The RAND Corporation
Trigger Price Regulation
Author(s): David J. Salant and Glenn A. Woroch
Source: The RAND Journal of Economics, Vol. 23, No. 1 (Spring, 1992), pp. 29-51
Published by: Blackwell Publishing on behalf of The RAND Corporation
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RAND Journal of Economics
Vol. 23, No. 1, Spring 1992
Trigger price regulation
David J. Salant*
and
Glenn A. Woroch *
We consider the difficulty of achieving efficient prices and investments when returns on a
public utility's projects are vulnerable to opportunisticratemaking. We model the long-term
relationship between a firm and its regulator as a time-dependent supergame in which the
regulator sets price ceilings to maximize surplus and the firm invests to maximize profit.
Wefind history-dependentstrategiesthat supportself-enforcing,mutually beneficialequilibria.
Equilibriumpayoffs are close to the planning solution provided interestrates are small enough
and capital depreciatesfast enough. We concentrate on "triggerprice regulation" where, in
response to inefficient behavior, the regulator cuts price down to operating cost and thefirm
curtails investment. This mechanism performs well even withproduction economies and with
restrictions on players' threats.
1. Introduction
* A long-term relationship develops between a public utility and its customers when each
must commit resources for an extended period. The firm invests in specialized, immobile
plant and equipment while utility customers make complementary investments of
their own.
Several authors observed that during the course of the relationship, returns on such
investments are vulnerable to "opportunistic behavior." I The firm may refuse to expand
capacity to meet growing demand or even to replace worn-out plant, allowing service quality
to deteriorate. Attempting to serve the interests of consumers, a regulatory agency may
disallow recovery of "imprudent" investments or engage in regulatory lag and excessively
slow depreciation.
* GTE Laboratories Incorporated.
We are grateful to Tom Lyon, Stan Reynolds, Bob Rosenthal, and Ingo Vogelsang for detailed comments.
Discussionswith Jim Friedman, Roy Radner,and Mike Toman were also valuable.Bill Taylorgave helpful comments
on an earlierversion that was presented at the biennial conference of the InternationalTelecommunications Society,
June 1988 in Cambridge, Mass. The views expressed in this article are those of the authors and should not be
attributedto GTE Corporation or any of its subsidiaries.
' Williamson ( 1975 ) and Klein, Crawford,and Alchian ( 1978 ) study the incentive to appropriatequasi-rents
in a world of perfect information. Opportunism also arises when, despite promises to the contrary, a regulator
incorporatesthe firm's private information about technology and demand revealed through its operation. For details
see Baron (1988).
29
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THE RAND JOURNAL OF ECONOMICS
Investment incentives might be restored if actions of the firm and the regulator are
policed by outside parties. Legislaturesand courts do constrain behavior of the participants,
but limited information makes their control highly imperfect. Whatever their range of discretion, all parties will continue to pursue immediate gains at the expense of long-term
efficiency.
The length of the firm-regulatorrelationship plays a crucial role in this story. With a
finite horizon, agreements will invariably unravel because the temptation for opportunism
just before the relationship dissolves is too strong. By induction, the same holds for each of
the preceding periods.
The conclusions change when the relationship lasts indefinitely. In that case, results
from the study of repeated games suggest that the firm and its customers can secure the
mutual benefits forgone through shortsightedness. The longer the relationship lasts, the
greaterthe potential losses and, therefore, the greaterthe rewardsto continued cooperation.
If players can credibly threaten to punish each other severely, then most any outcome
superior to the static equilibrium could be sustained without outside intervention.
Applying the insight of this Folk Theorem to the firm-regulator relationship is the
principal goal of this article. We seek price and investment plans that are mutually beneficial
for investors and consumers and at the same time self-enforcing.
To illustrate how this logic applies to regulation, consider a simple example in which
a one-time sunk investment will supply a service at any scale.2 Assume away physical depreciation and demand growth so that no additional capital outlays are necessary. Left to
its own, the firm will undertake this venture only as long as monopoly pricing over its
lifetime covers its capital cost.
The dead-weight loss that arisesjustifies some form of rate regulation. First-best prices
should equal marginal cost, which in this case is zero. To recoup the initial outlay, secondbest prices call for a constant mark-up each period.
It is unclear whether a regulatory agency will adhere to these prices as time unfolds.
Staff turnover and political crises weaken its resolve to fulfill past promises. The agency can
please current consumers by cutting price, perhaps to a level that just keeps the firm in
business. Exercising a little foresight, a profit-seeking enterprise will never invest in the
first place.
Prospects for better outcomes improve when this relationship continues indefinitely.
At the extreme, a regulator must consider the loss to an infinite generation of consumers
resulting from the firm's reaction to a price cut. This sum can be large relative to the efficient
level if the weight attached to distant generations is not too small. In the case of a perfectly
durable, lump-sum investment, the regulator does not care because the firm will continue
to produce at any nonnegative price.
Investment need not be so lumpy, however. When capacity can be added smoothly,
the firm and regulator could agree on a candidate price-investment path. By accumulating
capacity gradually, the firm holds back some investment. Should it ever reach the efficient
capacity, it would lose all leverage when capital is perfectly durable. To deter opportunism
by the regulator, it must approach a limit that falls short of the efficient level. In this way,
the firm and regulator strike a balance between the benefits of continued expansion and
the temptation to appropriate returns on existing capacity.
When capital depreciates, repeated investments are needed to replace worn-out plant
and equipment-even if investment is lumpy. Then, whether the efficient solution could
be supported as an equilibrium depends on the speed of depreciation. The faster capital
depreciates, the less the regulator can gain from confiscatory rates. Thus, neither the de-
2
This case is examined in closer detail in Salant and Woroch ( 1991 ).
SALANT AND WOROCH
/
31
preciation rate nor the discount factor can be too small if the firm and regulator are to
voluntarily cooperate.
We model this relationship as a time-dependent supergame played by a profit-maximizing enterpriseand a surplus-maximizingregulator.Startingwith an initial stock of capital,
the utility invests in durable plant and equipment to provide a single service. Capacity
additions incur an adjustment cost that favors gradual accumulation. Furthermore, all capacity costs are assumed to be sunk.
As a benchmark, we solve the planning problem of maximizing discounted consumer
surplus while ensuring the firm breaks even. The solution is a dynamic version of the
Ramsey-Boiteuxpricing rule together with an investment path approachinga limiting capital
stock. Much of the article is concerned with how closely this solution can be approximated
by a noncooperative equilibrium.
We first look at the set of equilibria when the firm and regulator must each commit at
the outset to complete investment and price paths. We find that the rigidity imposed by
such commitments prevents any benefits from cooperation being attained: at the unique
equilibrium the regulator never allows any profit, so no investment ever occurs.
We then consider the possibility that both sides can, at any date, alterprice or investment
plans. In that case, an equilibrium exists in which the regulator follows the efficient price
path as long as the firm invests efficiently. Any deviation "triggers" a punishment: the
regulatorcuts price down to operating expense and the firm curtails investment.
In general, the more that players can "punish" each other in this way, the greater is
their incentive to cooperate. Indeed, the lower the regulator can set price and the more the
firm can divest, the larger is the set of equilibrium outcomes that trigger strategies can
sustain. In particular, we find that, for a sufficiently small interest rate and depreciation
rate, trigger strategies support the planning solution.
A linear example illustrates the triggermechanism and partially characterizesthe equilibrium set. The example confirms that the effectiveness of trigger price strategies depends
on values of initial capacity in addition to the interest and depreciation rates.
Several novel implications follow from our study of the firm-regulatorrelationship that
apply to the design of regulatory institutions. A first step toward cooperation is the creation
of conditions conducive to free and active bargainingbetween the firm and regulator. Next,
by raisingthe damage from defecting from the negotiated outcome, regulatoryreform makes
better outcomes possible. This argues for such unorthodox prescriptions as a relaxation of
the ban on confiscatory rates and the removal of minimum standards for service quality.
2. The model
* Production and consumption occur at an infinite number of discrete points in time.
The firm produces a single, perishable product with a known technology. Except for the
capital accumulation equation, demand and cost conditions are unchanging and separable
across periods.3
0 Demand. Let Pt and qt be the actual price and quantity in period t. Price is bounded
above by a ceiling ^t. Stationary demand is Q(p), from which we derive the inverse demand
P(q). Consumer surplus is
S(p, q)
=
JP(x)dx-pq.
