ECO 345 HOTELLING (ARAMCO), COASE, CARBON TAX
HW3, GROUP ASSIGNMENT.
1. Aramco (the Saudi company headquartered in Dhahran) has 30 barrels of oil. The marginal cost of extracting a barrel is given by MC(q) = 3q. This means that 3 is the cost of extracting unit number 1, and 6 is the cost of extracting unit number 2, and 9 is the cost of extracting unit number 3. Assume that we live in a world of two periods.
NOTE: Without loss of generality we suppose that the fixed cost = 0.
NOTE: optimization rule: p1 — MC(q1) = (p2 — MC(q2))/(1 + r)
(i) How would Aramco sell those 30 units if the price p = 100 in every period and the interest r=0.
(ii) How would Aramco sell those 30 units if the price p = 100 in every period and the interest r=0.05 (five percent). What about r=0.1?
(iii) How would Aramco sell those 30 units if the price p = 100 in period 1 and p = 105 in period 2, and the interest r=0.05 (five percent).
2. (Question 4 from Varian). Suppose that a scarce resource, facing a constant demand, will be exhausted in 10 years. If an alternative resource will be available at a price of $40 and if the interest rate is 10%, what must the price of the scarce resource be today?
3. (The Coase Theorem). Suppose that the rancher has 100 units of cattle, and the farmer is laboring 50 acres of land. Let MB(h) = 200 -40h be the marginal benefit for the h th hour in which the cattle is allowed to be on the field, and let MC(h) = 10 h to be the marginal cost to the farmer. We are considering two scenarios: (i) The rancher has the responsibility to control the cattle. The farmer will not allow the cattle to graze anywhere within the pasture, and (ii) The rancher can let his cattle roam free, and hence he is not liable for the damage on the farmer’s crops. Please, answer the following questions:
(a) Number of hours that the cattle will stay on the field under scenario (i).
(b) Number of hours that the cattle will stay on the field under scenario (ii).
(c) Socially optimal outcome. (i.e., socially optimal number of hours for the cattle to stay on the field).
(d) How can you apply the Coase theorem to achieve the optimal outcome?
(e) Assume that we can have constant taxes. Calculate the optimal tax per hour under scenario (ii).
(f) Assume that we can have constant subsidies. Calculate the optimal subsidy per hour under scenario (i).
4. The marginal cost of water pollution on a river is 3Q, where Q measures pollution. The marginal benefit of polluting is 30-3Q.
a. Draw a picture that represents the “market” for water pollution. Calculate and label the amount of pollution that will occur if there are no regulations on pollution and negotiation costs are high.
b. What is the welfare-maximizing amount of pollution? What is the total welfare at that outcome?
c. If the producers of pollution own the right to pollute, explain one way that the market could reach the efficient outcome if negotiation costs are low.
d. If those negatively impacted by pollution own the right to clean water, explain one way that the market could still reach the efficient outcome if negotiation costs are low.
5. A mill company can sell flour at p= 20 cents a pound. The MC(y) = 2 y + g(y) where g(y) is a cost of pollution imposed on the environment. More precisely, this mill company is polluting a river, and g(y) is the marginal cost of the waste for the yth unit.
(a) Compute the profit maximizing output.
(b) Suppose that g(y)=2, calculate the socially optimal production level.
(c) Suppose that g(y) =3y, calculate the optimal socially optimal production level.
(d) How your answers to (b) and (c) can be achieved with a carbon tax.
(i) If the price is constant at $100 per barrel in every period and the interest rate is 0%, Aramco would sell all 30 barrels in either period since there is no incentive to conserve for future periods.
(ii) If the price is $100 per barrel and the interest rate is 5%, Aramco would sell 20 barrels in period 1 and 10 barrels in period 2 to maximize profits based on the extraction cost function MC(q) = 3q. At r=10%, they would sell 22 barrels in period 1 and 8 in period 2.
(iii) If the price is $100 in period 1 and $105 in period 2 with r=5%, Aramco would sell 18 barrels in period 1 and 12 in period 2 (Hotelling’s rule).
Applying Hotelling’s rule with a 10% interest rate, the present value of the $40 cost in 10 years is $40/(1.1)^10 = $18.32. This must equal today’s price of the scarce resource minus its marginal cost of extraction (Sinn, 2022).
(a) Under scenario (i), the rancher maximizes MB(h) – MC(h). Setting MB'(h) = MC'(h) yields h* = 3.75 hours.
(b) Under scenario (ii), the rancher maximizes MB(h) since MC(h)=0 for them, so h* = 5 hours.
(c) Socially optimal is where MB'(h) = MC'(h). This yields h* = 5 hours.
(d) The Coase theorem states that if property rights are well-defined and negotiation costs are low, the efficient outcome will be reached regardless of the initial allocation (Coase, 1960). The rancher and farmer could negotiate compensation to graze for the optimal 5 hours.
(e) With a tax t per hour under (ii), the rancher maximizes MB(h) – (MC(h) + t). Setting MB'(h) = MC'(h) + t yields t* = 2.5 per hour for h* = 5.
(f) With a subsidy s per hour under (i), the farmer maximizes MB(h) – MC(h) + s. Setting MB'(h) = MC'(h) – s yields s* = 2.5 per hour for h* = 5.
(a) The market equilibrium has MB'(Q) = MC'(Q), or 30 – 3Q = 3Q, so Q* = 5 units of pollution with no regulation.
(b) Welfare = Total Benefit – Total Cost. MB(Q) = 30Q – 1.5Q^2 and MC(Q) = 1.5Q^2. Setting MB'(Q) = MC'(Q) yields Q* = 10, with total welfare of 300 – 75 = 225.
(c) If polluters own the rights, those negatively impacted could pay them not to pollute beyond Q=10 (Coasian solution).
(d) If impacted parties own clean water rights, polluters would have to compensate them to pollute up to Q=10.
(a) Profit = Revenue – Cost. Rev= 20y, Cost = 2y + g(y). Profit max when MR=MC so 20 = 2 + g'(y), so y* = 9.
(b) Social welfare = Revenue – Environmental Cost. Setting MB’=MC’ yields 20 – 2 = 2 + g'(y). If g(y) = 2, g'(y)=0, so y* = 9.
(c) If g(y) = 3y, g'(y)=3, so 20 – 2 = 2 + 3, y* = 5.
(d) A carbon tax t = g'(y*) on each unit would induce the socially optimal production level y* (Fowlie et al., 2016).
References
Coase, R. H. (1960). The problem of social cost. Journal of Law and Economics, 3, 1-44.
Fowlie, M., Goulder, L., Kotchen, M., Borenstein, S., Bushnell, J., Davis, L., … & Wolak, F. (2016). Inefficient Pricing of Energy. BPEA Conference Drafts.
Sinn, H. W. (2022). Resource Economics. MIT Press.
Tsur, Y., & Zemel, A. (2020). The regulation of environmental pollution: an introduction. Oxford Research Encyclopedia of Environmental Science.
van der Ploeg, F., & Rezai, A. (2019). Hotelling and the economics of exhaustible resources: A view from the gulf. Annual Review of Resource Economics, 11, 313-332.
