Suppose that the person with the above utility function is struck by lightning. He survives, except now his utility function is u = β1 log(x1 − γ1) + β2 log(x2 − γ2). Will his consumption decisions change as a result of being struck by lightning? Briefly explain your answer. (c) Returning to the original utility function, assume that the consumer faces the budget constraint p1x1 + p2x2 = w, where p1 > 0, p2 > 0 are prices and w > 0 is wealth. 

Set up the Lagrangean and solve for the two first-order conditions. Then use the budget constraint to solve for the Walrasian demands of both goods. These should be functions of p1, p2, w, β1, β2, γ1, and γ2. Finally, use your answer to part (a) to simplify your answer. (d) Finally, turn to a more general utility function with L goods: u = Y L k=1 (xk − γk) βk , where βk > 0 for all k, which is maximized subject to the budget constraint PL k=1 pkxk = w. Using the same approach as in part (c), derive the Walrasian demands for the L goods. 

(Hint: first substitute out the Lagrangean multiplier using the first-order conditions for two goods, i and j, then find an expression for pixi and sum that expression over all i, using a normalization similar to the one used in part.

 

 

Consider an expenditure function that has the form log[e(p, u)] = X L k=1 αk log(pk) + u Y L k=1 p βk k . Don't worry that this is expressed as log[e(p, u)] instead of as e(p, u). After all, if we apply the exp(·) function to both sides, the left-hand side will become e(p, u). (a) Apply Shephard's lemma to this expenditure function to obtain the Hicksian demand for good i. (Hint: Differentiate with respect to log(pi) to obtain an elasticity that includes the Hicksian demand. Note also that p βk k equals (e log(pk) ) βk . The Hicksian demand should be a function of the α's, the β's, prices, and u.) (b) Derive the indirect utility function that is associated with this cost function. This is easier than part. Finally, use your answer to (b) to obtain the Walrasian demands. What is the name of the derivation that is used to obtain the Walrasian demands?

 

Consider a firm that produces a single output q ≥ 0 using inputs z1 ≥ 0 and z2 ≥ 0, where the input-requirement set is nonempty, strictly convex, closed, and satisfies weak free disposal. Assume the firm operates in competitive markets. The firm's profit function is π(r1, r2, p) = p α 4(r1 + √ r1r2 + r2) , where p > 0 is the price of output, r1 > 0 and r2 > 0 are the input prices, and α is a constant parameter. (a) What condition on α (if any) is required for π(r1, r2, p) to satisfy the price homogeneity property of a valid profit function? Justify your answer. (b) Use π(r1, r2, p) to derive the firm's unconditional supply and factor demands. (c) Derive the firm's conditional factor demands and cost function. (d) It is easy to verify that the conditional input demands in (c) are non-increasing in their own price. Show that this result holds in general for a cost function derived from a production possibility set with N inputs and M outputs.

 Consider a firm that produces output q ≥ 0 at a cost of c(q), where c 0 (q) > 0 and c 00(q) > 0. Also assume that there is a probability 1 > α > 0 that the firm experiences an equipment failure and incurs additional repair costs equal to cR > 0 per unit of output or cRq in total. The competitive price of output is p > 0. (a) Derive the firm's first-order condition for an interior solution assuming its objective is to maximize expected profit. What is the economic intuition of this condition? (b) Derive the firm's first-order condition for an interior solution assuming its objective is to maximize its expected utility of profit, where the strictly increasing and strictly concave function u(·) characterizes its risk preferences. (c) Will the firm produce more if its objective is to maximize expected profit or if its objective is to maximize the expected utility of profit? Justify your answer and provide the economic intuition for your result. 

Consider an expenditure function that has the form log[e(p, u)] = X L k=1 αk log(pk) + u Y L k=1 p βk k . Don't worry that this is expressed as log[e(p, u)] instead of as e(p, u). After all, if we apply the exp(·) function to both sides, the left-hand side will become e(p, u). (a) Apply Shephard's lemma to this expenditure function to obtain the Hicksian demand for good i. (Hint: Differentiate with respect to log(pi) to obtain an elasticity that includes the Hicksian demand. Note also that p βk k equals (e log(pk) ) βk . The Hicksian demand should be a function of the α's, the β's, prices, and u.) (b) Derive the indirect utility function that is associated with this cost function. 