(1)
I The analysis can accommodate systematic demand growth or technological progress or even small perturbations around the stationary path (see Salant and Woroch ( 1992)). Arbitraryswings in demand alter the nature
of the planning solution.
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THE RAND JOURNAL OF ECONOMICS
When the price charged exactly equals the demand price (i.e., p = P(q)), (1) becomes the
usual Marshallian consumer surplus, S(q)
S(P(q), q), where S'(q) = -qP'(q) and
S"(q) = - [P'(q) + qP"(q)] are both assumed to be nonpositive.
0
Production and costs. Before production can begin, the firm sinks an entry cost of F
that is independent of the scale of operation. This figure includes expenditures on planning,
site preparation, and any necessary permits and licenses.
Production requires variable factors and durable capital. For expositional simplicity
we assume a constant unit variable cost of v ? O.' Later we allow average variable costs to
fall. Either way, output is constrained by the amount of installed capacity: qt ? qt.
The firm begins with initial capacity & ? 0, which depreciates at a geometric rate 5.If
gross investment in period t is I,, next period's capacity is just qt = It + q't-1, where
y = I - 6 is the rate at which capital survives. Investment is irreversible (i.e., I, 2 0).
Construction costs in period t are C(It), where C(0) = 0 and C(* ) is a convex, increasing function. To rule out cases where investment is never worthwhile, we require that,
when starting from scratch, the marginal benefit of capacity exceeds its marginal cost:
S'(O) > C'(O).
Convexity of construction costs is not inconsistent with natural monopoly.5 Although
unit costs of building capacity may be rising, its durability reduces cost of production in
the future. Such scope economies give an incumbent firm an advantage in production over
multiple periods. Furthermore, the sunkness of investment confers a first-mover advantage
that may be insurmountable by subsequent entrants. Combined, sunkness and durability
of capital create a natural monopoly that warrants rate regulation.
o The objective functions. All parties-the firm, consumers, and the regulator-use a
constant market interest rate r to discount the future by the factor p = 1/(1 + r).
The firm's owners are solely interested in the stream of investment returns. In the short
run, the cash flow is just 7rt= (pt - v)qt - C(It). Over the long run, they evaluate the firm's
performance according to discounted profits:
00
ll(p,
q,
A;
r, 8)
-
= z
pt((pt
t=
- v)qt -
C( q~t-
yt-i)),
(2)
A
where we have made the substitution It = t- yqt-.
Net benefits to consumers is given by discounted consumer surplus:
00
S(p, q; r)
=
z ptS(pt, qt).
(3)
t= I
As a representative of consumers' interest, the regulatory agency adopts maximization of
this measure as its objective.
3 The planning solution. Here we describe and partiallycharacterizethe second-best priceinvestment path. The planner's problem is to maximize the surplus Q subject to the firm's
I
Since variable cost is independent of capacity, the model must be modified to accommodate cost-reducing
innovation. This could be done by letting v depend on cumulative R&D expenditures. This is part of the subject
of the companion article by Salant and Woroch ( 1992).
S Allowing for the possibility of increasing returns to scale in construction would complicate the analysis in
at least two ways: (i) timing of investment would matter, and (ii) the minimum size of a capacity increment relative
to depreciation and discount rates would have to be small enough for triggerpolicies to work. We believe several
important classes of natural monopoly are captured with the given assumptions that allow increasing returns in
production and unrecoverable set-up costs.
SALANT AND WOROCH
/
33
participation constraint: II 2 llo. The term llo includes the opportunity cost of the nextbest venture.
As written, the planner places no explicit weight on the firm's profit. This formulation
sacrificesno generality,however. First of all, a "fair"returnon investment could be imbedded
directly into 11o. Second, by varying ll1, the set of second-best optimal investment plans
can be derived. As long as the firm earns at least llo, it will voluntarily enter this industryeven with the prospect of regulation. When H1oexceeds unregulated monopoly profits, however, a subsidy must be paid for the firm to embrace regulation.
A solution to the planning problem (denoted by *) consists of a sequence of capacities,
price caps, and quantities { qt*, p^t*,q* } that solves
maximize 4V(p,q; r)
(PP)
subject to ll(p, q, q; r, 6) 2 11o+ F
Pt < P(qt),
qt < O~tt
qt
2
Yqt-l
for each t given an initial capacity No. Choice of price Pt has been suppressed because the
regulator does not gain from departing from the price ceiling Pt. Furthermore, a simpler
version of (PP) in which price exactly clears the market (i.e., 3t = P(qt)) yields the same
solution.6 A solution to (PP) occurs at a singularity of the Lagrangian:
00
pt{S[P(qt),
?C =
t] + X[[P(qt)
-
v]qt- C(qt-
yqIt-)] + /t(qt-
4t)}.
(4)
t=1
A solution to the problem exists and satisfies the usual Kuhn-Tucker conditions (see Luenberger ( 1969)).
Necessary conditions for closely related welfare problems have been studied on various
occasions.7 When there is no excess capacity (i.e., qt = t), the first-orderconditions reduce
to a period-by-period, inverse-elasticity rule:
where,as usual, Et
-P(qt))/P'(qt)qt
Pt- Mct
q1
Pt
Et
(5
is the price elasticity of demand and
=(X -
1)/XE
(0, 1)
is the "Ramsey number." Note that the value of X, and likewise 7, derives from the profit
constraint and, hence, depends on both a and r. Here MCt is a dynamic marginal cost:
MCt = v + C'(It)
-
ypC'(It+1).
(6)
The first two terms in (6) are simply marginal operating and investment costs; the last term
accounts for the savings of reduced investment in the following period from an additional
unit of current investment, after properly discounting and deducting for depreciation.
At the planning solution, capacity approachessome limiting level for any initial capacity.
This limit capacity,j(r, 6, 11o, &o),will dependon IIo, r, and 6 as well as the initial level,
6 Had we included the constraint p, < P(q,), the corresponding Kuhn-Tucker conditions would be
vP[P(qj)- pt] = 0, where v, is the associated multiplier. If p, < P(q,), then the complementary slackness conditions
imply pt(Xqt -qu) = vP'(q,) = 0, and so X = 1. When X = 1, pt drops out of all the first-orderconditions and
reduces to those derived from (4). Note too that when X = 1, setting unit price p, differentfrom P(qu), and allowing
the planner to make lump-sum transfers in the amount of [p, - P(q,)]q, from investors to customers would not
increase the value of the Lagrangian.
7Brock and Dechert ( 1985) establishthe necessity of a nearly identical condition. In their dynamic investment
problem, output is always equated to capacity.
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THE RAND JOURNAL OF ECONOMICS
q0. When initial capacity is small, the planning solution calls for capacity to gradually
increaseover time. But when initial capital stock is large,additional investment is unnecessary
until capacity depreciates to near its limit value. We derive other properties of the planning
solution as needed below.
O Steady states. The system is in steady state when capacity is unchanging, so that investment exactly replaces depreciated capital: It = bqt-I for all t 2 1. Bars over the variables
indicate steady-state values: q^t= c, pt = f = P(,j), and so forth.
Many initial capacities can be maintained as steady states, provided capacity starts at
that level. Of these, many also generate revenue that covers investment, production, and
opportunity costs, i.e., any q such that [P(q) - v]q - C(aq) 2 r(Ho + F). The largest such
capacity provides an important benchmark, since it maximizes consumer surplus over all
feasible steady states:
maximize S(q)
subject to [P(q) - v]q - Cbq)
?
r(IIO+ F).
Let q*(b, r, Hlo+ F) be the solution to (SS). As r -O 0,jqj*(b,r, IIo + F)
q*(b) and its associated price, f*(b) = P(q*(a)), satisfy
P*(b) = v + C[bq*(b)]/q*(b).
(SS)
-
*(a), where
(7)
A consequence of Lemma 1 and the proof of Corollary 2 is that the solution to planning
problem (PP) converges to the single steady state (ft*(b), j4(b)) as r -O 0 and t -s 00 for
any initial qc and IIo + F. Call the path pt = ft*( b), qt = j*( b) the steady-state planning
solution. When H1oand Fare both zero, this is the surplus-maximizing, steady-stateoutcome
for any positive r.
When the system does not start in a steady state, the planning solution has capacity
settle down to the limit capacity qj(b,r, H1o,do). Its value is closer to the steady-stateplanning
solution qi*( 6) the closer the interest rate r is to zero. For any fixed r, the limit capacity is
increasing in the initial capacity c%.
