Suppose that the demand function is Q= s/p, where Q is the total quantity demand

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

Ans a) Under the cournot competition the firms are choosing their quantities simultaneously.

Q = s/p ………………………………….. Market demand function

p = s/Q …………………………………..Inverse market demand function

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.

Ci = F + mqi .........................................Firm i’s cost function

where F be a firm's fixed cost, m be its constant marginal cost & qi is the firm i’s output.

Let there are n firms in the market which are engaged in cournot competition.

Let pi be the profit of ith firm in the market, the

pi = pqi - Ci

pi = [s/Q]qi - [F + mqi]

Total output can be expressed as Q = Q-i + qi where Q-i = Q – qi is the output of all firms except for firm i.

Pi = [sqi/( Q-i + qi)] - F - mqi

The first order condition with respect to qi :

pi/qi = [s(Q-i + qi) - sqi] / (Q-i + qi)2 – m = 0

There are n firms in the market & hence, n equations for n equilibrium quantities has to be solved. We know from cournot model of 2 firms that each firm in the end is producing the same output. Therefore, the Nash equilibrium involves equal quantities in n firms setup also because firms are symmetric. Symmetry implies that Q-i = Q – qi = n qi - qi = qi(n-1)

Now, equilibrium market price is:

p = s/Q

   = s/nqi*

  = s/n[(s/m) {(n-1) / n2}]

= mn/(n-1)

p* = m[1 + 1/(n-1)]   ……………………………………………(2)

Now, the profit of the ith firm is:

pi = p* qi* - F - m qi*

   = m[1 + 1/(n-1)] * (s/m) [(n-1) / n2] – F – m[(s/m) {(n-1) / n2}]

    = s/n – F – s(n-1)/n2

    = s/n – F – (s/n)[(n-1)/n]

    = (s/n)[1 – (n-1)/n] – F

    = (s/n)(1/n) – F

pi* = s/n2 – F ……………………………………………….(3)

Ans b) As n grows without bound, equations (1) – (3) gives us the perfectly competitive outcome. The output of each firm’s falls in the market, the market price falls to the constant marginal cost and hence each firm becomes the price taker. Moreover, the firm’s profits falls to zero and even below if the fixed cost is not recovered.

Ans c) As the size of the market (s) increases, there is an increase in the number of firms operating in the market .i.e. n increases and hence the concentration of the market (1/n) falls.

Ans d) As the size of the market (s) increases, the equilibrium size of the firm increases if the firm is operating at declining average cost or there is a restriction on the entry of new firms and the equilibrium size of the firm falls if there is no restrictions on the entry of new firms or the firm’s average cost is rising.

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