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Production externalities and long-run equilibria: bargaining and Pigovian taxation.



Economists have taken two distinct approaches to externality problems. In one, originating with Pigou [1920], the government imposes "corrective" taxes so that agents incorporate into private decisions the effects of their actions on others. Optimal corrective taxes induce efficient outcomes. Another approach, most closely associated with Coase [1960], argues that bargaining among agents over allowable levels of externalities achieves efficiency without government intervention. Coase asserted that, if bargaining is costless, efficiency obtains no matter who holds the property rights. Either approach can produce efficient outcomes. When bargaining is costless, the Coase Theorem appears to justify decentralized solutions to externality problems which do not require government action. Pigovian taxation appears needed only when bargaining is costly or infeasible.

Open questions about both approaches remain. For Pigovian taxes, two strands of literature that focus on separate requirements for sustaining Pareto efficiency yield apparently contradictory results. First, taxes must sustain not only the optimal level of externality in the short run, but also the optimal numbers of firms under free entry. In the long run, a linear tax on the externality, provided the proceeds are distributed as lump-sump grants to consumers, can support the optimal allocation. Other possible tax structures, such as subsidies for pollution abatement, sustain a short-run optimum but lead to the wrong numbers of firms in the long run.

Second, consider the short run with Pigovian taxes and bargaining. Either one separately can sustain the optimum. However, for consumption externalities Buchanan and Stubblebine [1962] and Turvey [1963] have shown that, when this tax revenue is distributed as lump sums, the outcome is inefficient if bargaining takes place. Since the tax revenue flows away from the parties to the externality, one gain from bargaining is to reduce the outflow of tax revenue. In effect, the externality damage is double-counted. Buchanan-Stubblebine and Turvey show that if the tax revenue is transferred directly to the damaged party, then the efficient solution is sustained, even if bargaining is costless. For production externalities, if short run shut down is possible, revenue transfer may not eliminate the bargaining inefficiency.

Pareto optimality requires both correct entry incentives and a need to prevent inefficiencies from bargaining. The requirements for the latter contradict those for the former. These two strands are reconciled in a model with negative producer-producer externalities and free entry into the polluting and damaged industries. Bargaining among firms is possible, but the government may be unable to observe whether it occurs. The efficient outcome can be sustained with a nonlinear Pigovian tax system. Taxing output of the damaging industry's firms only when it is in excess of the efficient level and using a franchise fee to tax away profits supports both the correct output and the correct number of firms. This scheme leads to an allocation within the core, so that bargaining among firms cannot raise joint profits. Thus, bargaining costs are irrelevant to the government's policy choice.

Coasian bargaining also has problems in the long run with free entry. If rights or liabilities accrue to an agent by entering an industry, efficiency fails. For example, a liability rule under which anyone harmed may collect compensation induces too much entry. Frech [1973; 1979] argues that this problem arises not from a bargaining failure, but from a failure to specify complete property rights. Such rights should be exogenously assigned and not be earned by entry. However, for several reasons discussed below, specifying complete property rights is not sufficient for bargaining outcomes to be efficient.

We show that the circumstances under which bargaining outcomes are efficient are quite limited. Several assumptions, other than zero bargaining costs, are needed. First, the property right must be very strong: its holder must be able to control not only the right to pollute but also the right to enter. Second, its holder must as as a price setter. The rights owner needs to negotiate directly with consumers, extracting all consumer surplus available by first-degree price discrimination.

Under both assumptions the full Coase Theorem holds, but the outcome is inefficient if either is violated. If firms take output prices as given and bargain only among themselves, then the objective function is not concave. This nonconcavity is both local and global and arises not just from the possibility of complete shutdown in one industry. If firms set prices but cannot bargain with consumers, then the standard monopoly inefficiency arises. If the property right does not control entry, then the rights holder is unable to collect bribes from the damaged firms, and the efficient outcome is not an equilibrium.

Although these results demonstrate a sense in which the Coase Theorem is valid in the long run, they reveal that bargaining is not a decentralized solution to the externality problem. Bargaining cannot be among only a small set of firms, taking output prices as given. Rather, bargaining must control both industries' outputs and involve all consumers. The property rights holder must effectively imitate a socialist planner in choosing output levels. Bargaining does not correct a difficulty in the operation of the price system on the margin but must replace it entirely.

