Suppose that the demand function is Q= s/p, where Q is the total quantity demand
ID: 1189045 • Letter: S
Question
Suppose that the demand function is Q= s/p, where Q is the total quantity demanded, s is a measure of the size of the market, and p is the price of the homogeneous good. Let F be a firm's fixed cost and m be its constant marginal cost. If n firms compete in a Cournot model, calculate the price, p, a typical firm's output, q, and a typical firm's profit, pi.
a) Prove that:
i) p= m{1+ 1/n-1}
ii) q= (s/m)(n-1/n^2), and
iii) pi = s/n^2 minus F.
b) If entry is free, what does n equal?
c)What happens to equilibrium concentration, 1/n, as s increases?
d)What happens to equilibrium firm size as s increases?
Explanation / Answer
Let the demand curve be Q = S/p where Q measures the quantity demanded of a particular communications service which for present purposes is assumed to be homogeneous; p measures the unit price of the product or service; and S measures total consumer expenditure on a product or service at a specific time and is independent of market price.[1] S also provides a measure of market size and quantity demanded for the market is simply Q = Sqi = qi×N, where N is the number of firms. Since this market demand function has a constant, unit own-price elasticity (the demand curve is isoelastic), it can be shown that the profit-maximizing monopoly price approaches infinity for any marginal cost greater than zero. For analytical convenience, it is assumed that sales fall to zero above some cut-off price pm.
Suppose N facilities-based carriers decide to enter the market in State 1 of the game. The profit function of a representative firm i in Stage 2 of the game is given by
(1)
where qi is firm i’s level of output and p is market price, which is a function of total market output {p = p(Q)}, and c is marginal cost. Differentiating equation (1) with respect to qi produces the first-order condition for firm i:
(2)
where marginal cost is assumed constant across all output levels
The conjectural variation term measures firm i’s guess regarding how other firms will react to its output changes and is a critical assumption in models of oligopolistic competition. Setting qi =q for all i (all firms are identical), equation (2) can be solved for the conjectural variation equilibrium price:
(3)
unless equation (3) exceeds pm, the price at which sales become zero, in which case p = pm (the monopoly price). Consistent with the typical expectations of increases in the number of competing firms, equation (3) shows that for any given f > 0, increases in the number of firms reduces price. In the limit, price approach marginal cost as the number of firms increases.
In the Cournot model, rival firms choose the quantity they wish to offer for sale. Each firm maximizes profit on the assumption that the quantity produced by its rivals is not affected by its own output decisions. In other words, the conjectural variation of the Cournot firm is equal to one (f = 1) so that p = c{N/(N 1)}. Note that Equation (3) is a Cournot Nash Equilibrium for f = 1. With Cournot competition, price approaches marginal cost as the number of rivals increases (p® c as N® ¥).
At equilibrium market price p, equilibrium output per firm is qi = S/Np. Firm i’s profit, therefore, is
. (4)
Assuming S or market size is constant, profits realized are clearly dependent on the number of competitors, N, that enter the market and the intensity of price competition (f). For a fixed level of the intensity of price competition, equation (4) shows that as the number of firms increases, the equilibrium level of profit approaches zero. Alternatively, holding N constant, an increase in the market size, S, will tend to increase the equilibrium level of profits. As expected, the more intense is price competition (the lower is f), other things constant, the lower is firm profit. Note that the intensity of price competition can be viewed as scaling market size, with more price competition being (mathematically) equivalent to a smaller market size.
Given the entry decisions of other firms, firm i incurs Fixed F in stage 1 upon entry. The net profit of firm i is
{fS/(M + 1)2} – F (5)
where M is the number of other firms choosing to enter. Entry is profitable if the expression in equation (5) is positive. Entry continues in Stage 1 of the game until profits just equal the sunk cost of entry, so that the number of firms in equilibrium is the integer part of
(6)
where N* is the equilibrium number of firms in the industry and 1/N* is the equilibrium level of concentration. Because we have assumed all firms are identical, 1/N* also is equal to the Herfindahl-Hirschman Index. Note that the equilibrium number of firms N* is expressed as a function of market size (S), the level of sunk entry costs (k), and the intensity of price competition (f). In the Cournot case, f = 1 and .
Entry Costs
Equation (6) shows that the number of firms in equilibrium is inversely related to the level of set-up cost, F. If set-up costs are trivial, the number of firms in equilibrium will be arbitrarily large and the equilibrium level of profit will approach zero.
Market Size
An increase in the size of the market, S, relative to the level of set-up costs would result in a less concentrated (or more fragmented) market structure.
Price Competition
Within the context of the two-stage game with sunk entry costs, the more intense is price competition the higher is industry concentration.
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