Competitive Market
1.) Consider the two-round bargaining game. The minimum the seller will sell his home for 188,000 and the maximum the buyer is willing to pay is $200,000. Both players know these two amounts and are bargaining over the difference (M=$12,00). Assume the disagreement values are zero for both players. Player 1 moves first by making a proposal and Player 2 can accept and reject. If player 2 rejects Players 1 proposal, then Player 2 gets to make a proposal, which Player 1 can reject and accept. The game is then over. Suppose the both players discount the future income at the rate d=0.2 per period. That is, $0.20 now is equivalent to $1,00 received next round. Find the equilibrium for this 2-round game. What is the sale price of the home? Which player gets the larger share of M?2.) In Depositor versus Saving & Loan Association (S&L), suppose that the S&L can invest $100,000 in a junk bond (the official name for this sort of security is high-yield bond) with the following rates of return: with high effort (H represents high research effort), the probability is 0,98 that the rate of return is 15% and 0.02 that the money vanishes; with low effort (L represents low research effort), the probability is 0.92 that the rate of return is 15% and 0.08 that the money vanishes. The rate of return on government bonds is 4%. After paying off the depositor, the S&L is left with $2000 minus research cost. Draw the extensive form and solve. Where should the depositor put the money?3.) Find time path (general solution) of P and U, given_P= (1/4) 2U + n dP/dt = 1/2(P-n) dU/dt = – (m-p)4.) Consider a model of a competitive market in which price is adjust to excess demand or supply and firms enter or exit the industry if profits or losses are being made. Price adjusts to excess demand according to ?=(qD-qS), >0 where = a+bp is the demand function, =mN is the supply function, p is price, N is the number of firms in the industry, is speed-of adjustment coefficient. Making these substitutions gives ?=(a+bp-mN). The number of firms adjusts according to ?=z (p ?), z>0 where ? is the fixed average cost of production. Firms enter (?>0) if price exceeds average cost (positive profits) and exit if price is less than the average cost (negative profit). Solve this system of differential equations for price and number of firms.5.) Suppose you have won $200,000 in the state lottery and you have decided to retire on your winnings. Suppose you deposit your winnings in a bank that pays 8% annual interest (compounded annually) and make yearly withdrawals of $30,000. (Hint: if periodic withdrawals, each of amount d, are deposited in a bank account whose initial amount is that pays interest at a rate r per period, then the difference equation that describes the total amount in the account after n periods is = (1+r) d)a. What are the difference equation and initial condition that describes the future value of your account?b. Will you ever run out of money? If so, when?6.) Consider the following model, = s , = k ( – ), = where s is the marginal propensity to save and k is the acceleration coefficient. The third equation states the equilibrium condition for the determination of national income (ex ante saving = ex ante investment). Derive the general solution for . Comment on your result. (Let = for t=0)
