i need detailed expanations ony

1. (36 points) Two firms, A and B, are competing in the production of a homogenous good. The good's marginal cost for both firms is equal, MC = $25. Assuming linear reaction functions, describe what would happen to output and price in each of the following situations if the firms are in (i) collusive equilibrium, (ii) Cournot equilibrium, (iii) Bertrand equilibrium. (a) (4 points for each of (i)-(iii)) The demand curve shifts to the left. (b) (4 points for each of (i)-(ii)) Because it invents a new and improved machine, the marginal cost at firm B decreases to $20. (c) (4 points for each of (i)-(iii)) Costs in the entire industry increase due to an increase in wages. Problem 1 by MIT OpenCourse Ware. 2. Problem removed due to copyright restrictions. This content is presented in audio form in the Solution Video for Problem Set 8, Problem 2 3. (28 points) Suppose a perfectly competitive labor market has a demand curve of [ = 120 - 2wr and a supply curve of " = Sw, where w is the wage rate is dollars and L is the quantity of labor in person-hours. (a) (2 points) What are the equilibrium values of the wage and employment? (b) (4 points) Suppose the government imposed a minimum wage of $14 per hour. Now what are the equilibrium values of the wage and employment? (c) (8 points) Repeat part (a), assuming now that the market is a monopsony. (d) (8 points) Repeat part (b), assuming now that the market is a monopsony. (e) (6 points) Does the imposition of the minimum wage decrease employment here under perfect competition? What about under monopsony? Give a brief intuitive explanation for your answer and why it may be different under the two different market structures. Problem 3 courtesy of William Wheaton. Used with permission. 4. (11 points) Suppose you face the following lottery. You can earn 1 of 3 possible grades in this class: an "A" , a "C", or an "F", with the following probabilities: 10' 10' IF = 10 Your current wealth (w) is $400. If you receive an "A", you gain (e-g. I pay you) $500. However, if you get an "F", you lose (e-g. you pay me) $300. If you receive a "C", you DO NOT GAIN OR LOSE anything. Assume your utility function, defined over wealth, is U(w) = v (w). (a) (6 points) What is your expected utility (EU)? [Hint: be sure to calculate your total wealth in each "state".| (b) (5 points) What is the certainty equivalent level of wealth (w*), that is, the guaranteed payoff at which a person is "indifferent" between accepting the guaranteed payoff and their expected utility from (a)?1. Consider an industry with 3 firms, each having marginal costs equal to 0. The inverse demand curve facing this industry is P(Q) = 60 - Q. where Q = q1 + 92 + 93 is total output. (a) If each firm behaves as a Cournot competitor, what is firm I's best response function optimal choice given other firms outputs? (b) Calculate the Cournot equilibrium. (c) Firms 2 and 3 decided to merge and form a single firm with marginal costs still equal to 0. Calculate new industry equilibrium. Is firm 1 is worse of or better of as a result? Was it a good idea for firms 2 and 3 to merge? Would it be a good idea for all three firms to organize the cartel? (d) Suppose firm 1 can commit to a certain level of output in advance. If the choice of firm I is qi, what would be the optimal choices of firms 2 and 3? (Hint: After observing q1 firms 2 and 3 would engage in (Cournot) duopolistic competition. What is the optimal level of q1? Calculate profits of firm 1, compare with (b). 2. Consider an economy with 3 firms and 2 consumers. Each consumer owns 10 units of Land. Firm 1 produces Food and Wood using technology (-L, F, W) = (-1, 1, 2). Firm 2 produces only Food with technology (-L, F) = (-2, 1) and firm 3 produces only Wood with technology (-L. W) = (-1, 1). Firms 2 and 3 are owned by consumer 1, firm 1 is owned by consumer 2. Consumers have identical utilities u(w, f) = vwf. Calculate Walrasian equilibrium. 3. There is one consumer and one firm. The firm may have a high quality indivisible product (with probability q) or a low quality product (with probability 1 -q). The firm knows the value of the product, while the consumer cannot observe it prior to the sale. The consumer's utility from a product of given quality is vi - p, where vy = 8, y = 4, and p is the price paid. Costs of production are ch = 3, q = 1. (a) Under what conditions on q consumers will be willing to buy the product at a prespecified price p? What qualities of the product would be sold? (Hint: Analyze it case by case. E.g., if both qualities are sold, what is the expected utility of the consumer? Would she by? Would both types of the firm sell?) (b) Suppose a firm can spend some money A on advertisement of its product, and A is observable by the consumers. Present a separating equilibrium, where the high quality firm advertises, A; > 0, and sells the product at price p = 8, while the low quality firm does not advertise, A; = 0 and charges p = 4. (It would suffice if you check incentive compatibility and individual rationality constraints for only two choices of A, A; and A;.)Problem 1 (30 minutes) Consider a consumer with a utility function w(x1,x2) = e(n+mm(xz])"" (3 points) a) What properties about utility functions will make this problem easier to solve? (3 points) b) Which of the non negativity input demand constraints will bind for small m? (10 points) c) Derive for the marshallian (uncompensated) demand functions and the indirect utility function. (3 points) d) Derive the expenditure function in terms of original utils u. (6 points) e) Suppose that there are 5 people in the economy each with endowments m', i = 1,2,3.4.5. i) Suppose that m' > p, Vi. Construct the aggregate demand function for x1 and x2.' What properties do the individual demand functions have that simplifty this problem? ii) Now suppose that m' p; fori = 3,4.5. Construct the aggregate demand for x1,*2 Problem 2 (25 minutes) Consider a firm with a production function of f(x1+12) = min(2x,*1 + x2) (5 points) a) Show that this function is homogeneous of degree 1. i) What does this imply about the structure of the cost function? ii) What does this imply about the returns to scale of the technology? (10 points) b) Derive the cost function for the firm. What are the conditional factor demands? (5 points) c) Suppose that p = 2.5, wj = 3, w2 = 1. If we introduce the constraint that x,