Question

3. Suppose a consumer's utility function is given by U(A, B) In(A)+In(B). Suppose the price of each apple (A) is €6, and the price of a loaf of bread (B) Is €6 and the consumer's income is €120 ) Write down the Lagrangian for this problem and solve for the optimal consumption of apples and (ii) Report and interpret your solution for the Lagrange multiplier. bread. i) Evaluate the marginal utility of bread and the marginal utility of apples at the optimal solution. Do these marginal utilities seem reasonable? (iv) Evaluate the marginal rate of substitution at the optimal levels of consumption. Does this marginal rate of substitution seem reasonable?

Answer 1

i. The Lagrangian is, L- InA+InB - [120- 64-6B] Differentiating, OL CA aL OB =120-64-6B-0 (3) Putting (1) and (2) into equation (3) 120-6*1/62-6*1 From (1) and (2), A-10 and B-10 .-0→120 _ 2 // = 0 λ = 1 i. For a constant optimal Lagrangelmultiplier-6o the utility is maximized for the budget constraint, 120 64+6B Marginal utility of apple is, Marginal utility of bread is, 1/B So for both apple and bread one additional unit gives less utility and the amount of extra utility is inverse of the unit of apple or bread consumed iv. Marginal rate of substitution is, L -1 where is the slope of the budgetconstraint iii. MU6/ So the slope of budget constraint is negative and reasonable and a 45 degree negative sloping straight line