RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES

RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES*

S. N. JACOB1

The University of Newcastle

INTRODUCTION The problem raised by the existence of technological externalities in an otherwise

competitive society has produced an extensive literature. Through successive refinements in taxonomy and analysis a rather cohesive theoretical picture has emerged.

The so-called pecuniary externalities originally suggested by Marshall and subsequently defined more specifically by J. Viner have been shown to be described best in the context of neoclassical markets, i.e. they are not externalities within the most useful definition of that term. Specifically, externalities arise when effects on production or welfare are not reflected fully in market prices [14].

All technological externalities that are Paretian-relevant, i.e. those which tend to produce an equilibrium away from the Pareto-optimal solution, can be described within the context of an unpriced factor [3, 131.

Within the general set of Paretian-relevant externalities two important taxonomical distinctions have been made : (1) between reciprocal (intra-industry) and non-reciprocal (inter-industry) exter-

(2) between input-generated and output-generated externalities [6, 10, 171. It has also been observed that there is an important asymmetry between external economies resulting in falling supply curves and external diseconomies resulting in rising supply curves [14].

The classical solution to obtaining Paretian optimality when externalities exist involved either taxes or subsidies such that private values of the marginal product were made equal to the social values of the marginal product [16] or revenue- maximizing rents based upon private ownership of the unpriced resource [ll]. That these two approaches produce identical results was subsequently demonstrated [5]. In the case of reciprocal externalities between firms, alternatives would be merger or collusion between the firms ; and it was suggested that mergers to produce “natural units” would be the best solution if competition were not reduced [4].

It was also established that the optimal output after adjustment for externalities will always be less than optimal output if no externalities existed [14].

However, the issue was subsequently raised as to whether the existence of an external diseconomy necessarily produced over-use of a resource’ and hence whether

* The author wishes to express his sincere gratitude to Dr. A. J. Guttman for his numerous comments and suggestions with respect to the mathematical presentations in the paper, in particular with respect to the appendices. The author takes full responsibility for all errors of analysis or interpretation which are necessarily his own.

nalities [4, 14, 191;

‘And conversely whether an external economy resulted in under-use of a resource.

1975 RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES 89

normal tax-subsidy policies would necessarily move the competitive equilibrium toward the Paretian-optimal equilibrium [l, 2,6, 181. The problem was subsequently formulated within the context of production possibility frontiers or transformation schedules and it was demonstrated that in the general case, externalities did not imply a movement along the given frontier (or schedule) upon which the Paretian- optimal equilibrium lay, but rather to a new frontier (or schedule) [6, 191. The two frontiers will coincide only in the case of reciprocal externalities between firms with identical marginal cost and externality-producing functions [ 191.

In virtually all of these discussions, the underlying assumptions were : perfect competition, divisibility of the scarce factor and hence the externality-producing activity, and monetization (at least to a numeraire) of the externality.2

A number of real world problems to which the above theoretical analysis has been applied do not exhibit these assumed characteristics. This paper will examine a paradigm based upon road congestion to bring forward several significant issues and difficulties facing the policy maker in dealing with reciprocal externalities in a real world context. These issues have not been made explicit in the theoretical literature to date.

The Paradigm A firm operates a mineral extraction plant at a remote inland site and transports

the mineral ore to the sea for transhipment by ship. The route to the sea is along a narrow road built by public authority to encourage development of the resource. The firm employs a number of trucks of identical capacity and performance on the road. Because of the narrowness of the road, the trucks must slow down and drive on the shoulder when they pass each other (obviously each truck shipment involves a return journey) with the effect that truck operating costs (driver’s salaries, vehicle operating costs, etc.) rise as the volume of shipments per unit of time, Q, rises. Hence, there is a rising marginal cost curve ( M C ) associated with the firm’s trucking activities. The M C curve represents a technology constraint due to the nature of the road itself.

For reasons wholly independent of the trucking costs, and in particular of the road congestion costs, the firm is faced with a declining marginal profitability for its mineral output, which can be directly related to a declining marginal profitability for truck shipments.

The intersection of the marginal cost curve for trucking congestion with the marginal profitability curve ( M P ) will represent the short run profit-maximizing equilibrium for the firm at Q1 (see Figure l).3

2Schall discusses briefly the effect of monopoly in the industry with externalities, but the discussion seems to do nothing more than verify Mishan’s observation that optimal output after adjustment will be less than optimal output without any externalities, and to suggest that with the monopolist’s internalization of externalities, the resulting industry output may be closer to the ideal equilibrium (i.e. no externalities) than with competition. With no congestion on the road (e.g. due to a wider road) the firm would operate at Q which would be the intersection of the firm’s marginal cost and marginal revenue curves excluding external effects. The M P curve is simply the difference between them.

