eco4

nq55678

1. Consider the following regulation problem. Let x be output, c be marginal cost of output and R be net return of production. The firm's net profit before regulatory payments is given by R = x cx, for 0 x 1. Note that x has to lie in the unit interval and that output has been normalized to equal revenue. Assume that the regulator does not know the firm's cost coefficient c, which can take on n values: 0 c1 ... cn 1. The regulator assesses probability pi 0 to each of the i possible marginal cost levels. The firm knows its marginal cost and chooses x. The regulator can tax the firm based on its realized revenue. a. Set up the program that solves for the regulator's optimal policy, expressed as a direct revelation mechanism. Assume that the regulator's objective is to extract as much surplus from the firm as possible, subject to the constraint that the firm can quit if profits are negative (and pay nothing). b. Draw a diagram that shows how the firm chooses its best response as a function of its cost. Prove that the firm's output choice x is a non-decreasing function of marginal cost. c. Show that the optimal policy takes the simple form: a fixed charge T for producing anything (hence, all firms with c T produce and those with c > T do not). It suffices that you give an argument for n = 3. *2.Consider a screening model of the following sort. The agent produces output q at a cost c(q,) = q, where q is output and is a cost parameter. The principal, who cannot observe either c or , offers to pay the agent the amount p(q) if the agent produces output q. Given this incentive scheme, an agent with cost parameter responds by producing the amount q(). What kind of payment scheme p(q) should the principal choose in order to implement the response function q() = 1/2 ?

Problems 11 to 23 and Case Problem 2 require the use of data mining software. If using R/Rattle to solve these problems, refer to Appendix: R/Rattle Settings to Solve Chapter 5 Problems. If using JMP Pro to solve these problems, refer to Appendix: JMP Pro Settings to Solve Chapter 5 Problems.

k-Means Clustering of Sandler Movies.

Attracted by the possible returns from a portfolio of movies, hedge funds have invested in the movie industry by financially backing individual films and/or studios. The hedge fund Star Ventures is currently conducting some research involving movies involving Adam Sandler, an American actor, screenwriter, and film producer. As a first step, Star Ventures would like to cluster Adam Sandler movies based on their gross box office returns and movie critic ratings. Using the data in the file Sandler, apply k -means clustering with k=3 to characterize three different types of Adam Sandler movies. Base the clusters on the variables Rating and Box. Rating corresponds to movie ratings provided by critics (a higher score represents a movie receiving better reviews). Box represents the gross box office earnings in 2015 dollars. Normalize the values of the input variables to adjust for the different magnitudes of the variables. Report the characteristics of each cluster including a count of movies, the average rating of movies, and the average box office earnings of movies in each cluster. How would you describe the movies in each cluster?

[9:25 PM, 10/24/2021] Flo: Tariffs never make small countries better off, but there are cases where they can make a large country better off. Draw side-by-side graphs for a good (call it "toothpicks") where the small country can produce the good at a lower price than the large country. In this problem: SA : P =5 + 1.1Q DA : P =14 1.1Q SB : P =2 + 1.1Q DB : P =10 1.1Q Note that there is a worked example similar to this in the lesson. DO NOT just copy what was done in the lesson. Think through the problem independently and be sure to explain your notation.

a) (5 points) Which country is the small country? How do you know?

b) (15 points) Show the domestic price and quantity in each market, free trade price (needs to be the same in both countries), and the imports and exports i... [10:53 PM, 10/24/2021] Flo: The firm has six current risk management options it can use to manage this risk:

[1] Insurance policy with a FA of $11,500, a premium of $1,900, and a deductible of 175. This option has a worry value of $550.

[2] Insurance policy with a face amount (FA) of $10,000, a premium cost of $1,500 and a deductible of 200. Implement a loss reduction program that costs $75 and changes the probability of a $0 loss to 72% and eliminates the potential of an $11,000 loss. All other loss probabilities stay the same. This option has a worry value of $500.

[3] Retention with a worry value of $750.

[4] Insurance policy with a face amount (FA) of $13,000, a premium cost of $1,750 and a deductible of 150. Implement a loss prevention program that costs $50 and changes the probability of a $0 loss to 75%, a $4,000 loss to 13%, a $8,000 loss to 9%, a $9,250 loss to 2%, and a $11,500 loss to 1%. This option has a worry value of $600.

[5] Insurance of $11,000 with a premium cost of $2,350.

[6] Insurance policy with a FA of $12,000, a premium of $2,000, and a deductible of 100. This option has a worry value of $440.

a. Construct a loss matrix. What is the expected cost for each option? (3 points each)

b. Assume that the risk manager makes decisions based on total cost. What option would she choose? (2 points)

*3.A seller may supply a single object to a buyer. Let x be 1 if trade takes place and 0 if not. Let tS be the amount of money that the seller receives and tB the amount that the buyer pays. Let v be the buyer's valuation and s be the seller's valuation of the object and assume preferences are quasi-linear. We can then normalize utilities so that the seller's utility is tS cx and the buyer's utility is vx tB. Assume the preference parameters v and c are independently drawn from a uniform distribution on [0,1]. The buyer knows v and the seller knows c. a. What is the efficient rule for trade? b. Let mS and mB be the seller's respectively the buyer's reported preference for the object. Determine the set of direct mechanisms (expressed as a function of the reports) that admit truth telling as a dominant strategy and implement efficient trade (ie. the set of Groves mechanisms). c. Show that there is a unique Groves mechanism that has the property that whenever trade does not occur, the transfer payments are set equal to zero (tB = tS = 0). Is this mechanism feasible? d. Show that there is no Groves mechanism for which the budget breaks even for all reported preferences. 4. Suppose there are a continuum of sellers and a continuum of buyers (where each continuum is normalized to one). Each seller has one unit of the good and has valuation vs drawn (independent from those of the other sellers and buyers) from the distribution Fs on [a,b]; similarly, each buyer has unit demand and has valuation vb drawn (independently) from the distribution Fb on [c,d]. Assume that b > c (not everyone ought to trade). Let ms(vs) and mb(vb) denote a seller's probability of selling and a buyer's probability of buying, respectively. Let ts(vs) and tb(vb) denote a seller's transfer and a buyer's transfer (from the planner), respectively. Among the many possible mechanisms that might be used, consider the "Walrasian mechanism":

ms(vs) =1 and ts(vs) =p if vs p and ms(vs) = 0 = ts(vs) if vs > p, mb(vb) =1 and tb(vb) = p if vs p and mb(vb) = 0 = tb(vb) if vb < p. a. Show that IC and IR are satisfied for any p. b. Show that there is a value of p such that the mechanism is balanced and trade is efficient. c. The Myerson-Satterthwaite Theorem states that it is impossible to achieve efficient and individually rational trading in a Bayesian NE between a buyer and a seller with private information about their valuations. How can this be reconciled with parts a and b above?