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Efficient user charges in a rent-seeking model.

EFFICIENT USER CHARGES IN A RENT-SEEKING MODEL

This paper examines efficient user charges on governmentally provided facilities in the presence of rent seeking. We find that the efficient user charge depends upon the relative slopes of the demand curve for the use of the facility and the corresponding marginal cost curve, as well as the level of rent seeking over the revenue raised. Except for a special case, the efficient user charge is found to differ from the charge indicated by the intersection of demand and marginal cost curves. Examples show that actual user charges on government facilities are often set at inefficient levels.

I. INTRODUCTION

Governments control access to many services, facilities, and resources for which either no charge, or only a token charge, is imposed on users. Under the prompting of concerns over budgetary deficits and utilization efficiency, there is increased interest in user charges on a wide array of government facilities.(1) By placing charges on government facilities, or by increasing existing charges toward efficient levels, significant amounts of revenue can be raised and consumption decisions made more responsive to cost conditions.

This paper concentrates on the efficiency justification of user charges. In particular, the question of what is the efficient user charge to impose on access to an existing facility controlled by government is addressed. Abstracting from problems associated with incomplete information on the relevant demand and cost conditions, the answer to this question may seem obvious. The standard analysis indicates that the efficient price for a good, whether imposed by market forces or by government authorities, is determined by the intersection of the demand curve and the marginal cost curve. This answer is clearly correct in the case of market prices. However, the intersection of the demand and marginal cost curves will determine the efficient governmentally imposed user charge only if there are no efficiency-relevant differences between government prices and market prices. By recognizing a fundamental difference between government prices and market prices, the analysis in this paper qualifies the standard view on efficient user charges.

The distinction between government and market prices arises from the fact that a market price generates revenue that is privately owned, while a government price generates revenue that is commonly owned. The claimants of revenue from a market price are generally well-specified, as is the extent of their claims. This is not the case with revenue from a government price. Additional revenue raised by government typically goes into the general fund, which is then allocated among rival claimants through competition for political influence. Even when government revenues are earmarked, they are typically earmarked to a function where specific claimants are not well specified. Earmarking may reduce, but surely does not eliminate, the competition for political influence.(2)

Competition for political influence requires the use of real resources and, to a large extent, represents pure waste since it is primarily concerned with distributing existing wealth, rather than creating new wealth. This waste is referred to as a rent-seeking cost and has to be recognized as inextricably associated with government revenue. Although some of what is perceived as cost by rent seekers is, from a social perspective, a costless transfer, the fact remains that raising government revenue with a user charge will motivate socially costly rent seeking, and the amount of this cost will be positively related to the amount of revenue raised.(3)

One might argue that an increase in government revenue from a user charge may not result in additional rent-seeking cost, since total government revenue may remain constant as other charges or taxes are reduced. Even if this is the case, however, interest groups will compete to have their charges or taxes reduced. Additional rent-seeking activity occurs whenever government revenues are created from any source, even if these revenues are offset by reductions in revenues from other sources.

The next section analyzes the implications of rent seeking for the efficiency of user charges.

II. THE ANALYSIS

Consider a facility available in some fixed quantity over which government controls access. Assume the facility is a standard good with P(U) representing the downward sloping inverse demand for the use, U, of the facility. Let C(U) represent the cost of facility use, with the cost increasing in U at an increasing rate; i.e., C [prime] > 0, C [double prime] [is greater than or equal to] 0. Equating the inverse demand to the user charge (K) determines the level of usage as a function of K, U(K), which declines as K is raised. Finally, assume that rent seeking motivated by the user-charge revenue creates a rent-seeking cost equal to some fixed proportion ([Alpha]) of the revenue, where 0 < [Alpha] [is less than or equal to] 1.

The efficient user charge maximizes total net benefits, which equal total benefits to users from the level of usage determined by the user charge, less the cost of providing that level of usage, less rent-seeking costs, or (1) [Mathematical Expression Omitted]. Restricting the user charge (K) to be non-negative, then the user charge which maximizes total net benefits in (1) must satisfy the first-order condition (2) {P[U(K)] - C [prime] [U(K)]}

U [prime](K) [is less than or equal to] [Alpha][KU [prime](K) + U(K)], where the strict inequality holds only if the optimal charge, [K.sup.*], equals zero.(4)

In the standard analysis, [Alpha] = 0 and the strict equality holds in (2) (inequality is still possible, but not under any interesting conditions when [Alpha] = 0) In this case, (2) becomes P(U) = C [prime](U), and the solution, K, is the user charge which equates marginal cost with marginal benefit. With positive rent seeking ([Alpha] > 0) the efficient user charge will in general, though not always, deviate from the efficient private market price (K [bar]).

