Incomplete ownership, rent dissipation, and the return to related investments.

Deacon, Robert T.

I. INTRODUCTION

When ownership rights to a resource are costless to enforce, trades occur and an equilibrium price emerges. In such cases it is a price and a transfer of wealth from one agent to another that rations use of the resource. Absent ownership no price is charged and use is rationed by a transaction cost. Acquiring a barrel of oil or an acre foot of water from a shared reservoir is a transaction--a resource in the common domain is transformed into property by the act of lifting it to the surface where ownership can be enforced. The relevant transaction cost in this case is the cost of pumping. In a fishery the relevant transaction cost is the cost of locating and capturing a portion of the unowned stock.

The marginal transaction cost limits each individual's use and accomplishes the necessary task of equating demand to the resource's supply. Users of the resource are in equilibrium when this marginal transaction cost equals the resource's marginal value. The transaction cost also eliminates all or part of the rent the resource would earn if owned, however, and the size of this loss depends on the relationship between average transaction cost and the resource's marginal value.

This suggests that the amount of rent an unowned resource will yield in equilibrium, after netting out relevant costs, depends on the shape of the cost function for acquiring it. If marginal cost is much greater than average cost, as in cases where marginal cost is steeply rising, the equilibrium rent will be relatively large.(1) While the shape of the cost function depends on the nature of the resource, it is not immutable. For example, it can be altered by imposing constraints on the inputs or the technology used. It follows that users of a resource would benefit collectively if the technology or inputs used were constrained in a way that causes the marginal cost of obtaining it to rise steeply. The optimal choice of such constraints is a well-formulated second-best policy problem.

This specification of the problem is incomplete, however, because the intensity of competition for the shared resource can also be changed by actions that affect it only indirectly. An innovation or investment that lowers the individual's cost of acquiring it, for example by reducing the price of an input used, has this effect. So does an action that raises the resource's marginal value product, possibly by affecting the price or availability of a substitute. Such actions increase competition for the shared resource and raise the equilibrium cost of actions taken to acquire it. In this sense the rent dissipation that is normally thought to characterize competition for shared resources can spread to related actions or investments.

Both of these considerations are pursued in what follows. Second-best policy toward a shared resource is examined by considering regulations that can control some but not all of the inputs needed to acquire it. It is shown that the optimal input constraint and the fraction of potential rent such a policy can capture depend on two determinants--the elasticity of substitution between, and relative price of, regulated and unregulated inputs.

It is also shown that competition for the shared resource can eliminate all or part of the return to related actions, i.e., actions that result in a shift in the relevant private cost or benefit function. The size of such losses depends in a simple way on the elasticities of relevant cost and benefit functions. These additional losses can occur even in cases where direct competition causes complete rent dissipation. As a result, the loss that results from free access can exceed the rent the resource would earn in a complete markets equilibrium. This phenomenon, termed excess dissipation, is illustrated with a number of examples. Investments that reduce the percolation of groundwater back into a common aquifer or expenditures to prevent fish from escaping from a net and returning to the natural population during the process of capture are shown to have this effect. Such conservation actions lower the cost of obtaining the resource and they can lower welfare because the individual's incentive to conserve is excessive in such circumstances. Examples of other actions, such as marginal cost pricing for a substitute resource and innovations that affect the productivity of the unowned resource, are also presented.

This work is related to several streams of literature. Some of the propositions developed on rent dissipation are reminiscent of findings in the rent-seeking literature developed by Gordon Tullock [1980a], Richard Posner [1980], and others. The excess dissipation result in particular is similar to a result presented by Tullock [1980b]. The questions raised and results obtained share similarities with analysis by Braverman and Stiglitz [1982] on share-cropping. In cases where the landlord cannot directly control the amount of effort the worker supplies, they show that the landlord can benefit by altering prices or taking other actions that appropriately shift the worker's private marginal cost and benefit schedules for effort. In the present context, users of the shared resource can benefit by mutually refraining from activities that lead to lower private marginal costs or higher marginal private benefits. Wijkander [1985] and Sandmo [1976] have studied the problem of regulating an externality indirectly, by altering prices of complements and substitutes to the item of direct interest. Finally, Deacon and Sonstelie [1991] have examined the losses that arise from price controls and rationing by waiting. Their analysis stresses the notion that privately rational adjustments to rationing by waiting can be socially self-defeating if they intensify competition for the price-controlled item.

While this paper shares common features with these works, it is directed toward different objectives and reaches several new conclusions. In particular, the finding that the loss from free access competition can exceed the rent a resource would earn with complete markets, and the observation that returns to privately rational actions may be dissipated by competition for a related resource that is unowned, appear to be new. Another novel feature is the precise characterization of factors that determine the efficacy of second-best input controls when the technology has constant elasticity of substitution.

The analysis begins by examining two extreme property rights regimes, complete ownership with costless transactions and free access. While these extremes may not accurately describe any actual resource, the examination of polar cases is often a productive way to gain insight on the more typical regimes that lie in between.(2) A class of intermediate cases is then developed by specifying limits on the use of some but not all of the inputs needed to acquire the resource in question. These limits amount to partial ownership, e.g., rights to operate a fishing vessel or a water well. The model is formulated in such a way that 'excessive' use of the shared resource degrades its quality or accessibility and increases the cost of acquiring it. This continues until the marginal cost and marginal benefit of obtaining the resource are equated at a level of use that does not cause further degradation. The following section lays out the notation and general structure of the model.

II. A MODEL ECONOMY WITH A SHARED INPUT

Consider an economy with N identical agents, each of whom is a self-sufficient producer-consumer. Each agent owns a firm that produces a consumption good and each consumes its own firm's output. Trade among individuals is possible, but none occurs in equilibrium because individuals are identical. A shared (unowned) input is necessary for production of the consumption good. To fix ideas, imagine an economy in which the shared resource is a stock of fish and where each individual fishes from the common stock and consumes his or her own catch. The number of firms, N, is large enough that all take prices as given and ignore the effect of their own actions on the quality or accessibility of the shared resource.(3)

The firm's production function is

(1)q = q(z)

where q is output of the consumption good per firm and z is the firm's use of the shared resource.(4) As a consumer, the individual's preferences are represented by

(2)w = u[q(z)] + x

where x is leisure, the numeraire. Both U(-) and q([center dot]) are assumed concave so W([center dot]) is concave as well. Where no confusion can arise utility is abbreviated to U(z) + x, the first term of which is the total value product of the shared resource, z.(5)

In order to obtain z the firm must use two ordinary inputs, energy (e) and capital (k). The reason for introducing two inputs becomes clear later. The amount of z obtained for given e and k depends on the quality or accessibility of the resource, D. By convention an increase in D makes the resource more difficult to acquire. For fish, D would be inversely related to the abundance of the stock, so the effort required for search and capture is greater for greater D. The production function for obtaining the shared resource is

(3)z = z(e/D, k/D)

where z([center dot]) is concave. Doubling D doubles the amounts of e and k needed to obtain a given level of z. D is common to all firms and each takes its level as given.

The individual is endowed with [Mathematical Expression Omitted] units of leisure per period. Leisure can either be consumed or transformed into e and k. The unit costs of e and k, [p.sub.e] and [p.sub.k], are assumed constant. The individual's consumption of leisure is thus constrained by

[Mathematical Expression Omitted].

