Consider the utility functionu(x1, x2) = (x1 - 2)(x2 - 3),with accompanying budget constraint p1x1 + p2x2 = m.
1. Draw an indifference curve for this utility function.
2. Form the associated Lagrange function and find the associated first-order conditions for maximizing this utility function subject to the constraint.
3. Find the Bordered Hessian for this constrained maximization problem. State the sufficient second-order conditions and verify that they hold.
4. Solve the first-order conditions to find the demand function for good x1. Are goods x1 and x2 substitutes or complements? (To answer this question, take the appropriate derivative and see whether it is positive or negative.) Does consumption of x1 increase or decrease in income. (Again, answer with a derivative.)