In order to specify the first-order condition in terms that can be illustrated in familiar diagrams, divide both sides of (2) by the negative term U'(K) to obtain (3) C [prime](U) - P(U) [is less than or equal to]

- [Alpha](K + U dK/dU). The left-hand side of (3) is the marginal value of reducing use of the government facility, which is positive when use is in excess of the usage level U [bar] (corresponding with K [bar]), and negative when use is less than U [bar]. The bracketed term on the righthand side of (3) is the standard expression for the marginal revenue (MR) from increasing usage. Multiplying this marginal revenue by -[Alpha] yields the marginal rent-seeking cost of reducing use, which is positive when the standard marginal revenue is negative, and negative when the standard marginal revenue is positive.

The degree of rent seeking ([Alpha]) and the slope of the marginal cost curve relative to the slope of the demand curve determine the efficient user charge with rent seeking. To demonstrate this point, first consider the case shown in Figure 1 where the marginal cost curve is less steeply sloped than the demand curve [Mathematical Expression Omitted]. Assuming [Alpha] [is less than or equal to] 1/2, then the second-order condition is satisfied (see footnote 4). Therefore if a user charge (K) exists which satisfies (3) as an equality, then it is efficient. It is likely, however, that no such interior solution exists. For example, if [C.sub.1] [prime] is the relevant marginal cost curve in Figure 1, and [Alpha] = 1/2, then [C.sub.1] [prime] - P < - MR/2 at U(O), and therefore the marginal rent-seeking cost of reducing use by increasing the user charge above zero exceeds the marginal value of reducing what is conventionally considered excessive use. Hence the efficient user charge is no user charge.

If the relevant marginal cost curve in Figure 1 is [C.sub.2] [prime], then [C.sub.2] [prime] - P > - MR/2 at U(0). The marginal benefit from reducing excessive use below U(0) exceeds the marginal rent-seeking cost of doing so, and therefore the efficient user charge ([K.sup.*]) is positive. As shown in Figure 1, the first-order condition (3) is satisfied as an equality at [U.sup.*] (the distances ab and cd are equal). Note that in this example, the efficient user charge ([K.sup.*]) is less than the price that would be determined by a private market (K [bar]). As rent-seeking inefficiency increases, the efficient user fee falls and is more likely to be zero.(5)

It is possible in Figure 1 for some 1/2 < [Alpha] < 1 that a solution to (3) as an equality occurs to the left of the intersection of the demand and marginal cost curves, but to the right of U = 0. This solution will be a local minimum, and the value of the objective function, (1), will increase as usage declines further toward zero.(6) The optimal usage in this case will be either zero or U(0). But since the total value of use will be positive at U(0) when marginal cost is less steeply sloped than the demand curve ([Mathematical Expression Omitted]), usage level U(0) will dominate, and so the efficient user charge is no charge. Therefore, whenever [Alpha] < 1 and marginal cost is flat relative to demand ([Mathematical Expression Omitted]), the efficient charge ([K.sup.*]) for a government facility is less than the private market price (K [bar]) and possibly equal to zero, depending upon the slope of the marginal cost curve relative to the demand curve.

Note that the more price inelastic the demand for use, the greater the range of marginal cost curves that will result in an efficient charge less than the conventional charge, and the more likely that the efficient charge is zero. This situation probably occurs on many urban roads used for commuting to work. Commuter travel on roads is a derived demand and hence is likely to be price inelastic, particularly where there are few public transportation substitutes.

Now consider the case shown in Figure 2 where the marginal cost curve bisects the demand curve ([Mathematical Expression Omitted]), and so, given the linearity assumptions, the intersection between P and C [prime] occurs where the price (user charge) elasticity of demand, [Epsilon], equals 1. Since marginal revenue equals zero when elasticity equals one, this intersection establishes a solution to (3), and when [Alpha] < 1, this solution is unique. Therefore, the efficient user charge is the same as that conventionally deemed efficient ([K.sup.*] = K [bar]). If [Alpha] = 1, then (3) is satisfied for all user charges from that which is prohibitive all the way down to zero. All user charges are equally efficient when marginal cost and demand have the same slope, but no matter what the charge is, the net social benefit from using the government facility equals zero. Although an interesting case, it is a highly unlikely that demand and marginal cost will have slopes identical in absolute value.