In section II, the model and the Pareto optimal conditions are presented. All proofs are deferred to the appendix. In section III, the impossibility of an efficient decentralized bargaining solution in the long run is shown. In section IV, a nonlinear Pigovian tax system is developed that yields efficient long-run outcomes. Section V contains some conclusions.


Two industries produce goods X and Y, with labor as the only input. Industry X damages industry Y as a by-product of its output, increasing Y's input requirements. The damage is atmospheric in nature; only total output X matters and not its distribution across firms.

Formally, n.sub.x is the number of identical firms in industry X and x=X/n.sub.x each firm's output. F(x) is the strictly convex labor requirement function of each firm. n.sub.y is the number of firms in industry Y, y=Y/n.sub.y is the output per firm, and G(y, X) is the labor requirement function of each firm with G strictly convex in y and X jointly.

All individuals have the same quasi-linear utility function with constant marginal disutility of labor, U(X, Y, L) = U(X, Y) - L. U(X, Y) is assumed to be strictly concave in X and Y. The first-best allocation can be found by choosing X, Y, n.sub.x and n.sub.y to maximize W = U(X, Y) - n.sub.x.F(X/n.sub.x.) -n.sub.y.G(Y/n.sub.y., X). The necessary conditions for the solution are *W/*X = U.sub.1.(X, Y) - F'(x) - n.sub.y.G.sub.2.(y, X) = 0 *W/*Y = U.sub.2.(X, Y) - G.sub.1.(y, X) = 0 *W/*n.sub.x = xF'(x) - F(x) = 0 *W/*n.sub.y = yG.sub.1.(y, X) - G(y, X) = 0.

The assumptions on U, F, and G guarantee that these are also sufficient.

Let p = (p.sub.x., p.sub.y.) be the producer price vector and q = (q.sub.x., q.sub.y.) the consumer price vector for X and Y. Labor is the numeraire commodity. Profits per firm in industry X are II.sub.x = p.sub.x.x - F(x) - E.sub.x., where E.sub.x is any franchise fee or entry tax. In the absence of bargaining, the first-order condition for profit maximization is p.sub.x = F'(x). Profits per firm in industry Y are II.sub.y = p.sub.y.y -G(y, X) -E.sub.y., where E.sub.y is a franchise fee or tax on entry, net of any damages paid firms in this industry. Profit maximization thus implies p.sub.y = G.sub.1.(y, X). T is the lump-sum transfer received by consumers.

Proposition 1. Let the first-best allocation be (X, Y, n.sub.x., n.sub.y.). If firms do not bargain with each other, then p.sub.x = F'(X/n.sub.x.), q.sub.x - p.sub.x = n.sub.y.G.sub.2.(Y/n.sub.y., X), q.sub.y = p.sub.y = G.sub.1.(Y/n.sub.y., X), T = t.sub.x.X, and E.sub.x = E.sub.y = 3 yield a market equilibrium which sustains the first-best allocation. If revenues are not distributed lump sum to consumers, then the first-best allocation is not sustained.

Modeling bargaining with a fixed number of firms is relatively straightforward in the Coasian tradition. No formal process of negotiation is specified. However, the process is assumed to operate costlessly and thus realizes all possible gains from trade. Here, the outcome will maximize joint profits. It is assumed that firms do not use externality negotiations as a way of obtaining monopoly profits. Thus, producer prices are taken as given. In the absence of taxation, the bargaining solution maximizes joint profits * [triple bond] p.sub.x.n.sub.x.x + p.sub.y.n.sub.y.y - n.sub.x.F(x) - n.sub.y.G(y,X) with respect to x and y.

The distribution of profits is not crucial. The legal structure and the specific process of bargaining will determine who gets the excess over profits which could be earned in the absence of bargaining. As long as * is fully distributed to the firms, the outcome is efficient in the short run.

Proposition 2. If n.sub.x and n.sub.y are fixed and the government does not impose corrective taxes, the efficient outcome is attained by costless bargaining among the firms.


Entry complicates bargaining over externalities. Whether the Coase Theorem continues to hold has been hotly debated. For example, Calabresi [1965], Mohring and Boyd [1971], Tybout [1972], and Schulze and d'Arge [1974] argue that in the long run efficiency may fail. In contrast, Frech [1973; 1979] argues that the theorem remains fully valid in the long run. The question of how entry affects entitlements has two parts. Only the first, how one acquires an entitlement to damage or to prevent damage, has been emphasized in the literature. Such rights might be obtained simply by entry into the industry or might be exogenously assigned and then acquired only by purchase from the owner. Frech [1979] denotes the former as a liability rule and the latter as a property right. Under a liability rule the Coase Theorem fails. If polluters have rights to collect bribes, then profits from the bribes induce excess entry. If the victims have the rights, any compensation collected induces entry into that industry. Frech argues that these problems do not arise with property rights. However, other problems arise, as shown below.

