1. Consider the demand function x = αm/p that emerges from a Cobb-Douglas tastes.
a. Derive the income elasticity of demand and explain its sign.
b. We know that Cobb-Douglas tastes are homothetic. In what way is your answer to (a) simply a property of homothetic tastes?
c. What is the cross-price elasticity of demand? Can you make sense of that?
d. Without knowing the precise functional form that can describe tastes that are quasilinear in x, how can you show that the income elasticity of demand must be zero?
e. Consider the demand function x1(p1, p2) = (αp2/p1)β. Derive the income and cross-price elasticity of demand.
f. Can you tell whether the tastes giving rise to this demand function are either quasilinear or homothetic?
2. Suppose demand and supply are given by xd = (A - p)/α and xs = (p - B)/β (assume that demand is equal to marginal willingness to pay.
a. Derive the equilibrium price p* that would emerge in the absence of any intervention.
b. Suppose the government imposes a price ceiling pc that lies below p*. Derive an expression for the disequilibrium shortage.
c. What is the deadweight loss from the price ceiling?
3. Suppose that labor demand is given by ld = (A/w)α and labor supply is given by ls =(Bw)β.
a. What is the wage elasticity of labor demand and labor supply?
b. What is the equilibrium wage in the absence of ant distortions?
c. What is the equilibrium labor employment in the absence of any distortions?
d. Suppose A = 24,500, B = 500, and α = β = 1. Determine the equilibrium wage* and labor employment l*
e. Suppose that a minimum wage of $10 is imposed. What is the new employment level lA and the size of the drop in employment (l*- lA)?
4. Suppose the domestic demand curve for bushels of corn is given by p =24 – 22.5(10)-10 x while the domestic supply curve is given by p = 1+2.5(10)-10
x. Suppose there is no income effect to worry about.
a. Calculate the equilibrium price p* (in the absence of any government interference). Assume henceforth that this is also the world price for a bushel of corn.
b. What is the quantity of corn produced and consumed domestically? (Note: the price per bushel and quantity produced is roughly equal o what is produced and consumed in the United States in an average year).
c. How much is the total social cost (consumer and producer) surplus in the domestic corn market?
d. Next, suppose the government imposed a price floor of pf = 3.5 per bushel of corn. What is the disequilibrium shortage or surplus of corn?
e. In the absence any other government program, what is the highest possible surplus after the price floor is imposed and what does this imply about the smallest possible size of the deadweight loss?
f. Suppose next that the government purchases any amount that corn producers are willing to sell at the price floor pf but cannot sell to domestic consumers. How much does the government have to pay?
g. What happens to consumer surplus? What about producer surplus?
h. What happens to total surplus assuming the government sells the consumer it buys on the world market at the price p*?
i. How much does deadweight loss jump under just the price floor as well as when the government purchasing program is added if pf = 4 instead of 3.5? What if it is 5?
5. Suppose your tastes over current consumption c1 and future consumption c2 can be modeled through the utility function u(c1, c2) = c1
αc2(1-α), your current income is m, and you will earn no income in the future. The real interest rate from this period to the future is r.
a. Derive your demand functions c1(r, m) and c2(r, m) for current and future consumption.
b. Define "saving" as the difference between current income and current consumption. Derive your saving - or capital supply- function ks(r, m). (Note: It turns out that this function is not actually a function of r).
c. Derive the indirect utility function V(r, m); that is, the function that gives us your utility for any combination of (r, m).
d. Next, derive your compensated demand functions c1c(r, u) and c2c(r, u) for current and future consumption.
e. Define the expenditure function E(r, u): that is, the function that tells us the current income necessary for you to reach utility level u at interest rate r.
f. Can you compare your answers by comparing V(r, m) with E(r, u)?
6. Suppose the market demand curve for good Z is given by Z =8Pz-2 Where Z = market quantity demanded for good Z in units per time period Pz = price of good Z in dollars per unit
a. Determine the value of the own price elasticity of demand for good Z.
b. Compute the marginal revenue function using MRz = Pz[1 - 1/ε), where ε is the price elasticity of demand.
c. Determine the value of marginal revenue when the price of good Z is $30.
d. At what price will marginal revenue equal $100?
7. Do problems 14.1, 14.3, 14.6, 14.7, 14.9, 15.3, 15.4, 15.5, 15.7 and 15.9 from the Workouts in Intermediate Microeconomics by Bergstrom and Varian. (Write your answers in separate sheets (not on the pages of the Workouts) and show all necessary steps. You will not get any points for just writing the final answer(s) - you need to show how you got the answer(s))