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Each individual consumer takes the prices as given and chooses her consumption bundle, (x1, x2) R 2 +, by maximizing the utility function U(x1, x2) =...

Each individual consumer takes the prices as given and chooses her consumption bundle, (x1, x2) ∈ R 2 +, by maximizing the utility function U(x1, x2) = ln(x1 · x2), subject to the budget constraint p1 · x1 + p2 · x2 = 1000.

(a) (3 points) Write out the Lagrangian function for the consumer's problem.

(b) (6 points) Write out the system of first-order conditions for the consumer's problem.

(c) (6 points) Solve the system of first-order conditions to find the optimal values of x1 and x2. Your answer might depend on p1 and p2.

(d) (Voluntary, 0 points) Check that the critical point satisfies the second-order condition. 

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