The second question is precisely what power entitlements encompass. The standard view is that the property right permits the owner to pollute or to prevent pollution; these are designated complete property rights. If they grant the further power to control entry into the damaged and damaging industries, the rights are designated ultra complete property rights. It is shown that ultra complete property rights are necessary for efficiency from bargaining and that complete property rights are inadequate.

With ultra complete property rights, the rights owner can control the number of firms in each industry and the level of operations of any firm. The owner collects an entrance fee from all firms. If the rights owner maximizes profit, his goal is to extract the maximum possible revenue from the firms in the industries, subject to the constraint that firms must achieve nonnegative profits. Implementing ultra complete property rights would often be impractical, but might be possible in some cases. Consider the difference between a limited watershed such as a lake, and a general airshed. If the lake has two types of users, a polluting chemical industry and a fishing industry harmed by the pollution, it is reasonable that a lake owner can control not only pollution but also benign use of the lake. Similar control of the airshed would require not only control over pollution but over any use of the atmosphere. Anyone who breathed would have to purchase that right from the owner. Clearly such control of an airshed is impractical.

Consider first the case in which the owner takes prices of outputs and inputs as given when determining the optimal use of the property right. This assumption, made by both Schulze and d'Arge [1974] and Frech [1979], is standard on both sides of the debate over the validity of the long-run Coase Theorem.

Let Z.sub.x and Z.sub.y be the fees the rights holder charges for entry into industries X and Y. Total revenue is n.sub.x.Z.sub.x + n.sub.y.Z.sub.y. Profits per firm after fees are *.sub.x = p.sub.x.x-F(x)-Z.sub.x and *.sub.y = p.sub.y.y-G(y,X)-Z.sub.y., in the absence of taxes. The requirements that *.sub.x [is greater than or =]0 and *.sub.y [is greater than or =]0 and that the rights holder maximizes revenue imply that Z.sub.x and Z.sub.y are set to make *.sub.x and *.sub.y each equal to zero. Hence, Z.sub.x = p.sub.x.x - F(x) and Z.sub.y = p.sub.y.y - G(y,X). The problem for the property rights holder is to max I [triple bond] n.sub.x[p.sub.x.x - F(x)] + n.sub.y[p.sub.y.y - G(y, X)]. x,y,n.sub.x.,n.sub.y

Proposition 3 shows that the solution to this problem will not be a social optimum with positive outputs for both industries.

Proposition 3. Assume that ultra complete property rights exist, that there are zero bargaining costs, and that the rights holder takes output prices as given. Then second-order conditions for the rights holder's objective function fail at an interior solution to the first-order conditions. Thus, the socially efficient outcome cannot be sustained.

At the first-best allocation, zero profits are earned in the damaged industry. To sustain this, the rights holder must set Z.sub.y = 0. Profits per firm in industry Y can be made positive by a slight reduction in x or n.sub.x. This allows a positive entry charge; increasing n.sub.y by even a small amount then increases the rights holder's total profits.

Price taking is crucial, since the rights holder must believe that entry rights can be sold to additional firms. Similarly, profits could be increased by a slight increase in x to raise profits for each firm in that industry. Selling more entry rights raises total profit. This could shut down the damaged industry, but the rights holder does not care since industry Y contributes no revenue.

This nonconcavity requires the presence of an externality and does not arise with two independent industries. Consider a watershed used by two industries, neither affecting the other, owned by a single individual who can sell entry to as many firms as desired. With constant returns to scale and unlimited capacity of the watershed, the only equilibrium prices are those equal to minimum long-run average cost in each industry. Maximum profits per firm are then zero and no entry fee can be charged. If either price is above this level, then the rights holder would desire to sell entry to an unlimited number of firms and earn infinite revenues.

