Research in Economics 61 (2007) 37–43
www.elsevier.com/locate/rie
A solution to the monopolist’s problem when demand is
iso-inelasticI
Vanda Tulli a,∗ , Gerd Weinrich b,1
a Dipartimento di Metodi Quantitativi per le Scienze Economiche e Aziendali, Università di Milano-Bicocca, Piazza dell’Ateneo Nuovo 1,
20126 Milano, Italy
b Catholic University of Milan, Largo Gemelli 1, 20123 Milano, Italy
Received 1 September 2005; accepted 10 October 2006
Abstract
Economic theory does not provide a solution to the monopolist’s price setting problem when the demand curve faced is isoelastic
with elasticity smaller than or equal to one. In the present paper it is argued that this is no longer true when uncertainty about demand
and risk aversion of the monopolist are admitted. In particular it is shown that a monopolist with constant relative risk aversion r
facing an expected demand curve with constant elasticity ε may be seen as behaving like a standard monopolist facing a demand
curve having elasticity (1 + r )ε.
c 2007 University of Venice. Published by Elsevier Ltd. All rights reserved.
Keywords: Monopolistic price setting; Inelastic demand curve; Uncertainty; Risk aversion
1. Introduction
One of the best known results in economic theory is that a monopolist’s price setting problem does not possess
a solution in the case of isoelastic demand curves with own-price elasticity (in absolute value) smaller than or equal
to one. However, many estimates of elasticities yield values not exceeding one.2 Moreover, the non-existence of a
solution is somewhat embarrassing for economic modelling since for example Cobb–Douglas utility functions, which
are probably the most basic and the most frequently adopted type of utility functions used by economists, yield unitelastic demand functions and thus cannot be employed in general equilibrium models with a monopolist.
The purpose of the present paper is to show that non-existence of an optimal price no longer necessarily holds
when the standard model is embedded in a larger structure which allows for uncertainty and risk aversion by the
I This paper has benefited from valuable comments and suggestions by an anonymous referee. All remaining shortcomings are the authors’
responsibility.
∗ Corresponding author. Tel.: +39 02 6448 3164.
E-mail addresses: [email protected] (V. Tulli), [email protected] (G. Weinrich).
URL: http://www.unicatt.it/docenti/weinrich (G. Weinrich).
1 Tel.: +39 02 7234 2728.
2 For example in Deaton (1975) the market demands of only 8 out of 37 goods investigated (admittedly not necessarily traded in monopolistic
markets) display elasticities larger than one. This refers to the numbers given in the column “CEM” (constant elasticity model), under the heading
Price elasticities, in Table 1, p. 266.
c 2007 University of Venice. Published by Elsevier Ltd. All rights reserved.
1090-9443/$ - see front matter doi:10.1016/j.rie.2006.10.002
38
V. Tulli, G. Weinrich / Research in Economics 61 (2007) 37–43
monopolist. The main result is that a risk averse monopolist having constant relative risk aversion r and facing an
isoelastic expected demand curve with elasticity ε can, under plausible circumstances, be seen as behaving like a
corresponding standard monopolist facing elasticity (1 + r )ε. Thus the assumption in economic theory of elasticity
larger than one can be justified by seeing it as a shortcut for the product of two factors, the true elasticity, which
may well be smaller than one, and a factor larger than one that reflects uncertainty and risk aversion. In particular,
when demand is unit-elastic, any positive risk aversion is sufficient. Thus the non-existence of a solution in the case
of Cobb–Douglas utility functions appears to be a rather special case, namely when there is no uncertainty and no risk
aversion at all. The reason that with risk aversion an optimal price exists is reminiscent of Bernoulli’s St. Petersburg
paradox: while with risk neutrality the expected payoff function is monotonically increasing in price, risk aversion
may eventually, that is when uncertainty about profit has grown sufficiently large, bend it down so that a maximum
exists.
In Section 2 we set out the model and in Section 3 we derive the results. Section 4 contains concluding remarks.
2. The model
Consider a monopolist who is uncertain about the demand function for his/her product. More precisely, for any
price p > 0 he/she believes that the quantity demanded y is a random variable that satisfies
(A1) y( p) = (a/ p ε )z( p), a > 0, ε > 0, for some random variable z( p) such that E[z( p)] = 1 for all p > 0.
