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Game Theory and Microeconomic Analysis Questions

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

- Derive the monopolistâ€™s profit-maximizing supply point, that is, the price and quantity () under the implicit assumption of no price discrimination. (7 marks)
- At the output level , what is the average cost paid by the monopolist? (5 marks)
- How high can fixed costs be and still permit the monopolist to make nonnegative profit? (7 marks)
- 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.

- Set up the maximization problem and derive the First Order Conditions. (3 marks)
- What is the economic interpretation of the equation you derive when you combine the first two FOCs? (3 marks)
- If , and Â find the actual consumption for the two goods and the associated utility for this bundle. (4 marks)
- Derive the demand equations for andÂ Â and present in a graph the optimal consumption bundle. (5 marks)
- 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.

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

- 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)