# Enforcing time-inconsistent regulation.

I. INTRODUCTION

Passage of legislation is merely the beginning of the regulatory task. The successful translation of policy from legislative debate into regulation is not guaranteed. Implementation may be especially problematic for regulations that are "time-inconsistent." Regulations have the potential for time-inconsistency if the government cannot credibly commit itself to enforcement of the original legislation at every relevant point in time.

Consider a law that sets a regulatory standard to be enforced several years in the future. The regulation's implementation may take the following course for any given enforcement period. First, firms are expected to engage in long-term investment to move towards meeting the standard. Second, after that investment has been completed, an enforcement agency reviews and has the opportunity to modify the standards, shortly before the date at which they are to be enforced. Finally, after the agency's review, firms make short-term investment to meet the standard. This process may be repeated for subsequent enforcement periods.

This sequence of events presents inherent problems for the enforceability of a regulation. During the long-term investment stage regulated firms know that the relevant administrative agency may be unable to credibly bind itself to future actions. Given this, firms may believe that the agency's enforcement posture depends on their own actions; firms may therefore be able to act strategically to induce the regulatory agency to lower the relevant standard before its enforcement date. Facing such incentives, firms may not undertake the desired long-term investments. Upon review, the enforcement agency could find that requiring firms to make short-term investments to meet the standard may involve substantial social costs. Facing such costs, the agency may grant regulatory relief and relax the standard.

In these circumstances, society's welfare may be greater if firms engaged in the necessary long-term investment to meet the standard prior to the enforcement date. In order to force firms to make the necessary long-term investment, however, the enforcement agency may need to threaten to punish firms by enforcing the original standard even if firms' long-term investments are below optimal levels.

The need to threaten this punishment places the agency in a difficult position. If firms do not undertake the desired long-term investments it may be in the agency's (and society's) interests to relax the initial standard to avoid the potentially higher costs associated with short-term investments. Firms will understand this, and thus make no ex ante investment towards meeting the standard. The regulation is therefore time-inconsistent and unenforceable.

In the case of automobile fuel economy regulations, for example, Congress in 1975 set standards for model years 1978 and afterwards, but provided the executive branch, in the form of the National Highway Traffic Safety Administration (NHTSA), with discretion to modify the standard for model years after 1984. Automobile manufacturers then undertook long-term product investment that may have been initiated in order to meet the standards. For model years 1985, 1986, and 1989, however, firms were able to gain regulatory relief from NHTSA at the start of the relevant model years. If relief had not been granted, there would have been insufficient time for further product innovation, and firms would have had to meet the standard by eliminating less fuel efficient vehicles from their fleets, a strategy that would have been costly not only to themselves, but to their employees and customers as well. (See Kleit |1990~ and Greene |1991~.)

On the other hand, in the 1970s the Congress passed laws to be enforced by the Environmental Protection Agency that required automobile companies to invest millions of dollars in pollution control devices. While automobile firms achieved some delay in the implementation of the law, the devices ultimately were installed. (See White |1981~.) Pollution control devices are thus an example of a time-inconsistent standard that was eventually successfully enforced. Indeed, the only apparent difference between the enforceability of automobile fuel economy standards and automobile emission standards was the agency chosen to administer them.

This article presents a game theoretic model that demonstrates how certain types of regulatory agencies can enforce time-inconsistent regulation.(1) Administrative procedures, as discussed by McCubbins, Noll, and Weingast |1987; 1989~, allow agencies to credibly commit to policies that are consistent with their bureaucratic missions. Agencies will be modelled in terms of whether their administrative procedures have made them "committed" to enforcing a particular time-inconsistent policy under any circumstances. Because firms realize that a particular agency might be forced by its own administrative procedures into acting against the public welfare, potentially time-inconsistent regulation may be enforceable. This model stands in contrast to the previous literature on regulatory enforcement, such as Laffont and Tirole |1986~, which does not deal with issues of commitment.

II. THE ADMINISTRATIVE INSTITUTIONS

The analysis here assumes that legislation is enacted setting a regulatory standard. The legislation, however, also gives the administrating agency authority to lower the standard should some unforeseen event occur. As McCubbins et al. |1987, 256-7 and 261; 1989, 449~ and Hahn |1988, 224-5~ point out, if a large degree of technological uncertainty is involved, the legislature may not desire to choose an exact standard to be reached, given the possibility that relevant information is not yet available. Thus, the legislature may grant the agency the discretion to modify future standards.

This leaves the legislature with the problem of how to constrain the agency to enforce its decisions. Even if the administrative agency agrees with the legislative branch, and neither one's opinions change over time, the temporal nature of the regulation, by itself, may generate time inconsistency and prevent the policy from being implemented. Therefore, the legislature must succeed in committing the agency to enforcing the standard even if the desired investments do not take place.

Conceptually, the legislature or the agency could solve this problem by creating for itself a reputation for "toughness" or "irrationality." Creating such a reputation, however, at least in the models of Kreps and Wilson |1982~ and Diamond |1989~, takes a good deal of time, given the low levels of irrationality expected from economic actors. Since elected government officials serve for a finite period, they may find that making the investment needed to create a reputation for toughness is not a viable option.

As McCubbins et al. |1987, 257-8~ describe, a method by which the legislature can bring an administrative agency under control and solve the implementation problem is through the use of administrative procedures. Procedural requirements, often quite complex, restrict the choices available to an administrative agency and ensure that legislative desires are implemented by creating a credible commitment, even without direct legislative supervision. In the context of time-inconsistent policy, administrative procedures are presented here as a solution to the traditional principal agent problem, not only across political actors (as McCubbins and his coauthors describe), but across time as well.(2)

Of course, all administrative agencies are not alike. In an agency's enabling statute the legislature may be able to implicitly choose its preferred level of administrative commitment. The legislature may also be able to select from among a variety of existing agencies in which to entrust the enforcement of a particular statute, depending on the level of commitment it desires. The commitment of a particular agency to a particular regulation can be thought of as a function of its own internal procedures, as well as how consistent the particular regulation is with that agency's political mission.(3) In turn, the agency's mission may be a function of the concerns of the agency's client interest groups (as in Stigler |1970~) or the political interests of the members of the relevant congressional oversight committees (along the lines of Weingast and Moran |1983~).(4) Thus NHTSA, as its name implies, may be more likely to be committed to regulations that improve automobile safety and less likely to be committed to regulations that do the reverse. Firms can be expected to understand this and to take it into account when making their regulatory investment decisions.

III. THE REGULATORY GAME

Prior to the start of the game, the legislature passes a law setting the standard at S. It will be assumed that S is set at the welfare-maximizing level (S*), given that the regulation is successfully enforced. To capture the often complicated administrative procedures established, the regulated firm is modeled here as unaware as to whether the relevant enforcement agency is "committed" (type = C) and thus unable to modify the standard, or "uncommitted" (type = N) and therefore able to do so. Under the bureaucratic "rules of the game," however, the firm knows the probability |p.sup.N~, 0 |is less than~ |p.sup.N~ |is less than~ 1, that the agency is uncommitted and willing to lower the standard. This probability is a function of how consistent the relevant regulation is with an agency's mission and how strongly the agency's administrative procedures commit it to that mission. The firm updates this probability during the game according to Bayes' Law.

