Firm Regulation and Profit-Sharing: A Real Option Approach


 

Firm Regulation and Profit-Sharing: A Real Option Approach

Michele Moretto,  Paola Valbonesi

2006

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This paper can be downloaded without charge at:

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The opinions expressed in this paper do not necessarily reflect the position of

Fondazione Eni Enrico Mattei

Corso Magenta, 63, 20123 Milano (I), web site:

 

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Firm Regulation and Profit-

Sharing: A Real Option Approach

Michele Moretto and Paola Valbonese

NOTA DI LAVORO 7.2006

JANUARY 2006

PRCG – Privatisation, Regulation, Corporate Governance

Michele Moretto,

 

Department of Economics, University of Brescia

Paola Valbones

 

i,

Department of Economics, University of Padova

Firm Regulation and Profit-Sharing: A Real Option Approach

Summary

To avoid high profit levels often experienced in countries where monopolies in public

utility sectors are regulated through price-cap mechanisms, several regulatory agencies

have recently introduced profit-sharing (PS) clauses aimed at obtaining price reductions

to the benefit of consumers. However, the implementation of these PS clauses has often

turned out to be severely con- trained by the incompleteness of the price-cap itself and

the non-verifiability of firms’profits. This paper studies the properties of a second-best

optimal PS mechanism designed by the regulator to induce the regulated monopolist to

divert part of its profits to custormers. In a dynamic model where a reg- ulated

monopolist manages a long-term franchise contract and the regulator has the option to

revoke the contract if there are serious welfare losses, we first derive the welfare

maximising PS mechanism under verifiability of prof- its. Subsequently, we explore the

sustainability of the PS mechanism under non-verifiability of profits. In a infinitehorizon

game, it is showed that the dynamic sustainability of the PS clause crucially

depends upon the magni- tude of the regulator’s revocation cost: the higher this cost, the

lower the profit shared and the less frequent the regulator’s PS introduction. Finally, we

present the endogenous and dynamic price adjustment which follows the adoption of the

investigated PS mechanism in a price-cap regulation setting.

Keywords:

 

Price-cap regulation, Profit-sharing, Real options

JEL Classification:

 

C73, L33, L5

We wish to thank A. Bennardo, G. De Fra ja, S. Martin, P. Régib eau, C. Scarpa, P.

Simmons and seminar participants at Naples, Brescia, Lusanne, Padua, Verona and

Warwick Universities for their useful comments. Both the authors acknowledge

financial support from the MIUR through the Cofin 2004 – No. 2004134814_005 grant.

Address for correspondence:

Michele Moretto

Department of Economics

University of Brescia

Via S.Faustino 74b

25122 Brescia

Italy

E-mail: moretto@eco.unibs.it

1 Introduction

Recent European liberalization and US experience in regulation of public util-

ities shows that price-cap regulation (PCR) allows prices to diverge greatly

from actual costs and often generate “abnormal” pro.ts for the .rm.

 

1

Reg-

ulators dislike high corporate pro.ts under PCR because they reduce the

welfare of consumers and – favouring the .rm – downgrade the regulator.s

reputation for being able to set the “right” price of the service. This is why,

in the last decade, PCRs have been modi.ed in a variety of ways in order to

induce the regulated .rms to rebate part of their pro.ts to customers. Pre-

cisely, regulators have often advocated “pro.t-sharing” (PS) – or “earning-

sharing” – schemes to make the regulated .rm share with its customers a

fraction of the pro.ts it generates beyond a certain level.

In the European experience of regulation, the textbook example for pro.t-

sharing refers to the price-cuts implemented by the British electricity regu-

lator in between 1994 and 1995, well before the o¢ cial price review in 1999:

since the initial price control for the electricity companies turned out to be

over-generous allowing high pro.ts, the regulator intervened reducing prices

and thus returning some of those “excess” pro.ts to consumers.

 

2

Sappington

(2002), among others, shows that these PS practices are usually introduced

in the US telecommunication industry by the regulator in the form of direct

payment to customers or reduced prices for key services.

 

3

The present paper

mainly deals with this latter form of PS.

A fundamental feature of these real-world PS mechanisms is, thus, the

discretion left in the hands of the regulator which entitles the regulator him-

self to adjust the PCR adopted ex-ante calling for an unilateral “renewal”

of the regulatory contract. By its own discretionary nature, though, the PS

mechanism allows for disputes and ex-post re-contracting between the regu-

1

 

This drawback of the price-cap as an incentive mechanism stems from its inability

to set a contingent price that incorporates all the uncertainties faced by the .rm in each

period of the regulatory contract.

2

 

Similar price-cuts have since been made by other regulators as British Gas Transco

(gas transmission and distribution), National Grid Company (electricity transmission)

(Green, 1997). Other often quoted European examples are about Oftel, the British tlc

regulator (Armstrong et al.(1994)), and Ofwat, the water industry regulator in England

and Wales (see the Ofwat.s home page for the current pro.t-sharing mechanism).

3

 

Reduction in prices has been widely employed to regulate intrastate accounting rates

a¤ecting earnings of telecommunication providers (Sappington, 2002).

2

lator and the regulated .rm about both the level of pro.t that should trigger

the sharing rule and the dynamic path that the regulated price should fol-

low. It has been informally argued that this may make it substantially more

di¢ cult to implement PS prescriptions.

 

4

The aim of this paper is to investigate the properties of second-best op-

timal PS mechanisms, designed by regulators to induce regulated .rms to

divert part of their “excessive” pro.ts to custormers. Speci.cally, we analize

a dynamic game with two players: a regulated monopolist who manages a

long-term franchise contract to provide a public utility service (e.g.: water

supply, waste managment, gas or electricity distribution, etc

 

.)

; and a welfare-

maximising regulator who has the right, during the contractual relationship,

to ask for price reductions from the regulated .rm, and to revoke the franchise

contract if he perceives the .rm.s pro.t as “excessively” high. We model the

regulator.s “outside” option to revoke the contract as a perpetual call option

where the regulator – considering the .rm.s pro.t as an underlying asset –

has to determine when to pay an exercise price to re-acquire managment of

the utility and to re-determine provision of the service. Speci.cally, the game

may last an in.nite number of periods, and ends once the regulator exercises

the option to revoke the franchise contract. Each period is divided into four

stages: at the .rst stage, nature chooses the realization of a random variable,

determining the .rm.s pro.t; at the second stage, after having observed the

.rm.s pro.t, the regulator decides whether or not to ask for a price reduc-

tion; at the third stage, the .rm decides whether or not to comply with the

regulator.s prescription; .nally, at the fourth stage, the regulator, based on

the price set by the monopolist, may revoke the contract.

It turns out that the assumption on pro.t veri.ability a¤ects the analysis

of the game beginning with the second stage: indeed, when the pro.t – and

the other regulatory variables – are observable but nonveri.able, the regulator

cannot force the .rm to cut “excessive” pro.ts as “no court or other third

party will accept to arbitrate a claim based on the value taken by these

variables”

 

5

. This implies that – when pro.t are not veri.able – the .rm can

e¤ectively choose whether to comply or not with the PS rule, retaining all

pro.ts above the pro.t threshold that triggers the PS until the regulator

revokes the contract. In contrast, when the veri.ability of pro.t is assumed,

4

 

As stressed by Green (1997), the PS would require audited cost information for cal-

culating allowable pro.ts (and prices) levels, information which are often very di¢ cult to

be collected by the regulator.

5

 

Salanié (1997, p. 177).

3

the regulator.s PS prescription reduces the game to a take-it-or-leave-it o¤er

to the .rm.

For the sake of clarity, the analysis is broken into two parts. As a bench-

mark, we .rstly investigate the simpler, though less realistic, case where the

.rm.s pro.t is veri.able and – consequently – the PS mechanism imposes

contractual obligations contingent on realized pro.ts; then, we move to the

more realistic case of pro.t.s non-veri.ability.

In the .rst part of the analysis, after having formally de.ned the PS clause

we investigate, we identify the pro.t threshold that determines the regulator.s

introduction of the welfare-maximizing PS rule. We then demonstrate that

at such a pro.t level the regulator is indi¤erent between contract closure and

imposing the PS. Hence, we formally show that in the unique equilibrium

of the game the monopolist will always comply with the regulator.s PS rule

and the contract will never be revoked; the regulator will impose the PS rule

whenever the pro.t is higher than the identi.ed trigger level.

In the second part of our analysis we turn to the case of pro.t non ver-

i.ability. The main di¤erence with the previous case is that a monopolist,

who decides not to comply with the PS rule, can – now – retain all the pro.ts

above pro.t.s threshold triggering PS until the regulator calls for contract

closure. The regulator will now revoke the contract, say at period

 

t

, only if

revocation at that period is e¤ectively his best reply. In other words, in the

absence of pro.t veri.ability, an incentive constraint imposing dynamic op-

timality of the revocation policy must be satis.ed in order to make credible

the regulator.s revocation threat.

In this setting, we formally show that for all the pro.t levels higher that

the pro.t threshold which makes the regulator indi¤erent between contract

closure and imposing the PS, it is sequentially optimal for the regulator to

revoke the contract, while revoking the contract for lower pro.t levels will

never satisfy the sequentiality. Hence, the perfect equilibrium of the game

with pro.ts non-veri.ability is also such that the .rm complies with the PS

rule chosen by the regulator in each period, as long as the revocation has not

been carried out.

