THE PIGOVIAN TAX-SUBSIDY POLICY WITH PRODUCTION EXTERNALITIES UNDER A SIMPLIFIED GENERAL EQUILIBRIUM SETTING* By MASAAKI HOMMA

Transcription

1 THE PIGOVIAN TAX-SUBSIDY POLICY WITH PRODUCTION EXTERNALITIES UNDER A SIMPLIFIED GENERAL EQUILIBRIUM SETTING* By MASAAKI HOMMA 1. Introduction The Pigovian tax-subsidy policy has been authorized as a means of attaining an efficient allocation of resources under a competitive equilibrium with externalities.1) However, there are, at least, three difficulties for the effective implementation of the policy. The first is related to the fact that in order for the Pigovian policy maker to be effectively enforced correct information on the economic environments with externalities must be obtained and optimal size of taxes and subsidies directly calculated by the policy maker. This has been pointed out by Coase (1960), Turvey (1963), and Davis and Whinston (1966). To cope with this difficulty, we have to design a successive adjustment system that leads the policy maker to optimal taxsubsidy structure. At least two tax-revision processes have been proposed. One procedure would revise "statically" the tax levied on each firm generating externalities so as to equate the current tax rate to the current marginal external effect as is assumed by Baumol (1972), while the other would adjust "adaptively" the tax in proportion to the difference between them as formulated by Homma (1973).2)The second difficulty stems from the fact that an efficient allocation of resources may be not attained through the Pigovian tax-subsidy policy because of the possible game theoretic situation. This problem has been dealt with by several authors. Wellisz (1964), Williams (1966), and Olson and Zeckhauser (1966) derived graphically a reaction curve process for two economic units. With regard to this problem, a matter of primary concern is how each firm behaves in the presence of externalities to realize the profit maximizing output. It is assumed in Baumol (1972) that each firm follows the "Marshallian" adjustment, i.e., each moves its own supply quantity in the same direction of marginal profit. More precisely, each firm adjusts its own supply in proportion to the excess current demand price minus current tax (subsidy) over supply price (private marginal cost with respect to its own output). The * The author is grateful to Professors K. Inada, K. Kuga, S. Royama, T. Watanabe of Osaka University for valuable suggestions and the anonymous referee of this journal for helpful comments. All remaining errors are of course his own. 1) Coase (1960), and Buchanan and Stubblebeine (1962) have cast doubts on the effective implementation of the policy. However, their understanding may be seriously confused. Their confusion has been cleared away by Negishi (1972), Shibata (1972) and Starrett (1972). 2) See also Okuguchi (1974). He analyzed global stability of a dynamic Pigovian policy

2 third difficulty is concerned with the distributive aspect of the Pigovian tax-subsidy policy. It is the question who pays the subsidy or to whom the tax proceeds are returned.3) It is a matter of value judgement what kind of the distributive policy should be associated with the Pigovian tax-subsidy policy. Within a general equilibrium framework, the Pigovian tax-subsidy policy never fails to entail some normative aspect in answerning such the question. In this paper we would like to focus our attention on a reconciliation of these problems mentioned above in a simplified general equilibrium setting. In so doing we shall follow the standard procedure as described below. In Section 2 we present our model which characterizes the presence of production externalities, and give the definition of the Pigovian tax-subsidy equilibrium. In Section 3 we formulate the model of a dynamic adjustment process, and examine the stability property of the system. In Section 4 concluding remarks will be briefly mentioned. 2. Notations and the Model 2.1 Throughout this paper we shall maintain the following notations: dji=amount of the j-th commodity by the i-th consumer; xj=amount of the output of the j-th industry; j=l,2,***, m, =amount of labor supplied by the i-th consumer; i=1, 2, ***,n, li =amount of labor employed by the j-th industry; j=1, *j 2,***, m, dj=*ni=1dji***total demand of the j-th commodity; j=1, 2, ***, m, pj=the market price of the j-th commodity; j=1, 2, ***, m, tj=a per unit tax rate (subsidy rate, if it is negative) levied on the j-th industry generating external diseconomies (economies) by the Pigovian policy maker; j=1, 2,***, m, Ti=a lump-sum subsidy (tax) paid to the i-th consumer from the Pigovian policy maker; i=1, 2, ***, n, j=profit of the j-th industry; * j=l, 2,***, m, =proportion of the profit of the j-th industry *ij distributed to the i-th consumer; i=1, 2, ***, n, j=1, 2,***, m, di=(d1i,***, dji, dmi, -li)***consumption vector of the i-th consumer; di=(d1i, ***, dji,***, dmj); i=1, 2, ***, n, i=1, 2,, n, x=(x1,***,xj,***, xm)***commodity vector, =(x1,***, xj-1, xj+1,***, xm)*** externality xj vector of the j-th industry suffering from other industries; j =1, 2,***, m, p=(p1,***,pi,***, pm)***market price vector, t= (t1, ***,tj,***,tm)*** tax rate vector for externality, 3) See, for instance, Turvey (1963)

