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Monopoly rationing revisited
Economics Letters Y ( 19X2) 17-2 I North-Holland Publishing Company
MONOPOLY
Rcccived
RATIONING
14 October
17
REVISITED
*
I YX I
This paper establishes that, whenever a simple pricing strategy fails to attain to the rationing outcome. a more complex pure price strategy may yet do so. It is alao shown that rising marginal cost with uncertain demand may make rationing bcncficial.
1. Introduction
Honkapohja (1980) demonstrates that a monopolist facing uncertain demand conditions may benefit from restricting the quantity that it will sell at its chosen price. His example hinges on a price-taking consumer whose random endowment is not known to the monopolist ex ante. Every example necessarily entails special assumptions, and Honkapohja’s is no exception. The unique feature that drives his result is the fact that the price takes is a net buyer of good 2 in one state but a net seller in the other. As a result, the monopolist will wish to set the price low in order to buy cheaply in one state, then add a quantity ceiling to limit its losses in the other state. This example raises several important questions that we shall explore here. Are there cases in which a monopolist is actually observed both to buy and to sell the same commodity? Is there an alternate strategy that attains to the rationing outcome for such a monopolist? Are there other, possibly more natural conditions that encourage a rationing strategy? Briefly, the answer to each question is yes. 2. The original example For the convenience * This paper embodies policy of the Federal
of the reader, we recount
Honkapohja’s
example.
the views of the author and doea not reflect the official views or Reserve System or the Federal Rescrve Bank of New York.
0 165- 1765/82/0000-0000/$02.75
0 1982 North-Holland
Let the utility
function
of the price taker be
ZJ. = In x,( + In x2( so that his demand z, =
for net trades z becomes
:w*,p- w,,./%
z2 =
:w,,/p - w2c/2,
where p denotes the ratio p2/p,. Assume that the price taker’s endowment wCwill take one of two values with equal probability, w,’ = (1, 1).
w,‘=(o,2),
Let the monopolist’s
endowment
be w,,, = (1, 1) and his utility
be
U,, = In Xlnl + In xZn,. Now a pure price policy yields EU,,z =+[ln(l
-p)
+ln
21 +i[ln(3/2-p/2)
+ ln(3/2-
1/2p)]
p of about 0.5574, gives
which, for the optimal
EU,, = iln 2 + iln 0.326. Imposing a non-negative nopolist faces the lottery [I -p,
2].withprob
:,
purchase
[l +pF,,
constraint
1 -T2]
Zz on good 2, the mo-
withprob
+
if the constraint is binding in the event wC= w,?. (It cannot be binding for wC = w,’ since the price taker will sell good 2 in that event.) The monopolist’s expected utility becomes
which, for p --t0 and L2 4 0, gives EU,,
arbitrarily
close to
iln 2 + iln
which exceeds the value obtained
under
1 a pure price strategy.
IO
3. Middlemen and markups There exists at least one broad class of economic agents who both buy and sell the same commodities in the normal course of business. These agents are generally termed ‘middlemen’: retailers, wholesale distributors, used car salesmen, and all others whose primary function is to store goods in the passage from one owner to the next. Occasionally such agents may also possess monopoly power. However, we rarely, if ever, observe middlemen to restrict their sales quantities (special promotional sales aside), nor do we observe them to sell at their purchase prices. Rather, they follow a strategy which intuition suggests would be at least as profitable as rationing: they establish different prices for the purchase and sale of each commodity. ‘Buy low, sell high.’ In terms of the original example, the monopolist’s problem is then to maximize
EU, =+[ln(l
-P)
+h12]
++[ln(3/2-p/2)
+ln(3/2-
1/2p)],
where P denotes the ratiop,/p, at which the monopolist will buy good 2 whilep denotes the ratio at which it sells good 2. By feasibility, 0 < P < 1 and l/3
aEr/,,/aP=O=:[l/(P-
l)],
aEun,/ap
= 0= +
r
-3+3/p’ 1 10 3p 3/p I I
while the second-order conditions indicate that the solution will yield a maximum. The outcome is then seen to be a corner solution for the buying price, P + 0; and an interior solution for the selling price, p = 1. At these values we obtain
Eut?, approaches
+ln 1 ++ln2++ln
1
which has the same limit as the rationing strategy. Thus the monopolist in this example can earn as high an expected utility under a pure price strategy as by rationing, provided that the buying and selling prices are set separately.
4. Rising marginal cost and rationing We turn, finally, to the issue of whether a monopolist could benefit from rationing if it is a net seller of its output in all states. Again by example, we shall demonstrate that rationing can dominate a pure price strategy under the simple conditions of random demand and rising marginal cost, when uncertainty in the level of demand does not influence its elasticity. Assume that the monopolist has a cost function C(y) with C’> 0, C” > 0. Let demand be a function 4 = q( e,p) with a~/&9 > 0, aq/ap < 0, where 8 is a random variable with density f( 0). Moreover, assume that changes in B do not alter the elasticity of q(0,p) for fixed p. The monopolist under a pure price strategy will maximize ET by choice of p, so that at the equilibrium pricep* its expected profits will be En-=p*
/= q(e.p*)f(e) -33
de-Jx
-‘x
c(q(e,p*))f(e)
de.