3. The strategic
approach
* The planning solution presumes the regulator has direct control over production and
investment. Invariably,a privatelyowned and managed firm prefershigher prices and smaller
capacities than the levels dictated by efficiency. Therefore, we inquire whether it is possible
for the regulator to sustain the planning solution when it sets price ceilings while the firm
controls investment and production.
o Strategies. Assume that, when selecting their actions, the firm and the regulator enjoy
perfect recall and perfect foresight. At the outset, the firm can elect not to enter the market
by withholding its resources from production. At the beginning of each period, the firm
invests in capacity and the regulator sets a price ceiling. Figure 1 diagrams the extensive
form of the game.
For much of what follows, these decisions are made simultaneously. Simultaneous
moves reflect an information lag: neither the firm nor the regulatorcan react instantaneously
to unexpected deviations by the other. This lag allows either party to "cheat" by deviating
from any anticipated path before it becomes known to the other. In the meantime, the firm
can earn a one-time profit by reducing, or perhaps ceasing, investment. The regulator can
augment its surplus for a single period by lowering the price ceiling.
Given a price ceiling and installed capacity, price-quantity pairs must meet three conditions:(i) pricecannotexceedthe ceiling,(ii) quantitycannotexceed capacity,and (iii)
SALANT AND WOROCH
/
35
FIGURE1
EXTENSIVEFORMOF THEGAME
F
Stay out
Enter
R
'I0
F
Pi,
R
P2
F
GA
/
S
quantity cannot exceed demand realized at the chosen price. All three constraintsare satisfied
in the shaded regions of Figure 2.
These constraints still leave a wide range of feasible price-quantity pairs. Each pair
yields a different payoff to the two parties. We resolve the indeterminancy by selecting the
price-quantity pair that maximizes short-run profit:
[p(
q,q)
qU,)] = argmax {(p - v)q: p
p, q < , q Q(p)}
(8)
We could have used other rules to select a feasible pair, but this one has several attractive
features. First, the firm's apparent discretion over short-run production and pricing makes
sense when it possesses superior or more timely cost and demand information. This information asymmetry makes direct control impractical for a regulator. Furthermore, we can
show that if price and quantity were at the firm's discretion, the firm would nevertheless
maximize short-run profit as in (8).'
8 The rule also limits the price-quantity pairs that are realized. Implicit in (8) is the assumption that the firm
will operate at full capacity as long as the regulator'schoice of a price cap allows it to cover its operating expenses.
Typically,regulatedfirmsareguaranteedsome minimalrateof return,andso the firmwill shutdownwhenprice
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THE RAND JOURNAL
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FIGURE2
DETERMINATION
OF FINALPRICEANDQUANTITY
PX
iL~~~~~~~~~
D~~~~~~~~~
q
q
(a)
(b)
Strategies are rules that determine actions at each date as functions of the history up
to that time. The history of the game up to period t is
ht = (qoP3l, cil
, Pt-,,
A strategy for the regulator is a sequence of functions 0
each period t as a function of the history
At: ht
=
{ /t } specifying a price ceiling in
At E
9
Similarly, a strategy for the firm is a sequence of functions
for each t as a function of the history
A1t:ht
(9)
qt-1)
>t
'y
i
=
{ At} specifying
1*
(10)
a capacity
_It E 9.(1
Notice how the assumption of irreversible investment constrains the firm's actions in each
period.
0 Equilibria. A subgame is the game remaining given a history ht. A subgame perfect
equilibrium is a pair of strategies (-/*, 1*) in which, given any period t 2 1 and history ht,
(i) 4j* (restricted to the subgame) maximizes the firm's discounted profit from period t on
when it expects the regulator to use 0*, and (ii) k* (restricted to the subgame) maximizes
the discounted surplus from period t on given the firm is using 41* to set capacities.
A steady-state equilibrium is a strategy pair, (/*, 4t*), satisfying (i) and (ii) above
such that capacities and price ceilings are constant along the equilibrium path:
= P(*)
for each t, where h* denotes histories in
,*(ht*) = cjo = * and /*(ht*) =
A
fails to cover at least some portion of its capital costs. Additionally, the firm may be able to reduce output below
capacity when the price ceilings do not allow adequate return on investment. Such considerations can easily be
included by reinterpretingv as the smallest mark-up over operating costs that will be acceptable to the firm and a
as the fraction of capacity the firm will use when it can no longer earn an adequate return on its capacity.
SALANT AND WOROCH
/
37
which qs = q* and i3s= 5* for all s < t. Capacity remains at its initial level do at a steadystate equilibrium.
Our formulation of the firm-regulatorrelationship has a couple of features that set it
apart from other applications of supergames. First, due to the irreversibilityof capital, the
sequence of constituent games is time dependent: each new play of the game depends on
the history of play through the current capital stock.9 Also, an inherent asymmetry exists
between the two players. Unlike oligopoly models, for instance, the strategiesand the payoffs
of the firm and regulator are qualitatively different.'0
4. Extreme equilibria
* This section characterizes equilibrium outcomes of the game, focusing on the one that
maximizes surplus. We show how optimal outcomes depend on interest and depreciation
rates. We also demonstrate how the difference between the surplus-maximizing equilibrium
and the planning solution vanishes as the interest rate goes to zero.
The information lag that forces the firm and regulatorto move simultaneously prevents
the planning solution from being an equilibrium outcome. Only when the regulator can
observe the firm's investment before setting price ceilings is the planning solution an equilibrium.
We begin by finding the equilibrium when players are restrictedto open-loop strategies.
In that case their actions at each date are independent of history, except for the irreversibility
constraint.
There are two reasons for considering this game. First, as in Reinganum and Stokey
(1985 ), commitment can be measured by the length of time over which players are unable
to deviate from planned actions. Open-loop strategies represent an extreme form of commitment, since players are never able to adjust their actions after the initial period.
Second, a unique equilibrium in open-loop strategies exists for any subgame. When
closed-loop strategies are available-so actions at each date can depend on the history to
that date-the open-loop equilibrium continues to be an equilibrium. Furthermore, it is
the minmax point of the game. This means that the joint minmax strategies are an openloop equilibrium in any subgame: neither side would ever have any incentive to deviate
from these strategies.
i3 The open-loop game. An open-loop strategy for the regulator is a sequence of price
ceilings {f1t:t = 1, 2, ...,
o } with fit2 0 for each t. Likewise, an open-loop strategy for
the firm is a sequence of capacities {cit: t = 1, 2,.. ., o } with cit2 ycqt- for all t. For any
initial capacity q0, this game has a unique Nash equilibrium: Pt = v and 4t = 'oy for
each t.
It is easy to verify that these strategies form an equilibrium. Let fit= v for each t. Then
the firm will never wish to invest, because it will not earn a return on its investment.
Obversely, if ct = y
q , the regulator forgoes some surplus if price exceeds unit cost.
To see that no other equilibria exist, suppose that for some r > 0 we have T> v. Then
the regulator can increase its period r payoff S(p37, qj) by simply reducing j3T.On the other
hand, when j. = 0, reducing fit to v has no effect on its payoff. More succinctly, A = v is a
9 Gilbert and Newbery ( 1989) construct a supergame model of the firm-regulatorrelationship without time
dependence. In Benhabib and Radner ( 1992), time dependence arises from a renewable resource. Our approach
is based on a variation of Friedman ( 1988).
10In this respect our setup is similar to Fernandez and Rosenthal ( 1990), who model sovereign-debt renegotiation as a time-dependent supergame played by a borrower nation and a lender bank.
l A Nash equilibrium is a pair of price and capacity sequences such that the firm (the regulator) cannot
increase discounted profit (surplus) by adjusting its capacity (price) sequence given that the regulator will adhere
to its price (capacity) sequence from the outset.
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THE RAND JOURNAL
OF ECONOMICS
best reply to qt 2 0 for each t. Further, if q > yqT-yfor some r, then the firm can increase
its profitsby reducing investment-unless the marginal revenue is positive at the depreciated
stock. We conclude that
Proposition 1. (i) For any initial capacity q10,there is a unique open-loop equilibrium:
At = V and (t = ytlq0 for each t 2 1. (ii) The unique steady-state, open-loop equilibrium
is At = v and 1t= 0 for all t.
