An Integrated Approach to the Theory of Externalities: An Exposition Pankaj Tandon Boston University One of the key topics in modern micro-economic theory is the theory of externalities. In a world where many believe that economic activity is leading to large changes in global climate, with possibly serious repercussions for billions of people, understanding the theory of externalities and the relevant policy interventions has become an important part of the economist's tool-kit. One central tenet of this understanding is that, although Pigou's original policy prescription for a negative externality was the imposition of a (Pigouvian) commodity tax, 1 it is now widely understood that an effluent charge or equivalent policy intervention in the effluent market is a superior approach. But traditional treatments of such policies focus on the market for effluents, thereby obscuring the effect of these policies on the original commodity markets. In this paper, I present a simple algebraic and graphical exposition of the theory of externalities that allows the commodity and effluent markets to be seen as an integrated whole. The effects of effluent charges on the underlying commodity markets can be clearly demonstrated in such an approach. The sense in which the optimal effluent charge is equivalent to an optimal commodity tax can also be clarified in this way. Traditional Approaches The starting point for the traditional approach is the basic competitive market (Fig. 1). 2 Suppose S(Q) and D(Q) represent the inverse demand and supply curves for the final output Q, and suppose (Q 0, P 0 ) at point A represents the quantity-price equilibrium under perfect competition. In the absence of any externalities, this equilibrium represents a Pareto-optimal solution to the allocation problem in this market. The curve D(Q) can be interpreted as the social marginal benefit curve and S(Q) can be interpreted as the social marginal cost curve; their intersection is the point at which the social marginal benefit and social marginal cost are equalized. $ SMC = PMC + MEC S(Q)=PMC p 1 * B p 0 A C D(Q) 0 Q Q 1 * Q 0 Figure 1: Simple Externality and the Pigouvian Tax 1 See Pigou (1920). 2 As examples of this approach, see Pindyck and Rubinfeld (2009), Figure 18.1 on p. 646, Perloff (2008), Figure 17.1 on p. 603, or Nicholson and Snyder (2008), Figure 19.1 on p. 676.
Now suppose the production of this commodity results in the discharge of pollutants into the environment. The curve S(Q) is no longer the social marginal cost of production; it is only the private marginal cost of production. The social marginal cost curve would lie above the curve S(Q), say at SMC, where the vertical distance between S(Q) and SMC is equal to the marginal external cost (MEC). Now the market optimum has shifted to point B, where the SMC curve intersects with D(Q), although the competitive outcome remains at A. The Pigouvian method for attaining the optimum is to impose a commodity tax equal to the size of the marginal external cost (= the distance BC in the figure), in order to force the quantity level down to Q 1 *. Price would then rise to p 1 *. The alternative to the Pigouvian commodity tax would be an effluent charge or other such intervention that would seek to control the level of emissions directly rather than indirectly through the level of production of the underlying commodity. Traditionally, this is analyzed by looking at the market for emissions. 3 Of course, there isn't truly a market for emissions until government intervention forces the creation of such a market. Figure 2 illustrates the approach. 2 $ MCA MDC f* 0 e e* ē Figure 2: The Traditional Approach to Optimal Abatement The curve labeled MDC represents the marginal damage cost of emissions. Note that this MDC is not the same as the vertical distance between S(Q) and SMC in Figure 1, because the units of analysis are different. The horizontal axis in Figure 1 represented the quantity of the final output, while the same axis in Figure 2 represents the level of emissions of the pollutant. Thus, the MDC here is the marginal external damage cost of emissions in response to changes in the level of emissions, while the MEC implicit in Fig. 1 was the marginal external cost of emissions in response to changes in the level of final output. The curve MCA in Figure 2 represents the marginal cost of abating or reducing emissions, but it is drawn in mirror image from what we might expect, since increasing abatement is a reduction in emissions. The point ē represents the level of emissions that would occur absent any abatement. For purposes of the analysis, this level 3 See, for example, Pindyck and Rubinfeld, op. cit., Figure 18.4 on p. 652, or Perloff (2008), Figure 17.2 on p. 606.
is treated as fixed, 4 even though in reality it could be a choice variable of the firms. The curve MCA then shows the marginal cost of abating or reducing the level of emissions being discharged into the environment, reading the level of abatement from right to left, starting at the point ē. The optimal levels of abatement and emissions are then easily seen in this context to be at the intersection of the MEC and MCA curves, since the MEC curve can be seen as the mirror image of the marginal benefit of abatement. This optimal solution can be attained either through the imposition of an effluent charge of f* or by setting emissions standards at e*. Thus the traditional approach allows us to analyze either the optimal (Pigouvian) commodity tax or the optimal effluent charge, but without any clarity of the relationship between the two markets. It allows us to see one or the other separately, but not both together. Since the units of analysis are different, it is hard to see intuitively what the relationship between the two really is. It is this disconnect that the integrated approach attempts to rectify. An Alternative Integrated Exposition I present the integrated approach algebraically first and then graphically. We start with the case of no externalities. As in Figure 1, let D(Q) and S(Q) represent the inverse demand and supply curves in the commodity market. For simplicity, and without loss of generality, I will treat the suppliers as a single price-taking firm. The inverse supply curve is then nothing but the firm's marginal cost curve and its integral is the firm's cost curve. One way to think about optimality in a competitive market is to maximize surplus, W: W =V p pq C Q (1) 3 where V(p) represents consumer surplus as a function of the price, p. Now V p = Q p dp (2) p so and therefore dv dp = Q p (3) dv dq = Q p ' Q (4) Differentiating (1), and using (4), we then have dw dq = Q p ' Q Q p ' Q p Q C ' Q =0 (5) 4 Pindyck and Rubinfeld, for example, say: To simplify, we assume that the firm's output decision and its emissions decision are independent although they also note that a firm can substitute among inputs by changing technologies in response to an effluent fee (op. cit., p. 651).