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FIGURE 1

Marginal cost of road congestion

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Thus the firm's congestion costs have become internal.

Now assume that a second firm opens a mineral extraction plant identical to the first with the same marginal profitability characteristic^.^

The Competitive Equilibrium for two Identical Firms Assume that the M P curves of the two firms are of the form

M P = a - b Q (1)

M C = cQ where a, b and c are positive constants. (2) and that the traffic congestion costs of the road yields an M C curve of the form5:

For a single firm, the equilibrium quantity of truck shipments is given by setting (1) equal to (2), or:

a Q I =b+c (3)

If a second firm, identical to the first, begins operations at the same site, there will now be much greater congestion on the road. The optimum quantity of truck shipments for the industry will obviously not be 2 x Q1 ; but rather it will be given by the intersection of the industry M P curve and the congestion M C curve. The former is given by adding together the two Q values at constant M P (horizontal summing), so that :

(4) b 2

M P I = a - - Q

4The justification for two firms operating at the same site with declining M P curves can be made in terms of the production process (rather than the markets for the goods), e.g indivisibility and declining efficiency of a particular input factor such as a dragline, i.e. it is assumed that the market rice of the mineral resource is unaffected by the output level of the firms. Free entry is assumed to & partially restricted by some external institutional constraint or by some minimum firm size due to the indivisibility just referred to.

'The linear form is selected for expositional convenience only. The conclusions for the general case in which, within the range of production possibilities considered in the analysis, MP is any monotonically decreasing function and MC is any monotonically increasing function are presented in Appendix A.

1975 RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES 91 from which the industry, and hence social, welfare optimum quantity will be found by setting (2) equal to (4), so that :

2a Q’=b+2c

The question is: what will the industry output be if both firms maximize profits independently, without merger or collusion? Remarkably, it will be the same quantity as given in (5).

The second firm, in considering the scale of its operations, will take the first firm’s operations as given. Thus the congestion cost of the first truck for the second firm will not be zero (as it was for the first firm) but will be approximately equal to cQ1 since Q1 trucks are already on the road. The net effect of this observation is that the MC curve for the second firm is shifted upward by the quantity cQ1 :

Equating (1) and (6) gives the profit-maximizing volume, Qz, for the second firm: MC2 = cQ1 +cQ (6)

Q i a C

Qz = b+c-b+c (7)

As soon as the second firm begins operation, the first firm will experience a shift in its apparent MC curve of cQZ, since it will be forced to accept the volume of trucks on the road due to the second firm, Qz, as given (i.e. beyond its control). If we denote each firm’s adjustment by a prescript which indicates the sequential number of the adjustment, then the new profit-maximizing volume of the first firm is given by the solution of:

a-bQ = cQ,+cQ which we denote lQ1, so that

Qz a C lQ1 =

b + c b + c

for notational simplicity, and substitute (3) into If we let x = -any y = ~

b + c b + c a C

(7) and then (7) into (8) we get : iQi = X - Y ( X - X Y )

= x ( l - y + y Z ) (9)

The adjustment process is not completed, however. The second firm will note the change in traffic volume and will adjust its own volume to a new profit-maximizing level which we denote lQz, and is therefore given by :

The iterative adjustment process will theoretically continue ad infiniturn until both firms are producing equal amounts of congestion.6 In general,

Zn Zn+ 1

k = O k = O ,,Q1 = x C (- l )kyk and ,,Qz = x C (- l )kyk

61t is easily seen, that .Q1 gets smaller and “Q, gets larger as n, the number of adjustments, gets larger. From (3) and (7) Q, < Q1 and from (3) and (8) lQ1 < Q1. This latter relationship in conjunction with (7) and (10) indicates Q, < 1Q2. From this procedure follow the inequalities:

QZ < ~ Q z < 2Q2 < ... Qi > 1 Q i > ZQI > ...

