Question 1:

In a perfectly competitive market for coal, consumers' benefit function from consuming coal Q, is given by:

B(Q) = - 0.25Q² + 240Q.

In addition, the coal producer has a cost function given by:

C(Q) = 0.1Q² + 2Q.

Suppose the government imposes an ad valorem tax on coal producers of 7% of the sale price.

a. What is the competitive equilibrium in the absence of the tax?

b. What is the competitive equilibrium with the ad valorem tax?

c. What is the change in consumer surplus as a result of the tax?

d. What is the change in producer surplus as a result of the tax?

e. How much revenue is generated by the tax?

f. What is the deadweight loss from the tax?

g. What is the change in total welfare due to the tax?

Question 2:

Suppose the demand for (blended) fuel F, can be given by the following demand function:

P_F = 200 - 4F.

Fuel retailers are indifferent between using corn derived ethanol E, and crude oil derived gasoline G, in the fuel they sell to consumers. To produce blended fuel, assume ethanol is splash blended in tanker trucks at no additional cost and there are no limits to how much ethanol can be blended into fuel. Thus fuel retailers are willing to supply fuel according to:

F = E + G.

Assume that fuel retailers are the only demanders for ethanol and gasoline. 

Ethanol supply is given by:

P_E = 1.45 + 6E.

Gasoline supply is given by:

P_G = 1.45 + 6E.

For simplicity suppose fuel, ethanol, and gasoline are all denominated in the above in energy equivalent gallons of gasoline (hearafter, simply "gallons"). Suppose all of the gasoline that is supplied came from crude oil imported from abroad. 

Suppose corn growers in Iowa convince the government to subsidize ethanol by $4.50/gallon. 

Relative to the no subsidy, competitive market equilibrium:

a. What is the change in producer surplus to ethanol producers due to the subsidy?

b. What is the change in producer surplus to gasoline producers due to the subsidy?

c. What is the change in fuel demander's consumer surplus due to the subsidy?

d. What is the final bill to taxpayers for the subsidy?

e. What is the change in total welfare due to the subsidy?

f. Iowa corn growers argued that the subsidy would help "reduce our dependence on foreign oil." This is the same thing as saying that they believe there is an additional societal benefit for each gallon of gasoline displaced by the subsidy relative to the competitive market equilibrium, given our prior assumptions. Suppose this is the case. Is the ethanol subsidy a good way to achieve this goal? Why or why not? Explain your reasoning.

g. How large would the benefits from reducing oil dependency (per gallon of gasoline displaced) need to be to just justify the welfare loss due to the subsidy?

Question 3:

Price taking consumers' receive a benefit from consuming electricity Q, according to the following function:

B(Q) = -Q² + 25000Q.

A single vertically integrated utility (VIU) has a cost function given by:

C(Q) = 0.4Q² + 50Q +306250

The VIU has convinced the public utility commission (PUC) that it is a natural monopoly and thus it has been granted an exclusive franchise to produce electricity. 

At the most recent rate-hearing, the PUC established a target price based on rate of return regulation. During the hearing, the VIU claimed that its rate-base was $4,875,000 and that the market rate of return it should be allowed should be 10%. It provided receipts to the PUC showing expenses of $262,500. Suppose the PUC has complete knowledge of market demand.

a. What is the competitive equilibrium in the absence of rate of return regulation?

b. What is the monopoly equilibrium in the absence of rate of return regulation?

c. What is the target price set as a result of the regulatory hearing? Use the quantity from part a to calculate.

d. What is the monopoly equilibrium under the rate of return regulation?

e. What is the change in consumer surplus from the rate of return regulation relative to the competitive equilibrium? Relative to the no regulation monopoly equilibrium?

f. What is the change in producer surplus from the rate of return regulation relative to the competitive equilibrium? Relative to the no regulation monopoly equilibrium?

g. What is the change in total welfare from the rate of return regulation relative to the competitive equilibrium? Relative to the no regulation monopoly equilibrium?

h. The government provided the monopoly franchise under the assumption that the VIU was a natural monopoly. Was their assessment correct? Did society gain or lose because of the government's decision? How much did they gain or lose? Please provide detailed reasoning for all three answers.

Question 4:

Consider a firm with average costs of producing electricity Q, given as follows:

ATC(Q) = 1000/Q + 100 if Q ≤ 1000.