 

6

In equilibrium, the expectation of being able to induce pro.t-sharing

makes it rational for the regulator not to exercise its option to revoke, and

this fact also makes it rational for the .rm to continue to comply with the

6

 

E¢ cient sub-game perfect equilibria in in.nite-horizon threat-games are investigated

in Klein and O.Flaherty, 1993; Shavell and Spier, 2002.

4

PS prescription. On the one hand, given that for the monopolist the loss

from revocation of the (in.nitely-lived) contract is greater than the expected

stream of pro.t cuts (prescribed by the PS rule), it will be e¢ cient for the

.rm to continuosly maintain pro.t at a level lower than the threshold that

triggers the PS. On the other hand, the regulator will revoke the contract for

any pro.t level higher than the pro.t threshold that triggers the PS.

Our model also shows that the sustainability of the PS mechanism cru-

cially depends upon the magnitude of the regulator.s revocation cost. Such

cost represents a form of capture of the regulator by the .rm

 

7

: the higher

the revocation cost, the lower the pro.t shared and the less frequent the

regulator.s PS prescription.

Finally, as in our model the .rm.s pro.t is stochastically determined, the

pro.t threshold that triggers the PS rule de.nes an (expected) time interval

after which the monopolist will be asked for the .rst time to reduce its price

and a time interval between each pair of regulatory reviews asking for price

reductions. We show that both the equilibrium trigger pro.t level as well as

the time interval between each pair of regulatory reviews (i.e. the regulatory

lag) positively depend on the regulator.s revocation cost.

The present paper is related to two di¤erent strands of literature. On a

formal level, our paper builds upon the literature on the stochastic control

techniques developed to identify optimal timing rules and optimal barrier reg-

ulations.

 

8

These techniques are widely used in the literature of irreversible in-

vestments,

 

9

and emphasize the option value of delaying investment decisions,

i.e. the value of waiting for better (although never complete) information on

the stochastic evolution of a basic asset.

In reference to the economic literature on the regulation of .rms, our pa-

per takes stock of the studies on drastic regulatory changes such as stochastic

regulatory review (Bawa and Sibley, 1984) and expropriation by the regula-

tor (Salant and Woroch,1992; Gilbert and Newbery, 1994). Bawa and Sibley

(1984) show that the .rm.s incentive to indulge in over-capitalization can be

tempered by the fact that this raises pro.ts and – consequently – makes it

more likely that the regulator will cut prices; in contrast to their approach,

which emphasizes strategic .rm behaviour

 

vis a vis

both the likelihood reg-

ulator review and price adjustment, we focus on the regulator.s decision to

7

 

For a discussion on the large literature on regulatory capture we refer to La¤ont and

Tirole (1994, chapt. 11).

8

 

See Harrison and Taksar (1983), Harrison (1985) and Dixit (1993).

9

 

Dixit and Pindyck (1994) is the seminal text in this area.

5

impose a regulatory review in the form of a PS rule and on the informational

conditions which makes it enforceable.

Salant and Woroch (1992) and Gilbert and Newbery (1994) present mod-

els on expropriation by the regulator where the price regulation occurs en-

dogenously as a self-enforcing and mutually bene.cial cooperative equilib-

rium. In both those discrete-time repeated game frameworks, the regulatory

lag does not a¤ect the players.behaviour. In contrast, in our continuous-time

repeated game model, the explicit unilateral approach to contract renewal

 

10

– i.e. the regulator sets the PS rule or calls for contract closure – allows us

either to determine the regulatory lags endogenously or to study its determi-

nants. Speci.cally, in our framework, the endogeneity of the regulatory lag

 

11

consistently belongs both to the level of the .rm.s pro.t and the regulator.s

revocation cost.

According to the approach of this paper, the regulator.s decision to intro-

duce the PS rule is based on welfare consideration and modelled along with

the option to revoke the contract: both these elements are absent – as far as

we know – from the literature on PS and expropriation by the regulator. In-

deed, these considerations allow us to recognize, .rstly, that the PS calls for

an intertemporal regulatory setting and, secondly, that market uncertainty

and the regulator.s credible threat to close the contract, represent relevant

issues in the PS de.nition and its enforcement.

Finally, we ought to mention a limit of the present analysis: we do not

consider in this paper the well-known trade-o¤generated by the introduction

of a PS rule between lowering extreme pro.ts and dulling the .rm.s incentive

for cost reduction.

 

12

This is because our model, though simple, focusses on

the essential features which characterize the regulator.s motivation and the

information setting for enforcing a PS rule, leaving the .rm the strategic

choice of compling with the PS rule or not complying.

The remainder of the paper is structured as follows. Section 2 presents

10

 

For an analysis of incentives to call for contract renewal from a bilateral perspective,

see Andersen and Christensen (2002).

11

 

As discussed in La¤ont and Tirole (1994, p.15), endogeneity of the regulatory lag –

which is in the essence of the most real-life adopted regulation mechanism – is especially

important when the incentive properties of regulation are investigated.

12

 

In this respect the literature has mainly stressed that compulsory sharing of pro.t

may: a) reduce the .rm.s incentive to minimize operating costs and increase revenue

(Lyon, 1996; Crew and Kleindorfer, 1996); b) provide an incentive to undertake projects

that are unduly risky (Blackmon, 1994); c) lead the utility to delay investment (Moretto

et al., 2003).

6

the basic model in which a PS rule is introduced. Section 3, derives the regu-

lator.s value of the option to revoke the contract as well as the optimal pro.t

threshold that triggers it (

 

Proposition 1

): when .rm.s pro.ts are veri.able,

this threshold indeed represents the optimal level at which to introduce the

PS rule (

 

Proposition 2

). Section 4 explores PS sustainability when the .rm.s

pro.t is nonveri.able (

 

Proposition 3

). Section 5 investigates the price ad-

justment which follows the introduction of a PS rule in a PCR setting, and

.nally Section 6 concludes with some implications of the study for policy and

a suggestion of how the model could be extended.

2 The model

We consider a risk-neutral pro.t-maximizing .rm managing a one-time sunk

indivisible project for the provision of a public utility under a long-term

franchise contract. We assume that no new investments

 

13

are undertaken

during the contract period; this assumption emphasizes that the focus of

our analysis is on the .rm-regulator relationship in a regulatory framework

designed to cope with high pro.ts, not with conduct that aims to conceal

high pro.ts.

Moreover, we assume that the in.nitely lived project produces a .ow of

pro.ts

 

t

which evolves over time according to a geometric Brownian motion,

with instantaneous rate of growth

 


>
0 and instananeous volatility  

0 :

d

 

t =

tdt + tdWt; 0 = 

(1)

where

 

dWt

is the standard increment of a Wiener process, uncorrelated over

time and satisfying the conditions that

 

E(dWt) = 0 and E(dW

2

t

 

) = dt:

Hence, under these assumptions the value of an in.nite project,

 

V ();

can

be expressed by (Harrison, 1985, pag.44):

V

 

() = E

0

Z

1

0



 

te􀀀tdt j 0 =





=





 

􀀀


(2)

where

 

 >
is the constant risk-free rate of interest14. As V

is a multiple of



 

, it also is a geometric Brownian motion with the same parameters

and

13

 

Moretto et al. (2003) investigate endogenous investment in pro.t-sharing regulation.

14

 

Alternatively, we could use a discount rate that includes an appropriate adjustment

for risk and take the expectation with respect to a distribution for

 



that is adjusted for

risk neutrality (see Cox and Ross, 1976; Harrison and Kreps, 1979).

7



 

:

dV

 

t =
V
tdt + VtdWt; with V0 = V

(3)

Although equation (2) is an abstraction from real projects, we can think

at

 



as the “reduced form” of a more complex model where the instantaneous

cash .ow

 

t = (zt) depends on a vector of variables zt

, which may include

the market price, the quality of the service, the .rm.s investments and market

shocks that account for some sources of uncertainty in consumer demand

and/or technological choice.

In our model, when the monopolist makes “huge” pro.ts, the regulator

can introduce a PS rule to divert these “excess” pro.ts to consumers, or

revoke the .rm.s contract to re-obtain responsability for managing the utility

and re-address the project.s pro.tability. In what follows, we .rstly model

how the PS rule is designed and, in the next section, how the contract closure

is modelled.

Among the many ways of introducing PS, the simplest one is the setting

of an upper bound on pro.ts by the regulator,

 

.15 In technical terms 

is

a re.ecting barrier, i.e. at

 

; a “pro.t cut” stops t from going above 

.

To set up an appropriate mathematical model representing the above PS

rule we are guided by the theory of optimal barrier regulations (Harrison and

Taksar, 1983; Harrison, 1985). In addition, as from (2) choosing

 



is equiva-

lent to choosing an upper limit to the value of the project

 

V 

, hereinafter we

take

 

Vt

as the primitive exogenous state variable for the regulatory process

we are considering. Thus, if the monopolist starts with the initial project.s

value

 

V0 < V , the regulator.s PS rule applies as follows:

16



 

for Vt < V ; let Vt

evolve on its own and follow the geometric Brownian

motion (3);



 

for Vt  V ; introduce the PS rule drt to stop Vt from going above V 

.