In the externality case, in the damaging industry X, excluding entry fees, there is a single long-run average cost curve and thus a unique price equal to minimum long-run average cost (*.sub.x). In the damaged industry, however, the average cost curve depends upon the output of the polluter. Average cost to a firm in industry Y rises with X.X = 0 leads to the lowest possible minimum of long-run average cost in industry Y(*.sub.y.). If output prices were *.sub.x and *.sub.y., then the return to the rights holder must equal zero and one industry must shut down. If n.sub.x is positive and x minimizes long run average cost, then at *.sub.y only negative profits could be earned in industry Y; hence n.sub.y = 0 must hold. If n.sub.x is zero and if y minimizes long-run average cost, then zero profits are earned in industry Y and n.sub.y can be positive.

If the price in an industry exceeds its minimum, then a price-taking rights holder thinks positive revenues can be obtained from that industry. Consider an efficient allocation with both X and Y positive. Then p.sub.x > *.sub.x and p.sub.y > *.sub.y must hold. Even though p.sub.y equals the minimum of average cost given the optimal level of X, it is above the minimum for zero output in X. A rights holder taking p.sub.y as given can lower X to reduce average cost below p.sub.y.. Thus, the interaction between X and minimum average cost in industry Y is crucial to the nonconcavity. Proposition 3 requires an externality interaction between firms. Ultra complete property rights and price taking are consistent with sustaining the efficient outcome without externalities, but not with externalities.

With only complete property rights, the efficient outcome will not be sustained by an output price-taking rights owner. Frech [1979] analyzed this, arguing that the outcome is efficient. He assumed the rights owner would charge the marginal damage per unit of pollution, rather than extracting all surplus available. This is not the rights holder's best strategy. Consider the case where the rights holder does not produce either good. His objective is to maximize revenue from payments from polluters for damage and from by bes by victims to prevent damage. At the efficient allocation, no bribes can be paid because those firms earn zero profits. Any tax, subsidy or franchise fee for Y firms causes (1b) and (1d) to be violated. Thus, no bribes by Y firms should be paid if the efficient solution is to occur. The rights holder will maximize revenue by disregarding the victims, that is maximizing n.sub.x.Z.sub.x = [p.sub.x.X - n.sub.x.F(x)]. This will not result in the first-best outcome. Similar problems result if the rights holder operates firms in one of the industries.

Clearly, price taking is a major assumption in Proposition 3. If the rights holder can control the number of firms, it seems especially unreasonable for him to think prices are independent of output levels.

Proposition 4. Assume that ultra complete property rights exist, that there are zero bargaining costs among firms but infinite bargaining costs with consumers, and that the rights holder takes demand functions as given. Then the socially efficient outcome cannot be sustained as an equilibrium.

That monopolistic behavior yields inefficient outcomes is straightforward and is recognized by French [1979] and others. For this reason, most research assumes output price-taking behavior. However, if the cost of bargaining with consumers is zero, then bargaining among all parties can yield the efficient outcome. Note that under ultra complete property rights, the rights holder has complete control over the levels of output of X and Y and thus can make all-or-nothing offers over outputs to consumers as well as to firms in industries X and Y. That is, the rights holder can act as a first-degree price discriminator.

Proposition 5. Assume that there exist ultra complete property rights and that the holder of the rights can bargain costlessly with all firms in industries X and Y and with all consumers of these products. Then the first best allocation is achieved regardless of who holds the property rights.

Proposition 5 shows that, with ultra complete property rights and zero bargaining costs with firms and consumers, a Coasian efficiency result obtains even under free entry. This does not contradict Schulze and d'Arge [1974] and others who argued that the Coase theoren fails in the long run since they made neither assumption.

Our result does not justify private bargaining as a decentralized process to correct inefficiencies. The bargaining is not just local between the parties involved in the externality, but is global, at least among all consumers of the goods produced by these industries. The process required to achieve efficiency might be called private socialism. The rights holder must have all the information a socialist government needs to run a planned economy and must make all-or-nothing offers to firms and consumers as in a command economy. The bargaining needed for efficiency in the long run is not a minor adjustment of the market mechanism, but a total replacement of it.

The assumptions are so extreme and unrealistic that the efficiency result cannot be taken even as a useful approximation for policy. Moreover, they cannot be significantly weakened. If the parties in an externality cannot bargain with consumers, then inefficiency results (propositions 3 and 4).

If the property right is only complete, even if costless bargaining with consumers is possible, an efficient outcome will not result. For example, if the property right does not give control over entry into the damaged industry, then the rights holder does not have control over the output of Y. Bargaining with consumers will involve only the level of x and n.sub.x.. Even though the bargainers may recognize the effects on the equilibrium levels of y and n.sub.y of changes in x and n.sub.x., they lack sufficient instruments to achieve the first-best outcome.

