prelims_Macro Prelim ANSWERS Sept 2007

prelims_Macro Prelim ANSWERS Sept 2007 - Answer Key for the...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern