Suppose a firm's marginal cost function is given by MC(q) = q+1, where q is the quantity produced. Problem 4.1 We will prove later that a profit maximizing firm facing market price p will choose the quantity q such that MC(q ) = p. Solve q for this firm when p = 15. Problem 4.2 Solve for the profit-maximizing quantity for any price p. In other words, derive the implicit function q (p) that solves p = MC(q (p)).

Problem 5 (2.10) Consider the Cobb-Douglass function f(x1, x2) = (x1) (x2) , where and are both constants strictly between 0 and 1. Problem 5.1 Pick any values for and such that + < 1, and pick a number c. Then graph the contour curve for f(x1, x2) = c. Try it by plotting 10 different (x1, x2) points that solve f(x1, x2) = c, and then connecting them with a smooth curve. Problem 5.2 Do the previous problem for two different values of and , but this time choose them so that + > 1