Equilibrium Px = $50, Equilibrium X = 50. CS = Shaded Triangle A, Whose Area Is $1,250

Dr. Donna Feir

Economic 313

Problem Set 7

Externalities

1. Suppose that the aggregate demand for good x is given by x = 100 – px, and aggregate supply is given by x = px.1[1]

a.  Draw a diagram illustrating the competitive equilibrium price and quantity in the market for good x. On this diagram, illustrate consumer surplus and producer surplus.

Equilibrium px = $50, equilibrium x = 50. CS = shaded triangle A, whose area is $1,250. PS = shaded area, whose area is $1,250.

Now suppose that the production and consumption of good x results in marginal external costs of $20. That, the marginal social cost curve will be parallel to the aggregate supply curve, but will lie $20 above aggregate supply.

b.  On your diagram from (a), draw the marginal social cost curve and identify the aggregate external cost, given the equilibrium quantity of x in your answer to part (a).

We know MPC = MRT = x, so MSC = MPC + MEC = x + 20. Given this, at x = 50, total external costs = $1,000, the shaded area.

Suppose that the government introduces a price floor in the market for good x.

Specifically, assume that the price of x cannot fall below $60.

c.  Illustrate the effects on consumer and producer surplus resulting from the price floor, relative to the unregulated equilibrium. By how much does consumer surplus plus producer surplus decrease as a result of the regulation?

First diagram: just the change in equilibrium price and quantity. New price is $60, so x = 40.

Second diagram: new CS = $800, shaded triangle below.

Third diagram: decrease in CS = shaded area = $450.

Fourth diagram: new PS= shaded area = $1,600.

Fifth diagram: remember old PS from (a) = area B, redraw below.

Sixth diagram: Change in PS has two components. First, a gain of the shaded rectangle: 40 units used to be sold at a price of $50, now they sell at a price of $60,for a gain of $400. Second, a loss of the shaded triangle. There used to be 50 units sold; now there are only 40. Those 10 units generated PS = $50, and this is now lost.

Overall the change on PS is thus + $400 - $50 = +$350. Which adds up since we know new PS = $1,600 while old PS = $1,250.

Overall effect? CS down by $450, PS up by $350. Net effect is thus - $100.

d.  By how much do aggregate external costs decrease, as a result of the regulation?

First diagram: new aggregate external costs is shaded area. Each unit of x still results in $20 of external costs, so 40 units in total result in $800 of aggregate external costs.

Second diagram: decrease in aggregate external costs is shaded area. 10 fewer units of x are produced, so the overall savings in external costs is $200 (each unit no longer produced lowers external costs by $20).

e.  Use your answers to parts (c) and (d) to demonstrate that this policy is NOT a Pareto improvement, but IS a potential Pareto improvement.

Market participants are worse off while those adversely affected by x are better off.

So this is not a PI. BUT, the gains in external costs ($200) more than offset the losses to market participants ($100), so the winners from the policy could - in principle - compensate the losers from the policy such that at least some people are better off and no-one is worse off. This is the definition of a PPI: a change that results in gains and losses but where the gains are such that, with appropriate compensation, we could turn the change into an actual PI. There are winners (who gain) and losers (who lose) from the policy, but we can see that gains more than offset the losses. In some sense (in the PPI sense), the policy is “worth it”, even though it is not a PI.

Now suppose that, instead of the price floor, the government introduces a $20 per unit tax on the production of good x.

f.  (Once again, demonstrate that this policy is not a Pareto improvement, but is a potential Pareto improvement. Remember (from Econ 103) that your social welfare analysis in the presence of a tax should account for the fact that tax revenues raised are a benefit to society.

The tax raises consumer price to 60, so CS decreases by areas a+b+c. The tax lowers producer price to 40, so PS decreases by areas d+e+f. Market participants, in aggregate are worse off by areas a+b+c+ d+e+f.

The government gains a+b+e+f in revenue and external costs fall by g+c+d. Gains from the tax are thus a+b+e+f + g+c+d.

Gains exceed losses by g, so the policy is a PPI. The net gain to social surplus is g,the DWL due to the uncorrected externality.