“Qi > nQz

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so that at the nth step of the iteration the first firm has its equilibrium volume decreased by an amount -xyZn-l +xy2", while the second firm has its equilibrium volume increased by y times this amount, that is, by ~ y ~ " - x y ~ " + ~ . The sum of the

series(1-y+yZ-y3+ ...) is wellknown tobe-forlyl < 1.Sincey =c/b+c,and

both b and c are positive, this condition on y is clearly satisfied. Hence the equilibrium volume for both firms Q* will be

1 l + Y

(11) X Q* =--

l + Y

The industry volume will be twice this amount or *. Substituting for x and y : 1+Y

This is identical to the social optimum given in (5 ) earlier.

The conclusion is that two firms inflicting reciprocal externalities upon each other will rationally adjust their outputs to the same optimal level that would pertain under a single merged firm or in an industry with collusion. No Pigovian tax or subsidy program is required.

It should be noted that the result is independent of the sequential arrival of the firms. If the two firms started operations simultaneously at Q1, they would discover that the road congestion was making them operate at a loss. If both firms react at the same time, they will both move to Qz, then to lQ1, then to zQl etc., converging in parallel to the same limit as given in (1 1).

It can also be demonstrated that the result will be achieved by firms of unequal size (different MP curves) and for the sequential entry of additional firms to an industry of N firms in equilibrium (see Appendixes B and C).

These conclusions are also independent of any non-commercial traffic which may be sharing the use of the road with the mining firms, since this additional traffic would merely represent an upward shift in the original MC curve of the first firm, i.e.

where k is the amount of congestion due to non-commercial traffic. MC1 = k + c Q

The Problem of Indivisibility It is quite obvious that if the transport operations of the firms were undertaken

by private trucking contractors, the results would be unchanged. The MC curves would intersect the $ axes at the average cost of operating the trucks independent of congestion, but the M P curves of the extracting firms would rise by an equal amount since they are no longer faced with these costs (see Figure 2).

However, the preceding analysis applies only if the individual firms (decision- makers) are able to adjust the volume of truck shipments to meet changing conditions, If each truck is owned by its driver and truck shipments are made on a fixed contract basis of P dollars per shipment, the analysis yields markedly different conclusions.

Each driver will perceive the congestion cost facing his own truck as marginal to himself (the difference between 0 and 1 truck trip); but this will be the auerage


cost (AC) to the trucking industry serving the road. The marginal cost ( M C ) to the industry will be the total congestion costs to all other truck drivers introduced by the marginal truck. The MC curve, which is identical to the function in the previous analysis since it is a technological constraint, is not perceived by any truck owner. With perfect competition and free entry to the trucking industry the contract price, P, and the quantity of truck shipments Q f , will be determined by the intersection of the AC and M P curve.

FIGURE 2

- e 5 .= ? z p g

.- E

w

Average cost/truck

Q, Q,Quantity of truck shipments

0

0' I I

The classic case of an external diseconomy has now been introduced to the trucking industry. Instead of the industry optimum Qr, given by the intersection of the M P and M C curves, there is over-utilization of the road which includes an excessive road use equal to (Q - Q I ) . There is no incentive for any truck owner to reduce the number of his truck trips (from 1 to 0) since only he would lose while all others would benefit. Nor would he be willing to reduce the frequency of his trips by Qr/Qr since this would mean that over time he would be foregoing income represented by the area of the triangle W Y Z (see Figure 3).

FIGURE 3

B .a c -5

.- AC B P - P

P'

0 Q, Q, Quantity of truck shipments

If all truck drivers could be induced to reduce the frequency of their trips by Qr/Q ,-, i.e. so that in the standard time period the number of truck trips were reduced to Qr, then there would be a net gain of producers surplus to the trucking industry (without changing the contract price), represented by the area V X W Y and an off-


setting net loss in consumer surplus to the mining firms represented by the area of the triangle X W 1’ The net gain to social welfare will be represented by the area of the triangle XVY:

In perfect competition the price after the reduction in trip frequency will move to P which will transfer the surplus represented by area PWXP‘ from consumer to producer surplus, but net social welfare will remain unchanged. It is obvious that it would pay the trucking industry (and society) to “bribe” the mining firms an amount equal to the area PYXP’ per time period to accept a higher price P‘ and a lower volume of shipping Qr, since they will still gain the net represented by the area XVY:

The simple fact that there is no incentive for any individual truck owner to move in this direction has led to the conclusion that one of the following solutions be employed : (a) Merger of all truck owners into one trucking firm; (b) Collusion between the truck owners in negotiating contracts with the mining

firms ; (c) A so-called Pigovian tax be levied equal to XZ such that the perceived cost to

each trucker lies on the MC curve ; (d) Quasi-private ownership of the road to charge rents for its use such that revenue

is maximized. (It has been demonstrated that the revenue-maximizing rent will be exactly equal to X Z . )

Solutions (a) and (b) will produce the desired optimum social surplus but at the expense of the mining firms since it is doubtful that the truckers will offer any “bribes” once given a monopolist’s position.

Solutions (c) and (d) have been shown to be identically equivalent, and that revenue, represented by the area P‘XZP”, will accrue to either the tax authority or the quasi- private firm as the case may be. The social surplus is also maximized, but again at the expense of the mining firms. Faced with the tax X Z , the trucking firms will raise their price to P‘ reducing consumer surplus to P‘XC. The calculation of the producers surplus is changed from PUO- U V Y to PITO - T Z X . However, since both expressions are obviously identically zero (since average cost equals average revenue at both Qs prior to the tax and Qr after the tax), the problem of effecting a Paretian optimum reduces to one of utilizing the tax revenues P‘XZP” to pay a subsidy equal to P‘XYP to the mining firms.

It is well-known that PWZP“ is equal in area to X V Y W and hence the full subsidy can be paid from tax revenues with the surplus XVY remaining. The subsidy to each firm could be based on the quantity of truck shipments made during any specified period without introducing any disequilibrating effects into the market.

‘At Qf the producers surplus for the trucking industry is given by PUO-UVY; consumer surplus is given by CPY; and the total social surplus is given by the sum: CPY + (PUO - UVY) = CPUX + UXY + PUO - XVY - UXY

= CPUX+PUO-XVY = cxo-XVY

At Q, the producers surplus is given by P U O - U X W ; the consumers surplus by CXWP and the total social surplus by : CXWP+(PUO-UXW)+ = CXUP+UWX+PUO-UXW

= CXUP+PUO = cxo


The tax must be stated in an appropriate way if it is to impact the frequency. A simple tax per truck will reduce the quantity of shipments; but, in all probability, it will do so by driving some truck owners off the road, rather than by affecting their frequency. This problem will be discussed in detail in the next section. A tax on the mining firms based upon their output would have the same effect. However, a tax based upon the number of trips per unit time period, where the time period is sufficiently great to permit “continuity” in the adjustment in the number of trips, would produce the desired reaction.

If there is an equally productive alternate use for the trucks, the “frequency tax” would, in effect, be a “square-wheel tax” as resources would not be transferred from the mining operation but merely be used less. A tax which tolled trucks from the road would then be more efficient, providing their alternate use did not increase congestion elsewhere. Of course, one might point out that if any alternate use existed with less congestion, profit-maximizing truck owners would have moved there even before the tax.

The same conclusions will also follow even if the truck owners also operate the mining equipment, i.e. if each of the N mining firms owns only one truck. Without a Pigovian tax, X Z , the firms will operate as an industry at Qr, with a net social surplus of ( C X O - X V Y ) . After the tax they will operate at Qr with a surplus ( C S P + P T O - X Z T ) . Again, this will be less than the pre-tax surplus by the area of P‘XYP.8 The receipts from the tax can be used to subsidise the industry by this amount leaving a net tax surplus of ( P W Z P ” - X Y W ) or X V X If the subsidy is distributed on the basis of a flat rate per truck owned, the equilibrium will still not be di~turbed.~

Thus with indivisibility in the decision variable as seen by the decision-makers, competition will not lead to a Paretian optimal equilibrium. However, an appropriate Pigovian tax exists which can induce the welfare optimizing equilibrium. Subsidies can be devised which will restore the pre-tax welfare of the firms affected by the tax, leaving a surplus which represents a net gain to social welfare.

The Problem of Discontinuity Over Time The preceding analysis assumed that there was a continuity of demand for trucking

over any period of time such that the frequency of truck trips could be adjusted by individual truck owners in response to a Pigovian tax. If, however, the paradigm is altered so that shipments are made only during prescribed intervals of sufficiently short duration such that only one trip can be made by each truck, the situation changes again.

