Econ 5021W - Winter 2023 Assignment 1
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Econ 5021W - Winter 2023
Assignment 1
1. (24 marks total) Consider the following modification to the one-period “micro-macro” model from Section 3 of Lecture Note 1 (LN1). First, let h 三 1 _ e denote the HH’s leisure, and replace any appearances of e in the model with 1 _ h. Next, let’s do away with the separability assumption, so that the HH utility function is given simply by U (c, h) (not u(c) ＋ν(h) as we assumed in LN1).
(a) (6 marks) Solve the HH problem of maximizing U (c, h) subject to the budget constraint c = w(1 _h)＋y to obtain an optimality condition governing the optimal trade-off between c and h (analogous to equation (3) from LN1). Give an economic interpretation for this conditions
(b) (6 marks) Assume henceforth that the form of U is given by
U (c, h) = ,．│ ┌ φc 
＋ (1 _ φ) h
┐ 
, if η 
1 ,
．．(c| hi2| , if η = 1
where φ e (〇, 1) and η > 〇 are parameters. Use this functional form in your answer to (a), and then solve this equation to obtain an explicit expression for c as a function of h.
(c) (6 marks) Totally differentiate the budget constraint c = w(1 _ h) ＋y and your answer to (b),1 yielding two new equations. Use your answer to (b) to eliminate any appearances of φ from these equations. Finally, solve explicitly for dc and dh as functions of dw and dy .
(d) (2 marks) For what values of η will c and h both be normal goods?
(e) (4 marks) Suppose for simplicity that y = 〇, so that, from the budget constraint, c/w = 1 _ h. For what values of η will c increase in response to an increase in the wage w? What about h?
2. (8 marks) Consider the problem of a firm that “lives” forever. The profits of the firm in period t are given by πt = F (Kt, Lt) _ wtLt, where Kt is a stock of capital owned by the firm at the beginning of t, Lt is the quantity of labour hired by the firm during t, F (K, L) is a production
function, and wt is the market wage for period t (taken as given by the firm). The wage wt is stochastic. Assume that F is twice continuously differentiable, and strictly increasing and concave in both arguments. The firm’s capital stock evolves according to the “law of motion”
Kt扌1 = (1 _ δ) Kt ＋ψ (It) ,
where δ e (〇, 1) is the depreciation rate, It denotes investment by the firm in its capital stock in period t, and ψ(.) is a twice continuously differentiable, strictly increasing and strictly concave function. The initial level of capital, K│, is exogenous. The firm’s profits in period t are split between its capital investment and dividends Dt that are issued to its owners, so that it faces the budget constraint It＋ Dt = πt in each period. We assume the firm wishes to maximize its expected value of the discounted stream of dividends,
/
R2tDt ,
t≥│
where R > 1 is the constant gross interest rate. Solve the firm’s date-t problem (analogous
to what we did for the HH problem on pp. 3-4 of LN2). Combine any conditions you get to eliminate any Lagrange multipliers.
3. (8 marks) Consider the following modification of the workhorse model of LN2. First, let’s do away with labour, so that the HH only receives the (stochastic) lump-sum income yt in period
t. Second, let’s now assume that period utility from consumption at t depends not only on its current consumption ct, but also on its consumption the previous period, ct21 . Specifically, period-t utility is assumed to be u(ct _ hct21), where h e [〇, 1) is a parameter. As usual, we assume u\ > 〇, u\\ < 〇. Thus, at date t, the household wishes to maximize the objective function2 /
Et βī u (ct扌ī _ hct扌ī21) .
ī ≥│
Write down the Lagrangian for the household’s problem at date t analogous to the one on p. 3 of LN2. Obtain the FOCs with respect to the household’s choice variables at t (be sure to state what these are). (NOTE: In this question, you don’t need to combine these conditions to eliminate Lagrange multipliers.)
2023-01-28