Question

Chester consumes only bread (b) and cheese (c); his utility function is U(b,c) = bc. In Chester’s town, cheese is sold in an unusual way. The more cheese you buy, the higher the price you have to pay. In particular, c units of cheese cost Chester c2 dollars. Bread is sold in the usual way (i.e., at a constant price) and the per-unit price of bread is pb > 0. Chester’s income is m dollars.

(a) Write down Chester’s budget constraint for bread and cheese and graph this constraint. In your graph, place b on the vertical axis and c on the horizontal axis. Label any interesting points (intercepts, kinks, etc.). Note that your labels should be given in terms of m and pb.

(b) Setup the Lagrangian for Chester’s utility maximization problem and write down the first-order conditions.

(c) Recall that in the usual utility maximization problem, at an interior solution, the ab- solute value of the marginal rate of substitution is equal to the ratio of the prices. For Chester’s unusual budget constraint, find the analogous condition here. That is, at the optimal consumption bundle, |MRSc,b| is equal to what?

(d) Solve for Chester’s optimal consumption bundle. Your answers should be in terms of m and pb.

(e) Draw a picture that includes Chester’s budget constraint and an indifference curve, which illustrates Chester’s optimal consumption bundle.

Answer 1

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