Stock externalities, Pigovian taxation and dynamic stability

Journal of Public Economics 33 (1987) 59-72. North-Holland

STOCK EXTERNALITIES, PIGOVIAN TAXATION AND DYNAMIC STABILITY

Dagobert L. BRITO

Department of Economics, Rice University, Houston, TX 77001, USA

Michael D. INTRILIGATOR*

Department of Economics, University c$ California, Los Angeles, CA 90024, USA

Received March 1985. revised version received October 1986

Stock externalities represent a type of consumption externality in which individual consumption of a particular good leads to production of a public bad, such as in the arms race and in the greenhouse effect. Certain of these stock externalities can be regulated by Pigovian taxation of the good that produces the externality. For others, however, depending on an ordinal property of preferences, the equilibrium becomes dynamically unstable, so it is necessary to use direct regulation to support an optimal equilibrium.

1. Introduction and purpose

Stock externalities represent a type of consumption externality in which individual consumption of a particular good leads to production of a byproduct. This byproduct may be a public bad which adversely affects all individuals. An example is the arms race. Individual nations acquire weapons to increase their security, but the very presence of weapons in other countries itself creates insecurity, resulting in further weapons acquisitions. The result of this stock externality is likely to be unstable, with increased weapons and increased tensions reinforcing one another, resulting in an arms race spiral. Another example is the greenhouse effect. Consumption of fossil fuels leads to the production of carbon dioxide, resulting in higher temperatures. If the higher temperatures lead to burning even more fossil fuels, e.g. due to greater use of air conditioning, then the situation could again be unstable, leading away from rather than towards an equilibrium.

*We would like to thank P. Diamond, A. Dixit, R. Filimon, R. Gretlein, T. Groves, J. Hamilton, W. Heller, Jr., T. Sandier, S. Slutsky and R. Starr for their comments. We have also benehted from conversations with Kenneth Arrow. Finally, we have benefited from very useful comments from the referees. This research was supported by grants for collaborative research from the National Science Foundation. D.L. Brito’s work on cotton production in northeast Mexico was supported by a grant from Rice University in 1970.

0047-2727/87/%3.50 0 1987, Elsevier Science Publishers B.V. (North-Holland)

60 D.L. Brito and M.D. Intriligator, Stock externalities

This paper analyzes these stock externalities to determine their effects and the ways in which they might be controlled. In particular, it addresses two questions concerning stock externalities. First, under what conditions can a stable optimal equilibrium be supported by Pigovian taxation of the good that produces the externality? Second, under what conditions is direct regulation, rather than taxation, necessary to enforce an optimal equilibrium? We derive a property of preferences which makes it possible to divide groups of consumers producing stock externalities into two categories: those which can be regulated by Pigovian taxes independently of group size and, by contrast, those for which the equilibrium is unstable and it is necessary to use direct regulation to support an optimal equilibrium. Thus, certain externalities can be regulated by taxation, whereas others require direct regulation.

We derive a necessary and sufficient ordinal condition on the preferences of agents for the stability of group behavior where consumption is regulated by Pigovian taxation. For groups in which agents’ preferences do not satisfy this condition, regulation of the stock externality by Pigovian taxation is not possible since regulation of the externality by Pigovian taxation is unstable.

This paper draws upon two apparently unrelated literatures. The first is the work of Buchanan and Kafoglis (1963) Diamond and Mirrlees (1973), Sadka (1978) Cornes (1980) and Sandmo (1980) which addresses the problem of consumption externalities in which the good consumed by one agent enters the utility function of other agents. The structure of these models is in many ways similar to those of the second literature, consisting of the papers of Brito (1972), Intriligator (1975), Intriligator and Brito (1976), Simaan and Cruz (1975), Gillespie and Zinnes (1975) and Brito and Intriligator (1985) on the arms race. This literature treats the arms race as the outcome of rational behavior in which arms lead to insecurity, which, as already noted, is a type of stock externality. Both literatures address the question of the interaction of consumption externalities and the level and stability of the resulting equilibrium. This paper differs from previous ones in that the externality produced is a stock, like the global level of armaments or of carbon dioxide, rather than a flow, like the increase in the level of weapons or of carbon dioxide produced by the current burning of fossil fuels, and in that the externality enters the utility function of all agents.

