Externalities in the Koopmans Diagram

Transcription

1 Externalities in the Koopmans Diagram Peter Norman Sørensen Institute of Economics, University of Copenhagen Translation of August 2000 Danish version March 23, Introduction The two Welfare Theorems are often taken to prove that a free market economy works well. In our note on distortional income taxes, interference with the free markets only reduced welfare. This could convey the impression, that economic policy damages efficiency and must be motivated by important distributional concerns. However, some simple and realistic changes to our equilibrium model easily turn around the argument: Walrasian equilibria can be inefficient, while market regulation serves to enhance efficiency. Externalities are considered the most important of such modifications one agent s choice of consumption or production now directly influences other agents utilities or profits. The resulting situation is distinct from our usual market model, in that the affected parties have no choice whether to buy or sell this external influence on their well-being. A main characteristic of externalities is precisely this, that they do not pass through the markets. The present note will present one of the key examples of external effects, that of polluting production. Hopefully, the example will illustrate why the theory on externalities and public goods forms a corner stone of modern environmental economics. In Varian s Intermediate textbook, Chapter 32 treats externalities. Section 32.1 analyzes an Edgeworth box, 32.3 analyzes a two firm model, and 32.6 deals with many firms. Our note complements Varian s chapter by introducing in Section 2 a Robinson Crusoe economy with one consumer and one polluting firm. Varian s Chapter 30 introduces the usual Robinson Crusoe economy absent externalities. Recall that this economy s alloca- The new version of Micro 3 needs this note in English. The translation is as literal as possible. 1

2 Externalities Micro 3 2 tions are illustrated in the Koopmans diagram, cf. Varian s Figures Below, Section 3 will identify our Walrasian equilibrium, and Section 4 will note that it is not Pareto optimal. Section 5 analyzes various welfare enhancing microeconomic policies. Some comments made in Section 5 refer to public goods, introduced by Varian in Chapter Model This economy is populated by one consumer and one firm, and features three commodities, time (labor time and free time), a consumption good, and smoke. Time and the consumption good are traded on markets, but there is no market for smoke. The firm can transform labor time to the good. Using labor time input l, its consumption good output is y = f(l). As usual f(0) = 0, f is increasing and concave with f (l) + as l 0, and f (l) 0 as l +. As an unfortunate by-product of production, the firm emits smoke in the amount z, and we assume the simple relationship z = y. Thus, smoke is emitted in exactly one unit per unit of produced consumption good. The consumer is initially endowed with the amount n of time. If n is supplied as labor time, the remaining n n is free time. The consumer likes free time and the good, with a usual utility function h over these two commodities. However, the consumer dislikes smoke, expressed in the final utility function u as follows. At labor n, consumption c, and smoke x, the utility is u(n, c, x) = h(n n, c) x. (1) Here, the utility function h over free time and the satisfies our usual assumptions: it is continuous, differentiable, strictly increasing, and has strictly convex indifference curves. Assume that the marginal utility of c tends to infinity as c tends to zero. The consumer owns the firm. This could be interpreted in several ways. In our leading Walrasian model, ownership means that the firm solves its profit maximization problem at given market prices, and the resulting profit directly enters the consumer s income.