Proposition 1 shows that when the firm must make simultaneous and independent
commitments over an indefinite horizon, both owners and ratepayers suffer. The reason
for this inefficiency is that the commitments are made unilaterally and noncooperatively.
In their widely accepted analysis, Kydland and Prescott (1977) reach what seems to
be the opposite conclusion. They argue that in dynamic policy games the outcome improves
when the planner commits to an entire future course of action and all other agents follow.
Kydland and Prescott then point out that the planner will typically wish to renege on its
promises. For this reason, full optimality may require binding, bilateral contracts.
In practice, such bilateral commitment is not always possible and, as we show, is generallyunnecessary.When neither the firm nor the regulatormakes any preplaycommitments,
efficient equilibrium can arise, although the no-investment-no-cost-recovery outcome of
Proposition 1 remains another equilibrium outcome. This latter outcome forms the conflict
point of the game when it is expanded to include closed-loop strategies.
Corollary 1. Starting with any history, hT for r > 0, the strategy pair qt(ht) = v and
1t(ht) = yq~t-,, for all t ? r and for any continuation game, (i) is a (subgame perfect) Nash
equilibrium and (ii) constitutes the minmax threats of the game.
Proof. See the Appendix.
o Optimal threats. So-called minmax threats support the maximal cooperation at equilibrium. The minmax threat for a playerin any continuation game is the equilibrium strategy
that holds its opponent to the lowest payoff among the set of equilibria strategy pairs. Here
we assume that the firm can, without penalty, stop investing and the regulatorcan deny the
firm any further capital recovery.
These threats are unrealistically severe. In practice, the firm faces an obligation to serve
and a regulator can limit but not forever deny all capital recovery. Even the most severe
credible threats are not likely to be carried through ad infinitum without the possibility of
renegotiation. Corollary 3 below establishes that our approach can accommodate threats
that are less severe or of shorter duration.
Corollary 1 reveals the simple stationary form that these extreme threats possess: no
investment and no profits. They yield a payoff of zero for the firm and
z
ptS(ytT+1qT)
t=T
for the regulator, where q^T
is the capacity at the date X- at which the threat is carried out.2
o Steady-state triggerprice equilibria.This section demonstratesthat the planning solution
is not an equilibrium when the firm and the regulator simultaneously choose capacity and
price at each date. Nevertheless, we can show that the planning solution can be closely
approximated at an equilibrium.
12
Farrell and Maskin ( 1989) and Bernheim and Ray ( 1989) have examined the effect that renegotiation has
on the equilibrium set in infinitely repeated games. Their work, which does not strictly apply to the time dependent
games considered here, suggest that efficient outcomes can still arise at equilibrium. Green and Porter ( 1984), and
Abreu, Pearce and Stacchetti ( 1986, 1990) have analyzed equilibrium in repeatedgames with imperfect monitoring,
i.e., where players cannot detect with certainty deviations by others. This work suggests that triggerprice regulation
can still be effective when the regulator cannot perfectly monitor the firm.
SALANT AND WOROCH
/
39
When the firm's opportunity cost, Ilo, and sunk entry costs, F, are both zero, the
interest rate, r, does not affect the solution to (SS). The following proposition establishes
that at = qi*(6) for each t 2 0 and 3tt= P(q*(6)) for each t 2 1 is not an equilibrium
outcome where
q-*(6)
max {q: [P(q) - v]q - C(6q) 2 0}
= argmax {S(q):
[P(q)
-
v]q - C(6q) 2 0}.
The reason lies in the fact that the firm could earn a one-time profit whereas (SS) has it
exactly breakeven.
Proposition 2. The steady-state planning solution qt = q-*(3) > 0 for each t is not a (closedloop) equilibrium outcome.
Proof See the Appendix.
Section 5 presents an example in which the planning solution is an equilibrium outcome
< q-*(5) and the interest rate is not too large (see Proposition 7 below). However,
whenever q^o
the steady-state optimum, qt* = 4i*(6) for all t, cannot be an equilibrium outcome.
For fixed r and 6, it is possible to characterize the maximum capacity that can arise at
a steady-state equilibrium. This capacity must provide sufficient profit to discourage the
firm from curtailing investment. Depreciation must be fast enough and discounting slow
enough that the regulator will not wish to deviate. Only in the absence of depreciation will
there be no loss in consumer surpluswhen the firm stops investing. The following proposition
characterizesthis maximum capacity.
Proposition 3. Let q( 6, r) be defined as the largest q for which
[P(q)
- v]q-
C(q)
-yrC(6q)/(
1 - yr)
(12)
and also
00
00
p ts[P(
t=T
)
2]-
P S( v, -y qr)
( 13)
t=T
for each Xr 1. Then (i) q^t= q(6, r) and it = P(q^(b,r)), t 2 1, is an equilibrium outcome
= q(6, r) and (ii) given any qo > q^(6,r), at = qo for all t, is not an equilibrium
provided q^%
outcome.
Proof See the Appendix.
Notice that as r -O 0,q( 6, r) q- qi*( 5). Thus, this proposition says that as the interest
rate tends to zero, the largest initial capacity that can be maintained at a steady-state equilibrium converges to the optimal steady-state capacity.
When capital is perfectly durable (i.e., 6 = 0), trigger strategies fail to support any
steady-stateoutcome with q-> 0. To see this, consider any q+ > 0 such thatp+ = P(q?) > v.
If the firm's trigger strategy calls for it to set q^= q+ whenever P5s= p+ for all s < t, then the
regulator will not wish to maintain ^ = p'. This is so because once steady-state capacity
is achieved, no future investment is needed when capital is perfectly durable. The regulator
can raise consumer surplus in every period without loss of service. For (p', q+) to be a
steady-state equilibrium outcome, ( 13) must hold. The nearer 6 is to 0, the smaller the
discount rate must be for the regulator to comply with any steady state 5* > v.
5. Other trigger price equilibria
* Proposition 3 indicates that there are steady-state equilibria that closely approximate
the surplus-maximizing, steady-state solution among the class of such paths satisfying the
40
/
THE RAND JOURNAL OF ECONOMICS
firm's break-even constraint. In this section we look at the performance of trigger strategies
in supporting non-steady-state, as well as steady-state, price-output paths.
First we allow for an arbitraryinitial capacity. Next we relax the assumption of constant
unit operating costs and entertain the possibility of increasing returns in production characteristic of natural monopoly. Finally we consider a variation in the order of moves in
which the regulator moves after the firm.
In each case we see that trigger strategies can improve upon static noncooperative
outcomes. These strategies have the firm and regulator respond instantly to a departure
from the planned path with their minmax threats. This applies as well to the defecting party,
since self-punishment not only issues a best reply to the opponent's punishment but also
penalizes an opponent for a delayed response to an infraction.
In addition to being extremely severe, the punishments continue forever. If the future
matters enough, the present value of forgone surplus will ensure cooperation. Nevertheless,
no matter how damaging the consequences of a breakdown in cooperation, the firm and
regulator are unable to attain the planning solution as long as there is some discounting.
o Nonstationary equilibria. Given any initial capacity, we construct an equilibrium path
with normalized payoffs arbitrarily close to the steady-state optimal payoffs for positive
depreciation rates and an interest rate sufficiently small. To describe the equilibrium set,
we must normalize the payoffs, since discounted profit and surplus both diverge as the
interest rate goes to zero. This is done by multiplying discounted surplus and profits by the
interest rate.
These equilibria result in capacity paths converging to q*( 5), the optimal steady-state
capacity, as r -O 0 for any initial capacity. Furthermore, any per-period payoff pair,
S = S(j) and ir = [P() - v]C(6), with j E [qm(, ), 4*(6)], can be approximated
arbitrarilyclosely by equilibrium-normalized payoffs for r sufficiently small. Here qm(6) is
the steady-state output that provides the firm with the largest per-period profit, that is,
qm(6)
argmax {[P(q)
-
v]q
-
C(6q)}.
Given a sequence 6 = { at: t = 1, 2, ..., 9o
the regulator can be defined as
} of capacities, the normalized payoff for
00
NS((Q, r) = r
ptS(q^t)= rg'(C; r)
z
t=1
and for the firm as
00
NII(CQ, r) = r
z
pt[[P(q^t)
- v]q^t-
C(qt-
yq^t)]
= rH(CQ; r, 6).
t=1
Recall that the planning solution solves (PP): it maximizes discounted surplus (3)
subject to the break-even constraint
00
z pt([P(qt)
-
v]qt - C(qt -
yqt-l))
? I-o
+ F.