110, if Q > 1000

A producer with this type of cost function is a firm that has a temporary natural monopoly (decreasing average total costs) for 0 ≤ Q ≤ 1000. After this, for Q> 1000, the firm has constant average total costs and thus is no longer a natural monopolist. In this case, the latter results because in the long run there is technological change that occurs to render the firm to no longer be a natural monopoly producer.

Suppose that in the present day demand is given by:

P = 600 - 2.25Q.

In the future, let demand be given by:

P = 5000-2.25Q

a. In the present day, identify the monopoly equilibrium in the absence of regulation.

b. In the present day, suppose a regulator imposes an average cost price ceiling and grants the firm an exclusive franchise to produce electricity. Identify the price ceiling and the resulting post-policy equilibrium in the present day. What is the change in consumer surplus relative to the monopoly equilibrium? What is the change in producer surplus? What is the change in total welfare?

c. Suppose the price ceiling and franchise identified above remain in effect until at least the future period. What is the post-policy equilibrium in the future?

d. What is the competitive equilibrium in the future? What is the change in consumer surplus relative to the future post-policy equilibrium? What is the change in producer surplus? What is the change in total welfare?

e. Given the results thus far, and under the assumption of no discounting, if the government had perfect foresight, should the government have intervened? Explain your reasoning.

Question 5:

Let X be the amount of electricity supplied to residential customers and Y be the amount of electricity supplied to industrial customers. Electricity producers can produce X alone, Y alone, or both, according to the following cost functions:

Cx = 750 + 12X,

CY = 900 + 12Y, and

C_XY = 1000 + 12X + 12Y.

Historically, the Tesla Electric Utility has individually supplied both customer classes, although other producers are not restricted from entering the market. During that time residential customers have accounted for of the total kilowatt-hours demanded. Tesla is wary of being brought before the state public utility commission for undue discrimination and has made the case that historical demand is a reasonable method for allocating common costs using fully distributed cost (FDC) pricing. The public utility commission has agreed. Residential customers have demand given by:

P_x = 30 - 0.25X,

and industrial customers have demand given by:

P_Y = 150 - 2Y.

Because Tesla is a regulated utility prices in both markets are determined by average cost pricing. Assume that fixed charges are not available.

a. What are the prices and quantities achieved in both the X and Y markets in equilibrium under FDC prices?

b. What is the total deadweight loss under FDC pricing?

c. Is the price for X under FDC pricing subsidy-free? Is the price for Y under FDC pricing subsidy-free? Given that the public utility commission would like to avoid incentivizing new entrants into this market, did the commission make the correct choice in permitting the current FDC pricing scheme?

d. Using Excel, across various values of P_X and P_Y, identify the second-best Ramsey prices. Report them and the total deadweight costs under Ramsey pricing. Do the Ramsey prices make sense? Are they subsidy-free (show mathematically)?

Question 6

In class, we discussed how firms that have decreasing average total costs can be classified as natural monopolies. The more precise definition of a natural monopoly is any industry that exhibits subadditive average total costs. Subadditivity refers to whether it is cheaper to have one firm produce total industry output or whether additional firms would yield lower total cost. Subadditivity is a weaker requirement than decreasing average total costs. Any firm that has decreasing average total costs also has subadditive average total costs (thus decreasing average total costs is a sufficient condition for subadditive average total costs), but some firms that have subadditive average total costs need not have decreasing average total costs (thus decreasing average total costs is not a necessary condition for subadditive average total costs).

Suppose output can only be added to an industry in lumpy amounts. That is, a single factory can be built according to the following average total cost function:

ATC(Q) = 0.02Q² - 2Q +70,

And two factories according to the following average total cost function:

ATC(Q) = 0.02Q² - 2Q + 220,

a. Suppose demand is given by: P = 300 - 7Q. Should one factory supply the market or two? Explain your reasoning.

b. Suppose demand is given by: P = 250 - 3Q. Should one factory supply the market or two? Explain your reasoning.

c. Suppose demand is given by: P = 200 - 2Q. Should one factory supply the market or two? Explain your reasoning.

d. Up to what quantity does the first factory exhibit increasing returns to scale? Constant returns to scale? Decreasing returns to scale?

e. Up to what quantity does the second factory exhibit increasing returns to scale? Constant returns to scale? Decreasing returns to scale?

For the decreasing average total cost industry, the firm exhibited increasing returns to scale for all output levels. This is not the case here, in which production follows the standard three stages of production, although new production can only be added in a lumpy way. Up to what quantity should one factory produce? Two factories produce? Explain your reasoning.