3 Externalities Micro Walrasian Equilibrium With only the two markets for time and the consumption good, a Walrasian equilibrium consists of an allocation {(n, c, x), (l, y, z)} and a price vector (w, p) jointly fulfilling: 1. the firm chooses l, y and z in order to maximize the profit π = py wl under the technological restrictions z = y = f(l) and l 0, 2. the consumer takes profit π and smoke amount x = z as given, and chooses n n and c 0 to maximize the utility u(n, c, x) under the budget constraint pc π +wn, 3. and markets clear: l = n and y = c. Observe some simple consequences of our assumptions. (i) Since there is no market for smoke, the firm bears no cost of pollution, so the choice of z does not directly enter is profit. (ii) The consumer is forced to take the smoke amount as given, again due to the absence of the smoke market. Taking the firm s chosen z as granted when maximizing the consumer s utility, corresponds to the Nash equilibrium principle that a best reply is calculated taking other players actions as given. (iii) Rewrite the budget constraint in this Robinson Crusoe model by adding the value of the consumer s initial time endowment on both sides: w(n n) + pc π + wn. It then resembles our usual Walrasian equilibrium constraint, as the consumer s income is the profit plus the value of initial endowments. The firm s problem is simple to solve, since z is of no importance. Using y = f(l), the chosen l maximizes pf(l) wl, with first order condition pf (l) w = 0 or f (l) = w/p. The marginal product equates the real wage, as in Varian s Figure The consumer must choose n and c to maximize h(n n, c) x, where x is a given. The consumer must then focus on maximizing h(n n, c), and must therefore behave just like our usual consumers. For the moment, let the variable t = n n denote free time and M = π + wn denote income. The consumer must choose (t, c) to maximize h(t, c) under

4 Externalities Micro 3 4 wt+pc M. The solution is to choose a bundle where the indifference curve through (t, c) is tangent to the budget line, i.e. where (minus) the marginal ratio of substitution equates the price ratio: h t (t, c)/h c (t, c) = w/p. This is well captured in Varian s Figure 5.1, but returning to the variable n = n t the solution is perhaps better illustrated in Figure y, c h = h W L budget line f(l) c W L n W L n l, n Figure 1: Walrasian Equilibrium In Walrasian equilibrium, the firm chooses (l, y) as in Figure 30.2, the consumer chooses (n, c) as in Figure 30.3, and the markets for time and the good are clearing. Altogether, this yields Figure 30.4, here rendered as our Figure 1. The unique equilibrium arises where the indifference curve is tangent to the production function. Let (n W L, c W L, x W L ) denote the equilibrium allocation the figure shows n W L, c W L, while x W L = c W L by implication. The firm s f = w/p with the consumer s h t /h c = w/p yield this equilibrium condition: f (n W L ) = h t(n n W L, c W L ) h c (n n W L, c W L ). (2) It may seem slightly surprising that we use our usual figure to illustrate Walrasian equilibrium in our model with externalities. But our model is quite special, since the

5 Externalities Micro 3 5 external effect x has no influence on the consumer s marginal trade-off of n for c. In more general utility functions u(n, c, x), the level of x would typically affect the shape of the (n, c) indifference curve. In the Koopmans diagram it would then appear as if the firm faced a different type of consumer whenever y = z(= x) was changed. That diagram would then no longer be of any help. So, given the utility function specified in (1), is the externality completely without consequences in this economy? No, the externality still influences the consumer s equilibrium utility level u = h x, and we will soon find that the First Welfare Theorem is defunct. 4 Optimality Since this is a one consumer economy, Pareto optimality reduces to the maximization of this consumer s utility subject to the technological constraints. This means that the consumer adopts direct control over the firm s decisions. We must then maximize u(n, c, x) = h(n n, c) x by choosing (n, c, x) fulfilling the constraints x = c = f(n) and n 0. Inserting x = c we see that we must maximize h(n n, c) c over (n, c) (0, 0) subject to c = f(n). Exercise 1 asks you to solve this using Lagrange s method, but here we take n 0 as the only free variable and maximize h(n n, f(n)) f(n). The necessary first order condition is h t + h c f f = 0. Let n P O denote the solution of this condition, and likewise let x P O = c P O = f(n P O ); then the condition is f (n P O ) = h t(n n P O, c P O ) h c (n n P O, c P O ) 1. (3) The marginal characterizations (2) of Walrasian equilibrium and (3) of optimality are not the same. The difference arises since only the optimality problem takes into account the consumer s disutility of smoke. Had we confined the optimality problem to the maximization of h(n n, c) rather than the complete utility function h(n n, c) x, we would

