Lecture 8: Consumption-Savings Decisions

Lecture 8: Consumption-Savings Decisions Florian Scheuer 1 Plan 1. A little historical background on models of consumption 2. Some evidence (!) 3. Studing the incomplete markets consumption model (which we saw last class as an example) with recursive tools 2 The Permanent Income Hpothesis ecall the consumption-savings problem under certaint s.t. and a No Ponzi condition max c t t=0 β t u(c t ) a t+1 = a t + t c t Assuming is constant, the NPV budget constraint is Euler equation t c t = t t t=0 t=0 u (c t ) = βu (c t+1 ) Special case: β = 1 1

Euler equation implies constant consumption Budget 1 1 1 c = t t t=0 c = 1 t t t=0 E.g. if t = constant, consume income ever period For later use, it is helpful to rewrite this as (exercise! hint: use the sequential budget constraint) c = 1 a t + which holds for an period t Now assume that: the income process is uncertain ] t+s s = ] 1 a t + t + s=0 t+s s, s=1 the household acts as though it wasn t: certaint equivalent behavior (CEQ) Certaint equivalent behavior is not optimal in general But it s optimal if preferences are quadratic - we ll see this later (idea: linear Euler equations) Consumption becomes c t = 1 ( )] a t + t + E t s t+s s=1 (1) Idea: each period, ou recompute the expected NPV of future income, forget it s uncertain and recalculate the optimal consumption path Set current consumption equal to annuit value of assets and expected future income This is called the permanent income hpothesis (PIH): given permanent income ( ) p t = t + E t s t+s, s=1 2

c t should be a function of p t and not independentl of current income t (Milton Friedman) Alternativel, we can again rewrite (1) using the sequential budget constraint as c t = 1 ( ]) E t s s s=0 Notice timing 2.1 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: c t = α 0 + α 1 t Kenesian marginal propensit to consume (MPC): α 1 < 1 PIH MPC Marginal propensit to consume out of wealth: same for MPC out of assets a t. c t (E t s=0 s s ]) = 1 0.04. small! Marginal propensit to consume out of a purel temporar income increase: 1 (holding E t s=1 s t+s ] constant) c t t = Marginal propensit to consume out of a permanent income increase c t ȳ = 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. 3

2.2 The random walk prediction Take first difference of (1) and use sequential budget constraint (exercise!): c t = c t c t 1 = 1 ( s (E t t+s E t 1 t+s ) s=0 ) (2) Change in consumption comes from revisions in permanent income (i.e. news arriving at t about the expected present value of future income) Therefore E t 1 ( c t ) = 0 Intuition: consumer has no access to insurance, so consumption smoothing implies minimizing c Testable! 1. egress c t on information known at t 1: should be unpredictable! 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 1 = 1. 3. elax certaint-equivalence Test Euler equation: ( u ) (c t ) E t 1 β t u = 1 (c t 1 ) CA case with riskless interest rate Assume values for σ egress c σ t ( ) ) σ ct E t 1 (β = 1 c t 1 c σ t on c σ t 1 and information known at t 1 4 = c σ t 1 (β) 1 + ε t

Evidence from Hall 1978]: If previous consumption was based on all information consumers had at the time, past income should not contain an additional explanator power about current consumption above past consumption. Supported in the data. As long as income is somewhat predictable, this distinguishes PIH from Kenesian or more generall from: c t = D(L) t 2.3 Test of consumption smoothing Start from (2), estimate stochastic process for income and check. Example: t follows an MA(1) process t = ε t + βε t 1 Using (2), c t = 1 = 1 = 1 ( s (E t t+s E t 1 t+s ) s=0 ( ) t E t 1 t + 1 (E t t+1 E t 1 t+1 ) ( ) 1 + 1 β ε t ) since E t t+1 = βε t and E t 1 t+s = 0 for all s 1 5

