Economics 210B Due: September 16, Problem Set 10. s.t. k t+1 = R(k t c t ) for all t 0, and k 0 given, lim. and
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1 Economics 210B Due: September 16, 2010 Problem 1: Constant returns to saving Consider the following problem. c0,k1,c1,k2,... β t Problem Set 10 1 α c1 α t s.t. k t+1 = R(k t c t ) for all t 0, and k 0 given, and lim T R T k T 0 a. Show that the constraint lim T R T k T 0 is equivalent to imposing a constraint that the present discounted value of lifetime consumption is less than or equal to initial wealth. b. Give an example of a (non-feasible) time path of consumption and capital which violates the last constraint. c. Write Bellman s equation for this problem, substituting in the law of motion for capital. (You can ignore the last constraint as you write Bellman s equation.) d. Guess and verify a solution of the form V (k) = Ak 1 α for some constant A. If possible, calculate A. d. State the policy functions in the forms c t = Bk t and k t+1 = Ck t, and calculate B and C, if possible. f. Show that for the optimal policies from (e), the final constraint on the problem is not violated. g. Explain the intuition behind the behavior of the model. Is there a steady state? What restriction on the exogenous parameters of the model would cause it to have a steady state? Problem 2: Optimal saving under exponential utility Consider an individual who imizes exponential utility over an infinite horizon: {c t,w t+1 } 0 β t ( 1 and s.t. α ) lim t e αct s.t. W t+1 = R(W t c t )+y t+1 W t R t 0; and W 0 given. Here β (0, 1) and α>0. W represents current wealth; c is consumption; y is income, which we will assume is constant: y t = y>0for all t. R>1isoneplusthe interest rate. a. Show that imposing the last constraint on the worker s problem is equivalent to imposing the following constraint: c t R t y t R t + W 0 t=1 b. Write a Bellman equation describing the individual s imization problem, writing the value function in terms of current resources W t. (Note: you may ignore the last constraint when writing the Bellman equation.) 1
2 c. Guess that the value function has the form V (W t )= Ae ZWt where A>0 and Z > 0 are unknown constants. Rewrite the Bellman equation in terms of this functional form. d. Find the policy functions for c t and W t+1 in terms of W t and in terms of the unknown constants A and Z. d. Show that the guess is correct. Find the constant Z. (You are not required to find A.) e. Prove that under the optimal policy, c t = R 1W R t + constant. Describe the time path of W t under the optimal policy. Does this optimal policy satisfy the last constraint on the problem? Problem 3: Climate change Consider a farmer who chooses consumption and saving over time, who is subject to random changes in climate which affect her productivity. Specifically, if productivity today is A>0, then next period productivity will be AG with probability one half, and A/G with probability one half, for some number G>1. Her imization problem is: c 0,k 1,c 1,k 2,... E 0 β t ln c t subject to: c t + k t+1 = A t k α t for all t 0 and A t+1 = A t G with prob. 1 2, A t with prob. 1 G 2 and k 0, A 0 given. The notation E 0 before the sum of utilities here represents the expectation" of utility. a. How many state variables are there? Are they exogenous or endogenous? b. Write Bellman s equation for this problem. Be sure to take into account the two possible future values of A. c. Let s guess that the value function takes the form C ln A + D ln k + E,whereC, D, and E are unknown constants. Rewrite the Bellman equation using this functional form. d. Solve the imization problem to find the policy functions in terms of the unknown parameters. e. Substitute the policy functions into the value function and check whether your guess is correct. f. Solve for C and D (you may also solve for E if you are really bored and have nothing else to do with your time.) g. Considering the law of motion for A and the policy functions, describe the behavior of the model as well as you can. 2
3 Some optional problems: Extra Problem 1: Guess and verify. Consider again the individual s optimization problem given by c t,k t+1,c t+1,k t+2,... s=0 β s t ln(c s ) s.t. c s +k s+1 = Ak α s for all s t, and k t given. Notice that this problem has been written as a decision problem beginning in period t, choosing consumption and capital for all periods s t. a. Define V t (k) to be the optimized value of this problem, starting at time t, given the capital available at time t. Definec t (k) and k t+1 (k) as the policy functions at time t,given capital at time t. Write a formula for V 0 (k) in terms of k 0 and the functions c t (k) and k t+1 (k) for all t 0. Note that this formula need not include the symbol. b. Write a formula for V 1 (k) in terms of k 1 and the functions c t (k) and k t+1 (k) for t 1. c. Show