6 Externalities Micro 3 6 have arrived at the first order condition h t + h c f = 0. This is (unsurprisingly) the marginal characterization (2) of Walrasian equilibrium, since the First Welfare Theorem would have been in force in an economy without externalities. In our special model, the Pareto optimum can be illustrated in a Koopmans diagram. Since x = c, we should really derive the consumer s preferences in our (n, c) diagram from the utility function h(n n, c) c. These new preferences are used when finding the Pareto optimum, while the old preferences defined the indifference curves in our original Koopmans diagram. Using the new indifference curves, we arrive at the new Koopmans diagram. The diagrams of Figures 1 and 2 are thus different. The unique Pareto optimum is found where the new indifference curve is tangent to the production function. y, c u = u P O u = u W L f(l) W L c P O n P O Figure 2: Pareto Optimum n l, n How do the allocations (n W L, c W L, x W L ) and (n P O, c P O, x P O ) differ? Intuitively, there is less pollution in optimum since we take into account the consumer s smoke aversion. Figure 2 confirms this intuition. Indifference curves are steeper in Figure 2 than in Figure 1, since the consumer of Figure 2 is less happy with c. Thus, the new indifference curve

7 Externalities Micro 3 7 passes through the Walrasian equilibrium as shown in the drawing, and therefore the more efficient allocations are to the left of the equilibrium allocation. More formally: Proposition 1 We have x P O < x W L. Proof: When solving the optimality problem, we maximize u(n, c, x) over a set of feasible (n, c, x). One feasible allocation is (n W L, c W L, x W L ), since x W L = c W L = f(n W L ) and n W L 0. Since (n W L, c W L, x W L ) satisfies (2), it cannot satisfy (3), and so it cannot be optimal. Thus u(n P O, c P O, x P O ) > u(n W L, c W L, x W L ). Suppose for a moment that we also have the undesired x P O x W L. Using (1), we then infer that h(n n P O, c P O ) > h(n n W L, c W L ). But this contradicts the well established First Welfare Theorem in the economy without externalities, for this economy would have the same equilibrium (n W L, c W L ), and this allocation should then maximize h(n n, c) under the technological constraints. The contradiction implies that the undesired x P O x W L must be false. As a direct consequence of x P O < x W L, also c P O < c W L and n P O < n W L. Respecting the impact of pollution on the consumer, the activity level of the economy is optimally reduced. Please notice carefully, that the Pareto optimum nevertheless involves a positive amount of pollution. With x = 0 the technological constraint would give c = 0, but the large marginal utility of c at c = 0 renders this sub-optimal. This idea, that the optimal pollution level may be strictly positive, defines a major difference between economists and many green activists. Economists bear in mind that environmental policies involve economic sacrifices, while others may be guided by the ideal world of zero pollution. 5 Microeconomic Policies We have now described an economy where the Walrasian equilibrium allocation is not Pareto optimal. This property is known as a market failure. Sub-optimality of the equilibrium provides a good rationale for not leaving the economy alone as an unregulated

