Example 22 Infinitely repeated version of Example 21
Let U x, y xy, amp 2, and T 24. We derived C amp 12, 24 with e amp 12 in the one-shot case Example 2.1 . At the partnership equilibrium, eP 8 with CP 16, 16 . Note that U C amp 288 and U CP 256. One Nash equilibrium for the infinitely repeated partnership has each partner supply 12 hours of effort in the first period and every subsequent period as long as the other partner supplied 12 hours in each previous period, but will supply 8 hours of effort in period t and every subsequent period if the...
Example 64 A chance to double your salary
If the individual's utility-of-wealth function is U w 4v w then the EU of 40,000 for sure is 4V40,000 800. The EU of a gamble thatyields 80,000 with probability i 2 or zero with the same probability is This individual prefers 40,000 for sure because ityields a higher level of expected utility than the gamble. We say that individuals are risk averse if they prefer having w for sure to an uncertain wealth level with an expected monetary value that is no higher than w. If asset A yields a high...
.bidder1haslearned That Bidder 2 Plans To Bid 50.
1. Explain why the first-price, sealed-bid auction is not outcome equivalent to the Vickrey auction. 2. Explain why the English auction is not outcome equivalent to the Dutch auction. 3. There are two bidders in a first-price, sealed-bid auction. Bidder 1 has learned that bidder 2 plans to bid 50. What is bidder l's payoff-maximizing response as a function of his or her reservation value 4. There are two bidders in a first-price, sealed-bid auction. Bidder 1 knows that individual 2 will submit...
Competition and reputation
The producer of a commodity has two sets of rivals. The obvious group consists of the other producers, but in a market economy consumers are the also firm's adversaries in the sense that, unless somehow constrained, the producer can increase profit by misleading its customers about product quality, in particular. Product quality has many dimensions durability, versatility, performance, operating cost, maintenance cost and there are many opportunities for the unscrupulous manufacturer to reduce...
Problem set
The first two questions assume that T 1 and each individual can set e 0 or e 1 but not any intermediate value. If e 0 then the probability of an accident is 1 2 but if e 1 the probability of an accident is 1 4. 1. B w 2.5v w, a 30, and z 72 for each individual. Find the competitive equilibrium and determine whether it is efficient. 2. Each individual's utility-of-wealth function is B w 5 ln w 1 . If there is an accident then the individual's wealth will be 40 but if there is no accident wealth...
Hostile takeovers
DianeCam corporation has two owners, Soren and Rosie, each of whom owns two shares in the firm. The current market value of a share is 10. Edie wants to acquire all four shares in the firm and then replace the current manager with a more efficient one. This will raise DianeCam's profit and hence the market value of a share from 10 per share to 18. Therefore, Edie could offer to buy the outstanding shares at 15 each. This would give Soren and Rosie a nice profit. But, why would they sell if the...
Example 52 The effort supply function
Let U x, y ax 1 2x2 y and R e, f fie f. Assume also that E f 0. Then E R e, f fie. If y 6R F then E y 6E R e, f F and we have E y 6fie F, and thus EU ax 2 x2 6fie F a T e 1 T e 2 6fie F because x T e. And a T e 1 2 T e 2 aT ae 1 2T2 Te 1 2e2, so we have The terms inside the square brackets are constant that is, independent of the manager's choice of e. We maximize EU by employing calculus or the formula for maximizing a quadratic. We get the effort supply function This does increase when 6, the...
Example 25 Corner points need not apply
Maximize U x, y x2y subject to 5x 2y 60. From the budget equation we have y 30 5x 2 30 2.5x. Substitute this for y in the utility function to obtain V x2 30 2.5x 30x2 2.5x3. We want to maximize this function of x subject to 0 lt x lt 60 5. V x 60x 7.5x2 and V x 60 15x. Setting the first derivative equal to zero yields Because U 0 if x 0 or y 0, utility will be maximized at a point where x is strictly greater than 0 and strictly less than 12. Therefore, V' x 0 at the solution to the consumer...
Consumer surplus
Now we show that if utility is quasi-linear then the demand function for commodity X can be used to estimate the utility function. Specifically, if U x, y B x y then the demand curve for X can be used to recover the benefit function B x . For convenience we assume that B 0 0. Quasi-linear utility also means that the area under the demand curve and above the line P pi, where pi is the given price of X, is equal to the utility gain from being able to purchase X at the price p1. We refer to this...
Examples of managerial shirking
Managers' immediate concern is their long-run well-being. Unless incentives or personal integrity take them in a different direction, their performance will be designed to enhance their present income, nonmonetary rewards, and future monetary rewards on the job involving the use of a company airplane, etc. , perceived value to other companies to enhance job prospects elsewhere , and retirement package. Studies of the agency problem have uncovered a long list of avoidable deviations from profit...
Example 22 The nonlinear fare motivates the agent to perform well
F 3 and c 1. Hence the fare is 3 D. As in Example 2.1, each trip is 5 miles long by the short route and 10 miles by the long route. The driver can make 30 trips a day of 10 miles each or 55 trips a day of 5 miles each. When the driver works efficiently her revenue is 55 x 3 55 x 1 x 5 440, but she if shirks her revenue is only 30 x 3 30 x 1 x 10 390. The driver's revenue is lower when she shirks. If the driver can make only 50 trips a day when she takes the short route, the revenue would only...
Example 51 Extensive form twoperson game
The game is represented as Figure 1.5 . At the first stage player A has a choice of moving left or right. If A moves left the game is over, and A's payoff is 1 and B's payoff is 5. If player A moves right at the first stage then player B has the next move and can go up or down. If B chooses up then each gets a payoff of 3, but if B moves down then A's payoff is 0 and B's payoff is 2. Consider the normal form representation of the same game displayed as Table 1.13. R U represents the strategy B...
Example 43 Dividing a cake
There are n individuals who are to share one cake. Assume that each person's preference scheme is independent of the amount of cake received by anyone else and that each person always prefers more to less. The feasible set consists of the different ways allocations of dividing a single cake among the n persons. Allocation x assigns the fraction xi of the cake to individual i. Of course, xi gt 0 for all i and xi xi x2 ---- xn lt 1. These are the feasibility conditions. Our assumption of...
Equilibrium theorem for infinitely repeated games
The argument of Section 7.2, showing that a wide range of strategy pairs can be sustained as a Nash equilibrium of the infinitely repeated prisoner's dilemma game, is easy to generalize to the infinite replication of any n-person game in which each person has a finite number of available actions. Let's quickly review the prisoner's dilemma case The strategy play D every period yields a payoff of at least d every period the payoff will be either d or h and thus yields an overall payoff of at...