The new “regulated” process

 

Vt 􀀀 rt

can be described by the following

15

 

For qualitatively analogous rules see Sappington and Weisman (1996).For a more

general discussion about pro.t-sharing rule adopted in the recent experience of public

utility regulation see Sappington (2002)

16

 

Really, this is a “value-sharing” rule: we call it PS as there is a one-to-one relationship

between the .rm.s value and pro.ts. See Moretto and Valbonesi (2000) for the explicit

model of the production decision of the .rm.

8

stochastic di¤erential equation

 

17

:

dV

 

t =
V
tdt + VtdWt 􀀀 drt; V0 = V; for Vt 2 (0; V ]

(4)

where the increment

 

drt

represents the .rm.s pro.t reduction between

time

 

t and t + dt

.

The PS rule is then a process proportional to

 

Vt

, parametrized by the ini-

tial condition

 

V ;

right-continuous, non-decreasing and non-negative, de.ned

as:

r

 

t = a(V )Vt if Vt  V ;

(5)

where

 

a(V )  [1 􀀀 infTv

t



V

 



V

 

v



]

 

; T = inf(t  T j Vt 􀀀 V  = 0+)

and

r

 

t = 0 for all t  T

(see Appendix A). As shown in Figure 1 below, the

PS de.ned in (5) increases to keep

 

Vt lower than V 

and is given by the

cumulative amount of pro.t control exerted on the sample path of

 

Vt

up to

t:

 

18

Figure 1 – about here –

17

 

Stochastic di¤erential equations such as (5) are a notational convenience, since only

their integral counterparts are well de.ned. The “impulse”

 

dr

must be interpreted as

potentially taking .nite values when a discrete jump occurs (Harrison and Taksar, 1983;

Harrison, 1985).

18

 

It is worth noting that the above setup allows us to deal with more complex pro.t-

sharing mechanisms. For example, suppose at

 

V 

the regulator introduces a PS rule

de.ned as a percentage cut of the .rm.s pro.ts: we can model this rule adding a new

stochastic di¤erential equation for the pro.t cut. That is:

dV

 

t =
V
tdt + VtdWt 􀀀 drt

;

and

dM

 

t = sdr

t

where

 

M is de.ned by giving up 1=s unit of V for each unit of M: Threfore, above V 

;

the PS rule can now be reformulated in terms of the new variable

 

Yt = Vt=Mt:

When the

existing combination of

 

(Vt;Mt) places Yt above s;

the regulator intervenes immediately

by cutting back on pro.ts (

 

drt > 0

). The amount of pro.ts cut is very small and is such

as to push the .rm.s value along a line of slope

 

1=s

.

9

3 The optimal PS rule

In the previous section we have modelled the PS rule (5) for a given exoge-

nous upper bound value

 

V 

. We now need to de.ne which value triggers

the regulator.s PS introduction as well as the regulator.s adoption of the

alternative strategy, the option to revoke the contract.

In this section we perfom our analysis under the simpler assumption of

the .rm.s pro.t veri.ability which – in this context – refers to the fact that

the pro.t level can be proved in court: this implies that the regulator.s threat

of contract closure becomes binding for every .rm.s pro.t level higher than

the optimal trigger.

We assume the regulator minimizes an intertemporal social welfare loss

function which is an increasing function of the .rm.s pro.t: indeed, an in-

crease in the monopolist.s pro.ts reduces the monetary value of consumer

welfare. In this perspective, if the .rm.s pro.ts becomes too high, that is, if

the social welfare loss is too large, the regulator adopts one of the following

alternative and equivalent strategies:

1. introducing a PS rule – de.ned as (5) – to divert pro.ts from the .rm

to consumers;

2. revoking the contract and re-determiing provision, thus re-addressing

the project.s pro.tability.

Contract closure is then an “outside” option the regulator can always

exercise. We model this option as a perpetual call option, with the project.s

value

 

V

as the underlying asset: thus, by considering a social welfare loss

function, the regulator revokes the contract if

 

V

exceeds a critical threshold

V

 



.

In this section we prove that (5) is – under the assumption of pro.t veri.-

ability – optimally determined by .xing

 

V  = V 

, which makes the regulator

indi¤erent between revoking the contract and applying the PS rule. We ob-

tain this result by .rstly determining the value

 

V 

that triggers revocation

(

 

Proposition 1), and then showing that V 

is indeed the regulator.s optimal

trigger to introduce the PS rule (5) (

 

Proposition 2

).

10

3.1 Social welfare and the option to revoke

The regulator.s intertemporal loss function when the contract is revoked at

time

 

T is:

19

L

 

(VT ; V0) + (I 􀀀 VT )

(6)

where

 

L(VT ; V0)

is the consumers.welfare reduction up to the revocation

time

 

T; V0 is the value of the project at time zero, and LV (VT ; V0) > 0

;

L

 

(V0; V0)  0: The term I 􀀀 VT

is the regulator.s (net) cost of revocation.

Indeed, revocation is costly as the contract closure determines that the man-

agement of the project is back in the regulator.s hands and this implies that

the regulator should implement the new utility provision (i.e.: through direct

management, privatization or contracting out to another .rm). Speci.cally,

this cost of revocation captures the regulator.s cost in .nding for a new fran-

chisee, or – in the case of direct provision of the service – in training and

hiring new personnel and/or adopting new technologies, or legal expenditure

if the .rm decides to sue the regulator, or, more generally, any cost belonging

from regulatory capture from the .rm.

 

20

Since minimizing (6) is equivalent to maximizing

 

VT 􀀀 I 􀀀 L(VT ; V0)

,

it is evident that rent extraction can be part of the regulator.s objective in

revoking the contract.

 

21

Exercising the option to revoke requires the payment

of the sunk cost

 

I plus the social cost L(VT ; V0). By the sunkness of I;

it is

never optimal to revoke when

 

VT 􀀀I 􀀀L(VT ; V0)

is equal to zero, that is, for

the regulator, it is better to wait until the value reaches a higher level.

Among the many possible ways of modelling the social welfare loss, we

adopt a utilitarian criterion and approximate

 

L(VT ; V0) as (VT 􀀀V0)

, where

0

 

   1

can be interpreted as the .scal distortion in raising public funds

if the service has to be run in

 

house.22 De.ning F(V )

as the value of the

19

 

Our results will still hold if the regulator.s preference also takes into account .rm.s

value, but give it less weight than public funds.

20

 

For a discussion about the di¤erent sources of regulatory capture from the .rm, see

La¤ont and Tirole (1994, chapter 11).

21

 

See Crew and Kleindorfer (1996, p. 218), for a discussion on rent extraction as included

in the regulator.s objective function.

22

 

According to the utilitarian criterion we can approximate the welfare function at time

T

 

by the weighted and discounted average of the net surplus of consumers K 􀀀(1+)V

T

and the value of the project

 

VT

. Hence, the social welfare loss is simply given by:

L

 

(VT )  K 􀀀 V0 􀀀 [K 􀀀 VT ] = (VT 􀀀 V0

)

11

option at

 

t = 0

, we get:

F

 

(V

) = max

T

E

 

0

 

(

 

VT 􀀀 I 􀀀 L(VT ))e􀀀T j V0 =

V



(7)

= max

T

E

 

0

h

(1

 

􀀀 )VT 􀀀 ^I)e􀀀T j V0 =

V

i

where

 

T(V ) = inf (t  0 j Vt 􀀀 V  = 0+)

is the unknown future time when

the option is exercised,

 

V 

is the threshold value that triggers that action

and

 

^I  I 􀀀V0

is the exercise price. The optimization is subject to (3) and

V

 

0

23

 

.

Note that

 

F(V )

is a perpetual call option. By using standard results in

the (real) option valuation (Dixit and Pindyck, 1994), the solution of (7) is

given by:

Proposition 1

 

The value of the regulator.s option to revoke at time t 

0

is given by:

F

 

(V

) =

8<

:

A

 

(V )V for all V < V



(1

 

􀀀 )V 􀀀 ^I for all V  V



(8)

where:

V

 



=

 

 

 

􀀀

1

1

1

 

􀀀



^

 

I;

with

 

 

 

􀀀

1

>

 

124

(9)

and:

A

 

(V 

) =

1

 

􀀀



 

(

 

V )1􀀀 > 0

(10)

Proof.

 

see Appendix B

Hence, the regulator.s optimal revocation rule can be expressed as: .

 

Revoke

the contract as soon as the value of the project exceeds the adjusted

break-even value

 

V 

..

Inspection of the opportunity cost

 



in (9) reveals that:

where

 

K

is the expected value of the consumers.willingness to pay for the service (La¤ont

and Tirole, 1994).

23

 

Moreover, for a consistent optimal revocation, we must also assume that ^I > 0

and

V

 

 􀀀 ^I > 0

.

24

 

> 1 is the positive root of the quadratic equation: ( ) =

1

2

 

2 ( 􀀀1)+
􀀀

= 0

12



 

as  ! 0;

i.e. the regulator becomes socially .indi¤erent.between the

direct management of the utility and the franchising contract to a .rm,

V

 

 drops to

 

 

􀀀1

I and the probability of revocation increases.



