I need help answering the parts to Problem 2.

Thanks!

, C AcrobathDc File Edit View Sign Window Help 0 6 65 M LII 4:? 1oo%|:- g Tue9:21 PM JannieYang Q 55 0 0 mm Home Tools hw.pdf x a G) I . likt'i'JElQ 4n'k@125%'v?'E/af> 62839 Q E2; Problem 2 (Module 2) [25 points total]. Robinson Crusoe (RC) lives alone on an E island. He has T > 1 number of hours each day that he can either devote to napping 1:: (n) or spend on his boat catching sh ((1) If he spends L hours catching sh, he will a: catch (1 = L sh, i.e. his production function for sh is q = f (L) = L. Since a day has a T hours, his time constraint can be written as E2, n + L = T. (1) El] As a consumer, RC's utility function over the two goods it and q is given by U (n, q) = I > lnn + (1. As a producer, RC runs his shing expeditions like a business, hiring labor L 4 E3 at wage rate it) (normalized to 1) and selling sh on the market at the market price p. a He makes prots 7r, which he pays out (to himself) as prot shares. Therefore, RC, the consumer's, income is m m = 7r + T O and his budget constraint is pq + n = m. E 4': 2.1 [4 points] Consider RC, the consumer. Solve for his demand functions for naptime n(m, p) and sh qd(m,p). [Although RC's income in can be written in terms of 7r and T, it will be easier for later parts of the question, to write the demand functions 7 as functions of m and p.] |-) C AcrobathDc File Edit View Sign Window Help 0 6 65 M LII 4.33100%|Z- g Tue9:22 PM JannieYang Q 55 , 0 0 hw.pdf Home Tools hw.pdf x a G) I . Ieae' 4/7'k@125%''?'E/af> 62889 2.2 Consider RC, the producer. Q (a) [3 points] Solve for RC's cost function. i.e. nd the cost 0 (q) of catching q E?) sh. [Hint: there is no need for complicated mathematics. Think of how many E1 hours of labor are needed to catch q sh and then calculate how much these ' hours cost in terms of wages] g (b) [3 points] RC has taken a course in microeconomics back in Britain and a decides that the best way to organize his little island economy is to act as a perfectly competitive rm. What are RC's prots 7r(p) and What is his supply E2) of sh, q5(p), both as a function of 1)? El] 2.3 [5 points] Given the supply of sh you calculated in 2.2(b) and the demand of sh E\" ' you calculated in 2.1, derive the competitive equilibrium price for sh in this island ' 5 economy. What are the equilibrium prots and income at that price? Calculate I a the equilibrium allocations of naptime (n*)7 labour (L*) and sh (q*). W 2.4 [3 points] Is the equilibrium allocation you calculated in 2.3 pareto efcient? " O 25 Consider RC, the rm, again. RC realizes that, as the sole producer of sh on the island, he can take advantage of his monopoly position and charge a higher price E for sh than the price he charges as a perfectly competitive rm (assume that he 1' continues to be a price taker in the labor market, i.e. he takes the wage rate as given). He reasons that higher prots will mean higher income for himself as a Acrobat Pro DC File Edit View Sign Window Help C VE M 100% 27 Tue 9:22 PM Jannie Yang Q ... hw .pdf Home Tools hw .pdf 5 17 + 125% Y consumer and that he will ultimately be better off if he behaves like a monopolist. He knows that the market demand function for fish is q(m, p), the demand function for fish derived in 2.1. (a) [1 point] Using the demand function for fish q(m, p) derived in 2.1 and the firm's cost function C(q) derived in 2.2(a), write down the monopoly profit TM(m, p). [Since the demand function is a function of the consumer's income m and the price of fish p, so will be the monopoly profit.] (b) [1 point] By maximizing profit with respect to p, find the monopoly price p in this market. [ The price will depend on income m] (c) [1 point] Use the demand functions for naptime n(m, p) and fish q(m, p) La derived in 2.1 to find the amount of naptime and fish RC, the consumer, demands at the monopoly price. [These will be functions of income m alone] (d) [1 point] How much labor does RC, the producer, need to hire in order to satisfy the market demand for fish you derived in 2.5.(c)? (e) [1 point] Using your answers to 2.5(c) and (d) and the time constraint in (1), solve for the equilibrium level of income m. Using your solution for my derive the equilibrium allocations of naptime (nM), fish (qM) and labor (LM). to 2.6 [2 points] How does the equilibrium allocation (n , q ) calculated in 2.5(e) com- pare to the equilibrium allocation (n*, q*) calculated in 2.3? Do you expect the monopoly allocation to be pareto efficient? What is your advice to RC; should he act as a perfectly competitive firm or as a monopolist on his island