# Taxation, fines, and producer liability rules: efficiency and market structure implications.

1. Introduction

The control of external diseconomies created by worker, consumer, and environmental exposure to hazardous substances is a pervasive problem. In many industries, consumers and workers are routinely exposed to health risks associated with radiation, cigarette smoking, diethylstilbestrol (DES), asbestos, saccharin, dioxin, vinyl chloride, beta-napthylamine, benzidine, coke-oven emissions, and various occupational carcinogens arising from the use of basic materials such as pesticides, petroleum, coal, paraffin, iron ore, nickel, and chromium. Other industries involved with petroleum production, nuclear power generation, metal mining and smelting, pulp milling, and solid waste disposal pose the threat of contamination through accidental "spills" into the environmental medium. The unifying feature of these problems is that the extent of external damages, whether embodied in the probability of product failure or in the likelihood of human and environmental exposure to hazardous substances, is determined jointly by a producer's choice of output and safety provision.

Two important forms of externality control in hazardous sectors of the economy are producer liability rules and direct regulatory control through the use of output (Pigouvian) taxation and fines on injuries. This paper provides a comparative analysis of the relative efficiency properties of producer liability rules and regulatory policy in both short-run and long-run competitive equilibria. The essential feature of the model is that the provision of product safety in the hazardous sector is endogenized as a choice variable of the firm. The extent or magnitude of external damages can be limited, for example, by removing carcinogens from consumer products, by following re-entry guidelines after the application of pesticides, by requiring safety gear for construction workers, or by implementing better containment measures for solid and liquid waste. Given the conceptual unity of worker, consumer, and environmental safety issues, producer liability rules and regulatory controls are nested in a general model that allows their comparative properties to be examined.(1)

The theoretical framework, which views producer care as a choice variable of the firm, falls within a family of papers in the liability literature (e.g., Hamada 1976; Shavell 1980; Landes and Posner 1985; Marino 1988). In each of these papers, producer (or strict) liability achieves social optimality in a long-run competitive equilibrium, provided that the extent or likelihood of damage is not correlated across firms (i.e., no externalities exist in the liability functions).(2) The present analysis supports the finding that tort liability achieves a first-best resource allocation in the long run but finds the choice of regulatory controls to be more problematic.

In an important contribution, two papers by Carlton and Loury (1980, 1986) discuss the limitations of Pigouvian taxes in long-run equilibria. For the case of unavoidable damages, they demonstrate that a Pigouvian tax, which does not affect the privately optimal scale of each competitive firm, is inefficient when the damage function does not depend multiplicatively on the item that is taxed. This finding, though certainly important, is limited by the fact that firms typically exert some control over the extent of external damages through their choice of safety provision. In the following section, the limitations of Pigouvian taxation are revisited in a model that allows competitive firms to invest in product safety measures that reduce the probability of accidental injury to workers, consumers, and the environment. It is shown that Pigouvian taxation fails to achieve a socially optimal outcome in a long-run competitive equilibrium in which both output and safety provision are taxed (subsidized) according to their marginal contributions to social damage. Conversely, an appropriately designed policy involving fines on accidents and subsidies on safety provision does achieve an efficient outcome; however, the optimal policy may involve the taxation, not the subsidization, of product safety.

This paper also addresses the effect of increased exposure to tort liability on market structure. The existing literature regarding the effects of producer liability on market structure has focused on the issue of solvency (e.g., Ringleb and Wiggins 1990; Boyd and Ingberman 1994; Watts 1998). In particular, there is some evidence that an increase in producer liability may have little effect on product safety when assets can be shielded through divestiture. In a recent paper, Ringleb and Wiggins (1990) examine a wide range of hazardous industries and find that increased exposure to tort liability tends to stimulate de novo entry.(3) Ringleb and Wiggins hypothesize that the entry of firms results through incomplete capitalization and/or through latent risks that allow small firms to cease production before claims are made. Such divestiture is liability reducing when small firms conducting the risky task have insufficient assets to pay damages and declare bankruptcy when suits are filed or, in the case of latent harm, exit the industry before injury emerges. It is shown here that, regardless of firm solvency, entry and loss of incumbent market share can occur purely through market forces following an increase in producer liability exposure. The finding that increased tort liability induces entry in hazardous sectors, therefore, is not sufficient evidence to support the hypothesis that firms respond to greater liability exposure by divesting risky activities.

The remainder of the paper is structured as follows. In the next section, a model is developed to address the comparative efficiency properties of tort liability and regulatory policy in short-run and long-run competitive equilibria. Section 3 considers the structural implications of increased producer liability exposure in a market setting with neither regulatory controls nor solvency issues, and section 4 provides concluding comments.

2. The Model

Consider a simple partial equilibrium model with n identical competitive firms. Each firm produces a homogeneous product with inverse demand given by P(Y), where Y = ny is total industry output. Production by each firm in the industry also imposes additional damages to society, g(y), in the event of an accident. The expected damage, D, created by a firm's production decision is given by D = ag(y), where a[Epsilon] [0,1] is the probability in which an accident occurs. In cases in which damages arise through worker exposure to toxic substances or through environmental "spills," a may be interpreted as the probability of product failure, as in Marino (1991). In cases in which external damages arise through health risk, a may be interpreted as an inverse measure of product safety, for example, the level of product carcinogenity or the degree of worker exposure to toxic substances. Throughout, the accident probability is designated quite generally as an inverse measure of product safety. To produce y units of output at safety level a, the representative firm incurs production costs of c(a,y), where dc(a,y)/dy [greater than] 0 and dc(a,y)/da [less than] 0.

The policy maker's problem is to choose industry output and the number of firms in a partial equilibrium setting so as to maximize social welfare. In this context, social welfare is defined as total consumer surplus net of production costs and external damages in the hazardous sector. In a short-run equilibrium, the number of firms is fixed and the socially optimal a and y can be completely characterized as the solution to

[Mathematical Expression Omitted].

Using the definition of the inverse demand curve, a and y satisfy the first-order conditions

P(ny) = dc/dy (a, y) + ag[prime](y) (1)

and

g(Y) = -dc/dy(a, y). (2)

Equation 1 equates the market price with marginal social damage, which is the sum of marginal private cost and marginal external cost. Equation 2 equates total external damage with the marginal cost of providing product safety; that is, it states that the socially optimal level of product safety occurs where the marginal cost of investing in safety measures is equal to total external damage at the equilibrium level of output.(4)

In the case of direct regulatory controls, the short-run competitive equilibrium (SRCE) is described with regard to three policy instruments: a tax on output, t; a subsidy on safety provision (i.e., a tax on "negative" safety), s; and a fine on accidents, f. The SRCE can be completely characterized as the solution to

[Mathematical Expression Omitted].

Using the definition of the inverse demand curve, a and y satisfy the first-order conditions

P(ny) = dc/dy(a, y) + af + t (3)

and

s + fy = -dc/da(a, y). (4)

This results in the following proposition.

PROPOSITION 1. For appropriately chosen policy pairs t,s or f,s, the short-run competitive equilibrium coincides with the short-run social optimum.

Proof. It is necessary to show that if [a.sup.*],[y.sup.*] are a short-run social optimum, then there exists a policy pair [t.sup.*],[s.sup.*] or [f.sup.*], [s.sup.*] such that [a.sup.*], [y.sup.*] are a short-run competitive equilibrium.

For the case of taxation, suppose that [a.sup.*],[y.sup.*] solve Equations 1 and 2 and define [s.sup.*] = g([y.sup.*]) and [t.sup.*] = ag[prime]([y.sup.*]). Then, when f = 0, it follows immediately from Equations 1 and 2 that [a.sup.*],[y.sup.*] also satisfy Equations 3 and 4.