The game itself has two periods, t = 1, 2. Allowing the game to be longer than one period generates incentives for an uncommitted agency to act strategically in order to generate a reputation for toughness. This in turn will be shown to generate a discontinuity in the firm's optimal strategy. Each period has two stages, permitting analysis of the two decisions made in each period, one by the firm and one by the agency. In the first stage of period t the firm

improves the regulated aspect of its product an amount |F.sub.t~ at some cost to itself. The firm chooses |F.sub.t~ to minimize the present value of its expected costs over the two-period game, given |p.sup.N~ and any previous events in the game. In the second stage, after examining the first-stage product improvements |F.sub.t~, the reviewing administrative agency sets a standard |S.sub.t~, which may be equal to or less than the mandated standard S*. If the agency is uncommitted, it sets the standard to maximize net welfare given first-stage investment, knowing that the firm will take the existence of an amended standard in period one into account when making its first-stage investment in period two. A committed agency sets |S.sub.t~ = S*. After the agency makes its decision, the firm has no choice but to improve its product |H.sub.t~ = |S.sub.t~ - |F.sub.t~ |is greater than~ 0 at some cost to reach the standard.

In terms of automobile fuel economy standards, first-stage improvements |F.sub.t~ can be thought of as long-term engineering innovation and investment affecting the fuel efficiency of individual automobiles. Second-stage improvements |H.sub.t~ can be thought of as short-term changes in the mix of vehicles offered for sale that generate relatively higher sales of the more fuel-efficient vehicles.

For simplicity the cost function to the (risk-neutral) firm in period t is assumed to be

|Mathematical Expression Omitted~

where in general b |is greater than~ a. The firm is assumed to gain no benefit from the production of this good, and hence without regulation it would not be produced.

Society is assumed to value the regulated good with constant marginal utility v. Solving for S* and dropping subscripts, the net total social welfare from reaching a level of the regulated good S* = F* + H* where F is generated from first-stage long-term investment and H is the second-stage short-term investment is maximized at

(2) F* = v/2a,H* = v/2b,S* = F* + H* = v(a+b)/2ab.

It is also assumed that the agency cannot observe |F.sub.t~ prior to the second stage of period t.(5) A sequential equilibrium to this game will be solved using backwards induction.

Second Period: Second Stage

If the agency is of type C, it is unable to modify the legislature's initial decision and |S.sub.2~ = S*. The firm then improves its product |H.sub.2~ = S* - |F.sub.2~. If the agency is of type N, it maximizes welfare over |S.sub.2~ given |F.sub.2~:

(3) W = v|S.sub.2~ - |a|F.sub.2~.sup.2~ - b|(|S.sub.2~ - |F.sub.2~).sup.2~.

Maximizing W with respect to S yields

(4) |S.sub.2~ - |F.sub.2~ = v/2b = |H.sub.2~ = H*,

with the value of |H.sub.2~ independent of |F.sub.2~. The firm then improves its product |H.sub.2~ = H*.

Second Period: First Stage

The goal of the regulated firm at this point is to minimize its expected costs over the probability that the agency is of type N. That probability, |p.sub.2~, is a function of |p.sup.N~, the a priori probability that the agency is of type N, and the events of the first period. If the agency did intervene in period one it has, in effect, "shown its hand" and demonstrated with probability 1 that it is of type N. Therefore, |p.sub.2~ is equal to 1 if intervention occurred in period one; otherwise |p.sub.2~ = |p.sub.2~(|p.sup.N~, |6F.sub.1~, |S.sub.1~) = |p.sub.2~ (|p.sup.N~, |F.sub.1~). (|S.sub.1~ equals S* if intervention did not occur.) The firm thus knows that given its first-stage improvements |F.sub.2~ it will have to improve its good H* with probability |p.sub.2~ and face probability 1 - |p.sub.2~ that its good will have to be improved S* - |F.sub.2~. Given this, the firm minimizes its expected costs over |F.sub.2~,

(5) Min E(|C.sub.2~) = |p.sub.2~(|a|F.sub.2~.sup.2~ + b|H*.sup.2~) + (1 - |p.sub.2~)||a|F.sub.2~.sup.2~ + b|(S* - |F.sub.2~).sup.2~~.

Minimizing (5) with respect to |F.sub.2~ and solving yields

(6) |F.sub.2~ = (1 - |p.sub.2~)bS*/|a + (1 - |p.sub.2~)b~ = F*(a + b)(1 - |p.sub.2~)/|a + b(1 - |p.sub.2~)~ = F*k(|p.sub.2~).

The function k is the fraction of optimal investment undertaken by the firm given the firm's posterior probability that the agency is type N (uncommitted) with k(1) = 0, k(0) = 1, and dk/d|p.sub.2~ |is less than~ 0.

First Period: Second Stage

A type C agency will set the standard at S*, and the firm will have no choice but to improve its produce S* - |F.sub.1~. If |F.sub.1~ |is less than~ F* and the agency is of type N, it will grant relief (setting the standard at |F.sub.1~ + H* = |S.sub.1~ |is less than~ S*) if the benefits of doing so outweigh the costs.

Define |p.sup.E~ = |p.sub.2~(|p.sup.N~, |F.sub.1~, No relief) as the equilibrium posterior probability that the agency is uncommitted given that the agency did not grant relief in the first period. Let |F.sup.N~ be the first-period, stage-one investment at which the agency is indifferent between granting relief and enforcing the original standard, given that |p.sup.E~ = |p.sup.N~. (See the appendix for the derivation of |F.sup.N~.) At |F.sup.N~ the firm is not "fooled" by enforcement of the original standard since the firm believes, given |F.sub.1~, that an uncommitted agency would not have granted relief, since doing so would have revealed to the firm that it was uncommitted.

Define region III as ||F.sup.N~, F*~. By definition if |F.sub.1~ is in region III, the agency will not grant relief, since the costs of granting relief are greater than the benefits. If relief is not granted when |F.sub.1~ is in region III, the firm has no reason to believe |p.sub.2~ |is greater than~ |p.sup.N~. Thus, if |F.sub.1~ is in region III, it always is optimal for a type N agency to mimic a type C committed agency and not grant relief.

Define region I as |0, |F.sup.C~~ where |F.sup.C~ is the maximum |F.sub.1~ such that the agency will want to grant relief, given that if relief is not granted the firm will have |p.sub.2~ = 0. (See the appendix for the derivation of |F.sup.C~.) By definition, if |F.sub.1~ is in region I, a type N agency will always grant relief. Here the benefits of granting relief are always greater than the costs, even though if relief is not granted the firm believes with probability 1 that the agency is of type C (committed).

Define region II as ||F.sup.C~, |F.sup.N~~. If |F.sub.1~ is in this region, no pure strategy equilibrium exists for an uncommitted agency. There is, however, a mixed strategy solution similar to the one derived by Kreps and Wilson |1982~ (see the appendix). Once the agency makes its decision, the firm improves its product |S.sub.1~ - |F.sub.1~.

First Period: First Stage

Given the strategies calculated above, the firm now minimizes its expected costs by choosing |F.sub.1~. If it chooses |F.sub.1~ in region III, it knows that relief will be granted with probability 0. If it chooses |F.sub.1~ in region II, it knows relief will be granted with probability |p.sup.N~L||p.sup.E~(|F.sub.1~), |p.sup.N~~. If it chooses |F.sub.1~ in region I, it knows relief will occur with probability |p.sup.N~. The firm thus faces the possibility frontier outlined in Figure 1.

A solution to the firm's cost-minimization problem can be generated by analyzing the lowest cost strategy for each of the three regions. Assume that the standard will be enforced with probability 1 - |p.sup.N~. The firm will thus minimize its expected costs over |F.sub.1~:

MinE|C(|F.sub.1~)~ = (1 - |p.sup.N~) ||a|F.sub.1~.sup.2~ + b|(S* - |F.sub.1~).sup.2~~ + |p.sup.N~)(|a|F.sub.1~.sup.2~ + b|H*.sup.2~) + R|(1 - |p.sup.N~)(a|F*.sup.2~ + b|H*.sup.2~) + |p.sup.N~b|H*.sup.2~~

where R is the private or market discount rate. Solving for |F.sub.1~ yields

(8) |F.sub.1~ = F*k(|p.sup.N~) = |F.sup.I~.