(14a)
Now consider a variant of this problem with a different constraint. Let
@**( r, 6, I-o, q^o)
=
{*
*
(r,
6,
HI, do): t = 1, 2, . . ., 00}
maximize (3) subject to
00
2: p't-+'([P(qt)
t=7
for - = 1, 2, ... , 0o .
-v]qt
-C(qt
- -qt-,))
2 Ho + F
(14b)
SALANT AND WOROCH
/
41
The solution to this second problem satisfies a stricterprofit constraint than the original
planning problem. ( 14b) requires that at any date r the value of the firm never falls below
lO, while ( 14a) merely requires that the initial market value of the firm exceeds nlo. Equivalently, (1 4b) implies the firm has a nonnegative cash flow. Which constraint is appropriate
depends on the circumstances facing the firm. The value of the firm's opportunity cost may
change over time. When initial investment represents a sunk cost, the firm's opportunity
cost jumps upon deciding to enter the industry. Nevertheless, the two paths converge when
there is no discounting.
A characterization of the constrained optimum path, Q* *( r, 6, I-o, q^o),is a first step
to find how closely the planning solution can be approximatedat an equilibrium. To simplify
notation, we include F in I-o for the remainder of this section.
Lemma 1. For any (r, 6, I-o, q^o)with 11o < max { [P(q) - v]q - C(6q) }, the capacity path
q **(r, 6, IIo, q^o)* q*(r, 6, I-o, q^o)as t -0o0 for some q*(r, 6, Ilo, '?o)< 00 .
Proof See the Appendix.
The next lemma illustrates an equilibrium strategy pair having a "trigger property"
and yielding normalized payoffs for the regulator arbitrarily close to the optimal payoffs
from ( 14b).
Lemma 2. Given any To? 0, ro > 0, Ho > 0, and e > 0, there is an r(E) > 0 such that for
any r < r(E) the following strategy pair is a subgame perfect equilibrium:
ht = h*
p[q*
[ *(ro 6, 1O, ?o)]
if
v
otherwise
=*ht
(15)
and
41*(ht)
rq**(ro,
t
if
6, IIo, o)
~~~~~~~~~~(16
otherwise,
Iqt-l
where h*= (o, P3 * *, q* ,...,
foreachs =1, 2 ..., t-1.
ht = h(
pt-, qt-),ps*
P(q**),andq**
=**(r,
6, no, do)
Proof. See the Appendix.
A direct consequence of Lemma 2 is the following proposition, which states that if the
interest rate is sufficiently small, equilibrium payoffs can approximate the payoffs of the
steady-state planning solution as closely as desired.
Proposition 4. Given e > 0, there is an r(E) such that the payoffs to the regulator from the
capacity path Q *( r) determined by the equilibrium strategy pair ( 15 ) and ( 16) satisfies
INS((cQ*(r),r)
-
S(q*(6))I
<e
whenever r < r( e).
Proof: See the Appendix.
Immediate from the proof of the above proposition is that, starting from any initial
capacity, a range of capacities can be reached and maintained at an equilibrium capacity,
provided the discount rate is sufficiently small.
Corollary 2. Let J E [qm(6),
Given
at(ht)
a-nd
-
Js
e
j*(6)),
where qm(6) = argmax {[P(q)
-
v]q
-
C(6q)}.
> 0 and q^o,there exists an r and a subgame perfect equilibrium (0, A) such that
J as t
INS
where,)ofor each t and s, ht = (do, p), Jl . . , pt-1, qt- ) ps = P(qs)
AShI
- Vt-I
andn INSTQf/-Y
I
e-
fovr anyn r E
(O. PA)
42
/
THE RAND JOURNAL OF ECONOMICS
Observe that unless qo _ q*( 5), the normalized profit and surplus of the equilibrium
outcomes are bounded away from 0 and S( q*( 6)), respectively. This is so because the cost
of building up capacity to 4j*( 5) can never be made up, even though its normalized value
tends to zero as r -- 0. Thus, S(q-*( 5)) must always exceed any equilibrium payoff when
qo <4(6)
Further, the normalized value of the firm's profits can never vanish at any date t when
qt_1> 0 and pt > v, as the firm can always earn a one-time profit by not investing and then
shutting down. However, as r -- 0, the firm's per-period profits along an equilibrium path
converging to some qc> 0 can be made to approach zero.
The punishments implied by the triggers in (15) and (16) are unrealistically grim.
Corollary 3 below follows directly from Lemma 2 and Proposition 4 and relies on less severe
or shorter punishments. Note that although less severe punishments imply a smaller equilibrium set for any interest rate, it remainstrue that equilibriumcan approximatethe planning
solution for sufficiently small interest rates.
Reducing the severity of the punishments requires only minor modifications in the
proof of Proposition 4. To see this, suppose the regulator can prevent the firm from abanwhere
doning its investment. In particular,suppose that the firm cannot invest less than 6cq^,
6 > 5', so that the firm must replace some but not all of actual depreciated capital. In
addition, assume the regulator cannot force price below v'> v.
The punishments 3t = v' and qt = qt-l)] qt-l = qtq^t-l + 6'q^tj = [1 - (6 outcomes
and
can
replace the punishments in the right-hand
then can support cooperative
side of both ( 15) and ( 16 ) for appropriately reduced discount rates. Summarizing,
Corollary 3. Given any qo > 0, ro > 0, Ho > 0, and e > 0, there is an r(E) > 0 such that for
any r < r(E) the following strategy pair is a subgame perfect equilibrium:
ht = h*
if
{P[q* *(r0, 6, Ho, '1o)]
=*ht
(15a)
otherwise
v'
and
fq*(ro,
6 Ho, q0)
1qt-
ht = h*
if
otherwise,
where v + 647*(5) > v'> v, 6 > 6Y> 0, and where h * is any history in which the last defection
occurred at date r and then not again for T periods. That is, the firm sets capacity at
q, = -yqs-l for each s = r + 1, ... ., r + Tand then qj* = q(r, 6, 110,q,+T) at all other dates
s, while the regulator sets the price ceiling at p3 = v' for each s
= X
+ 1, . . .,
X- +
T and
then p* = P(qs*) thereafter.
Proof: See the Appendix.
Since punishment occurs for T periods after the last defection, capital does not accumulate at the cooperative rate. Instead, the shortfall must be made up gradually once cooperation resumes.
o Increasing returns in production. Up to this point, production has exhibited constant
returns to scale. Yet natural monopoly is often associated with significant scale economies.
Proposition 5 generalizes Lemma 2 and Proposition 4 to allow for increasing returns to
scale in production up to capacity. In particular, we suppose that operating costs are of the
form
w0
h0a
where
v(0) = 0, v
V(qf q) =
A an
--..
O
v(q)
A'q<O
for
q?<
for
q>
(17)
SALANT AND WOROCH
/
43
This is equivalent to assuming a generalized Leontief production of the form
q = G[Min {aL, j3K}], where L and K are the amounts of variable and fixed inputs and
a and 3 are production coefficients. Scale economies are captured by G(O) = 0, G' > 0,
and G' > 0.
Proposition5. Suppose for any rHi0< a, for some II > 0,
q*( , r, Ho)
argmax {qP(q)
-
v(q, q) - C(q)
2
rHO:q -q
exists. Then the strategies
-
P[40(r,9 , ion, ^o)]
if
ht
v(q, yqyt-,)/'yqt-,
otherwise
(t1
and
Ot(ht) =
r4I'*(ro,
if
,HIIjO)
ht=hh*
otherwise
IYqt-i
(19)
form a subgame perfect equilibrium where q-t**(ro,6, I-o, do) maximizes (3) subject to
*
(14b) and ht* = (qo, qr P(4*),
...,
q-t*-, P(q4t*-l)).Also, the associated normalized
surplus, NS(&* *( ro, 6, Ho, &o)), converges to steady-state optimal surplus S(q*( 3)) as
ru- 0.
Proof. See the Appendix.
0
Equilibrium when the firm and regulator alternate moves. Often, regulators observe a
utility's investments before they set price ceilings. In some cases, they condition rates on
the level of investment the firm undertakes. Such cases demand a sequential move game.
To begin, let the firm first set capacity at and the regulator follow with price ceiling A.