There is some value of 0, given p*, such that marginal cost MC equals marginal revenue MR. Call it A. Note that, by rising marginal cost, Mc(q(e,p*))>MR(q(e,p*)) for all B>A since MR=(p(l + I/C) by definition, and 6 is independent of 19by assumption. Thus P*q(A.
P*) - c(qbL
for all B>A. Equilibrium
expected
p*>> ‘P*q(e.P*)
profits
- c(q(e,
may be rewritten
(1)
p*>)
as
Suppose now that the monopolist sets a quantity ceiling 4 = q( A.p*). Under this rationing strategy the expected profits become E7TR =p*Jll_q(e,p*)f(e)
a
c(q(e,p*))f(e)
de+p*~Xq(A.p*)f(B)
de-/mc(qI~~p*))f(~) A
de
de.
In comparing to sign
this outcome
with that of the pure price strategy.
ETR ~ Err =p* /%Lp*)f.(~)
de-p*/%b*)f.(@ de 4
A
=L=[P*Y-
we wish
c-(q) -p*dd.
P*)
+ C(YWP*>)] f(e) de.
By (1) the bracketed integrand is positive. Thus E.lrR > ET, and the monopolist benefits from a rationing strategy relative to a pure price strategy. This result is quite intuitive. It merely says that a monopolist which must set a price before its actual demand level is known will benefit from refusing to produce beyond the level at which MC = MR, if such a level exists. Otherwise, in the event that demand is higher, the firm would sustain MC> MR and would earn lower profits than in the rationed case.
5. Conclusions
A profit maximizing monopolist may indeed benefit from a rationing strategy relative to a pure price strategy, if it is subject to incomplete information. However, in some instances at least, a sufficiently clever price strategy can attain the same outcome associated with a single price plus rationing. This result was demonstrated for a monopolist which sells in some states of the world but buys in others. For the monopolist with rising marginal cost, it remains an open question whether a peak-load pricing scheme could perhaps duplicate the rationing outcome. More generally, given any set of conditions under which a monopolist will benefit from rationing, we may ask whether there exists a more complex pure price strategy which would be equally profitable.
Reference Honkapohja, 203-209.
S.. 1980, A note on monopolistic
quantity
rationing,
Economics
Lcttcrs
6.
MONOPOLY
Rcccived
RATIONING
14 October
17
REVISITED
*
I YX I
This paper establishes that, whenever a simple pricing strategy fails to attain to the rationing outcome. a more complex pure price strategy may yet do so. It is alao shown that rising marginal cost with uncertain demand may make rationing bcncficial.
1. Introduction
Honkapohja (1980) demonstrates that a monopolist facing uncertain demand conditions may benefit from restricting the quantity that it will sell at its chosen price. His example hinges on a price-taking consumer whose random endowment is not known to the monopolist ex ante. Every example necessarily entails special assumptions, and Honkapohja’s is no exception. The unique feature that drives his result is the fact that the price takes is a net buyer of good 2 in one state but a net seller in the other. As a result, the monopolist will wish to set the price low in order to buy cheaply in one state, then add a quantity ceiling to limit its losses in the other state. This example raises several important questions that we shall explore here. Are there cases in which a monopolist is actually observed both to buy and to sell the same commodity? Is there an alternate strategy that attains to the rationing outcome for such a monopolist? Are there other, possibly more natural conditions that encourage a rationing strategy? Briefly, the answer to each question is yes. 2. The original example For the convenience * This paper embodies policy of the Federal
of the reader, we recount
Honkapohja’s
example.
the views of the author and doea not reflect the official views or Reserve System or the Federal Rescrve Bank of New York.
0 165- 1765/82/0000-0000/$02.75
0 1982 North-Holland
Let the utility
function
of the price taker be
ZJ. = In x,( + In x2( so that his demand z, =
for net trades z becomes
:w*,p- w,,./%
z2 =
:w,,/p - w2c/2,
where p denotes the ratio p2/p,. Assume that the price taker’s endowment wCwill take one of two values with equal probability, w,’ = (1, 1).
w,‘=(o,2),
Let the monopolist’s
endowment
be w,,, = (1, 1) and his utility
be
U,, = In Xlnl + In xZn,. Now a pure price policy yields EU,,z =+[ln(l
-p)
+ln
21 +i[ln(3/2-p/2)
+ ln(3/2-
1/2p)]
p of about 0.5574, gives
which, for the optimal
EU,, = iln 2 + iln 0.326. Imposing a non-negative nopolist faces the lottery [I -p,
2].withprob
:,
purchase
[l +pF,,
constraint
1 -T2]
Zz on good 2, the mo-
withprob
+
if the constraint is binding in the event wC= w,?. (It cannot be binding for wC = w,’ since the price taker will sell good 2 in that event.) The monopolist’s expected utility becomes
which, for p --t0 and L2 4 0, gives EU,,
arbitrarily
close to
iln 2 + iln
which exceeds the value obtained
under
1 a pure price strategy.
