Show that f is increasing and concave in both th and x -

Question

Question 1. Intrahousehold allocations and specialization

Suppose a family consisting of a wife f and a husband m values a commodity Z, which is produced with inputs of home production t'_h and purchased goods x. The household- level utility function is U = U (Z), with Z = f (x, t'_h) = x^.4 t'^.6_h. Household members purchase input x in the market at price p using earned income w_ft_wf + w_mt_wm and unearned income v, where wi is the market wage of, and t_wi is the working time chosen by, member i ∈ {f, m}. Home production t'_h = w_hft_hf + w_hm t_hm, where w_hi is the home production efficiency of member i ∈ {f, m}. The couple therefore faces the following set of five constraints:

(a) wife's time: t_hf + t_wf = T
(b) husband's time: t_hm + t_wm = T
(c) income: w_f t_wf + w_mt_wm + v = px
(d) home production: w_hf t_hf + w_hmt_hm = t'_h

(e) commodity production: Z = f (x, t'_h )

(a) Show that f is increasing and concave in both t'_h and x.

(b) Let T = 100, v = 50 and p = 5. When the household members allocate their time optimally, calculate how much Z the household can produce if (i) w_hm = 10, w_m = 0, w_hf = 0, w_f = 10, and (ii) if w_hm = w_hf = 6 and w_m = w_f = 5. What does this imply about the optimal investments in human capital the spouses should make early in life?

(c) Suppose v increases from 50 to 100. What do we expect to happen to t_hf , t_wf , t_hm and t_wm? What about individuals' investments in w and wh at the beginning of life?

2. Monogamy and polygamy

(a) Suppose we have a society with N_m men and N_f women. Using the analysis of the monogamous marriage market from Chapter 3 (see slide 4 of the corresponding notes) where women supply themselves on the marriage market and men demand women as wives, show that if N_m > N_f , women will receive all the gains from marriage at the monogamous equilibrium point e_m.

(b) Show that if N_m > N_f polygamy doesn't raise women's payoff from marriage. Why not?

(c) Consider an economy with two types of men, type A and type B. All women are identical. Let MP_A1f > MP_A2f > MP_A3f > MP_1Bf > Z_f >
MP_A4f > MP_B2f , and 3N_m^A + N_m^B < N_f < 4N_m^A + N_m^B. What are the marriage market payoffs of each type of man and of women under monogamy? What are the marriage market payoffs of each type of man and of women under polygamy? (Use a graph it it helps.) Is polygamy likely to exist in this economy? Explain your reasoning why or why not.

(d) Why is there less polygamy worldwide today than 300 years ago?

Attachment:- economics-Lecture note.rar