In this game four participants play the role of four firms, the only firms that sell in a particular market. The firms produce an identical product, and they incur the same costs of production: it costs a firm $10 for each unit it produces. The firms will interact with one another in the market for 20 periods. In every period each of the firms will choose a quantity to produce, and all production by the four firms will be sold to buyers at whatever price will clear the market. The market demand in each period is given by the equation (i.e., the demand curve) P = 70 - (1/2)Q, where Q denotes the total quantity the four firms bring to the market, and P denotes the resulting market-clearing price -- i.e., the price at which buyers are actually willing to buy Q units. (Note the implicit assumption that the product cannot be stored from one period to the next. Any units not sold will spoil before the next period arrives.)
What outcome should we expect to see? How much should we expect each firm to produce, and what should we expect the price of the product to be? How much profit should we expect firms to earn?
Two different answers suggest themselves:
(a) Perhaps we should expect the firms to produce at a level that will maximize their joint profits. Because of the symmetery of the situation, and the constant per-unit cost, this could clearly be accomplished with all four firms producing the same quantity, in which case each firm would earn the same level of profit. It's not hard to show that this would entail each firm producing q = 15 units and earning $450 profit in every period, at a price of P = $40, with a total quantity of Q = 60 units being sold. Let's call this the Monopoly Outcome, since it's as if the firms are cooperating partners in a monopoly.
(b) Or perhaps we should expect the Competitive Outcome to prevail, in which the price is equal to the marginal cost of production, i.e., P = $10. The market quantity would therefore be Q = 120 units sold altogether in every period, and each firm would earn zero profit.
Game theory suggests a third possible outcome:
(c) If each firm believes it will not be able to affect the production levels of the other firms, and thus simply chooses its own production level to maximize its own profit, taking the other firms' production levels as given, then the outcome would be a Nash equilibrium. (This idea originated with Augustin Cournot, in 1838, and John Nash generalized it more than 100 years later, an accomplishment for which he was awarded the Nobel prize in Economics. Roger McCain's introduction to game theory has a chapter on Nash equilibrium. Sylvia Nasar's NY Times article about John Nash is extremely interesting.) It's relatively easy to show that the Nash equilibrium in our market game has each firm producing q = 24 units, for a total quantity of Q = 96 units sold in the market each period. The resulting market-clearing price is P = $22, and each firm earns profits of $288 per period.
Note that the Nash/Cournot outcome is intermediate between the Monopoly outcome and the Competitive outcome:
Total Q P Profit Monopoly: 60 $40 $ 1800 Nash/Cournot: 96 $22 $ 1152 Competitive: 120 $10 $ 0Our In-Class Game #1 was played in the Economic Science Laboratory's teaching lab. The class was divided into four different markets, each with four firms. (Only 19 terminals were available, so we could not run more than four markets. Thus, many of the firms were comprised of two students working together at a terminal.) Profits would be converted into quiz points.
Total Q P Profit Market #1 89 $25 $ 1199 Market #2 102 $19 $ 843 Market #3 100 $20 $ 964 Market #4 86 $27 $ 1202 C xx N x x M |---------|---------|---------|---------| $0 $10 $20 $30 $40 This diagram shows the price at the Competitive, Nash, and Monopoly outcomes, and the average prices observed in each of the four markets, denoted by x's.