Constrained Optimization; Derivation of Demand Functions
1. Find the bundle (x1; x2) that maximizes the function f(x1; x2) = x1x2 subject to x1 + x2 = 2. 2. Find the bundle (x1; x2) that maximizes the function f(x1; x2) = 3x1x 3 2 subject to p1x1 + p2x2 = I, where p1 = 5, p2 = 25, and I = 100. 3. Consider a consumer whose utility function is U(x1; x2) = 2p x1x2 with an income of $Y , where the price of good 1 is p1 and the price of good 2 is p2. Find the utility maximizing consumption bundle. 4. A consumer maximizes the utility function ln(x1) + ln(x2) + ln(x3) subject to the budget constaint p1x1 + p2x2 + p3x3 = I. Solve for the optimal consumption of x1, x2, and x3.
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