Hence this is an example where the stochastic process for income features serial correlation (if β > 0) and c t > 1 ε t MPC from innovation to current income depends on persistence of income process if we do not control for permanent income. Could compute this for more general AMA processes for income α(l) t = β(l)ε t Campbell and Deaton 1989]: estimate t = 8.2 + 0.442 t 1 + ε t with σ ε = 25.2 Income growth positivel correlated permanent income more volatile than current income. Using (2): c t = 0.558 + 1 ε t = 1.78ε t for = 1.01 quarterl. Standard deviation of c t should be 1.78 times σ ε. Data: 27.3 for total; 12.4 for nondurables, less than σ ε = 25.2: excess smoothness! One explanation: slow adjustment to innovations in permanent income 2.4 Test of Euler equation (without assuming values for σ) Derive log-linear version of Euler equation Hansen and Singleton, 1983]: E t ( ( ) ) σ ct+1 β t+1 = 1 c t 6

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 2σ log t+1 σ ε t+1 σ (3) 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 using lagged values of consumption or income growth or interest rates as instruments for log t+1. 7

Excess sensitivit to predictable current income 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)? Tests generall rel on aggregate data. Aggregation issues: across goods across agents: Euler equations non-linear (Attanasio and Weber) across time: data averaged over continuous time, which introduces serial correlation Carroll 1997]: don t ignore the v t term in (3)! Buffer-stock behavior predicts sstematic relation between v t and wealth/income ratio More generall: precautionar savings 3 Precautionar savings: two-period example No complete markets Onl riskless borrowing-saving 8

Uncertain future income FOC: max a u ( 0 a) + β Pr (s) u (a + 1 (s)) s u ( 0 a ) + β Pr (s) u (a + 1 (s)) = 0 (4) s Define â as the solution to ) u ( 0 â) + βu (â + Pr (s) 1 (s) s Is it the case that a = â (certaint equivalence)? Proposition 1. a > â if u ( ) is convex Proof. Assume the contrar: = 0 s a â u (a + 1 (s)) u (â + 1 (s)) Pr (s) u (a + 1 (s)) Pr (s) u (â + 1 (s)) s s ) > u (â + Pr (s) 1 (s) s = u ( 0 â) b definition β u ( 0 a ) (a â) β (u convex) which contradicts (4) Precautionar savings if u ( ) > 0 Inevitable as c 9

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 4 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 ) (5) s.t. x (s ) x c] + (s ) c x + b with equalit if c < x + b u (c) λ (s) µ = 0 s β Pr ( s s ) V(x (s ), s ) λ (s) = 0 x iid shocks and borrowing constraint not binding: u (c) β Pr(s s) V(x (s ), s ) s x u (c) = βev (x )] again: precautionar savings if V convex, but V is now endogenous! 10

result: u > 0 V > 0 (Sible 1975) Envelope condition V(x, s) x = u (c(x, s)) Euler equation: with equalit if c < x + b u (c) β s Pr(s s)u ( c(x (s ), s ) ) 4.1 The β = 1 case For β = 1, the Euler equation is so marginal utilit follows a supermartingale u (c) s Pr(s s)u ( c(x (s ), s ) ) 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] < + What does it mean for a sequence of random variables to converge a.s. to a random variable? 1. The sequence converges 2. The point towards which it converges is random, i.e. could be different for different realizations of the sequence 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. 11

B the Martingale Convergence Theorem, the marginal utilit of consumption converges almost surel to a finite number. (What finite number could depend on the sequence of income realizations). Suppose that number is greater than zero. Then consumption converges to a finite number. Since after an sequence s t the realization of t could be (s 1 ) forever, then the maximum constant consumption that is sustainable under all conceivable future income realizations after histor s t is c(s t+n ) = (s 1) + 1 x(st ) for all n 0, i.e. the annuit value of current assets and a stream of future incomes all equal to the lowest possible realization. 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 ) occurs, which contradicts optimalit. Hence, a time-invariant consumption level does not maximize utilit, and future consumption must var with income shocks. As a result, consumption cannot converge to a finite limit. 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: assets also diverge to Does not depend on u (c) (but in a wa it does) 4.2 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) Allow c < 0 Suppose is iid 12

