Suppose that a consumer wants to maximize his happiness as expressed by utility function u(c1, c2). Her income is M and the prices of goods 1 and 2 are p1 and p2, respectively. The consumer faces non-negativity constraints.
1) Write down the problem of the consumer and the first order necessary conditions for c1* , c2∗ and λ∗ .Suppose that u(c1, c2) = log c1 + A log c2. Use the first-order conditions found above to find optimal c1∗ , c2∗ as functions of p1, p2 and M. (USE Kuhn-Tucker Theorem)
2) Now suppose that u(c1, c2) = log(c1) +c2. Also, suppose that the consumer faces prices and income (p1, p2, m) = (1, 5, 3). The consumer is faced with a budget constraint and two non-negativity constraints, one for each good. Solve for c1∗and c2∗. Hint: The utility function guarantees that the demand for one of the goods is strictly positive. What does this imply about its multiplier?