Assignment:
Answer each part of each question. Show your work when appropriate. Partial credit will be awarded so attempt to answer each question, but please stay on topic with your answers.
1. Consider the market for dried beans in a small town of 9,000 consumers. Let each consumer's preferences over beans (B, in pounds) and other goods (C) be given by
U(B,G) = 12B1/2 +G
For the rest of this question, fix the price of other goods at PG = 1 and let each consumer have a total weekly budget of I = 100.
(a) Write the budget constraint for a consumer in terms of the price of beans, PB, and any other variables you need.
(b) What is the marginal rate of substitution between beans and other goods (∂G/∂B)?
(c) Use your answer to explain why each consumer's demand for beans is B* = 3∂/P2B.
(d) Would you describe beans as a normal good, an inferior good, or neither? Explain.
Suppose only 1,000 consumers show up to the store to buy beans in a typical week. Let the weekly supply of beans be perfectly inelastic at 9,000 as stores can't adjust their stock in such a short time period.
(e) What is the weekly market demand curve for beans when there are 1,000 consumers?
(f) Solve for the market-clearing price of beans.
(g) At this price, what is each consumer's optimal bundle? That is, how many pounds of beans does each consumer purchase, and how much do they spend on everything else?
(h) What is each consumer's utility in the market equilibrium?
An impending snow storm leads people to stock up on beans. Suppose all 9,000 consumers in this small town try to buy beans in the same week.
(i) What is the new market demand curve for beans?
(j) What is the competitive equilibrium price PB? Do markets clear at this price?
(k) Solve for the new optimal bundle and utility level for each consumer after the price change in a competitive equilibrium.
Anti-gouging laws are laws designed to protect consumers by limiting excessive price increases during a time of crisis Suppose the local government has an anti-gouging law that sets a price ceiling of PB = 3.
(l) What quantity of beans does each consumer demand at this price?
(m) Does the market for beans clear? If not, is there a surplus or a shortage? Explain.
(n) Suppose stores sell beans on a first-come-first-served basis. How many consumers will be able to buy their desired quantity before supply runs out?
(o) What is the utility level for these consumers with a price ceiling? What is their compen¬sating variation of the price ceiling relative to the snow-storm competitive equilibrium? That is, how much money would you have to give or take away to a consumer to get them to their old utility level at the new price? (Hint: Thy drawing a preference map. Note that IP doesn't change with income.)
(p) What is the utility level for the remaining consumers who cannot purchase beans? What is their compensating variation of the price ceiling relative to the snow-storm competitive equilibrium? (Hint: It is as though they are constrained by a ration of B = 0.)
(q) Given your answers to parts (o) and (p), how successful would you say the anti-gouging law was at protecting consumers?
Suppose stores decide to limit consumers to one pound of beans apiece to ensure that every consumer can purchase some. Let the price remain at the ceiling of PB = 3.
(r) What is the consumption bundle for each consumer under this ration? What level of utility does this consumption bundle deliver?
(s) Would you consider the ration to be a Pareto-improving policy among consumers relative to the first-come-first-serve allocation? Explain.
(t) Would you consider the ration with a price ceiling to be a Pareto-improving policy among consumers relative to the snow-storm competitive equilibrium? Explain.