We will show that if the good whose consumption produces the stock externality is a gross substitute with respect to the externality, then it is possible to use Pigovian taxation to sustain the equilibrium. However, if the good that produces the stock externality is a gross complement with respect to the externality, then it may not be possible to sustain the equilibrium with Pigovian taxation if the elasticity of demand of that good with respect to the externality is greater than one.

We do not usually expect to observe unstable equilibrium, however there

D.L. Brito and M.D. Intriligator, Stock externalities 61

are some examples. As already noted, if the greenhouse effect leads to greater fossil fuel consumption, then this will produce yet more carbon dioxide

which will further increase the greenhouse effect. The model thus suggests that there could be a potential instability that should be a subject of concern, and it implies that in such circumstances the situation calls for direct regulation rather than Pigovian taxation. The arms. race leads to a similar potential instability that must be treated via direct regulation. Yet another example of a stock externality was the introduction of sorghum in northeast Mexico. This was a very important cotton producing area, yet within five years of the introduction of sorghum the production of cotton had stopped. This was a puzzle, since cotton should have been a more profitable crop. The explanation, as given by entomologists, was that sorghum harbored certain insects that attacked the cotton crop, increasing the cost of insecticides so that cotton eventually became unprofitable. Thus, production of sorghum produced a stock externality, damage due to insects, which changed the relative costs in favor of the production of sorghum.

This paper investigates stock externalities for three types of maximizing behavior: static/nonstrategic behavior; dynamic/nonstrategic behavior; and dynamic/strategic behavior. In each of these three cases sufficient conditions for the stability of an equilibrium to be supported by Pigovian taxation are investigated.

2. The planner’s problem

Consider N identical individuals, indexed 1,2,.

x= f xi, i=l

(2.3)

62 D.L. Brito and M.D. Intriiigator, Stock externalities

so that

8= ~ [Uyi_6Xi]=U ~ yi-_X. (2.4) i=l i=l

Thus, in this model of stock externalities, increases in the level of the public bad stem from aggregate consumption of the y good, e.g. increases in insecurity stemming from aggregate purchases of weapons in the arms race and increases in carbon dioxide in the atmosphere stemming from aggregate

consumption of fossil fuels. The representative individual consumes yi and zjr and the utility function

ld = u(x, Yi, zi) (2.5)

gives utility as a function of the public bad and the consumption levels of both goods. u is assumed to be a continuously differentiable strictly concave function which depends negatively on pollution (u, ~0) and positively on the consumption of both goods (u,, u3 > 0).

The Pigovian tax r in the budget constraint (2.1) is assumed to be constant, and it is imposed on each unit of the good producing the stock externality. The tax could alternatively have been assumed to be a function of X.’ As an example, if y* is the level of consumption that is socially optimal, then imposing a Pigovian tax r = -k for all yi< y* and r = k for yi> y* would result in a stable equilibrium. Other schemes like this can be constructed; however, they are closer in spirit and in informational require- ments to direct regulation than to Pigovian taxes, requiring information on yi, etc2

A planner computing the optimal level of pollution for N identical agents will maximize, by choice of yi and zi:

i e-“Nu(X,yi,zi)dt (2.6)

subject to

(c+T)yi+Zi=l, (2.7)

_t= Nay,--6X. (2.8)

‘An example of a tax on yi based on X would be to vary the price of electric power as a function of air pollution X, which could be controlled by sending a signal to the electric power meter. In fact, this type of technology has been implemented in some locations for peak load pricing.

‘A related argument is that of a group smaller than the ‘minimum size for a viable coalition’ that would fmd it beneficial to choose an inferior option, e.g. consume goods that produce the public bad, as discussed in Schelling (1978, ch. 7).


The Hamiltonian associated with this problem is3

(2.9)

where q is the costate variable associated with eq. (2.8) and y is the Lagrange multiplier associated with the budget constraint given by (2.7).