“This is demonstrated simply: if x z difference in surplus, then

(1’)x = (CXO - XVY) -(CXP’ + P”TO - XZTO) = (CXP’ + P‘XUP+ PUTP + P’TO - xv Y) -(CXP + P”TO - xw U - U W Z T ) = P‘XUP+PUTP-XYY + X W U + U W Z T

But(2’) PUTP+UWZT = XVY+XYW :. (3’)x = P‘xUP+xvY+xYw-wvY+xwU

= P‘XUP + x W U + XY w + P’XYP Since the equality (2’) is true independent of the shapes of the curves, so is the result.

gNote that without the market to differentiate between the consumers and producers of congestion, the rationale of the subsidy must be changed. Were it defined as before by the quantity of shipments, the firms could enter the foreknowledge of the subsidy into their long-term profit calculations as a shift downward of the MC curve by the amount of the subsidy per shipment.


Presumably, the truck owners will operate their trucks along the coast until the shipping period approaches whereupon they will drive inland to pick up the shipments from the mining firms. Ignoring the marginal advantages of being first (avoiding meeting full trucks returning to the sea) or last (avoiding meeting empty trucks going to the mines), the analysis of competitive equilibrium is the same as in the previous section. The individual truck owner sees his marginal cost as the industry average cost with equilibrium at Qf.

However, the introduction of a Pigovian tax, introduces a new effect in this situation. Since no adjustment in trip frequency is possible, i.e. it is either “show” or “no- show”, the tax will effectively toll some trucks off the road. If the tax is X Z , there will be Q, - Q, fewer trucks each shipping period. With the normal assumptions of perfect competition, the result of the tax would be indeterminate since all the truck owners would presumably react the same, i.e. all would go or none would go. In a real world situation with uncertainty and differing utility given to risk, the determinate result could be expected in an otherwise “perfectly competitive” market.

The welfare results are ostensibly the same as before except that now a new element has been introduced: how to evaluate the net loss in welfare to those truck owners that were tolled-off. In the analysis in the previous section the change in the producers’ surplus to the truck owners did not affect the net welfare results, since the surplus at both levels of output was identically zero. In the case under consider- ation here, the identity is no longer sufficient as a measure of the net welfare change to the truckers. It merely indicates that the surplus accruing to those truck owners remaining in operation is equal to the total pre-tax surplus; but it begs the question as to the welfare effect upon those no longer willing (or able) to risk the journey and the loss that would result if too many trucks show up. If there were an equivalent alternate use for the trucks, the net welfare loss would be zero ; but as argued earlier, this appears to be an unwarranted assumption.

The issue is even clearer if the truck owners also operate the mines. Being tolled off the road is now equivalent to being driven out of the mining industry. Again the welfare analysis of the previous section pertains with respect to the industry before and after the tax. The subsidy P‘XYP is used to reimburse those firms remaining in the industry. The loss of welfare experienced by those tolled out of the industry is indeterminate within the analysis (recalling that all firms are equal except for the risk-taking propensities of the owners).

Even if this loss can be ascertained, the question arises as to how the subsidy can be administered. Quite simply, any subsidy of the types suggested in the previous section will encourage all truck owners in the economy to claim a subsidy for being tolled off the road. The criterion of actual participation is no remedy since any truck owner could “risk” one trip before being “tolled off the road” and thereby becoming eligible for the subsidy from then on.

The conclusion in the case of demand for the externality-producing activity being discontinuous over time is that the Pigovian tax in itself cannot be counted on to produce a Paretian optimal result. Certain members of the pre-tax community will be worse off, the degree of their net loss may be indeterminate, and no straight- forward subsidy program seems to exist which can provide equitable compensation.

Clearly the goal of improving the social welfare from the perfectly competitive equilibrium (at Qf) remains valid and is, on a priori grounds, possible. But the

1975 RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES 97 “classical” Pigovian solution no longer produces the expected (desired) results, nor are those results determinate beforehand in the context of Paretian-optimality.

Problem of Heterogeneity A final modification of the paradigm can be used to illustrate another problem

that frequently arises in situations involving reciprocal externalities. Returning to the case of N firms each operating one truck on a continuous basis, it shall now be assumed that each firm has a different M P curve (e.g. because of non-homogeneity in the mineral resource deposits) and a different M C curve (e.g. because of differences in driving ability).

Empirically the competitive equilibrium will provide some volume, Qf, which supposedly indicates the intersection of the industry’s M P curve with the industry’s AC curve at some point, Y (see Figure 4). However, the shapes of these curves can no longer be specified with any precision. The marginal activity does not produce a determinate effect upon either the M P or AC (or M C ) curves, e.g. it may involve a firm with high returns and high costs or low returns and low costs. The firms in the industry at equilibrium are those for which M P - A C > 0. In effect the industry is dealing with a single decision variable which might be called marginal net profit ( M N P ) . That is, the marginal activity is the one with the smallest contribution to M N P , i.e. with the minimum value for M P - A C .