Useful properties of CAA: Household solves 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) V(x) = max u(c) + β Pr(s )V( x c] + (s )) c s Guess: Using guess: V (x) = max c V(x) = Au( 1 x) 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 )]] Therefore ( A = max u( α) + βae u ( 1) α + 1 )]] α 13 (6)

which verifies the guess because A is a constant. Solve b taking FOCs: ( u ( α) + βa ( 1) E u ( u( α) = βa ( 1) E u ( 1) α + 1 ( 1) α + 1 )] )] = 0 = 0 (7) ecall (6): ( A = u( α) βae u ( 1) α + 1 )] and using (7): A = u( α) u( α) 1 = 1 u( α) 14

eplace back in the FOC (7) ( 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 ( 1) α = u (βe 1 u α = u 1 = 1 γ = 1 γ = 1 γ ( βe log βe ( )]) u 1 ( 1 )]] u 1 1 log (β) + log log (β) 1 u 1 ( )]] E u 1 1 ( ( )]) E u 1 1 (8) The polic function for consumption is therefore c(x) = 1 x α = 1 x 1 log (β) γ 1 + u 1 Special case of β = 1 and no uncertaint about : c(x) = 1 x + This is exactl what we found under the PIH! ( ( )]) E u 1 1 The term: adjusts for β = 1 1 log (β) γ 1 15

The (constant) term: adjusts for the precautionar savings effect. ( ( )]) u 1 E u 1 1 u concave means ( 1 E u ( 1 u (E 1 u )] )]) ( < u 1 ( 1 < u = 1 E ) E ( 1 u )) E Hence, the polic function for consumption has the same slop as under PIH, but a lower intercept. PIH would hold under risk-neutralit. Evolution of wealth: x t+1 = (x t c(x t )) + 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 (8) Implication: the variance of the wealth distribution in the population diverges to infinit! There is no invariant distribution. Consumption c t = 1 x t α is also a random walk with drift (since it is a linear function of a random walk process). Drift: consumption will tend to increase iff 1 γ log (β) 1 + u 1 ( ( )]) E u 1 1 c < 1 x + E < E It could drift off to infinit even with β < 1, if there is sufficient risk and risk aversion 16

Was out: Life ccle (OLG): people retire/die, no perfect inheritabilit. This limits the persistence of wealth and consumption inequalit. Infinite horizon but: other preferences borrowing constraints 4.3 β < 1 and decreasing absolute risk aversion ecall that the coefficient of absolute risk aversion is defined as γ(c) = u (c) u (c) Now assume that β < 1 and lim γ(c) = 0 c + Idea: large precautionar savings effect for low assets, but goes to zero for high assets. This allows for some upper bound on assets and thus a non-degenerate wealth distribution. 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 (5) 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. 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 ) )) = β s Pr(s )u (c ( x c(x)] + (s ))) u (c (x max(x))) u ( c ( x max(x) )) (9) 17

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))) u ( c ( x min (x))) u (c (x max(x))) = 1 ( u (c (x max(x))) u c ( x max(x) )) 1 c(x ˆmax(x)) c(x min (x)) u (c) u (c) dc c(x ˆmax(x)) u (c)dc c(x min (x)) The fact that lim c + γ(c) = 0 implies that eplacing (10) in (9) and taking limits: lim s Pr(s )u (c ( x c(x)] + (s ))) x u (c (x max(x))) = 1 (10) Since β < 1, this implies that for high enough x, lim u (c(x)) βu ( c(x x max(x)) )] = 0 u (c(x)) < u (c(x max(x))) and hence c(x max(x)) < c(x), so consumption will 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 If we think of the population as a collection of independent individuals, there is a stead state wealth distribution 18

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