Strategic Firm Behavior under a Dynamic Regulatory Adjustment Process
Author(s): David Sappington
Source: The Bell Journal of Economics, Vol. 11, No. 1 (Spring, 1980), pp. 360-372
Published by: RAND Corporation
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Strategic firm behavior under a dynanic
regulatory adjustment process
David Sappington*
The regulatory mechanism proposed by Vogelsang and Finsinger (V-F) will
induce the regulatedfirm to adopt behavior other than myopic profit maximization. Pure waste, inefficient factor utilization, excessive research and
development, and overinvestment in demand-increasing expenditures may be
employed by the firm to increase long-run profits. The particular type of
strategic behavior adopted, and the extent to which it is pursued, will depend
upon the firm's allowed rate of return on capital, its discount rate, and the
information it receives regarding the regulatory regime. Under reasonable
conditions, the strategic behavior induced by V-F regulation will cause a
significant reduction in consumers' surplus plus producer's profit below the
level achieved in the absence of any regulation.
1. Introduction
* An ingenious mechanism for regulating multiproduct monopoly firms has
recently been proposed by Vogelsang and Finsinger (V-F) (1979). The mechanism
enables the regulator to guide the prices set by a profit-maximizing monopolist
to their Ramsey levels, even though the regulator may possess virtually no
information regarding the cost and demand functions of the firm.
Despite its many attractive features, the V-F scheme is unable to detect
and control a wide range of strategic behavior by the regulated firm.1 This
paper explores such behavior and demonstrates that it should be of great concern
to any regulator who considers imposing V-F regulation.
One form of strategic behavior designed to enhance the long-run profits
of the regulated firm is termed "pure waste" and involves the purchase of
* Princeton University.
The inspiration, encouragement, and invaluable assistance provided by Robert D. Willig
deserve far more than the mere thanks that is offered here. The helpful suggestions and assistance
provided by Edward J. Green and Charles H. Berry are also gratefully acknowledged. A note of
thanks is also extended to Raymond D. Hill, Michael Leeds, and Tim Bresnahan along with the
other members of the Industrial Organization Workshop at Princeton University and to the
anonymous referees and the Editorial Board of this journal for their helpful comments and
suggestions. The author assumes sole responsibility for any remaining errors.
1 Vogelsang and Finsinger are aware that their scheme may induce the firm deliberately
to earn negative profits in some period to relax the regulatory constraint in future periods. They
also suggest a clever scheme to eliminate such behavior if it is detected. Consequently, the
strategic behavior considered in this paper is limited to cases in which the regulated firm earns
positive perofits in every period.
360
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SAPPINGTON / 361
inputs which have no productive value. We develop a set of conditions under
which a monopolist may engage in pure waste in Section 3. We also discuss the
role played by information in the decision to waste in that section, and show
that the same monopoly firm will not waste if it is surprised by the introduction
of the regulatory regime. We prove that pure waste is absent once demand and
prices have settled at stationary, steady-state levels.
In Section 4, by using dynamic programming, we analytically derive the
manner in which a regulated firm may optimally deploy pure waste to forestall
(and in an extreme case, prevent) attainment of the steady state, and thereby
increase its profits. Furthermore, we show that the magnitude of waste induced
by V-F regulation may be so great as to cause a substantial loss in social welfare,
as measured by consumers' surplus plus producer's profit, relative to the
unregulated situation.
In Section 5, we briefly discuss a number of alternatives to pure waste.
These "strategic activities," which generate more utility for the firm per dollar
of expenditure than pure waste, include the purchase of perquisites for
executives of the firm, the employment of productive factors in inefficient
combinations, and the allocation of excessive funds to research and development and demand-increasing expenditures. Finally, in Section 6, we present our
conclusions. First, though, we briefly review the fundamental properties of the
V-F regulatory scheme in Section 2.
2. Review of the V-F regulatory scheme
* The V-F regulatory mechanism is designed to maximize consumer surplus
subject to the requirement that profits of the regulated firm be nonnegative.
The scheme achieves its objective when:
(1) cost and demand fupctions do not change over time;
(2) intertemporal cost effects are absent;
(3) there are decreasing ray average costs in the regulated firm; and
(4) the firm, which is assumed to possess complete information regarding its
cost and demand functions, maximizes profit.