This is easier than part (a). (c) Finally, use your answer to (b) to obtain the Walrasian demands. What is the name of the derivation that is used to obtain the Walrasian demands?

 

The faculty at the Smith School have started sharing their business contacts with the administration and students to help in recruiting efforts. The hope is that these business contacts will provide job leads for students, potential students for the school and possibly donations to the school. The new dean of the Office of Career Management (OCM) surveyed MBA students, undergraduates, and administration officials to determine how much value they place on these faculty contacts. These three groups represents all the members of the Smith School who receive any value from these contacts. The survey revealed that full-time MBA students (as a whole) valued the senior faculty contacts at $100 each and the junior faculty contacts at $50 each. Senior faculty contacts are often higher up in an organization and thus potentially more valuable to students and administrators. The undergraduate students (as a whole) valued both junior and senior faculty contacts at $50 each. Finally, administrators (as a whole) valued senior faculty contacts at $50 each and junior faculty contacts at $30 each. The total number of senior faculty contacts available is 100. The total number of junior faculty contacts available is 50. The Smith School plans to make these contacts available to students and administrators at no charge. 

 

a. Draw a demand curve combining the demand curves of each of the three sets of consumers for all 150 contacts. 

 

b. Faculty claim that the opportunity cost of providing this contact information is $160 per contact. If faculty are paid $160 to provide these contacts, how many contacts should the Smith School purchase? N 

 

c. Suppose the administration asks for contributions from students who make use of the contacts in order to raise the money paid to faculty. Will enough money be collected to generate the optimal number of contacts? Explain. 

 

d. The companies that comprise the faculty contacts also value the distribution of their employee names to students. In the past this form of networking has been beneficial in finding the best new employees. These companies place a value of $50 on the distribution of each senior and junior faculty contact. Will the Smith School change the number of contacts it purchases given this new information? Explain. 

 

e. If the companies offered to contribute $25 per contact distributed by the Smith School, would this change your answer to part d? Explain.

 

Country A learns that country B is boosting its military in preparation for a possible invasion. A has a choice between preparing for an invasion, at a cost if 100 units of wealth, and not preparing, in full view of B's spies. In case of invasion, a prepared A can choose to destroy its entire infrastructure at an additional cost of 900 units of wealth, or not to do so. If an invasion occurs and A's infrastructure is not destroyed country B gains 100 units of wealth and A loses 100 units of wealth (in addition to the preparation cost of 100 units, if A has prepared). If an invasion occurs and A's infrastructure has been destroyed, B loses 5 units of wealth. If no invasion occurs, country B gets nothing.
(a) Describe this game using a tree, carefully labelling all its components.

(b) Solve this game using backward induction.

 
(c) Describe the game in strategic form, find all its pure-strategy Nash equilibria and indicate which of these is subgame perfect.

(ii) (a) State Zermello's Theorem for finite, two-player sequential games.

(b) Consider a game played with 10 dots drawn on a sheet of paper. Two players, Red and Blue, alternate in choosing a pair of dots which has not been chosen before by either player, and connecting the two dots with a line of their colour. Red moves first.
The game ends if either there are four dots pairwise connected with lines of same colour, in which case that colour is declared the winner, or if all pairs of points have been connected with a line, but there is no winner, in which case the game ends with a draw. Use a strategy stealing argument to prove that there is a strategy that guarantees a win or a draw to the first player to move.

 Suppose that  rm can perfectly price discriminate (rst degree price discrimination). 

How much will is produce? How much will its prots be?

What will be the equilibrium prices and quantities, if there are TWO  rms that choose quantities simultaneously? (Cournot Competition).
Now assume that the  rst  rm gets to choose quantity before the entrant.

 What are the quantities that these  rms will produce and the price in the market (Stackelberg Competition). Why are these quantities dierent?

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1 Perfect Price Discrimination by a monopoly firm The firm will produce at the efficient quantity Qe ... View the full answer