Suppose a profit maximizing automobile manufacturer produces its output in two plants, one in the U.S. and the other in Canada. The total costs of producing in the two plants are identical, except that the output produced in the U.S. is subject to a per unit tax, t. Suppose the two total cost functions are
TCus=(Qus^2)/2 + Qus +1 + tQus .
TCcan=(Qcan^2)/2 + Qcan +1
The firm’s demand function is P=26-Qr , where Qr is total output in U.S. and Canada.
i) Find the first-order conditions for this problem.
ii) Find the reduced form solutions for optimum values of Q US and Q CAN .
iii) Show that the second-order condition for this problem is satisfied.
iv) By partially differentiating your reduced form solutions, describe (both mathematically and in words) the effect of a change in per unit tax on optimum output in both countries.
v) Now find the same comparative static results by totally differentiating your first-order conditions, rearranging them, writing them in a matrix form, etc.