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Answer Key for the September 2007 Macro Prelim 1. Imagine a representative consumer with time separable, logarithmic utility, V = (1 ° ° ) E t X 1 j =0 ° j ln C t + j ; (1) where 0 < ° < 1 . Consumption is an element of an exogenous random vector X t which evolves according to X t +1 = ± + AX t + " t +1 ; (2) where " t ± iid N (0 ; °) : Suppose that A has a single unit eigenvalue, with all other eigenvalues less than 1 in magnitude. For convenience, assume that ln C t is the °rst element of the vector X t : Assume ±su¢ cient discounting.²(This is intentionally vague; the context should become clear as you develop your answer.) Derive the consumer²s value function. (Hint: There are at least two ways to solve this, a recursive and a nonrecursive approach. The recursive approach is a lot easier. The nonrecursive approach turns into an algebraic quagmire.) ANSWER: The Bellman equation is V ( X t ) = (1 ° ° ) U ( C t ) + °E t V ( X t +1 ) : The max operator drops out of the rhs because this is an endowment economy. From (1) and (2), it is obvious that the value function is linear in X t : This follows from the fact that expected utility is linear in expectations of ln C t + j and that those ex- pectations are linear in X t : Thus, I conjecture that the value function is linear in X t V ( X t ) = b 0 + b 1 X t ; where b 0 is a scalar and b 1 is a row vector conformable with X: We just need to solve for the undetermined coe¢ cients b 0 and b 1 : After substituting the conjecture into the Bellman equation, we get b 0 + b 1 X t = (1 ° ° ) U ( C t ) + °E t [ b 0 + b 1 X t +1 ] ; = (1 ° ° ) e 1 X t + ° [ b 0 + b 1 E t X t +1 ] ; where e 1 is a selector vector (i.e., a row vector with a 1 in the °rst position and zeros everywhere else). Next, substitute the expectation of next period²s X : b 0 + b 1 X t = (1 ° ° ) e 1 X t + ° [ b 0 + b 1 ( ± + AX t )] : After a bit of algebra, we get b 0 + b 1 X t = ° ( b 0 + b 1 ± ) + [(1 ° ° ) e 1 + °b 1 A ] X t : 1

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Equating powers on the left and right-hand sides implies b 0 = ° ( b 0 + b 1 ± ) ; b 1 = (1 ° ° ) e 1 + °b 1 A: Solve the second condition for b 1 : b 1 ° °b 1 A = (1 ° ° ) e 1 ; b 1 ( I ° °A ) = (1 ° ° ) e 1 ; b 1 = (1 ° ° ) e 1 ( I ° °A ) ° 1 : After substituting the solution into the condition for b 0 , we get b 0 = °e 1 ( I ° °A ) ° 1 ±: This veri°es the conjecture. Hence the value function is V ( X t ) = e 1 ( I ° °A ) ° 1 [ °± + (1 ° ° ) X t ] : 2. Consider a two-period overlapping generations model in which people work when young and retire when old. Consumers maximize ln( c 1 t ) + (1 + ² ) ° 1 ln( c 2 t +1 ) ; subject to the ³ow budget constraints c 1 t + s t = w t ; c 2 t +1 = (1 + r t +1 ) s t : c 1 and c 2 represent consumption when young and old, respectively, and w t is labor income, s t is savings, r t +1 is the real interest rate earned on saving, and (1 + ² ) ° 1 is the subjective discount factor. Firms are competitive and produce output by combining labor and capital using a Cobb-Douglas production function. Labor is supplied inelastically, so employment at t is the same as the number of young people, N t : The population grows at an exogenous rate n; so that N t +1 = (1+ n ) N t : Output per worker is Y=N = F ( K=N; 1) ; or y = f ( k ) : Equilibrium in the goods market requires that saving equal investment. Assuming
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