Thus the elasticity of E y = a/ p ε with respect to p is ε.3 Assuming constant marginal cost c > 0, any realization
y = (a/ p ε )z gives rise to the profit π = ( p − c)y = a( p 1−ε − cp −ε )z. Invoking the expected utility hypothesis
and denoting as f (y, p) and g(z, p) respectively corresponding densities of y and z given p, there exists a von
Neumann–Morgenstern utility function V (π ) so that the firm intends to maximize
Z
Z
U ( p) = V (( p − c)y) f (y, p)dy = V (a( p 1−ε − cp −ε )z)g(z, p)dz.
The standard case corresponds to V being linear so that the monopolist is risk neutral. Then his/her problem is to
maximize expected profit ( p − c)E[y( p)] = a( p 1−ε − cp −ε ) which yields the well-known formula for the optimal
price
ε
c.
(1)
ε−1
Obviously this is meaningful only if ε > 1.
The situation changes when risk aversion of the monopolist is admitted. Since, for ε ≤ 1, p 1−ε − cp −ε is strictly
increasing, the question then is whether there exist conditions on V (π ) and f (y, p) and g(z, p) respectively which
bend the expected utility function eventually down. Intuitively this requires that, as expected profit increases, so
does the variance of profit, and risk aversion is sufficiently large to give enough weight to this fact. We will provide
a solution for the case of CRRA utility functions and lognormal distributions. This is expressed in the following
assumptions.
p=
(A2) The firm’s utility function is Vα (π ) = (π α − 1)/α when α ∈ (−∞, 1] \ {0} and V0 (π ) = ln π. Thus the
firm’s relative risk aversion is r = 1 − α.
(A3) For any p > c the random quantity demanded y( p) as perceived by the firm is lognormally distributed with
parameters ν( p) and σ 2 ( p), where the function σ 2 (·) is differentiable.4
3. Results
A random variable y being lognormally distributed with parameters ν and σ 2 has expected value and variance
2
2
2
E y = eν+σ /2 ,
σ y2 = e2ν+σ eσ − 1 .
3 Strictly speaking the elasticity is −ε. We shall use the term “elasticity” throughout to mean its absolute value, taking for granted that it is
negative.
√
4 The lognormal distribution is given by the density function f (y; ν, σ 2 ) = y 2π σ 2 −1 e−(ln y−ν)2 /2σ 2 .
V. Tulli, G. Weinrich / Research in Economics 61 (2007) 37–43
39
Fig. 1. U (·, α) for, from top to bottom, α = 1, 0.25 and −0.25.
From (A1) this yields
ν( p) = ln a − ε ln p − σ 2 ( p)/2
σ y2 ( p) = (a/ p ε )2 [e
σ 2 ( p)
(2)
− 1].
(3)
Expected utility under (A2) and (A3) is, for α 6= 0,
Z
1
( p − c)α
y α f (y; ν( p), σ 2 ( p))dy − .
U ( p, α) =
α
α
R
2 2
Using the standard result5 that y α f (y; ν, σ 2 )dy = eαν+α σ /2 this yields
U ( p, α) =
( p − c)α α[ν( p)+ασ 2 ( p)/2] 1
e
− ,
α
α
and (2) implies
aα
U ( p, α) =
α
p−c
pε
α
e−α(1−α)σ
2 ( p)/2
−
1
.
α
(4)
Typical shapes of U (·, α) are shown in Fig. 1.6
The existence of a price that maximizes U (·, α) depends obviously on the behavior of σ 2 ( p) or, equivalently, since
by (3)
!
ε 2
p
2
2
σ ( p) = ln
σ y ( p) + 1 ,
(5)
a
on that of σ y2 ( p). While (3) does not imply any restriction on σ y2 ( p) in terms of being increasing or decreasing, we
can show the following result:
Lemma 1. Under assumptions (A1)–(A3) there exists a solution to the monopolist’s price setting problem for any
positive ε < 1 if the variance σ y2 ( p) of the quantity demanded and the coefficient of the monopolist’s relative risk
aversion r are such that
lim inf
p→∞
σ y2 ( p)
p −2ε [ p 2(1−ε)/r − 1]
> a2.
