4. Use the Kuhn-Tucker conditions to derive demand at general prices and wealth levels (i.e. pi , w should show up in your demand function) for the following utility maximization problems and discuss whether the Kuhn-Tucker sufficiency conditions are satisfied?

(a) u(x1, x2) = ln(x1 + 1) +ln(x2 + 1) subject to p1x1 +p2x2 m, x1 0, x2 0 where p1 > 0, p2 > 0 and m > 0.

(b) u(x1, x2) = x 2 1 + (x2 5)2 s.t. x1 0, x2 0, and 2x1 + x2 4 where p1 > 0, p2 > 0 and m > 0.

(c) u(x1, x2) = x1x2 subject to x1 + x2 100, x1 40, x1 0, x2 0 where p1 > 0, p2 > 0 and m > 0.