Whereas the steady-state planning solution could only be approached when moves were
simultaneous, we show that it can be achieved when the regulator enjoys a second-mover
advantage. 13
As before, a strategy for the firm is a rule determining the choice of capacity in each
period t as a function of the history through period t - 1. Now a strategy for the regulator
^
specifies the choice of as depending on ht and qt. the firm's capacity in period t. Let
h= [ht, t]. Consider the following strategy pair, (X*, l*):
0*(h+)
h+
t =(h*,') qt
Pt
if
Lv
otherwise
qt*
At*(ht) =(21
= yqt-l
if
~~~~~~~~(20)
ht=ht
otherwise,
)
where ht*= (do, qi,IP*,
Pt-,), the optimal path through period t -1. Given 6 < 1,
this strategypair is a subgame perfect equilibrium that attains the planning solution provided
r is not too large. The firm earns zero profit along the equilibrium path, and since it moves
first,it cannot earn more by setting off the trigger.But once a deviation has occurred, neither
player has an incentive to cooperate.
Proposition6. Given 3 < 1, there is a ro > 0 such that the planning solution is a subgame
perfect equilibrium outcome whenever 0 < r < ro.
13
Where pitis set after the firm has committed to qt, then the planning solution will always emerge as a
subgameperfectequilibriumoutcomewithlinearinvestmentcosts.
44
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THE RAND JOURNAL OF ECONOMICS
The remaining possibility has the regulator commit to a price ceiling for each period
before the firm determines its capacity. In that case we find the same "regulatorylag" as in
the simultaneous-move game. As in Proposition 2, this lag implies that the planning solution
is not an equilibrium outcome because the firm is able to earn a one-time profit by defecting.
6. A linear example
* A simple example illustrates how triggerprice regulation can sustain win-win outcomes.
Consider the case of linear investment cost:
(22)
C(I) = I,
and let both unit operating cost v and the fixed entry cost F be zero. In this case, dynamic
marginal cost in (6) is constant and increasing in both the interest and depreciation rates:
MCt = fp(b + r).
(23)
Substituting (23) into the pricing rule (5) and then rearrangingyields
t - fp( + r) -X
Pt
(24)
Et
This expression characterizes second-best prices when there is no excess capacity, which
holds as long as 'yqOc j*. Since it is independent of qj,-,, its solution is a steady state:
=ft* for all t ? 1. Thus, the planning solution requires investment
qt= qt= 4* and Pt
of q* - 'yqoto bring capacity immediately up to its limit value 11*;thereafter, investment
maintains this level by covering physical depreciation, &j*.
The participation constraint in (PP) in the linear case is
00
-10 c z pt[P(qt)qt
-
f(q't -
yqt-1)]
t= 1
=
(p*
-
#6)4*/r - f3yp(4* - 4o)
(25)
Given capacitiessatisfying(24) and (25), we constructthe followingtriggerstrategies:
* =
*
p
if
qs=qs I
vV
otherwise
lzyqt-l
if *vs = *,
otherwise.
S
Ps=P*,
..,t
1
(26)
Ps =P
,
s=
1, ..,t-
1
(27)
We now want to comparepayoffsalong the cooperativepath with payoffsto defectingfirstfor the firm,and then for the regulator.A tilde(-) distinguishesvaluesalongthe defection path.
a Defectionby the firm.We firstshow when the firmwill not wish to defectin the first
A trustingregulatorwill set the price
period.The firm entersperiod 1 with capacity oyqo.
ceilingat Pi = p*, but shouldthe firmdeviate,pricewill be cut to operatingcost:Pt = v,
t
?
2. In that event, its best reply is to cease all investment: qt = ytq0,t 2 2.
In period 1, a defectingfirm selectsql to maximizeone-periodprofit.Generally,this
will call for no investment(i.e., ql = 'yqo)unlessmarginalrevenueis positiveat that productionlevel. Therefore,defectionin period 1 savesthe firm up to fl(q*- qo), the cost
in revenuesthat period,so thattotal profit
of plannedinvestment.It collectsat least qy0*jO
from defection (discounted to period 0) is bounded below: P[Pi*p - l(i q-yo)] ? oypj5q^o.
Suppose,instead,that the firmadheresto the optimalpath.Then its first-periodprofit
equals p
(*-
=*
-
_yo),
which is negative whenever initial capacity is less than the
SALANT AND WOROCH /
45
limit capacity qO< 4*. Cooperation yields profits discounted to period 0 of Ho. The firm
will cooperate provided
10 > yppj*qo.
(28)
In subsequent periods t ? 2, the firm's payoff along the candidate path (discounted to
date t) is (f5* - #6)j*/ r, which from (25) is at least flpy(c* - go) + Ho. Defection yields
at most a one-time profit (discounted to date t) of ypy*qj*.
3 Defection by the regulator. We now solve ( 13 ) to calculate the values of r and 6 for
which the regulator will not defect. The regulator'sdiscounted payoff from cutting price to
Pt = 0 for each t 2 1 is
pS(O,ci*) + z2 ptS(0, lyt-lq*)
(29)
2
In comparison, the surplus along the planning solution is
00
c*) = S(U*, j*)Ir.
z ptS(ft*,
(30)
Cooperation dominates defection whenever (30) exceeds (29):
00
S(f *, 4*)Ir
2
pS(O,
oa*)+ 2: pts(ol 1Yt-lq*)-
(31)
2
Note that the right side of (31 ) tends to 0 as r tends to 0.
Proposition 7. Suppose that qo < 4* and that I-o is sufficiently large so that (28) and (31)
hold. Then, for r sufficiently small, the planning solution is a subgame perfect equilibrium
outcome with trigger strategies described in (26) and (27) when the regulator and the firm
move simultaneously. When the firm chooses capacity before the regulator sets the price
ceiling, the planning solution is once again an equilibrium outcome if the interest rate is
not too large.
The conditions imposed on (28) and (31 ) in this proposition govern how much profit
and surplus the firm and regulator must receive to induce them to cooperate. Inspection of
the participation constraint (25) reveals that as r -- 0, smaller values of 5* will still allow
(28) to be satisfied. Condition (31) indicates that as the interest rate gets small, average
surplus need only be positive to be guaranteed that the regulator will wish to continue
cooperating.
o Special case of linear demand.A more complete characterizationof equilibriais possible
if we take demand to be linear:
P(q)=a-bq
(32)
for 0 < q c a/b, so that the price elasticity is
Et
=
(a - bqt)/bqt.
(33)
Consumer surplus then becomes
S(p, q) = aq - bq2/2 - pq,
(34)
which equals bq2/2 whenever p = a - bq. We require sufficient demand to cover average
total cost in a steady state, or a - bq > #3 for some q.
Solving (24) yields the steady-state capacity:
-*
a - flp(b + r)
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THE RAND JOURNAL
OF ECONOMICS
Substituting (35) into (25) and rearrangingresults in a quadratic in 1 + 7:
(1 + 7)2bba2/r + (1 + n)[p1'ycxa- (a - f35)a/r] + (Ho - pfryqo) = 0.
(36)
The firm prefers to cooperate when (28) holds, which in this case is
10?2 ypco[na + 3p(b+ r)]/(l
+
(37)
).
The regulator prefers to cooperate when
00
z
p'b4 2/2 = bj*2/2r
cai*/(1
+ r - Ay)- bi*2/[2(1
+ r-
_2)]
(38)
1=1
or equivalently,
4* > [2a(1 + r -
2)r]/[b(
+ 2r - 2)
(39)
Expressions such as (37) and (39) sweep out an entire range of discounted profit and
surplus that can be supported when interest rates are small. On the frontier, each level of
surplus is associated with a differentprofit. Which of the many profit-surpluspairs is selected
must be decided by the players. Furthermore, triggerstrategiessuch as (26) and (27) specify
a particular price-investment path to be used. In fact, other combinations of prices and
investments could yield exactly the same payoff levels. Aware of this indeterminacy, the
firm and regulator must jointly decide on one such path, or risk a loss of efficiency.
7. Policy
implications
* Opportunism constrains the growth and operation of a regulated utility. Efficiencyminded policymakers cannot ignore its effects. Full commitment to a policy is not a solution,
however. On the contrary, flexibility of response raises a strong deterrent to inefficient
behavior.
With this in mind, we find that trigger price regulation produces some unorthodox
prescriptionsfor the design of policies toward natural monopoly. These lessons apply equally
well to other forms of monopoly regulation,including rate-of-returnand price-capregulation.
