Exam V - Svar till tenta ligger under filnamnet: Svar till Exam V.pdf PDF

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Svar till tenta ligger under filnamnet: Svar till Exam V.pdf...

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Thomas Giebe

Intermediate Micro HT18-1NA072

Exam 25 Jan 2019 Instructions: This exam has 11 Questions (check!). You can earn a total of 57 points. Always show how you solved the questions; do not give a result without any computation or illustration. This does not apply to Exercise 1. Question 1 (Economic Concepts) Give a (very) brief explanation of the following economic concepts. 1. maximum principle [1 point] 2. price-taking [1 point] 3. price leadership (game) [1 point] 4. dynamic efficiency [1 point] 5. tragedy of the commons [1 point] 6. sunk cost [1 point] 7. positive analysis [1 point] 8. perfect complements (as a form of consumer preferences) [1 point] 9. positive externality [1 point] 10. price elasticity of demand [1 point] Question 2 (Consumer) A consumer consumes two goods in quantities x1 and x2 . The market prices per unit of those goods are p1 = 4 and p2 = 2. The consumer’s income (wealth) is m = 20. 1. Write down the consumer’s budget equation. [1 point] 2. Suppose now that there is a quantity tax of t = 3 per unit of good 2 that the consumer has to pay to the government. What is the maximum quantity of good 2 the consumer can afford to buy before and after the introduction of the tax (everything else is unchanged)? [2 points] 1

Question 3 (Producer) 1

2

A firm has the production function y = f (z1 , z2 ) = z13 z23 where y is the output quantity and z1 and z2 are the two inputs’ quantities. The unit prices of these inputs are w1 = 1 and w2 = 2. Compute the firm’s cost function c(y ) for these given input prices. [6 points] Question 4 (Equilibrium) Consider a perfectly competitive market for some consumption good. Inverse market demand is pD (x) = 10−2x where x is the demanded quantity. Inverse market supply is pS (x) = 4x. Compute the market equilibrium (x∗ , p∗ ), i.e., the market-clearing price and the corresponding quantity. [2 points] Question 5 (Monopoly) A monopolist serves two markets with inverse demand functions p1 (x1 ) = 30 − x1 and p2 (x2 ) = 10 − x2 . The monopolist’s cost function is c(x) = 2x where x = x1 + x2 is the total quantity produced. Compute the monopolist’s profit-maximizing quantities (x13D, x3D 2 ) assuming that the monopolist uses third-degree price discrimination. [4 points] Question 6 (Oligopoly (Bertrand)) Consider a duopoly of firms 1 and 2. Market demand has the form x(p) = 20 − p and, similar to the lecture, we assume that the firm with the lower price faces this demand function, while the firm with the higher price has zero customers. If both firms have the same price, they each serve one half of that market demand. Firm 1 has marginal cost MC1 = 4 and firm 2 has marginal cost MC2 = 6. 1. Is (p1 , p2 ) = (6, 6) a Bertrand equilibrium? Argue. [4 points]

2

Question 7 (Game Theory) Consider a market-entry game. Player 2 (P2) is currently a monopolist. Player 1 (P1) thinks about entering the market. If P1 enters the market, then P2 can either fight or not fight. If P1 does not enter the market, P2 remains a monopolist. The payoffs for these three situations are given in the game tree (and the following payoff matrix) below. For example, if P1 does not enter the market, the payoffs are (0, 4), which means that P1 gets zero payoff and P2 gets a payoff of 4. P1 In

Out

P2 Fight

No Fight

−3, −1

P1

In Out

0, 4

2, 1 Fight -3, -1 0,4

P2 No Fight 2, 1 0, 4

1. Solve the game (game tree above) by backwards induction, answering the following questions. [2 points] (a) Suppose P1 has already entered the market. What will P2 do? (b) Given your answer on the previous question, will P1 enter the market? 2. Using the payoff matrix above, find all Nash equilibria (in pure strategies) of this game. [2 points] Question 8 (Welfare) Consider a market for some consumption good. Inverse demand is pD (x) = 20 − x and the inverse supply function is pS (x) = x. Suppose the quantity x = 6 is being traded (sold and consumed) at price p = 14. 1. Compute the consumer surplus (CS). [3 points] 2. Compute the producer surplus (PS). [3 points] (Hint: You may solve this by computation or by drawing an appropriate figure and computing the areas geometrically.) 3

Question 9 (Externalities) Consider a steel factory (S) and a fishery (F) who use water from the same river. The steel factory’s profit is a function of the quantity of steel, s, and the river’s water temperature, t. The steel factory can change (i.e. choose) the water temperature as part of its production process. The fishery’s profit is a function of the quantity of fish, f , and the water temperature. The fishery cannot change the water temperature. The two profit functions are π S (s, t) = s −

s2 t2 + 5t − , 2 2

π F (f, t) = f −

f2 + 7t − t2 . 2

(Hint: The above profit functions already include all revenues and cost.) 1. Compute the maximum profits that the two firms achieve if they decide about their production independently. [4 points] 2. Suppose now that the two firms merge. What is the maximum profit they can achieve after the merger? Compare with your result from 1. and give a (very) short intuitive explanation. [3 points] Question 10 (Public Goods) Two pensioners, A and B, together own a garden. They independently choose how many hours each of them works in the garden. The number of hours is denoted by xA and xB . Their payoffs (utilities) from their work in the garden are described by 1 uA = 4(xA + xB ) − x2A , 2

2 uB = 2(xA + xB ) − xB

For example, in A’s payoff function, the term 4(xA + xB ) describes the utility from seeing a beautiful garden (The garden’s beauty is a function of the total hours worked by both pensioners), and the term − 12 x2A describes the cost of feeling tired from the garden work. 1. Compute how many hours, xA and xB , each of them will work in the garden if they decide independently. [3 points] 2. Compute the number of hours each of them would have to work in the garden to maximize both pensioners’ total welfare (Pareto-optimal decision). Compare with your result from 1. and give a (very) brief intuitive explanation for the difference. [3 points] 4

Question 11 (Asymmetric Information) Consider a used-car market as discussed in the lecture and exercises. As usual, sellers know the quality of their car while buyers do not. Sellers and buyers are risk-neutral (i.e., they only care about expected payoff if there is uncertainty). The following table contains the reservation values of buyers and sellers for the two qualities of cars (good=plum, bad=lemon).

Seller Buyer

plums 350 500

lemons 50 100

1. Suppose the buyers think that in the total ‘population’ of cars, there are 60% good cars and 40% bad. Do you expect that good cars will be traded under these conditions? Argue. [3 points] 2. How large would the share of good cars in the population have to be at least, such that good cars will be traded? [2 points]

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