FIGURE 4

Q I f ‘truck shipments

Faced only with the M N P curve and the volume Qf it is no longer possible to construct a relevant Pigovian tax or to derive the welfare implications of any tax- subsidy program. For example, a tax on truck shipments may well serve to toll off the road good drivers with poor mining leases with very little effect upon congestion costs although reducing road usage. On the other hand if it removed the poor drivers with good mining leases the tax could lead to increased road usage with reduced congestion.

The conclusion is simply that when the population of decision-makers producing reciprocal congestion has unequal marginal profitability and externality-producing functions, the structure and effect of a Pigovian tax is indeterminate as are its implications with respect to Paretian optimality.”

‘OThese suggestions were implicit in the articles by Schall [19] and Goetz and Buchanan [ 6 ] , but it ma) serve some purpose to represent them in the context of the analytic device used in this paper.


Implications and Conclusions The condition that has been examined within this paper falls into the classification

of reciprocal output-generated diseconomies. While the paradigm has been cons- tructed within the framework of competitive equilibrium between firms, the conclusions can apply to a wide range of real world problems: traffic congestion in general, peak-load crowding of public parks and facilities, and industrial pollution of a shared resource.

If the decision variable which produces the externality is divisible to each decision- maker, then profit (or welfare) maximizing behaviour by the externality-producing population will result in the Paretian-optimal output for the group as a whole without the necessity of merger, collusion, or Pigovian tax-subsidy. This conclusion holds for any number of firms (as long as all are responsive to the changes of output of the externality by the other firms) of differing size and with differing marginal profitability functions. To the degree that this adjustment process can be effected in practice, one might argue that commercial firms be exempt from road congestion taxes (if they exist) on the grounds that they will already have adjusted to the Paretian-optimal output.

If the decision variable is indivisible to the decision-maker then profit maximizing behaviour by all the firms will not lead to Paretian-optimal output. However, a Pigovian tax can be successfully employed to induce the firms to operate in such a way as to produce the Paretian-optimal output for the industry. If the externality- producing activity is continuous over time, the tax can be applied in such a way as to reduce the frequency of the activity for all decision-makers. This will provide a clear-cut welfare gain with a precise theoretical value.

If the externality-producing activity is discontinuous in time, the Pigovian tax will produce a reduced output level which will appear to coincide with the Paretian- optimal solution; but the welfare conclusions will now be ambiguous since there is no direct way to evaluate the losses in welfare to those tolled off the road. Further, there seems to be no way to devise a subsidy to reimburse those tolled off, hence the Paretian-optimal conditions cannot be attained. While the paradigm in the text may appear to be far-fetched, it has implications with respect to home-to-job peak hour traffic. This type of activity produces a reciprocal externality and is discontinuous, i.e. one must “show” or “not-show’’ at his place of work, rather than varying his income by the frequency of his trips. Road usage taxes frequently recommended in the literature [e.g. 9, 12, 211 will provide equally unclear welfare results.

Finally, if the marginal values and costs of the externality producing activity are heterogeneous over the population, the Pigovian tax cannot be specified and the welfare gains (or losses) from such a tax will be wholly indeterminate. If one challenges the common assumption of constant and/or uniform value of time across society [e.g. 7,8,15] the concept of road usage taxes as a means for reducing traffic congestion is again shown to be difficult to defend theoretically.

It would appear that the conclusions presented above might, with imagination, be extended to more diverse areas of economic activity beyond that of the simplistic paradigm and its obvious extension to peak load traffic congestion.

1975 RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES 99 APPENDIX A

Assume two firms with identical M P curves of the form

M P , =f(Q) ( i = 1,2) ( 1 4 wheref is a continuous, non-increasing function of Q for Q 2 0 satisfying f(0) = K, z 0, and an M C curve for congestion costs

M C = s(Q) ( 2 4 where g is a continuous, non-decreasing function of Q for Q 2 0 satisfying g(0) = K,, where 0 < K, i K,.

