UNIVERSITY OF SOUTHAMPTON ECON6008W1 SEMESTER 2 2020-21 ECON6008 Industrial Economics Duration: long take-home assignment This paper contains 3 questions Answer ALL three questions. Question 1 carries 40 marks and each other question carries 30 marks for this assessment. Copyright 2021 v01 c University of Southampton Page 1 of 3 2 ECON6008W1 1. Consider a Cournot oligopoly with three
rms i = 1; 2; 3. All
rms have the same constant marginal cost c = 1. The inverse demand function of the market is given by P = 9 Q, where P is the market price, and Q = P3 i=1 qi is the aggregate output. (a) Solve for the Nash equilibrium of the game including
rm out- puts, market price, aggregate output, and
rm pro
ts (Hint: the NE is symmetric). [20] (b) Now suppose these three
rms play a 2-stage game. In stage 1, they produce capacities q1, q2 and q3, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices p1, p2 and p3. The marginal cost for each
rm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for
rm i is Di (pi; p i) = 8<: 9 pi P j 6=i qj if pi > pj for all j 6= i 9 pi 3 if pi = pj for all j 6= i 9 pi if pi < pj for all j 6= i : (Note, here we assume that the e¢ cient/parallel rationing ap- plies). Prove that it is a Nash equilibrium of the second stage subgame that each
rm charges the market clearing price p = 9 q1 q2 q3. [20] 2. Consider an in
nitely repeated Bertrand oligopoly game with dis- count factor < 1. The unit cost of production is a constant c = 0:2 and the same for all n > 2
rms. There are no
xed costs. Describe a form of triggerstrategies that can facilitate tacit collu- sion in pricing. Determine the condition under which such strategies can sustain the monopoly price in each of the following cases: (a) The market demand in each period is D (p) = 1 p. (Calculate [15] the monopoly price and pro
t explicitly in your answer.) Copyright 2021 v01 c University of Southampton Page 2 of 3 3 ECON6008W1 (b) At the end of each period, the market ceases to exist with prob- [15] ability . 3. Consider a duopoly market, where two
rms sell di¤erentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two
rms choose prices p1 and p2 simultaneously. The derived demand functions for the two
rms are: D1 (p1; p2) = S2 + p2 p1 2t and D2 (p1; p2) = S2 + p1 p2 2t , where S > 0 and the parameter t > 0 measures the degree of product di¤erentiation. Both
rms have constant marginal cost c > 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and pro
ts of the two
rms. [10] (b) From the demand functions, qi = Di (pi; pj) = S2 + pj pi 2t , derive the residual inverse demand functions: pi = Pi(qi; pj) (work out Pi(qi; pj)). Show that for t > 0, Pi(qi; pj) is downward-sloping, i.e., @Pi(qi;pj)@qi < 0. Argue that, taking pj 0 as given,
rm i is like a monopolist facing a residual inverse demand, and the optimal qi (which equates marginal revenue and marginal cost) or pi makes Pi(qi; pj) = pi > c, i.e.,
rm i has market power. [10] (c) Calculate the limits of the equilibrium prices and pro
ts as t! 0. What is Pi(qi; pj) as t ! 0? Is it downward sloping? Ar- gue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p1 = p 2 = c) holds only in the extreme case of t = 0. [10] END OF PAPER Copyright 2021 v01 c University of Southampton Page 3 of 3
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