Kapitel 2: Consumption Decision

Question 1

Consumer’s problem

Answer 1

Question 2

Corner solution

Answer 2

I We say that the consumer’s problem is on a corner solution if the best feasible bundle contains zero amounts of some goods.
I If the marginal utility of expenditure on some goods is systematically larger than the marginal utility of expenditure on other goods, the latter will not be purchased in an optimal bundle.
I In this example, the consumer’s marginal willingness to pay for an additional unit of x1 in sacrificed units of x2 (MRS = |slope| of IC) is greater than the cost of x1 in units of x2 (|slope| of budget line).
I Therefore the consumer chooses x2 = 0 and spends all her income on x1 = M/p1.

Question 3

Lagrange method

Answer 3

foc = first order conditions

Question 4

Properties of the solution to the Lagrange problem

Answer 4

Question 5

Properties of the solution of the consumer’s optimization problem

Answer 5

I Sufficiency of FOC: It follows from the convexity of preferences that the utility function is quasi-concave. This guarantees that the FOC are not only necessary but also sufficient for a maximum.

I Uniqueness of the solution: Uniqueness requires “more convexity” than required so far. The solution to the consumer’s problem is unique if convex combinations of indifferent bundles are strictly better than any of the two bundles.

I Continuity of the solution: Continuity of u(x) and compactness of B(p, M) guarantees that that optimal consumption x
∗ changes continuously with prices and income.