 

as  ! 1

, i.e. the opportunity cost of direct management by the

regulator rises,

 

V  ! 1

and the regulator never revokes.

To interpret (8), let.s rewrite it in the following form:

F

 

(V

) =

h

(1

 

􀀀 )V  􀀀 ^I

)

i

V

V

 





 

(11)

=

h

(1

 

􀀀 )V  􀀀 ^I

)

i

E

 

0

 

e

 

􀀀T



Maximizing (7) means maximizing the expected discounted value of the

net bene.t

 

(1 􀀀 )V  􀀀 ^I when the utility is expropriated at time T

, where

E

 

0

 

e

 

􀀀

T



=

􀀀

 

V

V

 





 

<

 

1

is the expected discount factor. Then, maximizing

(11) with respect to

 

V 

gives the optimal revocation trigger as in (9).

3.2 Revocation vs Pro.t-Sharing

Since for

 

Vt > V 

it is optimal for the regulator to revoke the contract

to minimize social welfare loss, in this section we demonstrate that

 

V 

is

indeed the optimal re.ecting barrier for the regulator.s introduction of the

PS mechanism (5). That is, by setting

 

V  = V 

, the regulator is indi¤erent

between applying the PS rule and revoking the contract.

First, we .nd the regulator.s welfare loss function when the PS has been

adopted. Denoting the expected value of future cumulative pro.t reduction

due to (5) by

 

R(VT ; V ); the regulator.s loss function at T

becomes:



 

(VT 􀀀 V0) + (I 􀀀 VT ) + (1 􀀀 )R(VT ; V )

(12)

where

 

V 

is a generic re.ecting barrier. Next, the value of the regulator.s

option to revoke (7) at zero is now:

F

 

r(V

) = max

T

E

 

r

0

h

(1

 

􀀀 )VT 􀀀 ^I 􀀀 (1 􀀀 )R(VT ; V ))e􀀀T j V0 =

V

i

(13)

where the superscripts refer to the PS rule (5) and the optimization is subject

to (4) and

 

V0

. Solution of (13) shows that:

13

Proposition 2

 

i) The regulator.s optimal revocation trigger once the PS is

adopted is equal to:

V

 



=

 

 

 

􀀀

1

1

1

 

􀀀



^

 

I

(14)

ii) If

 

V  = V ;

the PS rule (5) keeps the regulator indi¤erent to revoca-

tion, i.e.:

F

 

r(Vt) = 0 for t  0

(15)

Proof.

 

see Appendix C

Proposition 2 ascertains the optimality of the sharing rule (5) with re-

spect to the regulator.s alternative equivalent strategy, that is, the optimal

regulator.s contract closure: if the .rm keeps pro.ts below

 

V ;

revocation

is never optimal.

Proposition 2 provides further considerations. First, the optimal revoca-

tion trigger under PS is equal to the optimal revocation trigger without PS

as in (9). This is just an application of the dynamic programming principle

of optimality: if at

 

t = 0 the regulator sets V 

as the optimal revocation

trigger, this should be optimal for any

 

t > 0;

independently of any future

policy after

 

V 

.

Second, if the regulator sets

 

V  = V 

as a re.ecting barrier, the value of

its option to revoke is always equal to zero. The intuition behind this relevant

result is a straightforward implication of the barrier control

 

rt

applied to the

process

 

Vt:

Indeed, the true cost of exercising the option for the regulator is

not just equal to the strike price

 

^I

, but also includes the future pro.t cuts

R

 

(Vt; V ) and the value of the forgone option Fr(Vt)

. Thus, the net expected

present value of optimal exercise at time

 

t  0

is:

E

 

r

t

n

[(1

 

􀀀 )V  􀀀 ^I 􀀀 Fr(V )]e􀀀(t􀀀T

)

o

􀀀

 

(1􀀀)R(Vt; V ) = 􀀀Fr(V 

)



V

 

t

V

 





 

(16)

where the last equality follows from

 

R(Vt; V ) = [V  􀀀 ^

I

1

 

􀀀

]

􀀀

 

V

t

V

 





 

(see Ap-

pendix C). Maximizing (16) with respect to

 

V 

gives:

 

F

 

r(V 

)

V

 

 􀀀 Fr0(V ) = 0

(17)

Since to avoid arbitrage at

 

V 

the second term of (17) must be equal to

14

zero, we get (15).

 

25

That is, from (16), the regulator is indi¤erent between

introducing the PS rule and revoking the contract when the expected bene.ts

from pro.t regulation exactly o¤set the expected social welfare loss due to

the monopolist.s excess pro.ts.

Finally, although the PS rule (5) is simply proportional to the project.s

value, several novel implications follow:



 

rt is parametrized by the initial condition V 

which, in turn, depends

on the revocation cost

 

I and on the opportunity cost parameter :

An

increase in

 

I and  decreases rt

:



 

rt

is non-decreasing and is given by the cumulative amount of pro.t

cuts exerted on the sample path of

 

Vt up to t: Thus, rt

relates to past

realizations of

 

Vt

, which makes the PS time-dependent.

4 E¢ ciency of the PS

An important facet of the PS rule – analysed in the previous sections – is

its dynamic sustainability when the .rm.s pro.t are observable, but nonver-

i.able. In this case, the regulator cannot force the .rm to cut “excessive”

pro.ts as “no court or other third party will accept to arbitrate a claim based

on the value taken by these variables” (Salanié, 1997, p.177). Then, given

the assumption of pro.t.s non-veri.ability, is still the regulator.s threat of

contract closure in itself su¢ cient to induce the .rm to comply with (5) as

V

 

t intersects V 

? In this section we formally demonstrate that the proposed

PS rule sustains a perfect equilibrium for the repeated continuous time reg-

ulatory relationship that starts at

 

T

: we do this by showing that any .rm.s

deviation from (5) makes contract closure worthwhile for the regulator. In

addition, since

 

Vt

is a Markov process, it is easy to ascertain that the equi-

librium is also sub-game perfect.

25

 

The function Fr(Vt)

is de.ned as the expected value of the regulator.s net bene.t when

the utility is expropriated at time

 

T:

As the net bene.t is a continuous function of the

primitive process

 

Vt, also Fr is a continuous function except perhaps when Vt = V 

and

the pro.t sharing rule

 

rt is applied. The behavior around Vt = V 

is given by expanding

F

 

r(Vt)

as:

F

 

r(V ) = Fr(V  􀀀 dr) = Fr(V ) 􀀀 Fr0(V )

dr

which yields

 

Fr0(V ) = 0 :

This condition holds at any re.ecting barrier without any

optimization being involved (Dixit, 1993).

15

The regulatory game we consider here lasts a possibly in.nite

 

26

number

of periods, and ends once the regulator exercises the option to revoke the

franchise contract. Each period is divided in four stages: at the .rst stage,

the regulated monopolist is delegated to manage the supply of a public utility

and nature chooses a parameter determining the pro.t of the .rm. At the

second stage, after having observing the .rm.s pro.t, the regulator decides

whether or not to ask for a PS: if the regulator perceives that the monop-

olist is making “excessively” high pro.ts, he sets a pro.t ceiling, say

 

V 

,

according to (9), above which the PS rule (5) applies; the regulator accom-

panies its announcement with a threat to revoke the contract if the .rm does

not comply.

 

27

. At the third stage, the monopolist decides whether or not

to comply with the regulator.s prescription (i.e.: that is, whether or not to

start with stream of payment

 

rt  0)28

. At the fourth stage, the regulator,

conditional on the price sets by the monopolist, decides whether or not to

revoke the contract. If the regulator does revoke, the monopolist su¤ers the

loss

 

Vt and the regulator obtains Vt 􀀀 I

(i.e.: the net gain from revocation).

If the regulator does not revoke, the game goes ahead to the next period and

it is repeated.

However, without a binding commitment by the regulator, any .nite num-

ber of .rm pro.t reductions will be ine¢ cient. In fact, the regulator.s prob-

lem is that for any

 

t  T

he has an incentive to carry out his threat, even

if the monopolist reduces its pro.ts. Since this means that the monopolist

will not ward o¤ the threat by reducing its pro.ts, the monopolist will not

reduce them. Thus, the unique sub-game perfect equilibrium is ine¢ cient:

revocation is carried out regardless of the monopolist.s positive net present

value. To avoid this ine¢ ciency the .rm must continuously “control” its

pro.ts; that is, for

 

t  T, the monopolist should consider V 

as the ceiling

not to be crossed, and reduce its expected pro.ts just enough to keep

 

Vt

<

V

 

 to prevent contract closure:29

To summarize:

26

 

Owing to uncertainty, neither .rm nor regulator can perfectly predict Vt

each time.

As

 

Vt follows a random walk, for each time interval dt

there is a constant probability of

moving up or down. Therefore, the game ends in .nite (stochastic) time with probability

one, but everything proceeds as if the horizon were in.nite.

27

 

By the Markov Property of (22), in our model it is not important when the regulator

announces

 

V  as long as it is between zero and T

.