3 August 1975 M. Homma: The Pigovian Tax-Subsidy Policy with Production Externalities Additional notations will be introduced at relevant places. Let * stand for any of the variables defined above. *(*) denotes the value of the variable at time *. When there is no misunderstanding, all variables are denoted without explicitly referring to time *. 2.2 Let us give the outline of the economy. The economic system to be considered consists of n consumers indexed by i(i=1, 2, ***, n), m industries indexed by j (j=1, 2, ***,m), and the Pigovian policy maker. There are (m+1) commodities, among which the last one is the primary factor called labor (numeraire) whose price set at unity without loss of generality. For simplicity's sake, each industry is assumed to be composed of a representative firm. Each produces its own single output by making use of labor and generates externalities to all other industries by way of the output in the process of production.4) Under this circumstance, we can assume that the production set of the j-th commodity is formalized as the function Fj is assumed to be twice-continuously differentiable and Fj/**j is assumed to be positive. By the implicit function theorem, we can assert * the existence of a function Vj such that The i-th consumer's utility is represented by the function Ui is twice-continuously differentiable. Assuming that the social welfare function W is an individualistic, we shall let W take on the special form ai's are positive fair shares, and *ni=1*i=1. In equilibrium all markets must be clear. That is, we have and The planning problem, therefore, is to maximize social welfare function (4) subject to the production function (2) and the market clearing conditions (5) and (6). We write the Lagrangian of the problem as and obtain the following first order conditions: 4) Murakami and Negishi (1964) showed that the technological efficiency of individual firms and the optimal allocation of resources may be mutually inconsistent when output of a firm generates external diseconomies and input effects external economies. To escape from this difficulty, we assume that input effects no externality

4 pi, * and *j denote the Lagrange multipliers associated with the constraints. From the first order conditions (8)-(11), we can derive the following Pareto optimality conditions: It is noted that the first term of R.H.S. corresponds to the marginal external effect (the marginal net social cost) of the k-th industry and the second term corresponds to the marginal private cost of the k-th industi y. 2.3 We call a tuple (d*, x*, p*, t*) the Pigovian tax-subsidy equilibrium configuration if it satisfies the following conditions: Condition M (market equilibrium conditions)5) Condition C (the equilibrium of consumers) is a maximum vector of the utility *i of the current consumptions, subject to the budget equation Under this regime, the demand price will satisfy Condition F (the equilibrium of representative firms) xj is a maximum point of the profit, subject to the production possibility set (2) and the suffering externality level xj. The solution to this problem can be shown to satisfy the following necessary conditions:6) 5) In our model, Walras' Law is expressible by the following identity: Taking Walras' Law into consideration, we need not explicitly refer to the equilibrium condition for labor market