The V-F regulatory scheme restricts the vector of prices (pj) that the
monopolist can charge in period j to lie within the region
Rj-l = {p |p X(pj31) - C[X(p-M)] C ?},
where
x(pj-1) = demand at prices Pj-l in periodj - 1,
C[x(pj-,)] = cost of producing x(pj1).
In other words, the firm can charge any prices for its products as long as the
revenues generated by selling last period's outputs at these prices would not
have exceeded last period's total costs. For a monopolist who markets only
one product, the restriction requires that price in any period not exceed the
average cost of production in the preceding period.
3. Informational asymmetries and the problem of waste
* One of the most attractive features of the V-F regulatory scheme is the
limited amount of information it requires for implementation. The regulator
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362 / THE BELL JOURNAL OF ECONOMICS
must know only the magnitude of the output of the firm in the preceding
period and the total cost of producing that output, along with the fact that
the firm's technology is characterized by decreasing ray average costs.2 However, because the regulatory agency knows so little about the firm's cost
structure, the agency is unable to determine whether the expenses it observes
reflect the true costs of efficient production. In this section we describe situa-
tions in which it is profitable for the firm to expend resources on unproductive
inputs (i.e., waste) to raise its costs and relax the regulatory constraint in
subsequent periods. We also present counterexamples in which the regulated
firm will choose to refrain from such strategic behavior.
Consider, first, the situation in which the V-F regulatory mechanism is
most effective in the absence of strategic behavior. If the technology of the
regulated producer of a single product is characterized by constant average
costs, V-F regulation will drive profits to zero and welfare to its maximum level
in only one period, regardless of where on its cost curve the firm chose to
operate in the preceding period. However, the elegant power of the scheme may
be destroyed by strategic behavior as is suggested in
Proposition 1: Consider a monopoly firm with a cost function C[x] = c x facing
a downward sloping demand curve x(p). If the firm is aware that V-F regulation
will be instituted in the second of the next two periods of its operation, then
it will find waste in the first period to be profitable if its discount factor (/3) is
sufficiently large, specifically, if /3 > x(p1)/x(p2).
Proof: The Lagrangian function defining the firm's problem is:
L = p1x(p1) - [c + w]x(p1) + 3[p2X(P2) - CX(P2)]
+ f4(C + W)X(p 1)P-2X(P1
where w = waste per unit of output in the first period.
The first-order conditions for a maximum include:
[Pl - - - - 2)(c + W)][Ox/0pl] + Xl = 0. (la)
3{(P2- c)[Ox/0p2] + x2} - X= 0. (lb)
+ ' 1; w[- 1 + 4)] = O. (Ic)
P2 ' c + w; k[c + w- P2]= (Id)
From (ic) and (Id) we have w = ow= 04P2 - c]. Rearranging (la) and using
(Id), we can show that Pi > c + w ? P2, so that 4[P2 - c] = w : P2 - c.
Because 4 < 1, this condition implies that w > 0 wheneverp2 > c. Using (lb),
we can show that /3 > x1/x2 is a sufficient condition for P2 > c. Q.E.D.
The specification of the cost function in Proposition 1 (and also in Propositions 2 and 3 which follow) assumes that the allowed rate of return on capital
(s) is identical to the true cost of capital (r) for the firm and that productive factors
are employed in their most efficient combinations. In fact, when s = r the firm
2 Although these requirements appear minimal when compared with the information needed
to implement other regulatory schemes, they are not insignificant. The definition of a product is
often less than precise, as products which are otherwise identical may differ in such characteristics
as the time of day, season, or location in which they are produced.
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SAPPINGTON / 363
is indifferent among combinations of productive capital, productive variable
inputs, and unproductive inputs that satisfy the production constraint and are
of equal cost. To facilitate the interpretation of waste as expenditures in excess
of the minimum amount required to produce a given level of output and for
analytic convenience, it is assumed throughout the discussion that when the
firm is indifferent among input combinations, it employs productive inputs
efficiently and relies on pure waste to relax the regulatory constraint. When
s differs from r, though, pure waste will never occur (as is proved in Proposition 4 below).