For the case of a fine-subsidy pair, suppose that [a.sup.*],[y.sup.*] satisfy Equations 1 and 2 and choose [f.sup.*] = g[prime]([y.sup.*]) and [s.sup.*] = [y.sup.*](g([y.sup.*])/[y.sup.*] - g[prime]([y.sup.*])). For such a [f.sup.*], it is clear from Equation 1 that [a.sup.*],[y.sup.*] will also satisfy Equation 3 when t = 0. To see that [a.sup.*],[y.sup.*] also satisfy Equation 4, rewrite Equation 2 as

g([y.sup.*]) + [s.sup.*] - [s.sup.*] + [f.sup.*][y.sup.*] - [f.sup.*][y.sup.*] = -dc/da ([a.sup.*], [y.sup.*])

or

[s.sup.*] + [f.sup.*][y.sup.*] = -dc/da ([a.sup.*], [y.sup.*]). (5)

It is obvious from Equation 5 that [a.sup.*],[y.sup.*] satisfy Equation 4. Q.E.D.

The result in Proposition 1 is fairly transparent: With two distortions, it is possible to achieve the social optimum with two policy instruments. However, the case of fines on accidents is of some independent interest. If the purpose of a fine on accidents is to achieve economic efficiency rather than to serve as a compensatory mechanism for injured parties, the optimal per unit fine is assessed according to the marginal damage, not the average damage, associated with product failure. Moreover, when fines are assessed on accidents, the socially optimal policy control may involve the taxation, not the subsidization, of product safety. If the marginal damage of each firm's output exceeds the average damage at the optimal point, then the first-best policy pair involves a fine on accidents equal to marginal damage and a tax on product safety equal to the difference between marginal and average damage.

An alternative to direct regulatory control is the implementation of a producer liability rule. In the case of tort liability, the SRCE is completely characterized as the solution to

[Mathematical Expression Omitted].

Using the definition of the inverse demand curve, a and y satisfy the first-order conditions

P(ny) = dc/dy (a, y) + ag[prime](y)

and

g(y) = -dc/da(a, y),

which coincide directly with Equations 1 and 2. Thus, a producer liability rule achieves the first-best social outcome in a short-run competitive equilibrium.

In the long run, entry (exit) may occur and the number of firms is endogenous.(5) Using the definition of the inverse demand curve, the long-run social optimum (LRSO) can be characterized as the solution to

[Mathematical Expression Omitted].

The first-order conditions are

P(ny) = dc/dy(a, y) + ag[prime](y), (6)

g(y) = dc/da(a, y), (7)

and

yP(ny) - c(a, y) + ag(y). (8)

With respect to the regulatory controls (t,f, and s), the long-run competitive equilibrium (LRCE) can be completely characterized by the conditions

P(ny) = dc/dy(a, y) + af + t, (9)

s + fy = -dc/da(a, y), (10)

and

yP(ny) = c(a, y) + yt + as + ayf. (11)

It is now possible to prove the following proposition.

PROPOSITION 2. There exists no t,s pair such that the long-run competitive equilibrium coincides with the long-run social optimum.

Proof. It is necessary to show that if t is the tax on output and s is the subsidy on product safety, then if [a.sup.*],[y.sup.*],[n.sup.*] satisfy Equations 6-8, they will not also satisfy Equations 9-11. First notice from Equations 6 and 7 that t,s must meet the conditions of Proposition 1 for an LRSO. To complete the proof, one must show that t = [t.sup.*] and s = [s.sup.*] will not satisfy Equation 11 for [a.sup.*],[y.sup.*], [n.sup.*].

Suppose that [a.sup.*], [y.sup.*], [n.sup.*] satisfy Equation 9-11 and let [s.sup.*] = g([y.sup.*]) and [t.sup.*] = ag [prime]([y.sup.*]). For such a [s.sup.*], [t.sup.*] it is clear from Equations 9 and 10 that [a.sup.*], [y.sup.*], [n.sup.*] will also satisfy Equations 6 and 7. However, for such a tax-subsidy pair, [a.sup.*], [y.sup.*] will not satisfy Equation 8. To see this, rewrite Equation 8 as

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [a.sup.*]g(.sup.*]) + [y.sup.*] [t.sup.*] - [y.sup.*][t.sup.*] + [a.sup.*] - [a.sup.*][s.sup.*]

or

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [y.sup.*][t.sup.*] + [a.sup.*][s.sup.*] - [a.sup.*]g[prime]([y.sup.*]). (12)

From Equation 12 it is apparent that [a.sup.*],[y.sup.*],[n.sup.*] will not satisfy Equation 11 when f = 0. Q.E.D.

The intuition behind this result is familiar: The number of distortions exceeds the number of policy instruments. In an LRCE, the level of output, the degree of product safety of an individual firm, and the number of firms in the industry each contribute to external damages. The standard prescription requires three separate Pigouvian taxes, where the missing instrument, the "entry tax" of Carlton and Loury (1980), is effectively a tax on the number of firms. For cases in which producer care affects external damages, however, Proposition 2 demonstrates that the failure of Pigouvian taxation is not a potential outcome, as in the case of exogenous product safety, but a fundamental policy result. A nontrivial damage function does not exist for which a tax-subsidy pair leads to coincident outcomes in long-run competitive and socially optimal equilibria. The intuition for this result is as follows. Consider, as in Carlton and Loury (1980, 1986), the case in which the level of producer care does not affect the level of social damage. Here, if the marginal damage of an additional unit of output equals the average damage (e.g., external damages depend only on total industry output), then the optimal tax on entry is zero and a single Pigouvian tax on output is an appropriate fiscal policy. Next, consider the case in which producer care affects social damage. To achieve a regulated social optimum here, marginal damage must equal average damage with respect to each partial effect: the level of output and the degree of product safety. Clearly, for any tax-subsidy pair, a damage function exists that satisfies the correct entry incentive with respect to output, and a damage function also exists that satisfies the correct entry incentive with respect to product safety. Proposition 2 reveals the fact that these two functions cannot coincide.

PROPOSITION 3. There exists a f,s pair such that the long-run competitive equilibrium coincides with the long-run social optimum.

Proof. It is necessary to show that if [a.sup.*],[y.sup.*],[n.sup.*] are a long-run social optimum, then there exists a policy pair [f.sup.*],[s.sup.*] such that [a.sup.*],[y.sup.*][n.sup.*] are a long-run competitive equilibrium.

Suppose that [a.sup.*],[y.sup.*],[n.sup.*] satisfy Equations 6-8 and choose [f.sup.*] = g[prime]([y.sup.*]) and [s.sup.*] = [y.sup.*](g([y.sup.*])/[y.sup.*] - g[prime]([y.sup.*])). For such a [f.sup.*],[s.sup.*] it is clear from Equations 6 and 7 that [a.sup.*],[y.sup.*],[n.sup.*] also satisfy Equations 9 and 10 when t = 0. To see that [a.sup.*],[y.sup.*],[n.sup.*] also satisfy Equation 11, rewrite Equation 8 as

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [a.sup.*]g([y.sup.*]) + [a.sup.*][y.sup.*][f.sup.*] - [a.sup.*][y.sup.*]f[*.sup.*]+ [a.sup.*][s.sup.*] - [a.sup.*][.sup.s*]

or

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [a.sup.*][s.sup.*] + [a.sup.*][y.sup.*][f.sup.*] (13)

Clearly, Equation 13 coincides with Equation 11 when t = 0. Q.E.D.

Corollary. If marginal social damage equals average social damage at [y.sup.*], then a fine of [f.sup.*] = g[prime]([y.sup.*]) on accidents will alone achieve both the short-run and the long-run social optimum.

Proof. Obvious.

In Proposition 3 and its corollary, a joint policy that involves fines on accidents and subsidies on safety provision is capable of achieving a first-best outcome. These results illuminate an important point that the "failure" of a Pigouvian tax, in many instances, may arise quite simply from the fact that the supposed Pigouvian tax is a tax not on the externality but on a proxy for external damages. In the case in which external damages occur through accidental "spills" into an environmental medium, the source of the externality is the accident itself and depends only indirectly on the inefficiency of the associated output and safety levels of the firm.