8 Externalities Micro 3 8 (laissez-faire) market economy. We will now analyze the application of some microeconomic policy instruments. This is roughly along the lines of Varian s Section In our model, the simplest instrument gives the consumer direct control over the firm. But many real world firms affect the lives of many heterogeneous consumers. It is then less obvious how to implement direct control. The following Subsections therefore ignore direct control, but bear in mind that some firms are in fact controlled by the government. 5.1 Regulation Pollution with a Quota A planner could mandate that the firm is permitted to pollute no more than the amount z MAX, thereby influencing the firm s profit maximization problem. In case z MAX < z W L, the new restriction is binding, and the firm must reduce its activity level. This reduction will have an impact on the markets equilibrium prices. The Walrasian equilibrium is again defined as in Section 3, except that point 1 is changed to: 1. the firm chooses l, y and z in order to maximize the profit π = py wl under the technological restrictions 0 z = y = f(l) z MAX. If the planner happens to identify the truly optimal x P O, it will be an ideal solution to fix the smoke quota at z MAX = x P O. For then the equilibrium allocation is precisely our optimal (n P O, c P O, x P O ). The equilibrium price ratio is w/p = h t (n n P O, c P O )/h c (n n P O, c P O ), so that the consumer s problem is certainly solved at (n P O, c P O ). From (3) we also know that (n P O, c P O, x P O ) satisfies f = h t /(h c 1) w/p, so it might at a first glance appear as if the firm s problem has not been solved. But clearly h t /(h c 1) > h t /h c, so the firm actually has f > w/p. Marginal productivity exceeds marginal costs, so the firm would profit from an increased production level. But the new pollution restriction is already binding the firm at (n P O, c P O, x P O ), for the quota is precisely x P O. So, after all, the firm maximizes profits at (l, y, z) = (n P O, c P O, x P O ).

9 Externalities Micro Taxing the Externality A planner could decide to put a tariff (price) on smoke emission. The firm will then experience that pollution is costly, and it will therefore tend to reduce its optimal x. Letting τ denote the tariff, the firm s problem is now changed to: 1. the firm chooses l, y and z in order to maximize the profit π = py wl τz under the technological restrictions 0 z = y = f(l). As z = y, the profit can be rewritten as (p τ)y wl. In our simple model, the tariff is thus equivalent to a sales tax on the consumption good. The equilibrium revenue T = τz is transferred to the consumer as a lump sum income, so the consumer s problem becomes: 2. the consumer takes profit π, revenue T = τz, and smoke amount x = z as given, and chooses n n and c 0 to maximize the utility u(n, c, x) under the budget constraint pc π + T + wn. A tariff (e.g. customs duty or value added tax) is typically defined as a percentage price increase. Instead, our planner sets a nominal price τ which must be adjusted according to the equilibrium price level. Still, our model connects pollution and output so simply, that the tariff is equivalent to a certain percentage increase to the output price. The tariff can guide the new Walrasian equilibrium into the Pareto optimal allocation in our model, let τ = p/h c (n n P O, c P O ). The consumer chooses such that w/p = h t /h c. The firm acts with price p τ on y and therefore satisfies f = w/(p τ). These two conditions combine with the definition of τ to our optimum characterization (3): f = w p τ = w 1 p (1 1/h c ) = h t 1 h c (1 1/h c ) = h t h c 1.

10 Externalities Micro 3 10 y, c iso-profit h=h P O f(l) budget c P O n P O n l, n Figure 3: Regulation with a tariff Figure 3 illustrates how the Pareto optimum is achieved. The relative prices w/p determine the slope of the consumer s budget line which is tangent to the indifference curve for (n, c) at the Pareto optimal allocation. The iso-profit lines of the firm are steeper, as the firm sees relative prices w/(p τ). The ideal tariff level guides the firm s profit maximizing choice to the Pareto optimal allocation. The firm s iso-profit line and the consumer s budget line intersect the c-axis at different points. This may seem surprising in the Koopmans diagram. However, the consumer receives both the firm s profit and the tax revenue as income the tax revenue accounts for the intersection difference. The optimally determined τ = p/h c is called a Pigou tax. Since w/p = h t /h c, we also have τ = w/h t. Both expressions reflect that the tariff is the marginal cost associated with compensating the consumer for a marginal smoke increase. Observe that the planner can determine the ideal tariff when knowing merely h c or h t the firm s production function is irrelevant. In a world with multiple firms, the tariff system has the further advantage over the quota system, that all firms arrive at the same marginal pollution cost. This means

