Problem Set 3. Consider a closed economy inhabited by an in ntely lived representative agent who maximizes lifetime utility given by. t ln c t.

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1 University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Problem Set 3 Guess and Verify Consider a closed economy inhabited by an in ntely lived representative agent who maximizes lifetime utility given by X t ln c t The agent has access to a production technology given by t=0 y t = kt ; 2 (0; ) where y t denotes output produced by an agent with access to k t units of capital is period t. Assume that k 0 is given, i.e., the agent is born with an initial ednowment of k 0 units of capital. Capital accumulates according to the transition equation k t+ = i t + ( ) k t (a) Formulate this problem as a dynamic programming problem by specifying the value function and the Bellman equation. Be sure to specify what the choice variables and state variables are in this problem. (b) Characterize the optimality conditions for this problem and derive optimal consumptioninvestment decision (the Euler equation). Please show and explain the steps you take and your results. (c) Now suppose you have to derive the value function for the agent by using the Guess and Verify technique. Assume that =. Show that the value function is given by V (k t ) = A + B ln k t by solving for the coe cients A and B as function solely of the parameters of the model. (d) Derive the policy functions for c t and k t+ for this problem under the same assumption of =.

2 Answers: (a) Bellman equation is: V (k t ) = max fln(c t ) + V (k t+ )g fc t;k t+ g subject to the constraints c t + k t+ y t + ( )k t y t = kt k t+ 0 and k 0 given Choice variables: c t and k t+ and state variable: k t b) Rewrite Bellman s equation: V (k t ) = max fln [kt + ( )k t k t+ ] + V (k t+ )g fk t+ g Get the rst-order condition w.r.t. k t+ Get the derivative of V (k t ) w.r.t. to k t (k t + ( )k t k t+ ) + dv (k t+) dk t+ = 0: dv (k t ) = k t + ( ) + [ u 0 (c dk t kt t ) + V 0 (k t+ )] dk t+ + ( )k t k t+ dk t Using Benveniste-Scheinkman condition and forwarding one period we get: Plugging this into the F.O.C.gives dv (k t ) = k t + ( ) dk t kt + ( )k t k t+ (kt + ( )k t k t+ ) = k t+ + ( ) (kt+ + ( )k t+ k t+2 ) which simpli es to = k t+ + ( ) c t c t+ 2

3 c) Plug your guess into Bellman s equation: Take F.O.C.(k t+ ) and get: V (k t ) = max fln(kt k t+ ) + [A + Blnk t+ ]g fk t+ g k t+ = B + B k t Substitute this back in the Bellman s equation and rearrange to get: V (k t ) = ln(k t )( + B) ( + B) ln( + B) + A + B ln(b) Equating coe cients from the initial guess gives A = ( + B) ln( + B) + A + B ln(b) B = ( + B) Rearrange to get: A = ln() + ln( ) B = d) The policy functions are: k t+ = k t c t = ( )k t 2 Consumption and Saving Suppose preferences of a representative agent in an in nitely-lived economy are given by V = E P t=0 t u (c t ) where E is the expectation operator and The agent s budget constraint is u(c) = c ; > 0: A t+ = R t+ (A t c t ) ; A 0 given 3

4 where R t is identically and indepependently distributed with ER (a) Write down the Bellman equation for this problem. t < =. (b) Derive the Euler equation to characterize optimal consumption and saving. (c) Derive the optimal solution for c as function of A by using the guess and verify method, i.e., derive the optimal policy function for c. Use the guess c t = A t. [You need to solve for as a function of the exogenous parameters of the model.] (d) What is the e ect of an increase in the volatility of interest rates on consumption? Explain your result. Answers a) Bellman equation is: V (A t ; R t ) = max fc tg t + E tv (A t+ ; R t+ ) c Choice variable: c t ; State variables: A t, R t. Note that A t+ is the outcome of a stochastic process so you cannot choose it. b) F.O.C.(c t ): Get derivative of V w.r.t. A t : dv ct (At+ ) E t R t+ = 0 da t+ dv (A t ) da t = E t dv (At+ ) R t+ = ct da t+ Using Benveniste-Scheinkman condition and forwarding one period we get: Plug back into F.O.C. to get dv (A t+ ) da t+ c t = c t+ Rt+ = E t c t+ c) Plug your guess into Euler equation (notice that we use the guess at a di erent place than in previous problem): (A t ) = E t R t+ (A t+ ) 4

5 Plug your guess into the budget constraint: A t+ = R t+ (A t A t ) Substitute the budget constraint into the Euler equation to get rid of A t+ : (A t ) = E t R t+ (R t+ A t ( )) Isolate : This implies that c t = = n Et R Et R t+ t+ o A t d) An increase in volatility of R t+ has a di erent e ect on consumption depending on its e ect on E t Rt+. This can be broken into two parts. First,the e ect on E t R t+ and second the induced e ect on E t R = t+ It is easy to check that the function R is concave or convex in R as is < or > than. If > then E t R t+ > fe t R t+ g in which case greater volatility of R raises E t R t+. In turn, E t R = t+ must rise as well since > 0. Thus, consumption must fall when R becomes more volatile if >. The opposite occurs when <, i.e., greater volatility of R raises consumption. In the special case of log utility ( = ) consumption becomes independent of R and hence in changes in the volatility of R as well. 5