Problem Set #4 Answer Key

Transcription

1 Problem Set #4 Answer Key Economics 808: Macroeconomic Theory Fall 2004 The cake-eating problem a) Bellman s equation is: b) If this policy is followed: c) If this policy is followed: V (k) = max {log c + βv (k c) c [0,k] k t = β t k 0 c t = ( β) β t k 0 d) The value function is calculated by simply substituting the sequence {c t into the utility function: ( ) V (k) = β t log ( β) β t k = = ( ( β t log ( β) β t) ) + log k ( β t log ( β) β t) + = constant + β log k β t log k e) The new policy function c (k) is the value of c that solves: { c (k) = arg max log c + β c [0,k] β log (k c) Taking first order conditions, the new policy function is c (k) = ( β)k f) Since c and c 0 are identical, we have already found the optimal policy function: c(k) = ( β)k

2 ECON 808, Fall g) First we find V by solving the maximization problem: The first order conditions are: Solving for c, we get: Then we substitute back in to get V : V (k) = max c [0,] {log c + βv 0(k c) = max {log c + β log(k c) c [0,] c β k c = 0 c = + β k V (k) = ( ) ( ) β log + β k + β log + β k = constant + ( + β) log k Since V and V 0 are not identical, we continue. Skipping through the algebra: V 2 (k) = constant + ( + β + β 2 ) log k V 3 (k) = constant + ( + β + β 2 + β 3 ) log k h) The pattern should be clear at this point: i V i (k) = constant + β j j=0 log k i) Since i j=0 β j = β, the limit is: V (k) = constant + β log k j) Not too surprisingly, the optimal policy under this value function is: c(k) = ( β)k 2 Optimal growth with Cobb-Douglas utility a) The Bellman equation can be written: or similarly. V (k t ) = max {log (Ak α t k t+ ) + βv (k t+ ) ()

3 ECON 808, Fall b) The way that I ve set up the Bellman equation, the necessary conditions are: Ak α t k t+ = βv (k t+ ) V (k t ) = αak α t lim t βt V (k t )k t = 0 Ak α t k t+ c) The Euler equation is: βαak α t+ c t+ = c t d) The steady-state capital, output, and consumption are: k = (αβa) e) The steady-state savings rate is: y = Ak α = A (αβa) α f) When α =, the Euler equation becomes: c = y k = A (αβa) α (αβa) s = y c y = (αβa) A (αβa) α βαa c t+ = c t We can rearrange so that: c t+ = βαa c t ( In general, the growth rate of a variable x t is x t+ x t (by definition) or log xt+ approximation), so the growth rate of consumption is: x t ) (a convenient βαa or log (βαa)

4 ECON 808, Fall Optimal growth with CRRA utility a) The Bellman equation can be written: { (Ak α V (k t ) = max t k t+ ) σ σ + βv (k t+ ) (2) or similarly. b) The way that I ve set up the Bellman equation, the necessary conditions are: (Akt α k t+ ) σ = βv (k t+ ) V (k t ) = αakt α (Akt α k t+ ) σ lim t βt V (k t )k t = 0 c) The Euler equation is: βαakt+ α c σ t+ = c σ t d) The steady state is exactly the same as in the Cobb-Douglas case. e) Since steady-state output, capital and consumption are the same as in the Cobb-Douglas case, so is the steady state savings rate. f) The Euler equation can be rearranged so that: ( ) σ ct+ = αβa c t So the growth rate is: or g c = (αβa) /σ g c = log αβa σ 4 Optimal growth with linear utility a) Because of the production technology, there is a maximum possible level of capital, k max = A Since capital cannot be bigger than that number, consumption cannot be bigger than Akmax α = A. Since ct < A, it must be that: β t c t < β t A < A β In other words, any feasible consumption stream provides finite utility. b) Let k be the usual steady state capital stock, i.e., f (k ) = /β. Suppose you are deciding whether to consume a particular unit of output or save it for tomorrow. Today, it will give you

5 ECON 808, Fall unit of utility. Tomorrow, it will give you βf (k t+ ) units of utility. If βf (k t+ ) >, then you will save 00% of output. If you save 00% of your output, then k t+ = f(k t ), so 00% saving is optimal if and only if βf (f(k t )) >, or equivalently if f(k t ) < k. If βf (k t+ ) <, you will consume 00% of your output. If this policy were pursued, then we would have k t+ = 0, and βf (0) = >. So consuming 00% of output can never be optimal. If βf (k t+ ) =, or equivalently if k t+ = k, you will be indifferent between saving and spending. Any consumption allocation is optimal. This means that the capital stock will follow the law of motion: k t+ = min {f(k t ), k = min {k t α, (αβa) In other words, the planner will choose zero consumption until the steady-state capital level is reached. If the planner can reach steady-state consumption starting in period s, then c s = f(k s ) k, and c t = f(k ) k = c for all t s. Between these three problems, we see how the shape of the flow utility function u affects the behavior of the model. I mentioned in class that Cobb-Douglas utility is a special case (the case that σ = ) of CRRA utility, and so is linear utility (σ = 0). The parameter σ measures a person s preference for smoothing consumption over time. The characteristics of the steady state are unrelated to σ. However, the speed at which the steady state is reached is closely related to σ. When σ is low (linear case), consumers will tolerate low early consumption to get to the steady state quickly. When σ is higher (Cobb-Douglas case), consumers will not tolerate low early consumption, and will reach the steady state more slowly. c) As I said before, the consumer will choose to save if βf (k) >, and spend if βf (k) <. When α =, the consumer will choose to save if βa > and spend if βa <. If βa <, there is a solution to the planner s problem - consume everything right away (c 0 = Ak 0 ). If βa >, there is no solution to the planner s problem. You didn t have to prove this, but I will. Suppose that there is a solution {c t which gives utility level U. We can prove such a solution does not exist by finding a feasible allocation that improves on it. A proposed solution will have the characteristic that there exists some time T such that c T > 0. I propose a new solution {ĉ t c t for all t / {T, T + ĉ t = c T if t = T c T + + A if t = T + The utility from this proposed solution is Û = U + βt + A β T. Since βa >, Û > U. Therefore U could not have been a solution.