The point |F.sup.I~ can be in any of the three regions. If |F.sup.I~ is in region I, that is the minimum cost point for the firm. The situation is more complicated if |F.sup.I~ lies in regions II or III. In that case, dE(C(|F.sub.1~)/d|F.sub.1~ |is greater than~ 0 for all |F.sub.1~ in region I and the firm prefers |F.sup.C~ to any other point in region I. To determine the equilibrium solution, however, it is necessary to compare the costs of |F.sup.C~ to the costs of the lowest cost points in regions II and III.

Assume that the standard will always be enforced. Then the lowest cost strategy for the firm is |F.sub.1~ = F*k(0) = F*. Thus, for region III, the lowest cost strategy is F*. It can be shown with some difficulty that there is no minimum cost point in the interior of region II. Thus, if |F.sup.I~ is greater than |F.sup.C~, the firm will choose between |F.sup.C~ and F*. This implies the firm will choose either to meet the standard or miss it by a great deal. It also implies that if the firm does not choose to meet the standard it will pick |F.sub.1~ low enough so that a type N agency will always grant relief.

IV. NUMERICAL RESULTS

The model of section III implies that credible enforcement of a time-inconsistent regulation requires a certain probability that the administering agency is committed. Table I presents the maximum |p.sup.N~ allowable for the policy to be enforced. The coefficient of first-period costs, a, is set at 1 as the numeraire. The marginal utility of the regulated good, v, is set at 1 in the computer program, although the results are invariate to all positive values of v. The private discount factor, R, equals .960.

The results in Table I indicate that |p.sup.N~ must be at a fairly low level (never higher than 0.545) for the policy to be efficiently implemented. Not surprisingly, the higher the discount rate of the agency (|R.sup.g~), the harder it is to enforce the regulation.(6) For instance, reducing |R.sup.g~ by 20 percent (from .960 to .768) reduces the maximum |p.sup.N~ from .545 to .484.

The results also show that as b, the factor of second stage costs, increases from 1 to 6 the maximum |p.sup.N~ for |R.sup.g~ = .96 decreases from .545 to .426. As b rises past 6, however, |p.sup.N~ declines only slightly. Increasing b has two effects on the viability of the regulation. Higher second-stage costs raise the costs to firm of not meeting the standard in the first stage. These higher costs, however, also make it more painful for the agency to enforce the standard should the level of first-period investment be suboptimal.

V. CONCLUSION

A large degree of administrative commitment is necessary for the effective implementation of time-inconsistent regulation. Intuitively, the time-inconsistency problem is mitigated by decreasing the potential discretion of the agency that enforces the standard. Effective enforcement of time-inconsistent regulatory standards therefore may require legislatures to generate strong administrative procedures.

Legislatures must also have the willingness to assign such policies to committed agencies. Congress may not have had that willingness in 1975 when it entrusted the regulation of automobile fuel economy standards, a regulation that appears to reduce automobile safety (see Crandall and Graham |1989~), to an agency (the National Highway and Traffic Safety Administration of the Department of Transportation) whose mission is to improve automobile safety. In light of their institutional missions, the Environmental Protection Agency or the Federal Energy Administration (the predecessor of the Department of Energy) might have been expected to be less willing to grant automobile companies regulatory relief.

It is also shown here that if a firm is not going to meet the initial part of a regulation it will miss it by a great deal to force an uncommitted agency into granting relief. There is no advantage in missing the standard by a small amount, as in that case even an uncommitted agency will enforce the standard. Firms may instead invest an intermediate amount (here defined as |F.sup.C~) and see if they can gain relief from an uncommitted agency. Thus, firms can be expected to choose to either meet time-inconsistent standards or to miss them by a great deal, holding themselves, their employees, and their customers hostage to a potentially committed administrative agency.

APPENDIX

The analysis in this appendix refers to portions of the first and second stages of the first period of the game discussed in section III.

First Period, Second Stage: Derivation of |F.sup.N~ and |F.sup.C~

The benefits from relief are realized in the first period because the reduced costs from the lower standard b|(s* - |F.sub.1~).sup.2~ - b|H*.sup.2~ are greater than the loss from the lower standard v|S* - (H* + |F.sub.1~)~ = v(F* - |F.sub.1~) if |F.sub.1~ |is less than~ F*.

(A1) Benefit of Relief = |B.sub.A~(|F.sub.1~) = |b|(S* - |F.sub.1~).sup.2~ - b|H*.sup.2~~ - |v(F* - |F.sub.1~)~.

The costs from granting relief in the first period are realized in the second period. By granting relief in the first period the agency reveals that it is uncommitted, and the firm will undertake no investment to meet the standard in the first stage of the second period (p = 1, thus (6) implies |F.sub.2~ = 0). Recall that |p.sup.E~ = |p.sub.2~(|p.sup.N~, |F.sub.1~, No relief) is the equilibrium posterior probability that the agency is uncommitted given that the agency did not grant relief in the first period. Since the costs of granting relief are not realized until the second period, they are discounted by |R.sup.g~, the agency's discount rate. The costs to the agency of granting relief, CA, are

(A2) |C.sub.A~(|p.sup.E~) = |R.sup.g~|v(F*k(|p.sup.E~) - F*k(1)~ - |R.sup.g~|a|(F*k(|p.sup.E~)).sup.2~ - a|(F*k(1)).sup.2~~ = |R.sup.g~|vF*k(|p.sup.E~) - a|(F*k(|p.sup.E~)).sup.2~~.

Setting (A1) equal to (A2) and solving for |F.sub.1~ yields the point |F.sup.N~ for which the agency is indifferent as to whether or not to show itself as type N and grant relief, given |p.sup.E~. This solution yields a quadratic, F*, plus or minus a constant. Since |F.sub.1~ is never greater than F*, the larger of the solutions for |F.sup.N~ can be disregarded and

(A3) |F.sup.N~ = F* - ||(C(|p.sup.N~) + b|F*.sup.2~ + b|H*.sup.2~ - b|S*.sup.2~ + vF*)/b~.sup..5~ = F* - |(C(|p.sup.N~)/b).sup..5~

To solve for |F.sup.C~, assume that if relief is not granted the firm believes with probability 1 that the agency is of type C. Setting (A1) equal to (A2) with |p.sup.E~ = 0 yields |F.sup.C~ = F* - |(C(0)/b).sup..5~, with |F.sup.C~ |is less than~ FN if |p.sup.N~ |is greater than~ 0.

Only a mixed strategy equilibrium in Region II:

The proof is as follows: Take any point |F.sub.1~ in region II. If the agency's strategy is to mimic a committed agency and not grant relief, then |p.sub.2~ = |p.sup.N~, as the firm knows that the type N agency will try to mimic the type C agency. If, however, |p.sub.2~ = |p.sup.N~, then by the definition of region II it is optimal for the type N agency to grant relief. In that case, however, |p.sub.2~ = 0. No pure strategy is an equilibrium for the type N agency because once it adopts that strategy the firm will have expectations in period two that make the strategy non-optimal.

