Free Economics Dissertations - Since Firms In This Model Are Competing On Quantities, Each Firm Charges The
Since firms in this model are competing on quantities, each firm charges the market price. This is assumed to be determined from the inverse demand function (or perhaps an auctioneer) as the price that equates the amount demanded by producers to the amount produced by the industry.
The inverse demand function gives price p charged by each firm as a function of total industry output Q. Each firm also faces a cost function C(qi), where qi is the quantity chosen by firm i. Thus the profits of each firm, given by ?i, can be calculated as total revenue pqi minus total cost C(qi)qi. Each firm attempts to maximise profits. An equilibrium consists of a set of quantity choices by each firm such that no firm can increase profits, given the quantities chosen by other firms.
A simple version of the Cournot model is presented below. Here, there are two firms firm i and firm j. Each firm faces a linear demand function and a cost function which equals constant marginal cost c.
p = a Q
Q = qi + qj
Since marginal cost equals c, it follows that ?i = pqi - cqi
We assume that dqj/dqi = 0 and that profits are maximised where d?i /dqi = (a-c) - 2qi - qj = 0.
Therefore firm i will set quantity qi = ((a-c) - qj)/2
This is known as firm i’s ‘best-response function’. It expresses the profit-maximising quantities of the firm as a function of quantities chosen by its rival. Since firm j faces the same costs, its best-response function, by symmetry, is qj = ((a-c) qi)/2. By substituting the best-response function of firm j into that of firm i, and vice- versa, we can find quantities which are best responses to each other, and therefore reach an equilibrium. In this case, a unique equilibrium is characterised by each firm setting their quantities equal to (a - c)/ 3.
The monopoly quantity for firm i (where the firm captures the industry) is found by setting qj equal to zero in the firm i best-response function. Thus the monopoly quantity for each firm can be ascertained, and is given by (a c)/2. Note that if each firm cooperated, they would produce (a c)/4 and therefore share the industry. However, this outcome is not considered an equilibrium of the game. In the ‘competitive’ outcome, firms would set price equal to marginal cost. Thus p would equal c, resulting in Q = a c. Each firm would produce half of the competitive output level, equivalent to (a c)/2. This outcome is socially optimal, since the value placed on the last unit sold by the producer is the same as that placed on it by the consumer.
The model can be extended to become a multi-period model, in which each firm’s quantity choice eventually converges to the Cournot equilibrium. It can also be extended to more than two firms, which predictably results in a convergence to the competitive equilibrium, with falling profits and prices for each firm and increasing industry output.
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