When firms fail to recognize the external damages associated with product failure, three sources of inefficiency arise through separate distortions in output level, safety provision, and the equilibrium number of firms. The use of accident fines allows the regulator to effectively control three distortions with only two instruments. The intuition behind this result is that a fine, which is assessed on a multiplicative relationship between output and safety provision, creates a nonlinearity between policy instruments. An increase in the fine affects both the level of output and the degree of producer care for the representative firm, whereas an increase in the subsidy changes only the marginal valuation of product safety. The number of firms in an LRCE is determined jointly by the subsidy payment and by the level of the fine on accidents. Thus, for any given revenue transfer between the firms and the regulator, it is possible for the regulator to induce different output-safety combinations through the choice of relative weights on the fine and subsidy instruments. Proposition 3 reveals the fact that relative weights on fine and subsidy instruments exist that satisfy the optimal output-safety combination and simultaneously meet the revenue transfer necessary to achieve the optimal number of firms.

For completeness, the long-run efficiency properties of tort liability are examined next. With respect to a producer liability rule, the LRCE can be completely characterized by

P(ny) = dc/dy(a, y) + ag[prime](y), (14)

g(y) = dc/da(a, y), (15)

and

yP(ny) = c(a, y) + ag(y). (16)

Equations 14-16 coincide with Equations 6 - 8. It follows directly that the LRSO and LRCE coincide under a system of producer liability rules; thus, a producer liability rule leads to first-best resource allocations for both short-run and long-run competitive equilibria.(6)

3. The Structural Implications of a Change in Producer Liability Exposure

This section considers the effect of a change in producer liability exposure on market structure. To simplify the analysis, the effect of increased liability exposure is considered in the absence of regulatory policy. Unlike previous papers that have focused on the structural implications of producer liability when firm solvency is important (e.g., Ringleb and Wiggins 1990; Boyd and Ingberman 1994; Watts 1998), attention is confined here to the case of a fully capitalized industry without divestiture incentives. The results described below demonstrate that de novo entry in the face of increased producer liability, as observed by Ringleb and Wiggins (1990) in the 1967-80 period surrounding rapid changes in U.S. liability law, is consistent with an alternate hypothesis of purely structural change.

Let [Theta] be a shift parameter in the liability function of the representative firm such that g = g(y;[Theta]). Without loss of generality, an increase in producer liability is represented by the condition [g.sub.[Theta]](y;[Theta]) [greater than] 0, whereas an increase (resp. decrease) in the marginal injury relation is represented by the condition [g.sub.y[Theta]](y;[Theta]) [greater than] 0 (resp. [less than] 0).(7) It is also helpful to define for future reference the ratio of change in the marginal and average injury relation, [[Epsilon].sub.[Theta]] = [g.sub.y[Theta]]y/[g.sub.[Theta]], as the elasticity of the shift in liability structure. Denoting the second-order partials associated with Equations 14 and 15 as [[Pi].sub.yy], [[Pi].sub.aa], and [[Pi].sub.ay] yields the following proposition.

PROPOSITION 4. In a long-run competitive equilibrium, a change in producer liability exposure affects the level of output and safety provision of a representative firm as

dy/d[Theta] = -[g.sub.[Theta]] (a[[Pi].sub.aa] (1 - [[Epsilon].sub.[Theta]]) + y[[Pi].sub.ay])/y([[Pi].sub.yy][[Pi].sub.aa] - [([[Pi].sub.ay]).sup.2]), (17)

da/d[Theta] = [g.sub.[Theta]](y[[Pi].sub.yy] + a[[Pi].sub.ay](1 - [[Epsilon].sub.[Theta]]))/y([[Pi].sub.yy][[Pi].sub.aa] - [([[Pi].sub.ay]).sup.2]). (18)

A change in liability exposure affects the number of firms as

dn/d[Theta] = [ag.sub.[Theta]]/P[prime][y.sup.2] + [ng.sub.[Theta]](a[[Pi].sub.aa](1 - [[Epsilon].sub.[Theta]]) + y[[Pi].sub.ay])/[y.sup.2]([[Pi].sub.yy][[Pi].sub.aa] - [([[Pi].sub.ay]).sup.2]). (19)

Proof Perturbing the conditions for an LRCE in Equations 14-16 and making use of the envelope theorem yields

[Mathematical Expression Omitted], (20)

where [[Pi].sub.ay] = [[Pi].sub.ya], = -([c.sub.ay] + [g.sub.y]) and where [[Pi].sub.yy] = -([c.sub.yy] + [ag.sub.yy]) [less than] 0, [[Pi].sub.aa] = - [c.sub.aa] [less than] 0, and [[Pi].sub.yy] [[Pi].sub.aa] - (([[Pi].sub.ay]).sup.2] [greater than] 0 by the Routh-Hurwicz stability conditions. Solving the system of Equation 20 and collecting terms completes the proof. Q.E.D.

In general, the effects of a change in liability exposure on market structure in Equations 17-19 are ambiguous and depend on the value of the shift elasticity parameter, [[Epsilon].sub.[Theta]], and on the cross-effect between output and safety provision, [[Pi].sub.ay]. A testable implication of the divestiture hypothesis therefore occurs when the change in liability structure increases the average injury relation to a greater extent than the marginal injury relation ([[Epsilon].sub.[Theta]] [less than or equal to] 1) and when higher investments in product safety improve the marginal profitability of output, [[Pi].sub.ay] [less than or equal to] 0. That is, for the special case in which [[Pi].sub.ay] [less than or equal to] 0 and [[Epsilon].sub.[Theta]] [less than or equal to] 1, greater producer exposure to tort liability (at least weakly) increases the output of the representative firm, enhances product safety, and precipitates the exit of firms through a purely structural effect in the market. Only under these somewhat restrictive circumstances might one conclude that de novo entry in a hazardous sector is motivated by entirely non-structural factors, such as the level of firm solvency.

In many instances, increased exposure to tort liability can stimulate de novo entry in a hazardous sector through market forces and regardless of the solvency of firms. To further clarify the structural implications of tort liability, it is helpful to consider a special case of the model that suppresses the cross-effect between output and safety provision. For example, consider, as in Shavell (1980) and Marino (1991), the cost function c(y,a) = yc(a), where [c.sub.a] [less than] 0 and [c.sub.aa] [greater than] 0. In this case, production costs increase linearly with output, which implies that [[Pi].sub.ay] = 0. Denoting the (inverse) demand elasticity as [Eta] = -P[prime]]Y/P and the elasticity of the slope of the damage function as [Zeta] = [g.sub.yy]/[g.sub.y] yields the following proposition.

PROPOSITION 5. If c(y,a)/y = [c.sub.y](y,a), increased producer exposure to tort liability (i) increases product safety, (ii) increases the output of a representative firm if and only if [[Epsilon].sub.[Theta]] [less than] 1, and (iii) increases the number of firms if and only if [[Epsilon].sub.[Theta]] - 1 [greater than] ([Zeta]/[Eta])(ag/Py).

Proof Making the appropriate substitutions in Equations 14-16 and simplifying yields [g.sub.y] = g/y = -[c.sub.ay]. It follows immediately that [[Pi].sub.ay] = 0. The second-order condition associated with Equation 14 implies [g.sub.yy] [greater than] 0; thus, Equations 17-19 reduce to

dy/d[Theta] = [g.sub.[Theta]](1 - [[Epsilon].sub.[Theta]])/[g.sub.yy]y, (21)

da/d[Theta] = -[g.sub.[Theta]]/[c.sub.aa], (22)

and

dn/d[Theta] = [g.sub.[Theta]][[ag.sub.yy] - nP[prime](1 - [[Epsilon].sub.[Theta]])]/P[prime][g.sub.yy][y.sup.2], (23)

respectively, where [g.sub.[Theta]] [greater than] 0 for an increase in liability exposure. Inspection of Equations 21 and 22 completes parts (i) and (ii). Noting that the denominator in Equation 23 is negative, entry occurs following increased liability exposure if and only if [Xi]a[g.sub.y] + [Eta]P [less than] [Eta]P[[Epsilon].sub.[Theta]]. Substituting [g.sub.y] = g/y and simplifying completes the proof. Q.E.D.