Recall that |p.sup.E~ is the equilibrium value of |p.sub.2~(|F.sub.1~). In region II that equilibrium exists when the cost of relief for a type N agency is equal to the benefits of relief. As shown above, |p.sup.E~(|F.sup.C~) = 0 and |p.sup.E~(|F.sup.N~) = |p.sup.N~. The derivative of the costs with respect to |p.sup.E~, d|C.sub.A~/d|p.sup.E~, is |is less than~ 0 and d|B.sub.A~/d|F.sub.1~ |is less than~ 0. Since the cost of granting relief equals the benefits, by the definition of |p.sup.E~ in region II, it implies that d|p.sup.E~/d|F.sub.1~ |is less than~ 0 and 0 |is less than~ |p.sup.E~(|F.sub.1~) |is less than~ |p.sup.N~ if |F.sub.1~ is in the interior of region II. Solving for k using (A1) and (A2) yields

(A4) k = 1 - {1 - ||4aB(|F.sub.1~)/|v.sup.2~|R.sup.g~~.sup..5~},

and (6) implies that |p.sup.E~ =|a + b)(1 - k)~/(b(1 - k) + a).

Assume that if |F.sub.1~ is in region II the type N agency adopts a mixed strategy, granting relief with probability L(|p.sup.E~, |p.sup.N~) = (|p.sup.N~ - |p.sup.E~)/||p.sup.N~(1 - |p.sup.E~)~. If relief is not granted, the firm updates its belief according to Bayes' Law and concludes that the agency has probability |p.sup.E~ of being type N. Thus, the firm has no incentive to switch out of this strategy and an equilibrium is reached.

While a mixed strategy equilibrium is the logically correct solution for region II, it is not clear what it means in this context. A mixed strategy equilibrium implies that before the game starts, the agency is able to commit itself to granting or not granting relief at random should a region II outcome occur. Thus, while the firm does not know what type the agency is, it does know that a type N agency will have already committed itself to a mixed strategy. It is not clear that this is a realistic equilibrium, but given the absence of any other solution to this problem, it is used here.

First period, first stage: If |F.sup.I~ is in Region I, that is the minimum cost point for the firm.

The proof is as follows: Given that the probability of relief being granted is |p.sup.N~, the firm will prefer |F.sup.I~ over all other points in region I. Take any other |F.sub.1~ in regions II or III. The probability of relief being granted at any such investment |F.sub.1~ is less than |p.sup.N~. Since dE|C(|F.sub.1~)~/d|p.sup.N~ |is less than~ 0, such a point has higher costs associated with it than if it had probability of relief |p.sup.N~. By construction of |F.sup.I~, however, even if the probability of relief were |p.sup.N~ at some |F.sub.1~ in regions II and III, the firm's expected costs would be lower at |F.sup.I~. Thus, the firm will select |F.sup.I~ since it dominates any other |F.sub.1~ in regions II or III.

No minimum cost point in the interior of Region II:

The outline of the proof is as follows: The shape of the possibility frontier in region II is oval and convex to the origin, which implies the derivative of expected costs with respect to |F.sub.1~ at |F.sup.C~ is positive infinity. It also implies that dp/d|F.sub.1~ at |F.sup.N~ is zero, and thus the derivative of expected costs at that point is positive (this comes from the analysis of all points in region I). Thus, the point in region II at which the derivative of expected costs with respect to |F.sub.1~ equals zero is a local maximum and not the |F.sub.1~ which minimizes costs in that region. The two candidates for cost minimizing points in region II therefore are |F.sup.C~ and |F.sup.N~. But since |F.sup.C~ is in region I, it is weakly dominated from the firm's point of view by |F.sup.I~. Since |F.sup.N~ is in region III, it is weakly dominated by F*.

1. This type of question has also been dealt with in a macroeconomic setting. For a review of the relevant literature, see Chari and Kehoe |1990~.

2. A time-inconsistent regulation could also be undermined by the repeal of the regulation by the legislature if the desired long-term investments have not been made. The existence of complicated legislative procedures (Shepsle |1979~ and McCubbins et al. |1989, 435-44~), however, may make it very difficult to form the voting coalition necessary to repeal such a regulation. In effect, time-inconsistent regulations can be thought of as being protected from the legislature by legislative procedures in the same way they are protected from an administrative agency by administrative procedures.

3. McCubbins et al. |1989, 470-2 and 479-81~ discuss two examples of Congress choosing between agencies with different regulatory agendas. In the first instance, in 1976, the House of Representatives unsuccessfully attempted to insulate the Environmental Protection Agency (EPA) from the influence of the less regulatory oriented Department of Justice by eliminating the requirement that the Justice Department represent the EPA in federal court. This would have removed the Justice Department's de facto veto power over EPA enforcement decisions. In the second instance, in 1977, the House of Representatives wanted to give the Federal Trade Commission veto power over the EPA rules on automobile warranties. (The Trade Commission was expected to resist such warranties on antitrust grounds.) The Senate, on the other hand, desired the EPA to have full authority over the warranties in order to generate more stringent regulatory provisions. The final compromise gave the EPA full authority but limited the length of the warranties that could be required by the EPA.

4. see Weingast |1984~ for a discussion of how both these factors influenced policy at the Securities and Exchange Commission from 1955 to 1975.

5. For instance, a regulatory agency may have a great deal of difficulty evaluating the effect of new fuel-saving technologies created by automobile companies before those technologies are actually put into place.

6. The agency may or may not value the future at the same rate as private agents. For instance, if an election is imminent, the regulatory agency's political clients (the legislature and the head of the executive branch) may be much more concerned about satisfying voters today than voters tomorrow.

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Kleit, Andrew N. "The Effect of Annual Changes in Automobile Fuel Economy Standards." Journal of Regulatory Economics, June 1990, 151-73.

Kreps, D.M., and R. Wilson. "Reputation and Imperfect Information." Journal of Economic Theory, 27(2), 1982, 253-79.

Laffont, J.J., and J. Tirole. "Using Cost Observations to Regulate Firms." Journal of Political Economy, June 1986, 614-41.

McCubbins, Matthew D., Roger G. Noll, and Barry R. Weingast. "Administrative Procedures as Instruments of Political Control." Journal of Law, Economics, and Organization, Fall 1987, 243-77.

-----. "Structure and Process, Politics and Policy: Administrative Arrangements and the Political Control of Agencies." Virginia Law Review, 75(2), 1989, 431-82.

Shepsle, Kenneth A. "Institutional Arrangements and Equilibrium in Multidimensional Voting Models." American Journal of Political Science, February 1979, 27-59.

Stigler, George. "The Theory of Economic Regulation." Bell Journal of Economics, Spring 1970, 3-21.

Weingast, Barry. "The Congressional Bureaucratic System: A Principal Agent Perspective (with Applications to the SEC)." Public Choice 44(2), 1984, 147-91.

Weingast, Barry, and Mark Moran. "Bureaucratic Discretion or Congressional Control? Regulatory Policymaking by the FTC." Journal of Political Economy, October 1983, 765-800.

White, Lawrence. The Regulation of Air Pollution Emissions from Motor Vehicles. Washington, D.C.: American Enterprise Institute, 1981.

Passage of legislation is merely the beginning of the regulatory task. The successful translation of policy from legislative debate into regulation is not guaranteed. Implementation may be especially problematic for regulations that are "time-inconsistent." Regulations have the potential for time-inconsistency if the government cannot credibly commit itself to enforcement of the original legislation at every relevant point in time.

Consider a law that sets a regulatory standard to be enforced several years in the future. The regulation's implementation may take the following course for any given enforcement period. First, firms are expected to engage in long-term investment to move towards meeting the standard. Second, after that investment has been completed, an enforcement agency reviews and has the opportunity to modify the standards, shortly before the date at which they are to be enforced. Finally, after the agency's review, firms make short-term investment to meet the standard. This process may be repeated for subsequent enforcement periods.