The industry MPI curve is given by adding together the two Q values at constant MP, so that

MPI = f (Q/2) (3'4)

f(Q/,) = g(Q) which *QI in equilibrium. ( 4 4 The quite general conditions onfand g insure that such an intersection takes place, i.e. that there exists a Q, 2 0.

and the industry optimal (Paretian optimal) solution is given by :

If the first firm is in operation its profit maximizing behaviour is given by f(Q) = g(Q) which *Q1, its equilibrium solution. ( 5 4

f (Q) = dQi + Q) which * Q2 ( 6 4

The alternative adjustments by the two firms is given in the notation previously adopted in the text:

f (Q) = d Q 2 + Q) * iQi ( 7 4 f (Q) = diQi + Q ) a 1Q2 (8.4)

f (Q) = d.-iQ2+Q)*.Qi ( 9 4 f ( Q ) = d.Qi +Q)*.Q2 ( 1 0 4

Lt (nQ1-nQ2)-+0 and Lt "Q1 = Lt "Q2 = Q* (11A)

f(Q*) = &Q*) (12A)

When the second firm commences operations it will maximize its profits where:

...

As n + w the two sequences {"Q,} and {"Q,} will converge to the same limit, say Q*. That is,

n- t W n-cc n- tm

From (10A) we obtain on taking the limit, n -+ m :

Comparing this to (4A), it is clear that the two equations are identical with Q* = Q,/, or QI = 2Q*.

The general proof can be extended to a new firm entering an initial industry with N firms, providing the conditions for convergence of the iterative process (as described in Appendix C) are met (4A) becomes :

( 1 3 4

(14.4)

( 15'4)

f(Q/N + 1) = g(Q) a Qr in equilibrium. and (12A) becomes:

so that by comparing (13A) and (14A) we obtain: f (Q*) = d N Q * + Q*) = d ( N + 1)Q*l

Q * = Qr/N+1 Or Q r = ( N + l ) Q *

APPENDIX B Assume two firms have M P curves of the form :

M P , = a,-b,Q and

M P , = a, - b2Q where ala2, b, and b, are positive constants.

The M C curve, representing a purely technological constraint based upon the road dimensions, remains the same for the two firms. That is, M C = cQ. Then the industry M P curve can be derived.

100 AUSTRALIAN ECONOMIC PAPERS

By rearranging (1B) and (2B):

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Since MP, is obtained by summing Q, and Q2 at constant MP, so that:

Q I - - %+%- ('+'> MP, which becomes b, b2 bl b2

Mp, = a l b 2 + a 2 b 1 Q, b1 +b2 61 + b2

Setting (4B) equal to MC, the social optimum output for the industry will be

The equivalent forms of equations (3), (7) and (8) (in the main text) can be easily derived by equating MC = cQ with (1B) to obtain:

When the second firm takes into account the presence of the first firm it follows that:

Q, and, further that (7B)

Q2 (8B)

Q a2 c

,Q, = a1 c

2 - b2+c b2+c

b,+c b,+c

and y2 = If we let x, = -, x2 = -, y, = C

The general form of the two series of successive adjustment outputs becomes, for the first firm:

a1 a2 b ,+c b2+c b, + c b2+c

.Qi = ( x i - x 2 ~ i ) ( l + ~ i ~ 2 + ~ : ~ 3 ... Y ~ Y ? ) (9B) and for the second firm :

.Q2 = ( x ~ - x ~ Y ~ ) ( ~ + Y ~ Y ~ + Y : Y : . . . Y Y Y ~ (W As this adjustment process continues and n + co, we have

provided that ly,y2 I i I, which is clearly satisfied. Therefore :

The industry volume in competitive equilibrium is the sum of these two quantities, viz.:

This is identical to the expression for the social optimum output given in (5B). Thus it is shown that profit-maximizing behaviour in a situation involving reciprocal externalities produces optimal output without merger, collusion, or Pigovian taxes (subsidies) regardless of the size of the firms, under the assumptions made.

1975 RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES

APPENDIX C

101

If there are N identical firms with the following M P and M C curves (the M C curve is still cQ, but each firm sees that section of it which includes all the other firms activities as given)

M P , = a-bQ(i = 1,2 ... N) M C i = c(N - l)Qi + cQ where a, b and c are positive constants.