Although our stylization of price control lacks some basic features of the typical price-cap
scheme, it captures the essential spirit of that scheme. 14
Furthermore, modification of our setup to have the regulator set the allowed return on
investment instead of a ceiling price is straightforward.Gilbert and Newbery (1989) carry
out such an exercise when capital is not durable. How well a particular regulatory program
performs relative to another will turn on the characteristics of the particular market. It
clearly is a subject that warrants further analysis.
Whatever the institutional framework, several novel implications follow from our analysis of trigger price regulation.
Facilitate negotiations between the firm and regulator. We have seen that, if they cooperate with one another, investors and consumers can realize huge benefits. Mutual gains
require a climate conducive to cooperation and coordination that can be achieved only
through free and active negotiation. Restrictions on the exchange of offers, and on the terms
of agreement, only prevent the bargaining process from pricing and investing efficiently.
This observation is not new. Coase ( 1960 ) pointed out the value of costless negotiation
in a general setting of bilateral externalities. We expand on Coase's point by demonstrating
that, due to the multiplicity of equilibria, the attainment of a win-win outcome requires a
degree of coordination that is possible only if negotiation costs are not too high.
O
14 The first implementation of price caps was for British Telecom and had its roots in Littlechild ( 1983). The
U.S. version applied to AT&T is codified in Federal Communications Commission (1988). Linhart, Radner, and
Sinden ( 1983) and Vogelsang( 1989) offerearlyspecificationsof the price-capmechanism and analyzeits performance.
SALANT AND WOROCH
/
47
On the other hand, policies that facilitate bargainingcan yield unwanted consequences.
Specifically, face-to-face negotiations raise the temptation for managers of the firm and staff
at the commission to strike a deal at customers' expense. Free access to negotiations should
eliminate much corruption, but then the administrative process becomes slow and cumbersome when every interest group is allowed to intervene.
O Raise the conflict point. The equilibriawere supportedwith triggerstrategiesthat credibly
threaten to halt any capital recovery and all investment. We recognized that in practice,
the use of such grim strategies is unlikely. As we show, less damaging threats might still
support the same payoffs but will invariably shrink the equilibrium set.
The degree to which negotiation can improve on static, noncooperative behavior depends on how damaging both parties find a breakdown of the bargaining process. Paradoxically, measures that make the threat point more distasteful may actually improve and
enhance the stability of the regulatory climate.
This suggests that each party should be given expanded freedom to respond to opportunistic behavior. By relaxing its obligation to serve, for instance, a firm can react to unrenumerative rates by reducing service quality or by abandoning a service altogether. Symmetrically, a regulator could mete out,a more severe punishment for inadequate investment
if the constitutional ban on confiscatory rates were eased. It might also be allowed to cut
rates in ancillary markets served by the firm.
The method used to finance investment can have important strategic effects as well.
Specifically, the larger the fraction of capital financed by debt, the less shareholders stand
to lose if the regulator drives the firm into bankruptcy. Limits on the ability of a public
utility to use debt financing therefore make cooperative outcomes more likely and more
stable.
Policies that reduce payoffs associated with the threat point, while desirable from a
bargaining perspective, are not without tradeoffs. A more severe threat point enlarges the
set of equilibria to include both better and worse outcomes. And the damage done is greater
if the trigger is set off by mistake or during a brief lapse of rationality.
Outside intervention by courts and legislaturesmay directlylimit opportunisticbehavior,
lessening the need for noncooperative punishments. But whatever the powers of these third
parties,there invariablyremains some opportunity for improvement. Triggerprice regulation
offers an avenue to exploit these mutual gains.
o Raise the discount factors. The more weight the regulator and the firm attach to the
future, the higher will be the costs of defection. Lengthening the tenure of commissioners
and their staffs, or instituting a system of overlapping terms, would encourage them to
internalize the impact of their actions. Permitting regulatory staff limited employment opportunity in the industry after leaving the agency is another way of increasing the costs of
defection. Similarly, managers will place more weight on the firm's future by using properly
designed compensation schemes or by opening up post-retirementemployment opportunities
to regulatory officials and staff."5
Allowing such post-retirement opportunities has both benefits and costs. While such
positions serve to mitigate opportunisticbehavior, corruption may result if these employment
opportunities are used as bribes.
O Restrict the firm's capital structure. Our results establish the connection between equilibrium and the firm's capital structure. In particular, excessively durable capital exposes
the firm to expropriation should it adhere to efficient accumulation rules. Durability raises
15
Salant ( 1991 ) illustrates how overlapping terms can result in cooperative outcomes in an infinitely repeated
game with finitely lived players.
48
/
THE RAND JOURNAL OF ECONOMICS
the regulator's temptation to renege on any cooperative plan. Proposition 4 and Corollaries 2 and 3 illustrate the role played by depreciation.
To capture the benefits of their relationship, both parties would like to limit the commitment the firm undertakes when it invests. Institutions could prohibit adoption of technologies that require highly sunk expenditures on plant and equipment. Compromising
production efficiency may be a small price to pay for strategic stability.
8. Conclusion
* We model the regulation of a public utility as a long-term relationship between a firm
and its regulator.After formulating a time-dependent supergame,we extend a Folk Theorem
to show that when the participants are sufficiently patient and capital is not too durable,
the planning solution can be approximated arbitrarilyclosely at an equilibrium. We propose
"trigger price regulation," in which the regulator, responding to any deviation from the
efficient path, cuts price down to operating cost and the firm curtails investment. This policy
continues to perform well for differentspecificationsof the technology, sequencing of moves,
and limitations on noncooperative behavior. Elsewhere we have demonstrated that its attractive properties hold when the firm invests in cost-reducing innovations or product enhancements, and when capital is lumpy. Its effectiveness over a wide range of environments
positions trigger price regulation as an attractive option for regulatory reform.
APPENDIX
*
Proofs of Corollaries 1 and 3, Propositions 2 through 5, and Lemmas 1 and 2 follow.
Proof of Corollary 1. (i) If the regulatoranticipates the firm will set q, = -yd,- for each t ? r, then consumer surplus
is maximized by settingfi, = v for each t. This choice of a price ceiling has no effect on future investment. Obversely,
if the firm believes the regulatoris going to set fi, = v for t ? r, then it has no incentive to incur additional investment
expenses,and so c, = -y4q, for t -r.
(ii) To see that these strategiesthreaten minmax penalties, notice that they provide the firm with zero profit
for the remainder of the game, and the regulator'spayoff is limited to the maximum surplus that can be amassed
utilizing existing capacity. The firm can always choose to exit and so cannot be held down to any lower payoff.
And the regulator cannot be held to any lower payoff if, as we assume, it can set rates at operating costs that keep
the firm's existing capacity running. Q.E.D.
Proof of Proposition 2. Let ' and t be any strategy pair having the equilibrium property that fi, = f*(6) and
q, = j*( 6) for each t. Notice that at this steady-state optimum, p~q,= j*( 6)j*( 6) = vq*( 6) + C(bq*( 6)), making
profit zero in each period.
Suppose defection occurs in period 1. The firm sets capacity at 41 = -yqO= -y,*( 6) provided static monopoly
output is less than -y'2o.Assuming the regulator sets price at f*( 6), the firm earns a profit in that period equal to
-yC(6q*(6)). Further, by ceasing all subsequent investment, the firm avoids any future losses regardless of the
regulator'sactions. Therefore, this defection gives the firm greaterprofits than does A, which is zero. Q.E.D.
Proof of Proposition 3. Let q = d(6, r) be as in Proposition 3:
-(q) = [P(V)-v]7-
C(6o) 2 yrC[64]/(1 - yr),
(Al)
for each r 2 1 let
00
Z, p'S[P(4),
00
4] 2 Z pIS(V, _y'-7),
(A2)
and let j3(6, r) = P[c2(6, r)] be the corresponding price. Given 6 < 1, both inequalities (Al) and (A2) are eventually satisfied as r approaches zero, since the right side of (Al) goes to zero as r -> 0, and for fixed 6 < 1, (A2)
is satisfied for r sufficiently small. (In general, one of (Al) and (A2) will bind while the other is slack.) Thus,
- C(64) -> 0 and so d(6, r) -> #*(6), the optimal steady-state capacity. In the following, fix
[P() -v]
6 < I and letq* =*(b).