In response to an increase in liability exposure, firms that internalize tort liability as a component of production costs change their operating scale to equate marginal and average cost inclusive of the increased injury expense. When there are no cross-effects between output and safety provision, increased exposure to tort liability has no effect on the productive scale of a representative firm only if the change in the marginal injury relation exactly coincides with the change in the average injury relation ([[Epsilon].sub.[Theta] = 1]).(8) Conversely, the level of output associated with minimum average cost decreases for a competitive firm whenever the upward shift in the average cost curve exceeds the change in marginal cost, as in the case of increased producer liability and [[Epsilon].sub.[Theta]] [greater than] 1. For a sufficiently large value of the shift elasticity, the reduction in productive scale by incumbent firms makes entry attractive despite the increased liability exposure in the industry. Entry is more likely to occur in a hazardous sector following an increase in tort liability when inverse demand is price elastic, the marginal damage function is price inelastic, and the increase in marginal damage is large relative to the change in average damage. Furthermore, de novo entry is more likely when total revenue is large relative to the total injury associated with product failure, a condition under which firm solvency is not likely to be an issue.

Part (iii) of Proposition 5 provides an intuitive condition for increased liability exposure to induce entry through a purely structural effect. As the following example demonstrates, the magnitude of this effect can be substantial.

Example

Consider a market with linear demand, P(ny) = A - [Alpha]ny, and a quadratic damage function, g = b + [Beta][y.sup.2]. Increased exposure to tort liability is represented by an increase in either b or [Beta]. Suppose that each of n competitive firms has the cost function c(a,y) = cy/a. Solving Equations 14-16 for the equilibrium level of output and safety provision for each firm yields [y.sup.*] = [(b/[Beta]).sup.0.5] and [a.sup.*] = [(c/2).sup.0.5][(b[Beta]).sup.0.25]. It is immediately apparent that the equilibrium output level of each firm is increasing in b and decreasing in [Beta], whereas the probability of product failure is decreasing in both b and [Beta]. Thus, regardless of the nature of the increase in producer liability, the level of safety provision increases. Using the definition of inverse demand, the equilibrium number of firms is [n.sup.*] = ((A - [P.sup.*])/[Alpha])[([Beta]/b).sup.0.5], from which it follows directly that entry occurs whenever the ratio [Beta]/b increases in response to the change in liability structure.

The above example indicates that, in general, no clear correspondence can be drawn between de novo entry and the level of firm solvency. Specifically, divestiture incentives are not a necessary condition for entry to occur in a hazardous industry. The entry of new firms in response to increased liability exposure leads to qualitative predictions regarding divestiture only in the case in which the average injury relation increases to a greater extent than the increase in the marginal injury relation. Moreover, the implications of entry differ markedly, depending on whether entry is induced through divestiture or through purely structural forces. Quite unlike the implication of entry in the case considered by Ringleb and Wiggins (1990), entry in the present model, when it occurs, is consistent with the long-run socially optimum number of firms.

4. Concluding Remarks

This paper has compared taxation and producer liability rules in a model that allows competitive firms to invest in product safety measures that lower the probability of worker, consumer, and environmental damages. The paper has shown that Pigouvian taxation fails to achieve a socially optimal outcome in a long-run competitive equilibrium in which both output and safety provision are taxed (subsidized) according to their marginal contributions to social damage. The analysis demonstrated that a system of fines on accidents and subsidies (taxes) on product safety provision are capable of achieving first-best resource allocations. However, the optimal policy pair assesses a per unit fine on marginal, not average, damages and potentially involves the taxation, not the subsidization, of product safety investments. The analysis further revealed that tort liability achieves a first-best resource allocation in both long- and short-run equilibria, which highlights the appeal of legal controls in hazardous sectors of the economy.

This paper also addressed the effect of increased exposure to tort liability on market structure. The results demonstrated that entry and loss of incumbent market share can occur purely through market forces following an increase in producer liability. The implication of this finding is that a divestiture incentive is not a necessary condition for small-firm entry to occur in response to increased exposure to tort liability. Entry (exit) of competitive firms generally occurs following nonuniform changes in the marginal and average components of the liability function, even when solvency is not an issue. In particular, entry occurs when the increased exposure to tort liability sufficiently increases the marginal injury relation relative to the change in the average injury relation. The implications of this result contrast sharply with that of divestiture-induced entry, as the entry of firms following increased producer exposure to liability is associated with first-best levels of output and safety provision in the hazardous sector. A more thorough consideration of the structural effects of tort liability is necessary to understand the underlying motivations and the commensurate welfare implications of entry in hazardous sectors of the economy.

This research has benefitted from helpful comments by Joseph Farrell, Jonathan Hamilton, Michael Hanemann, Theodore Keeler, Jeff Perloff, Dave Sunding and two anonymous referees.

1 Note that producer liability and direct regulatory controls may be viewed as complements rather than substitutes under uncertainty. In stochastic environments, the distinction between ex ante and ex post means of control is particularly important (see Kolstad, Ulen, and Johnson 1990).

2 Marino (1988) demonstrates that the long-run optimality of producer liability fails to hold when the probability of harm depends nonlinearly on cumulative output in the industry. In this case, arguments in the expected damage functions of individual firms are interdependent, and an externality is created by the imposition of the liability rule.

3 For a comprehensive treatment of the entry incentives created by different liability rules when solvency is important, see Watts (1998).

4 A similar result is derived by Marino (1991).

5 For the long-run model of producer liability, the common law doctrine of "coming to the nuisance" is observed; that is, attention is confined to cases in which pollutees cannot become victims by choice. For a thorough analysis on the efficiency of bargaining outcomes when liability rules and property rights accrue to new entrants, see Hamilton, Sheshinski, and Slutsky (1989).

6 This result is also derived by Hamada (1976), Shavell (1980), and Landes and Posner (1985).

7 From an initial zero-liability position, [Theta] may represent a movement to a producer liability system, as occurred in the early 1970s in the United States. To avoid complications resulting from discontinuity in [Theta], one can think of the initial level of liability exposure, g(y), as being infinitesimally small.

8 Such a result is familiar to competitive models in which firms choose a single variable (e.g., output) subject to a linear penalty schedule (e.g., a unit tax).

References

Boyd, James, and Daniel E. Ingberman. 1994. Extending liability: Should the sins of the producer be visited upon others? Resources for the Future Discussion Paper 95-05, Washington, DC.

Carlton, Dennis W., and Glenn C. Loury. 1980. The limitations of Pigouvian taxes as a long-run remedy for externalities. Quarterly Journal of Economics 95:559-66.

Carlton, Dennis W., and Glenn C. Loury. 1986. The limitations of Pigouvian taxes as a long-run remedy for externalities: An extension of results. Quarterly Journal of Economics 101:631-4.

Hamada, Koichi. 1976. Liability rules and income distribution in products liability. American Economic Review 66:228-34.

Hamilton, Jonathan H., Eytan Sheshinski, and Steven M. Slutsky. 1989. Production externalities and long-run equilibria: Bargaining and Piguvian taxation. Economic Inquiry 27:453-71.

Kolstad, Charles D., Thomas S. Ulen, and Gary V. Johnson. 1990. Ex post liability for harm vs. ex ante safety regulation: Substitutes or complements? American Economic Review 80:888-901.

Landes, William M., and Richard A. Posner. 1985. A positive economic analysis of products liability. Journal of Legal Studies 14:535-68.

Marino, Anthony M. 1988. Products liability and scale effects in a long-run competitive equilibrium. International Review of Law and Economics 8:97-107.

Marino, Anthony M. 1991. Market share liability and economic efficiency. Southern Economic Journal 57:667-75.

Ringleb, Al H., and Steven N. Wiggins. 1990. Liability and large-scale, long-term hazards. Journal of Political Economy 98:574-95.

Shavell, Steven. 1980. Strict liability versus negligence. Journal of Legal Studies 9:1-25.

Watts, Alison. 1998. Insolvency and division of cleanup costs. International Review of Law and Economics. In press.