(1C) (2C)

The situation in equilibrium is given by extension of (11) and (12)

N Na b+Nc QX = C Qi =-

i = 1

When the (N + l ) Ih firm (identical to the first N firms) enters the industry it perceives its M C curve as :

MCN+1 = cQz +cQN+~ and hence (5C)

The equilibrium assuming that the N firms react in collusion would be given by the analysis in Appendix B : (N + l)a

b+(N+l)c QI =

If the N firms react independently two results are possible: that given in (7C) or an exploding oscillation

When the (N + l)'h firm enters the industry and begins operations per (6C), the N firms react by cutting

in both the industry's and the individual firms' output.

back their operations to a new cost level IQi given by: a c

b+c b+c ,Qi = ---[(N-l)Qi+QN+l] (i = 1,2 ... N)

The value of (8C) will always be less than (3C) and greater than (6C), which can be verified by simple algebraic manipulation.

The form of the solution now depends upon the assumptidn as to how the firms react at this point. If only the (N+I)'* firm reacts to the first round reaction of the N firms and then the N firms react to the (N+l)"' firm (and so on), the two sequences InQi} and {nQN+l} approach the same limit, Q*, as n - + 03.

or in general :

As n becomes large we have thc following results : n-m Lt ( n Q i - n - l Q 3 = ,f;;(.Q~+i-.-iQ~+i) = n t t , ( m Q i - n Q N + l ) = O

and, letting Q* be the firm optimum; Q* = Lt .Qi = Lt .-1Qi = Lt "QN+I = ,& " - i Q ~ + i

n- x n-m n-m

102 AUSTRALIAN ECONOMIC

Then equations (11C) and (12C) become in the limits as n -P co

Q* = a b + ( N + 1)c

and for the N + 1 firms :

PAPERS

( N + 1)a Q r = b + ( N + l ) c

which is identical to (7C).

However, if on the second round of adjustments, after the N firms react to the initial entry of the (Nf l ) lh firm, all the ( N + 1 ) firms react at the same time to the new situation, (1OC) and (12C) become:

(16C) a c

nQi = ~ - b f r [ ( N - l ) , - i Q i + . - z Q ~ + i ] (i = 1,2 ... N )

There is no longer direct convergence since zQi is now greater than ,Qi as can be seen by comparing (15C) and (8C) since lQi < Q i . On the next round of adjustments, the ( N + l)lh firm reacts to the N firms increase by decreasing its output and the original N firms react to each other’s increase by also decreasing their output. Thus there is parallel oscillation in the output of all firms over successive rounds. The condition for convergence (damped oscillation) is that zQi < Qi, i.e. the total increase in output of the N firms in reaction to their own first round adjustment is less than their initial reaction to the entry of the ( N + l)lh firm. This can be shown as follows: Suppose zQi > Q,. This implies that 2QN+1 < QN+l from (11C). It is always true that l Q N + 1 > Q N + 1 ,

. ’ . 3Qi < l Q p Again by (11C) 1 Q N + 1 < a Q N + 1 but if lQi > 3Qi and Q N + 1 > 2 Q N + 1 this implies zQi < 4Qi and 4QN+1 < Z Q N + l . Thus the oscillation of each sequence (and hence the industry output) explodes with ever incre?sing amplitude, as shown by the following inequalities :

. . . ~ Q N + I < Z Q N + I < Q N + I < i Q ~ + i < ~ Q N + I ...

... 3Qi < 1Qi < Qi < ZQi < 4Qi ’.. In the linear case, the condition for convergence is b > (N - 2)c. Thus it is always true for a third firm

entering the industry, and will be true for a 4th firm if b > c. This result can be demonstrated explicitly by comparing (3C) and (15C). Qi > zQi is true if

+ (b+c)’(b+Nc) ( b + c ) ( b + N c )

L>---- 1 b + N c b + c b + c [ bZ + bc + c3

( N - 1 )

where the right-hand side of the inequality is obtained by substituting (4C) into (6C), then (3C) and (6C) into (8C), and then (6C) and (8C) into (15C).

Rearranging :

1 1 1 N b 2 + N b c + ( N - l ) c Z

1 ( N - l ) ~ Nb2 + Nbc + ( N - 1 ) ~ ’

NbZ + Nbc + ( N - 1)c2 ( b + c)’

( N - 1 ) <

( N - l ) b z + 2 ( N - l ) b c + ( N - 1)~’ < N b 2 + N b c + ( N - 1)~’ 0 < b Z - ( N - 2 ) b c or ( N - 2 ) c < b

Since cases of wildly explosive oscillation do not occur in the real world it is doubtful that this type of decision making algorithm is of theoretical value. The existence of a lag in the adjustment process can explain the empirically observed stability in the general case.

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RECIPROCAL DISECONOMIES AND PIGOVIAN TAXES