5 August 1975 M. Homma: The Pigovian Tax-Subsidy Policy with Production Externalities Condition P (the behavior of the Pigovian policy maker) The Pigovian policy maker chooses the corrective tax rate t so as to match with the marginal external effect: Condition B (the balanced budget of the government) The tax proceeds collected from firms are fully redistributed to consumers in a lump-sum fashion: A mention may be needed that the Pigovian tax-subsidy equilibrium (d*, x*, p*, t*) satisfies all the requirements of Pareto optimality condition (12). 3. The Dynamic System and its Stability 3.1 We turn now our attention to the dynamic adjustment system that reflects the price adjustment of the market, supply quantity move of each firm, tax revision of the Pigovian policy maker, and the income redistributive formula. Our dynamic system consists Assumption M of the following set of assumptions: The price of the commodity moves in the direction of and proportion to currentt excess demand. This is nothing but a simplified version of a modern formulation of the "Walrasian" tatonnement which represents the well-known "the law of supply and demand": *j's are positive adjustment coefficients and the dot over variables denotes its time-derivative. Assumption C Each consumer realizes instantaneously the utility maximizing demand vector d* in response to the change in prices and income. Under this circumstance, the following relations will be derived from differentiating the budget equation (15) and the demand price (16): 6) We should note here that no firms can control the suffering externality level xj, which are taken as parameter. It then follows that the profit maximizing output depends not only on the price minus tax rate but on the suffering externality vector from other firms

6 Rjgi=*Rji/*dgi, Rjli=*Rji/*(-li) and the like. Assumption F Each firm adjusts its actual output "adaptively" in the same direction of and proportion to the marginal profit, assuming that the suffering externality from other firms will remain the same level as in the immediately proceeding period: Assumption B The Pigovian policy maker redistributes the tax proceeds collected from firms generating externalities so as to keep all the consumers equitable in the process of dynamic adjustment in the sense that the enjoyable utility levels of consumers move in the same direction. In order to meet such a requirement, the tax proceeds are distributed to each consumer by the following rule; We should add some comments on the dynamic system being composed of Assumptions M, C, F, P and B. It should be noted that all the consumers are kept equitable in the sense stated above. In order to see this, it is enough to show that the sign of the utility-change is identical to all consumers throughout the dynamic adjustment process. Differentiating (14) with respect to time r and substituting (16), we obtain Ui=Ui/Uli with U,li**Ui/*(-li). By making use of (22), (23), (25) and (27), the relation (28) can be rewritten as -126-

7 August 1975 M. Homma: The Pigovian Tax-Subsidy Policy with Production Externalities We can easily check that the sign of Ui is independent of i, and thus obtain the desired result. It should be, moreover, emphasized that our dynamic system is compatible with Walras' Law throughout the adjustment process. Substituting (27) into (23) and summing over individuals gives Differentiating (2) with respect to time *, we obtain It follows from these two relations (30) and (31) that We are in a position to assert the following theorem: Theorem (The local quasi-stability of the Pigovian tax-subsidy equilibrium) If an economy satisfies Assumptions I and II, then the Pigovian tax-subsidy equilibrium (d*, x*, p*, t*) is locally quasi-stable in the sense that the time path described by the dynamic system consisting of Assumptions M, C, F, P and B, converges to any of the 7) The convexity of individual production set implies the aggregative convexity, but not conversely. This exactly corresponds to the relation obtained by differentiating Walras' Law with respect to time *. 3.2 We shall now examine the (local) quasi-stability of the dynamic adjustment system associated with the Pigovian tax-subsidy equilibrium. For this purpose, let us introduce the following assumptions on the utility function and the aggregative production set. Assumption I Each consumer has a quasi-concave utility function with Uli>0. Assumption II The aggregative convexity of the production set is satisfied, i.e., the functional matrix is positive quasi-definite for any x,7)

