ECON20120/ECON30320
Mathematical Economics I
Section A
Answer all questions.
Question A.1 (Word Limit: 100 words)
For this exercise s=1+the 4th digit of your student number. Consider the following game in normal form.
Pl. 2
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L
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M
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R
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U
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3+s,3+s
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1+s,2+s
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2+s,4+s
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C
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2+s,1+s
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2+s,s
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5+s,2+s
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D
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4+s,5+s
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3+s,4+s
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3+s,2+s
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(i) If the game is played with simultaneous moves, identify all the pure strategy Nash equi- libria (if any).
(ii) If the game is played with simultaneous moves, identify all the mixed strategy Nash equilibria (if any).
(iii) If the game is played with sequential moves, where player 1 moves first, identify all the subgame perfect Nash equilibria (if any).
(iv) If the game is played with sequential moves, where player 2 moves first, identify all the subgame perfect Nash equilibria (if any).
(v) What do you conclude from your answers to (i), (iii) and (iv)? [10 marks]
Question A.2 (Word Limit: 300 words)
For this exercise s=1+the 4th digit of your student number.
In an industry there are three firms, f1 , f2 and f3 , producing a homogeneous product with total costs C(qi) = 30qi + F , where qi is the output level of firm fi , i = 1, 2, 3 and F is a fixed cost that each firm must pay to access the market. The inverse demand curve for the industry is p = 150 - Q, where Q = q1 + q2 + q3 .
(i) Find the Cournot equilibrium output and profit (as a function of F) for each firm. Show your calculations in deriving the equilibrium.
(ii) Firms f1 and f2 decide to merge and become firm fA. The resulting market structure consists of two firms: fA and f3 and each of them has to pay the fixed cost F. Calculate the Cournot competition output and profit level of firm fA (as a function of F). Show your calcu- lations in deriving the equilibrium.
(iii) The merger agreement stipulates that firms f1 , f2 split the profits of the new firm fA , according to the outcome of bargaining with alternating ofers, in which f1 will make the first ofer and will last four rounds, after which both get nothing. If F = 232 × s and both firms have a discount factor δ = 3/1, explain carefully whether or not firms f1, f2 benefit from merging in firm fA . [20 marks]
Question A.3 (Word Limit: 300 words)
For this exercise s=2+the 3rd digit of your student number.
In an industry there are two firms, f1 and f2, producing a homogeneous product in quantities q1, q2 with constant marginal costs c1 = 90 and c2 = 30 respectively. The inverse demand curve for the industry is p = 130 - 2Q, where Q = q1 + q2 . The two firms are in Stackelberg competition where Firm 1 moves first.
(i) Compute the equilibrium quantities and price in this market.
(ii) Suppose that this market takes place every period t = 1, 2 . . . and the marginal cost of firm 1 evolves over discrete time according to the diference equation:
c1,t = 90 − s/1c1,t−1, c1,0 = 90 (1)
By studying the above dynamic system, compute the long run equilibrium quantities and price in this market.
(iii) From your solution in (ii), find for how many periods this market will remain monopo- listic. [20 marks]
Section B
Answer either Question B.1 or Question B.2.
Each question is worth 50 marks.
Question B.1 (Word Limit: 300 words) For this exercise s=1+the 5th digit of your student number.
Two firms 1, 2 compete with diferentiated products each facing demand:
D1(p1 , p2 ) = 4 - 2p1 + p2
D2(p1 , p2 ) = 7 + p1 - p2
where Di(p1, p2 ) and pi are respectively the quantity produced and the price charged by firm i = 1, 2.
Both firms face constant marginal costs c1 = 2 and c2 = 5 respectively.
(i) Compute the equilibrium prices and quantities pi and Di(p1 , p2 ), i = 1, 2 when the two firms engage in simultaneous price setting competition.
(ii) Suppose that firm 2 can be informed about the price set by firm 1 before it decides its own and this information comes at a cost f. Firm 1 will be informed whether or not firm 2 paid the cost f, i.e., it will know whether or not firm 2 will learn the price p1 before it sets its own price p2 .
(a) Model this situation
(b) Suppose that the cost f is a given amount f = 4/1 × s. Compute the equilibrium prices and quantities pi and Di(p1, p2), i = 1, 2 in this case.
(c) Suppose instead that the cost f is determined by alternating ofers bargaining between the informant and the manager of firm 2, over the value of the information for firm 2. The bargaining lasts two rounds, the informant makes the first ofer and both have a discount fac- tor δ = 10/9. Compute the equilibrium prices and quantities pi and Di(p1, p2 ), i = 1, 2 in this case.
(d) Explain whether or not the manager of firm 1 might be inclined to volunteer this infor- mation. [50 marks]
Question B.2 (Word Limit: 300 words) For this exercise s=2+the 5th digit of your student number.
Two Cournot competitors face an inverse demand curve p = 14 - Q, where Q is the combined output of the two firms. Both firms have constant marginal cost of c = 2. Let BR1(q2 ) and BR2(q1 ) denote the best response functions of firm 1 and firm 2 respectively, in the simultaneous move game.
(i) Derive the expressions for BR1(q2 ) and BR2(q1 ) and find the Cournot equilibrium price and quantity.
(ii) When out of equilibrium the two firms adjust their outputs as follows:
dt/dq1 = 2 (BR1(q2) − q1) + 4 (q2 − BR2(q1))
dt/dq2 = BR2(q1) − q2
where BR1 and BR2 are the functions found in (i).
(a) Compute the long run equilibrium price in this market. How does it compare to the Cournot equilibrium price?
(b) Compute the time path of the price in this market.
(c) If at some instance t you observe p(t) = 3s, explain and carefully justify whether or not you would buy the commodity or rather wait to buy it at a later time. [50 marks]