The individual, as owner of a representative firm, uses amounts of e and k that minimize [p.sub.e]e + [p.sub.k]k, the cost of acquiring z, subject to (3). The resulting cost function,

(5) C = h([p.sub.e],[p.sub.k], z) D,

is homogeneous of degree one in D and exhibits non-decreasing marginal costs for z. The homogeneity property occurs because doubling D doubles the amounts of e and k needed to obtain a given z. Note that [C.sub.ZD] [is greater than] 0, so increasing D raises the marginal cost of acquiring the shared input. Where no confusion can result the cost function is abbreviated to h(z)D.(6)

While each individual regards D as given, its level is determined by the economy-wide use of z. If use is high, quality or accessibility declines and D increases. The resource is capable of natural regeneration, however, so D will fall if use is low enough. This process is represented by

(6) D = g(Nz - NR)

= f(z - R),

where R is natural regeneration per firm, assumed fixed. The fixed number of firms, N, has been suppressed. It is further assumed that f[prime] [is greater than] 0, f(0) = 0, and D [is greater than or equal to] [Delta], where [Delta] is a positive constant. The rationale for bounding D away from zero is explained later. In a steady state z = R, and any D [is greater than or equal to] [Delta] can be sustained as a steady state so long as the individual chooses z=R at the value of D specified.(7)

Combining (2), (4), and (5), utility can be written

[Mathematical Expression Omitted],

which is the utility from consuming q(z) plus that portion of the leisure endowment not used to acquire z. The next section examines steady-state utility under alternative ownership regimes. Since z = R in a steady state, (7) implies that the only variable that can vary across alternatives is D, the quality or accessibility of z.

A few stylized examples will make the model more concrete. In each case increased use of the resource impairs its availability and increases the cost of acquiring it. This increased cost, in turn, rations use. For the fishery, z is catch, which serves as an input to the production of food, q. Fishing inputs are vessel capital (k) and labor (e). For mnemonic purposes, let D be the 'dispersion' of the stock and let R be 'recruitment'. Fishing disperses the stock, raising the effort required to make a given catch, and natural growth replenishes it.

For application to groundwater, interpret z as water used to irrigate a farm output, q. Each grower uses a pump (k) and fuel to run the pump (e) to acquire z. The accessibility of groundwater is determined by the 'depth' of the aquifer, D, and the greater the depth the greater is the cost of pumping a unit of water to the surface. Depth is increased by pumping and reduced by 'recharge', R, the natural inflow of water.

A third example involves firms that withdraw water from a common source, perhaps a lake, and return equal volumes of effluent to it. Here, z is clean water needed to produce an output, q. Since water obtained from the source is polluted it must be treated before use, and k and e represent inputs used in treatment. The quality of the resource is measured by D, an index of how 'dirty' the source of water is, and the cost of treatment is positively related to D. Using more of the lake's water results in increased emissions and increases in D. The lake is capable of natural 'regeneration', however, which cleanses it of R units of effluent per period.

While obviously simplified, the model captures a phenomenon that is important for many actual resources. Use of a resource beyond some level impairs its quality, making it more costly to acquire for those who use it, and this cost, in turn, rations use.

III. STEADY-STATE WELFARE AND EXCESS DISSIPATION

The objectives of this section are twofold, to show how the degree of rent dissipation under free access is related to the elasticity of the cost function for acquiring the resource and to demonstrate that the losses that result from free access can spread to related investments. To examine the rent dissipation issue it is first necessary to determine the rent the resource would earn with ownership. This rent is identified by exploiting the fact that complete ownership results in an allocation that maximizes the present value of utility. Attention is focused exclusively on steady states to provide a clear exposition of the rent dissipation issue.

The first step is to characterize the efficient steady-state allocation. Starting in a steady state, consider a temporary departure that marginally increases z in one period only. The consumer values the temporary increase in z at [U.sub.z](z) - [h.sub.z](z)D. Temporarily increasing z also increases D by f[prime](0), however, and this effect persists in all future periods. The consequent increase in present value extraction cost is h(R)f[prime](0)/r, where r is the rate of interest. If the economy is in an efficient steady state, the marginal benefit and marginal cost of temporarily increasing z are equal, which implies(8)

(8) [U.sub.z](R) - [h.sub.z](R)D = h(R)f[prime](0)/r.

Equation (8) implicitly gives the optimal steady-state value of D:

(9) D* = [[U.sub.2](R) - h(R)f[prime](0)/r]/[h.sub.2](R).

If the resource were owned and its flow sold competitively, the owner would charge a price equal to the opportunity cost of withdrawing a marginal unit. This opportunity cost is given by the right-hand side of (8). With complete markets, then, the rent is [Rho]* = h(R)f[prime](0)/r per unit z. Since the resource yields R units of z per year, its total steady-state rent would be

(10) [Rho]*R = Rh(R)f[prime](0)/r

per year.

Absent ownership each individual considers the quality or accessibility of the resource as fixed and chooses a level of z that maximizes utility. The steady-state equilibrium with free access is thus found by maximizing (7) with respect to z taking D as given and setting z = R in the result. The first-order condition and steady-state D are

(11)[U.sub.z](R) - [h.sub.z](R)D = 0

(12)[D.sup.c]= [U.sub.z](R)/[h.sub.z](R).

Notice incidentally that (9) and (12) coincide if the interest rate is infinite--in this case the free access solution is efficient.(9)

Steady-state welfare in the two ownership regimes can be obtained from (7) by setting z=R and alternately substituting (9) and (12) for D. Letting W* and [W.sup.c] denote steady-state welfare in the two regimes, the dissipation due to free access is

(13) W* - [W.sup.c] = ([D.sup.c] - D*)h(R)

= [Rh(R)f[prime](0)/r][h(R)/R[h.sub.z](R)].

The first bracketed term is the steady-state rent the resource would earn with complete markets. Let [Sigma][is equivalent to]R[h.sub.z](R)/h(R) denote the output elasticity of the cost function for acquiring z evaluated at the steady state, and note that convexity implies [Sigma][is greater than or equal to]1. Then (13) may be summed up as follows.

PROPOSITION 1: The steady-state welfare loss that results from free access use of the shared input is W* - [W.sup.c]= [Rho]*R/[Sigma], where [Rho]*R is the steady-state rent the shared resource would earn with complete markets and [Sigma] is the output elasticity of the cost function for acquiring it.

This establishes the promised link between rent dissipation and the elasticity of the cost function.(10)

Figure 1 illustrates the model and provides some intuition for the preceding proposition. The downward sloping curve labeled private marginal value is [U.sub.z], the marginal value product of z. The curve labeled social marginal value is [U.sub.z] - hf[prime]/r, the marginal value product of z less the associated opportunity cost of degrading the resource. The upward sloping curves give marginal and average costs of z for different values of D. If the resource were owned, the efficient steady-state would be established with resource quality D*, which equates the marginal cost of z to its social marginal value at z=R. The resource earns a steady-state rent equal to area abcd.