2. A soot-spewing factory that produces steel is next to a laundry. We will assume that the factory faces a prevailing market price of P=$40. Its cost function is C=X2, where X is the steel output. The laundry produces clean wash, which it hangs out to dry. Suppose each unit of steel produced produces one unit of soot (S), that is, X = S. The soot from the steel factory smudges the wash, so that the laundry has to protect the laundry from the soot of the factory and this increases its costs of producing clean clothes. The cost function of the laundry is C = Y2 + ½ S, where Y is pounds of laundry washed. A pound of clean laundry sells for $10. Both firms face a competitive market.

a.  What outputs X and Y would maximize the sum of the profits of these two firms? How big is the joint profit?

Maximizing joint profits: Max{X,Y, S} 40 X +10 Y - X2 - Y2 - .5S but X=S

Or equivalently Max{X,Y} 40 X +10 Y - X2 - Y2 - .5X

Profit maximizing conditions: Price of output equals marginal cost of that output: with respect to X: 40 =2X + .5

…with respect to Y: 10 = 2Y

Hence X* = 39.5/2 = 19.75, Y* = 5.

Total profit: 50+790 – (25 + 19.752 + .5*19.75) = 415.06.

b.  If each firm individually maximizes its own profit, what will be the output of each firm? How big is each firm’s profit?

Steel Factory: Max{X} 40 X - X2

Profit maximizing conditions, Price of output equals marginal cost of that output: 40 = 2X. Hence X = 20.

Profit of steel factory: 800 – 400 = 400

Laundry: Max{Y} 10 Y - Y2 - .5S

Profit maximizing conditions, Price of output equals marginal cost of that output: 10 = 2Y so Y = 5.

Profit of laundry: 50 – 25 - 10 = 15

Total profit: 50+800 – (25 + 400 + .5*20) = 415. This is less than in a).

c.  What per-unit tax would we need to set on soot to obtain the outputs found in Part a) of this problem? What is the government’s revenue from this tax? What is the profit of each firm?

Steel Factory: Max{X,S} 40 X - X2 – t*S but S=X

Or equivalently, max{X} 40 X - X2 – t*X

Profit maximizing conditions, Price of output equals marginal cost of that output: 40 =2X + t

Comparing this with the optimality conditions from a) with respect to X where 40 =2X + .5, we must have t = .5. In this case X = 19.75 and therefore S = 19.75. Tax revenue is .5*19.75 = 9.875.

The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the profit of the laundry is: 10*5 – 52 -.5* 19.75 = 15.125.

d.  Draw the marginal benefit curve (marginal profit curve of steel factory) of soot and the marginal external cost curve of soot in a diagram with soot on the x-axis and $ on the y-axis. Indicate the socially optimal amount of soot and mark the government’s tax revenue from an optimal tax on soot.

In order to find the steel factory’s marginal profit as a function of soot write down the profit of the steel industry as a function of soot:

Profit = 40 X - X2 but we also know that S=X

so equivalently profit = 40 S - S2

Marginal profit as a function of soot is given by the derivative of the profit as a function of soot, that is MP = 40 –2S.

In order to find the marginal external cost of soot, first write down the external cost of soot: e(S) = .5S. Then marginal external cost of soot is given by e’(S) = .5.

The socially optimal amount of soot is where the marginal benefit of soot to society (given by the marginal profit of the steel industry as a function of soot) equals the marginal external cost of soot. That is, 40 S - S2 = .5, and therefore S = 19.75. The Pigouvian tax (optimal tax on soot) is equal to the marginal external cost of soot at the socially optimal level of soot, that is t = . 5. Government revenues from this tax is equal to the area of the rectangular with height .5 and length 19.75.

e.  Suppose the laundry has the right to clean air and is willing to let the steel factory pollute for a price of q per unit of soot. What is the equilibrium price of soot and how much revenue does the laundry get from selling its rights to clean air? What is the profit of each firm?

Steel Factory:

Max{X,S} 40 X - X2 – q*S but S=X and so can write profit in terms of S: 40 S - S2 – q*S

Set marginal benefit of soot equal to its marginal cost. Marginal benefit of soot to steel factory is how much marginal profit is created: 40 –2S, marginal cost of soot is equal to q. Thus optimal amount of soot is found by setting marginal benefit equal to marginal cost: 40 – 2S = q. The firm’s inverse demand for pollution rights is q = 40 –2S.

