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Game Theory and Microeconomic Analysis Questions

## Question 1: Ultimatum Game

Suppose two players are asked to split Â£1000Â in a way that is agreeable to both. Suppose Ayo is the one that splits the Â£1000Â and Silvie is the one who decides to accept or reject. For instance, if Ayo says Â£200, they are offering Silvie a split of the Â£1000Â that gives Â£800Â to Ayo and Â£200Â to Silvie. After an offer has been made by Ayo, Silvie simply chooses from two possible actions: either Accept the offer or Reject it. If Silvie accepts, the Â£1000Â is split in the way proposed by Ayo; if Silvie rejects, neither player gets anything. A game like this is often referred to as an ultimatum game.

Ayo thinks there is a pretty good chance that Silvie is the epitome of a rational human being who cares only about walking away with the most they can from the game. Ayo doesnâ€™t know Silvie that well and thinks there is some chance ?Â that she is a self-righteous moralist who will reject any offer that is worse for her than a 50-50 split. Assume throughout that pounds can be split into infinitesimal parts.

a)Structure this game as an incomplete information game. (5 marks)

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b) What are the pure strategy equilibria? (15 marks)

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c) What would happen if Silvie, as a self-righteous moralist which she is with probability ? rejects all offers that leave her with less than Â£100? (5 marks)

QUESTION 2:Â Suppose demand for zÂ is characterized by the demand curve

1. Derive the monopolistâ€™s profit-maximizing supply point, that is, the price and quantity () under the implicit assumption of no price discrimination. (7 marks)

2. At the output level , what is the average cost paid by the monopolist? (5 marks)

3. How high can fixed costs be and still permit the monopolist to make nonnegative profit? (7 marks)

4. If this monopolist is threatened by a possible competitor, how much will they be willing to pay their lawyers to get copyright protection? (6 marks)

QUESTION 3:Â An individual has a utility function defined over two goods as

The price of is and the price of is and income equals I.

1. Set up the maximization problem and derive the First Order Conditions. (3 marks)

2. What is the economic interpretation of the equation you derive when you combine the first two FOCs? (3 marks)

3. If , and Â find the actual consumption for the two goods and the associated utility for this bundle. (4 marks)

4. Derive the demand equations for andÂ Â and present in a graph the optimal consumption bundle. (5 marks)

5. How would your answer change if ? How much more money would you need to keep the previous bundle? Show in the graph the substitution and income effects (10 marks)

QUESTION 4:Â Consider two firms that compete in quantities. The (inverse) demand function is given by P(Q)=3 ? Q, where Q = q1 + q2. Consider the following set up:

Firm 1 decides whether to double their research and development budget before they decide how much to produce. If firm 1 decides not to double the R&D budget, it pays nothing and incurs a marginal cost of 1. If firm 1 decides to invest, it pays F > 0 and incurs a marginal cost of 0. In any event, firm 2â€™s marginal cost is 1.

1. Compute the best response functions when (i) firm 1 does not increase the R&D budget and (ii) firm 1 increases the R&D budget. What is firm 1â€™s profit in each case? (10 marks)

2. Given your answer in part (a), when will firm 1 decide to increase the budget? (8 marks)

3. How does this budget decision affect firm 2â€™s output and profit levels? (7 marks)

QUESTION 5:Â Suppose your firm has a decreasing returns to scale, Cobbâ€“Douglas production function of the form

Where A is technology, l is labour and k capital, while ?Â and ?Â are positive numbers.

1. Calculate input demand and output supply functions. You can do so directly using the profit maximization problem, or you can use the cost function below and Shephardâ€™s Lemma. (8 marks)

a)Derive the profit function. (7 marks)

b)Derive the conditional input demand functions, either by setting up the cost minimization problem, or you can employ Shephardâ€™s Lemma and use the cost function given in part a. (5 marks)

c)Consider a tax on labour that raises the labour costs for firms to (1+ t)w. How does this affect the various functions for the firm? What would happen if instead of a labour tax, a tax on capital raises the capital cost for the firm to (1 + t)r ? (5 marks)