28

 

In our in.nite-lived project without investment, the .rm.s dominant strategy is not

to make the payment

 

rt  0

, that is not to comply.

29

 

There are many e¢ cient sub-game perfect equilibria where the threat of revocation

induces an in.nite .ow of payments by the .rm to prevent contract closure (see Shavell

16

Proposition 3

 

The following strategy represents a sub-game perfect equilib-

rium:

i) As long as

 

Vt < V  nothing is done. As soon as Vt crosses V 

from

below, the monopolist reduces its pro.ts by (5) and the regulator does not

revoke.

ii) Pro.t regulation ends in .nite (stochastic) time with probability one.

Proof.

 

see Appendix D

As both the players expect an in.nite repetition of their relationship,

their choices in each period will depend on the previous moves. The players.

strategy for each period

 

t  T

can be described as follows: the monopolist

observes

 

Vt and chooses to impose a pro.t reduction rt

or not; the regulator

.Does not Revoke.if the .rm has followed the rule

 

rt to keep Vt < V 

for all

t

 

0 < t30

. On the contrary, the regulator .Revokes.if the .rm has deviated

from

 

rt at any t0

< t:

Our stochastic-continuous time framework calls for an instantaneous re-

ply by the regulator when the monopolist departs from the PS rule (5), that

is, the regulator adopts the most severe punishment: revocation.

 

31

The reg-

ulator believes that this mechanism, from the initial date and state

 

(T; V )

;

will be retained for the whole (stochastic) planning horizon. Since the project

we model here is in.nitely-lived, the present value of forgone pro.ts will in-

deed always ensure participation by the .rm, and the expectation of future

pro.t regulations keeps the regulator from carrying out his threat.

 

32

Finally, the second part of the

 

Proposition 3

says that with probability

one the pro.t regulation ends in the (stochastic) .nite interval. Intuitively,

although the .rm prefers to cut pro.ts rather than terminate the contract

(i.e. the loss from closure is greater than the expected pro.t cuts), it always

and Spier, 1996, Proposition 2).

30

 

In our continuous time setting we assume, without any loss of generality, that when

the regulator is indi¤erent between revoking the contract and not revoking, he does not

exercise the option.

31

 

In continuous time repeated games there is no notion of last time before t

, so induction

cannot be applied. For examples on how to represent continuous time as a sequence of

discrete-time grids that becomes in.nitely negligible, we refer the reader to Simon and

Stinchcombe (1989) and Bergin and MacLeod (1993).

32

 

Considering a long-term, but .nite, franchise contract, the .rm.s opportunistic be-

haviour in the last period of the contract should be taken into account: in this case, the

.rm.s incentive into comply with the regulator.s PS prescription are di¤erent from that

we have modelled here.

17

prefers to stop payment if the threat of revocation is not carried out. The

.rm “regulates” pro.ts until

 

Vt  V according to rt; but when Vt

reaches,

for the .rst time after

 

T; the trigger V 

again, it ceases regulation. That

is, at

 

T0 = inf(t  T j Vt 􀀀 V  = 0􀀀), if the .rm sets rT0 = 0

, the regulator

will face a jump from zero to

 

F(V )

but revocation is not carried out (See

Figure 1). The game then starts afresh.

5 PCR and dynamic price adjustment

Let.s conclude showing how the above PS rule can be implemented in a PCR

setting. To do this we introduce a reduced form for the .rm.s pro.t function

(1) that depends only on the price of the service and a demand shock, i.e.

z

 

t = (pt; t)

. That is, we assume that:

1. The market demand at time

 

t

is a constant-elasticity function of the

price

 

pt

:

D

 

(pt) = tp􀀀

t

 

(18)

with

 

2 (0; 1)

:

2. The random parameter

 

t

follows a trendless, geometric Brownian mo-

tion, with instantaneous volatility

 

 > 0;

i.e.:

d

 

t = tdWt; with 0 = 

(19)

where

 

dWt is the standard increment of a Wiener process33

.

3. No operating costs are present but there is a per period .xed cost

 

c34

.

Then, the monopolist.s project gives a pro.t .ow at each time

 

t

equal

to

 

35

:



 

(pt; t) = v(pt; t) 􀀀 c  p1􀀀

t

 

t

􀀀 c: (20)

33

 

By the Markov Property of (19), the quality of all subsequent results does not change

if we assume a trend for demand to increase.

34

 

The .xed costs we consider here are, as standard in the literature, .ow .xed costs

of production: that is, we assume that the .rm begins the .rst period endowed with a

technology whose operation entails a .ow cost

 

c

per unit of time.

35

 

We avoid, for simplicity, considering operating options such as reducing output or even

shutting down that incresaes the value of the .rm (MacDonald and Siegel, 1986; Dixit and

Pindyck,1994, chs. 6 and 7).

18

4. The monopolist is subject to price regulation. The price

 

p

is allowed to

increase by the di¤erence between expected in.ation (the Retail Price

Index,

 

RPI

) and an exogenously given expected increase in productiv-

ity over time (

 

x

):

dp

 

t = (RPI 􀀀 x)ptdt; with p0 = p

(21)

These assumptions enable us to reduce the model to one dimension. Ex-

panding

 

d(pt; t)

and applying the Itô.s lemma it is easy to show that

v

 

(pt; t)

is driven by the following geometric Brownian motion:

dv

 

t =
v
tdt + vtdWt with v0 = v;

(22)

with:

 

 

 (1 􀀀 )(RPI 􀀀 x

)

The drift parameter of the process

 

vt

is a linear combination of the corre-

sponding parameters of the primitive process

 

t

and of the price-cap rule (21).

Finally, since the monopolist is risk-neutral, using the simpli.ed expression

for the pro.t function (22), the market value of the project becomes:

V

 

=

v



 

􀀀

􀀀

c



(23)

As far as the price-cap revision is concerned, in the event of the .rm.s

pro.ts going beyond a “pre-determined” level, the PS rule requires the

 

x

factor to be automatically adjusted upward, making the price-cap adjustment

rate

 

RPI 􀀀x

more stringent (Sappington, 2002). According to this practice

we can rewrite (4) as:

dV

 

t = (1 􀀀 )(RPI 􀀀 x0)Vtdt + VtdWt; V0 = V; for V 2 (0; V ]

(24)

where

 

x0 = x

􀀀

d

 

inf

0

 

v

t

(

 

V =Vv)

=dt

(1

 

􀀀

) inf

0

 

v

t

(

 

V =Vv) > x

is the new price decrease factor which

stops the process

 

Vt from going any higher than V : How the x0

factor works

seems intuitively appealing. As the numerical value for

 

V 

is known, by (23)

the optimal revocation trigger (9) can be written as

 

p1􀀀

t

 

t =

 

 

􀀀

1

1

1

 

􀀀





 

􀀀




 

(c+



 

^I); from which the boundary value for 

is determined by:



 

(pt

) =

 

 

 

􀀀

1

1

1

 

􀀀





 

􀀀



c

 

+ ^

I

p

 

1􀀀

t

(25)

19

For any given value of the price

 

pt; random .uctuations of t

move the

point (

 

t; pt

) horizontally left or right. If the point goes to the right of the

boundary, then a price reduction is made immediately shifting the point

down to the boundary. If

 

t

stays to the left of the boundary, no new price

reduction is applied. Thus, price reduction proceeds gradually to maintain

(25) the equality. To illustrate, suppose

 

RPI 􀀀 x = 0 so that pt = p0

for

all

 

t; by inverting (25) we obtain the optimal boundary function p(t)

which

determines the optimal price regulation as a function of the sole state variable



 

t

:

p

 

t = p

0





 





 

t



 

1=1􀀀

with

dp

 

t

d

 

t

<

 

0

(26)

Futhermore, higher costs shift the boundary (25) to the right,

 

0 > 



and determine .rm.s smaller price reduction to comply with the PS rule

 

36

.

The boundary function for this case is shown in Figure 2 below.

Figure 2 – about here –

6 Final Remarks

In this paper we employ a real options approach to investigate the properties

of a second-best optimal pro.t-sharing (PS) mechanismimposed by a welfare-

maximising regulator. We consider a dynamic setting where a regulated

monopolist is delegated to manage a long-term franchise contract to supply a

public utility. If the monopolist.s pro.t becomes “excessively” high, that is, if

the social welfare loss is too large, we have assumed that the regulator always

has the possibility to adopt one of the following alternative and equivalent

strategies:

– introducing a PS mechanismto divert pro.ts fromthe .rmto consumers;

– revoking the contract and re-determing provision, thus re-addressing the

project.s pro.tability.

Speci.cally, we have modelled the regulator.s option to revoke as a per-

petual call option which is a function of the .rm.s pro.ts, social welfare and

the regulator.s cost of revocation.

36

 

As matter of fact, cost padding by the franchisee is another strategy that might be

used to avoid the appearance of excess pro.t. It would be possible to model the franchisee

as reporting costs and the regulator as employing auditors to determine the accuracy of

cost reports, but this is not our topic here.