for a welfare maximum. 5 This simplifies to the familiar condition that price must equal marginal cost: p Q =C ' Q (6) 4 Now let us introduce a negative externality. Suppose the production process in this industry leads to the creation of a toxic substance according to the effluent production function : e=e Q, e ' Q 0 (7) where e represents the quantity of effluent produced. Emission of this substance causes damage to the environment according to the damage function: D=D[e Q ], D ' e 0 (8) Social surplus is now W =V p pq C Q D[e Q ] (9) and the social optimum can be found by differentiating (9) and setting it equal to zero: dw dq = Q p' Q Q p' Q p Q C ' Q D ' [e Q ] e' Q =0 (10) This reduces to p Q =C ' Q D ' [e Q ] e' Q (11) This condition is the familiar one that says the optimal level of output occurs where the price is equal to the marginal social cost of production. The first term on the right hand side of (11) is the private marginal cost of production and the second term is nothing but the marginal external cost of production. This second term has been given some greater structure here through the recognition that production of the pollutant is a side-product of the production process for the final product, and damage to the environment is caused by the level of pollutants emitted. This recognition is crucial to the integrated approach, as will become clear as we consider the level of optimal abatement. Before turning to that, we should note that the optimal Pigouvian tax would simply be equal to the marginal external cost of production, the second term on the right hand side of (11): t * = D' [e Q ] e ' Q (12) 5 Throughout the paper, I assume the second-order conditions for maximization are satisfied.
5 Turning now to the possibility of abating the effluent, suppose the representative firm could abate, at a cost of A, an amount a of the effluent before emitting it into the environment, according to an abatement cost function: A= A a, A' a 0, A (a)>0 (13) Presumably, this function would be convex. Further, if the abatement level is a, the environmental cost to society of the emissions will be D= D[e Q a ] (14) since now only e(q)-a of the effluent is emitted into the environment. Social surplus is now W =V p pq C Q D[e Q a] A a (15) and the social problem is to maximize this by suitable choice of Q and a. The first order conditions for this maximization are: W Q = Q p ' Q Q p ' Q p Q C ' Q D ' [e Q a] e' Q =0 (16) and W a =D ' [e Q a] A' a =0 (17) (16) reduces to p Q =C ' Q D ' [e Q a] e ' Q (18) which is similar to (11) except that the marginal damages are now evaluated at the net emission level [e(q)-a] rather than simply e(q) in order to allow for the abatement level. (18) is the condition for the optimal output level, which is at the point where price is equal to the marginal social cost. Simultaneously, (17) specifies the optimal level of abatement, which is at the point where the marginal cost of abatement A'(a) is equal to the marginal benefit of abatement, which is simply the marginal damage averted D'[e(Q)-a]. The essential contribution of this alternative approach is precisely this simultaneous determination of the optimal level of abatement and the optimal output level. Thus the two markets which can be examined only separately in the traditional approach are examined together in this approach. As a result, we can see that setting the optimal effluent fee automatically results in the optimal output level as well. From (17), we see that the optimal effluent fee is
f * = D' [e Q * a * ] (19) 6 where the asterisks represent the optimal levels of the respective variables. Substituting this in (18) we get the condition for the optimal output level: p Q =C ' Q f * e' Q (20) This is precisely the level of output that firms will choose if faced with the effluent fee f*. If, for simplicity and without loss of generality, we treat the producers of the final output as a single price-taking firm, its problem will be to choose Q and a in order to maximize profits: = p Q C Q A a f * [e Q a] (21) Its first order conditions for this maximization will be: Q = p C ' Q f * e' Q =0 (22) and a = A' a f * =0 (23) Taking into account (19), we easily see that (22) and (23) are precisely the same as the social optimality conditions (20) and (17) respectively. Thus the setting of the optimal effluent fee leads the price-taking firms to choose the level of output and the level of abatement optimally. How might we represent this integrated approach graphically? The first step is to construct what might be called the integrated marginal damage cost curve. This is the marginal damage cost of emissions, inclusive of any abatement costs borne, assuming that optimal abatement has taken place. The curve may be better explained by how it is derived, which is by looking at the effluent market in a new way. Instead of finding the optimal effluent fee or abatement level by the usual method outlined in Figure 2, we can do so by the process outlined in Figure 3. In panel (i) we have the marginal cost of abatement curve, A'(a), and in