Proposition 3 shows that the lack of property rights is not the cause of inefficiencies from externalities. Even if one agent has complete control over outputs, if the individual is a price taker, then the efficient outcome is not sustained. Thus, completely specified property rights fail to lead to efficiency in the presence of externalities.

Jointness of supply seems to be the important factor, as argued by Baumol and Oates [1978]. In the long run with ultra complete property rights, the number of firms in each industry is under the rights owner's control. The nonconcavity arises because, at the first best, even under constant returns to scale, neither industry is producing at its lowest possible average total cost including fees. Thus the rights holder can manipulate outputs so that firms in one industry earn positive profits at current prices. These positive profits can then be earned by every entrant, and total revenue to the rights holder will increase beyond what is earned at the efficient level. The interrelation of costs across industries, even when under the control of one person, creates the possibility of jointly supplying entry at positive profits to any number of firms, thus causing the problem.

To amplify this, consider Knight's [1924] road example. Assume there are two classes of traffic, trucks and cars, and that driving by trucks effects the ability of cars to drive, but not other trucks, and driving by cars has no effect on either. This situation fits the model, so even if the road were privately owned and the owner were able to control entry of each type, the efficient outcome would not be sustained. On the other hand, if all traffic were identical and the externality from driving were reciprocal, the nonconcavity need not arise and, depending upon the nature of the external effect, the first best might be sustainable.


Given the impossibility of a decentralized bargaining solution, we turn in this section to analyze Pigovian taxation. If taxes are levied, it is assumed that the government is unaware whether or not bargaining takes place. The government directly observes damage and imposes a tax on the damaging activity. Firms remain price and tax takers--they assume the marginal tax will not change as they adjust production levels which changes the marginal effect of the externality activity. If the government pays compensation to the damaged industry, firms recognize that its level will change as they change their outputs by bargaining.

With Pigovian taxes, the natural assumption is that the damaging industry has the right to pollute if it pays the tax. Thus, the damaged indusry Y must compensate firms in industry X to induce further reductions in output of X. In addition, it is assumed that any firm which enters the damaging industry acquires the right to pollute (and pay the taxes). The existing firms cannot prevent entry into either industry. Existing firms bargain only among themselves, taking prices and the number of firms as given. Thus, their goal as specified above in section II is to maximize * = p.sub.y.n.sub.x.x + P.sub.y.n.sub.y.y - n.sub.x.F(x) - n.sub.y.G(y,X) wif respect to x and y.

Consider the short run with bargaining and a Pigovian tax on X's output. If the tax revenue is transferred lump sum to consumers, then firms can gain by bargaining to adjust output of the damaging commodity to reduce the revenue outflow. With appropriate side payments, both industries attain higher profits. Thus bargaining leads to an inefficient outcome. On the other hand, if the tax revenue is transferred non lump sum to the damaged industry so that firms recognize the transfer, then the outcome is efficient. No bargaining will take place. This result of Buchanan and Stubblebine [1962] and Turvey [1963] is stated as proposition 6.

Proposition 6. For fixed n.sub.x and n.sub.y., let (q*, p*, x*, y*) be an equilibrium with individuals maximizing utility taking prices as given, firms maximizing joint profits taking prices as given, and the government choosing q* and p* assuming that the resulting supplies arise from each industry separately maximizing profits.

A. If government revenue is transferred to consumers as a lump sum payment (t.sub.x = n.sub.y.G.sub.2.(y*, n.sub.x.X*), t.sub.y = 0, T = t.sub.x.n.sub.x.X*, and E.sub.x. = E.sub.y = 0, then x* and y* differ from the first best allocation.

B. If government revenue is explicitly transferred to industry Y (t.sub.x. = n.sub.y.G.sub.2.(y*, n.sub.x.X*), t.sub.y = 0, E.sub.x = T = O, and E.sub.y = - t.sub.x.n.sub.x.X*/n.sub.y.), then x* and y* are the first-best allocations if bargaining cannot reduce the number of firms.

In A, the externality is incorporated twice, in the tax and in the victims' costs. In B, after the revenue transfer, the externality appears in the first-order conditions only though the victims' costs. The nonconcavity problem of the previous section would also arise here; profits would rise for the group if a victim could be bribed to shut down and polluters then increased output.