5 See e.g. Meyer (1970).
6 The graphs are obtained setting a = c = 1, ε = 0.75 and σ 2 ( p) = 1.5 ln p.
(6)
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V. Tulli, G. Weinrich / Research in Economics 61 (2007) 37–43
A necessary condition for this to hold is
lim p 2 σ y2 ( p) = ∞.
(7)
p→∞
In the case ε = 1, (7) is necessary and sufficient for any r > 0. If, for some σ02 > 0 and p0 > 0, σ y2 ( p) ≥ σ02 for all
p ≥ p0 , then (7) is satisfied while (6) holds whenever r > (1 − ε)/ε.
Proof. Consider first the case α > 0. Then from (4) U ( p, α) > U (c, α) = −1/α for all p > c. For ε < 1,
α
2
p−c
=: s( p) tends to infinity, so e−α(1−α)σ ( p)/2 =: t ( p) necessarily has to tend to zero if a solution is to exist.
pε
2
[ln( p − c) − ε ln p] tends to infinity, which
Now s( p)t ( p) → 0 when p → ∞ iff, taking logarithms, σ 2 ( p) − 1−α
2
2
is equivalent to σ ( p) − 1−α (1 − ε) ln p → ∞ as lim p→∞ [ln p − ln( p − c)] = 0. Using 1 − α = r , this proves
existence when
lim inf σ 2 ( p)/ ln p > 2(1 − ε)/r
p→∞
while σ y2 should satisfy (6) from (5). If ε = 1, then it is sufficient that t ( p) → 0 which is equivalent to σ 2 ( p) → ∞
for p → ∞, and hence to (7) by (5). Since the expression on the left hand side of (6) is equal to
p 2 σ y2 ( p)
p 2(1−ε) [ p 2(1−ε)/r − 1]
,
(8)
(6) implies (7), as the denominator of (8) tends to infinity for p → ∞ whenever ε < 1.
Consider now the case α < 0. Then U ( p, α) > U (c, α) = −∞ for all p > c, and a maximum exists if
U ( p, α) → −∞ for p → ∞, which is equivalent to ln(−U ( p, α)) → +∞ for p → ∞. From (4) and since α < 0
2
this is true when α[ln( p−c)−ε ln p]−α(1−α)σ 2 ( p)/2 → ∞ or, equivalently, σ 2 ( p)− 1−α
[ln( p−c)−ε ln p] → ∞,
and thus we are back to what we have had before. Also the case ε = 1 is analogous to what has been argued when
α > 0.
Regarding the case α = 0, by differentiability of U in p the existence of a maximum when (6) holds implies that,
in both cases, α > 0 and α < 0, there exists a pα such that ∂U ( pα , α)/∂ p = 0. As Vα0 (π ) = π α−1 is continuous in α
for all α, the same must be true in the case α = 0.
Finally, if σ y2 ( p) ≥ σ02 for all p ≥ p0 for some σ02 > 0 and p0 > 0, (7) holds trivially, while (6) holds whenever
2ε > 2(1 − ε)/r , or r > (1 − ε)/ε, because in that case the denominator in (6) tends to zero. The term p 2 σ y2 ( p) in condition (7) is the variance of revenue. That the “riskiness” of total revenue increases in
price is a standard assumption in models of the firm’s behavior under uncertainty (see e.g. Leland (1972, p. 279));
here it has to tend to infinity (which implies the same for the variance of profit, σπ2 ( p) = ( p − c)2 σ y2 ( p)). Sufficient
for this to hold is that σ y2 ( p) be bounded away from zero, which is one of the conditions expressed in Lemma 1.
Maximizing the firm’s expected utility we obtain the following first-order condition:
Lemma 2. If a solution exists, it must satisfy
p=
r τ ( p) + ε
c
r τ ( p) + ε − 1
(9)
where
τ ( p) = p
d 2
σ ( p)/2.
dp
(10)
Proof. From (4)
"
#
d 2
∂U
p − c α−1 −α(1−α)σ 2 ( p)/2 1
d p σ ( p)
α
=a
e
p − ε( p − c) − p( p − c)(1 − α)
∂p
pε
2
p ε+1
which is zero iff the term in square brackets is. Using 1 − α = r , this is equivalent to (9). Since Vα0 (π ) is continuous
in α, this derivation extends to the case V0 (π ) = ln π.