If ownership were abolished and replaced by free access each individual would rationally take D as given and choose z to equate marginal cost to private marginal value. Starting where D=D* this causes withdrawal to exceed natural replenishment, so D increases and the marginal cost schedule shifts up. The free access equilibrium is established at point b, where D has risen to [D.sup.c] and the private marginal benefit and marginal cost of z are equal at z = R.

When ownership is abolished, the amount by which marginal private cost rises at z = R just equals the resource's rent. At the margin, then, rent is exactly dissipated. Exact dissipation need not apply to inframarginal units, however. Average dissipation equals the increase in average cost at z=R, so total dissipation is the shaded area fghi. This equals 1/[Sigma] times the complete markets rent.(11) If the unit cost of z were constant ([Sigma] = 1) marginal and average cost would increase by the same amount, eliminating the entire rent. Cost functions with high values of [Sigma] exhibit steeply rising marginal costs and small degrees of dissipation, and functions with low values, the converse.

The discussion turns next to the question of how free access can affect the return to related investments. Consider two kinds of costly actions. The first kind reduces the marginal cost of acquiring the resource. One specific example is effort spent inventing a more efficient groundwater pump. Development of a new fishing method or a sensing device that reduces the search cost needed to locate a stock of fish is another. The per period amount spent on such actions is denoted [Phi] and the effect on cost is captured by writing the cost function as Dh(z, [Phi]), where [h.sub.[Phi]][h.sub.z[Phi]] [is less than] 0. The second kind of action raises the marginal value product of z. An investment that lowers the cost of transporting fish to market is one example because it raises the value of fish at the dock. Another is a farmland improvement that allows the production of crops irrigated by groundwater. This second kind of action costs 0 per period and its effect on demand for z is captured by writing U = U(z,[Theta]), where [U.sub.[Theta]], [U.sub.z[Theta]] [is greater than or equal to] 0.

With these modifications utility becomes

[Mathematical Expression Omitted].

Free access use of z implies [U.sub.z](z,[Theta]) = D[H.sub.z](z, [Phi]), so steady-state D is

(15) D([Phi], [Theta]) = [U.sub.z](R,[Theta])[h.sub.z](R, [Phi]).

It can be verified that [Delta]D/[Delta][Theta] [is greater than or equal to] 0 and [Delta]D/[Delta][Phi] [is greater than or equal to] 0, so steady-state D increases if either action is taken. In words, actions that either reduce the marginal cost or raise the marginal value of the unowned resource intensify competition for it and further degrade its quality.

To determine the size of such effects consider a cost reducing action first and let [W.sup.[Phi]] and [D.sup.[Phi]] be the corresponding steady-state utility and resource quality with free access. Using superscript c to denote free access values without the action, the steady-state net return to [Phi] is

(16) [W.sup.[Phi]] - [W.sup.c] = [D.sup.c]h(R) - [D.sup.[Phi]]h(r,[Phi]) - [Phi].

Using (12) and (13) this can be written

(17) [W.sup.[Phi]] - [W.sup.c] = [U.sub.z](R){[h(R)/[h.sub.z](R)] -[h(R, [Phi])/[h.sub.z](R, [Phi])]} - [Phi]

= R[U.sub.z](R)[(1/[Sigma]) - (1/[[Sigma].sup.[Phi]])] - [Phi],

where [[Sigma].sup.[Phi]] is the elasticity of the cost function given the action [Phi], evaluated at z = R. The return to action [Phi] under free access can now be summarized as follows.

PROPOSITION 2: Let [Phi] be expenditure on an action that lowers the marginal cost of acquiring the shared resource. The steady-state gross return to this action is zero unless the action changes the elasticity of the total cost function, evaluated at z = R. If the elasticity is changed by the action, then the gross return to [Phi] is positive if the elasticity rises and negative if it falls.

When action [Phi] is available, the loss due to free access can exceed the rent the resource would earn with complete markets, a phenomenon termed excess dissipation. If the resource were owned it would earn [Rho]*R per period. Suppose, instead, that it is allocated by free access and that [Sigma] = 1, but action [Phi] is not available. By Proposition 1 the rent [Rho]*R is exactly dissipated. If action [Phi] then becomes available and the elasticity of total cost is unchanged, steady-state utility falls by [Phi] as Proposition 2 states. In total, the loss due to free access exceeds, by [Phi], the rent the resource would earn if owned.(12) The action temporarily earns a positive return during the transition between old and new steady states, but this return necessarily vanishes in the long run.

If [Phi] changes cost elasticity, its steady-state gross return will either be positive or negative. There is no compelling reason to expect [Sigma] to change one way or the other, however, and under fairly weak conditions it necessarily remains constant. For example, if the production function for acquiring the resource is homothetic then the cost function takes the form C = Df([p.sub.e], [p.sub.k])m(z), where m([center dot]) is a monotone increasing function. In this case actions that reduce input prices [p.sub.e] and [p.sub.k] have no effect on [Sigma]. Investments that yield Hicks-neutral technological improvements also leave [Sigma] unchanged in this case.(13)

While the return to action [Phi] either partially or completely vanishes in the long run, it is important to note that the individual was not myopic in pursuing it and would not choose to undo the action once the new equilibrium is reached. To the individual, who properly takes D as given, some positive expenditure on action [Phi] is rational so long as [Delta]W/[Delta][Phi] [is greater than] 0 when evaluated at the free access steady state. This condition is met if

(18) -[D.sup.c][h.sub.[Phi]](R,0) [is greater than] 1,

and nothing in the model rules this out.(14) When undertaken such actions cause D to increase and steady-state utility to fall. The individual has no incentive to undo the action when the new equilibrium is reached, however. Rather, condition (18) is more likely to be met after D has been raised by the action, that is, [Phi] looks more attractive in hindsight than it did when first considered.

Figure 2 illustrates the intuition of Proposition 2 for the case where the shared resource is acquired at constant cost. The free access equilibrium without [Phi] occurs at point a, with D = [D.sup.c]. The marginal cost of z is MC([D.sup.c], 0), which just equals its private marginal value at z = R. A device is then invented that costs [Phi] per period and lowers the marginal and average cost of acquiring z. Given [D.sup.c], marginal cost falls to MC([D.sup.c], [Phi]) and the individual who adopts the device stands to gain a per period amount equal to the shaded area less [Phi]. Assume this is positive so the device is adopted. The lower marginal cost is not sustainable because use of the resource exceeds natural replenishment. This causes D to increase and marginal cost to shift up. In the new equilibrium D = [D.sup.[Phi]] so marginal cost, now MC([D.sup.[Phi]], [Phi]), again limits use of z to R. The long-run gross return to adopting the device is zero, so steady-state utility is reduced by [Phi].

Turning next to an action that raises the private marginal value of z, let [W.sup.[Theta]] and [D.sup.[Theta]] denote steady-state utility and resource quality under free access if action [Theta] is undertaken. The action's steady-state net return is

(19) [W.sup.[Theta]] - [W.sup.c] = {[U(R, [Theta] - [D.sup.[Theta]]h(R)] - [U(R) - [D.sup.c]h(R)]} - [Theta] ={U(R, [Theta]) - [R[U.sub.z](R, [Theta])/[Sigma]]} -{U(R) - [R[U.sub.z](R)/[Sigma])]} - [Theta]

where (12) and (15) have been used. A useful way to express this is

(20) [W.sup.[Theta]] - [W.sup.c] = R{[U(R, [Theta]) - U(R)/[Sigma] -[[U.sub.z](R, [Theta]) - [U.sub.z](R)]/[Sigma]} - [Theta].