Finally, consider the case shown in Figure 3 where the slope of the marginal cost curve is greater than the slope of the demand curve ([Mathematical Expression Omitted]). Any solution to (3) as an equality will yield the efficient user charge.(7) Clearly, the distance between the marginal cost and demand curves is greater than -[Alpha]MR for all levels of rent seeking and levels of usage to the right of the intersection of demand and marginal cost. Therefore, the marginal benefit from reducing usage remains in excess of the marginal rent-seeking cost of doing so until usage is reduced below the level conventionally considered efficient. This move is into the region where the marginal rent-seeking cost of reducing use is negative, since increasing the user charge reduces revenue as soon as the price elasticity of demand becomes less than unity. It pays then to reduce use of the facility into the region where the marginal value of usage exceeds its direct marginal cost. In the case where [Alpha] = 1/2, as illustrated in Figure 3, the equality C [prime] - P = -[Alpha]MR is uniquely satisfied at [U.sup.*] < U [bar] (where ab equals cd). As long as [Alpha] < 1, this equality will be satisfied at some positive U less than U [bar]. Therefore, the efficient user charge will be less than prohibitive, but will reduce usage below that considered efficient by conventional analysis. If [Alpha] = 1, C [prime] - P = - [Alpha]MR at zero usage, which, by virtue of the second-order condition, is a unique solution and the efficient solution. The efficient user charge in this case is one that is prohibitive.

Examples of this case include facilities for which demand is typically very elastic with respect to price (and hence [Mathematical Expression Omitted]). Any facility provided by government which is a substitute in consumption with facilities produced by the private market is likely to fit this scenario. In the case of Amtrak and many other pork-barrel projects in the U.S., actual pricing policy is opposite that suggested here; i.e., actual user charges are typically set below that indicated by the intersection of demand and marginal cost.(8)

Up to this point the analysis has considered only the rent seeking aimed at capturing the revenue from a user charge. Conceivably, there may also be costly rent seeking over the level at which the user charge is set, in addition to the competition for revenue raised by the user charge. For example, users will want the charge set as low as possible, while those seeking the resulting revenue will want the charge to be set as close to the revenue maximizing position as possible. It is possible to make some general comments about the effects of this additional cost without incorporating the cost associated with the rent seeking by users into a formal analysis.

First, in the case represented by Figure 1, the efficient user charge is less than that which maximizes revenue. In this situation, revenue rent seekers and user rent seekers will apply opposing pressure for a change in the user charge, with the former wanting to increase the user charge and the latter wanting it lowered. The relevant question here is whether revenue rent seekers reduce their rent seeking over the level of the charge more if the user charge is increased than user rent seekers will increase their rent seeking. If so, then efficiency will be enhanced by increasing the user charge somewhat above the level we have specified as efficient ([K.sup.*]). If not, then the efficient charge will be at or below [K.sup.*].(9)

If the situation is that described in Figure 2, then revenue is being maximized and the reasonable assumption is that the greatest rent seeking reduction will come from responding to the pressure from users to reduce the charge. This suggests that the efficient charge will be somewhat less than that determined by the intersection between the demand and marginal cost curves.

Finally, the situation described in Figure 3 finds both users and revenue seekers desiring a lower charge that the one that is efficient in our analysis. Therefore, when rent seeking over the level of the charge is considered, the efficient user charge will be somewhat below the one we have determined to be efficient, and closer to the conventional market price.

III. CONCLUSIONS

The preceding analysis has demonstrated that, with the possible exception of a special case, considerations of rent-seeking costs will alter estimates of efficient user charges from those suggested when rent-seeking costs are ignored. The case of highways is interesting to consider in connection with the above analysis for several reasons. User charges have often been advocated for highways; e.g., Roth [1967] and Henderson [1985]. Congestion costs have been widely studied for highways, and statistical techniques for estimating the magnitude of congestion costs are readily available. Furthermore, highway facilities are typically provided by government authorities, although they cannot be considered pure public goods in an economic sense. As noted earlier, demand for travel to work is a derived demand and consequently is likely to be relatively price inelastic. Therefore the demand curve will be steeply sloped. Estimates of the highway speed-flow relationship, such as Inman [1978], indicate that marginal congestion costs increase slowly until very busy levels of traffic are reached. On many roads, these very heavy levels of traffic occur only infrequently. In such cases, it is likely that demand is steeper than marginal cost ([Mathematical Expression Omitted]) for the relevant range of usage. Consequently, the efficient user charge is less than that suggested by the conventional analysis, and perhaps equal to zero. In cases where very heavy levels of traffic do occur, perhaps during only part of the day, it is conceivable that marginal cost is steeper than the demand curve ([Mathematical Expression Omitted]). In such cases the efficient toll in the presence of rent seeking is greater than suggested by the conventional analysis.