8 equilibria as time tends to inifinity, starting from any initial point (d0, x0, p0, t0) sufficiently close to it. Proof. Let us define N as a Lyapunov function by the following: Differentiating (34) with respect to time * and taking (20), (22), (25) and (26) into account, we obtain We shall now focus our attention on the first term of R.H.S. in (35). We can derive the scalar product *pjdji within the bracket by the following procedure. It r follows from substitution of (28) into (24) and rearrangement that Here, and Premultiplying (36) by row vector di', we obtain This corresponds to the scalar product *pjdji in the bracket of the first term. 1 Substituting (39) into (35) leads to It can be easily shown that N is always negative in the neighborhood of the Pigovian tax-subsidy equilibrium value. First, it follows from Assumption I, i.e., quasi-concavity of utility function that the functional matrix Ai is negative-definite owing to the well-known result on the theory of related goods.9) This implies that the first term within the bracket is always negative unless di=0. Next, the second term within the bracket is negligible in the neighborhood of the Pigovian tax-subsidy equilibrium value, i.e., pj=sj=0. This follows from the relation (29).10) Last, it can be seen from Assumption II, i.e., the aggregative convexity of the production 8) A prime after vector denotes its transpose. 9) See, for instance, Morishima (1955). 10) It is noted that the first term within the bracket of (40) corresponds to the substitution effect, and the second term corresponds to the income effect

9 August 1975 M. Homma: The Pigovian Tax-Subsidy Policy with Production Externalities set that [-V(x)] is negative quasi-definite and thus the second term in (40), i.e., x'[-v(x)]x is always negative unless x=0. Thus d, x, p, and t locally approach d*, x*, p* and t* that fulfill all the conditions of the Pigovian tax-subsidy equilibrium, as time tends to infinity. Q.E.D. 4. Concluding Remarks In this paper we have formulated the Pigovian tax-subsidy policy with production externalities under a simplified general equilibrium setting, and investigated the dynamic adjustment system associated with the policy. The analysis suggests two areas for further research. Firstly, we are only concerned with the case in which each economic agent forms the "static" expectation on the suffering externality level. Alternative assumptions on expectation formation will be introduced into the model. Secondly, our attention is confined to the specified redistribution formula closely related to that proposed by Malinvaud (1972). The enjoyable utility levels of all economic units depend on what kind of redistribution formula prevails. It would be, however, difficult to reach definite conclusion in determining the general redistribution formula, for value judgement may outweigh in this problem. (Osaka University) REFERENCES [1] Baumol, W. J., "On Taxation and the Control of Externalities," American Economic Review, 67 (June, 1972). [2] Buchanan, J. M. and W. C. Stubblebeine, "Externality," Economica, 29 (November, 1962). 3] Coase, R. H., "The Problem of Social Cost," The Journal of Law and Economics, [ 3 (October, 1960). 4] [ Davis, O. A. and A. B. Whinston, "On Externalities, Information and the Government-assisted Invisible Hand," Economica, 33 (August, 1966). [5] Homma, M., "A Dynamic Pigovian Policy with Production Externalities," Economic Studies Quarterly, 24 (August, 1973). [6] Malinvaud, E., "Prices for Individuals Consumption, Quantity Indicators for Collective Consumption," Review of Economic Studies, 34 (October, 1972). [7] Morishima, M., "A Note on Definitions of Related Goods," Review of Economic Studies, 23 (2, 1955). [8] Murakami, Y. and T. Negishi, "A Note on a Formation of External Economy," International Economic Review, 5 (September 1964). [9] Negishi, T., General Equilibrium Theory and International Trade, North-Holland Publishing Co., [10] Okuguchi, K., "Global Stability of a Dynamic Pigovian Policy," Economic Studies Quarterly, 25 (August, 1974). [11] Olson, M. and R. Zeckhauser, "An Economic Theory of Alliances," Review of Economics and Statistics, 48 (August, 1966). [12] Shibata, H., "Pareto-Optimality, Trade and the Pigovian Tax," Economica, 39 (May, 1972). [13] Starrett, D. A., "Fundamental Non-Convexities in the Theory of Externalities," Journal of Economic Theory, 4 (August, 1972). [14] Turvey, R., "On Divergences between Social Cost and Private Cost," Economica, 30 (August, 1963)