1. (40%) Suppose that the consumer in a continuous-time life-cycle model chooses the consumption path and retirement age to maximize (1) subject to the equation of motion a(x) = ra(x]ty-c(x) (2) if one is working, or a (x) = ra (x) - c(1) (3) if one has retired. The boundary conditions are a (0) = 0, a (7) 20, where the notations are same as in the lecture notes. Conditional on a particular retirement age, it can be shown (as in Assignment 2) that e(x) = el-pic (0). (4) From now on, assume further that p = 0. When p = 0, express (4) as c (:, R) = ("Ic (0, R) , (5) so that the dependence of consumption at age a on the retirement age R is expressed explicitly. Now, consider the optimal choice of retirement age. (a) Based on the conditional consumption choices derived above, consider V (R) = In(e(x. R))dr - / od (6) as a function of retirement age (R). Obtain the first-order condition characterizing the optimal retirement age. Give an interpretation of the derived equation. b) Express the optimal retirement age in terms of the parameters. (Hint: first use the lifetime budget constraint, which is given by Jereydr = [ e -"ze(x, R) dix, (7) to obtain c (0, R), as in Assignment 2.) (c) Compare your answer in part (b) with equation (7) of Kalemli-Ozcan and Weil (2010). d) Would an increase in 7 lead to an increase or a decrease in the optimal value of R in this model? Show your steps. (e) (Bonus: 5%) Comment on whether the procedure in (a) is valid or not in deriving the optimal retirement age for this model.\f. Lifetime budget constraint -H e e (x, R) dx = ex yalx (39) . Differentiating it with respect to R, we obtain endc (x. R) -ddx =e my aR Meaning: . Substituting the above equation into V (R), we obtain V (R) = 1 e thy _e pho (0. R) Corner solutions of retirement age are not very interesting ( "deathbed retirement" is not common in modern economy) * Thus, the first-order condition (V (R*) =0) for an interior solution of optimal retirement age (R* ) is: 1 (40) RHS: marginal cost (measured in utility units) of increasing retirement age, discounted back to time 0 LHS: marginal benefit (measured in utility units) of increasing retirement age, discounted back to time 0: increase in w. discounted (by /) to time 0, multiplied by marginal utility of consumption at time 0