Macroeconomic Theory II Homework 2 - Solution Professor Gianluca Violante, TA: Diego Daruich New York University Spring 204 Problem The household has preferences over the stochastic processes of a single consumption good that are ordered by E 0 t=0 βt ln(c t ), where β (0, ) and E 0 is the mathematical expectation with respect to the distribution of the consumption sequence of a single nonstorable good, conditional on the value of the time 0 endowment. The consumer s endowment is drawn from the distribution P rob(y t = 2) = π, P rob(y t = ) = π, where π (0, ). At all times t 2, y t = y t. At each, t 0, the household can lend, but not borrow, at an exogenous and constant risk-free one-period net interest rate of r that satisfies ( + r)β =. The consumer s budget constraint at t is a t+ = ( + r)(a t c t ) + y t+, subject to the initial condition a 0 = y 0. One-period assets carried (a t c t ) over into period t + from t must be nonnegative, so that no-borrowing constraint is a t c t. At time t = 0, after y 0 is realized, the consumer devises an optimal consumption plan. Part a Draw a tree that portrays the possible paths for the endowment sequence from date 0 onward. The four possible paths are: {y 0 =, y =, y 2 =, y 3 =,..., y t =,...} {y 0 =, y = 2, y 2 = 2, y 3 = 2,..., y t = 2,...} {y 0 = 2, y =, y 2 =, y 3 =,..., y t =,...} {y 0 = 2, y = 2, y 2 = 2, y 3 = 2,..., y t = 2,...} as displayed in Figure. The solution is based on an earlier version by Dan Greenwald and Miguel de Faria e Castro, NYU. Comments and errors should be addressed to ddaruich@nyu.edu.
Figure : Endowment process possible paths (given y 0 ). Part b Assume that y 0 = 2. Compute the consumer s optimal consumption and lending plan. Let V (a, y) be the value function of an agent at time t for t, who receives income y forever, and enters the period with assets a. The Bellman equation is subject to and the usual non-negativity constraints on a and c which now reads V (a, y) = max ln(c) + βv (a, y ) () c,a a = ( + r)(a c) + y (2) y = y (3) y a + r 0. Substituting the constraint into the Bellman equation, we obtain V (a, y) = max ln (a + y ) a + βv (a, y) a + r which is now subject only to non-negativity constraints. The FOC is and the envelope condition is c + r = βv (a, y) + λ V (a, y) = c. Combining the two we get c β( + r) c where c and c are the optimal choices of consumption. Applying the assumption β( + r) = now yields c c. (4) We next want to show that (4) holds with equality. To see this, assume false, so that c c. By the complementary slackness condition, this implies that c = a, and that a = y, which implies c y since (adding t for clarity) c t = a t = ( + r)(a t c t ) + y t y t 2
since by the no-borrowing constraint at t, a t c t 0. Also, by the borrowing constraint at t + c t+ y t+ = y t. Therefore, c t c t+. This implies c > c, which contradicts (4). (ALTERNATIVE: Assume it does not bind (if the solution I find is non-binding we are fine), then c = c = c.) Therefore, (4) holds with equality, and so the consumer s path is to maintain constant consumption after receiving the y income draw. Let c be this optimal constant level. Then from the budget constraint (2), we can rearrange (and introduce lag operator notation) to obtain ( + r L )a = c + r y. Inverting the lag polynomial and applying transversality and no Ponzi conditions, we obtain a = ( ( + r L ) c ) + r y ( ) j ( = c ) + r + r y j=0 a = + r c r r y (5) which implies that asset holdings are also constant after the agent receives the y income draw. Since the level of asset holdings at time is given by a = ( + r)(y 0 c 0 ) + y, we can substitute into (5) and rearrange to obtain c = r(y 0 c 0 ) + y. which is less than a = a, (Alternative: our guess that it wasn t binding would hold), if y 0 c 0 which is imposed by the borrowing constraint at time 0. Equation () now becomes V (a, y) = ln (r(y 0 c 0 ) + y ) + V (a, y) + r and so V (a, y) = + r ln (r(y 0 c 0 ) + y ). (6) r Now consider the time 0 problem. Since the