The expression in brackets {[center dot]} is the gross return per unit z to action [Theta], and it can be explained as follows. The action raises the value of output, and the size of the increase per unit z is the increase in average value product. This increase is the first fraction in brackets. The action also raises the marginal value product of z, however, and this causes competition for z to intensify and D to rise. The cost of this competition is greater the greater is the increase in the marginal value product of z, and the smaller is the elasticity of the cost function for acquiring z. The second fraction in brackets gives the steady-state cost of this increased competition. Recalling that [Sigma] [is greater than or equal to] 1, the gross return is necessarily positive if average value product rises more than marginal value product.

The net return can also be written

(21) [W.sup.[Theta]] - [W.sup.c] = U(R, [Theta]) {1 - [[Eta].sup.[Theta]]/[Sigma]]} - U(R){1 - [[Eta]/[Sigma]]} - [Theta], where [Eta] and [[Eta].sup.[Theta]] are elasticities of U(z) and U(z, [Theta]) with respect to z, evaluated at z = R. Concavity and convexity assumptions guarantee 0 [is less than or equal to] [Eta]/[Sigma] [is less than or equal to] 1 and 0 [is less than or equal to] [[Eta].sup.[Theta]]/[Sigma] [is less than or equal to] 1. In some instances [Theta] may shift U([center dot]) without changing its elasticity at z = R. (This is necessarily true if the effect of [Theta] is multiplicative, i.e., if U(z, [Theta]) = g([Theta])U(z).) In this case (21) takes a simple form and can be interpreted as follows.

PROPOSITION 3: Let [Theta] be expenditure on an action that raises the total and marginal value product of a free access resource, z. If the action does not change [Eta], the elasticity of U([center dot]) evaluated at z = R, then the net return to [Theta] is [W.sup.[Theta]] - [W.sup.c] = [U(R, [Theta]) - U(R)][1 - ([Eta]/[Sigma])] - [Theta]. This return approaches the limiting value of -[Theta], indicating complete dissipation, as [Eta]/[Sigma] approaches 1. This is the case of constant cost and constant marginal value product of z. Alternatively, the return approaches the limiting value of U(R, [Theta]) - U(R) - [Theta], or no dissipation, as [Eta]/[Sigma] approaches 0. The zero dissipation limit is approached if the marginal cost schedule is nearly vertical or the total value product schedule is relatively flat, when evaluated at z = R.

No figure is presented to illustrate this result, although the effect is easily visualized. Starting from the free access equilibrium where private marginal value equals marginal cost and D = [D.sup.c], action [Theta] raises the private marginal value schedule. If D remained fixed the value of the action would equal the increase in area under the private marginal value schedule for z [is less than or equal to] R. The increase in marginal value causes z to rise above R, however, so D increases and the marginal cost schedule shifts up. This process stops only when D has risen to [D.sup.[Theta]] where marginal cost equals the new private marginal value schedule at z = R. This cost increase eliminates at least part of the value [Theta] would otherwise earn.

IV. EXAMPLES AND EVIDENCE

Four practical examples involving groundwater are used to illustrate the finding that privately rational actions can be socially self-defeating if they intensify competition for a shared resource. The first two examples are cost reducing: improvements in groundwater pumping technology and investments in water conserving irrigation systems. The next two examples involve increased marginal values: changes in irrigation technology and increased prices for surface water.

Before proceeding, a brief comment on groundwater institutions and the applicability of the model is in order. In this paper free access is defined to exist when any user can pump as much water as he or she wishes, without quantitative restrictions or controls on the method of withdrawal. Among the legal doctrines applied to groundwater, this most closely resembles 'absolute ownership'. As described by Tarlock [1989, 4.04] and Aiken [1980, 923-24], absolute ownership applies the rule of capture to groundwater and anyone owning a parcel of land overlying an aquifer is eligible to participate. Absolute ownership was inherited from the English common law and is still in general use in the U.K. Most arid states in the U.S. have departed from it to varying degrees, although it is still practiced in Texas. Outside of the west, absolute ownership is more common; Connecticut, Louisiana, Maine, Rhode Island, and Indiana still practice it.

The doctrine of 'beneficial use' is a modified rule of capture. Tarlock [1989, 4.05] and Aiken [1980, 924-25] explain that it limits the use of water to applications on land overlying the aquifer and prohibits patently wasteful or malicious uses. The uses described in the stylized model of this paper would not be judged per se 'unreasonable' by these criteria, so the model is generally applicable in areas where this doctrine is applied. This includes the states of Alabama, Florida, Kentucky, Maryland, New York, North Carolina, Tennessee, Illinois, and Oklahoma. In addition, Nebraska and Arizona apply reasonable use outside of specially designated groundwater management areas.(15)

'Prior appropriation', the other major doctrine, is the most common rule applied in western states. Under prior appropriation a right to withdraw water from an aquifer is established when a permit is granted by the relevant government agency and water is actually withdrawn and used, as Aiken [1980, 926-27] explains. Whether states practicing this doctrine are subject to the rent dissipation characterized in the model depends on how the appropriation rule is applied. If the courts interpret such rights quantitatively, and either limit use efficiently or assess damages accurately, the result will approximate the complete markets outcome.(16) While the degree to which this occurs is not known exactly, correct assessment of damages seems unlikely and quantitative limits on use are not the general rule in practice. For correct damage assessment, a pumper withdrawing an acre foot of water must be liable for damages equal to the present value of all future pumping cost increases imposed on other users--and the charge must be assessed on each acre foot withdrawn. At a minimum this would require knowledge of how much is withdrawn by individual users. By contrast, Bowman's [1990] survey of groundwater policy in the U.S. identified only twenty states in which governments even collect information on water use, either by metering or self-reporting, and fewer yet that place any quantitative controls on amounts withdrawn. Absent quantitative limits on amounts used or accurate damage assessments, the value of groundwater in situ will not be a part of the opportunity cost faced by the individual user. If the user faces no charge, explicit or implicit, the free access model portrayed earlier applies.

With these qualifications, then, groundwater appears an appropriate vehicle for illustrating the model's implications.

Pumping Technology, Pumping Costs and Groundwater Levels

The model predicts that improvements in pumping technology or reductions in prices of pumping inputs will increase the depth to water, dissipating at least part of the return that would otherwise result. There is much evidence in agreement with this prediction.(17) The deep-well turbine pump was introduced for use in irrigation in the 1920s. Andrews and Fairfax [1984, 164, 165, 170], Dunbar [1977, 663; 1983, 153], and California Department of Water Resources [1982, xix, 9, 28] all implicate this innovation, together with other technical improvements and declining energy prices, for the rapid growth in groundwater use and consequent decline in water levels that occurred after 1930 in Texas, California, and other parts of the western U.S. Prior to 1930 the use of groundwater for irrigation was negligible. By 1945 groundwater accounted for 21 percent of all water used for irrigation and by 1975 the share had risen to 39 percent, as Frederick [1982, 71-73] notes. This growth was accompanied by a general decline in water levels. Cooper [1968, 116] reports reductions as great as ten to thirty-five feet per year in some parts of California. Mapp and Eidman [1976, 9] report that water levels fell 2.06-3.6 feet per year in the central basin of the Ogallala formation during the 1960s and 1970s.

