Question

) A consumer maximises their utility by choosing how many t− shirts to buy and how many limited-edition sweaters to buy. If they buy m t-shirts and n sweaters, then their utility will be u(m,n)=m 51n54

. Each t-shirt costs 1 and each sweater costs p. The consumer aims to spend exactly 150 on these t-shirts and sweaters. Given the limited-edition nature of the sweaters, the store allows each shopper to buy a maximum of 5 sweaters. a) (4 points) Formally write the consumer's maximisation problem. b) (4 points) Write the problem's Lagrangian and first-order condition(s). c) (4 points) Identify the cases from complementary slackness, along with their associated equation and check. d) (6 points) Sketch the consumer's demand function for sweaters as a function of their price p, labelling any key points. Justify your answer mathematically. e) (2 points) Interpret your previous answers with an economic rationalisation.

Answer 1

The student's question involves setting up and solving a consumer utility maximization problem with constraints, determining the Lagrangian and first-order conditions, exploring complementary slackness, sketching the demand function for sweaters, and providing an economic rationale for the consumer's decisions.

Step-by-step explanation:

Formal Maximization Problem

The consumer's maximization problem is to choose values of m (number of t-shirts) and n (number of sweaters) to maximize their utility u(m,n) = m^0.51 * n^0.54, subject to the budget constraint 1*m + p*n = 150 and the constraint that n ≤ 5.

Lagrangian and First-Order Conditions

The Lagrangian for this problem is L = m^0.51 * n^0.54 + λ(150 - m - p*n), where λ is the Lagrange multiplier representing the shadow price of the budget constraint. The first-order conditions are derived by taking the partial derivatives of L with respect to m, n, and λ and setting them to zero.

Complementary Slackness

In cases of complementary slackness, either the slack variable is zero, or the corresponding dual variable (the Lagrange multiplier) is zero. This implies that if the sweater limit is binding (n=5), then the Lagrange multiplier associated with this constraint will be non-zero.

Consumer's Demand Function Sketch and Mathematical Justification

The consumer's demand function for sweaters as a function of price p needs to consider both the budget constraint and the utility maximization condition. Key points to label would be the maximum number of sweaters (n=5) and the point where the budget constraint is tangent to an indifference curve. Mathematical justification comes from solving the first-order conditions.

Economic Rationalization

The economic rationalization for the consumer's behavior is grounded in the principle of marginal utility. The consumer will purchase goods to the point where the last unit of currency spent on each good yields the same marginal utility per currency spent, which is the condition for maximization of utility.