The conditions on distribution of the revenue from the Pigovian tax for correct long-run entry as given in proposition 1 and to prevent bargaining inefficiencies as given in proposition 6 appear to be inconsistent. It is impossible for a linear Pigovian tax system to achieve efficiency in these circumstances.

Proposition 7. In the long run, with the right to damage acquired by entry into industry X, if the government imposes optimal linear Pigovian taxes and if costless bargaining occurs, then regardless of the rule dividing joint profits or of how the government distributes the revenue raised form the Pigovian tax, the outcome differs from first best.

At the first-best number of firms, there would be excess profits either from the tax revenue distributed to those damaged or to the gain from bargaining if revenues are given to consumers. These excess profits induce too much entry.

To avoid both problems simultaneously, a more complex Pigovian tax system is needed. To prevent excess entry into Y, no revenue must flow there. To prevent excess entry into X, profits from the ability to damage must be taxed away. By taxing only output of firms above the first-best level, the output tax raises no revenue at the first-best allocation. Proposition 8 establishes that there exists a tax system whose allocation is robust to bargaining.

Proposition 8. Assume that firms take prices and the number of firms as given, whether maximizing profits jointly or separately, and that entry occurs in any industry with positive profits. The government chooses a tax rate on output by damaging firms above the optimum quantity assuming that supply is determined from firms separately maximizing profits, charges a lump-sum franchise tax for entry into the damaging industry equal to what the tax would have been if charged on output below the optimum level, and transfers the franchise tax revenue as a lump-sum payment to consumers. That is, T.sup.x = max[0, t.sub.x.* (x - x)], t.sub.x.* = n.sub.y.*.G.sub.2.(y*, n.sub.x.*.sub.x.*.), where revenues of each x firm are q.sub.x.x - T.sup.x., t.sub.y.* = 0, E.sub.x = T.sub.x.*x, E.sub.y = 0 and T = n.sub.x.*t.sub.x.*x, where x is the first-best output per firm in industry X.

Then the first-best allocation is an equilibrium if firms maximize profits separately. In addition, taking as given prices, taxes and numbers of firms, the first-best allocation is in the core: no group of producers can raise the profits of all its members by changing quantities and redistributing profits.

This avoids the Buchanan-Stubblebine problem since the Pigovian tax raises no net revenues. The firms perceive no net outflow and thus cannot gain by altering the level of the damaging activity. There is no entry problem because the extra revenue retained by the damaging firms (as compared to a linear Pigovian tax) is extracted by the lump-sum franchise fee.

In the short run, Buchanan and Stubblebine's cure for inefficiency is to transfer the Pigovian tax revenue to the damaged firms. It seems plausible that augmenting that solution with a franchise tax in the damaged industry (equal to each firm's share of the tax revenue) could restore efficiency in the long run. While this prevents bargaining which changes aggregate output, firms in industry Y could decrease their tax bill by shutting down some firms to lowr franchise fee payments and increasing production in the remaining ones to keep revenue constant. The increase in average cost is a second-order effect and is dominated. Since closing down firms raises the revenue transfer to the remaining ones, such an approach is generally feasible. Because increasing x increases tax liabilities, closings to avoid franchise fees will not be desirable under the scheme in proposition 8. The firms perceive that, if some firms shut down, new firms will enter unless the existing firms expand output to maintain total output. The marginal tax on increases in output of polluters prevents shutting down some of these firms to lower franchise tax payments. It also inhibits a strategy of shutting down victim firms to reduce real damage costs and then expanding output of polluters. Thus the non-concavity problem of proposition 3 does not arise here. Avoiding any explicit transfer of revenue from the corrective tax to other firms or to consumers is the additional requirement imposed on the tax system, if the first best is to be sustained when bargaining is feasible.

Use of property rights instead of liability rules does not greatly simplify the problems of Pigovian tax systems. Complete property rights have the same efficiency problems as the liability rule in the presence of Pigovian taxes. With ultra complete property rights, the nonconcavity problem arises if prices and taxes are taken as given. At the social optimum, shutting down one polluting firm permits bribes to be collected from Y firms.


A decentralized efficient bargaining solution to production externalities with free entry does not exist. If firms are price takers, a nonconcavity in the objective function of the rights holder upsets the first-best allocation. If the rights holder takes demand functions as given, a standard monopoly inefficiency results. Only with ultra complete property rights is the first best sustained by bargaining, and then only if the rights holder bargains with all relevant consumers as well as firms. This is not a decentralized process, but involves one agent extracting all surplus and determining production and consumption. In effect, the rights holder manages a command economy.

