Optimal Pigouvian Taxation when Externalities Affect Demand

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1 Optial Pigouvian Taxation when Externalities Affect Deand Enda Patrick Hargaden Departent of Econoics University of Michigan Version of August 2, 2015 Abstract Purchasing a network good such as a cell phone generates the positive externality of aking phones ore useful for others. Should we subsidize cell phones? If pollution/fues fro cars ake couting by foot less desirable, people becoe ore willing to pay road tax. How then should we tax car pollution? To address these questions, I extend and generalize the standard optial coodity tax odel to let the deand for an externality-generating good depend on society s total consuption of that good. The optial tax rate depends on three factors: the deand elasticities of the good, the arginal social cost, and the response of consuption to the externality. These factors are additive and separable. I find that if a negative externality affects consuption enough, the optial policy can be to subsidize it. 1 Introduction This paper derives the optial tax rate for a coodity whose deand is affected by its own externality. Exaples of such goods include autoobiles, fashionable attire, credit card achines, languages, cell phones, and office software. The contribution of this paper is a generalization of the result of Sando 1975, who in turn generalized Pigou 1920 and Rasey When the governent needs to raise revenue, how should we tax externality-generating goods? Sando showed the solution was an additive and separable weighted average of Pigouvian taxation and a for of the Rasey inverse-elasticity rule. This paper extends that odel, generalizing it to allow those goods whose deands are affected by externalities, and again finds that the solution has can be considered an additive and separable weighted average. 1

2 Many externalities do not directly affect deand. If a widget factory pollutes a lake, the deand for widgets does not change. However this general principle is inapplicable to any goods and services. The deand for cars is affected by traffic congestion. How fashionable a particular shoe is depends on how any others purchase the sae style of shoe. There is little benefit to being the sole Esperanto-speaker in the city, or the only person with a fax achine. More generally, the deand for an iportant set of goods including but not liited to network goods depends on the consuption behavior of other individuals. To y knowledge, the optial policy in relation these goods has not been investigated. The taxation of these goods is thus of interest to public econoists. However, this paper is not of solely theoretical interest. I argue that this odel is applicable to a large class of goods. Society has observed the rise of fax achines, telephones, coputer operating systes, and the internet. With technological progress, the relevance and proinence of network goods in particular will certainly significantly increase. Many will becoe part of the tax base. For this reason, this paper is relevant to policy as well as to public econoic theory. 2 The Model Consistent with the literature I odel a governent optiizing the unweighted su of utilities for n identical consuers subject to a governent budget requireent T. Lup su taxes are infeasible, leisure is the untaxable good, and consuers purchase goods based on tax-inclusive prices. Each consuer chooses labor effort x 0, the wages of which act as a nueraire, and the copleent of labor is leisure. The consuer s proble is thus to axiize u i = u1 x 0, x 1,..., x, α 1 L = u. + λ x 0 P i x i 2 In addition to labor, there are taxable coodities in the econoy. Coodity generates an externality α, which for siplicity we will think of as total consuption of x. I denote u i as the derivative of the utility function with respect to x i, and therefore denote the derivative of utility with respect to α as u +1. If we are considering a negative externality, then u +1 < 0. I assue that consuers do not consider their own effect on the externality, and that the usual conditions for an interior axiu hold. 2

3 The solution of the consuer s proble requires the following two first-order conditions: F OC 1 : u 0 = λ 3 F OC 2 : u i = λp i, i = 1,..., 4 Note also fro the budget constraint that x 0 + P i x 0 + x k = 0 x k = x 0 P i. I follow convention by peritting governents to adjust a price vector P to axiize society s indirect utility V P : V P = u [1 x 0 P, x 1 P,..., x P, αp, αp ] 5 Note that this forulation perits both that deand for x be a function of α, and that α directly affects utility. The case of cars and air pollution ay be a useful exaple. The welfare effect of adjusting the price of good k is: 1 V P x 0 = u 0 + Substituting in 3, 4: = u 0 x 0 + x u i + u u i + u = λ 0 + λp i + u P k x 0 = λ P i And using the fact x k = x 0 V P P i = λx k + u + + u, we conclude that Equation 10 will be useful shortly. Define the governent s proble as the axiization of V P subject to raising a budget of at least T. Define t i, the tax on good i, as the difference between the final price and the producer price: t i = P i p i. Iplicitly this is assuing perfectly copetitive production arkets. This is an unreasonable assuption for any of 3

4 the goods considered, but generalizing the optial coodity tax to address for arket iperfections is not the focus of this paper. The governent axiization proble can be suarized as L = nv P β [ n ] P i p i x i T Now using Equation 10, we can see that a necessary condition for the optial coodity tax rate is: L = λx k + u β This can be siplified. Noting that = n, t i = λ + β β [ t i + x k ] 11 = 0 12 x k + n x u +1 + u 13 β Using the terinology of the existing literature, let the coefficient atrix on t i the transpose of the acobian of the taxable goods deand functions be denoted. Further let det and denote ik as the cofactor of the eleent in row i, colun j of. Then, applying Craer s Rule: [ ] λ + β t k = x + n β β u +1 + u P i ik 14 As per Sando 1975, it can be shown that: P i ik = 0 for k for k = 15 Consequently, λ + β t k = x + n β β t k 1 λ = P k β + 1 x + n λ 1 β λ P k u +1 + u P k 1 {k=} 16 1 {k=} 17 u +1 + u Defining θ i as the tax rate on good i, i.e. θ i t i /P i and µ as the negative of the ratio of 4