The control of external diseconomies created by worker, consumer, and environmental exposure to hazardous substances is a pervasive problem. In many industries, consumers and workers are routinely exposed to health risks associated with radiation, cigarette smoking, diethylstilbestrol (DES), asbestos, saccharin, dioxin, vinyl chloride, beta-napthylamine, benzidine, coke-oven emissions, and various occupational carcinogens arising from the use of basic materials such as pesticides, petroleum, coal, paraffin, iron ore, nickel, and chromium. Other industries involved with petroleum production, nuclear power generation, metal mining and smelting, pulp milling, and solid waste disposal pose the threat of contamination through accidental "spills" into the environmental medium. The unifying feature of these problems is that the extent of external damages, whether embodied in the probability of product failure or in the likelihood of human and environmental exposure to hazardous substances, is determined jointly by a producer's choice of output and safety provision.

Two important forms of externality control in hazardous sectors of the economy are producer liability rules and direct regulatory control through the use of output (Pigouvian) taxation and fines on injuries. This paper provides a comparative analysis of the relative efficiency properties of producer liability rules and regulatory policy in both short-run and long-run competitive equilibria. The essential feature of the model is that the provision of product safety in the hazardous sector is endogenized as a choice variable of the firm. The extent or magnitude of external damages can be limited, for example, by removing carcinogens from consumer products, by following re-entry guidelines after the application of pesticides, by requiring safety gear for construction workers, or by implementing better containment measures for solid and liquid waste. Given the conceptual unity of worker, consumer, and environmental safety issues, producer liability rules and regulatory controls are nested in a general model that allows their comparative properties to be examined.(1)

The theoretical framework, which views producer care as a choice variable of the firm, falls within a family of papers in the liability literature (e.g., Hamada 1976; Shavell 1980; Landes and Posner 1985; Marino 1988). In each of these papers, producer (or strict) liability achieves social optimality in a long-run competitive equilibrium, provided that the extent or likelihood of damage is not correlated across firms (i.e., no externalities exist in the liability functions).(2) The present analysis supports the finding that tort liability achieves a first-best resource allocation in the long run but finds the choice of regulatory controls to be more problematic.

In an important contribution, two papers by Carlton and Loury (1980, 1986) discuss the limitations of Pigouvian taxes in long-run equilibria. For the case of unavoidable damages, they demonstrate that a Pigouvian tax, which does not affect the privately optimal scale of each competitive firm, is inefficient when the damage function does not depend multiplicatively on the item that is taxed. This finding, though certainly important, is limited by the fact that firms typically exert some control over the extent of external damages through their choice of safety provision. In the following section, the limitations of Pigouvian taxation are revisited in a model that allows competitive firms to invest in product safety measures that reduce the probability of accidental injury to workers, consumers, and the environment. It is shown that Pigouvian taxation fails to achieve a socially optimal outcome in a long-run competitive equilibrium in which both output and safety provision are taxed (subsidized) according to their marginal contributions to social damage. Conversely, an appropriately designed policy involving fines on accidents and subsidies on safety provision does achieve an efficient outcome; however, the optimal policy may involve the taxation, not the subsidization, of product safety.

This paper also addresses the effect of increased exposure to tort liability on market structure. The existing literature regarding the effects of producer liability on market structure has focused on the issue of solvency (e.g., Ringleb and Wiggins 1990; Boyd and Ingberman 1994; Watts 1998). In particular, there is some evidence that an increase in producer liability may have little effect on product safety when assets can be shielded through divestiture. In a recent paper, Ringleb and Wiggins (1990) examine a wide range of hazardous industries and find that increased exposure to tort liability tends to stimulate de novo entry.(3) Ringleb and Wiggins hypothesize that the entry of firms results through incomplete capitalization and/or through latent risks that allow small firms to cease production before claims are made. Such divestiture is liability reducing when small firms conducting the risky task have insufficient assets to pay damages and declare bankruptcy when suits are filed or, in the case of latent harm, exit the industry before injury emerges. It is shown here that, regardless of firm solvency, entry and loss of incumbent market share can occur purely through market forces following an increase in producer liability exposure. The finding that increased tort liability induces entry in hazardous sectors, therefore, is not sufficient evidence to support the hypothesis that firms respond to greater liability exposure by divesting risky activities.

The remainder of the paper is structured as follows. In the next section, a model is developed to address the comparative efficiency properties of tort liability and regulatory policy in short-run and long-run competitive equilibria. Section 3 considers the structural implications of increased producer liability exposure in a market setting with neither regulatory controls nor solvency issues, and section 4 provides concluding comments.

2. The Model

Consider a simple partial equilibrium model with n identical competitive firms. Each firm produces a homogeneous product with inverse demand given by P(Y), where Y = ny is total industry output. Production by each firm in the industry also imposes additional damages to society, g(y), in the event of an accident. The expected damage, D, created by a firm's production decision is given by D = ag(y), where a[Epsilon] [0,1] is the probability in which an accident occurs. In cases in which damages arise through worker exposure to toxic substances or through environmental "spills," a may be interpreted as the probability of product failure, as in Marino (1991). In cases in which external damages arise through health risk, a may be interpreted as an inverse measure of product safety, for example, the level of product carcinogenity or the degree of worker exposure to toxic substances. Throughout, the accident probability is designated quite generally as an inverse measure of product safety. To produce y units of output at safety level a, the representative firm incurs production costs of c(a,y), where dc(a,y)/dy [greater than] 0 and dc(a,y)/da [less than] 0.

The policy maker's problem is to choose industry output and the number of firms in a partial equilibrium setting so as to maximize social welfare. In this context, social welfare is defined as total consumer surplus net of production costs and external damages in the hazardous sector. In a short-run equilibrium, the number of firms is fixed and the socially optimal a and y can be completely characterized as the solution to

[Mathematical Expression Omitted].

Using the definition of the inverse demand curve, a and y satisfy the first-order conditions

P(ny) = dc/dy (a, y) + ag[prime](y) (1)

and

g(Y) = -dc/dy(a, y). (2)

Equation 1 equates the market price with marginal social damage, which is the sum of marginal private cost and marginal external cost. Equation 2 equates total external damage with the marginal cost of providing product safety; that is, it states that the socially optimal level of product safety occurs where the marginal cost of investing in safety measures is equal to total external damage at the equilibrium level of output.(4)

In the case of direct regulatory controls, the short-run competitive equilibrium (SRCE) is described with regard to three policy instruments: a tax on output, t; a subsidy on safety provision (i.e., a tax on "negative" safety), s; and a fine on accidents, f. The SRCE can be completely characterized as the solution to

[Mathematical Expression Omitted].

Using the definition of the inverse demand curve, a and y satisfy the first-order conditions

P(ny) = dc/dy(a, y) + af + t (3)

and

s + fy = -dc/da(a, y). (4)

This results in the following proposition.

PROPOSITION 1. For appropriately chosen policy pairs t,s or f,s, the short-run competitive equilibrium coincides with the short-run social optimum.

Proof. It is necessary to show that if [a.sup.*],[y.sup.*] are a short-run social optimum, then there exists a policy pair [t.sup.*],[s.sup.*] or [f.sup.*], [s.sup.*] such that [a.sup.*], [y.sup.*] are a short-run competitive equilibrium.

For the case of taxation, suppose that [a.sup.*],[y.sup.*] solve Equations 1 and 2 and define [s.sup.*] = g([y.sup.*]) and [t.sup.*] = ag[prime]([y.sup.*]). Then, when f = 0, it follows immediately from Equations 1 and 2 that [a.sup.*],[y.sup.*] also satisfy Equations 3 and 4.

For the case of a fine-subsidy pair, suppose that [a.sup.*],[y.sup.*] satisfy Equations 1 and 2 and choose [f.sup.*] = g[prime]([y.sup.*]) and [s.sup.*] = [y.sup.*](g([y.sup.*])/[y.sup.*] - g[prime]([y.sup.*])). For such a [f.sup.*], it is clear from Equation 1 that [a.sup.*],[y.sup.*] will also satisfy Equation 3 when t = 0. To see that [a.sup.*],[y.sup.*] also satisfy Equation 4, rewrite Equation 2 as

g([y.sup.*]) + [s.sup.*] - [s.sup.*] + [f.sup.*][y.sup.*] - [f.sup.*][y.sup.*] = -dc/da ([a.sup.*], [y.sup.*])

or

[s.sup.*] + [f.sup.*][y.sup.*] = -dc/da ([a.sup.*], [y.sup.*]). (5)

It is obvious from Equation 5 that [a.sup.*],[y.sup.*] satisfy Equation 4. Q.E.D.