The condition that guarantees that waste is profitable for the firm, x2/xl > 1/,8,
has an intuitive interpretation. A decision to waste in period 1 increases the
level of cost per unit of output to the firm. It also raises the period 2 price
ceiling by an identical amount. Hence, waste will only be profitable for the firm if
the number of units sold tomorrow sufficiently exceeds the number of units on
which the waste is spent today, i.e., if the rate of increase in output (x2/x1)
exceeds the rate at which the value of future revenues declines (1/f.3
The same logic explains why the regulated firm will refrain from pure
waste in the steady state where its prices are stationary. When prices are
constant, demand does not vary from one period to the next, i.e., xi+1/xi = 1,
so that waste will not be profitable for the firm regardless of its discount
factor (,l3 ? 1). This result is generalized to the case of the multiproduct firm
and is stated more formally as Proposition 3 in the Appendix.
Proposition 2 describes another situation in which the regulated firm will
refrain from pure waste.
Proposition 2: Consider a firm characterized by the cost and demand functions
described in Proposition 1. If V-F regulation is imposed on the unsuspecting
firm during the next two periods of its operation (periods 1 and- 2), then the
firm will not waste. (Note: "Unsuspecting" implies that the firm was not deliberately wasting in the period prior to the first (period 0).)
Proof: The firm's problem is to choose Pi, P2, and w to maximize
P1X(PJ - [w + c]x(pJ) + ,0{p2x(p2) - CX(P2)}
subject to
P1x(Po) ? cx(po)
and
P2X(P1) ' [C + w]x(p1).
Letting Xi be the Lagrange multiplier associated with the constraint on the price
set in period i, the first-order conditions for a maximum are:
[Pl - 02P2 - (1 - 02)(C + W)][Ox/Xp1] + X1 - = 0; (2a)
:{[2- c][Ox/0p2] + X2} - Xl= ; (2b)
02?1;
w[-1+02]=0;
(2c)
Pi c; 02[c - pi] =; (2d)
P2 '< c + w; 02[C + W - P21 = ? (2e)
3Note that if the demand curve faced by the profit-maximizing monopolist is linear, the ratio
x2/x, is always 1/2. Thus, waste may occur in this case whenever the firm's discount factor
exceeds .5.
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364 / THE BELL JOURNAL OF ECONOMICS
Rearranging (2a) and using (2d) and (2e), we have
01 > [pi - (w + c)][Ox/0p] ? 0 so thatPt = c.
If waste is to occur, 02 must be equal to 1 from (2c), so that
(i) 8[P2 - C][Ox/Xp2] = Xi 2 X2 (from 2b));
(ii) P2 = c + w (from (2e)).
These conditions, along with the fact that Pi = c, imply that P2 = c, so that
w = 0 from (ii). Q.E.D.
The fact that the unsuspecting monopoly reveals its true costs to the
regulator mandates that the firm depicted in Figure 1 produce at output level x1
in period 1. If it decides to waste, the firm incurs a loss equal to w x1 in this
period. The waste permits a rise in the price ceiling from c to c + w in period 2,
but even by eliminating all of its waste in period 2, the firm can earn at most
w x2 (<w x1). Thus, even with a discount factor equal to unity, the monopoly
will suffer a loss in discounted profits if it decides to waste.
An analogous argument holds true in a T-period model with T ? 2. Even
for T very large and ,8 = 1, the firm could do no better than to reduce unit waste
by a very small amount in each period after the first. Such a policy, though,
would generate at most p2ADc (<p2BDc = w xi) in profits over the remaining
T - 1 periods, so without warning of the impending regulation, the firm would
choose not to waste.
Yet Proposition 1 suggests that if V-F regulation is anticipated, deliberate
waste by the regulated firm may diminish the efficacy of the V-F scheme before
the steady state is attained. The extent to which the scheme's effectiveness is
diminished is the subject of Section 4.
4. The implications of waste for social welfare
* This section derives the optimal pattern of waste for a regulated monopoly
which produces a single product at constant marginal cost and faces a linear
FIGURE 1
V-F REGULATION IN THE "UNSUSPECTING" MONOPOLY CASE
P,$
Ad
p0
w{
wP:-_________
I
\
I
0~~~~~~~~~~~~~~~~~~~~~~
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SAPPINGTON / 365
demand function. We also measure the extent of the welfare loss stemming
from this waste.