that there is a relationship between the function V 0 (k) and the function V 1 (k). This relationship is called the Bellman equation. Write the Bellman equation for time 0. d. Explain why, in this problem, the function V 0 (k) is the same as the functions V 1 (k), V 2 (k), andv t (k) for all higher t. Hence we can ignore the time subscript on the value function and simply call it V (k). (Similarly, we can write the consumption policy function as c(k) and the policy function for next period s capital in terms of this period s capital as ˆk(k).) Rewrite the Bellman equation to take this new notation into account. e. Guess that the value function has the form V (k) =C ln k + D, for some constants C and D. Rewrite the Bellman equation in terms of this functional form. f. Solve the imization problem stated in the Bellman equation from (e); obtain policy functions c(k) and k(k) interms oftheunknownconstants C and D. g. Substitute the policy function c(k) into the value function. By looking at the resulting equation, state whether or not the guess was valid. h. If the guess was valid, solve for C and D in terms of the underlying parameters A, α, and β. i. State the policy functions in terms of the underlying parameters. Also solve for the steady states c and k. j. Calculate the first and second derivatives of the value function and the policy functions. What are their signs? Do your answers make sense? Extra Problem 2: Recursive solution (working backwards) a. Now consider an alternative solution method for the previous problem. Suppose that instead of being infinitely lived, the imizer (known as Robinson Crusoe) realizes that he will die at the end of period T. Hence it is reasonable to define V T +1 (k T +1 ) 0 for all asset levels k T +1. Write the Bellman equation for period T and solve it (trivially!) to find V T (k T ), c T (k T ),andk T +1 (k T ). b. Now work backwards to solve the Bellman equations for periods T 1 and T 2 (unlike period T, these equations are not trivial to solve.) c. You should notice a pattern. What are c (k ), T s T s k T s+1 (k T s ),andv T s (k )? T s Show that the limits of these formulas as s arethe infinite horizon functions V (k), c(k), andˆk(k) that you derived in Problem 4. 3
4 Extra Problem 3: Linear disutility of labor Part I: Social planner Consider a social planner who chooses consumption c, labor h, and capital k to imize the following social welfare function. {c t,h t,k t+1 } β t (ln c t Bh t ) subject to: k t+1 = Ak α t h1 α c t t for all t 0, and k 0 given. a. Let V (k t ) be the value function associated with this problem. Write the Bellman equation for the social planner s problem. b. Use the Bellman equation to derive the Euler equation system for this problem. Simplify it to obtain a system of three autonomous difference equations that characterize c, h, and k. c. Insteadoftryingtosolvethesedifferenceequations, let sguess that the value function takes the form V (k) =G ln k + H. Rewrite the Bellman equation in terms of this guess. Find the first-order conditions, and find the policy functions in terms of the unknown constants G and H. d. Plug the policy functions back into the objective function to check your guess. Calculate G and H, and rewrite the policy functions and value function in terms of the true values of G and H. Comment on the optimal policy. Part II: Market economy Consider instead the following market economy. Workers choose consumption and labor supply, and save by investing in capital: {c t,h t,k t+1 } 0 β t (ln c t Bh t ) subject to: k t+1 = w t h t + q t k t + d t c t for all t 0 and k 0 given. where w is the wage rate, q is the rental price of capital, and d is dividend income. Firms only exist for one period. A firm in period t rents capital and hires labor to imize profits: K t,h t π t AK α t H 1 α t w t H t q t K t Prices in this economy must adjust so that the following aggregate consistency conditions are satisfied. Goods supply must equal goods demand: AK α t H1 α t = c t + k t+1 Labor supply must equal labor demand: h t Capital supply must equal capital demand: = H t k t = K t Dividends received by workers must equal profits earned by the firms: d t = π t 4
5 e. Write a Bellman equation for the worker s problem. (Hint : the worker s problem is not autonomous, so don t forget to put a time subscript on the value function.) f. Find the Euler equation system for the worker s problem. g. Find the first-order conditions for the firm s problem. h. Show that the equilibrium of this market economy implies exactly the same real allocation (consumption, labor, and capital) as the social planner s economy. In other words, show that by eliminating most of the equations and variables in the market economy, you can get back to exactly the same system of difference equations that you found in part (b). 5
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