Question: 1 the inverse demand function for a good takes the...
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.