20

Under the assumption of veri.ability of pro.ts, we have endogenously

determined the pro.t threshold that triggers revocation (

 

Proposition 1

) and

proved that this threshold keeps the regulator indi¤erent between revoking

the contract and applying the PS rule (

 

Proposition 2

). We formally show

that in the unique equilibrium of the game the monopolist will always comply

with the regulator.s PS rule and the contract closure will never occur; the

regulator will impose the PS rule whenever the pro.t is higher than the

identi.ed trigger level.

Under the more realistic assumption of pro.t.s non-veri.ability, we have

then investigated the dynamic sustainaibility of the PS clause. We formally

proved that – for all the pro.t levels higher that the pro.t threshold which

makes the regulator indi¤erent between contract closure and imposing the

PS – it is sequentially optimal for the regulator to revoke the contract: that

is, we showed that any .rm.s deviation from the PS rule makes revocation

worthwhile for the regulator. Hence, the perfect equilibrium of the game with

pro.ts non-veri.ability is also such that the monopolist complies with the

PS rule chosen by the regulator in each period, as long as the revocation has

not been carried out (

 

Proposition 3

). The price adjustment which follows

results thus endogenously and dynamically determined as the monopolist.s

best response to the regulator.s choice.

We have also showed that as the regulator.s contract closure can be very

costly – i.e. it could imply costs belonging from any form of capture of the

regulator by the .rm – a considerable regulatory lag can occur before a PS

rule is introduced: the higher the revocation cost, the lower the pro.t shared

and the less frequent the regulator.s PS prescription. This conclusion sug-

gests a theoretical reason why the PS mechanism in its dynamic application

would tend to be unsuccessful in the real world: speci.cally, as long as the

regulator.s threat to revoke the contract becomes not credible, the regulated

.rm is no longer rewarded for comply with the adopted PS rule.

Finally, introducing a reduced form of the .rm pro.t function, we have

provided the

 

x

factor adjustment in a price-cap regulation setting: this appli-

cation illustrates how the parameters which characterize the pro.t function

a¤ect the PS e¤ectiveness.

To close, let us brie.y suggest a possible extension of our analysis which

recalls the main symplifying assumption we adopted in the paper. The eco-

nomic literature on PS regulation generally holds that pro.t-sharing rules re-

duce the .rm.s incentive to invest (Lyon, 1996; Crew and Kleindorfer, 1996;

Sappington, 2002; Moretto et al., 2003). In contrast with this literature,

21

our analysis – which address speci.cally the unilateral regulator.s bargaining

position in the regulatory contract – has been carried out under the assump-

tion of no .rm investment. Thus, a natural extention of our model could

include the .rm.s choice of investment – both irreversible and reversible – to

assess the e¤ects of contract closure by the regulator on such .rm.s strategic

decision.

22

Appendix

A. The regulation mechanism

We de.ne the regulation which follows the introduction of the PS rule as the reduction

dV

 

t needed to keep Vt at V :

This is represented by a one-sided non-decreasing adapted

control process (as in Harrison, 1985) on the state variable

 

V

which is right-continuous

and non-negative. Then, the control policy consists of a process

 

Z = fZt; t  0g

and a

regulated process

 

V r = fV

r

t

 

; t  0g

such that:

V

 

r

t

 

 VtZt; for V

r

t

 

2 (0; V 

]; (27)

where:



 

i) Vt

is a geometric Brownian motion, with stochastic di¤erential as in (3);



 

ii) Zt is a decreasing and continuous process with respect to Vt

;



 

iii) Z0 = 1 if V0  V ; and Z0 = V =V0 if V0 > V  so that V

r

0

 

= V 

;



 

iv) Zt decreases only when V

r

t

 

= V 

.

Applying Ito.s lemma to (27), we get:

d V

 

r

t

 

=
V

r

t

 

dt + V

r

t

 

dWt + V

r

t

dZ

 

t

Z

 

t

; V

 

r

0

 

2 (0; V 

]

where

 

V

r

t

dZ

 

t

Z

 

t  VtdZt = 􀀀drt

is the in.nitesimal level of value forgone by the .rm. In

terms of the regulated process

 

V

r

t

 

, we can write:

r

 

t  r(Vt) = Vt 􀀀 V

r

t

 

 (1 􀀀 Zt)Vt

; (28)

Although the process

 

Zt may have a jump at time t = 0;

it is continuous and keeps

V

 

t below the barrier V 

exercising the minimum amount of control: in fact, control is

exercised only when

 

Vt crosses V 

from below with a probability one in the absence of

regulation. Therefore, in the case of

 

V0 < V ; we get V

r

t

 

 Vt;

with the initial condition

V

 

r

0

 

 V0 = V; and Zt

= 1:

At

 

T  T(V ) = inf(t  0 j Vt 􀀀 V  = 0+)

control starts so as to maintain

V

 

r

t

 

= V 

:

23

The .rm.s values are adjusted downward by the amount

 

rt = Vt􀀀V

r

t

 

 0

every time

V

 

 is hit. The same conditions (i)􀀀(iv) uniquely determine Zt

with the representation

form (Harrison,1985; Proposition 3, p. 19-20):

 

37

Z

 

t



(

min(1

 

; V =V0) for t

= 0

inf

0

 

v

t

(

 

V =Vv) for t  0

(29)

B. Proof of Proposition 1

The function

 

F(Vt) is de.ned as the expected value at time t

of the regulator.s net

bene.t when the utility is expropriated at time

 

T:

As the net bene.t is a continuous

function of the primitive process

 

Vt , also F is a continuous function of Vt

. Then, by a

short arbitrage argument (Cox and Ross, 1976; Harrison and Kreps, 1979), applying Ito.s

lemma to

 

F

, the value of the regulator.s option to revoke becomes the solution of the

following di¤erential equation (Dixit and Pindyck, 1994, p. 147-152):

1

2



 

2V

2

t

 

F00(Vt) +
V
tF0(Vt) 􀀀 F(Vt) = 0; for Vt 2 (0; V 

]; (30)

where

 

F(Vt)

must satisfy the following boundary conditions:

lim

V

 

t!

0

F

 

(V t) = 0

(31)

F

 

(V ) = (1 􀀀 )V 􀀀^I

(32)

F

 

0(V ) = 1 􀀀 

(33)

If the value of the utility goes to zero, so does the option. Conditions (32) and (33)

imply respectively that, at the trigger level

 

V 

, the value of the option is equal to its

liabilities where

 

^I

indicates the sunk cost of revocation (matching value condition) and

suboptimal exercise of the option is ruled out (smooth pasting condition). By the linearity

of (30) and using (31), the general solution is of the form:

F

 

(Vt) = AV

t

 

; (34)

37

 

This is an application of a well-known result of Levy (1948), for which the process:

ln

 

V

r

t

 

 ln Vt + lnZt  ln Vt 􀀀

inf

0

 

v

t

(ln

 

Vv 􀀀 ln V 

)

has the same distribution as the .re.ected Brownian process.

 

j ln Vt 􀀀 ln V  j

:

24

where

 

A is a constant to be determined and > 1

is the positive root of the quadratic

equation:

(

 

) =

1

2



 

2 ( 􀀀 1) +
􀀀  = 0

(35)

As (34) represents the option value of optimally revoking the contract, the constant

A

 

must be positive and the solution is valid over the range of Vt

for which it is optimal

for the regulator to keep the option alive

 

(0; V ]

. By substituting (34) for (32) and (33)

we get:

V

 



=

 

 

 

􀀀

1

1

1

 

􀀀



^

 

I ;

(36)

and:

A

 

(V )



h

(1

 

􀀀 )V  􀀀 ^

I

i

(

 

V )􀀀



1

 

􀀀



 

(

 

V )1􀀀 > 0:

(37)

This concludes the proof.

C. Proof of Proposition 2

We prove that when the regulator uses (28), its option to revoke is always equal to

zero.

Cost of regulation

Let.s denote by

 

R(V

r

t

 

; V )

the expected value of future cumulative pro.t cuts. At

t

 

= 0

this is given by:

R

 

(V0; V )  E

r

0

Z

1

0

e

 

􀀀tdr(Vt) j V

r

0

 

 V0 2 (0; V 

]



(38)

=

 

􀀀E

r

0

Z

1

0

e

 

􀀀tVtdZt] j V

r

0

 

 V0 2 (0; V 

]



where

 

V  is the (generic) upper re.ecting barrier de.ned in (27). Since V

r

t

 

is a Markov

process in levels (Harrison, 1985, Proposition 7, p. 80-81), we know that the foregoing

conditional expectation is in fact a function of the starting state alone.

