Let Q(p,) be a downward-sloping demand curve, let be a parameter of the demand curve, and suppose that Q is differentiable in p and strictly positive in the relevant region of prices. (In this question, it might be helpful to note that maximizing a strictly positive function f is equivalent to maximizing log f).

a) Suppose the firm solves maxp(pc)Q(p,) . Derive a sufficient condition for p to increase (weakly) with .

b) It is often assumed that the demand function is concave in order to validate the approach based on first-order conditions. Show that concave demand is sufficient to ensure that the profit function (from part a) is concave. Can you give an economic justification for concave demand? Is it necessary for the comparative statics prediction?