Problem Set

Q1. The market for paper in a particular region is characterized by demand function Q^D = 160000 - 2000P and supply function Q^S = 40000 + 2000P. Waste products from the production of paper are discharged into streams and rivers. The marginal external cost associated with the production of paper is mc^ext = .0006Q.

(a) Calculate the output and price of paper if it is produced under competitive conditions and no attempt is made to regulate the dumping of effluent.

(b) Calculate the external costs associated with the competitive outcome from (a).

(c) Determine the socially efficient output of paper, Q^so.

(d) A proposal is made to tax the sale of paper in order to bring about the socially efficient level of production. Determine the size of the specific tax on paper, t that will correct the externality.

(e) Determine the size of the ad valorem tax, τ that will correct the externality.

Q2. The marginal damages associated with an air pollutant are MD = 3/5 E. A single firm is the sole source of the pollutant. The marginal abatement costs of the firm are MAC = 200 - 2/5 E.

(a) Determine the socially optimal level of emissions for the firm and describe both (i) an Emission Standard, E^{^} and (ii) an Emission Tax, tE that will bring about the social optimum.

(b) Suppose that if the firm undertakes research and development there is a very high likelihood that a new process/technology will permit the firm to lower emissions and lower the cost of abating emissions. The marginal abatement costs with the new technology in place will be MAC' = 180 - 2/5 E. If the regulatory regime is an Emission Standard, E^{^} (as specified in a:(i) above), determine how much the firm might be willing to invest in order to develop the new process.

(c) If the regulatory regime is an Emission Tax, t_E (as specified in a:(ii) above), determine how much the firm might be willing to invest in order to develop the new process. Explain why there might be a difference under the two regimes if they both achieve the same (socially optimal) level of emissions.

(d) Suppose that the abatement costs are MAC = 200 - 2/5 E (no other abatement technology exists). MD^est = 3/5 E is the regulator's estimate of the marginal damages function. Calculate the Social loss that results from (i) an Emission Standard, E^{^} and (ii) an Emission Tax, tE that are based on MD^est when the true marginal damages are in fact MD^true = ¾ E.

Q3. Consider an exhaustible resource with total supply Q^- = 500. The social discount rate/opportunity cost of funds is r = 10%. The marginal cost of extracting the resource is constant mc^ext = $20.

(a) If the demand in each period is perfectly inelastic at a price P ≤ $100 at a quantity q = 50 each period determine number of years to exhaustion, P_T and the price of the resource at t = 0, P₀.

(b) Determine the rate of growth of the price, g. It may be helpful to use the following relations: P_T = P₀(1 + g)^T so that

P_T = P₀(1 + g)^T

P_T/P₀ = (1 + g)^T

T ln(1 + g) = ln P_T - ln P₀

ln(1 + g) = (ln P_T - ln P₀)/T

1 + g = exp(ln P_T - ln P₀/T)

(c) Suppose that the price grows at the rate g but just before price is set at t = 5 a renewable backstop technology becomes available at price P_b = 80. This alters the demand for the exhaustible resource so that it is now perfectly inelastic at q = 50 for any P ≤ P_b = 80. Explain what will happen to the price of the exhaustible resource at t = 5.

(d) Suppose that demand for the exhaustible resource is not perfectly inelastic. The demand is downward-sloping Q = 100 - P. Use a 4-quadrant diagram to qualitatively show how the price and extraction paths of the resource for the cases (i) no backstop technology and (ii) renewable backstop technology at P_b = 80.

Q4. For a particular fishery the quantity of fish that are caught can be represented by the function:

q = 20h - h²

where q is the quantity and h is the effort level (number of boats). The marginal catch (rate of change of total catch as h changes) is described by the function:

mq = dq/dh = 20 - 2h

The price/value of a unit of catch is constant, p = $100. The cost of operating a boat (cost of effort) is constant, c = $500.

(a) Write an expression for the Value of the landed catch as a function of the effort (number of boats).

(b) Write an expression for the Total Cost as a function of the effort (number of boats).

(c) At what level of effort (number of boats) is the gross value of the catch maximized? What is the net value after accounting for the costs related to the fishing fleet?

(d) If the fishery resource is treated as a common property/access resource, how many boats will operate in the fishery? What is the gross value of the catch? What is the net value of the catch after accounting for the costs related to the fishing fleet?

(e) At what level of effort will the net value of the catch be maximized? If the fisheries manager wishes to maximize the net value of the fishery how many boats should be licenced to operate in the fishery?