Thus, some government intervention is needed to attain the first best. A tax system exists to accomplish this, even if bargaining is possible, if property rights are limited as under a liability rule. If more complete property rights exist, just as under bargaining alone with price-taking firms, the first best cannot be achieved using the tax systems considered here. Thus, the inefficiencies from externalities in the long-run context do not arise from insufficiently complete property rights. Actually, the reverse holds since completely specified property rights prevent the government from being able to sustain te optimum.

When entry is free, we have shown that bargaining does not solve externality problems unless the market system is replaced by economy-wide negotiations. Under limited property rights, the first-best allocation can be sustained under a tax scheme which, however, is complicated by the requirement that it eliminate any possible gains from bargaining. In the circumstances considered here, it is better to limit property rights and discourage bargaining than to try to make them as complete as possible and encourage bargaining.

Of course, the advantages of the tax system depend upon the government having sufficient information to calculate correctly the optimal taxes. If the government lacks adequate information, then the relative desirabilities of the bargaining and tax outcomes depend upon comparisons of second-best outcomes.


Proof of proposition 1. See proposition 4, chapter 4 of Baumol and Oates [1975, 43].

Proof of proposition 2. The first-order conditions for joint profit maximization are [derivative]II/[derivative]x = n.sub.x.[p.sub.x - F'(x) - n.sub.y.G.sub.2.(y,X)] = 0 (A1) [derivative]II/[derivative]y = n.sub.y.[p.sub.y - G.sub.1.(y, X)] = 0. (a2)

Since q = p, (A1), (A2), and zero profits imply that (1a) and (1b) are satisfied. It is straightforward to check that second-order conditions are satisfied. Q.E.D.

Proof of proposition 3. The first-order conditions for this problem are [derivative]I/[derivative]x = n.sub.x.[p.sub.x - F'(x) - n.sub.y.G.sub.2.] = 0 (A3) [derivative]I/[derivative]y = n.sub.y.(p.sub.y - G.sub.1.) = 0 (A4) [derivative]I/[derivative]n.sub.x = p.sub.x.x - F(x) - n.sub.y.xG.sub.2 = 0 (A5) [derivative]I/[derivative]n.sub.y = p.sub.y.y - G(y, X) = 0. (A6)

If the producer prices are at the correct market clearing levels, then (A3)-(A6) imply that 1(a-d) are satisfied and the first best would seem to be supported. Taking the second derivatives of I with respect to x, y, n.sub.x., and n.sub.y and simplifying using the first-order conditions (A3) and (A4) yields the Hessian of the objective function:

Due to the zero on the diagonal (H.sub.44 = 0), the 2 x 2 principal minors found by deleting the first and second or second and third rows and columns are negative. Similarly, the 3 x 3 principal minors found by deleting either the first row and column or the third row and column are positive. The full determinant equals -n.sub.x.n.sub.y.F"G.sub.11.(xG.sub.2.).sup.2 < 0. Each of these signs violates the second-order conditions for concavity. Any interior solution to the first-order conditions is thus a saddlepoint, not a maximum. Hence, such a point will not be sustained by a profit maximizing rights holder. Since the social optimum satisfies first order conditions (A3)-(A6), it cannot be sustained. Q.E.D.

Proof of proposition 4. Without taxes, px = U1(X, Y) and py = U2(X, Y). Thus, the rights holder's revenue is I = nx[U1(X, Y)x - F(x)] + ny[U2(X, Y)y - G(y, X)]. At the first best, using equations 1(a-d), the first-order conditions for the rights holder's problem reduce to XU.sub.11 + YU.sub.21 = 0 (A7) XU.sub.12 + YU.sub.22 = 0. (A8)

These two equations imply that the Hessian of U is singular, violating strict concavity of U. Therefore, the first-order conditions cannot be satisfied at the first best. Even if second-order conditions were satisfied, the first-best allocation would not be the solution. Q.E.D.