5 Lagrangian ultipliers, i.e. µ λ/β, 1 θ k = 1 µ x P k Finally, substituting fro 4 and rearranging, we have: 1 nµ u +1 + u 1 {k=} 18 λp [ 1 θ k = 1 µ x ], k = 1,..., 1 19 P k [ 1 θ = 1 µ x ] [ ] [ ] i i u+1 x µ n µ n 20 P This solves for the optial tax rates. Equation 19 shows that the tax rate on the 1 typical goods is a for of the Rasey inverse elasticity rule, consistent with the findings of the previous literature. Equation 20 defines the optial rate for the th good. It shows that the tax coprises three additively separable coponents: the first, a Rasey-like factor which decreases in the elasticity of the good; the second, a Pigouvian factor increasing in the agnitude of the direct externality; and the third, an adjustent for how consuption responds to the externality. The departure of Equations 19 and 20 with previous research is the third consuption response coponent. If consuption does not depend on the externality, e.g. when the deand for widgets in unaffected by pollution in a lake, the consuption response coponent is zero and the optial tax rate collapses to that found by Sando This can be shown ore clearly by grouping the final ters in Equation 20 together: [ 1 θ = 1 µ P x ] i i u [ u+1 + µ n + x ] u With this forulation, we can interpret the result as a weighted average with weight µ of Rasey taxation and adjusted-externality taxation. There ay be disutility caused by α, but the extent to which α increases consuption of x can itigate that negative effect. Indeed, if x > u +1 u, then the optial policy is to effectively subsidize the negative externality. 2.1 Soe Illustrative Exaples The optial tax rate forula derived in this paper can be applied to a variety of contexts. In this section I illustrate the coponents of the odel with three exaples: vehicles and air pollution, vehicles and traffic congestion, and cell phones. Although this odel is static, I also show how the intuition could be interpreted in an interteporal setting. 21 5

6 Vehicles and Air Pollution The eission of particulate atter and nitrogen oxides by cars and trucks are textbook exaples of negative externalities. Let us assue that the direct effect of increased air pollution is to lower society s welfare, i.e. u +1 < 0. Indirectly, perhaps beyond a certain threshold, air pollution would discourage couting by foot. This would ake car usage ore appealing. Stated in the language of the odel, x > 0. Applying this logic to Equation 20, we note that the negative sign on u +1 tends to increase the tax on cars, but that is counterbalanced by the positive sign on x. Whether the tax rate θ is larger or saller than the standard Rasey rate depends on whether u +1 > x or u +1 < x. In the latter case, the optial tax rate is lower than the Rasey rate: the optial policy is to lower taxes i.e. effectively subsidize the negative externality. Vehicles and Traffic Congestion Traffic congestion, like air pollution, negatively affects social welfare. However it diverges fro air pollution in that congested roads akes car ownership less desirable. Therefore in the language of the odel, u +1 < 0 and x < 0. Here both coponents of the externality are undesirable and the tax rate unabiguously rises above the Rasey rate. Cell Phones For this illustration, we assue that the use of cell phones generates no direct externalities. 1 That is, one person s cell phone use does not directly affect another s utility. However the act of joining a cell phone network has the effect of aking cell phone usage ore desirable for others. In this case, u +1 = 0 and x treat the x rate accordingly. > 0. Fro Equation 20 the optial policy is to coponent identically to a typical positive externality, and to lower the tax Dynaic Effects The odel of this paper is static. ust as the logic of hoogenous agent odels can be extended to odels of heterogenous agents, discussion of dynaics is not precluded by the fact that this odel is static. If the intuition of the odel can be extended to a ulti-period setting then we can ake inferences about the interteporal iplications of the odel. This allows us to consider the entire dynaic path of taxation, rather than siply the rate at any tie t. 1 It has been suggested that there are negative epideiological consequences of increased electroagnetic radiation, but let us oit that for the sake of arguent. 6

7 Suppose a network good is developed, that initially the good has only a few users, and thus deand that is quite elastic. It is plausible in this case that the principal/largest effect in the optial tax forula is the x ter. Then, by Equation 20, the good should initially be subsidized. If after a few years growth the x tends to zero, the principal effect ay well be the Rasey taxation coponent. Under these conditions, the good should then be taxed. This intuition can clarify the counterintuitive logic that governents should subsidize early adoption of cell phones, and tax the when they becoe popular enough. 3 Conclusion The priary contribution of this paper is the derivation of optial tax rate for network goods. I developed a odel where the deand for an externality-generating good is affected by its own externality. This setting is applicable quite broadly: to odern network goods like phones and operating systes, but also to a variety of other iportant non-network products like autoobiles and clothing. Indeed any coponent of the utility function which is affected the society s total level of consuption can be investigated with this odel. The optial tax rate generalizes previous results fro the literature, including those of Pigou 1920, Rasey 1927, and Sando The tax rate coprises three additively separable factors related to substitution elasticities, the agnitude of the direct externality, and the effect of the elasticity on consuption behavior. Extending the odel to heterogenous consuers is left for future work. References Pigou, A. C The Econoics of Welfare, London: Macillan and Co. Rasey, F. P A contribution to the theory of taxation, Econoic ournal, pp Sando, A Optial taxation in the presence of externalities, Swedish ournal of Econoics, pp

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