This sequence of events presents inherent problems for the enforceability of a regulation. During the long-term investment stage regulated firms know that the relevant administrative agency may be unable to credibly bind itself to future actions. Given this, firms may believe that the agency's enforcement posture depends on their own actions; firms may therefore be able to act strategically to induce the regulatory agency to lower the relevant standard before its enforcement date. Facing such incentives, firms may not undertake the desired long-term investments. Upon review, the enforcement agency could find that requiring firms to make short-term investments to meet the standard may involve substantial social costs. Facing such costs, the agency may grant regulatory relief and relax the standard.

In these circumstances, society's welfare may be greater if firms engaged in the necessary long-term investment to meet the standard prior to the enforcement date. In order to force firms to make the necessary long-term investment, however, the enforcement agency may need to threaten to punish firms by enforcing the original standard even if firms' long-term investments are below optimal levels.

The need to threaten this punishment places the agency in a difficult position. If firms do not undertake the desired long-term investments it may be in the agency's (and society's) interests to relax the initial standard to avoid the potentially higher costs associated with short-term investments. Firms will understand this, and thus make no ex ante investment towards meeting the standard. The regulation is therefore time-inconsistent and unenforceable.

In the case of automobile fuel economy regulations, for example, Congress in 1975 set standards for model years 1978 and afterwards, but provided the executive branch, in the form of the National Highway Traffic Safety Administration (NHTSA), with discretion to modify the standard for model years after 1984. Automobile manufacturers then undertook long-term product investment that may have been initiated in order to meet the standards. For model years 1985, 1986, and 1989, however, firms were able to gain regulatory relief from NHTSA at the start of the relevant model years. If relief had not been granted, there would have been insufficient time for further product innovation, and firms would have had to meet the standard by eliminating less fuel efficient vehicles from their fleets, a strategy that would have been costly not only to themselves, but to their employees and customers as well. (See Kleit |1990~ and Greene |1991~.)

On the other hand, in the 1970s the Congress passed laws to be enforced by the Environmental Protection Agency that required automobile companies to invest millions of dollars in pollution control devices. While automobile firms achieved some delay in the implementation of the law, the devices ultimately were installed. (See White |1981~.) Pollution control devices are thus an example of a time-inconsistent standard that was eventually successfully enforced. Indeed, the only apparent difference between the enforceability of automobile fuel economy standards and automobile emission standards was the agency chosen to administer them.

This article presents a game theoretic model that demonstrates how certain types of regulatory agencies can enforce time-inconsistent regulation.(1) Administrative procedures, as discussed by McCubbins, Noll, and Weingast |1987; 1989~, allow agencies to credibly commit to policies that are consistent with their bureaucratic missions. Agencies will be modelled in terms of whether their administrative procedures have made them "committed" to enforcing a particular time-inconsistent policy under any circumstances. Because firms realize that a particular agency might be forced by its own administrative procedures into acting against the public welfare, potentially time-inconsistent regulation may be enforceable. This model stands in contrast to the previous literature on regulatory enforcement, such as Laffont and Tirole |1986~, which does not deal with issues of commitment.

II. THE ADMINISTRATIVE INSTITUTIONS

The analysis here assumes that legislation is enacted setting a regulatory standard. The legislation, however, also gives the administrating agency authority to lower the standard should some unforeseen event occur. As McCubbins et al. |1987, 256-7 and 261; 1989, 449~ and Hahn |1988, 224-5~ point out, if a large degree of technological uncertainty is involved, the legislature may not desire to choose an exact standard to be reached, given the possibility that relevant information is not yet available. Thus, the legislature may grant the agency the discretion to modify future standards.

This leaves the legislature with the problem of how to constrain the agency to enforce its decisions. Even if the administrative agency agrees with the legislative branch, and neither one's opinions change over time, the temporal nature of the regulation, by itself, may generate time inconsistency and prevent the policy from being implemented. Therefore, the legislature must succeed in committing the agency to enforcing the standard even if the desired investments do not take place.

Conceptually, the legislature or the agency could solve this problem by creating for itself a reputation for "toughness" or "irrationality." Creating such a reputation, however, at least in the models of Kreps and Wilson |1982~ and Diamond |1989~, takes a good deal of time, given the low levels of irrationality expected from economic actors. Since elected government officials serve for a finite period, they may find that making the investment needed to create a reputation for toughness is not a viable option.

As McCubbins et al. |1987, 257-8~ describe, a method by which the legislature can bring an administrative agency under control and solve the implementation problem is through the use of administrative procedures. Procedural requirements, often quite complex, restrict the choices available to an administrative agency and ensure that legislative desires are implemented by creating a credible commitment, even without direct legislative supervision. In the context of time-inconsistent policy, administrative procedures are presented here as a solution to the traditional principal agent problem, not only across political actors (as McCubbins and his coauthors describe), but across time as well.(2)

Of course, all administrative agencies are not alike. In an agency's enabling statute the legislature may be able to implicitly choose its preferred level of administrative commitment. The legislature may also be able to select from among a variety of existing agencies in which to entrust the enforcement of a particular statute, depending on the level of commitment it desires. The commitment of a particular agency to a particular regulation can be thought of as a function of its own internal procedures, as well as how consistent the particular regulation is with that agency's political mission.(3) In turn, the agency's mission may be a function of the concerns of the agency's client interest groups (as in Stigler |1970~) or the political interests of the members of the relevant congressional oversight committees (along the lines of Weingast and Moran |1983~).(4) Thus NHTSA, as its name implies, may be more likely to be committed to regulations that improve automobile safety and less likely to be committed to regulations that do the reverse. Firms can be expected to understand this and to take it into account when making their regulatory investment decisions.

III. THE REGULATORY GAME

Prior to the start of the game, the legislature passes a law setting the standard at S. It will be assumed that S is set at the welfare-maximizing level (S*), given that the regulation is successfully enforced. To capture the often complicated administrative procedures established, the regulated firm is modeled here as unaware as to whether the relevant enforcement agency is "committed" (type = C) and thus unable to modify the standard, or "uncommitted" (type = N) and therefore able to do so. Under the bureaucratic "rules of the game," however, the firm knows the probability |p.sup.N~, 0 |is less than~ |p.sup.N~ |is less than~ 1, that the agency is uncommitted and willing to lower the standard. This probability is a function of how consistent the relevant regulation is with an agency's mission and how strongly the agency's administrative procedures commit it to that mission. The firm updates this probability during the game according to Bayes' Law.

The game itself has two periods, t = 1, 2. Allowing the game to be longer than one period generates incentives for an uncommitted agency to act strategically in order to generate a reputation for toughness. This in turn will be shown to generate a discontinuity in the firm's optimal strategy. Each period has two stages, permitting analysis of the two decisions made in each period, one by the firm and one by the agency. In the first stage of period t the firm

improves the regulated aspect of its product an amount |F.sub.t~ at some cost to itself. The firm chooses |F.sub.t~ to minimize the present value of its expected costs over the two-period game, given |p.sup.N~ and any previous events in the game. In the second stage, after examining the first-stage product improvements |F.sub.t~, the reviewing administrative agency sets a standard |S.sub.t~, which may be equal to or less than the mandated standard S*. If the agency is uncommitted, it sets the standard to maximize net welfare given first-stage investment, knowing that the firm will take the existence of an amended standard in period one into account when making its first-stage investment in period two. A committed agency sets |S.sub.t~ = S*. After the agency makes its decision, the firm has no choice but to improve its product |H.sub.t~ = |S.sub.t~ - |F.sub.t~ |is greater than~ 0 at some cost to reach the standard.