 

38

Keeping the

38

 

For V0 = V > V  optimal control would require Z

to have a jump at zero so as to

ensure

 

V

r

0

 

= V :

In this case the integral on the right of (38) is de.ned to include the

control cost

 

r0 incurred at t = 0;

that is (see Harrison 1985, p. 102-103):

Z

1

0

e

 

􀀀tdrt  r0

+

Z

(0

 

;1

)

e

 

􀀀tdr

t

where

 

r0 = V 􀀀 V

r

0

 

:

25

dependence of

 

R on V

r

t

 

active and assuming that it is twice continuously di¤erentiable,

by Itô.s lemma we get:

dR

 

= R0dV

r

t

 

+

1

2

R

 

00(dV

r

t

 

)2

(39)

=

 

R0(ZtdVt + VtdZt

) +

1

2

R

 

00Z

2

t

 

(dVt)

2

=

 

R0(
V

r

t

 

dt + V

r

t

 

dWt + V

r

t

dZ

 

t

Z

 

t

) +

1

2

R

 

00Z

2

t

 

2

dt

=

1

2

R

 

002V r

2

t

 

dt + R0
V

r

t

 

dt + R0V

r

t

 

dWt + R0V

r

t

dZ

 

t

Z

 

t

where it has been taken into account that for a .nite-variation process like

 

Zt; (dZt)2

= 0:

As

 

dZt = 0 except when V

r

t

 

= V 

we are able to rewrite (39) as:

dR

 

(V

r

t

 

; V 

) = [

1

2



 

2V r

2

t

 

R00(V

r

t

 

; V ) +
V

r

t

 

R0(V

r

t

 

; V 

)]dt (40)

+

 

V

r

t

 

R0(V

r

t

 

; V )dWt 􀀀 R0(V ; V )dr(Vt

)

This is a stochastic di¤erential equation in

 

R: Integrating by part the process Re􀀀rt

we

get (Harrison, 1985, p. 73):

e

 

􀀀tR(V

r

t

 

; V ) = R(V 0; V 

)+ (41)

+

Z

 

t

0

e

 

􀀀

s



1

2



 

2V r

2

s

 

R00(V

r

s

 

; V ) +
V

r

s

 

R0(V

r

s

 

; V ) 􀀀 R(V

r

s

 

; V 

)



ds

+

 



Z

 

t

0

e

 

􀀀sV

r

s

 

R0(V

r

s

 

; V )dWs 􀀀 R0(V ; V 

)

Z

 

t

0

e

 

􀀀sdr(Vs

)

Taking the expectation of (41) and letting

 

t ! 1;

if the following conditions apply:

(a)

 

lim

l

 

!

0

Pr[

 

T(l) < T(V ) j V0 2 (0; V ]] = 0 for l  V

r

t

 

< V  < 1;

where

T

 

(l) = inf(t  0 j V

r

t

 

= l) and T(V ) = inf(t  0 j V

r

t

 

= V 

);

26

(b)

 

R(V

r

t

 

; V )) is bounded within (0; V 

];

(c)

 

e􀀀tV

r

t

 

R0(V

r

t

 

; V ) is bounded within (0; V 

];

(d)

 

R0(V ; V 

) = 1;

(e)

 

1

2

 

2V r

2

t

 

R00(V

r

t

 

; V ) +
V

r

t

 

R0(V

r

t

 

; V ) 􀀀 R(V

r

t

 

; V 

) = 0;

we obtain

 

R(V r; V )

as indicated in (38). Condition (a) says that the probability of the

regulated process

 

V

r

t

 

reaching zero before reaching another point within the set (0; V 

]

is zero. As

 

V

r

t

 

is a geometric type of process this condition is, in general, always satis.ed

(Karlin and Taylor, 1981, p. 228-230). Furthermore, if condition (a) holds and

 

R(V r; V 

)

is bounded, then conditions (b) and (c) also hold. According to the linearity of (e) and

using (d), the general solution has the form:

R

 

(V0; V ) = B(V )(V0) ;

(42)

with:

B

 

(V 

) =

1

 

(

 

V )1􀀀 > 0

(43)

As for

 

V0  V ; Z0 = 1 and V

r

0

 

 V0 = V; then R(V0; V ) = R(V ; V ):

On the

other hand, if

 

V0 > V ; we get Z0 = V =V0; so that V

r

0

 

= V  and R(V

r

0

 

; V 

) =

R

 

(V ; V )

:

The value of revocation under pro.t control

Indicating by

 

Fr(V )

the regulator.s value of the option under pro.ts control, this can

be expressed, at time zero, by:

F

 

r(V

) =max

T

E

 

r

0

h

((1

 

􀀀 )VT 􀀀 ^I 􀀀 (1 􀀀 )B(V )V

T

 

 

)e􀀀T j V0

= V

i

(44)

As in (34) this takes the form:

F

 

r(V ) = AV

 

If the regulator decides for revocation, the optimal threshold, say

 

V ;

must satisfy

the two familiar conditions:

A

 

(V ) = (1 􀀀 )V  􀀀 ^I 􀀀 (1 􀀀 )B(V )(V )

(45)

27

A

 

(V ) 􀀀1= (1 􀀀 ) 􀀀 (1 􀀀 ) B(V )(V ) 􀀀1

(46)

These two equations can be solved for the trigger

 

V  and for the constant A:

Simple

algebra shows that

 

V  is independent of B and then of the barrier V 

. The solution is:

V

 



=

 

 

 

􀀀

1

1

1

 

􀀀



^

 

I

and the constant

 

A

is equal to:

A

 

= 􀀀(1 􀀀 )B(V 

) +

(1

 

􀀀 

)

 

(

 

V )1􀀀

= (1

 

􀀀 

)



1

 

(

 

V )1􀀀

􀀀

1

 

(

 

V )1􀀀



Therefore setting

 

V  = V  the constant A = 0

and the option value is identically

equal to zero.

Finally, as

 

rt depends only on the primitive exogenous process Vt;

the .regulated.

process

 

Vt 􀀀 rt

is also a Markov process in levels (Harrison, 1985, Proposition 7, p. 80-

81). Thus, any option value beginning at a point

 

t

at which revocation has not taken

place has the same solution. This concludes the proof of Proposition 2.

D. Proof of Proposition 3

We prove that the regulatory scheme proposed in Proposition 2 is also optimal when

the regulator cannot force the .rm to adopt it. We proceed in the following way. First,

since by Proposition 2 the sharing rule

 

rt

makes the regulator.s option to revoke the

contract always equal to zero, it is also a good candidate for supporting a long-run equi-

librium of the threat-game. Next, we prove that this is indeed the case by applying a sort

of one-stage-deviation principle and showing that any deviation from

 

rt

makes revocation

worthwhile (the non-decreasing property of

 

rt

is crucial to this result). Finally, the Markov

property of the “regulated” process

 

Vt 􀀀 rr

makes the equilibrium sub-game perfect.

Revocation strategy and perfect equilibrium

It is well known that in.nitely repeated games may be equivalent to repeated games

that terminate in .nite time. At each period there is a probability that the game continues

one more period. The key is that the conditional probability of continuing must be positive

(Fudenberg and Tirole, 1991, p. 148). This is indeed our case, neither player can perfectly

predict

 

Vt

at any date and the sharing rule (28) with form (29) is viewed by both agents

28

as a stationary strategy for evaluating all future pro.t reductions.

 

39

In the strategy space

of the regulator it appears as:

 

(Vt; rt

) =

8>>>><

>>>>:

Do not revoke at

 

t = T

if the .rm

plays the rule

 

rt = (1 􀀀 Zt)Vt for t0

< t

Revoke if the .rm deviates from

r

 

t = (1 􀀀 Zt)Vt at any t0

< t

where

 

(Vt; rt) is the strategy at t with history (Vt;Zt)

. The regulator.s .threat.

strategy is chosen if the .rm deviates by adjusting

 

Vt less than rt

or by abandoning

r

 

t = (1 􀀀 Zt)Vt

as a rule to evaluate future adjustments. The regulator must believe

that the regulation, from the initial date and state (

 

T; V );

will be kept in use for the

whole (stochastic) planning horizon. If the .rm deviates, the regulator believes that the

.rm intends to switch to a di¤erent rule in the future and knows for sure that the regulator

will revoke immediately thereafter. The regulator does not revoke at

 

t if rt0  Vt0􀀀V

r

t

 

0

for

39

 

Integrating the di¤erential form (3), the geometric Brownian motion can be expressed

as:

V

 

t+dt = VtedY

t

where

 

dYt = dt+dWt and  =
􀀀

1

2

 

2: The di¤erential dYt

is derived as the continuous

limit of a discrete-time random walk, where in each small time interval of length

 

t

the

variable

 

y either moves up or down by h

with probabilities (Cox and Miller, 1965, p.

205-206):

Pr(

 

Y = +h

) =

1

2

 

1 +



 

p

t



!

;

 

Pr(Y = 􀀀h

) =

1

2

 

1

 

􀀀



 

p

t



!

or de.ning

 

h = pt

:

Pr(

 

Y = +h

) =

1

2



1 +



 



h



 

2



;

 

Pr(Y = +h

) =

1

2



1

 

􀀀



 



h



 

2



That is, for small

 

t; h is of order of magnitude O(pt)

and both probabilities become

1

2

 

+ O(pt); i.e. not very di¤erent from

1

2

 

:

Furthermore, considering again the discrete-

time approximation of the process

 

Yt; starting at V e+h;

the conditional probability of

reaching

 

V 

is given by (Cox and Miller, 1965, ch.2):

Pr(

 

Yt = 0 j Yt = 0 + h

) =



1

 

if  

0

e

 

􀀀2h=2 if  >

0

which converges to one as

 

h

tends to zero.

