Problem III.2 [26 points]

We assume that the private market for refined petroleum products is competitive. The supply and demand relationships are given by:

- Supply: $ P = 10 + 0.075Q $
- Demand: $ P = 42 - 0.125Q $

where $ P $ = price per barrel, $ Q $ = thousands of barrels per day. The supply curve also represents marginal private costs (MPC) of production, and the demand curve also represents marginal private benefits (MPB).

III.2.A [5 points]

- Graph the demand and supply curves.
- Find $ P $ and $ Q $ at equilibrium.
- Calculate consumer surplus, producer surplus, and total surplus.

III.2.B [4 points]

Now suppose that the production of refined petroleum products generates air and water pollution. The marginal external cost (MEC) function is given by:

- $ MEC = 0.05Q $

To achieve allocative efficiency, external costs should be taken into account in the social equilibrium values of $ P $ and $ Q $. They are at the intersection of the MSC (marginal social cost) and MSB (marginal social benefits) curves, where:

- $ MSC = MPC + MEC $
- $ MSB = MPB + MEB $ (here $ MEB = 0 $)

The last equation indicates that the MSB curve is the same as the supply curve. Draw a graph, find $ P $ and $ Q $ at the social equilibrium, and briefly discuss the results.

III.2.C [6 points]

Suppose that the market ignores air and water pollution so that the market equilibrium is given by the solution to III.2.A. In that case, calculate:

- Consumer surplus
- Producer surplus
- Total external costs (the area under the MEC curve up to $ Q $ solution of III.2.A)

Add all three (CS + PS - TEC).

III.2.D [8 points]

Assume that the optimum social equilibrium in the market for refined petroleum products is obtained by taxing refined petroleum products. What is the value of the optimum tax? After imposing this tax, which should lead to the optimum social solution calculated in III.2.B, find:

- Consumer surplus ($ CS' $)
- Producer surplus ($ PS' $)
- Government revenue (GR) from the tax
- Total external costs (TEC)

Then calculate $ CS' + PS' + GR - TEC $.

III.2.E [3 points]

Compare $ CS + PS - TEC $ from III.2.C with $ CS' + PS' + GR - TEC $ from III.2.D. What do you conclude?

When the optimal tax solution is used, the total surplus is 4561.626, while it is 2556 when the market ignores air and water pollution. The market solution causes a social welfare loss of $146.691.

Part III.2.A: Graph the demand and supply curves. Find P and Q at equilibrium and calculate consumer surplus, producer surplus, and total surplus. To graph demand and supply curves, follow the following procedure. Substitute P = 10 + 0.075 Q into the demand equation and simplify it to obtain QD. Substituting P = 42 – 0.125 Q in the supply equation, we get QS. Demand and supply curves can be graphed as follows using the obtained QD and QS equations.

[tex]\frac{Graph}{}[/tex] Equilibrium price is the price at which demand and supply intersect. Equilibrium quantity is the amount that is demanded and supplied at the equilibrium price. Qs=Qd 10 + 0.075 Q = 42 - 0.125 Q 0.2Q = 32 Q = 160 Equilibrium price is the price when demand and supply intersect, so the value of P is determined by substituting Q = 160 into P = 10 + 0.075Q. P = 10 + 0.075 (160) = 22 Consumer Surplus (CS) is the difference between the willingness to pay of a consumer and the price actually paid.

The amount of consumer surplus in this case can be calculated as follows. CS = 1/2(PD – Pe) Q CS = 1/2(42 – 22) (160) CS = 1600 Producer Surplus (PS) is the difference between the price received by a producer and the marginal cost of production. The amount of producer surplus in this case can be calculated as follows. PS = 1/2(Pe – PS) Q PS = 1/2(22 – 10) (160) PS = 960 Total Surplus is the sum of consumer surplus and producer surplus. Total surplus can be calculated as follows. TS = CS + PS TS = 1600 + 960 TS = 2560

Thus, at the competitive market equilibrium, equilibrium quantity is 160 thousand barrels per day, equilibrium price is 22, consumer surplus is 1600, producer surplus is 960, and total surplus is 2560.Part III.2.B: Draw a graph, find P and Q at the social equilibrium, and briefly discuss the results.The equation for marginal external costs is MEC = 0.05Q. By adding the MEC and MPC curves, we can find the marginal social costs (MSC) of production. Similarly, by adding MEB (which is 0) and MPB curves, we can find the marginal social benefits (MSB). We will find equilibrium price and quantity by finding the intersection of the MSC and MSB curves. [tex]\frac{Graph}{}[/tex]

Thus, the equilibrium price is 31.9 and the equilibrium quantity is 185.3 thousand barrels per day at the social equilibrium. Part III.2.C: In the given problem, the market ignores air and water pollution, so market equilibrium is obtained using the solution of 2.A. To calculate consumer surplus, producer surplus, and total external costs, the MEC curve should be plotted. To get the area under the MEC curve up to the Q solution of 2A, integrate the equation for MEC. The area under the curve up to Q = 160 can be calculated as follows. TEC = ∫0Q=160 MECdQ TEC = ∫0Q=160 0.05QdQ = 4 Consumer surplus is 1600, producer surplus is 960, and total external costs are 4. Total surplus is 2556.

Part III.2.D: The optimum social equilibrium in the market for refined petroleum products is obtained by taxing refined petroleum products. The optimal tax value is the difference between the MSC and the MPC at the social equilibrium. MSC = MPC + MEC = 10.75 + 0.05Q MSC = 10.75 + 0.05(185.3) = 19.035 Optimal tax value = MSC – MPC = 19.035 – 10.75 = 8.285. Thus, a tax of $8.285 per barrel of refined petroleum products should be imposed. The price received by producers after the imposition of a tax of $8.285 per barrel is P – 8.285. Equilibrium quantity is still 185.3 thousand barrels per day. PS’ = 1/2 (P’ – MPC) Q PS’ = 1/2 (31.9 – 10.75) (185.3) PS’ = 2115.

6 Consumer surplus can be calculated using the following formula. CS’ = 1/2(PD’ – P’)Q CS’ = 1/2 (42 – 31.9) (185.3) = 921.185 GR = Tax * Quantity = 8.285 * 185.3 = 1534.106 Total external costs can be calculated by calculating the area under the MEC curve up to the equilibrium quantity. The area under the MEC curve up to the equilibrium quantity is 9.265. CS’ + PS’ + GR – TEC = 921.185 + 2115.6 + 1534.106 – 9.265 = 4561.626. The optimal tax solution yields a higher total surplus.Part III.2.E: The social welfare loss, which is the difference between the total surplus in parts 2.A and 2.B, can be calculated as follows.

Social Welfare Loss = CS + PS + TEC – CS’ – PS’ – GR Social Welfare Loss = 1600 + 960 + 4 – 921.185 – 2115.6 – 1534.106 Social Welfare Loss = -146.691 When the optimal tax solution is used, the total surplus is 4561.626, while it is 2556 when the market ignores air and water pollution. The market solution causes a social welfare loss of $146.691.

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