11 Externalities Micro 3 11 that pollution is allocated efficiently among firms. This last feature is true even when the tariff is not optimal. This line of argument has had some success in the environmental policy debate, implying the gradual move from quotas to tariffs. Our note on taxation in the Koopmans diagram highlighted the distortional effect of taxes in a perfectly competitive economy. Instead, we have now studied a case where a tax nicely guides the economy towards the Pareto optimum. The market failure is the main characteristic of this case the externality is a first distortion, rendering the unregulated Walrasian equilibrium sub-optimal. The tariff s ability to mitigate this distortion reflects a general principle of economics: K distortions to the economy may result in a Pareto better equilibrium than only J < K distortions. This principle requires J 1, since the equilibrium in an undistorted economy is Pareto optimal (First Welfare Theorem). Governments typically levy taxes to finance public spending. The Combination of a new Pigovian tax with a reduction in an existing distorting tax will mitigate the harmful effects of both the externality and the existing tax. We say that the government reaps a double dividend when introducing the Pigou tax. This logic supposes that the two taxes have no interrelated effects, i.e. we assume that our two partial analyzes sum up to the general analysis. A more correct general equilibrium analysis may uncover that the dividend is not always doubled. 5.3 Creating a Market for the Externality In reality, an economic planner cannot possibly know all the consumers true utility functions and all the firms true production functions. Without such knowledge, the planner cannot always determine the optimal parameter values for z MAX and τ in the above two regulation policies. The planner could try to ask the involved parties about the true relationships, but the firm s profit is increasing in z MAX and decreasing in τ, and so the firm has an incentive to lie about the extent of pollution. Likewise, in a world of many

12 Externalities Micro 3 12 consumers, each consumer would be incited to claim that he/she is particularly hard hit by pollution. Alas, the planner cannot expect to hear the truth. The least demanding task for the planner may then be the creation of a new market for x this solution is emphasized in Varian s book. Let us thus augment our model with a new market in pollution permits, with initial endowments naturally defined by the planner as follows. The consumer has a right to zero pollution, but can choose to sell a permits (anticipating the degree x = a of pollution). Initially, the firm has no right to pollute, but can choose to buy b permits granting the right to pollute up to the amount b. Due to our model s simplicity, it is not too hard to analyze the Walrasian equilibrium of the augmented three-market economy. Let s > 0 denote the price of permits. The new definition of Walrasian equilibrium says that the prices (w, p, s) must satisfy: 1. the firm chooses l, y, z and b in order to maximize the profit π = py sb wl under the technological restrictions 0 z = y = f(l) b, 2. the consumer takes the profit π as given, and chooses n n, c 0, and a 0 to maximize the utility u(n, c, a) under the budget constraint pc π + sa + wn, 3. and markets clear: l = n, y = c and b = a. In the new Walrasian equilibrium, the price s will turn out to equate the ideal tariff τ levied on the firm in the previous Section. The equilibrium is thus Pareto optimal, illustrating that the First Welfare Theorem is back in force. With the new permits market, the economy perfectly fulfills the usual neoclassical assumptions. Clearly, the firm will not pay for unused pollution permits, so b = z. Now, the firm must choose l, y and b to maximize py sb wl subject to b = y = f(l) and l 0. Since y = b, this corresponds to maximizing (p s)y wl subject to y = f(l), and this is the standard problem of the firm. Its solution satisfies f = w/(p s).

