Answer the following

a. Suppose the bank kiiows whether the rm built the new facility or not in the rst period when it offers a contract. Draw the game tree of this game. b. Find all sequential equilibria of the game in [a]. c. Suppose from now on that the rm's possible investment in the rst period is in software and the bank does not know whether or not the rm made the investment when it offers the contract. Draw the game tree of this game. d. What is the optimal contract offer of the bank as a function of its beliefs? The bank believes that the rm invested in the software with probability p. e. Find all sequential equilibria of this game. 4. Consider a rm with the technology {3 = {311:2}? that purchases inputs 3:1 and an in competitive factor markets at the prices url = mg 2 1. Initially? the rm faces the demand curve qu} = 12D p for its output. a. Determine the rm's optimal supply decision and prots. b. Next} suppose the rm's prots were taxed at the constant rate of 20%. How would this affect the rms optimal decisions? How much would the rm pay in taxes and what would be its post tax prots? c. Suppose that? in addition to producing :13, the rm could also contribute money? y? to charity and thereby reduce its overall tax rate. The rate would then become y) 2 mas-{\"02 DlyL D}. Assuming the contribution is from pretax earnings} i.e.,. it can be treated as an additional cost, determine the optimal supply decision and charitable contri bution of the rm. What are the rms prots? lCompare the amount of taxes paid in part {b} to the amount of taxes and charity paid here. Also compare the rm's post tax prots. d. Suppose again that the tax rate is xed at t = [1.2. Now. while charitable contributions do not affect the tax rate? they garner publicity for the rm and increase demand for its a. Suppose the bank kiiows whether the rm built the new facility or not in the rst period when it offers a contract. Draw the game tree of this game. b. Find all sequential equilibria of the game in [a]. c. Suppose from now on that the rm's possible investment in the rst period is in software and the bank does not know whether or not the rm made the investment when it offers the contract. Draw the game tree of this game. d. What is the optimal contract offer of the bank as a function of its beliefs? The bank believes that the rm invested in the software with probability p. e. Find all sequential equilibria of this game. 4. Consider a rm with the technology {3 = {311:2}? that purchases inputs 3:1 and an in competitive factor markets at the prices url = mg 2 1. Initially? the rm faces the demand curve qu} = 12D p for its output. a. Determine the rm's optimal supply decision and prots. b. Next} suppose the rm's prots were taxed at the constant rate of 20%. How would this affect the rms optimal decisions? How much would the rm pay in taxes and what would be its post tax prots? c. Suppose that? in addition to producing :13, the rm could also contribute money? y? to charity and thereby reduce its overall tax rate. The rate would then become y) 2 mas-{\"02 DlyL D}. Assuming the contribution is from pretax earnings} i.e.,. it can be treated as an additional cost, determine the optimal supply decision and charitable contri bution of the rm. What are the rms prots? lCompare the amount of taxes paid in part {b} to the amount of taxes and charity paid here. Also compare the rm's post tax prots. d. Suppose again that the tax rate is xed at t = [1.2. Now. while charitable contributions do not affect the tax rate? they garner publicity for the rm and increase demand for its 2. Consider a competitive economy with two produced goods and two nonproduced inputs, labor and capital. There are two consumers. A worker (W) initially owns one unit of labor and no capital. A capitalist {C} initially owns it or 0 units of capital and no labor. Consumer i {i = WC} has the utility function manila'2, where Is: is that consumer's consumption of produced good if and where D t: c: v: 1. Let L: 3 CI and K: 2 ll be the total amounts of labor and capital used to produce good E. The total output of produced good 1 is min{L1, K1} and the total output of produced good 2 is L2 + ch, where c L} o. Let pg be the price of produced good E and let to and r be the prices of labor and capital respectively. a. Interpret the (inferences between the production functions for the two outputs. b. Show that in every competitive equilibrium positive amounts of both output goods are produced. c. Show that in every competitive equilibrium all the prices are positive. :21. What are the industry prot levels in competitive equilibrium? Find a competitive equilibrium when or = If? and c: it: = 1. For the rest of the problem, return to the general case in which the only restrictions on or, c, and la are those in the rst paragraph of this problem. e. Consider a competitive equilibrium in which both labor and capital are used in the production of good 2 (L; :> I] and K2 2: o). How are the output and input prices related among themselves? Be as complete and specic as possible. Find the total quantities of the produced goods as functions of the parameters or. c, and h. {For partial credit, write equations that determine these quantities as functions of the parameters.) Write two inequalities that the parameters must satisfy so that L2 r: U and K; \"a CI in equilibrium. f. In part c, if c were slightly higher, what would be the effect on the equilibrium ratio is r? 1What would he the silent on the worker's equilibrium utility level? ' g. Suppose that the parameters are such that no labor is used in the production of good 2 in equilibrium. If e were slightly higher, what would be the safest on the equilibrium ratio is as? What would be the eifect on the worker's equilibrium utility level? (Hint: What is known about L1 and the total output of good 1 in this case?) urn-usual 1.11.1\"! I- .- I u.\" \"11.. Pluwullll-J anus-ulna\": J- 1"! In run-u. u. c. Now. suppose that efforts of the researchers are not directly observable and not contractible. The researchers choose their labor inputs independently. but they can only do joint work and this can be enforced Derive the necessary conditions for strictly positive Nash equilibrium eort levels. Compare these effort levels with the effort level derived in b for general probability function p[-}. Compute eort levels r.- = 1: 2 for the probability function e}. cl. 'Ir'ir'hat is the eccpected number of successfully completed projects for the two re- searchers together in cases a_. b and c_. respectively when the probability function is given by f{-}'? Interpret the results. e. Suppose in part c. the researchers can also work on the side on their own if they so choose. Researcher 11 works r.- units of time for the partnership and e,- on his own private work. The effort inputs are not directly observable or contractible and the scholars choose their levels independently. i. \"That are the rst order conditions for a symmetric {i.e.: el = ea: r1 = 1'1} and strictly positive Nash equilibrium under the general probability function pH when [I e: s c l? . c v: s: ii. Suppose. instead: that the probability function is given by: 9H} = { l :2: D E? I 1 - Compute a syn'unetric Nash equilibrium in this case