Proof of proposition 5. Given the quasilinear utility function, how much it can extract from consumers is easy to determine. Let l denote the time endowment of the consumer and let R denote leisure hours. Let U.sup.N be the total utility level achieved by the consumer if the rights holder acts as an ordinary monopolist. The restriction on the bargaining outcome between the rights holder and consumers is that U(X, Y) + R - l >= U.sup.N. If the consumer must pay a fee to the rights holder to keep prices at the socially optimal levels, the budget constraint becomes pxX + p.sub.x.X + p.sub.y.Y + R <= l - Z.sub.c., which substituted in the utility function yields U(X, Y) - p.sub.x.X - p.sub.y.Y - Z.sub.c. The profits of the rights holder bargaining with itself, making the solution indeterminate. The rights holder bargaining with itself, making the solution indetermine. The rights holder can offer triples X, Y, and Z.sub.c., which will be preferred to the monopoly outcome if U(X, Y) - p.sub.x.Y - p.sub.y.Y - Z.sub.c >= U.sub.N. Hence, Z.sub.c <= U(X, Y) - p.sub.x.X - p.sub.y.Y - U.sub.N. (A9)

Obviously, Z.sub.c is maximized if (A9) holds with equality. The total revenue of the rights holder is now I = n.sub.x.Z.sub.x + n.sub.y.Z.sub.y + Z.sub.c. Substituting the values of Z.sub.x., Z.sub.y. and Z.sub.c found earlier, I = U(X, Y) - U.sub.N - n.sub.x.F(x) - n.sub.y.G(y, X).

The rights holder's goal is to choose x, y, n.sub.x., and n.sub.y to maximize I which equals social welfare W minus U.sup.N. Since U.sup.N is constant, maximizing I is identical to maximizing W. The rights holder might seek to increase I by threatening a lowr utility if consumers do not accept a proposed bargain. However, if there are sequential choices with U.sup.N determined as the utility which would arise if no bargain were reached, then such threats would not be subgame perfect, so they would not be credible. Hence, U.sup.N can be taken as fixed. Q.E.D.

Proof of proposition 6.

(A) With taxes, U1(X, Y) = qx holds in any equilibrium. Substituting this into [derivative]II/[derivative]X = 0, the first order condition for output of industry X, yields U1 - F' - 2n.sub.y.G.sub.2 = 0, which violates the social optimum condition (1a).

(B) This is precisely analogous to Turvey's [1963] result with consumer externalities. Q.E.D.

Proof of proposition 7. If the government imposes corrective taxes and transfers the revenue to consumers, then the bargaining outcome has inefficient quantities as shown in proposition 6(A). If the government imposes corrective taxes and transfers revenue to the damaged industry, then joint profits equal q.sub.x.n.sub.x.x + p.sub.y.n.sub.y.y - n.sub.x.F(x) - n.sub.y.G(y, X). Whatever the division rule, free entry will drive * to zero. Hence, q.sub.x.n.sub.x.x + p.sub.y.n.sub.y.y - n.sub.x.F(x) - n.sub.y.G(y, X) = 0 (A10)

Imposing the condition for joint profit maximization on (A10) yields n.sub.x.[F'(x)x + n.sub.y.G.sub.2.(y, X) - F(x)] + n.sub.y.[G.sub.1.(y, X)y - G(y, X)] = 0. (A11)

Given the term n.sub.x.n.sub.y.G.sub.2.(y, X) in (A11) it is impossible to satisfy (1c) and (1d) without violating (A11). Hence, the first best cannot be attained. Q.E.D.

Proof of proposition 8. At the first best, T.sup.x = 0. Utility maximization determines *. At *, the price p.sub.x is not defined, but a drop in x below * lowers revenues by q.sub.x which exceeds the drop in revenues from a linear tax on x. Hence, the firm does not desire to lower x. For increases in x, revenue increases by *, the same as with the linear tax, so an increase in x above * is not desirable. Since * and *, profit maximization by Y firms satisfies (1b). Equations (1) then imply zero profits. The first-best allocation is an equilibrium under separate profit maximization.

Joint profits for any subset * are

The first-order conditions with respect to x and y are where (A12) is for increases in x above * and (A13) is for decreases in x below *. Since *, then, from X firms' supply decisions * satisfying (A12). Given * and *, * satisfying (A13). (A14) is satisfied from Y firms' supply decisions. Since profits also equal zero, no entry into or exit from the group is desirable. Q.E.D.
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Title Annotation:taxation and bargaining as strategies to modify behavior of private firms
Author:Hamilton, Jonathan H.; Sheshinski, Eytan; Slutsky, Steven M.
Publication:Economic Inquiry
Date:Jul 1, 1989
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