The result in Proposition 1 is fairly transparent: With two distortions, it is possible to achieve the social optimum with two policy instruments. However, the case of fines on accidents is of some independent interest. If the purpose of a fine on accidents is to achieve economic efficiency rather than to serve as a compensatory mechanism for injured parties, the optimal per unit fine is assessed according to the marginal damage, not the average damage, associated with product failure. Moreover, when fines are assessed on accidents, the socially optimal policy control may involve the taxation, not the subsidization, of product safety. If the marginal damage of each firm's output exceeds the average damage at the optimal point, then the first-best policy pair involves a fine on accidents equal to marginal damage and a tax on product safety equal to the difference between marginal and average damage.

An alternative to direct regulatory control is the implementation of a producer liability rule. In the case of tort liability, the SRCE is completely characterized as the solution to

[Mathematical Expression Omitted].

Using the definition of the inverse demand curve, a and y satisfy the first-order conditions

P(ny) = dc/dy (a, y) + ag[prime](y)

and

g(y) = -dc/da(a, y),

which coincide directly with Equations 1 and 2. Thus, a producer liability rule achieves the first-best social outcome in a short-run competitive equilibrium.

In the long run, entry (exit) may occur and the number of firms is endogenous.(5) Using the definition of the inverse demand curve, the long-run social optimum (LRSO) can be characterized as the solution to

[Mathematical Expression Omitted].

The first-order conditions are

P(ny) = dc/dy(a, y) + ag[prime](y), (6)

g(y) = dc/da(a, y), (7)

and

yP(ny) - c(a, y) + ag(y). (8)

With respect to the regulatory controls (t,f, and s), the long-run competitive equilibrium (LRCE) can be completely characterized by the conditions

P(ny) = dc/dy(a, y) + af + t, (9)

s + fy = -dc/da(a, y), (10)

and

yP(ny) = c(a, y) + yt + as + ayf. (11)

It is now possible to prove the following proposition.

PROPOSITION 2. There exists no t,s pair such that the long-run competitive equilibrium coincides with the long-run social optimum.

Proof. It is necessary to show that if t is the tax on output and s is the subsidy on product safety, then if [a.sup.*],[y.sup.*],[n.sup.*] satisfy Equations 6-8, they will not also satisfy Equations 9-11. First notice from Equations 6 and 7 that t,s must meet the conditions of Proposition 1 for an LRSO. To complete the proof, one must show that t = [t.sup.*] and s = [s.sup.*] will not satisfy Equation 11 for [a.sup.*],[y.sup.*], [n.sup.*].

Suppose that [a.sup.*], [y.sup.*], [n.sup.*] satisfy Equation 9-11 and let [s.sup.*] = g([y.sup.*]) and [t.sup.*] = ag [prime]([y.sup.*]). For such a [s.sup.*], [t.sup.*] it is clear from Equations 9 and 10 that [a.sup.*], [y.sup.*], [n.sup.*] will also satisfy Equations 6 and 7. However, for such a tax-subsidy pair, [a.sup.*], [y.sup.*] will not satisfy Equation 8. To see this, rewrite Equation 8 as

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [a.sup.*]g(.sup.*]) + [y.sup.*] [t.sup.*] - [y.sup.*][t.sup.*] + [a.sup.*] - [a.sup.*][s.sup.*]

or

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [y.sup.*][t.sup.*] + [a.sup.*][s.sup.*] - [a.sup.*]g[prime]([y.sup.*]). (12)

From Equation 12 it is apparent that [a.sup.*],[y.sup.*],[n.sup.*] will not satisfy Equation 11 when f = 0. Q.E.D.

The intuition behind this result is familiar: The number of distortions exceeds the number of policy instruments. In an LRCE, the level of output, the degree of product safety of an individual firm, and the number of firms in the industry each contribute to external damages. The standard prescription requires three separate Pigouvian taxes, where the missing instrument, the "entry tax" of Carlton and Loury (1980), is effectively a tax on the number of firms. For cases in which producer care affects external damages, however, Proposition 2 demonstrates that the failure of Pigouvian taxation is not a potential outcome, as in the case of exogenous product safety, but a fundamental policy result. A nontrivial damage function does not exist for which a tax-subsidy pair leads to coincident outcomes in long-run competitive and socially optimal equilibria. The intuition for this result is as follows. Consider, as in Carlton and Loury (1980, 1986), the case in which the level of producer care does not affect the level of social damage. Here, if the marginal damage of an additional unit of output equals the average damage (e.g., external damages depend only on total industry output), then the optimal tax on entry is zero and a single Pigouvian tax on output is an appropriate fiscal policy. Next, consider the case in which producer care affects social damage. To achieve a regulated social optimum here, marginal damage must equal average damage with respect to each partial effect: the level of output and the degree of product safety. Clearly, for any tax-subsidy pair, a damage function exists that satisfies the correct entry incentive with respect to output, and a damage function also exists that satisfies the correct entry incentive with respect to product safety. Proposition 2 reveals the fact that these two functions cannot coincide.

PROPOSITION 3. There exists a f,s pair such that the long-run competitive equilibrium coincides with the long-run social optimum.

Proof. It is necessary to show that if [a.sup.*],[y.sup.*],[n.sup.*] are a long-run social optimum, then there exists a policy pair [f.sup.*],[s.sup.*] such that [a.sup.*],[y.sup.*][n.sup.*] are a long-run competitive equilibrium.

Suppose that [a.sup.*],[y.sup.*],[n.sup.*] satisfy Equations 6-8 and choose [f.sup.*] = g[prime]([y.sup.*]) and [s.sup.*] = [y.sup.*](g([y.sup.*])/[y.sup.*] - g[prime]([y.sup.*])). For such a [f.sup.*],[s.sup.*] it is clear from Equations 6 and 7 that [a.sup.*],[y.sup.*],[n.sup.*] also satisfy Equations 9 and 10 when t = 0. To see that [a.sup.*],[y.sup.*],[n.sup.*] also satisfy Equation 11, rewrite Equation 8 as

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [a.sup.*]g([y.sup.*]) + [a.sup.*][y.sup.*][f.sup.*] - [a.sup.*][y.sup.*]f[*.sup.*]+ [a.sup.*][s.sup.*] - [a.sup.*][.sup.s*]

or

[y.sup.*]P([n.sup.*][y.sup.*]) = c([a.sup.*], [y.sup.*]) + [a.sup.*][s.sup.*] + [a.sup.*][y.sup.*][f.sup.*] (13)

Clearly, Equation 13 coincides with Equation 11 when t = 0. Q.E.D.

Corollary. If marginal social damage equals average social damage at [y.sup.*], then a fine of [f.sup.*] = g[prime]([y.sup.*]) on accidents will alone achieve both the short-run and the long-run social optimum.

Proof. Obvious.

In Proposition 3 and its corollary, a joint policy that involves fines on accidents and subsidies on safety provision is capable of achieving a first-best outcome. These results illuminate an important point that the "failure" of a Pigouvian tax, in many instances, may arise quite simply from the fact that the supposed Pigouvian tax is a tax not on the externality but on a proxy for external damages. In the case in which external damages occur through accidental "spills" into an environmental medium, the source of the externality is the accident itself and depends only indirectly on the inefficiency of the associated output and safety levels of the firm.

