One of the conditions we identified as important to the first welfare theorem is that there are no externalities. One of the most important externalities in the real world is pollution from production (which we will explore in detail in Chapter 21).
A: Suppose that we consider the production of some good x and assume that consumers have tastes over x and a composite good y where x is quasilinear.
(a) Illustrate the market equilibrium in a graph with x on the horizontal and the price p of x on the vertical axis. Assume that the supply curve is upward sloping — either because you are considering the short run in the industry or because the industry is composed of firms that differ in their cost curves.
(b) On your graph, indicate the consumer surplus and producer profit (or producer surplus).
(c) In the absence of externalities, why is the market equilibrium output level the same as the output level chosen by a social planner who wants to maximize social surplus?
(d) Now suppose that, for every unit of x that is produced, an amount of pollution that causes social damage of δ is emitted. If you wanted to illustrate not just the marginal cost of production (as captured in supply curves) but also the additional marginal cost of pollution (that is not felt by producers), where would that “social marginal cost” curve lie in your graph?
(e) In the absence of any non-market intervention, do firms have an incentive to think about the marginal cost of pollution? Will the market equilibrium change as a result of the fact that pollution is emitted in the production process?
(f) Would the social planner who wishes to maximize social surplus take the marginal social cost of pollution into account? Illustrate in your graph the output quantity that this social planner would choose and compare it to the quantity the market would produce.
(g) Re-draw your graph with the following two curves: The demand curve and the marginal social cost curve (that includes both the marginal costs of producers and the cost imposed on society by the pollution that is generated). Also, indicate on your graph the quantity x∗ that the social planner wishes to produce as well as the quantity xM that the market would produce. Can you identify in your graph an area that is equal to the deadweight loss that is produced by relying solely on the competitive market?
(h) Explain how pollution-producing production processes can result in inefficient outcomes under perfect competition. How does your conclusion change if the government forces producers to pay δ in a per-unit tax?
B: In exercise 15.2, you should have derived the aggregate demand function XD (p) = 250, 000/p2 from the presence of 10,000 consumers with tastes that can be represented by the utility function u(x, y) = 10x0.5 + y. Suppose that this accurately characterizes the demand side of the market in the current problem. Suppose further that the long run market supply curve is given by the equation XS (p) = 250, 000p.
(a) Derive the competitive equilibrium price and quantity produced in the market.
(b) Derive the size of consumer surplus and profit (or producer surplus).
(c) Consider a social planner who wants to maximize the social surplus. How would this planner arrive at the same output quantity as the market?
(d) Now suppose that each unit of x that is produced results in a pollution cost to society of $0.61. What would be the market outcome in the absence of any non-market intervention?
(e) Verify that, when each unit of x results in $0.61 pollution cost, the social planner would choose x = 160, 000 as the optimal output quantity.
(f) Calculate the total social cost of pollution under the competitive market outcome. How much is social surplus reduced from what it would be in the absence of pollution?
(g) Calculate the overall social surplus (including the cost of pollution) under the social planner’s preferred outcome.
(h) What deadweight loss is produced as a result of the market’s overproduction?
SOLUTION
A:
(a) This is illustrated in panel (a) of Graph 15.6 where the demand curve D is also equal to the MWTP curve because of the quasilinearity of x. The market will produce output quantity xM and price pM.
(b) Consumer surplus is indicated as CS and producer profit — or producer surplus — is indicated as area PS in panel (a) of the graph.
(c) In the absence of externalities, the demand curve represents the marginal benefit society gets from each unit of x and the supply curve represents the marginal cost incurred by society. The difference between these is positive for all x less than xM — implying that social surplus is produced for each unit of output until we reach xM. For all output units above xM , however, the marginal social benefit (as measured by the MWTP curve) is less than the marginal social cost (as measured by the supply curve) — which implies we would incur a negative social surplus for any unit of output above xM. Thus, a social planner that wishes to maximize social surplus would choose to produce xM.
(d) This is illustrated in panel (b) of Graph 15.6 where the social marginal cost curve SMC lies δ above the supply curve — because, for every unit of output, society now incurs not only the marginal cost faced by producers (as represented in the supply curve) but also the additional marginal cost δ of pollution.
(e) Firms have no incentive to take the cost of pollution into account because they do not have to pay for it. Thus, the market equilibrium will remain unchanged and will continue to occur at the intersection of supply and demand — resulting in output xM.