V. Tulli, G. Weinrich / Research in Economics 61 (2007) 37–43
41
From inspecting Eq. (9) it is obvious that it is a generalization of (1); indeed, it becomes that when the firm is
risk neutral or when there is no uncertainty. An important special case in which (9) can be solved explicitly for p is
provided by the following assumption.
(A4) For p > c the second moment of quantity demanded is constant, E[y 2 ( p)] =: k.
Then we obtain the following result:
Proposition 1. Assume (A1)–(A4), r > (1 − ε)/ε and k ≥ (a/cε )2 . Then the monopolist’s optimal price exists and
is given by
p=
(1 + r )ε
c.
(1 + r )ε − 1
(11)
Proof. From σ y2 ( p) = E[y 2 ( p)] − (E[y( p)])2 , E[y 2 ( p)] = k, (3) and (A1) we obtain
k = (a/ p ε )2 [eσ
Solving for
σ 2 ( p)
2 ( p)
− 1] + (a/ p ε )2 = (a/ p ε )2 eσ
2 ( p)
.
yields
σ 2 ( p) = ln((a/ p ε )−2 k) = 2ε ln p − 2 ln a + ln k
(12)
which is positive for p > c by the assumptions made. Therefore by (10) τ ( p) = ε and thus by Lemmas 1 and 2 we
get the result. Formula (11) generalizes (1) and can be interpreted as saying that a monopolist facing a random demand function
with elasticity of expected demand ε and having relative risk aversion r takes the same decision that a risk neutral
firm facing elasticity (1 + r )ε would take. In this sense, risk aversion can simply be seen as increasing the elasticity
by the factor 1 + r .
Assumption (A4) implies that σ y2 ( p) is increasing towards a finite limit, which does not appear unreasonable. The
condition k ≥ (a/cε )2 is a consequence of assumptions (A1), (A3) and (A4) in the sense that only if it is met are
these assumptions jointly consistent. More precisely, together with p > c it avoids the occurrence of small values
of p(< (a 2 /k)1/(2ε) ) that would make σ 2 ( p) negative in (12) and thus contradict (A3). Note that the optimal price
according to (11) is independent of both a and k, and hence rescaling these parameters does not change it. Furthermore
ε 2
(1+r )ε
it can be shown that the above condition on k can be weakened to k ≥ a/ (1+r
c
by relaxing (A3) and (A4)
)ε−1
to hold for p > max{(a 2 /k)1/(2ε) , c} only and assuming non-randomness (i.e. y( p) = a/ p ε ) otherwise. In a case
where the weakened condition is still violated, the optimal price still exists but is equal to (a 2 /k)1/(2ε) . Finally,
Proposition 1 can be generalized also by assuming that some (any) nth moment of y( p), n ≥ 2, is constant or that
τ ( p) is any constant τ > 0, in which cases results similar to (11) can be obtained.7
Example. To illustrate the usefulness of the above results we apply them to a Robinson Crusoe economy with the
producer a monopolist and the consumer having a Cobb–Douglas utility function. Since the latter gives rise to unitelastic demand functions, in the standard model existence of general equilibrium fails. This is somewhat disturbing
as Cobb–Douglas utility functions are among the most basic and frequently used modelling tools in economic theory.
We now show that this problem can be overcome by introducing uncertainty and risk aversion of the monopolist.
More precisely, the consumer has a utility function U (y, `) = y h (` − `)1−h , 0 < h < 1, for consumption y
and labor `, with ` indicating total labor endowment. Denoting her/his income by I = w` + π, where w is the
wage rate, she/he wants to consume y = h I / p, with p the price of the consumption good. To this end she/he offers
`s = ` − (1 − h)I /w of labor services.
The firm rightly expects the consumer’s demand curve “on average” to be of the form y( p) = a/ p, but is not sure
that this is an exact relationship. Therefore it works with a random demand such that E[y( p)] = a/ p. Assuming
7 For this point see Tulli and Weinrich (2005).
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V. Tulli, G. Weinrich / Research in Economics 61 (2007) 37–43
Fig. 2. General equilibrium allocations with a Cobb–Douglas consumer and a monopolist whose risk aversion varies between zero and infinity.
moreover that it has a production function y = b`, b > 0, and fulfills the conditions of Proposition 1, it sets its price,
for any given w, according to (11) as
1 w
.
p ∗ (r ) = 1 +
r b
This yields a profit π = (wy)/(r b), depending on any y realized. Note that here, since ε = 1, any r > 0 is sufficient
for obtaining a finite optimizing price.