Martin and Archer [1971] compiled data on pumping costs and groundwater levels in Arizona between 1891 and 1967 and Figure 3 displays some of this information. The scale on the vertical axis is for the pumping cost per unit lift, expressed in 1967 cents per acre foot of water, per foot of lift.(18) Prior to 1920 the power sources used were varied and often primitive, involving mules in some cases, and pumping cost data for these years are unreliable. Prices of energy and pumping units fell dramatically after about 1920 while the durability and reliability of pumps increased, as Martin and Archer [1971, 27] explain. This is reflected in the steady reduction in pumping cost per unit lift, from $.20 per foot in 1925 to less than $.04 in 1967.

Mean depth levels rose rapidly, from 55 feet in 1925 to about 350 feet by 1967. Since the increase in depth approximately canceled the reduction in cost per unit lift, the pumping cost per unit water (the product of depth and pumping cost per unit lift) remained roughly constant. Aside from a jump in the early 1940s, it only varied from $11-$14 per acre foot.(19) In summary, the decline in water levels and pumping cost per unit lift, at the same time and approximately the same rate, clearly agrees with the model's prediction.(20)

Conserving Water Lost in Conveyance and Application

Only a fraction of the water pumped from an aquifer for irrigation actually is consumed by crops. For the nation as a whole the U.S. Soil Conservation Service [1976, 13, 14] estimates that about 59 percent of the water pumped from aquifers and diverted from streams is lost to seepage back into aquifers and rivers, evaporation, and consumption by unwanted flora and fauna. These losses can be cut substantially by investing in water-conserving systems for conveying water on the farm and applying it to crops. The following discussion evaluates such investments.(21)

One leading conservation strategy is replacement of unlined distribution ditches with lined channels or pipes. A second is installation of tail-water recovery systems. Some of the water applied using furrow irrigation collects in pools at the ends of furrows and is lost to evaporation or percolation into the ground. Installation of pumps and pumping equipment allows the farmer to recover and reapply this water. Land leveling is a third method, since it reduces runoff and prevents irrigation water from pooling on the land where it can evaporate or seep back into the ground. Where the irrigation source is groundwater, most of the water saved would otherwise percolate back to the aquifer.

The model must be modified slightly to evaluate such actions. Let [Mu] denote the fraction of pumped groundwater that the crop consumes. Since the level of farm output depends on water consumption, utility is U([Mu]z) + x. The pumping cost function is unchanged. Let [Pi] represent the per period cost of the system that distributes water and applies it to crops. With these modifications, welfare is

[Mathematical Expression Omitted].

Next, assume the water lost in distribution, (1-[Mu])z, percolates into the soil and eventually re-enters the aquifer.(22) Since this return flow is an additional source of recharge, the equation governing D becomes

[Mathematical Expression Omitted].

Thus, when return flow is allowed, changes in depth are determined by water consumption rather than the rate of withdrawal and the steady-state condition becomes [Mu]z = R.

Suppose there are several ways to distribute irrigation water from the well to the field, indexed by i. An alternative with a high [Mu] is a water-conserving strategy and is relatively costly, so [[Pi].sub.i] and [[Mu].sub.i] are positively related. Given any distribution system, the farmer chooses an extraction rate that maximizes utility given D. This implies U[prime]([[Mu].sub.i]z)[[Mu].sub.i] = D[h.sub.z](z), or

(24) U[prime]([[Mu].sub.i]z) = D[h.sub.z](z)/[[Mu].sub.i]

under alternative i, where U[prime] is the first derivative of U([[Mu].sub.i]z). This condition requires equality between the marginal benefit and marginal cost of water consumed by the crop.

Evaluating (24) at the steady-state where [[Mu].sub.i]z = R gives the steady-state depth in implicit form. Substituting this into (22) yields the following expression for steady-state utility.

(25) [Mathematical Expression Omitted], or [Mathematical Expression Omitted]

where [[Sigma].sup.[Mu]] is the elasticity of the cost function evaluated at the steady state. If the cost elasticity is independent of the pumping rate then it is also independent of [Mu]. This, in turn, implies that equilibrium welfare is independent of [Mu], the fraction of pumped water that is consumed by the crop. In this case the gross return to an investment that improves the system's effectiveness in conserving water is zero in the long run, so the net return is -[[Pi].sub.i]. If [Sigma] varies with the pumping rate, the gross social return to such investments may be positive or negative.

The intuition of this result can be grasped by imagining an aquifer with a distribution system so leaky that two units of water are pumped for each unit consumed by the crop. In long-run equilibrium the marginal cost of an acre foot of water consumed by the crop equals the marginal cost of pumping two acre feet out of the ground. If this system were replaced by one that eliminated distribution losses, the cost of water applied to the crop would fall by half even though the cost of pumping is not changed. Given the initial depth farmers would choose to increase water consumption, causing D to increase until the marginal cost of water consumed rises to its former level.

During the 1970s the U.S. government undertook a major effort to identify and evaluate strategies for reducing losses of irrigation water in distribution and conveyance. A federal interagency task force of the U.S. Department of the Interior [1979] identified strategies such as lining distribution ditches, tailwater recovery, and land leveling and estimated that $14.6 billion (1977 dollars) would be needed to install state-of-the-art measures.(23) The report estimated that these measures would save 38.6 million acre feet of water per year. Of this total, however, true losses to the system due to evaporation, consumption by unwanted plants, and so forth, amounted to only 3.3 million acre feet, or about 8.5 percent of the estimated gross saving. The balance, 35.3 million acre feet, would have returned to aquifers in the absence of conservation measures. Irrigation water that percolates back into an aquifer must, of course, be lifted to the surface again before it can be used and this requires the expenditure of energy. As described by the U.S. Department of the Interior [1979, 95], the proposed investments would have reduced re-pumping and an anticipated energy saving was the primary benefit attributed to the project. Reducing the cost of water available for consumption by crops causes users to increase consumption, however, and the water level to decline. In the long run the apparent energy saving that water conservation yields by avoiding re-pumping is dissipated by an increase in the distance that each unit of groundwater consumed is lifted.

Although the long-run social return to such investments is zero, they may well pass a private benefit cost test and hence be undertaken. The individual's incentive to conserve water, to prevent it from percolating back into the aquifer, is socially excessive. Any water that seeps away truly is lost to the farmer who pumped it, but it remains valuable to the community because it eventually will be available for reuse. Hence, the private benefit to conserving water exceeds the social benefit.(24) A similar conclusion applies to the fishery. When the net is being hauled the individual fisher's incentive to prevent fish from escaping is excessive from a social perspective, since some of those that escape will eventually be caught by someone else.

Development of Improved Irrigation Technologies and Rural Transportation Systems

To illustrate actions that intensify competition and cause dissipation by raising demand for z, consider improvements in irrigation technology. In 1949 Frank Zybach of Columbus, Nebraska developed the center pivot sprinkler, an innovation that eventually revolutionized irrigation in the Midwest. As Aiken [1980, 949-58] explains, this capped a decade of sprinkler technology development that led to extensive application in the 1950s. Before sprinklers, irrigation was by gravity methods and worked only on level land. The new technologies allowed use of rougher land without costly leveling operations and increased the demand for groundwater and surface water.

