Question

1. Consider a utility-maximizing price-taking consumer in a two good world. Denote her budget constraint by p1x1 + p2x2 = w, p1, p2, w > 0, x1, x2 ≥ 0 (1) and suppose her utility function is u (x1, x2) = 2x 1/2 1 + x2. (2) Since her budget set is compact and her utility function is continuous, the Extreme Value Theorem tells us there is at least one solution to this optimization problem. In fact, demand functions, xi(p1, p2, w), i = 1, 2, exist for this example.

(i) Use Lagrange’s Method to find the consumer’s demand functions when neither x1 nor x2 equals zero. Label this solution type interior and identify the inequality involving p1, p2 and w under which it applies.

(ii) Look at your answer to (i) and identify the corner (or boundary) solution and the inequality involving p1, p2 and w under which it applies. (iii) Interpret the Lagrange multiplier in the interior and corner solutions.

Answer 1