Assume the firm maximizes the present value of its profit stream subject
to the V-F regulatory constraint that price (p) in any period not exceed the level
of perceived average cost in the preceding period. Perceived average (marginal)
cost is the sum of true average (marginal) cost (c) and unit waste (w). The
monopoly's problem can be stated as:
T
Maximize I pi-l{pix(pi) - [c + wi]x(pi)} (3)
p1,wi-?, i=1
subject to
Pi = c+ wi1 i =2, ...,9T,
where
x(pj) = a - bpi = demand in period i;
r' = interest rate facing the firm; and
/3 = 1/(1 + r') = firm's discount factor.
The solution to this problem is given by equations (4) and (5):
-1 -dk[ 3 (a -bc) k =0,1, ...,9T- 29 (4)
WT-k = gk WT-(k+l) kb
Wi = [fTl1(oo) - ?2]-l[pe -? - /3(1 - /3)dT-2] (a - bc) (5)
b
where
f (x) = 2/3 - p[x]1',
n
f n(x) =...... (f (x)) .. .) f applied to x, n times,
gk = fk(C),)
k
dk =I 8 i Hl g-lj = g-1[I + f3dkJ1.
i=O j=O
Equations (4) and (5) are derived in the Appendix. The derivation assumes
that waste is nonnegative in every period in the unconstrained solution to (3).
The amendments to the solution procedure that are required when this assumption is violated are described in Lemmas 1 through 3 and Theorem 1, which
also appear in the Appendix.
These results were used to calculate the entries in the first three columns
of Table 1. Assuming that V-F regulation is anticipated in period 0, the table
shows the first period (t) in which the optimal value of waste for the firm is
zero for selected. interest rates (r') and associated discount factors (f3). The
table indicates that the less the firm values future profits relative to current
profit (i.e. , the lower is p) , the fewer the number of periods in which the firm will
waste. This is a plausible result, since waste entails the sacrifice of current
profit for future returns. For the firm which employs waste optimally, the
actual magnitude of waste in each period is presented in Table 2 for various
discount factors.4 The entries are expressed as a fraction of the distance,
4In this example and all subsequent ones, it is assumed that the firm's discount factor
and society's are identical.
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366 / THE BELL JOURNAL OF ECONOMICS
TABLE 1
COMPARISON OF V-F REGULATION AND NO REGULATION FOR SELECTED INTEREST RATES
PERFORMANCE OF V-F REGULATION PERFORMANCE OF NO REGULATION
OVER 10-YEAR PERIOD OVER 10-YEAR PERIOD
r3
t
--
_
CONSUMER PROFIT SOCIAL CONSUMER SOCIAL
SURPLUS WELFARE SURPLUS PROFIT WELFARE
0.00
1.000
0.01
.990
T
1.59
0.45
2.05
1.25
2.50
15
1.76
0.43
2.19
1.20
2.40
3.75
0.03
.971
9
2.31
0.39
2.70
1.10
2.20
3.29
0.05
.952
7
2.46
0.37
2.83
1.01
2.03
3.04
0.07
.935
6
2.43
0.35
2.79
0.94
1.88
2.82
0.10
.909
5
2.27
0.34
2.61
0.84
1.69
2.53
0.15
.870
4
1.98
0.32
2.30
0.72
1.44
2.16
0.33
.750
2
1.30
0.28
1.59
0.47
0.95
1.40
1.00
.500
1
0.62
0.25
0.87
0.25
0.50
0.75
3.59
t = FIRST PERIOD IN WHICH OPTIMAL VALUE OF WASTE IS ZERO IN T-PERIOD MODEL (T > t).
r'= REAL MARKET RATE OF INTEREST; ,3= DISCOUNT FACTOR.
(a-bc)2
ENTRIES IN COLUMNS 4-9 REPRESENT PRESENT VALUES AND ARE EXPRESSED AS FRACTIONS OF b
(NUMBERS MAY NOT ADD DUE TO ROUNDING).
(a - bc)lb, between the marginal cost curve and the vertical intercept of the
demand curve.