When firms fail to recognize the external damages associated with product failure, three sources of inefficiency arise through separate distortions in output level, safety provision, and the equilibrium number of firms. The use of accident fines allows the regulator to effectively control three distortions with only two instruments. The intuition behind this result is that a fine, which is assessed on a multiplicative relationship between output and safety provision, creates a nonlinearity between policy instruments. An increase in the fine affects both the level of output and the degree of producer care for the representative firm, whereas an increase in the subsidy changes only the marginal valuation of product safety. The number of firms in an LRCE is determined jointly by the subsidy payment and by the level of the fine on accidents. Thus, for any given revenue transfer between the firms and the regulator, it is possible for the regulator to induce different output-safety combinations through the choice of relative weights on the fine and subsidy instruments. Proposition 3 reveals the fact that relative weights on fine and subsidy instruments exist that satisfy the optimal output-safety combination and simultaneously meet the revenue transfer necessary to achieve the optimal number of firms.

For completeness, the long-run efficiency properties of tort liability are examined next. With respect to a producer liability rule, the LRCE can be completely characterized by

P(ny) = dc/dy(a, y) + ag[prime](y), (14)

g(y) = dc/da(a, y), (15)

and

yP(ny) = c(a, y) + ag(y). (16)

Equations 14-16 coincide with Equations 6 - 8. It follows directly that the LRSO and LRCE coincide under a system of producer liability rules; thus, a producer liability rule leads to first-best resource allocations for both short-run and long-run competitive equilibria.(6)

3. The Structural Implications of a Change in Producer Liability Exposure

This section considers the effect of a change in producer liability exposure on market structure. To simplify the analysis, the effect of increased liability exposure is considered in the absence of regulatory policy. Unlike previous papers that have focused on the structural implications of producer liability when firm solvency is important (e.g., Ringleb and Wiggins 1990; Boyd and Ingberman 1994; Watts 1998), attention is confined here to the case of a fully capitalized industry without divestiture incentives. The results described below demonstrate that de novo entry in the face of increased producer liability, as observed by Ringleb and Wiggins (1990) in the 1967-80 period surrounding rapid changes in U.S. liability law, is consistent with an alternate hypothesis of purely structural change.

Let [Theta] be a shift parameter in the liability function of the representative firm such that g = g(y;[Theta]). Without loss of generality, an increase in producer liability is represented by the condition [g.sub.[Theta]](y;[Theta]) [greater than] 0, whereas an increase (resp. decrease) in the marginal injury relation is represented by the condition [g.sub.y[Theta]](y;[Theta]) [greater than] 0 (resp. [less than] 0).(7) It is also helpful to define for future reference the ratio of change in the marginal and average injury relation, [[Epsilon].sub.[Theta]] = [g.sub.y[Theta]]y/[g.sub.[Theta]], as the elasticity of the shift in liability structure. Denoting the second-order partials associated with Equations 14 and 15 as [[Pi].sub.yy], [[Pi].sub.aa], and [[Pi].sub.ay] yields the following proposition.

PROPOSITION 4. In a long-run competitive equilibrium, a change in producer liability exposure affects the level of output and safety provision of a representative firm as

dy/d[Theta] = -[g.sub.[Theta]] (a[[Pi].sub.aa] (1 - [[Epsilon].sub.[Theta]]) + y[[Pi].sub.ay])/y([[Pi].sub.yy][[Pi].sub.aa] - [([[Pi].sub.ay]).sup.2]), (17)

da/d[Theta] = [g.sub.[Theta]](y[[Pi].sub.yy] + a[[Pi].sub.ay](1 - [[Epsilon].sub.[Theta]]))/y([[Pi].sub.yy][[Pi].sub.aa] - [([[Pi].sub.ay]).sup.2]). (18)

A change in liability exposure affects the number of firms as

dn/d[Theta] = [ag.sub.[Theta]]/P[prime][y.sup.2] + [ng.sub.[Theta]](a[[Pi].sub.aa](1 - [[Epsilon].sub.[Theta]]) + y[[Pi].sub.ay])/[y.sup.2]([[Pi].sub.yy][[Pi].sub.aa] - [([[Pi].sub.ay]).sup.2]). (19)

Proof Perturbing the conditions for an LRCE in Equations 14-16 and making use of the envelope theorem yields

[Mathematical Expression Omitted], (20)

where [[Pi].sub.ay] = [[Pi].sub.ya], = -([c.sub.ay] + [g.sub.y]) and where [[Pi].sub.yy] = -([c.sub.yy] + [ag.sub.yy]) [less than] 0, [[Pi].sub.aa] = - [c.sub.aa] [less than] 0, and [[Pi].sub.yy] [[Pi].sub.aa] - (([[Pi].sub.ay]).sup.2] [greater than] 0 by the Routh-Hurwicz stability conditions. Solving the system of Equation 20 and collecting terms completes the proof. Q.E.D.

In general, the effects of a change in liability exposure on market structure in Equations 17-19 are ambiguous and depend on the value of the shift elasticity parameter, [[Epsilon].sub.[Theta]], and on the cross-effect between output and safety provision, [[Pi].sub.ay]. A testable implication of the divestiture hypothesis therefore occurs when the change in liability structure increases the average injury relation to a greater extent than the marginal injury relation ([[Epsilon].sub.[Theta]] [less than or equal to] 1) and when higher investments in product safety improve the marginal profitability of output, [[Pi].sub.ay] [less than or equal to] 0. That is, for the special case in which [[Pi].sub.ay] [less than or equal to] 0 and [[Epsilon].sub.[Theta]] [less than or equal to] 1, greater producer exposure to tort liability (at least weakly) increases the output of the representative firm, enhances product safety, and precipitates the exit of firms through a purely structural effect in the market. Only under these somewhat restrictive circumstances might one conclude that de novo entry in a hazardous sector is motivated by entirely non-structural factors, such as the level of firm solvency.

In many instances, increased exposure to tort liability can stimulate de novo entry in a hazardous sector through market forces and regardless of the solvency of firms. To further clarify the structural implications of tort liability, it is helpful to consider a special case of the model that suppresses the cross-effect between output and safety provision. For example, consider, as in Shavell (1980) and Marino (1991), the cost function c(y,a) = yc(a), where [c.sub.a] [less than] 0 and [c.sub.aa] [greater than] 0. In this case, production costs increase linearly with output, which implies that [[Pi].sub.ay] = 0. Denoting the (inverse) demand elasticity as [Eta] = -P[prime]]Y/P and the elasticity of the slope of the damage function as [Zeta] = [g.sub.yy]/[g.sub.y] yields the following proposition.

PROPOSITION 5. If c(y,a)/y = [c.sub.y](y,a), increased producer exposure to tort liability (i) increases product safety, (ii) increases the output of a representative firm if and only if [[Epsilon].sub.[Theta]] [less than] 1, and (iii) increases the number of firms if and only if [[Epsilon].sub.[Theta]] - 1 [greater than] ([Zeta]/[Eta])(ag/Py).

Proof Making the appropriate substitutions in Equations 14-16 and simplifying yields [g.sub.y] = g/y = -[c.sub.ay]. It follows immediately that [[Pi].sub.ay] = 0. The second-order condition associated with Equation 14 implies [g.sub.yy] [greater than] 0; thus, Equations 17-19 reduce to

dy/d[Theta] = [g.sub.[Theta]](1 - [[Epsilon].sub.[Theta]])/[g.sub.yy]y, (21)

da/d[Theta] = -[g.sub.[Theta]]/[c.sub.aa], (22)

and

dn/d[Theta] = [g.sub.[Theta]][[ag.sub.yy] - nP[prime](1 - [[Epsilon].sub.[Theta]])]/P[prime][g.sub.yy][y.sup.2], (23)

respectively, where [g.sub.[Theta]] [greater than] 0 for an increase in liability exposure. Inspection of Equations 21 and 22 completes parts (i) and (ii). Noting that the denominator in Equation 23 is negative, entry occurs following increased liability exposure if and only if [Xi]a[g.sub.y] + [Eta]P [less than] [Eta]P[[Epsilon].sub.[Theta]]. Substituting [g.sub.y] = g/y and simplifying completes the proof. Q.E.D.