Evidence compiled by Aiken [1980, 949-58] indicates how this increased competition for groundwater and led to greater depths. Following the introduction of early sprinkler irrigation, the number of irrigation wells installed in Nebraska during the 1950s was four times higher than in the preceding decade and the number of acres irrigated with groundwater rose from 500,000 to two million. The center pivot technology began widespread application in the 1960s and the acreage irrigated by groundwater rose again, to over 3.5 million acres in Nebraska. Spurred partly by these developments and partly by high farm prices, the acreage irrigated by groundwater continued to rise through the early 1970s, reaching 7.4 million acres, or almost fifteen times the acreage involved in 1950. The consequent decline in groundwater levels was sufficient to spur some changes in regulation of groundwater use. Aiken [1980, 957] notes, however, that most parts of the state still do not place quantitative limits on amounts used by individuals.

A similar analysis applies to the development of rural transportation systems. The farmer must transport the crop from the field to the point of final consumption. The per unit transport cost acts much like a tax on the final product, and it can be avoided or reduced by improving a road, buying a new truck, and so forth. Each such investment involves a per period cost and, in return, reduces transport costs. Reducing the transport cost raises the net price received for the crop, however, and thereby raises the derived demand for groundwater. Water use increases, D increases, and marginal cost shifts up until it intersects the new higher derived demand schedule at z = R. Part of the return the improved transport system would otherwise earn is thereby dissipated. These actions will, of course, earn temporary returns during the transition between pre-investment and post-investment steady-states, and the period involved may be long in some instances.

Pricing Surface Water at Marginal Cost

The final example illustrates a completely different kind of demand-increasing action. Irrigation water often can be obtained from both ground and surface sources. Assume this is the case and that the two are perfect substitutes. Absent distortions elsewhere in the economy, efficiency in the allocation of resources requires that surface water be priced at its marginal cost. In practice, however, surface water usually is offered to irrigators at a price below marginal cost.(25) Suppose this is true here and that farmers are allowed to purchase as much surface water as desired at price [Mathematical Expression Omitted].

One might anticipate a welfare gain from raising the price of surface water to equal its marginal cost. Groundwater and surface water are perfect substitutes, however, so raising the surface water price shifts up the demand for groundwater and increases the rate of withdrawal. Starting from a free access steady state this increases D and marginal cost shifts up. Equilibrium is re-established only when the depth to water has increased to a point where the marginal cost of pumping R units equals the higher price of surface water. The steady-state total cost of pumping groundwater is increased as a result, eliminating at least part of the welfare gain expected from 'correctly' pricing surface water.

Figure 4 illustrates this example. The private marginal value of water is downward sloping and depends on the sum of ground and surface water use, [z.sub.g] + [z.sub.s]. Surface water is initially priced at [Mathematical Expression Omitted]. The marginal cost of surface water is MCS and is assumed constant. Given any D, the unit cost of pumping groundwater is assumed constant. Two conditions describe 'a steady-state equilibrium. First, farmers are indifferent between a marginal unit of water obtained from either source, which implies [Mathematical Expression Omitted] where [D.sup.0] is the depth to groundwater in the initial equilibrium. Second, groundwater consumption equals natural recharge, R. Thus the use and marginal cost of groundwater in the initial equilibrium are indicated by point a, and total water use is shown by

point b.

Absent distortions elsewhere one would anticipate a welfare gain equal to area A from raising the price of surface water to [Mathematical Expression Omitted], its marginal cost. The two sources of water are perfect substitutes, however, so the demand for groundwater is perfectly elastic at the price charged for surface water. Increasing the surface water price from [Mathematical Expression Omitted] to [Mathematical Expression Omitted] raises groundwater consumption above R causing D to increase. Groundwater use in the new equilibrium is at point c, where the depth to water has risen to [D.sup.1] and marginal cost is [Mathematical Expression Omitted]. The increase in steady-state pumping cost equals area B and the change in welfare from pricing surface water at marginal cost is A-B, which may be either positive or negative. In summary, pricing surface water at marginal cost is not optimal when irrigators have access to both surface water and unowned ground-water.(26)

V. THE VALUE OF SECOND-BEST REGULATION

Proposition 1 stated that the rent dissipation caused by free access is inversely proportional to [Zigma], the elasticity of cost with respect to z. This suggests that those who use the resource might gain by agreeing to a regulation that causes this elasticity to increase. This can be accomplished by fixing some of the inputs used to acquire the resource--in effect, forcing users to operate along a relatively steep 'short-run' marginal cost schedule. This is formalized in what follows by assuming k can be regulated but e cannot. Actual regulation of resource use often works in just this way. In a fishery it is common to limit the number of vessels (k) but not the number of days each operates (e). With groundwater, the number of wells permitted is often limited (k), but not the rate at which each pumps (e). Regulation may take this form because k is easier to observe and control than e. Clearly, controlling both k and e would be equivalent to exercising control over z directly.

The analysis starts by deriving a short-run or 'restricted' cost function with k fixed and examining when a marginal reduction in k will improve resource quality and raise welfare. The technology for acquiring z is hereafter assumed to be linearly homogeneous. This implies f(e/D, k/D) = f(e,k)/D = z, which can be inverted to obtain the input requirement function e = e(k,zD). The restricted cost function for z, given fixed k, can now be written as [p.sub.k]k + [p.sub.e] e(k,zD). Choosing units so that [p.sub.e] = 1, this is written [p.sub.k]k + V(k,zD), where V([center dot]) is variable cost. Given that the marginal productivities of e and k in acquiring z are positive and diminishing, one can verify that [V.sub.k] [is less than] 0 and [V.sub.z], [V.sub.zz] [is greater than] 0, where [V.sub.z] denotes the partial derivative of V([center dot]) with respect to its second argument.

For given k, utility is

[Mathematical Expression Omitted]

and free access use of z implies

(27) [U.sub.z](z) = [V.sub.z](k,zD)D.

Setting z=R in (27) gives D(k), the steady-state quality of the resource as a function of k, in implicit form: [U.sub.z](R) = [V.sub.z](k,RD(k)) D(K). Differentiating with respect to k yields

(28) [[V.sub.z](k,RD) + RD[V.sub.zz](k,RD)]dD/dk + D[V.sub.zk](k,RD) = 0.

This implies that dD/dk and [V.sub.zk] have opposite signs. It can be verified that [V.sub.zk] [is less than] 0 unless k is an inferior input, and inferior inputs have been ruled out by assuming f(e,k) to be linearly homogeneous. Hence dK/dk [is less than] 0, which is summed up as follows.

PROPOSITION 4. Let k be the regulated level of an input used to acquire the shared resource. Starting from a steady-state equilibrium in which all inputs are unregulated, a marginal reduction in k improves the steady-state quality or accessibility of the resource.

One can now derive the welfare effect of a marginal reduction in k evaluated at the unregulated equilibrium. Set z = R in (26) and differentiate with respect to k:

(29) dW/dk = -[p.sub.k] - [V.sub.k](k,RD) - [V.sub.z](k,RD)RdD/dk = -[V.sub.z](k,RD)RdD/dk,

since -[p.sub.k] - [V.sub.k](k,zD)=0 in the unregulated equilibrium. Recalling [V.sub.z] [is greater than] 0, as well as Proposition 4, this result can be described as follows.