In terms of automobile fuel economy standards, first-stage improvements |F.sub.t~ can be thought of as long-term engineering innovation and investment affecting the fuel efficiency of individual automobiles. Second-stage improvements |H.sub.t~ can be thought of as short-term changes in the mix of vehicles offered for sale that generate relatively higher sales of the more fuel-efficient vehicles.

For simplicity the cost function to the (risk-neutral) firm in period t is assumed to be

|Mathematical Expression Omitted~

where in general b |is greater than~ a. The firm is assumed to gain no benefit from the production of this good, and hence without regulation it would not be produced.

Society is assumed to value the regulated good with constant marginal utility v. Solving for S* and dropping subscripts, the net total social welfare from reaching a level of the regulated good S* = F* + H* where F is generated from first-stage long-term investment and H is the second-stage short-term investment is maximized at

(2) F* = v/2a,H* = v/2b,S* = F* + H* = v(a+b)/2ab.

It is also assumed that the agency cannot observe |F.sub.t~ prior to the second stage of period t.(5) A sequential equilibrium to this game will be solved using backwards induction.

Second Period: Second Stage

If the agency is of type C, it is unable to modify the legislature's initial decision and |S.sub.2~ = S*. The firm then improves its product |H.sub.2~ = S* - |F.sub.2~. If the agency is of type N, it maximizes welfare over |S.sub.2~ given |F.sub.2~:

(3) W = v|S.sub.2~ - |a|F.sub.2~.sup.2~ - b|(|S.sub.2~ - |F.sub.2~).sup.2~.

Maximizing W with respect to S yields

(4) |S.sub.2~ - |F.sub.2~ = v/2b = |H.sub.2~ = H*,

with the value of |H.sub.2~ independent of |F.sub.2~. The firm then improves its product |H.sub.2~ = H*.

Second Period: First Stage

The goal of the regulated firm at this point is to minimize its expected costs over the probability that the agency is of type N. That probability, |p.sub.2~, is a function of |p.sup.N~, the a priori probability that the agency is of type N, and the events of the first period. If the agency did intervene in period one it has, in effect, "shown its hand" and demonstrated with probability 1 that it is of type N. Therefore, |p.sub.2~ is equal to 1 if intervention occurred in period one; otherwise |p.sub.2~ = |p.sub.2~(|p.sup.N~, |6F.sub.1~, |S.sub.1~) = |p.sub.2~ (|p.sup.N~, |F.sub.1~). (|S.sub.1~ equals S* if intervention did not occur.) The firm thus knows that given its first-stage improvements |F.sub.2~ it will have to improve its good H* with probability |p.sub.2~ and face probability 1 - |p.sub.2~ that its good will have to be improved S* - |F.sub.2~. Given this, the firm minimizes its expected costs over |F.sub.2~,

(5) Min E(|C.sub.2~) = |p.sub.2~(|a|F.sub.2~.sup.2~ + b|H*.sup.2~) + (1 - |p.sub.2~)||a|F.sub.2~.sup.2~ + b|(S* - |F.sub.2~).sup.2~~.

Minimizing (5) with respect to |F.sub.2~ and solving yields

(6) |F.sub.2~ = (1 - |p.sub.2~)bS*/|a + (1 - |p.sub.2~)b~ = F*(a + b)(1 - |p.sub.2~)/|a + b(1 - |p.sub.2~)~ = F*k(|p.sub.2~).

The function k is the fraction of optimal investment undertaken by the firm given the firm's posterior probability that the agency is type N (uncommitted) with k(1) = 0, k(0) = 1, and dk/d|p.sub.2~ |is less than~ 0.

First Period: Second Stage

A type C agency will set the standard at S*, and the firm will have no choice but to improve its produce S* - |F.sub.1~. If |F.sub.1~ |is less than~ F* and the agency is of type N, it will grant relief (setting the standard at |F.sub.1~ + H* = |S.sub.1~ |is less than~ S*) if the benefits of doing so outweigh the costs.

Define |p.sup.E~ = |p.sub.2~(|p.sup.N~, |F.sub.1~, No relief) as the equilibrium posterior probability that the agency is uncommitted given that the agency did not grant relief in the first period. Let |F.sup.N~ be the first-period, stage-one investment at which the agency is indifferent between granting relief and enforcing the original standard, given that |p.sup.E~ = |p.sup.N~. (See the appendix for the derivation of |F.sup.N~.) At |F.sup.N~ the firm is not "fooled" by enforcement of the original standard since the firm believes, given |F.sub.1~, that an uncommitted agency would not have granted relief, since doing so would have revealed to the firm that it was uncommitted.

Define region III as ||F.sup.N~, F*~. By definition if |F.sub.1~ is in region III, the agency will not grant relief, since the costs of granting relief are greater than the benefits. If relief is not granted when |F.sub.1~ is in region III, the firm has no reason to believe |p.sub.2~ |is greater than~ |p.sup.N~. Thus, if |F.sub.1~ is in region III, it always is optimal for a type N agency to mimic a type C committed agency and not grant relief.

Define region I as |0, |F.sup.C~~ where |F.sup.C~ is the maximum |F.sub.1~ such that the agency will want to grant relief, given that if relief is not granted the firm will have |p.sub.2~ = 0. (See the appendix for the derivation of |F.sup.C~.) By definition, if |F.sub.1~ is in region I, a type N agency will always grant relief. Here the benefits of granting relief are always greater than the costs, even though if relief is not granted the firm believes with probability 1 that the agency is of type C (committed).

Define region II as ||F.sup.C~, |F.sup.N~~. If |F.sub.1~ is in this region, no pure strategy equilibrium exists for an uncommitted agency. There is, however, a mixed strategy solution similar to the one derived by Kreps and Wilson |1982~ (see the appendix). Once the agency makes its decision, the firm improves its product |S.sub.1~ - |F.sub.1~.

First Period: First Stage

Given the strategies calculated above, the firm now minimizes its expected costs by choosing |F.sub.1~. If it chooses |F.sub.1~ in region III, it knows that relief will be granted with probability 0. If it chooses |F.sub.1~ in region II, it knows relief will be granted with probability |p.sup.N~L||p.sup.E~(|F.sub.1~), |p.sup.N~~. If it chooses |F.sub.1~ in region I, it knows relief will occur with probability |p.sup.N~. The firm thus faces the possibility frontier outlined in Figure 1.

A solution to the firm's cost-minimization problem can be generated by analyzing the lowest cost strategy for each of the three regions. Assume that the standard will be enforced with probability 1 - |p.sup.N~. The firm will thus minimize its expected costs over |F.sub.1~:

MinE|C(|F.sub.1~)~ = (1 - |p.sup.N~) ||a|F.sub.1~.sup.2~ + b|(S* - |F.sub.1~).sup.2~~ + |p.sup.N~)(|a|F.sub.1~.sup.2~ + b|H*.sup.2~) + R|(1 - |p.sup.N~)(a|F*.sup.2~ + b|H*.sup.2~) + |p.sup.N~b|H*.sup.2~~

where R is the private or market discount rate. Solving for |F.sub.1~ yields

(8) |F.sub.1~ = F*k(|p.sup.N~) = |F.sup.I~.

The point |F.sup.I~ can be in any of the three regions. If |F.sup.I~ is in region I, that is the minimum cost point for the firm. The situation is more complicated if |F.sup.I~ lies in regions II or III. In that case, dE(C(|F.sub.1~)/d|F.sub.1~ |is greater than~ 0 for all |F.sub.1~ in region I and the firm prefers |F.sup.C~ to any other point in region I. To determine the equilibrium solution, however, it is necessary to compare the costs of |F.sup.C~ to the costs of the lowest cost points in regions II and III.

