Question

Consider the utility function U(x1,x2)=(x1−2)(x2−3), again with the budget constraint p1x1+p2x2=I

(A) Draw an indifference curve for this utility function. Form the associated Lagrangian and find the associated first-order conditions for maximizing this utility function subject to the constraint.
(B) 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.)

Answer 1

The utility-maximizing choice is found by forming a Lagrangian and solving the first-order conditions to obtain the demand function for good x1. By analyzing derivatives, we can assess whether goods are substitutes or complements and how consumption varies with income.

Step-by-step explanation:

To find the utility-maximizing choice along a budget constraint, we form a Lagrangian: L = U(x1, x2) + λ(I - p1x1 - p2x2), where U(x1, x2) = (x1 - 2)(x2 - 3), I is income, p1 and p2 are prices, and λ is the Lagrange multiplier.

The first-order conditions (FOCs) are obtained by taking derivatives of L with respect to x1, x2, and λ, and setting them to zero, giving us the system of equations to solve for the optimal quantities of x1 and x2. Solving these equations with respect to x1, we get the demand function for good x1.

By analyzing the cross-price derivative of the demand function, we can determine if goods x1 and x2 are substitutes or complements. If the derivative is positive, they are substitutes; if negative, they are complements. Similarly, by looking at the income derivative, we can tell if consumption of x1 increases or decreases with income. A positive derivative implies that consumption increases with higher income.