1. Assume the price elasticity of demand for some good is εD = -2 and the price elasticity of supply is εS = 0.2.
a) Suppose one point on the demand curve is (P,Q) = (10000,200) and another is (P,Q) = (11000, ^QD). Further, suppose one point on the supply curve is (P, Q) = (50000, 100) and another is (P, Q) = (60000, ^QS ). Find values for ^QD and ^QS, and use them to find equations for demand and supply.
b) Find the equilibrium price and quantity using your answer from part (a). Decimal approximations are fine.
2. Suppose that, for a particular consumer, the utility from consuming goods a and b is given by
U(a,b) = (3a^1/3)(b^2/3)
so MUa = (a^-2/3)(b^2/3) and MUb = (2a^1/3)(b^-1/3)
a) If Pa = 1 and Pb = 1, calculate the utility-maximizing bundle when the consumer's income is $6. How much utility does the consumer get from this optimal bundle?
b) If the price of b decreases to Pb = 0.5 but Pa and income remain the same as in part (a), find the new utility-maximizing bundle. What is the new level of utility achieved?
c) Find the income and substitution effects resulting from this change in the price of good b. Because this problem and the previous problems involve cubic roots, decimal approximations are fine.
d) Find the compensating variation and equivalent variation for this price change.
3. A firm's production function is given by
Q(K, L) = (4K^3/4)(L^1/4)
which implies MPL = (K^3/4)(L^-3/4) and MPK = (3K^-1/4)(L^1/4). Initially, the per-unit cost of labor is w = 1 and the per-unit cost of capital is r = 1. Then, the cost of capital increases to r = 3.
a) Find the cost-minimizing amount of labor and capital required to produce
Q = 1000 units of output both before and after the increase in the cost of capital. Decimal approximations are fine.
b) Use your results from (a) to find the firm's price elasticity of demand for capital between r = 1 and r = 3. Is the firm's demand for capital elastic or inelastic?
4. Suppose that, in a perfectly competitive market consisting of 2000 identical firms, the demand curve is
Q^D =1200-2P
Suppose all costs are sunk so each firm has the short-run total cost curve
ST C(Q) = 100Q^2
which implies the short-run marginal cost curve for each firm is given by
SMC(Q) = 200Q
Further, the STC curve implies each firm's average variable cost curve is
AV C(Q) = 100Q
Note that AV C is minimized when Q = 0, which means the firms will continue to
produce as long as P > 0.
a) Calculate the short-run equilibrium price in the market.
b) How much output will each firm produce individually? How much output will be produced by all firms combined?
c) Find the short-run profit per firm. Based on your answer, will the long-run equilibrium have more than, less than, or exactly 2000 firms?
5. Let the inverse demand in a market be given by
P(Q) = 10 - 4Q
and let the inverse supply curve be given by
P(Q) = Q
a) Find the equilibrium price and quantity in this market.
b) Find the producer surplus and consumer surplus.
c) Suppose now the government imposes a $1 tax on suppliers so the new inverse supply curve is
P = 1 + Q
Answer the following. (Hint: Refer back to the chapter 10 notes to make sure you know how to visualize this problem)
i. Looking only at the slopes of the supply and demand curve and remembering what they imply about elasticity, explain who will have the greater tax burden (tax incidence).
ii. What price will consumers pay for the good?
iii. What price will producers receive?
iv. What is the new consumer surplus and producer surplus?
v. How much will be collected in taxes?
vi. What is the deadweight loss? You can check that you have the correct values for parts (iii) through (v) by adding them together. They should sum to the same amount as PS + CS in part (b).
6. Consider a monopoly that faces the demand curve P = 20 - Q, and has the marginal cost curve MC = 2.
a) Use the demand curve to find the equation of the marginal revenue curve. (Hint: Use the same formula you used to find marginal revenue in the quiz.)
b) Find the profit-maximizing price and quantity for this monopoly if the monopoly uses uniform pricing. What is the producer surplus?
c) Now suppose the monopoly wants to increase profits using block pricing. The total cost the monopoly incurs is TC = 2Q. Use the steps outlined in the notes to find the optimal quantities, Q1 and Q2, and their corresponding optimal prices, P1 and P2 that maximize profits using a two-block pricing scheme. (Hint: You don't necessarily need to use a calculator or computer like you did in the quiz, but you can if you want.) What is the new producer surplus of the monopoly?
7. Suppose the inverse demand for an agricultural fertilizer is given by P (Q) = 48 - Q, and the inverse supply curve (the marginal private cost curve) is MPC(Q) = 4 + Q.Assume that the firm emits one unit of pollution for every unit of fertilizer produced. This pollution doesn't bother those around the firm as long as there are less than 4 units of pollutants emitted. When 4 or more units of pollution are emitted, the marginal external cost is
MEC = -4 + Q. The government wants to place a tax (emission fee) of $t per unit of pollution so that the market equilibrium is socially efficient. Using the graphs and tables in the notes above as a guide, perform the following. Draw a graph and make a table that compares the market equilibrium (with no emission fee) and the socially-optimal equilibrium:
a) Graph supply (with no emission fee), demand, MEC, and MSC = MPC + MEC. Label the point that corresponds with the equilibrium price and quantity of the fertilizer when there is no emission fee. Also, label the point that represents the amount of the fertilizer the firm will supply in a socially-optimal equilibrium.
b) Graph the new supply curve after the government imposes an emission fee that leads to the socially-optimal equilibrium. Indicate the price consumers will pay and the price that the firm will receive. Indicate the emission fee (i.e. value of t) that leads to the socially-optimal equilibrium.
c) Calculate the following for the case without the emission fee and with the emission fee: consumer surplus, private producer surplus, externality cost (the cost of pollution to society), government revenue, net social benefits, and deadweight loss.