Assume that the standard will always be enforced. Then the lowest cost strategy for the firm is |F.sub.1~ = F*k(0) = F*. Thus, for region III, the lowest cost strategy is F*. It can be shown with some difficulty that there is no minimum cost point in the interior of region II. Thus, if |F.sup.I~ is greater than |F.sup.C~, the firm will choose between |F.sup.C~ and F*. This implies the firm will choose either to meet the standard or miss it by a great deal. It also implies that if the firm does not choose to meet the standard it will pick |F.sub.1~ low enough so that a type N agency will always grant relief.

IV. NUMERICAL RESULTS

The model of section III implies that credible enforcement of a time-inconsistent regulation requires a certain probability that the administering agency is committed. Table I presents the maximum |p.sup.N~ allowable for the policy to be enforced. The coefficient of first-period costs, a, is set at 1 as the numeraire. The marginal utility of the regulated good, v, is set at 1 in the computer program, although the results are invariate to all positive values of v. The private discount factor, R, equals .960.

The results in Table I indicate that |p.sup.N~ must be at a fairly low level (never higher than 0.545) for the policy to be efficiently implemented. Not surprisingly, the higher the discount rate of the agency (|R.sup.g~), the harder it is to enforce the regulation.(6) For instance, reducing |R.sup.g~ by 20 percent (from .960 to .768) reduces the maximum |p.sup.N~ from .545 to .484.

TABLE I Maximum |p.sup.N~ That Enforces Standard a=1, v=1, R=.96 Agency Discount Rate |R.sup.g~ .960 .768 .576 .384 .192 1.0 .545 .484 .411 .320 .200 2.0 .466 .414 .351 .273 .168 3.0 .443 .393 .333 .258 .159 4.0 .433 .384 .326 .252 .154 5.0 .428 .380 .322 .249 .152 6.0 .426 .379 .321 .248 .151 7.0 .426 .378 .320 .247 .150 8.0 .425 .378 .320 .247 .150 9.0 .425 .378 .320 .247 .150 10.0 .426 .379 .321 .247 .149 Factor of 3rd Stage Costs(b)

The results also show that as b, the factor of second stage costs, increases from 1 to 6 the maximum |p.sup.N~ for |R.sup.g~ = .96 decreases from .545 to .426. As b rises past 6, however, |p.sup.N~ declines only slightly. Increasing b has two effects on the viability of the regulation. Higher second-stage costs raise the costs to firm of not meeting the standard in the first stage. These higher costs, however, also make it more painful for the agency to enforce the standard should the level of first-period investment be suboptimal.

V. CONCLUSION

A large degree of administrative commitment is necessary for the effective implementation of time-inconsistent regulation. Intuitively, the time-inconsistency problem is mitigated by decreasing the potential discretion of the agency that enforces the standard. Effective enforcement of time-inconsistent regulatory standards therefore may require legislatures to generate strong administrative procedures.

Legislatures must also have the willingness to assign such policies to committed agencies. Congress may not have had that willingness in 1975 when it entrusted the regulation of automobile fuel economy standards, a regulation that appears to reduce automobile safety (see Crandall and Graham |1989~), to an agency (the National Highway and Traffic Safety Administration of the Department of Transportation) whose mission is to improve automobile safety. In light of their institutional missions, the Environmental Protection Agency or the Federal Energy Administration (the predecessor of the Department of Energy) might have been expected to be less willing to grant automobile companies regulatory relief.

It is also shown here that if a firm is not going to meet the initial part of a regulation it will miss it by a great deal to force an uncommitted agency into granting relief. There is no advantage in missing the standard by a small amount, as in that case even an uncommitted agency will enforce the standard. Firms may instead invest an intermediate amount (here defined as |F.sup.C~) and see if they can gain relief from an uncommitted agency. Thus, firms can be expected to choose to either meet time-inconsistent standards or to miss them by a great deal, holding themselves, their employees, and their customers hostage to a potentially committed administrative agency.

APPENDIX

The analysis in this appendix refers to portions of the first and second stages of the first period of the game discussed in section III.

First Period, Second Stage: Derivation of |F.sup.N~ and |F.sup.C~

The benefits from relief are realized in the first period because the reduced costs from the lower standard b|(s* - |F.sub.1~).sup.2~ - b|H*.sup.2~ are greater than the loss from the lower standard v|S* - (H* + |F.sub.1~)~ = v(F* - |F.sub.1~) if |F.sub.1~ |is less than~ F*.

(A1) Benefit of Relief = |B.sub.A~(|F.sub.1~) = |b|(S* - |F.sub.1~).sup.2~ - b|H*.sup.2~~ - |v(F* - |F.sub.1~)~.

The costs from granting relief in the first period are realized in the second period. By granting relief in the first period the agency reveals that it is uncommitted, and the firm will undertake no investment to meet the standard in the first stage of the second period (p = 1, thus (6) implies |F.sub.2~ = 0). Recall that |p.sup.E~ = |p.sub.2~(|p.sup.N~, |F.sub.1~, No relief) is the equilibrium posterior probability that the agency is uncommitted given that the agency did not grant relief in the first period. Since the costs of granting relief are not realized until the second period, they are discounted by |R.sup.g~, the agency's discount rate. The costs to the agency of granting relief, CA, are

(A2) |C.sub.A~(|p.sup.E~) = |R.sup.g~|v(F*k(|p.sup.E~) - F*k(1)~ - |R.sup.g~|a|(F*k(|p.sup.E~)).sup.2~ - a|(F*k(1)).sup.2~~ = |R.sup.g~|vF*k(|p.sup.E~) - a|(F*k(|p.sup.E~)).sup.2~~.

Setting (A1) equal to (A2) and solving for |F.sub.1~ yields the point |F.sup.N~ for which the agency is indifferent as to whether or not to show itself as type N and grant relief, given |p.sup.E~. This solution yields a quadratic, F*, plus or minus a constant. Since |F.sub.1~ is never greater than F*, the larger of the solutions for |F.sup.N~ can be disregarded and

(A3) |F.sup.N~ = F* - ||(C(|p.sup.N~) + b|F*.sup.2~ + b|H*.sup.2~ - b|S*.sup.2~ + vF*)/b~.sup..5~ = F* - |(C(|p.sup.N~)/b).sup..5~

To solve for |F.sup.C~, assume that if relief is not granted the firm believes with probability 1 that the agency is of type C. Setting (A1) equal to (A2) with |p.sup.E~ = 0 yields |F.sup.C~ = F* - |(C(0)/b).sup..5~, with |F.sup.C~ |is less than~ FN if |p.sup.N~ |is greater than~ 0.

Only a mixed strategy equilibrium in Region II:

The proof is as follows: Take any point |F.sub.1~ in region II. If the agency's strategy is to mimic a committed agency and not grant relief, then |p.sub.2~ = |p.sup.N~, as the firm knows that the type N agency will try to mimic the type C agency. If, however, |p.sub.2~ = |p.sup.N~, then by the definition of region II it is optimal for the type N agency to grant relief. In that case, however, |p.sub.2~ = 0. No pure strategy is an equilibrium for the type N agency because once it adopts that strategy the firm will have expectations in period two that make the strategy non-optimal.