(f) Since the social planner wants to maximize the total social surplus, he would want to take into account the cost of pollution. Thus, the social planner would want to produce so long as SMC is below MWTP — or until x∗ in panel (b) of Graph 15.6.
(g) This is drawn in panel (c) of Graph 15.6. If the social planner’s production level x∗ is produced, the total surplus would be equal to area a. If the market produces x M instead, we would still get the area a of social surplus but we would incur a negative social surplus for the output units between x∗ and x M. That negative area is equal to b in the graph — giving us an overall surplus under the market of (a − b). Thus, the deadweight loss from the overproduction in the market is equal to area b.
(h) Pollution producing production processes result in inefficient market outcomes if firms are not forced to face the marginal cost of causing pollution. If the government charges the firms δ per output unit, it is in effect forcing firms to take the cost of pollution into account. As a result, the supply curve would shift up by δ (because the marginal cost of production has increased by δ). As a result, the new supply curve would intersect demand at x∗ — thus restoring efficiency in the market.
B:
(a) The competitive equilibrium occurs where demand intersects supply — i.e. where
Solving for p, we get the market price pM = 1. At that price, both the supply and demand functions tell us that the competitive market output will be xM = 250, 000.
(b) The consumer surplus is
The producer surplus (or profit) is equal to total revenues minus costs. Total revenues are 1(250, 000) = $250, 000, and total costs are half that (given the linear supply curve). More generally, the costs are just the area under the supply curve — which is
Producer surplus is then equal to 250, 000 − 125, 000 = $125, 000.
(c) The social planner would want to find the production level at which the marginal social benefit of the output equals the marginal social cost. In the absence of externalities, the marginal social cost curve is given by the curve that captures the marginal costs of produces — i.e. the supply curve. The marginal social benefit is given by the aggregate marginal willingness to pay curve which in turn is equal to the demand curve given that x is quasi- linear. Thus, the social planner solves the same problem we solved in part (a) — except that the social planner does not have to solve for the price since he is just interested in finding out the optimal quantity to produce.
(d) The market outcome would be unchanged since producers do not have to confront the pollution cost of production. Thus, xM = 250, 000 and pM = 1 remains the competitive market equilibrium.
(e) The social planner would again want to determine where the social marginal benefit of production equals the social marginal cost. The social marginal benefit is unchanged and given by the marginal willingness to pay function (which is equal to the demand curve given the quasilinearity of x). The demand curve is the inverse of the demand function XD (p) = 250, 000/p2 which is MWTP (x) = 500/x0.5. The social marginal cost, however, is now $0.61 higher than the marginal cost incurred by producers. The supply function is given as XS (p) = 250, 000p — which implies that the underlying marginal cost function for production is the inverse; i.e. MC (x) = x/250, 000. Adding the social marginal cost of pollution associated with each unit of output, the social marginal cost function is therefore SMC (x) = (x/250, 000) + 0.61. The social planner then needs to solve the problem
At x = 160, 000, both the right and left hand side of this equation are equal to 1.25 — which verifies x∗ = 160, 000 as being the socially optimal output level.
(f) In the market outcome, xM = 250, 000 — and society incurs a social marginal cost of $0.61 for each output unit produced. Thus, the social cost of pollution is 250, 000(0.61) = $152, 500. We previously calculated producer surplus of $125,000 and consumer surplus of $250,000 — for a combined producer and consumer surplus of $375,000. We now need to deduct the social cost of pollution — which implies an overall surplus of 375, 000−152, 500 = $222, 500 when the market produces 250,000 units of x.
(g) Under the social planner’s preferred outcome, x∗ = 160, 000. The benefit to consumers is then the area under the MWTP curve up to x∗ — i.e.
The cost to producers is the area under the marginal cost curve — i.e.
The overall surplus without taking into account the pollution cost is therefore 400, 000 − 51, 200 = $348, 800. The pollution cost is $0.61 for each of the 160,000 output units produced — for a total of 0.61(160, 000) = $97, 600. Subtracting this from the overall surplus in the absence of pollution, we get 348, 800 − 97, 600 = $251, 200.
(h) The deadweight loss is then the difference between the social surplus under the social planner and the social surplus at the market outcome — or 251, 200−222, 500 = $28, 700.