Where tolls are actually imposed on highways in the United States and other countries, they are invariably of constant value, regardless of the level of congestion. In periods of low congestion, the tolls tend to overcharge drivers relative to the efficient toll suggested by the conventional analysis, while during periods of heavy congestion, they tend to undercharge drivers relative to the efficient toll suggested by the conventional analysis.(10) The analysis in this paper indicates that this is opposite the desired pricing policy. Similar observations may be made for any government-owned facility which experiences peaks of congestion and on which a constant user charge, calculated as an average of the efficient charges suggested by conventional analysis, is imposed. [Figures 1 to 3 Omitted]

(1)Henceforth the term "facilities" will be substituted for the more cumbersome phrase, "services, facilities, and resources." (2)Also, when revenues from user charges are earmarked for building, repairing, or replacing facilities, rent seeking can lead to an excessive supply. See Hillman [1988]. The connection between the user charges imposed on a facility and the size of the facility, in the case of earmarked funds, has implications for the efficient user charge that deviate from the standard analysis, but which are not examined here. (3)Until recently, the political transfer of wealth from one group to another was considered to be socially costless. What one group lost the other gained. Tullock [1967] was the first to observe that resources would be expended in competition for government transfers and to analyze the efficiency implications of this competition. Krueger [1974] labeled this competition rent seeking. Articles on rent seeking are collected in Buchanan et al. [1980] and Rowley et al. [1988]. (4)Solutions to (1) must also satisfy the second-order condition which requires that the second derivative of (1) with respect to the user charge, K, be negative. Assuming the inverse demand curve and the marginal cost curve are linear, the second-order condition may be written as 1 - 2[Alpha] > C [double prime]/P [prime] < 0, where C [double prime]/P [prime] is a negative constant. Since satisfaction of the second-order condition does not depend on the value of K, any existing strict local extremum must be unique, since if K were a local maximum (minimum), a second local extremum would have to be a local minimum (maximum), which would violate the second-order condition. (5)If [Alpha] > 1/2, then unless the slope of the marginal cost curve (C [double prime] in Figure 1 is sufficiently close to the slope of the demand curve (~P [prime]~), the second-order condition may not be satisfied. In this case, [K.sup.*] = 0. If the second-order condition is satisfied, then there may or may not be a K which satisfies (3) as an equality. If not, then [K.sup.*] = 0; otherwise, 0 < [K.sup.*] < K [bar].

If [Alpha] = 1, the second-order condition is not satisfied, since C [double prime]/P [prime] is is greater than -1 , and 1 - 2[Alpha] = -1. Any value of K that satisfies (3) as an equality minimizes (1). Therefore, the efficient user charge in Figure 1 is [K.sup.*] = 0 if [Alpha] = 1. (6)If the solution were a maximum then it would have to be the case that C [prime] - P > [Alpha]MC at U = U(0), in which case the solution to (3) as an equality would have to occur to the right of the intersection between P and C [prime]. (7)The second-order condition is always satisfied when C [double prime] > ~P [prime]~. (8)Production of facilities by government likely does not preclude production of similar facilities by the private sector, which may be substitutes in consumption. See Roberts [1989] for a formal discussion of this point. (9)Users and revenue seekers are assumed to reduce their rent seeking over the level of the charge when the user charge is moved closer to their preferred level. (10)See, for example, Inman [1978] and Wilson [1988].

REFERENCES

Buchanan, James M., Robert D. Tollison, and Gordon Tullock, eds. Towards a Theory of the

Rent-Seeking Society. College Station: Texas A&M University Press, 1980. Henderson, J. Vernon. Economic Theory and the Cities. New York: Academic Press, 1985. Hillman, Arye L. "Tariff-Revenue Transfers to Protectionist Interests: Competition for Reduced

Protection or Supplementary Reward for Successful Lobbying?" Public Choice, August 1988, 169-72. Inman, Robert P. "A Generalized Congestion Function for Highway Travel." Journal of Urban Economics,

January 1978, 21-34. Krueger, A. O. "The Political Economy of the Rent-Seeking Society." American Economic Review, June

1974, 291-303. Roberts, Russell D. "The Interaction Between Public and Private Spending when Government is

Responsive to the Preferences of Citizens." Working Paper, Washington University, 1989. Roth, Gabriel. Paying for Roads: Economics of Traffic Congestion. London: Penguin Press, 1967. Rowley, C.K., Robert D. Tollison, and Gordon Tullock, eds. The Political Economy of Rent Seeking.

Boston: Kluwer Academic Publishers, 1988. Tullock, Gordon. "The Welfare Costs of Tariffs, Monopolies and Theft." Western Economic Journal,

June 1967, 224-34. ______. "Efficient Rent Seeking," in Toward a Theory of the Rent-Seeking Society, edited by J.

Buchanan, R. Tollison, and G. Tullock. College Station: Texas A&M University Press, 1980,

97-112. Wilson, Paul W. "Welfare Effects of Congestion Pricing in Singapore." Transportation, March 1988,

191-210.

DWIGHT R. LEE and PAUL W. WILSON, Professor and Holder of the Ramsey Chair of Private Enterprise Economics, The University of Georgia, and Assistant Professor, Department of Economics, University of Texas, Austin. The authors would like to thank an anonymous referee for helpful suggestions while accepting full responsibility for any errors in the analysis.
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Author:Lee, Dwight R.; Wilson, Paul W.
Publication:Economic Inquiry
Date:Apr 1, 1991
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