Recall that |p.sup.E~ is the equilibrium value of |p.sub.2~(|F.sub.1~). In region II that equilibrium exists when the cost of relief for a type N agency is equal to the benefits of relief. As shown above, |p.sup.E~(|F.sup.C~) = 0 and |p.sup.E~(|F.sup.N~) = |p.sup.N~. The derivative of the costs with respect to |p.sup.E~, d|C.sub.A~/d|p.sup.E~, is |is less than~ 0 and d|B.sub.A~/d|F.sub.1~ |is less than~ 0. Since the cost of granting relief equals the benefits, by the definition of |p.sup.E~ in region II, it implies that d|p.sup.E~/d|F.sub.1~ |is less than~ 0 and 0 |is less than~ |p.sup.E~(|F.sub.1~) |is less than~ |p.sup.N~ if |F.sub.1~ is in the interior of region II. Solving for k using (A1) and (A2) yields

(A4) k = 1 - {1 - ||4aB(|F.sub.1~)/|v.sup.2~|R.sup.g~~.sup..5~},

and (6) implies that |p.sup.E~ =|a + b)(1 - k)~/(b(1 - k) + a).

Assume that if |F.sub.1~ is in region II the type N agency adopts a mixed strategy, granting relief with probability L(|p.sup.E~, |p.sup.N~) = (|p.sup.N~ - |p.sup.E~)/||p.sup.N~(1 - |p.sup.E~)~. If relief is not granted, the firm updates its belief according to Bayes' Law and concludes that the agency has probability |p.sup.E~ of being type N. Thus, the firm has no incentive to switch out of this strategy and an equilibrium is reached.

While a mixed strategy equilibrium is the logically correct solution for region II, it is not clear what it means in this context. A mixed strategy equilibrium implies that before the game starts, the agency is able to commit itself to granting or not granting relief at random should a region II outcome occur. Thus, while the firm does not know what type the agency is, it does know that a type N agency will have already committed itself to a mixed strategy. It is not clear that this is a realistic equilibrium, but given the absence of any other solution to this problem, it is used here.

First period, first stage: If |F.sup.I~ is in Region I, that is the minimum cost point for the firm.

The proof is as follows: Given that the probability of relief being granted is |p.sup.N~, the firm will prefer |F.sup.I~ over all other points in region I. Take any other |F.sub.1~ in regions II or III. The probability of relief being granted at any such investment |F.sub.1~ is less than |p.sup.N~. Since dE|C(|F.sub.1~)~/d|p.sup.N~ |is less than~ 0, such a point has higher costs associated with it than if it had probability of relief |p.sup.N~. By construction of |F.sup.I~, however, even if the probability of relief were |p.sup.N~ at some |F.sub.1~ in regions II and III, the firm's expected costs would be lower at |F.sup.I~. Thus, the firm will select |F.sup.I~ since it dominates any other |F.sub.1~ in regions II or III.

No minimum cost point in the interior of Region II:

The outline of the proof is as follows: The shape of the possibility frontier in region II is oval and convex to the origin, which implies the derivative of expected costs with respect to |F.sub.1~ at |F.sup.C~ is positive infinity. It also implies that dp/d|F.sub.1~ at |F.sup.N~ is zero, and thus the derivative of expected costs at that point is positive (this comes from the analysis of all points in region I). Thus, the point in region II at which the derivative of expected costs with respect to |F.sub.1~ equals zero is a local maximum and not the |F.sub.1~ which minimizes costs in that region. The two candidates for cost minimizing points in region II therefore are |F.sup.C~ and |F.sup.N~. But since |F.sup.C~ is in region I, it is weakly dominated from the firm's point of view by |F.sup.I~. Since |F.sup.N~ is in region III, it is weakly dominated by F*.

1. This type of question has also been dealt with in a macroeconomic setting. For a review of the relevant literature, see Chari and Kehoe |1990~.

2. A time-inconsistent regulation could also be undermined by the repeal of the regulation by the legislature if the desired long-term investments have not been made. The existence of complicated legislative procedures (Shepsle |1979~ and McCubbins et al. |1989, 435-44~), however, may make it very difficult to form the voting coalition necessary to repeal such a regulation. In effect, time-inconsistent regulations can be thought of as being protected from the legislature by legislative procedures in the same way they are protected from an administrative agency by administrative procedures.

3. McCubbins et al. |1989, 470-2 and 479-81~ discuss two examples of Congress choosing between agencies with different regulatory agendas. In the first instance, in 1976, the House of Representatives unsuccessfully attempted to insulate the Environmental Protection Agency (EPA) from the influence of the less regulatory oriented Department of Justice by eliminating the requirement that the Justice Department represent the EPA in federal court. This would have removed the Justice Department's de facto veto power over EPA enforcement decisions. In the second instance, in 1977, the House of Representatives wanted to give the Federal Trade Commission veto power over the EPA rules on automobile warranties. (The Trade Commission was expected to resist such warranties on antitrust grounds.) The Senate, on the other hand, desired the EPA to have full authority over the warranties in order to generate more stringent regulatory provisions. The final compromise gave the EPA full authority but limited the length of the warranties that could be required by the EPA.

4. see Weingast |1984~ for a discussion of how both these factors influenced policy at the Securities and Exchange Commission from 1955 to 1975.

5. For instance, a regulatory agency may have a great deal of difficulty evaluating the effect of new fuel-saving technologies created by automobile companies before those technologies are actually put into place.

6. The agency may or may not value the future at the same rate as private agents. For instance, if an election is imminent, the regulatory agency's political clients (the legislature and the head of the executive branch) may be much more concerned about satisfying voters today than voters tomorrow.

REFERENCES

Chari, V. V., and Patrick J. Kehoe. "Sustainable Plans." Journal of Political Economy, August 1990, 783-802.

Crandall, Robert W., and John D. Graham. "The Effect of Automobile Fuel Economy Standards on Safety." Journal of Law and Economics, April 1989, 97-118.

Diamond, Douglass W. "Reputation and Acquisition in Debt Markets." Journal of Political Economy, August 1989, 828-62.

Greene, David L. "Short-Run Pricing Strategies to Increase Corporate Average Fuel Economy." Economic Inquiry, January 1991, 101-14.

Hahn, Robert W. "An Evaluation of Options for Reducing Hazardous Waste." Harvard Environmental Law Review 12(2), 1988, 201-30.

Kleit, Andrew N. "The Effect of Annual Changes in Automobile Fuel Economy Standards." Journal of Regulatory Economics, June 1990, 151-73.

Kreps, D.M., and R. Wilson. "Reputation and Imperfect Information." Journal of Economic Theory, 27(2), 1982, 253-79.

Laffont, J.J., and J. Tirole. "Using Cost Observations to Regulate Firms." Journal of Political Economy, June 1986, 614-41.

McCubbins, Matthew D., Roger G. Noll, and Barry R. Weingast. "Administrative Procedures as Instruments of Political Control." Journal of Law, Economics, and Organization, Fall 1987, 243-77.

-----. "Structure and Process, Politics and Policy: Administrative Arrangements and the Political Control of Agencies." Virginia Law Review, 75(2), 1989, 431-82.

Shepsle, Kenneth A. "Institutional Arrangements and Equilibrium in Multidimensional Voting Models." American Journal of Political Science, February 1979, 27-59.

Stigler, George. "The Theory of Economic Regulation." Bell Journal of Economics, Spring 1970, 3-21.

Weingast, Barry. "The Congressional Bureaucratic System: A Principal Agent Perspective (with Applications to the SEC)." Public Choice 44(2), 1984, 147-91.

Weingast, Barry, and Mark Moran. "Bureaucratic Discretion or Congressional Control? Regulatory Policymaking by the FTC." Journal of Political Economy, October 1983, 765-800.

White, Lawrence. The Regulation of Air Pollution Emissions from Motor Vehicles. Washington, D.C.: American Enterprise Institute, 1981.

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Author: | Kleit, Andrew N. |
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Publication: | Economic Inquiry |

Date: | Oct 1, 1992 |

Words: | 5941 |

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