PROPOSITION 5. Let k be the regulated level of an input used to acquire the shared resource. Starting from a steady state in which all inputs are unregulated, a marginal reduction in k raises steady-state utility.

Figure 5 illustrates the way regulation works. The unregulated equilibrium is at point a. The horizontal line MC([D.sup.c]) is marginal cost and [D.sup.c] is the equilibrium quality or accessibility of the resource, absent regulation. A regulation is imposed that fixes capital at [k.sup.r], below the level used without regulation. Given [D.sup.c] the effect is to make marginal cost less elastic and to shift it leftward to MC([k.sup.r][D.sup.c]). The immediate outcome is at point b. Since z is less than R, D falls and marginal cost declines. The regulated equilibrium is established at point a, when D has fallen to [D.sup.r].

The quality of the resource in the regulated equilibrium is improved to [D.sup.r] [is less than] [D.sup.c], in keeping with Proposition 4. It is not clear from the figure whether total cost has been raised or lowered by the regulation. While the area under MC([D.sup.c]) is expenditure on e and k, the area under MC([k.sup.r], [D.sup.r]) is expenditure on e alone because expenditure on k is a fixed cost. Proposition 5 guarantees, however, that a marginal reduction in k lowers total cost and raises welfare.(27)

Given that utility generally can be raised by restricting an input needed to acquire the resource, it is natural to ask how large such gains might be. The following analysis shows that the size of the gain and the k constraint that achieves it depend on the shape of the production function for acquiring z because this is what determines the elasticity of the restricted cost function, and on the relative prices of constrained and unconstrained inputs.

The general second-best problem of finding a k regulation policy that maximizes the present value of utility is beyond the scope of this paper, but the issues involved can be illuminated by solving a related problem--choosing a k constraint that maximizes steady-state utility. The first-best solution to this problem involves setting D as small as possible and attains utility [Mathematical Expression Omitted], where [Delta] is a physically determined lower limit on D.(28) Absent a positive lower bound on D, extraction cost would be zero for an unexploited resource, a feature that does not appear to characterize actual resources of any importance. For simplicity in describing the results obtained, the lower bound is taken to be arbitrarily small in what follows, although the reader can substitute different values and obtain more general results. With [Delta] arbitrarily small, the steady-state utility attainable from controlling z directly is [Mathematical Expression Omitted], and in what follows this is compared to steady-state utilities attainable by controlling k alone. The technology for acquiring the resource is assumed to be CES and the text illustrates three special cases, the Cobb-Douglas, Leontief, and linear technologies. A general treatment appears in the appendix.

If the elasticity of substitution is unity, the production function takes the Cobb-Douglas form, z = [e.sup.[Alpha]][k.sup.1-[Alpha]]. Let [k.sup.r] be the constrained level of k. The restricted total and marginal cost functions in this case are:

(30) C = [p.sub.e][([k.sup.r]).sup.1-[Gamma]][(zD).sup.[Gamma]] + [p.sub.k][k.sup.r],

(31) [C.sub.z] = [Gamma][p.sub.e][([k.sup.r]).sup.1-[Gamma]][(zD).sup.[Gamma]-1]D,

where [Gamma] = 1/[Alpha]. Notice that these functions are defined only if [k.sup.r] [is greater than] 0.

Steady-state D is established where [U.sub.z](R) = [C.sub.z]R. Making this substitution and solving for D yields [D.sup.c]([k.sup.r]) = [{[U.sub.z](R) [([k.sup.r]/R).sup.[Gamma]-1]/[Gamma][p.sub.e]}.sup.1/[Gamma]]. Substituting [D.sup.c]([k.sup.r]) into the cost function, the equilibrium value of total cost is R[U.sub.z](R)/[Gamma] + [p.sub.k][k.sup.r] and steady-state utility is

[Mathematical Expression Omitted].

Setting [k.sup.r] arbitrarily small maximizes steady-state utility and in the limit attains(29)

[Mathematical Expression Omitted].

The shared resource would earn the rent R[U.sub.z](R) with complete markets, and (33) indicates that the share [Alpha] is dissipated under second-best regulation.(30) Equivalently, the share that is captured (not dissipated) equals (1-[Alpha]), the share of total cost that would be spent on k in the unregulated equilibrium. In the Cobb-Douglas case, then, the steady-state welfare gain from second-best regulation of an input is equal to the unregulated expenditure on that input. This result clearly extends to cases where several inputs are used to obtain the resource and more than one can be controlled.

Consider, next, the Leontief case where the elasticity of substitution is zero. Intuitively, controlling a single input allows one to control both since inputs are used in fixed proportions, so controlling k alone should suffice to maximize steady-state utility. The following analysis shows that this is true.

The production function in this case is z = min(e, k)/D. Setting k=[k.sup.r] results in a marginal cost function that is horizontal at [p.sub.e]D per unit for the first [k.sub.r]/D units of z and then vertical. If D rises, this schedule shifts upward and to the left, and conversely if D falls. Given [k.sub.r], D is in a steady-state equilibrium when the vertical segment of marginal cost crosses the marginal value schedule at z=R. This implies R=[k.sup.r]/D, or RD=[k.sup.r]. Since e and k are used in fixed proportions, total cost is ([p.sub.e] + [p.sub.k])[k.sup.r]. Accordingly, steady-state welfare is

[Mathematical Expression Omitted],

which is maximized by setting [k.sup.r], and hence D, as small as possible. In the limit utility is

[Mathematical Expression Omitted].

In the Leontief case controlling k alone preserves the resource's entire rent--no dissipation occurs. The appendix shows that this zero dissipation result extends to any elasticity of substitution less than unity. In all such cases the regulated input is essential in that a specific minimum level is needed to obtain a given value of zD. By setting [k.sup.r] appropriately, the marginal cost schedule can be made to approximate the schedule for the Leontief technology.(31)

If the elasticity of substitution is infinite, the technology is linear: z=(e+k)/D. Regulating k clearly accomplishes nothing if [p.sub.k] [is less than] [p.sub.e], since only the cheaper of the two inputs will be used. In this case exact dissipation occurs--total cost equals RU[prime](R), the resource's entire rent.

Assume, therefore, that [p.sub.k] [is less than] [p.sub.e]. With k = [k.sup.r] capital expenditure is a fixed cost and the marginal cost of z is a step function. Given D, marginal cost is zero for the first [k.sup.r]/D units and is constant at [p.sub.e]D thereafter, and the vertical segment is a discontinuity. Since [p.sub.e] [is greater than] [p.sub.k] it is socially desirable to avoid using e in the steady state. An equilibrium in which k alone is used must satisfy two conditions. The first is [k.sup.r]/D = R, which ensures that k is large enough to obtain R units of the resource. The second is [p.sub.e]D [is greater than or equal to] [U.sub.z](R), which implies that the marginal cost of obtaining additional units of z by using e is at least as great as the marginal value of z. This guarantees that resource users have no incentive to use input e in the steady state. The first condition implies the steady-state cost level [p.sub.k][k.sup.r] = [p.sub.k]RD, so minimizing cost is equivalent to minimizing D. The second condition implies that the minimum sustainable value of D is [D.sup.r] = [U.sub.z](R)/[p.sub.e].