In the first period, the firm sets its price at the point on the demand
curve corresponding to the quantity produced when marginal revenue is equated
with perceived marginal cost. In subsequent periods price is restricted to
the level of perceived average cost in the preceding period. As a point of
reference in interpreting Table 2, the monopoly price set by an unregulated
firm exceeds marginal cost, c, by O.5[(a - bc)lb].
The implications of such behavior for social welfare (defined here as the
sum of consumers' surplus and profit) are significant. An illustrative example
is presented in columns 4 through 9 of Table 1. Assuming that the cost and
demand functions are stable over a ten-year period so that V-F regulation (as
TABLE 2
MAGNITUDE OF WASTE (W) IN PERIODS 1 THROUGH 7 OF A T-PERIOD MODEL (T > 7)
FOR SELECTED INTEREST RATES (r') AND ASSOCIATED DISCOUNT FACTORS (,3)
|
rI
|
3
W,
W2
|W3
|W4
W5
|W6
X7
0.00 1.000 T-1 T-2 T-3 T-4 T-5 T-6 T-7
T+1 T+1 T+ 1 T+1 T+1 T+1 T+1
0.05
0.952
0.10
0.15
.63
0.909
.51
0.870
0.25
1.00
.45
0.800
.29
.43
0
.16
.11
.19
.33
0.500
.29
.01
.03
.08
0
.06
0
0
0
0
0
0
0
.01
0
0
0
0
0
0
0
0
0
0
ENTRIES IN COLUMNS 3 THROUGH 9 REPRESENT FRACTIONS OF a-bc
b
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0
SAPPINGTON / 367
outlined in Section 2) can be employed during this time without the fear of
forcing the firm into bankruptcy, we show that unless the real rate of interest
faced by the firm exceeds 7 percent5, the imposition of V-F regulation will
reduce social welfare below the level that can be achieved in the absence of
all regulation.
The findings are, of course, sensitive to the length of the period for which
costs and demand are assumed to remain stationary. If they are forever stable
as Vogelsang and Finsinger assume, then the temporary welfare loss induced
by strategic behavior under V-F regulation will be inconsequential when
compared with the perpetual stream of benefits derived once the steady state is
reached.6 If the period of stability is shorter, however, V-F regulation cannot
continue forever as an uninterrupted series of price revisions without the benefit
of additional information, and unless such information regarding the shifts in the
cost and demand functions is sufficiently accurate, the regulator may have to
begin the regulatory process anew at the end of the stable period, and thus
lead to the foreboding results of the type presented in Table 1. For shorter
periods of stability, the waste induced by V-F regulation will incur social losses
that are even greater.
It should also be noted that one's evaluation of V-F regulation will be sensitive to the welfare index employed. The entries in Table 1 demonstrate that,
for our example, although no regulation may lead to a higher level of social
welfare than that achieved by V-F regulation, the latter will increase consumers' surplus above its level when the monopolist is unregulated. Thus, if
the maximization of consumers' surplus is the regulator's mandate from
society (as postulated in the V-F model), waste can be regarded as a cost of
transferring wealth from the monopoly to consumers. So long as the imposition
of V-F regulation increases consumers' surplus above the level achieved in the
absence of regulation by an amount which exceeds administrative costs, the
scheme is of net benefit to society. Of course, a final evaluation of V-F regula-
tion requires that its costs and benefits be carefully compared with those of
alternative regulatory schemes.
5. Alternative forms of strategic behavior
* The type of strategic behavior considered in the preceding sections is rather
specific in nature. In this section we briefly discuss alternatives to pure waste.
The importance of such a discussion is evidenced, in part, by Proposition 4,
which proves that waste will never occur in the presence of the regulatory
bias discussed by Averch and Johnson (1962).
5The real rate of interest is the relevant rate, because it is assumed that the regulator can
infer from available data the extent to which costs incurred in the preceding period have risen
as a result of inflation. If factors of production are affected differentially by inflation, though,
and the firm can freely substitute among factors as their real prices change, reliable inferences
cannot be made. In this case, the cost function cannot be assumed to be stationary for any
appreciable length of time.
6 In the extreme case in which /8 = 1, the firm will adopt a pattern of waste that
prevents attainment of the steady state and causes the welfare loss to increase as the period for
which V-F regulation is in effect grows longer. As T approaches infinity, the level of social welfare
achieved under the V-F scheme shrinks to 4/9 of the level obtained in the absence of regulation.
