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?