Combining these results, the optimal capital regulation is [k.sup.r] = R[D.sup.r] = R[U.sub.z](R)/[p.sub.e] and the minimum steady-state cost is ([p.sub.k]/[p.sub.e])R[U.sub.z](R). Recalling that exact dissipation results if [p.sub.e] [is less than] [p.sub.k], second-best regulation attains welfare

[Mathematical Expression Omitted]

in the steady state.

Figure 6 illustrates this case. The marginal cost curve under the optimal policy is abcd. Expenditure on [k.sup.r] is a fixed cost given that k is constrained. Its magnitude is shown as the diagonally lined area. The amount of rent captured (not dissipated) is shaded.(32)

This figure illustrates for a special case the way that second-best regulation of k works more generally. Absent regulation, marginal cost would be horizontal with equilibrium established at point c. In the unregulated equilibrium both inputs earn only their opportunity costs, [p.sub.e] and [p.sub.k], and the rent the resource would earn if owned is exactly dissipated. Fixing k alters the shape of the marginal cost function, forcing users onto a restricted marginal cost schedule that is less elastic than the unregulated schedule. The constrained input, k, then earns a rent, the shaded area, that partially offsets the rent the common property input would earn with complete markets. In effect, second-best regulation works by transferring some or all of the rent the resource would earn if owned to the regulated input.

The appendix presents more general results on second-best regulation with the CES technology. These results can be summed up as follows.

PROPOSITION 6. Let [k.sup.r] be the regulated level of an input used to acquire the shared resource and let the goal of second-best regulation be maximization of steady-state utility. Assume the technology exhibits constant elasticity of substitution, [Sigma], and that D can be made arbitrarily small.

a. If [Sigma] [is less than] 1 the second-best policy sets [k.sup.r] arbitrarily small and achieves the welfare level attainable with complete markets.

b. If [Sigma] = 1 the second-best policy sets [k.sup.r] arbitrarily small and raises welfare by an amount that equals expenditure on k in the unregulated equilibrium.

c. If [Sigma] [is greater than] 1 both the second-best capital constraint and the second-best welfare level depend on [Sigma] and [p.sub.k]/[p.sub.e], i.e., [k.sup.r]=[k.sup.r] ([Sigma], [p.sub.k]/[p.sub.e]) and W = W([Sigma], [p.sub.k]/[p.sub.e]). Given [Sigma], both [k.sup.r] and W are decreasing in [p.sub.k]/[p.sub.e]. Given [p.sub.k]/[p.sub.e], [k.sup.r] is single-peaked and W is U-shaped when plotted as functions of [Sigma].

d. If the extraction technology is linear and [p.sub.k] [is less than] [p.sub.e] the second-best policy sets [k.sup.r] = R[U.sub.z](R) /[p.sub.e] and captures the fraction (1 -[p.sub.k]/[p.sub.e]) of the shared resource's complete markets rent.

A common sense interpretation of Proposition 6 runs as follows. The benefit of constraining k lies in the fact that it limits competition for z by making the cost schedule for acquiring it less elastic. Loosely speaking, the degree of cost inelasticity achieved for a given k constraint is greater when the elasticity of substitution between controlled and uncontrolled inputs is low than when it is high. Constraining k also has a cost, however; it distorts the mix of inputs used so the cost of obtaining a given z is not minimized. When [Sigma] [is less than] 1 the benefit of limiting k dominates, and in the stylized model examined here maximum steady-state utility is attained. If [Sigma] [is greater than or equal to] 1, however, the regulated input is nonessential and the efficacy of a capital constraint in limiting competition for z and raising welfare falls discontinuously. In such cases substantial dissipation can persist even when the capital constraint is optimized and the amount lost depends on the exact level of [Sigma] and on the relative prices of controlled and uncontrolled inputs. The appendix illustrates these relationships graphically.

VI. CONCLUSIONS

The act of withdrawing an acre foot of groundwater from an aquifer is a transaction. Lifting it to the surface transforms it from a shared resource, accessible to anyone who owns a parcel of land and requisite pumping equipment, into property, a good for which rights are defined, enforceable, and vested in a specific individual. The relevant transaction cost is the cost of inputs used to withdraw it. Absent any price for water in the reservoir, this transaction cost is the only device that rations demand and equilibrium is established only when it is high enough at the margin to limit demand to the available supply. A similar characterization applies to the fishery, where the relevant transaction cost is the expense of inputs needed to find and catch fish, and to water withdrawn from a source that is polluted by use, where withdrawals are limited by the cost of treatment.

Innovations or other actions that either lower the cost of obtaining the resource or raise its marginal value product once acquired need not improve social welfare, even if they pass a seemingly sensible benefit cost test.(33) Rather, they intensify competition for the resource and further degrade its quality. Under plausible conditions their long-run net social return is negative even though they are rational to the individuals who undertake them. The same general conclusions appear applicable to the government goods and privileges that are prominent in the rent-seeking literature. The items at issue typically are in the 'government domain', available to those who can compete most effectively. The cost of acquiring rights to such privileges, through lobbying, contributions, and so forth is the factor that limits demand. The argument advanced here suggests that improvements in the technology of lobbying, e.g., better communication methods, more sophisticated methodologies for research aimed at influencing policy, and more efficient fund-raising techniques, may be socially self-defeating.(34)

Corrective policies often limit some but not all of the inputs needed to acquire an unowned resource. These policies can improve welfare by causing the marginal cost function to become steeper, so rent can be received for inframarginal units. This rent is attributable to the input fixed by regulation, so the policy effectively transfers all or part of the rent the resource would earn if owned to the regulated input. Such policies tend to avoid dissipation when the elasticity of substitution between controlled and uncontrolled inputs is low and when the relative price of the controlled factor is low.

These results extend the analysis of rent dissipation for unowned resources begun by Gordon [1954] and further developed and applied by Cheung [1970], Johnson and Libecap [1982] and others. Casting the problem in an explicit general equilibrium framework is found to be advantageous because it enables exact characterization of the rent the resource would earn with complete markets and the degree to which it is dissipated when ownership is absent. Others have claimed that dissipation is exact when agents using the resource are homogeneous. The introduction of costly complementary actions--actions that either raise the marginal value or lower the private marginal cost of the resource--shows that dissipation can either exceed or fall short of the rent the resource would earn with complete ownership. In particular, the finding that dissipation can exceed what the resource would earn if owned contrasts rather sharply with the received literature. In addition, characterization of the social gains to partial control of how the resource is used and the finding that these gains are related in a rather simple way to elasticities of substitution and relative prices, is new. Finally, the examples used to illustrate these results indicate that these principles are not understood in policy circles, at least among those concerned with policy toward groundwater. It seems likely that, with a bit of effort, examples of socially self-defeating innovations and government initiatives toward fisheries, degradable water courses, and other resources could be found as well.

APPENDIX

The first section of this appendix presents the dynamic optimization problem and the second section examines the second-best regulation problem when the technology for obtaining the shared input exhibits constant elasticity of substitution.