IO
3. Middlemen and markups There exists at least one broad class of economic agents who both buy and sell the same commodities in the normal course of business. These agents are generally termed ‘middlemen’: retailers, wholesale distributors, used car salesmen, and all others whose primary function is to store goods in the passage from one owner to the next. Occasionally such agents may also possess monopoly power. However, we rarely, if ever, observe middlemen to restrict their sales quantities (special promotional sales aside), nor do we observe them to sell at their purchase prices. Rather, they follow a strategy which intuition suggests would be at least as profitable as rationing: they establish different prices for the purchase and sale of each commodity. ‘Buy low, sell high.’ In terms of the original example, the monopolist’s problem is then to maximize
EU, =+[ln(l
-P)
+h12]
++[ln(3/2-p/2)
+ln(3/2-
1/2p)],
where P denotes the ratiop,/p, at which the monopolist will buy good 2 whilep denotes the ratio at which it sells good 2. By feasibility, 0 < P < 1 and l/3
aEr/,,/aP=O=:[l/(P-
l)],
aEun,/ap
= 0= +
r
-3+3/p’ 1 10 3p 3/p I I
while the second-order conditions indicate that the solution will yield a maximum. The outcome is then seen to be a corner solution for the buying price, P + 0; and an interior solution for the selling price, p = 1. At these values we obtain
Eut?, approaches
+ln 1 ++ln2++ln
1
which has the same limit as the rationing strategy. Thus the monopolist in this example can earn as high an expected utility under a pure price strategy as by rationing, provided that the buying and selling prices are set separately.
4. Rising marginal cost and rationing We turn, finally, to the issue of whether a monopolist could benefit from rationing if it is a net seller of its output in all states. Again by example, we shall demonstrate that rationing can dominate a pure price strategy under the simple conditions of random demand and rising marginal cost, when uncertainty in the level of demand does not influence its elasticity. Assume that the monopolist has a cost function C(y) with C’> 0, C” > 0. Let demand be a function 4 = q( e,p) with a~/&9 > 0, aq/ap < 0, where 8 is a random variable with density f( 0). Moreover, assume that changes in B do not alter the elasticity of q(0,p) for fixed p. The monopolist under a pure price strategy will maximize ET by choice of p, so that at the equilibrium pricep* its expected profits will be En-=p*
/= q(e.p*)f(e) -33
de-Jx
-‘x
c(q(e,p*))f(e)
de.
There is some value of 0, given p*, such that marginal cost MC equals marginal revenue MR. Call it A. Note that, by rising marginal cost, Mc(q(e,p*))>MR(q(e,p*)) for all B>A since MR=(p(l + I/C) by definition, and 6 is independent of 19by assumption. Thus P*q(A.
P*) - c(qbL
for all B>A. Equilibrium
expected
p*>> ‘P*q(e.P*)
profits
- c(q(e,
may be rewritten
(1)
p*>)
as
Suppose now that the monopolist sets a quantity ceiling 4 = q( A.p*). Under this rationing strategy the expected profits become E7TR =p*Jll_q(e,p*)f(e)
a
c(q(e,p*))f(e)
de+p*~Xq(A.p*)f(B)
de-/mc(qI~~p*))f(~) A
de
de.
In comparing to sign
this outcome
with that of the pure price strategy.
ETR ~ Err =p* /%Lp*)f.(~)
de-p*/%b*)f.(@ de 4
A
=L=[P*Y-
we wish
c-(q) -p*dd.
P*)
+ C(YWP*>)] f(e) de.
By (1) the bracketed integrand is positive. Thus E.lrR > ET, and the monopolist benefits from a rationing strategy relative to a pure price strategy. This result is quite intuitive. It merely says that a monopolist which must set a price before its actual demand level is known will benefit from refusing to produce beyond the level at which MC = MR, if such a level exists. Otherwise, in the event that demand is higher, the firm would sustain MC> MR and would earn lower profits than in the rationed case.
5. Conclusions
A profit maximizing monopolist may indeed benefit from a rationing strategy relative to a pure price strategy, if it is subject to incomplete information. However, in some instances at least, a sufficiently clever price strategy can attain the same outcome associated with a single price plus rationing. This result was demonstrated for a monopolist which sells in some states of the world but buys in others. For the monopolist with rising marginal cost, it remains an open question whether a peak-load pricing scheme could perhaps duplicate the rationing outcome. More generally, given any set of conditions under which a monopolist will benefit from rationing, we may ask whether there exists a more complex pure price strategy which would be equally profitable.
Reference Honkapohja, 203-209.
S.. 1980, A note on monopolistic
quantity
rationing,
Economics
Lcttcrs
6.