In response to an increase in liability exposure, firms that internalize tort liability as a component of production costs change their operating scale to equate marginal and average cost inclusive of the increased injury expense. When there are no cross-effects between output and safety provision, increased exposure to tort liability has no effect on the productive scale of a representative firm only if the change in the marginal injury relation exactly coincides with the change in the average injury relation ([[Epsilon].sub.[Theta] = 1]).(8) Conversely, the level of output associated with minimum average cost decreases for a competitive firm whenever the upward shift in the average cost curve exceeds the change in marginal cost, as in the case of increased producer liability and [[Epsilon].sub.[Theta]] [greater than] 1. For a sufficiently large value of the shift elasticity, the reduction in productive scale by incumbent firms makes entry attractive despite the increased liability exposure in the industry. Entry is more likely to occur in a hazardous sector following an increase in tort liability when inverse demand is price elastic, the marginal damage function is price inelastic, and the increase in marginal damage is large relative to the change in average damage. Furthermore, de novo entry is more likely when total revenue is large relative to the total injury associated with product failure, a condition under which firm solvency is not likely to be an issue.

Part (iii) of Proposition 5 provides an intuitive condition for increased liability exposure to induce entry through a purely structural effect. As the following example demonstrates, the magnitude of this effect can be substantial.

Example

Consider a market with linear demand, P(ny) = A - [Alpha]ny, and a quadratic damage function, g = b + [Beta][y.sup.2]. Increased exposure to tort liability is represented by an increase in either b or [Beta]. Suppose that each of n competitive firms has the cost function c(a,y) = cy/a. Solving Equations 14-16 for the equilibrium level of output and safety provision for each firm yields [y.sup.*] = [(b/[Beta]).sup.0.5] and [a.sup.*] = [(c/2).sup.0.5][(b[Beta]).sup.0.25]. It is immediately apparent that the equilibrium output level of each firm is increasing in b and decreasing in [Beta], whereas the probability of product failure is decreasing in both b and [Beta]. Thus, regardless of the nature of the increase in producer liability, the level of safety provision increases. Using the definition of inverse demand, the equilibrium number of firms is [n.sup.*] = ((A - [P.sup.*])/[Alpha])[([Beta]/b).sup.0.5], from which it follows directly that entry occurs whenever the ratio [Beta]/b increases in response to the change in liability structure.

The above example indicates that, in general, no clear correspondence can be drawn between de novo entry and the level of firm solvency. Specifically, divestiture incentives are not a necessary condition for entry to occur in a hazardous industry. The entry of new firms in response to increased liability exposure leads to qualitative predictions regarding divestiture only in the case in which the average injury relation increases to a greater extent than the increase in the marginal injury relation. Moreover, the implications of entry differ markedly, depending on whether entry is induced through divestiture or through purely structural forces. Quite unlike the implication of entry in the case considered by Ringleb and Wiggins (1990), entry in the present model, when it occurs, is consistent with the long-run socially optimum number of firms.

4. Concluding Remarks

This paper has compared taxation and producer liability rules in a model that allows competitive firms to invest in product safety measures that lower the probability of worker, consumer, and environmental damages. The paper has shown that Pigouvian taxation fails to achieve a socially optimal outcome in a long-run competitive equilibrium in which both output and safety provision are taxed (subsidized) according to their marginal contributions to social damage. The analysis demonstrated that a system of fines on accidents and subsidies (taxes) on product safety provision are capable of achieving first-best resource allocations. However, the optimal policy pair assesses a per unit fine on marginal, not average, damages and potentially involves the taxation, not the subsidization, of product safety investments. The analysis further revealed that tort liability achieves a first-best resource allocation in both long- and short-run equilibria, which highlights the appeal of legal controls in hazardous sectors of the economy.

This paper also addressed the effect of increased exposure to tort liability on market structure. The results demonstrated that entry and loss of incumbent market share can occur purely through market forces following an increase in producer liability. The implication of this finding is that a divestiture incentive is not a necessary condition for small-firm entry to occur in response to increased exposure to tort liability. Entry (exit) of competitive firms generally occurs following nonuniform changes in the marginal and average components of the liability function, even when solvency is not an issue. In particular, entry occurs when the increased exposure to tort liability sufficiently increases the marginal injury relation relative to the change in the average injury relation. The implications of this result contrast sharply with that of divestiture-induced entry, as the entry of firms following increased producer exposure to liability is associated with first-best levels of output and safety provision in the hazardous sector. A more thorough consideration of the structural effects of tort liability is necessary to understand the underlying motivations and the commensurate welfare implications of entry in hazardous sectors of the economy.

This research has benefitted from helpful comments by Joseph Farrell, Jonathan Hamilton, Michael Hanemann, Theodore Keeler, Jeff Perloff, Dave Sunding and two anonymous referees.

1 Note that producer liability and direct regulatory controls may be viewed as complements rather than substitutes under uncertainty. In stochastic environments, the distinction between ex ante and ex post means of control is particularly important (see Kolstad, Ulen, and Johnson 1990).

2 Marino (1988) demonstrates that the long-run optimality of producer liability fails to hold when the probability of harm depends nonlinearly on cumulative output in the industry. In this case, arguments in the expected damage functions of individual firms are interdependent, and an externality is created by the imposition of the liability rule.

3 For a comprehensive treatment of the entry incentives created by different liability rules when solvency is important, see Watts (1998).

4 A similar result is derived by Marino (1991).

5 For the long-run model of producer liability, the common law doctrine of "coming to the nuisance" is observed; that is, attention is confined to cases in which pollutees cannot become victims by choice. For a thorough analysis on the efficiency of bargaining outcomes when liability rules and property rights accrue to new entrants, see Hamilton, Sheshinski, and Slutsky (1989).

6 This result is also derived by Hamada (1976), Shavell (1980), and Landes and Posner (1985).

7 From an initial zero-liability position, [Theta] may represent a movement to a producer liability system, as occurred in the early 1970s in the United States. To avoid complications resulting from discontinuity in [Theta], one can think of the initial level of liability exposure, g(y), as being infinitesimally small.

8 Such a result is familiar to competitive models in which firms choose a single variable (e.g., output) subject to a linear penalty schedule (e.g., a unit tax).

References

Boyd, James, and Daniel E. Ingberman. 1994. Extending liability: Should the sins of the producer be visited upon others? Resources for the Future Discussion Paper 95-05, Washington, DC.

Carlton, Dennis W., and Glenn C. Loury. 1980. The limitations of Pigouvian taxes as a long-run remedy for externalities. Quarterly Journal of Economics 95:559-66.

Carlton, Dennis W., and Glenn C. Loury. 1986. The limitations of Pigouvian taxes as a long-run remedy for externalities: An extension of results. Quarterly Journal of Economics 101:631-4.

Hamada, Koichi. 1976. Liability rules and income distribution in products liability. American Economic Review 66:228-34.

Hamilton, Jonathan H., Eytan Sheshinski, and Steven M. Slutsky. 1989. Production externalities and long-run equilibria: Bargaining and Piguvian taxation. Economic Inquiry 27:453-71.

Kolstad, Charles D., Thomas S. Ulen, and Gary V. Johnson. 1990. Ex post liability for harm vs. ex ante safety regulation: Substitutes or complements? American Economic Review 80:888-901.

Landes, William M., and Richard A. Posner. 1985. A positive economic analysis of products liability. Journal of Legal Studies 14:535-68.

Marino, Anthony M. 1988. Products liability and scale effects in a long-run competitive equilibrium. International Review of Law and Economics 8:97-107.

Marino, Anthony M. 1991. Market share liability and economic efficiency. Southern Economic Journal 57:667-75.

Ringleb, Al H., and Steven N. Wiggins. 1990. Liability and large-scale, long-term hazards. Journal of Political Economy 98:574-95.

Shavell, Steven. 1980. Strict liability versus negligence. Journal of Legal Studies 9:1-25.

Watts, Alison. 1998. Insolvency and division of cleanup costs. International Review of Law and Economics. In press.

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Author: | Hamilton, Stephen F. |
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Publication: | Southern Economic Journal |

Date: | Jul 1, 1998 |

Words: | 5757 |

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