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1. Calculate beliefs p(a) for a seller using Definition 10 on p. 13 in Gjerstad and Dickhaut (1998) and the data in Figures 4 (c) and 4 (d) in Gjerstad (2007). Give exact values for the seller's belief at asks in the set A = {91, 92, 93, 94, 95, 96, 97, 98, 99). Use the graph in Figure 4 (a) to verify that your answers are plausible. (By 'plausible' I mean, for example, that an answer of p(95) = 0.5 is implausible since it is clear from the graph in Figure 4 (a) that the value of p(95) is greater than 0.5.) (5 points)

2. Calculate expected surplus E[S(a)] = (a -c) * p(a) for a seller with cost c = 90 at each of the points in the set A in Problem 1 for thel seller with beliefs p(a) as calculated in Problem 1. Which ask hasl the highest expected surplus for a seller with cost 90 and what isl the maximum expected surplus for that seller? (4 points)

3. Repeat Problem 2 for a seller with cost 80. related to the seller's cost in this example? How is the ask of a seller (3 points)

4. Gjerstad (2007) modifies the model in Gjerstad and Dickhaut (1998) in several ways. One of the modifications is that it takes a root of the belief function at each value where the belief is defined.

(a) What is the modified belief p* (a) at each value in A for a sellerl when the root is 5 for a seller? [Hint: Start by taking p*(91) = p(91)^0.2, p*(92) = p(92)^0.2, etc.] (3 points)

(b) For a seller with cost c = 90, what is the expected surplus relative to this modified belief p* at each value in A where the belief isl defined as in Part (a) of this problem? (3 points)

(c) How do the surplus maximizing asks differ between Problem 4 (b) and problem 2? Stated differenty, what effect on does taking a root in the belief function have on the surplus maximizing ask?l (3 points)

5. Assume that a seller has the utility function

u(x, y) = (x^(-1) + (250 y)^(-1))^(-1).

The utility for this seller if he has 760 units of X and 10 units of Y is approximately 116.564. Find the change in the utility level when he sells 'n' units of Y and obtains 95 units of x for each unit that he sells, for n = 0, 1, 2, .., 6. (3 points)

6. suppose that this seller has the base belief function in Problem 1.

(a) What will the belief function be for this seller if it is modified by taking the 10th root of the base belief? (3 points)

(b) suppose that the seller is considering selling one more unit. What ask price will lead to the highest expected increase in surplus, with the expectation taken relative to the belief function that you calculated in part (a) of this problem? [Hint: The change in utility is (u(760 + a, 9) - u(760, 10)).] (3 points)