We will only consider cases where the planner has an interior solution. If the planner’s solution were such that the optimal solution were at X=0, then the solution to the planner’s problem would be the prohibition of the consumption of y. A possible example is the recent prohibition of the consumption of asbestos in the United States. If the planner has an interior solution the first-order conditions are

Nu,+qNa-y(c+T)=o, (2.10)

Nu,-1’=0, (2.11)

tj-(r+d)q= -Nu,. (2.12)

In the steady state 4=0, so

q=NUl r+6’

and the following Samuelson efficiency condition must be satisfied:

(2.13)

(2.14)

The Pigovian tax 7 is chosen so that individuals satisfy (2.14).

3. The simple static/nonstrategic model

The simple model is one in which individuals act myopically at each point in time as static price-takers, so that the consequences for the society as a whole can be obtained from the behavior of a representative individual.

Consider now the case of a representative individual consuming yi and zi and myopically maximizing at each instant of time the utility function:

I,4 = u(x~ Yi, zi), (3.1)

‘The Hamiltonian is strictly concave in X, and if S>O a steady state solution exists and is unique. See Arrow and Kurz (1970, p. 51).

64 D.L. Erito and M.D. Intriligator, Stock externalities

where u is assumed to be a continuously differentiable strictly concave function which depends negatively on pollution (ur ~0) and positively on the consumption of both goods (u,, uJ > 0). The representative individual’s static problem of maximizing u by choice of yi and zi subject to the budget constraint (2.1) leads to the Lagrangian:

L(J1i, Zjr n, = U( X, yi, Zi) + n( 1 - (C + Z)yi - Zi), (3.2)

where 3. is the Lagrange multiplier. The first-order conditions are then

u,-k(c+z)=O, (3.3)

u,-i.=O, (3.4)

1 -(C+T)y-zj=o. (3.5)

Differentiating the first-order conditions with respect to the level of the public bad X and solving for Zyi/ZX yields:

(7X D D ’ (3.6)

where D is the determinant of the bordered Hessian matrix, which is positive. This result will be used to study the stability of the differential equation (2.4).

The differential equation will be stable at an optimal X if

32 (7 -=iXINayi-SX]<O, C’X

or, using (3.6)

(3.7)

(3.8)

Since N, a, uj, D and 6 are all positive, this term will be negative and the equation will be stable if

&(!z)_&(e&o. (3.9)


The term ~?(az/ay)/aX is the change in the marginal rate of substitution

between the private goods as the externality increases, and its sign is an ordinal property of the utility function. If this term is negative then an increase in the stock externality will increase the demand for y, so the good is a gross complement with respect to the externality. This can happen if X and yj are complements or if X and zi are substitutes. If

(3 dz - (-)>O; (7x (?y

so the good is a gross substitute with respect to the stock externality, then the equation will be stable.

A good being a gross substitute is a sufficient but not necessary condition for stability. We will show that a necessary and sufficient condition for the equilibrium to be stable is that the elasticity of yi with respect to X, given by (Sy,/ZX)(X/y,), is less than 1 at the equilibrium.4 This is also a condition that depends only on ordinal properties of preferences.

If we divide eq. (2.8) we get:

8 Nayi -=---- -6. x x

If we differentiate (3.10) and collect terms we get:

(3.10)

(3.11)

Equation (3.11) will be stable if the term [(ayJ~YX)(x/y,)- l] is negative. ??y,/?X<O is a sufficient condition for stability, but it is clearly not necessary.

Stability does not depend directly on any of the technical parameters of the problem such as a, N or 6. However, these parameters may influence stability through the solution of the planner’s problem as the elasticity of y with respect to X may change as the parameters vary. How these parameters influence the optimal level of X does not depend on any general properties of the utility function and would have to be computed using the ‘five equations that characterize the planner’s optimal steady state. If a and 6 are large enough, the rate at which X is created and decays could be high enough so that the stock externality would approximate a flow externality. The necessary and sufficient conditions for stability in the case of a stock would

4Conversations with S. Slutsky were very useful in noticing that the necessary and sufficient condition for stability involved the elasticity of y with respect to X.


then be the necessary and sufficient conditions for stability in the case of a flow externality.