For /3 < 1, however, waste will be driven to zero in some finite period of time, and the welfare
losses are smaller.
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368 / THE BELL JOURNAL OF ECONOMICS
Proposition 4: A monopoly firm subject to V-F regulation will not engage in
pure waste if its true cost of capital (r) differs from the allowed rate of return (s).
Proof: Consider a monopolist who employs capital (K) at a per-unit cost of
r and a vector of variable inputs (L) at unit costs z to produce a vector of
outputs x( ) which are sold at prices p. The technology of the firm is represented
by F(x,K,L) - 0. The problem facing the firm can be expressed as:
T
Maximize E i-1{pix(p) - rKi - zLi -W ,
i=l1
subject to
F(x(pi),Ki,Li) ' 0,
and
pix(pi-1) ' sKi-, + zLi-, + Wi-1, i = 1, . .. , T
where Wi is lump-sum waste in period i. Letting Xi and Oi be the Lagrange
multipliers associated with the production and V-F constraints, respectively,
in period i, the first-order conditions include:
r = Oi+1s + 4J&F/&Ki]; (6)
z = Oi+lz + 4j&F/oLj]; (7)
WiJ-1 + Oi+1] = 0, -1 + Oi+1 ' 0; (8)
Xi '- 0, oi ' 0, i = 1, ... ., T, 0T+1 = ?From (8), Wi can be strictly positive only if Oi+1 = 1.
Suppose s > r. Then if 0i+1 = 1, r = s + 4i[&F/0Ki] from (6). But this is
impossible, since 4J&F/&Ki] ' 0, so 6i+1 / 1.
Suppose s < r. Then if Oi+1 = 1, Xi = 0 from (7). But Xi = 0 implies that s = r
by (6), so 0i+1 / 1. Q.E.D.
There are two reasons the firm will choose to use excess productive
capital rather than hire other productive inputs when s exceeds r. First, each
dollar per unit of output spent on any type of capital raises the next period's
price ceiling by slr (> 1) dollars. Second, productive capital can be substituted
for other productive inputs that do not offer the same compensation in excess
of actual payments to the firm.
Even when s = r and overcapitalization is not especially attractive, pure
waste may be superseded by other "strategic expenditures." For example, the
regulated monopolist might pursue research and development projects to an
extent that would be excessive (and inefficient) in an unregulated environment.
Because the firm is fully compensated for the expense of such projects, if they
result in any reduction in production costs, they will be profitable for the firm
although pure waste may not be. The welfare implications of such expenditures
are ambiguous, though, because if the monopoly were conducting too little
R&D before the imposition of V-F regulation, the extra expenditures induced
by the regime could be socially beneficial. However, a full welfare evaluation
would have to account for the fact that the firm is not forced to bear any of the
social costs of its research nor does the V-F scheme induce the firm to pass on
the benefits of its research to consumers immediately.
Another type of strategic expenditure that can be profitable to the regulated
firm even after the gains from pure waste have been exhausted involves ex-
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SAPPINGTON / 369
cessive outlays aimed solely to increase demand (perhaps through advertising)
in subsequent periods.7 Such expenditures are especially profitable to the
regulated firm, because their current unit costs can later be recovered on the
larger volume of sales they create. Again, the welfare implications of such
expenditures are less clear than those of pure waste, since advertising may
have socially redeeming attributes. However, to the extent that advertising
provides no social utility and may therefore be regarded as wasteful, the waste
induced by V-F regulation in this form of strategic expenditure can exceed the
levels of pure waste examined in Section 4.
In effect, just as rate-of-return regulation can alter the shadow price of
capital to the firm, so, too, will V-F regulation lower the shadow price of
such activities as research and development and demand-increasing experi-
ments, even when s = r. In addition, the shadow price of perquisites for
corporate executives will also be lowered, so that such discretionary expendi-
tures are more likely to be observed than is pure waste. Consequently, it must be
concluded that a number of biases other than the A-J bias may be engendered
by V-F regulation, and without more precise information regarding the internal
operations of the firm, the regulator will be unable to detect and eliminate the
induced inefficiencies.
