PS2 IO 2021 - SEMINARI 2 IO PDF

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SEMINARI 2 IO...

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Industrial Organization Problem Set 2 Christian Michel The problem set consists of four problems.

1. (Hotelling duopoly – Note: for this exercise a separate solution manual will be handed out after the seminars have taken place) Consider a linear Hotelling duopoly, where consumers are uniformly distributed over a line of length 1, and two firms A and B are initially located at the ends of the line. A consumer of type x derives utility 1 −tx−p1 if she purchases product A (location 0) and utility 0.8 −(1−x)t−p2 if she purchases product B (location 1). Suppose that parameter values are such that in the equilibrium to be characterized below, the market is fully covered (You can assume that 0.25 > t > 0.1). You can assume marginal costs of 0 (but c > 0 works, too). Firms set prices simultaneously. [Note: If you prefer, you can compute the equilibrium for t = 0.2 with a slight drop in points instead] (a) Derive i. The location of the consumer indifferent between A and B, depending on the prices of the both firms ii. the equilibrium prices and quantities of both firms (hint: given (i), set up the maximization problems first, then derive the reaction functions and solve for the equilibrium prices) iii. equilibrium profits (b) Give 2 examples from the real world not discussed in the lecture for which the Hotelling model could be applied, and explain (max 4 sentences for each example) (c) Could you think of some limitations of this model to model real world product differentiation? (max 5 sentences) 2. (Vertical Product Differentiation [Hint: All necessary information to compute the solution is within the product differentiation lecture note]) Consider the model of vertical product differentiation that we discussed in class. That is, consumer i’s indirect utility from consuming good j is given by   θ z −p i j j vij =  0

if i buys good j if she buys nothing

where θ is the individual preference parameter and zj is the vertical quality characteristic. Suppose now, however, that there are four products, and that pairs of z and p are given as follows: (z1 , p1 ) = (5, 5) , (z2 , p2 ) = (3, 1) , (z3 , p3 ) = (5, 2) , (z4 , p4 ) = (8, 4) , Assume also that θ is uniformly distributed between 0 and 1, and the total mass of consumers is 1. (a) By looking at the price-quality combinations of each product, can you already rule out a firm from selling a product? State your reasoning. 1

(b) Which product will the consumer with the highest willingness to pay choose, i.e. those with the highest θ values? What would be the second best option these consumers consider? Which consumer type (θ) is indifferent between these two options? (c) Where is the consumer located that is indifferent between buying a product and rather choosing no product at all (which generates 0 utility)? Which product is he considering choosing? (d) Calculate the market share of each good and the “outside option.” (The outside option market share is defined as the share of of the potential consumers who do not end up choosing one of the products 1 − 4, as in (c). (Collusion with delay in monitoring [Hint: All necessary information to compute the solution is within the Dynamic Oligopoly Lecture note]) Consider a game with two firms and two actions. Firms simultaneously choose their action. Each firm can either collude or deviate. if both firms choose collude, each of them receives a payoff of 4. If both firms deviate, each of them receives 2. If one firm colludes but the other firm deviates, the firm which colludes receives 0, while the firm which deviates receives 6. 1. What is the equilibrium of the one-shot game? 2. Suppose the game is repeatedly played for an overall number of T = 10 periods. The payoff for a player i (i=1,2) is given by Πi =

9 X

δ t πi (t),

t=0

where πi (t) is the stage payoff in period t, and δ ∈ (0, 1) is the discount factor. For which values of δ can firms sustain a subgame perfect Nash equilibrium that consists of trigger strategies (i.e. always play collude at t if no firm has deviated until period t − 1, and play deviate otherwise)? Discuss (maximum 3 sentences).

3. Suppose now the game is repeated an infinite number of times. The payoff is given by Πi =

∞ X

δ t πi (t),

t=0

where πi (t) again is the stage payoff in period t, and δ ∈ (0, 1) is the discount factor. For which values of δ can firms sustain a subgame perfect Nash equilibrium that consists of trigger strategies (i.e. always play collude at t if no firm has deviated until period t − 1, and play deviate otherwise)? 4. The game remains repeated for an infinite number of times. However, now suppose that each firm cannot immediately observe the actions of its competitor (and therefore also not its own payoff). Instead, it only observes this data after a one period delay, i.e. after the game has already been played another time. (a) Assume firms are still considering trigger strategies (i.e. always play collude at t if no firm has deviated until period t-2, and play deviate otherwise). How do profits from colluding and profits from deviating now change compared to (c)? Also discuss (maximum 2 sentences).

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(b) For which values of δ can firms sustain a subgame perfect Nash equilibrium that consists of trigger strategies (i.e. always play collude at t if no firm has deviated until period t-2, and play deviate otherwise)? Compare your solution to the one in (c) and discuss (maximum 2 sentences). (Collusion under Cournot and Bertrand [Hint: All necessary information to compute the solution is within the Dynamic Oligopoly Lecture note]) Consider a Cournot game with two players and homogeneous products. The demand is given by p = a − q1 − q2 . Assume that production is costless. 1. Calculate the equilibrium profit for each firm under the one-shot Cournot game. 2. Find the outcome (output level, price, and profit) that maximizes the joint profit of two firms (collusion). 3. Assuming that firm 2 sticks to the collusive output level that you found in (b), what is firm 1’s optimal choice (we are still considering the one-shot Cournot game so no retaliation will follow)? 4. Now we consider an infinitely repeated game and the trigger strategy as we learned in class. Find the range of ρ such that the outcome where both firms play their trigger strategies is a SPE. 5. Consider again an infinitely repeated game and the trigger strategy as we learned in class, but now assume that competition is Bertrand in every stage game. Find the range of ρ such that the outcome where both firms play their trigger strategies is a SPE. Compare the result with your answer in (d). 6. From a competition policy standpoint, are price comparison websites always good for consumers? If yes, why? If not, give an example. (maximum 3 sentences)

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