Question

1. The inverse demand function for a good takes the constant elasticity form p(Q) = Qβ , −1 < β < 0, which is a commonly used simple functional form. The good is produced by n identical firms with a cost function c(qi) = cqi . Note that c 0 (qi) = c and c 00(qi) = 0; i.e., there are constant marginal costs. A specific tax of t per unit is imposed on the production of the good.

(a) Show that the (inverse) elasticity of demand is constant and equal to (dp/dQ)(Q/p) = β.

(b) Write down the profit maximizing problem for a representative firm i. Determine the first-order condition (the profit maximizing condition) for this firm by taking the derivative of profits wrt qi and setting it equal to zero. (NOTE: You do NOT impose symmetry at this point).

(c) Determine the symmetric Cournot-Nash equilibrium output of each firm by imposing symmetry on the profit maximizing condition: i.e., set qi = q(t) and solve for q(t), which will be a function of t. Note that Q = nq and p(Q) = Qβ

(d) Solve for the equilibrium price as a function of t and determine dp/dt. Is there over shifting in this case (i.e., does the equilibrium price paid by consumers increase by more than the increase in the tax)? What happens to dp/dt as n (an indicator of the “degree of competition”) increases, in particular what happens as n approaches infinity (which is the case of perfect competition where each firm is “atomistic” or “infinitely small” so as not to affect the market price)? Explain what is happening in the limiting case of perfect competition.

Answer 1

Solution:

Denoting beta by b for ease of writing. Inverse demand function: p(Q) = Q^b, -1 < b < 0

A firm's cost function: c(qi) = c*qi; so, marginal cost, c'(qi) = c and the double derivative of cost function: c''(qi) = 0

Specific tax of t per unit of production

(a) Inverse elasticity of demand, e = $(\partial p(Q)/\partial Q)*(Q/p)$

$\partial p(Q)/\partial Q$ = b*Q^b-1

So, e = (b*Q^b-1)*(Q/Q^b) = b*Q^b/Q^b = b, which is a constant. Hence, proved

(b) Profit maximizing problem for a representative firm: a firm i produces qi units of output, and so total output = $\sum_{i=1}^{n}qi$ = Q

Profits = total revenue - total cost

Total revenue, TR = price*quantity = p(Q)*qi

Total cost (with specific taxation on production) = c*qi + t*qi

So, Profit, L = Q^b*qi - c*qi - t*qi

And profit maximizing problem becomes:

maximize (Q^b*qi - c*qi - t*qi) such that $\sum_{i=1}^{n}qi$ = Q

First order condition: $\partial L/\partial qi$ = 0

$\partial L/\partial qi$ = b*Q^b-1*qi + Q^b - c - t

So, with FOC we have: b*Q^b-1*qi + Q^b - c - t = 0

(c) Using symmetry on profit maximizing condition, due to symmetry in cost and revenue functions, we know output produced by each firm is equal, so, qi = q(t) = Q/n (since Q =$\sum_{i=1}^{n}qi$ = n*q ( with n firms))

Then, from part (b), using the FOC, we have = b(n*q)^b-1*q + (n*q)^b - c*q - t*q = 0

So solving this, q^b = (c+t)/(n^b((b/n) + 1))

And, q = (1/n)*[(c+t)/(b/n + 1)]^1/b

This is the Cournot-Nash equilibrium.

(d) So, Q = n*q = n*((1/n)*[(c+t)/(b/n + 1)]^1/b) = [(c+t)/(b/n + 1)]^1/b

And, thus equilibrium price p(Q) = Q^b = ( [(c+t)/(b/n + 1)]^1/b)^b

p = (c+t)/(b/n+1)

And thus, $\partial p/\partial t$ = 1/(b/n+1) = n/(b + n)

We know that -1 < b < 0, so n - 1 < b+n < n

This implies that 1/(n-1) > n/(b+n) > 1

That is $\partial p/\partial t$ is greater than 1. So, clearly increase in tax, increases price by more than the increase in tax amount, so yes there is over shifting in this case.

Now, $\partial p/\partial t$ = 1/(b/n+1)

So, as n tends to infinity, (b/n) tends to 0, and so $\partial p/\partial t$ tends to 1. This means that change in tax increases the price by same amount as increase in tax, as number of firms become infinitely large.