panel (ii) we have the marginal damage cost of emissions D'(e). The horizontal sum of these two curves is the integrated marginal damage cost curve, IMDC, seen in panel (iii). It is akin to the firm-level marginal cost curve for a multi-plant firm. If a firm has two plants, we know the optimal division of output between the two plants will always involve setting the marginal costs of production in the two plants equal. In the same way, the optimal level of abatement always involves setting the marginal cost of abatement equal to the marginal benefit of abatement (i.e., the marginal damage from emissions). Summing the two curves from panels (i) and (ii) of Figure 3 will achieve precisely that. We simply plot the summation of the two curves against the level of emissions ē to obtain the marginal cost of emissions at the optimum. Reading back to panels (i) and (ii) gives us the optimal level of abatement, a*, and the optimal level of emission discharge, e*. Of course the optimal effluent fee f* would just be the optimal marginal damage cost at the optimum, as shown
by (19). And if a quantity intervention were preferred to the fee, the abatement level could be set at a* or the emission standard could be set at e*. Thus this approach solves the problem of optimizing the level of abatement; Figure 3 replaces Figure 2. 7 $ $ $ A'(a) D'[e(Q)] IMDC 0 a* a 0 e* e 0 ē e,a (i) Abatement (ii) Emissions (iii) The IMDC curve Figure 3: The Integrated Approach to Optimal Abatement How does this affect the output market? This was a question that the traditional approach was unable to answer easily, but the integrated approach answers this straightforwardly. The IMDC curve can be converted to an integrated marginal external cost curve IMEC (to replace the MEC in Figure 1) by simple multiplication by the function e'(q). Equation (18) had outlined the first-order condition for the output choice: price had to equal private marginal cost plus the marginal external cost D'[e(Q)-a] e'(q). This term, D'[e(Q)-a] e'(q), is the IMEC, because it includes the effect of optimal abatement. Since the IMDC curve is D'[e(Q)-a], multiplying it by e'(q) converts it into the IMEC curve. This IMEC curve is everywhere below the MEC curve of Figure 1, to reflect the use of optimal abatement at all times. The social marginal cost curve SMC in Figure 4 is the vertical sum of the private marginal cost and the IMEC. $ SMC=PMC+IMEC S(Q)=PMC p 2 * p 0 B' C' A D(Q) 0 Q 2 * Q 0 Quantity Figure 4: The Output Market in the Integrated Approach
On the surface, Figure 4 looks very much like Figure 1, which showed the output market in the traditional approach with no abatement. However, Figure 4 represents a substantively different situation. The IMEC curve takes into account the optimal abatement taking place in the effluent market. The distance B'C' looks superficially similar to the distance BC in Figure 1. But they are different. BC in Figure 1 was equal to the optimal Pigouvian tax t*. This was calculated in (12) to be: t * =D ' [e Q ] e' Q 8 B'C' in Figure 4 is a variant of the optimal effluent fee, adjusted for the units of measurement. Equation (19) told us the optimal effluent fee, which was equal to the IMDC: f * = D' [e Q * a * ] And we know the distance B'C' is the IMEC. Therefore, B ' C '= f * e' Q (24) This could be thought of as the effective Pigouvian commodity tax rate. The optimal effluent fee does change the equilibrium in the output market. The optimal output level falls from Q 0 to Q 2 * and the price will rise from p 0 to p 2 *. Thus the effluent fee affects the output market as if there had been a Pigouvian commodity tax of f* e'(q). Note that, because the IMEC is everywhere lower than the MEC from Figure 1, this effective tax is smaller than the optimal Pigouvian tax we saw in Figure 1; the commodity market is distorted less because of the presence of abatement activity. This is why it is better to intervene in the effluent market rather than the commodity market: it results in a lower distortion. But the output market is distorted, something we see clearly here, but which was not apparent in the traditional approach. Thus the integrated approach allows us to easily see the interaction between the effluent and commodity markets. We can now perform comparative statics exercises, such as studying the effects on output and emissions of improved abatement technology or increases in the damage caused by pollution or increases in the level of pollution. In this way, the integrated approach gives us a new tool with which to study and analyze clearly the effects of policy in the presence of externalities.
9 References Nicholson, Walter and Snyder, Christopher (2008), Microeconomic Theory, Basic Principles and Extensions (10 th edition), Thomson Southwestern. Perloff, Jeffrey M. (2008), Microeconomics, Theory & Applications with Calculus, Boston: Pearson Addison Wesley. Pigou, A.C. (1920), The Economics of Welfare, London: MacMillan. Pindyck, Robert S. and Rubinfeld, Daniel L. (2009), Microeconomics (7 th edition), Upper Saddle River: Pearson Prentice-Hall.