6. Conclusions
* It has been demonstrated that the most attractive feature of the V-F regulatory mechanism may, in fact, prove to be its tragic flaw. Although the
mechanism is ideal in the sense that it does not require the regulator to possess
detailed information regarding the demand and cost parameters in the industry,
it is flawed because it cannot prevent the regulated firm from exploiting the
regulator's ignorance to its own advantage. In some cases, the undetectable
strategic behavior adopted by the firm can cause the imposition of V-F regulation
to reduce consumers' surplus plus profit to a level below that achieved in the
absence of regulation.
The results in this paper were derived for the case of a single-product
monopoly firm producing with constant costs. It can be shown, though, that
strategic expenditures remain profitable for the regulated monopolist of a single
product in the presence of strictly increasing returns to scale. In fact, pure waste
is sometimes profitable for such producers, even after they have revealed their
true costs of production in an earlier period.
Although the multiproduct firm is also likely to engage in strategic
behavior under V-F regulation, the extent to which it will undertake expenditures of the form described in Sections 3 through 5 is not clear a priori.
Because the multiproduct firm is granted more leeway in its pricing decisions
than the single-product firm under V-F regulation8, it may be able to manipulate
its prices without wasting so as to reap the benefits the single-product firm
7 For example, with a linear demand curve and /8 = 1/2, the firm may advertise to an
excessive degree, although it will not engage in pure waste under these conditions. Furthermore,
expenditures that increase demand (however slightly) will often be profitable to the firm, even
after its true production costs have been revealed in an earlier period.
8 The multiproduct firm need not price each product at its average cost in the preceding
period as the single product firm must do under V-F regulation. Note that product-specific
average costs are well defined in the absence of cost complementarities.
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370 / THE BELL JOURNAL OF ECONOMICS
can gain only through waste. On the other hand, because the constraint on a
multiproduct firm is less restrictive, the firm may waste more freely and to a
greater extent. The issue deserves further research.
One other issue that merits further investigation concerns the timing of
regulatory hearings. The analysis in Section 4 assumed that the prices of the
regulated firm were revised at yearly intervals. The length of the regulatory
lag, though, can be flexible under V-F regulation. Because both the relevant
discount factor of the firm and the speed of convergence toward the steady
state are affected by the length of the lag, the strategic timing of price
revisions can be advantageous to the regulator and society.
In conclusion, it should be noted that because the scheme proposed by
Vogelsang and Finsinger can offer considerable improvement over existing
schemes in the absence of strategic play by the regulated firm, research that
leads to the discovery of effective methods for controlling such behavior may
prove to be of major importance for future regulatory policy.
Appendix
* Proposition 3. In the steady state where prices are stationary, the multiproduct monopoly subject to V-F regulation will not waste (whatever the value
of its discount factor, ,B < 1).
Sketch of proof: The first-order conditions of the associated problem state
that Oi+1 must equal unity for waste to be strictly positive in period i (where
Oi is the multiplier associated with the V-F constraint in period i). However,
0i+1 = 1 can be shown to imply that x(pi) = 0 for all i in the steady state, which
completes the proof by contradiction.
El Lemma 1. If the solution to (3) has wi = 0 for some i < T, then wt = 0
for i ? t ? T.
The proof of this lemma follows directly from Proposition 2.
D Lemma 2. If the firm is allowed to underreport its costs (so that waste is
negative), it will not do so.
Proof: If the possibility of negative waste is admitted (i.e., the firm is allowed to
underreport its true costs), then the dynamic program is of the form:
Vi(pi) = max {[pi - (c + max [0,wi])][a - bpi] + /Vi+l(pi+l)},
where Vi(pi) is the optimal value function in period i.
Now, suppose wi < 0. If wi < 0, then the first-order conditions for a maximum
state:
0 + 13{[pi+l - (c + max [0,wi+1])](-b) + a - bpi+1} = 0,
and
pi+1 = c + Wi.
Solving for wi yields wi = 1?2[(a - bc)lb] + 1/2 max [O,wi+1] > 0. Hence, by
contradiction, wi - 0 for all i. Q.E.D.
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SAPPINGTON / 371
D Lemma 3. The optimal solution to (3) is of the form
wi > 0 for i = 1,.. .,k,
wi = 0 for i = k + 1, . . ., T k < T.
The proof of Lemma 3 follows directly from Lemmas 1 and 2.