Consider the following closed-loop strategieswhich have a triggerproperty:
p(b, r)
0d*(hi) =
v
if
h, = h*
oherwise
(A3)
SALANT AND WOROCH
/ 49
h,=hh*
rq(6,r)
if
b4-_
otherwise,
(A4)
,*(h)=
whereh * is the sequencePT= f (6, r) and q,
t - 1. Thenthe strategypair(A3),
4(6, r) for each r = 1, 2,
(A4) form a (subgame perfect) Nash equilibrium if at any point in the game, and for any history up to that point,
both the regulator and the firm would wish to continue using those strategies (assuming the other one will do so
as well).
Now, given no previous defection up through date r, (A3) and (A4) imply that defection by the regulator
does not pay, provided
2
=
r),
ptS[pi(b
q4(b
2 ptS[v,
z
r)]
t=Tr
(A5)
r)].
t-yTq(b,
t=Tr
But this is guaranteed by (A2).
Once a deviation from h* has occurred, Corollary 1 says that the firm will never wish to incur additional
investment expenditures. Given h, = h*, the largest one-period profit the firm can obtain by defecting is
[fi(r, 6)
-
v]-y4(6, r)
=
-
yC[64(6, r)]/(l
(A6)
yr),
where, for r sufficiently close to zero,
pf(r, 6)
v + C[64(6, r)]/[(l
=
-
(A7)
yr)4(6, r)],
as in (Al). Equivalently, there is a p close to one satisfying (A7), so that the profit from continued cooperation is
co
2:
p
t
[pi(b
r)
-
4(b
v]
r)
-C[
64(b
r)
(A8)
]}-
t=I
Then, for p sufficiently near 1, (Al) implies that (A8) can be rewritten as
(A9)
-yC[64(6, r)]
- yr
1
which is just (A6). Therefore, the firm cannot gain from defection when p is sufficiently large. Observe that for
largervalues of q, (A l ) would be violated for sufficientlylarge p. Thus, no largersteady-statecapacity is sustainable
as a (subgame perfect) Nash equilibrium outcome, making the strategies (0*, P*) result in the surplus-maximal
steady-stateoutcomes.
Finally, it is important to note that our trigger strategies call for minmax punishments of defections. As a
result, outcomes supported by these strategies cannot be Pareto dominated by an equilibrium supported by any
other pair of closed-loop strategies. Summarizing these arguments gives
Proposition A. Given any 6, 0 < 6 < 1, there exists an F> 0 such that for r <
r-, the triggerstrategies(0*, P*) given
by (A3) and (A4) form a steady-state (subgame perfect) Nash equilibrium providing the regulatorwith the largest
surplus from among the set of steady-state outcomes.
q E [qm(6),
Proof An immediate implication of the above is that for any
qm(b) = argmax{[P(q)
-
v]q
-
4(6, r)]
where
C(6q)}
is the profit-maximizing steady-statecapacity, strategiesof the form (A3) and (A4) can be used to support a steadystate outcome with qt = qjand pt = P(4c) for all t. This follows because the above arguments in no way depend on
the steady-state capacity being 4*; we merely require that [P(j)-v]
-C(&q) > 0. Q.E.D.
Proof of Lemma 1. We show that for any '2o the sequence { q* *(r, 6, Ho,0o): t = 1, 2, .
and strictly monotone. Such a sequence must then have a unique limit point. For do
[P(qj*(r, 6, Ho)) - v]1*(r, 6, HO)- C(64*(r, 6, Ho)) = rHo, it mustbe the casethat
4*(r, 6, HO) > q**(.)
<
oo } is both bounded
i*( r, 6, Ho) where
> q*i*(.)
for each t, from the assumptions that S'(0) > C'(0) and that S" < 0 and C" > 0. Similarly, concavity of S( *) and
whenever qo> 4*(r, 6, Ho). Q.E.D.
convexity of C(*)imply that i*(r, 6, HO) q,**(.)
<q*-*(*)
_
Proof of Lemma 2. First notice that for the firm, continued play of (0*, P*) yields a normalized profit that remains
* * ) -* roHo as t -* oo, and
- C(q,* - -yq
strictlypositiveas r - 0. This is because(i) 7r* = [P(q)* * )-v]q*
therefore (ii) NIH = lim,0 r{ z H,/( 1 + r)'} = lim,0 r[(ro/r)Ho] = r0H0> 0. Defection yields a one-period
payoff of no more than II = max { [P(q)
tendingto zeroas r -> 0.
-
v]q: q 2 0 } and nothing thereafter, which has a normalized profit
50
/
THE RAND JOURNAL OF ECONOMICS
Further, adherence to 0* results in a payoff for the regulator converging to S[qc*(6)] as r -> 0. In contrast,
deviation from 0* at date t generates a normalized surplus of no more than
r 2i S1-y-t)
(
which tends to zero as r -> 0. Thus, for r sufficientlysmall, deviation from 0* does not pay for the regulator. Q.E.D.
Proof ofProposition 4. By Lemma 1, for rHo< HI
max { [P(q)
path (Q**(r6, HO,co) = {q**(r, 6, Ho, do)} satisfies
-
v]q
-
C(6q): q 2 0 }, the constrained optimal
q*(r, 6, HO,40) as t --> 0
q,**(r, 6, HO,qO)
for some q*( r, 6, Ho, co). Also notice that
4q*(b)
q*(r, 6, HO,Jo)
as
r - 0.
If not, then it must be the case that q*( r, 6, Ho, co) -> q' as r -> 0 for some q' # j*( 6). And it must also be the
case that q' < j*( 6), as 7*( 6) is the maximal capacity level that can be sustained by the firm while still earning
revenues in each period exceeding the total of rHoplus operating and depreciation costs. But for r sufficientlysmall,
one can always construct a path satisfying the profit constraint ( 14b) that provides the regulator a greater surplus
than any path converging to q' < j*( 6). Consider the steady-state capacity
qr = Max {q: [P(q)
-
v]q
-
C(6q) > r[Ho + C(ci*(6))]}.
Observe that as r - 0, qr-- *( 6). Let Q(r) be the "fastest," i.e., shortest number of periods, path to qrsatisfying
constraint ( 14b). (Note that if c^o< qr, this path reaches qrin one period.) Then as r -> 0,
NS(Q(r), r)
->
S(i*(6)) > S(q').
Thus, q*(r, 6, 11o, go) -> 7*(6) as r -> 0 for any So > 0.
Then for ro such that S[ q*( r, 6, Ho, 'o) ] > S[ q*( 6)I - , the strategypair ( 15 ) and ( 16) results in a capacity
path with the desired properties. Q.E.D.
Proof of Corollary 3. First, for T large, the punishment payoffs are approximately the same as is the case with ( 15)
and ( 16). Also, suppose for any 0 < r < ro, Ho in (I14b) satisfies
r0i0/r > ro max
{qt}
z
qtpt[P(qt) - v'] - C(qt - yq,).
t=I
Also, notice that continued play of (0*, ip*) yields a normalized payoff for the firm that approaches r0110as
r -> 0. And defection by the firm gives it a normalized payoff that the above inequality implies is strictly less than
that. So the firm would not wish to defect from ( 16a).
Further, the regulatorwill adhere to ( 15) and collect a normalized payoff converging to S[c*( 6)] as r -> 0,
whereas defection produces a payoff of S(q') for some 0 c q' < 4*( 6). Note that once a defection occurs, neither
side will wish to reward the other for T periods. Thus, after the punishment period ends, the incentives are as if
there were no defections. Q.E.D.
Proof ofProposition 5. First, notice that an argument identical to the one in Lemma 1 can be used to establish that
the path4t(r, 6, Ho, 2o)convergesto somelimit value,4*(r, 6, Ho, qo).Also,as r -* 0, 4*(r, 6, Ho, qo)-* q*(6).
Second, the incentives for the firm to defect from the candidate equilibrium path are the same as in the first
part of the proof of Lemma 2, and an identical argument implies that A is always a best reply to k.
That the regulator would find 4 a best reply to A for any history follows from the second half of the proof of
Lemma 2. The regulator's surplus from a deviation from ' is strictly smaller, given ht when v" < 0, than was the
case in Lemma 2, where v" = 0. Given a defection has already occurred, incentives are the same as in Lemma 2.
Finally, that 4*( r, 6, Ho, qo)must converge to i*(6) as r -> 0 also follows from the same argument given in
Proposition 4. Q.E.D.
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