Strategy thinking

Dr. Vidya Atal Strategic Thinking & Game Theory

Module 10 Assignment

Answer all the questions. Remember that every part of the question has points assigned; by just submitting an answer to each part you can get 50% submission credit, but if you do not submit an answer to a part, you get 0. So play strategically! Please upload the images of your answers on Canvas before the due date.

1. CEO is hiring Manager to supervise a project. M can put High effort or Low effort. If L, project is successful with probability 1

4 and revenue to P is 1(million). If H, project is successful with probability 1

2 . If project not

successful, revenue is 0. Manager is risk averse: u(m) = √ m, cost of high effort is disutility of 0.1. Manager’s

outside option salary is $160000 = 0.16(million) (hence utility u = √ 0.16 = 0.4). CEO is risk-neutral. Suppose

effort is not observable.

(a) What should be the salary (x) if project is not successful and the salary plus bonus (y) if project is successful if the CEO wants only low effort from the manager? Show your work. What is the expected profit to the company when it induces low managerial effort? (20 points) [Hint: Call

√ x = X,

√ y = Y.]

(b) What should be the salary (x) if project is not successful and the salary plus bonus (y) if project is successful so that the CEO can ensure high effort provision and maximize his own profit? Show your work. What is the expected profit to the company when it induces high managerial effort? (20 points)

(c) Which level of effort does the company want to induce from its manager? Why?

2. You are Greenland’s Minister for Peace and it is your job to purchase war materials for your country. The net

benefit, measured in Greenland’s dollars, from quantity Q of these materials is ( 2Q

1 2 −M

) , where M is the

amount of money paid for the materials. There is just one supplier Cameron Armaments (CA). You do not know their cost of production. Everyone knows that cost per unit of output is constant and that it is equal to 0.10 (low cost) with probability 0.4 and equal to 0.16 (high cost) with probability 0.6. Only CA knows its true cost type with certainty. You decide to offer a menu of two possibilities:

Contract 1 - Supply us Ql and we shall pay you Ml.

Contract 2 - Supply us Qh and we shall pay you Mh.

The idea is to set (Ql,Ml,Qh,Mh) such that low-cost CA will find contract 1 more profitable and high cost CA will find contract 2 more profitable.

(a) Write expressions for the profit of a low-cost and high-cost firm when it supplies quantity Q and is paid M. (5 points)

(b) Write the incentive compatibility constraints to induce a low-cost CA to select contract 1 and a high-cost CA to select contract 2. (5 points)

(c) Give the participation constraints for each type. (5 points)

(d) Assuming that each type chooses the contract designed for it, write the expression for Greenland’s expected net benefit. (5 points)

(e) Assume that Ql > Qh and further assume that constraints ICl and PCh bind, that is, they will hold with equality instead of weak inequalities. Use these constraints to derive lower bounds on your feasible choices (Ml,Mh) in terms of (Ql,Qh) . (5+5 = 10 points)

(f) Show that when ICl and PCh bind, ICh and PCl are automatically satisfied. (5 points)

(g) Substitute out for (Ml,Mh) using the expressions found in (e) to express your objective function in terms of (Ql,Qh) . Write the first order conditions for the maximization and solve them. (5+5 = 10 points)

(h) What is Greenland’s expected net benefit from offering this menu of contracts? (5 points)

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