13 Externalities Micro 3 13 The consumer maximizes u(n, c, a) = h(n n, c) a given the budget pc sa wn π. Trading c for n, the consumer can improve the utility unless the marginal trade-off equals the price ratio. So, a necessary condition is again u n /u c = w/p, or h t /h c = w/p. But the consumer can also trade c for a, so a new necessary condition is u x /u c = s/p, reducing to 1/h c = s/p. Therefore s = p/h c. The resulting price on the externality, s = p/h c, perfectly corresponds to the optimally designed tariff on the firm s pollution, derived in our previous Subsection. Repeating the computation from there confirms that our Walrasian equilibrium now has f = h t /(h c 1), implying optimality of the equilibrium. This conclusion does not depend on the initial allocation of pollution permits. The equilibrium is Pareto optimal by the First Welfare Theorem. After we created a market for the externality, the economy perfectly fits into the usual neoclassical theory. In general terms, however, the initial endowment allocation determines which of many Pareto optima is selected in equilibrium, and so the initial allocation has distributional consequences. A special feature of the Robinson Crusoe economy is the solitary consumer/firm-owner, rendering void any distributional concerns. By now, the market might seem the perfect solution to our problems, achieving Pareto optimality even when the planner knows little about production and preferences. Unfortunately, polluting smoke should really be viewed as a public good (bad). A firm s smoke affects many consumers in a non-depletable fashion, as one consumer is not perceptibly helped by the fact that other consumers also live in a smoggy world. As illustrated in Varian s Chapter 35, even the market solution cannot easily deal with public goods. Smoke permits sold by one consumer to the firm will have external effects on other consumers, and the market failure continues. In this situation, quotas and tariffs come back into play as the better instruments for regulation of pollution towards an efficient level.

14 Externalities Micro Compensation Claims One important instrument of pollution regulation is unfortunately hard to represent in our model, so here follows only a short discussion. Over the past centuries, many firms have polluted our Nature. Some pollution comes as a flow associated with current production, but other pollution materializes in isolated events such as oil spills. In many cases, particularly in the US, polluting firms have subsequently been taken to court, facing claims for very large sums in damages. European courts typically award smaller damages than American courts. The huge amounts reported from the US are typically awarded as punitive damages, meaning that the amount paid by the firm contains some punishment. The purpose of large damages is to induce firms to behave cautiously when deciding how much to pollute. In Europe, greater emphasis is placed on the direct regulation of pollution through quotas and tariffs. Quotas and tariffs that directly affect a firm s current earnings is a form of ex ante regulation. It takes place before the damaging consequences of pollution can be accurately assessed, and is based on mere presumptions about the welfare loss arising from the pollution. On the contrary, punitive damages provide a tool for ex post regulation. This tool is applied after the damage is done, and requires detailed evidence of the precise amount of harm inflicted by the pollution. An important disadvantage of ex post regulation is that the polluting firm may very well be unable to pay out the damages when the pollution can be finally assessed by then, the earned profits may have been long since paid back to the shareholders, who have limited liability. Which system fares better is a major topic for political and moral disputes. The different principles may partly explain the trade war between Europe and the US over genetically modified foods.

15 Externalities Micro Exercises 1. Derive the marginality condition (3) by applying Lagrange s method to maximize h(n n, c) c subject to c = f(n). 2. Assume that it is technologically possible to completely eliminate smoke emission. Paying the development cost K, the firm can acquire this technology which lets z = 0 at all levels of activity y = f(l). Show that our firm of Section 2 has no interest in buying this technology whenever K > 0. On the other hand, argue that the technology should be developed in Pareto optimum, provided K is sufficiently small. 3. The figures of this note are drawn on the basis of an explicit example using the functional forms f(l) = l and h(t, c) = 2 tc. Assume that n = 1. (a) Find the Walrasian equilibrium, and verify n W L = 1/3. Calculate the utility u W L. (b) Looking for the optimum, we let the utility be u(n, c) = h(n n, c) c. Show that u n < 0, but also that u c > 0 holds only when n n > c. So, at large values of c indifference curves bend backwards, as in Figure 2. Explain this backward bending intuitively. Explain why an optimum cannot be located on the backward bending part of an indifference curve. (c) Derive the optimality conditions, and show that c P O solves 9c 4 +c 3 6c 2 c+1 = 0. This equation is hard to solve, but try (e.g. using a computer) to graph the function 9c 4 + c 3 6c 2 c + 1, and eyeball that the equation has only two positive solutions, c 0, and c 0, Recalling c = n, show that n n > c while n n < c. Conclude that c P O 0, (d) Use the numerical solution from (c) to calculate the ideal tariff τ.