1 HOMEWORK #5 Suggested Solutions 1 If people can ship the good from California to New York, the firm could not charge a price in New York that exceeds the one in California by more than $1. It could still price discriminate between the two markets by charging different prices – i.e. ncpp≠ - but it is restricted in the extent that it can do so as we should have:1cncppp<≤+. To see this, consider the firm charging a price 1ncpp>+ in New York. Let ncppc=+ with 1c>. Then you could buy the good in California for cp per unit and ship it to New York for $1 per unit. The total cost of supplying it to the New York market would be 1cp+. Hence, you could sell the good in New York, undercutting the firm, by selling for just undernp. Because you offer a lower price than the firm, everyone should be buying from you and you would be making a profit of just under 1c− per unit. Therefore, the firm could not sustain sales in the New York market if it were to charge any price above 1cp+. In reality, of course, the wedge between the two prices ,ncpp could be sustained at more than $1 because the shipping costs are typically not the only transaction costs associated with re-sales. In the above example of profiteering by buying the good in California and shipping and putting it up for sale in New York, one should take into account the economic costs of finding the customers (i.e. sale-outlets, advertising, distribution etc). If all these costs come to, say, m per unit then re-sales are profitable only if 1ncppm>++. Hence, the firm can maintain a price-discriminating wedge of 1m+ in this case1. 2 It does not pay to collude with other firms in a repeated game if: 1. The game is to be played for a known, finite number of times T. This argument is thoroughly explained in your textbook (pp. 423). 2. The number of firms in the game is very large. In a large game, the quantity produced by an individual firm as part of any equilibrium will be small compared to the total equilibrium quantity. This means that, if an individual firm deviates from its assigned quantity under the collusion agreement, and produces something different, it will be rather hard for the rest of the firms to detect its deviation. The 1 One could argue that these type of transaction cost (outlets, advertising etc) that I am referring to, are also incurred by the firm in the first place. Thus, they ought to be included already in the price that the firm charges np. But the firm is very likely to be much more cost efficient in these undertakings than someone who just wants to set up a temporary business to take advantage of the price difference between the two markets. In this case, m is the extra amount that these transaction activities would cost per unit to you over and above what it costs per unit to the firm.2 new total quantity might not be significantly different from the collusion quantity for the other firms to realize that someone has deviated or to be able to figure out who the culprit is. Thus, co-operation is much harder to sustain. 3. When the interest rates are high, the firms will discount the future more heavily. Current payoffs become much more valuable than payoffs to be received in the future. The latter is, however, exactly what co-operation does. It tries to enforce firms accepting lower payoffs now for a future stream of higher payoffs. Hence, if interest rates are high, the players will have a higher preference for receiving payoffs now rather than later. Deviating and getting the higher than co-operative payoff now seems more attractive rather waiting to receive the benefits in the future. Similarly, the costs of punishment that deviation will bring about will also come tomorrow and, therefore, will be discounted. These costs appear smaller compared to the current benefits of deviating now. Again, deviation seems more attractive since its costs appear to be smaller. 3 Consider the payoff matrix of this game. Firm 2 H L H 5, 5 0, 0 Firm 1 L 0, 0 0, 0 Each player in this game has two strategies: • You can charge a high price (H) or • You can charge a low price (L). The payoffs are as shown in the matrix above (recall that, in each cell, the first entry shows the payoff of the row-player). Solving for the Nash Equilibrium (in pure strategies) Consider firm 1 choosing to price H. Firm 2’s best response is to choose H (and get a payoff of 5 rather than 0). In other words, given that 1 is playing H, 2’s best response is H. To check whether we can have a NE when firm 1 is playing H, we need now to check if play of H by firm 1 is also a best response to firm 2 playing H. And it is so, if firm 2 is playing H, firm 1’s best response is to choose H. Hence, (H,H) is a NE. Consider now firm 1 choosing to price L. Then, firm 2 is indifferent between choosing either H or L (bth strategies give the same payoff of 0 against play of H by firm 1). In other words, given that 1 is playing H, 2’s best response is any of H or L. To check whether we can have a NE when firm 1 is playing L, we need now to check if play of L by firm 1 is also a best response to whatever of H or L that firm 2 might be playing. But firm 1’s best response can be L ONLY if firm 2 is playing L. Hence, only (L,L) is a NE. Note:3 (L,H) is not because although we have just argued that H is a best response for firm 2 when it believes that firm 1 will play L, L is Not a best response for firm 1 when it believes that firm 2 is playing H. This game, therefore, has two NE in pure strategies (H,H) and (L,L)2. 4 Efficiency Explanations 1. The “meet or release” provision operates like an “insurance” contract for potential customers. Instead of engaging in costly searches for the cheapest supplier of a given commodity, the customer can enter into the contract knowing that if, in the future, he comes across another supplier that offers a lower price, he can switch over without being bound by the current contract. In the absence of such a provision, customers would have to be more careful and devote more time and resources into searching for the cheapest supplier at present as well as in trying to estimate potential changes in suppliers’ prices in the future. In particular, in markets where suppliers’ costs are uncertain and changing over time (e.g. due to weather conditions in agricultural product markets, technological changes or changes in input prices for long-term service contracts etc.) the
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