29

all

 

t0  t; because pro.t cuts are expected to continue with the same rule and Fr

(V ) = 0

for all

 

t  T. If rt0 < Vt0 􀀀 V

r

t

 

0 for some t0 < t

the regulator expects a di¤erent rule

and carries out the threat, switching from

 

Fr(Vt) = 0 to F(Vt)  V  􀀀 I:

The game

is over. To prove this, we .rst need to prove the following Lemma:

Lemma 4

 

For each t0 > T

we get:

R

 

(Vt0 ; V ) = ( 􀀀
)E

r

t

 

0

Z

1

t

 

0

e

 

􀀀(s􀀀t0)rsds

(47)

Proof.

 

Let.s consider R as in (38). For each t0 > T;

integration by parts gives:

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)VsdZs =

(48)

e

 

􀀀(t􀀀t0)VtZt 􀀀 Vt0Zt0 +



Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)VsZsds

􀀀

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)ZsdV

s

Taking the expectations of both sides and using the zero-expectation property of the

Brownian motion (Harrison, 1985, p. 62-63), we have:

E

 

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)VsdZs = E

r

t

 

0 [VtZte􀀀(t􀀀t0)]􀀀Vt0Zt0+(􀀀
)E

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)VsZs

ds

(49)

By the Strong Markov property of

 

V

r

t

40

 

; it follows that E

r

t

 

0 [VtZte􀀀(t􀀀t0)] = E

r

t

 

0 [VtZt]E

r

t

 

0 [e􀀀(t􀀀t0)

] =

V

 

E

r

t

 

0 [e􀀀(t􀀀t0)] ! 0 almost as surely as t ! 1;

so that:

E

 

r

t

 

0

Z

1

t

 

0

e

 

􀀀(s􀀀t0)VsdZs = 􀀀Vt0Zt0 + ( 􀀀
)E

r

t

 

0

Z

1

t

 

0

e

 

􀀀(s􀀀t0)(Vs 􀀀 rs)

ds

Since

 

􀀀Vt0Zt0 + ( 􀀀
)E

r

t

 

0

R

1

t

 

0 e􀀀(s􀀀t0)Vsds = 0;

substituting (38) and rearranging

we get:

R

 

(Vt0 ; V ) = ( 􀀀
)E

r

t

 

0

Z

1

t

 

0

e

 

􀀀(s􀀀t0)rs

ds

We now prove that

 

rt

is sub-game perfect by showing that the .rm cannot gain

by deviating from

 

rt in an arbitrarily short interval and conforming to rt

thereafter. In

40

 

The Strong Markov Property of regulated Brownian motion processes stresses the

fact that the stochastic .rst passage time

 

t and the stochastic process V

r

t

 

are independent

(Harrison, 1985, Proposition 7, p. 80-81).

30

particular, let us assume

 

(t0; t) an interval in which rs is kept .at at rt0 so that V

r

s

 

 V 

;

and

 

t is the .rst time in which dZt > 0

.

Considering the decomposition (49) we can write (47) as:

R

 

(Vt0 ; V ) = ( 􀀀

)



E

 

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)rsds + E

r

t

 

0

Z

1

t

e

 

􀀀(s􀀀t0)rs

ds



= (

 

 􀀀

)



E

 

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)rsds + E

r

t

 

0



e

 

􀀀(t􀀀t0

)

Z

1

t

 

0

e

 

􀀀(s􀀀t0)rs

ds



where we have de.ned

 

V r s = V

r

t

 

+s and rs = rt+s 􀀀 rt for t0  t:

Applying, again, the

Strong Markov Property of

 

V

r

t

 

we get:

R

 

(Vt0 ; V ) = E

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)rsds + E

r

t

 

0



e

 

􀀀(t􀀀t0)E

r

t

 

0

Z

1

t

 

0

e

 

􀀀(s􀀀t0)rs

ds



= (

 

 􀀀
)E

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)rsds + E

r

t

 

0

n

e

 

􀀀(t􀀀t0)R(Vt0 ; V 

)

o

= (

 

 􀀀
)E

r

t

 

0

Z

 

t

t

 

0

e

 

􀀀(s􀀀t0)rsds + R(Vt0 ; V )E

r

t

 

0

n

e

 

􀀀(t􀀀t0

)

o

where the second equality follows from the assumption that

 

rs = rt0  Vt0 􀀀 V

r

t

 

0

for all

s

 

2 (t0; t): Finally, noting that e􀀀(t􀀀t0) 1􀀀(t􀀀t0)

and

R

 

t

t

 

0 e􀀀(s􀀀t0)ds (t􀀀t0

)

we can simplify the above expression as:

R

 

(V t0 ; V )

(

 

 􀀀

)



r

 

t0



(

 

 􀀀

)



(

 

V t0􀀀V

r

t

 

0

) (50)

From (50), any application of controls

 

rt0 < Vt0 􀀀 V

r

t

 

0

leads to a reduction of (47)

for all

 

t  t0 and, then, by Proposition 2, to Fr(Vt; V ) > 0

which triggers revocation

by the regulator.

Furthermore, the .rm does not adjust by more than

 

rt

since doing so would not

increase the probability of delaying revocation. It does not pay less, since

 

rt < Vt 􀀀 V

r

t

induces closure making it worse o¤, i.e.

 

0 < Vt

:

Finally, as

 

V

r

t

 

 Vt 􀀀 rr

is a Markov process in levels, it is clear from (47) that any

sub-game that begins at a point at which revocation has not taken place is equivalent to

the whole game. The strategy

 

is e¢ cient for any sub-game starting at an intermediate

date and state

 

(t; Vt)

. This concludes the .rst part of the Proposition.

Non-decreasing path of

 

rt within [T; T0)

:

31

So far we have implicitly assumed that, once started at

 

T;

the pro.t-sharing goes on

forever

 

:

Earlier interruptions are not feasible as long as the threat of revocation is credible.

Credibility relies on the fact that the agency.s option to revoke if the .rm deviates from

r

 

t is always worth exercising at Vt > V :

However, in a Brownian path there is a positive

probability of the primitive process

 

Vt crossing V 

again starting at an interior point of

the range

 

(V ;1):

In this case, the .rm may be willing to stop cutting pro.ts. That

is, the .rm “regulates” pro.ts until

 

Vt  V according to rt; but when Vt

reaches, for

the .rst time after

 

T; a predetermined level, say V 0  V ;

it ceases regulation. The

regulator will then face a jump from zero to

 

F(V 0)  F(V )

making the threat of

revocation no longer credible.

To see how this happens let.s assume that the .rm stops adjusting pro.ts at time

 

T

0

with

 

T < T0 < 1; and T0 = inf(t  T j Vt  V 0 and V 0  V ); i.e. T0

is

the .rst time that the primitive process

 

Vt reaches V 0  V 

with pro.t regulation under

way. Then the value of the revocation option starting at any

 

t 2 [T;1) with t < T

0

can be expressed as:

~

 

F(V t; V

r

t

 

; V 0) = P(V 0; V t)E

r

t

 

[F(V T0)e􀀀r(t􀀀T0)

]+ (51)

(1

 

􀀀 P(V 0; Vt))E

r

t

 

[Fr(VT0)e􀀀r(t􀀀T)

]

where

 

P(V 0; Vt) is the probability of the primitive process Vt reaching V 0  V 

starting

at an interior point of the range

 

(V ;1);

which is equal to (Cox and Miller, 1965, p.

232-234):

Pr(

 

T0 < 1 j Vt)  P(V 0; Vt

) =



V

 

t

V

 

0



􀀀

 

2=

2

with

 

 = (
􀀀

1

2

 

2)41 . As the starting point is now any t 2 [T;1);

we can immediately

see in (51) the dependence on both

 

V

r

t

 

and Vt

:

Since the option value under pro.t regulation is zero, if

 

V 0

is never reached we get

F

 

r(VT0) = 0: On the contrary, if V 0 is reached and the contract is revoked;

it is simply

F

 

(VT0) = F(V 0);

and:

~

 

F(Vt; V 0) = P(V 0; Vt)Et[F(V 0)e􀀀r(T0􀀀t)

]

According to the Strong Markov Property of

 

Vt

equation (51) becomes:

41

 

This probability is P(V 0; Vt) = 1 for   0

:

32

~

 

F(Vt; V 0) = P(V 0; Vt)F(V 0

)



V

 

t

V

 

0



(52)

where

 

< 0 is the negative root of (35). Since at t the primitive process Vt

is greater

than

 

V 0 and P(V 0; Vt

)

􀀀

 

V

t

V

 

0



=

􀀀

 

V

t

V

 

0



 

􀀀2=

2



 

1; we obtain ~ F(Vt; V 0)  F(V 0)

for

all

 

t 2 [T; T0);

which implies that the following inequality holds:

~

 

F(Vt; V 0) = F(V 

)



V

 

0

V

 





 



V

 

t

V

 

0



 

􀀀2=

2



 

F(V )

(53)

Since

 

~ F(Vt; V 0)  F(V ) for all t 2 [T; T0);

the regulator.s optimal strategy is to

revoke immediately at

 

T

. To prevent revocation the pro.t adjustment must continue

until time

 

T0  T0(V ) = inf(t  T j Vt 􀀀 V  = 0􀀀)

when the trigger value

V

 

0 = V  is reached for the .rst time after T:

The game ends and can then be started

afresh. This concludes the second part of the Proposition

 

.

33

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37

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