This claim can be checked by noticing that the maximization problems faced by a planner or agent maximizing with respect to a flow externality are mathematically equivalent to the problems faced by a planner or agent maximizing with respect to a stock externality in a steady state. In a steady state eq. (2.8) implies that

NaYi X=-- 6 ’

(3.12)

which would be the equation for the production of X if it were a flow externality.

Theorem 1. If agents adopt nonstrategic behavior and if the good producing the stock externality is a gross substitute with respect to the externality, then the Pigovian tax equilibrium is stable.

Theorem 2. A necessary and sufficient condition for the optimal equilibrium to be stable is that the elasticity of y with respect to X, (~YyJ~?x)(x/y~) be less than 1 at the optimal equilibrium level of X.

The intuition behind these results is easy to see. If the good producing the stock externality is a gross substitute with respect to the level of the public bad X, then an increase in X will cause the indifference curves to shift so the

demand for y will be reduced. Conversely, if y is a gross complement with respect to X, then an increase in X will cause the indifference curves to shift so the demand for y will be increased.

In this simple case Pigovian taxation can always be used to regulate the externality if the good is a gross substitute with respect to the externality. If the good is a gross complement with respect to the stock externality, however, then Pigovian taxation cannot necessarily be used to regulate the externality. Control of the stock externality would in such a case require use of direct regulation rather than Pigovian taxation. The next section shows that similar results hold with regard to Pigovian taxation in the presence of stock externalities if agents are maximizing in a less naive fashion.

4. The dynamic/nonstrategic model

The dynamic model drops the assumption of static maximization of the previous section to consider dynamic optimization. In the dynamic model of stock externalities, individual i maximizes, at time s = t in [0, m) the objective


functional

(4.1)

As before, u is a continuously differentiable strictly concave utility function which decreases with the level of pollution X and increases with consump-

tion of each of the two private goods xi and yi. Here, however, individual i maximizes Ji by choice of paths for the levels of the private goods yi(s) and zi(s) over the periods from t to co, subject to the budget constraint at time s:

(C+T)yi(S)+Zi(S)=l, O~S<cO. (4.2)

Individuals are therefore continually revising their consumption levels at each instant t in [0, co), maximising Ji relative to the initial point t.

Each individual must make some assumption as to the behavior of other individuals since the maximization of Ji by choice of vi(s) and zi(s) depends on X and hence on the behavior of other individuals. It will be assumed in this section that individual i is nonstrategic in forming his or her expectations. Let y;(t) be the ith individual’s expectation of the consumption of the representative other individual. Thus, using (2.4), at time s:

8;=(N- l)ay;+ay,-6X;, (4.3)

where XF is the expected change on the part of the ith agent in the level of total pollution stemming from all other individuals consuming y; and the ith individual consuming yi.

The Hamiltonian for the optimization problem faced by the ith agent at time t, given his or her expectation yr, is

H=u(X;, yi,Zi)+pi[(N-l)~y~+Uyi-6X;]+~i[l-(C+T)yi-~i], (4.4)

where pi is the costate variable associated with the differential equation (4.3). The first-order conditions for the solution to this problem of stock

externalities in this dynamic/nonstrategic case are:

Uz(X;, yi, Zi) +piU-~~i(c+ T)=O, (4.5)

U3( xp, yi, Zi) - ii = 0, (4.6)

Pierpi= -U,(Xr, 4’i, Zi) +6pi. (4.7)

Define p = r + 8. Since fii = 0 at the steady state, the first-order conditions can be written:

U2(Xp> yi, zi) + UU,(Xr, _Vi, zi)/p - %i(C + z, = O, (4.8)

UJX;, yi, ii) - %, = 0, (4.9)

(c+r)yi+zi=l. (4.10)

There is no reason to expect that in the absence of some form of intervention the solution is optimal, but a Pigovian tax can be used to set the equilibrium level of the externality at the optimal level. The first-order condition for the steady state solution of the agent’s problem is

where r is chosen to satisfy the first-order condition:

!+++!E !!!. , P 0 u3

(4.11)

(4.12)

as in (2.14). The question that has to be answered, however, is whether this equilibrium is stable. Eqs. (4.2), (4.5), and (4.6) can be solved for y(X, p). Differentiating the first-order conditions with respect to X and solving for CyJ?X yields:

1 (4.13)

where ap/?X is understood to be along the stable trajectory of the optimal control problem and where D is the determinant of the Hessian of the Hamiltonian. Eq. (4.11) can be used to substitute for (c+r). Using p=u,/p from (4.7) yields:

(4.14)

The term 8(u,/u,)/~YX is familiar from the static myopic case. It describes the change in the marginal rate of substitution between y and z as X changes. The term UC!QI/U,)~X can also be given an economic interpretation. If I is the


income of the ith agent then uJr = i/r = aJi/i?l. By definition p = aJ,/aX. Thus

the term d(p/u,)/aX can be written:

(4.15)

where ?Z/aX refers to movement along an indifference curve in I-X space. CI(p/u,)/aX ~0 implies that a(aZ/aX)/aX>O, which can be interpreted to mean that nonpollution is a superior good.

The sign of a(p/uJaX is difficult to compute directly. Let X*[s, X(T)] be the solution to this problem given initial conditions X(t). Now replace p in eq. (4.13) by its value at the steady state, tr,/p, and let X[s, X(t)] be the suboptimal solution that results from solving the first-order conditions with

this substitution. Let

3,(X(t))= 4 emrcser) 4% X(t)), 9h X(O), @, X(t))lds, (4.16) f

JT(x(t)) = 7 eprcs-‘) uCX*(s, x(t)), Y*(s, x(t)), z*(s, x(t))lds, I

(4.17)

and X** be the solution to the steady state given by the Samuelson efficiency condition (2.14). Then for all X 2 X** and t:

+[_Ti(x)- ~;(x**)]* &[J:(x)- ~:(x**)]*. (4.18)

If the left-hand side of (4.18) is a Lyapunov function for X2X**, then the right-hand side of (4.18) is a Lyapunov function. Since ayJdX is a con- tinuous function of X, the left-hand limit is equal to the right-hand limit. Thus, sufficient conditions for the stability of 8 are sufficient conditions for

the stability of X*. A sufficient condition for X to be stable is that

The ordinal conditions

~0 and & 2 ~0 0 3

(4.19)

are sufficient for inequality (4.19).

70 D.L. Brito and M.D. IntriliRator, Stock elvternalitirs

Thus in this less naive model of nonstrategic behavior in a dynamic context it follows, as in the previous static case, that regulation by Pigovian taxation is possible if the good producing the stock externality is a gross substitute with respect to the externality.

5. The dynamic/strategic model

The assumption that agents behave in a nonstrategic fashion is reasonable for very large groups. For small groups, however, the assumption is questionable. In this section it will be assumed that agents behave in a

strategic fashion and adopt a Nash strategy.’ In this context the assumption that all agents play a Nash strategy is an assumption that can be justified because it is reasonable to assume that no agent is large enough to become a dominant Stackelberg leader. The differential equation for the stock externality is

(5.1)

where the sum 1 is understood to be from j = 1 to n, but where j # i. The ith agent’s Hamiltonian is

U(X,Yi,Zi)+~i[a(yi+~~j)-~x]+~i[l-(C+~)yi-Zi]~

The first-order conditions for the ith agent are:

u,(X,yi~Zi)+piU-j~i(c+T)=O,

UJ( X, yi, Zi) - 3., = 0,

(5.2)

(5.3)

(5.4)

(5.5)

(C+T)yi+Zi=l, (5.6)

where individual i takes into account his or her indirect effect on the yj through X. Except for the term x(ayj/2X), accounting for these indirect effects, these equations are similar to the nonstrategic case. However, (5.1) and (5.5) are a set of nonlinear partial differential equations with a two-point boundary condition and are very difficult to solve.

