Problem 2 (A Review of Microeconomics, 30%) Consider the following optimization prob- lem: max u (x) + Ou(y) x,y s.t. Pxx + Py y = Pxx + Pyy, where x and y are endowments, and 0 e (0, 1) is a constant. 1. Derive the first order condition via the method of Lagrange multipliers. 2. Derive the first order condition via an unconstrained optimization problem. 3. Derive the first order condition via intuition. 4. Now suppose that u(x) = log x, and u(y) = logy, where log denotes natural logarithm. Solve for the optimal choices x* and y*. 5. Determine if 2x* zo? Opx In this case, does the substitution effect or income effect dominate? 6. Determine if dx * zo? apy In this case, does the substitution effect or income effect dominate