continuation value is given by (6), we obtain max ln(c 0 ) + { } π ln (r(y 0 c 0 ) + 2) + ( π) ln (r(y 0 c 0 ) + ) (7) c 0 r s.t. the no-borrowing constraint. The first order condition with respect to c 0 is c 0 π r(y 0 c 0 ) + 2 + π + r(y 0 c 0 ) (with equality if c 0 < y 0 ). Consider the case y 0 = 2. I claim that in this case, (8) holds with equality. To see this, assume false. Then by complementary slackness, we must have c 0 = y 0 = 2, and so (8) becomes 2 > π 2 + π which is clearly a contradiction. Therefore, (8) holds with equality, and we obtain c 0 = π r(y 0 c 0 ) + 2 + 2 π + r(y 0 c 0 ). This is a quadratic equation in c 0 which will yields the optimal choice of time-0 consumption, c 0 < y 0. Intuitively, the agent knows he will be worse tomorrow (or that at least he cannot be better) so he would like to save just in case. Hence, his borrowing constraint does not bind. 3 (8)
Part c Assume that y 0 =. Compute the consumer s optimal consumption and lending plan. When y 0 =, I claim that the optimal time-0 consumption follows a corner solution, with c 0 = y 0 =. To see this, assume false, that c 0 <. Then (8) must hold with equality, and so we have < c 0 = π r( c 0 ) + 2 + π r( c 0 ) + < which is clearly a contradiction. Therefore, we must have c 0 =, and so the agent consumes his or her entire endowment at time 0 if he is poor. Intuitively, the agent knows he will be better tomorrow (or that at least he cannot be worse) so he would like to borrow. However, he is not allowed to do so, so he will consume as much as possible and save nothing. Part d Under the two assumptions on the initial condition for y 0 in the preceding two questions, compute the asymptotic distribution of the marginal utility of consumption u (c t ) (which in this case is the distribution of u (c t ) = V t (a t ) for t 2), where V t (a) is the consumer s value function at date t). When y 0 = 2, we have c = r(2 c 0 ) + y. Therefore, marginal utility in the limit is given by { u (c (r(2 c 0 ) + 2) w.p. π ) = (r(2 c 0 ) + ) w.p. π. When y 0 =, we have c = y, and so marginal utility in the limit is given by { u (c ) = 2 w.p. π w.p. π. So our result is that the agent self-insures only in the case y 0 = 2. Part e Discuss whether your results in part d conform to Chamberlain and Wilson s application of the supermartingale convergence theorem. The sufficient condition in Chamberlain and Wilson for consumption to diverge is that there exists an ε > 0 such that for any α R Pr (α y t α + ε I t ) < ε for all I t and all t 0. Here, this condition is not satisfied. For t and α = y, the above probability is for any ε > 0. Intuitively, there is not enough stochasticity. 4
Problem 2 A consumer chooses her consumption stream by solving max {c t} E 0 s.t. β t u (c t ) t=0 c t + a t+ = ( + r) a t + y t, a t+ φ The income process y t Y = {y, y 2,..., y N } follows a Markov chain π (y, y) = Pr (y t+ = y y t = y). The consumer also faces a natural borrowing constraint, i.e. φ = y /r, where r is the constant real interest rate. Part a Can you think of a joint condition on the period-utility u and on the transition matrix π such that the natural borrowing constraint is never binding for the agent (i.e. the solution of the problem above is always interior for all t)? A simple joint condition that achieves this result is that P i > 0 for all i, and that utility satisfies Inada conditions lim c 0 u (c) =. If in agent were at his or her borrowing limit, then this condition on the transition matrix ensures a positive probability of zero consumption in the next period, which cannot be optimal given our assumption on utility. Why do we need both? Clearly, if a t = φ and the income realization is equal to y, the consumer is forced to choose c t = 0 (check this using the budget constraint and the borrowing constraint). The Inada Condition implies that the consumer will never find it optimal to set c t = 0. This means that as long as there is a positive probability, from the current state i, of