In general equilibrium employment ` has to coincide with the consumer’s supply `s , and therefore it is determined
by ` − (1 − h)(` + r1 `) = `. This yields the transaction levels
hr
hr
b`.
`,
y ∗ (r ) =
1−h +r
1−h +r
Note that p ∗ (0) = ∞ and `∗ (0) = y ∗ (0) = 0 which corresponds to the result of non-existence of general equilibrium
in an economy with Cobb–Douglas consumers and a standard monopolist. Moreover, p ∗ (∞) = w/b, `∗ (∞) = h`
and y ∗ (∞) = hb`, which means that, as risk aversion tends to infinity, price and transaction levels converge to those
of the perfectly competitive equilibrium. In Fig. 2 the equilibrium allocations are shown as the part of the ray y = b`
where 0 ≤ ` ≤ h`. Varying r from zero to infinity permits us to connect the standard monopolistic equilibrium,
E 0 , with the competitive equilibrium, E ∞ , where the case of non-existence of the standard monopolistic equilibrium
appears now as a degenerate and limiting case of monopolistic equilibria with risk aversion. In particular this illustrates
most strikingly the inefficiency of monopoly.8
`∗ (r ) =
4. Concluding remarks
In this paper it has been shown how the introduction of uncertainty and risk aversion may overcome the problem of
non-existence of an optimal price for a monopolist facing a demand curve with elasticity smaller than or equal to one.
Under the conditions spelled out here there exists for any non-zero value of elasticity a degree of relative risk aversion
r such that non-existence of an optimal price disappears. In the special case of constant unit elasticity any r > 0 is
sufficient.
A main implication of the present analysis is that r > 0 may be seen as increasing the monopolist’s perception
of elasticity by the factor 1 + r . This provides in fact a way of rationalizing the assumption that elasticity ε exceed
one: while the true (under certainty) elasticity, say b
ε, may well be smaller than one, ε can be thought of as a perceived
8 In a similar way it can be shown (see Tulli and Weinrich (2005)) that, for fixed r > 0, by varying uncertainty one obtains the whole piece of
line E 0 − E ∞ , where E 0 corresponds to zero uncertainty and E ∞ to “infinite” uncertainty.
V. Tulli, G. Weinrich / Research in Economics 61 (2007) 37–43
43
or subjective value being obtained by assuming uncertainty and a degree of relative risk aversion r so that (1 + r )b
ε
equals ε.
The above result has been shown to hold for the case of lognormal density functions regarding the distribution of
quantity demanded at any price and utility functions displaying constant relative risk aversion. Considering that the
point of departure – constant elasticity of demand – is itself a specific (though very important) case, these specifications
seem the most natural ones for the problem at hand. A further question is whether the present results can be generalized
to hold for a whole class of distributions and utility functions. Although this is likely to be the case, it should be quite
clear that any specific formula like (11) requires concrete specifications of the functional forms involved.
Regarding the two basic assumptions in this paper, uncertainty and risk aversion, the first one should not be
controversial. Risk aversion, on the other hand, can be supported on the grounds of both empirical evidence and
theoretical reasoning based on capital market imperfections or on the impact that performance-based compensation
schemes have on risk averse managers.
References
Deaton, A.S., 1975. The measurement of income and price elasticities. European Economic Review 16, 261–273.
Leland, H.E., 1972. The theory of the firm facing uncertain demand. American Economic Review 62 (3), 278–291.
Meyer, P.L., 1970. Introductory Probability and Statistical Applications, 2nd edition. Addison Wesley, Reading, Massachusetts.
Tulli, V., Weinrich, G., 2005. A solution to the monopolist’s problem when demand is iso-inelastic, with an application to a general equilibrium
model. Rapporto di ricerca n. 94 del Dipartimento di Metodi Quantitativi per le Scienze Economiche e Aziendali, Università di Milano-Bicocca.
Paper downloadable at http://www.dimequant.unimib.it/pubblicazioni prof/qua2005/isowpds1.pdf.