D Theorem 1. The solution to (3) can be derived as follows:
(1) Find the solution (w1, ... , WT) to the unconstrained version of (3) by using
equations (4) and (5). If w1, . . ., WT satisfy Lemma 3, they constitute the
solution to (3).
(2) If wi < 0 for some i, then solve the corresponding unconstrained (T - 1)period problem. Let these values of waste be represented by wt, i = 1,
..T - 1.
(3) If wi*, i = 1, . .. , T - 1, satisfy Lemma 3 (with T replaced by T - 1),
then w,i = 1,.. ,T - 1, and WT =0 solve (3).
(4) If, on the other hand, w* < 0 for some t < T - 1, then solve the
corresponding unconstrained (T - 2)-period problem.
(5) Continue as in steps (3) and (4), decreasing the time horizon of the problem
by one period in each (of j) iteration(s) until the values of waste w*,
i = 1, . . . , T - j, generated by the unconstrained solution to (3) satisfy
Lemma 3 with T replaced by T - j.
The solution to (3) is:
wi = w1*, i =1 ... . T-j
wi = 0 i =T-j + 1, ... ,T.
Proof of Theorem 1.
(1) Suppose that afterj iterations, the procedure outlined in Theorem 1 first
produces a unique set of values for waste which satisfy Lemma 3, i.e.,
wi = wi* > 0, i = 1, T ..,-j,
=0,
i=T-j+
1,...,T.
It must be shown that these values constitute the solution to (3).
(2) Any set of values
wi = wi* > O, i = 1, ... ., T-j + kg
=0, i= T-j +k +1, ...,T,
for 1 ? k < j cannot solve (3) because if they did, the procedure would have
yielded these values in the j - k)th iteration.
(3) Any set of values
wi = wr* > 0, i = 1,.. ., -j-k,
=0 i = T-j-k + 1, ... .,T,
for 1 ? k <j cannot solve (3) because these values, although feasible,
were not selected in the (T - j)th iteration, and therefore cannot be more
profitable than the values stated in step (1) above.
(4) Any other set of values w1, . .. ., WT would not satisfy Lemma 3. Q.E.D.
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372 / THE BELL JOURNAL OF ECONOMICS
El Proof of Equations (4) and (5): In the terminal period waste is of no value
to the firm so WT = 0:
VT-1(PT-1) = max {PT-1X(PT-1) - [C + WT-1]XT-1 + f3VT(PT)},
WT-1
where V,(pt) is the optimal value function in period t. The first-order condition states that
a - bpT-1 = 3{PT[aXIaPT] + XT - C[VXIOPT]}
from the fact thatPT = C + WT-1 SubstitutingPT-1 = c + WT-2, XT = a - bPT,
and OX/OPT = -b, it follows that
WT 1 1 T2-IH-0 (a -bc)
2f3 2fb
Hence, equation (4) holds for k = 1. Assume that it holds for general k, and
prove that it holds for k + 1, i.e.,
Prove:
= 1 - (a - WT-k-1 = gk+1WT-k-2
-dk+1[1
3]
b c)
First,
VT-k-1PT-k-l) = max {[PT-k-1 - C - WT-k-1][a - bPT-k-1] + PVT-k(PT-k)}
From the first-order condition for this problem and the hypothesis, substituting
as above, we have:
[2 - k 1WT-k-1 = WT-k-2 - [ - [3dk][l - /] (a - bc)
which is readily solved to give the desired result, where the new notation is as
defined in the text following equations (4) and (5). Equation (5) is derived by
solving (for p 1 and w 1):
Vl(p1) = max {[Pi - c - w1][a - bpl] + /V2(p2)}
from the fact that W2 = fT-2(oo)W1 - dT-2[1 - 8][(a - bc)lb]. Q.E.D.
References
AVERCH, H. AND JOHNSON, L.L. "Behavior of the Firm under Regulatory Constraint." American
Economic Review, Vol. 52, No. 5 (December 1962), pp. 1053-1069.
VOGELSANG, I. AND FINSINGER, J. "A Regulatory Adjustment Process for Optimal Pricing by
Multiproduct Monopoly Firms." Bell Journal of Economics, Vol. 10, No. 1 (Spring 1979),
pp. 157-171.
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