getting y in the following period, the agent will never choose to set a t+ = φ in the current period. If we did not have the second condition (on π), the consumer could find it optimal to hit the borrowing constraint in states for which there was a zero probability of transitioning to state. Part b Suppose that the agent does not face any borrowing constraint. What joint condition on u and on the income process (i.e., on Y and on π) do we need to impose to insure that a t > 0 at every t? Assuming that a 0 > 0, a joint sufficient condition is that. β( + r) 2. u is decreasing and convex and satisfies Inada conditions. 3. y t follows a supermartingale, so that y t E t [y t+ ]. In the Markov chain case we are considering, this implies that y is an absorbing state. 5
To show sufficiency, note that the Euler equation for the unconstrained problem (i.e. no borrowing constraint) is given by u (c t ) = β( + r)e t [ u (c t+ ) ]. By Assumption, this implies u (c t ) E t [ u (c t+ ) ]. By Jensen s inequality and the assumed convexity of u, this implies and since u is decreasing, we obtain u (c t ) u (E t c t+ ) c t E t c t+. (9) From the budget constraint, we can invert the lag polynomial, and apply transversality and no-ponzi conditions to obtain ( ) j ( a t = c t+j ) + r + r y t+j. Taking expectations, we now have a t = j=0 j=0 Similarly, since a t+ is known at time t 2 we have ( a t+ = + r j=0 ( ) j ( E t c t+j ) + r + r E ty t+j. ) j ( E t c t++j + r E ty t++j ). But by the law of iterated expectations, Assumption 3, and (9), we have E t y t++j = E t [ E t+j y t++j ] E t y t+j E t c t++j = E t [ E t+j c t++j ] E t c t+j and so we obtain a t+ j=0 ( ) j ( E t c t+j ) + r + r E ty t+j = a t. Assuming a 0 > 0, this shows that the assumptions above are sufficient to ensure that a t > 0 for all t. Intuitively, if the agent is prudent and the stream of income is shrinking in expectation, he or she will save in order to smooth the consumption path. Problem 3 Assume that the consumer has CRRA period utility over consumption given by u (c t ) = c γ t γ 2 Under this formulation of the budget constraint, a t is assets held at the end of period t, or savings. 6
with γ > 0, discounts the future at rate β (0, ) and faces a constant interest rate R. Assume that consumption is conditionally log-normal with mean E t ln (c t+ ) = µ t and variance v t. Part a Show that the optimal consumption path follows E t ( ln c t+ ) = γ ln (βr) + 2 γv t. (0) The Euler equation for this problem is given by [ ] c γ t = βre t c γ t+ = βre t exp { γ ln c t+ }. By assumption, we have γ ln c t+ N( γµ t, γ 2 v t ), and so the standard properties of the log-normal distribution imply that E t exp { γ ln c t+ } = exp { γµ t + 2 } γ2 v t. Substituting into the Euler equation, c γ t = βr exp [ γµ t + 2 ] γ2 v t, taking logs γ ln c t = log(βr) γµ t + 2 γ2 v t, and Using the fact that E t ln c t+ = µ t E t ln c t+ = µ t ln c t, we obtain which is the desired result. E t ln c t+ = γ ln(βr) + 2 γv t () Part b Based on the equation above, does the agent display precautionary saving behavior? Based on the equation above, the agent does display precautionary savings behavior, since expected consumption growth is increasing in the variance of the consumption process, and so the agent saves more as volatility increases. 7
Part c Suppose we tested the Permanent Income Hypothesis, in particular the statement that obly news accruing between t and t + affects the change in consumption c t+ c t by running the regression ln c t+ = α 0 + α y t + ε t+, where y t is past income and we found that α is significantly different from zero. Based on your analysis above, could you say that this result means necessarily a rejection of the PIH? There are two ways to approach this question: the answer depends on whether the question relates to a strong PIH (in which consumption growth should not depend on any t-information set variable) or weak PIH (consumption growth does not depend on current income y t ). If it is