Some conditions for local stability can be derived if it is assumed that in the neighborhood of the equilibrium the utility functions of agents can be

‘The use of a Nash equilibrium in an arms race is discussed in Simaan and Cruz (1975).


approximated by a quadratic utility function. ‘Then, for the infinite horizon case the Ricatti equation is algebraic and the term 8yj/aX is a constant. Define p = r + 6 -1 (LYyj/aX). Then if pi =0 at the steady state the first-order conditions can be written:

u2(x?Yi>ziJ +au,(X, .Yi>zi)/p-;li(C+Z) =O, (5.7)

u,(X, yi, zi) -ii = 0, (5.8)

(c+z)yi+zi= 1. (5.9)

If p>O, these conditions are identical to conditions (4.Q (4.9) and (4.10), and the local stability properties of the model in this case of Nash strategic behavior are identical to those in the case of nonstrategic behavior. The sign of p, however, depends on the sign and magnitude of x(8yj/aX).

Theorem 3. If a Nash equilibrium exists then p>O, and the stability pro-

perties are us in Theorems I and 2.

Proof: Suppose p ~0. Then

u2(x,Yi3 zi) -“3(c + z, <Ov (5.10)

which implies that the ith agent can unilaterally improve his or her welfare by consuming more zi and less yi. Furthermore, p ~0 only if c i3yj/8X>0. Reducing yi will reduce X which will reduce yj for all other agents and therefore result in the further reduction of X. Thus p<O is not consistent with a Nash equilibrium.

From Theorem 2, if a Nash equilibrium exists, then the Pigovian taxation equilibrium is stable if the elasticity of the good producing the externality is less than 1. A sufficient condition is that the good producing the stock externality is a gross substitute with respect to the externality.

6. Conclusions

This paper has addressed the problem of the stability of equilibria in the presence of a stock externality, such as that of the arms race or the greenhouse effect for three types of behavior: static/nonstrategic, dynamic/ nonstrategic and dynamic/strategic. An ordinal property of preferences has been derived which makes it possible to divide groups of consumers into two categories: those which can be regulated by Pigovian taxes and those in which the equilibrium is unstable.

A sufficient condition on preferences of agents for the stability of groups


where consumption is regulated by Pigovian taxation is that the good be a substitute with respect to the externality. A necessary and suffkient condition for the optimal equilibrium to be stable is that the elasticity of y with respect to X be less than 1 at the optimal equilibrium of X. This condition also depends on ordinal properties of preferences.

References

Arrow, K.J. and M. Kurz, 1970, Public investment, the rate of return, and optimal tiscal policy (Johns Hopkins Press, Baltimore and London).

Brito, D.L., 1972, A dynamic model of an armaments race, International Economic Review 13, 3599375.

Brito, D.L. and M.D. Intriligator, 1985. Conflict, war and redistribution, American Political Science Review 79, 9433957.

Buchanan, J.M. and M.Z. Kafoglis, 1963, A note on public goods supply, American Economic Review 53, 403-414.

Cornes, R., 1980, External effects: An alternative formulation, European Economic Review, 3077 321.

Diamond, P.A. and J.A. Mirrlees, 1973, Aggregate production with consumption externalities, Quarterly Journal of Economics 87, l-24.

Gillespie, J. and D. Zinnes, 1975, Progressions in mathematical models of international conflict, Synthese.

Intriligator, M.D., 1975. Strategic considerations in the Richardson model of arms races, Journal of Political Economy 83, 339-353.

Intriligator, M.D. and D.L. Brito, 1976, Strategy, arms races, and arms control, in: J. Gillespie and D. Zinnes, eds., Mathematical systems in international relations research (Praeger, New York).

Sadka, E., 1978, A note on aggregate production with consumption externalities, Journal of Public Economics 9, 101~105.

Sandmo, A., 1980, Anomaly and stability in the theory of externalities, Quarterly Journal of Economics 94, 799-807.

Schelling, T.C., 1978, Micromotives and macrobehavior (W.W. Norton & Co., New York). Simaan, M. and J. Cruz, 1975, Formulation of Richardson’s model of arms race from a

differential game viewpoint, Review of Economic Studies 42, 67777.

Documents

Stock externalities, Pigovian taxation and dynamic stability