Laundry:

Max{Y,S} 10Y – Y2 - .5*S + q*S

Profit maximizing with respect to Y: 10 =2Y, so Y = 5.

with respect to S, marginal cost of soot given by.5 needs to equal marginal benefit of soot given by q (now that steel factory has to pay $q to laundry for each unit of soot), so .5 = q. The firm’s inverse supply of pollution rights to the steel factory is therefore q = .5.

Setting inverse demand equal to inverse supply, 40 –2S = .5, we find S = 19.75.

The laundry gets .5*19.75 = 9.875 from the steel factory.

The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the total profit of the laundry (profit from laundering plus revenues from selling rights to pollute to steel factory) is: 15.125 + 9.875 = 25.

f.  Suppose the steel factory has the right to pollute the air up to S’ = 20. The steel factory is willing to cut down its pollution for a price of q per unit of soot abated. What is the equilibrium price of soot abated and how much revenue does the steel factory get from selling its rights to pollute? What is the profit of each firm?

Steel Factory: Max{X,S} 40 X - X2 + q*(20 – S) but S=X

Or equivalently, max{S} 40 S - S2 + q*(20 – S)

Profit maximizing condition: marginal benefit of producing one more unit of soot needs to equal marginal cost. Marginal benefit given by marginal profit, marginal cost given by lost opportunity to collect q from laundry for reducing soot. Thus, 40 –2S = q.

Laundry: Max{Y, S} 10Y – Y2 - .5*S - q*(20 – S)

Profit maximizing condition: marginal benefit of producing one more unit of laundry needs to equal marginal cost. With respect to Y: Marginal benefit given by marginal revenue of Y, marginal cost given by marginal production cost of Y. Thus 10 =2Yand Y=5.

With respect to S: Marginal benefit given by saving $q that would otherwise have to be paid to steel factory and marginal cost given by increase in cost of producing laundry due to increase in soot: q = .5

From q = 40 –2S and q = .5, we find S = 19.75.

The steel factory gets .5*.25 = .125 from the laundry.

Total profit of the steel factory (profit from producing steel plus revenue from abating pollution) is: 790 – 19.752 + .125= 400.06; profit of the laundry is: 15.125 - .125 = 15.

In all the solutions to overcome the externality problem, we see that the efficient amount of pollution is achieved. We also see that depending on the solution (i.e. who owns the rights) we have different distributional effects.

Calculus Version of answer to question 2)

a.  Maximizing joint profits: Max{X,Y, S} 40 X +10 Y - X2 - Y2 - .5S s.t. X=S

Or equivalently Max{X,Y} 40 X +10 Y - X2 - Y2 - .5X

First order conditions: with respect to X: 40 –2X - .5 = 0

……with respect to Y: 10 – 2Y = 0

Hence X* = 39.5/2 = 19.75, Y* = 5.

Total profit: 50+790 – (25 + 19.752 + .5*19.75) = 415.06.

b.  Steel Factory: Max{X} 40 X - X2

First order conditions: 40 –2X = 0

Hence X = 20

Profit of steel factory: 800 – 400 = 400

Laundry: Max{Y} 10 Y - Y2 - .5S

First order conditions: 10 – 2Y = 0

Y = 5.

Profit of laundry: 50 – 25 - 10 = 15

Total profit: 50+800 – (25 + 400 + .5*20) = 415. This is less than in a).

c.  Steel Factory: Max{X,S} 40 X - X2 – t*S s.t S=X

Or equivalently, max{X} 40 X - X2 – t*X

First order conditions: 40 –2X – t = 0

Comparing this with the optimality conditions from a) with respect to X where 40 –2X - .5 = 0, we must have t = .5. In this case X = 19.75 and therefore S = 19.75. Tax revenue is .5*19.75 = 9.875.

The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the profit of the laundry is: 10*5 – 52 -.5* 19.75 = 15.125.

d.  In order to find the steel factory’s marginal profit as a function of soot write down the profit of the steel industry as a function of soot:

Profit = 40 X - X2 but we also know that S=X