considered as written, that consumption growth does not depend on any information in the time-t dataset, then this model does not satisfy the PIH, since expected consumption growth depends on v t. Instead, I will consider the PIH to be the statement that expected consumption growth does not depend on y t. In this case, we get an interesting result, which is that a model may satisfy the PIH even if the test for α = 0 is rejected at a high level of significance. To see this, recall that the estimating the regression c t+ = α 0 + α y t + ε t+ will produce consistent estimates of (α 0, α ) only if E[ε t+ y t ] = 0. For our model, () implies ε t+ = 2 γv t. Therefore, if y t and v t are correlated, we may have E[ε t+ y t ] 0. In this case, our estimate for α is inconsistent, and may converge to something other than 0, even though our model satisfies the PIH. Problem 4 Consider the income fluctuation problem of an agent with CARA utility. Precisely, assume that the agent is infinitely lived, discounts the future at the factor β, faces i.i.d. income shocks y t, can save/borrow through a risk-free asset with constant gross interest rate R (ignore borrowing limits), and has period utility u (c t ) = σ e σct. Guess that the optimal consumption allocation takes the following form c t = B (Ra t + y t ) + D where B and D are constants and have to be determined. Solve for the consumption allocation in closed form (i.e., determine the two constants) and argue that the precautionary saving motive is constant across all agents, i.e., it is independent of the individual pair of state 8
variables (a, y). Explain your answer. The Euler equation for this problem is given by Applying our guess, we obtain { [ exp σ B(Ra t + y t ) + D e σct = βre t e σc t+. { [ = E t exp ln(βr) σ B(Ra t+ + y t+ ) + D. (2) From the budget constraint and our guess for the form of c t, we obtain a t+ = Ra t + y t c t a t+ = ( B)(Ra t + y t ) D. Substitution into (2) now yields (dropping the t in the expectations because y is iid) { [ exp σ B(Ra t + y t ) + D = E [ e σby ] { [ t+ exp ln(βr) σ RB( B)(Ra t + y t ) + ( R)D and rearranging, we obtain = E [ e σby t+ ] exp { [( ) ln(βr) σ RB( B) B (Ra t + y t ) + ( R)D. Since this must hold for all a t, y t, we must have RB( B) = B, and so we have Similarly, we must have and so B = R. = E [ e σby t+ ] exp {ln(βr) σ( R)D} { [ D = ln(βr) + ln E e σby t+. σ(r ) There are two things to notice about precautionary savings:. For an argument that the precautionary savings motive that is constant across agents, note that the multiplicative term, B, is identical across agents. This term determines the proportion of wealth saved by the agent, and therefore the agent s precautionary savings behavior. Since it does not depend on a or y, this is the desired result. 2. You can see the precautionary savings motive more clearly through constant D. Note that this constant depends negatively on the term E exp( σ(r )y/r) and is the same for all agents, where y is the iid random variable that corresponds to income. Let µ denote the mean of y and ν its standard deviation. Then, we can take a second order approximation around µ of the term inside the expectation to obtain exp( σ(r )y/r) exp( σ(r )µ/r) σ R exp( σ(r )µ/r)(y µ)+ R + ( ) R 2 2 σ2 exp( σ(r )µ/r)(y µ) 2 R 9
Now take expectations on both sides to obtain E exp( σ(r )y/r) exp( σ(r )µ/r) + ( ) R 2 2 σ2 exp( σ(r )µ/r)ν 2 R Note that the first-order term cancels out and we are left with a constant plus something that depends positively on the variance of the income process. If we apply a mean-preserving spread to income (fix µ and raise ν), then the expectation term increases. Thus D, and consumption decreases. Since D is the same for all agents, consumption decreases by exactly the same amount regardless of the individual states (a, y). This is a very powerful result, as it tells us that all agents respond to the same change in the distribution of income by raising their savings by the exact same amount! 0