) x ^ x x' 45 x'(x,s n ) x'(x,s 1 ) x* x eferences S ao Aiagari. Uninsured idiosncratic risk and aggregate saving. The Quarterl Journal of Economics, 109(3):659 84, August Truman Bewle. The permanent income hpothesis: A theoretical formulation. Journal of Economic Theor, 16(2): , December John Y Campbell and Angus Deaton. Wh is consumption so smooth? eview of Economic Studies, 56(3):357 73, Jul John Y Campbell and N Gregor Mankiw. Permanent income, current income, and consumption. Journal of Business & Economic Statistics, 8(3):265 79, Jul

16 Christopher D Carroll. Buffer-stock saving and the life ccle/permanent income hpothesis. The Quarterl Journal of Economics, 112(1):1 55, Februar Gar Chamberlain and Charles A. Wilson. Optimal intertemporal consumption under uncertaint. eview of Economic Dnamics, 3(3): , Jul obert E Hall. Stochastic implications of the life ccle-permanent income hpothesis: Theor and evidence. Journal of Political Econom, 86(6):971 87, December Lars Peter Hansen and Kenneth J Singleton. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Econom, 91(2):249 65, April