2. several market segments. We
show that aggregate demand inherits existence properties from
the individual market
segments but this is not true for uniqueness properties. The
problem of using a limited
price menu to price multiple market segments is analyzed.
Using a single price for all
the market segments and a different price for each market
segment are two extreme
strategies that provide us with lower and upper bounds on
profits. We next consider
bounds and heuristics to design a menu of J > 1 prices for M > J
market segments
for a variety of demand functions including linear, log-linear
demands and for demands
governed by the multinomial logit model (MNL). Existence and
uniqueness results for
multiple products are provided for a variety of commonly used
demand models.
1 Introduction
We are concerned with the following static pricing problem:
r(z) = sup
p∈ X
(p−z)d(p) (1)
where z is the marginal cost of capacity, d(p) is the demand at
price p and X is the set of
allowable prices. Economists are usually interested in the more
general problem where costs
1
3. are non-linear. Our interest in the simpler problem with linear
costs stems from dynamic
pricing where problem (1) arises with z equal to the marginal
value of capacity. Readers not
interested in the connection to dynamic pricing can skip to
Section 2.
To see the connection with dynamic pricing consider the
problem of maximizing the ex-
pected revenue that can be obtained from finite, non-
replenishable, capacity c over a finite
horizon [0,T] assuming zero salvage value. Gallego and van
Ryzin [5] show that when demand
arrives as a Poisson process with intensity dt(p), then the value
function V (t,x), representing
the maximum expected revenue when the time-to-go is t and the
remaining inventory is x,
satisfies the Hamilton Jacobi Bellman (HJB) equation:
∂V (t,x)
∂t
= sup
p∈ X
(p− ∆V (t,x))dt(p), (2)
where ∆V (t,x) = V (t,x)−V (t,x−1) is the marginal value of the
xth unit of capacity, and the
conditions are V (t, 0) = V (0,x) = 0. Equation (2) requires
continuity of dt(p) with respect
to t. If dt(p) is piecewise continuous then the HJB equation (2)
holds over each subinterval
where dt(p) is continuous where the boundary condition is
4. modified to be the value function
over the remaining time horizon.
Notice that the optimization in (2) is of the form (1) with z =
∆V (t,x). If a maximizer,
say pt(z), exists for each z ≥ 0 and each t ∈ [0,T], then an
optimal solution to the dynamic
pricing problem (2) is to set price P(t,x) = pt(∆V (t,x)) at state
(t,x). There are two sources
of price variation in dynamic pricing. The first source is
variations due to state dynamics as
the marginal cost ∆V (t,x) changes with the state (t,x). Gallego
and van Ryzin show that
∆V (t,x) is increasing in t, so the marginal value of the x unit is
more valuable if we have more
time to sell it, and decreasing in x, so the marginal value
decreases with capacity. We will later
show that pt(z) is increasing in z, so a sale at state (t,x) causes
the price to instantaneously
increase to P(t,x− 1) > P(t,x). The second source of price
variation is changes in pt(z) due
to changes of demand dt(p) over time t. If pt(z) = p(z) is time
invariant and then P(t,x) is
increasing in the time-to-go t, since ∆V (t,x) is increasing in t.
This means that prices decline
in the absence of sales to stimulate demand. If pt(z) changes
with time then P(t,x) can either
increase or decrease over time as the forces of state dynamics
may be in conflict with changes
in willingness to pay.
Quasi-static pricing policies are heuristic pricing policies of the
form
Ph(t,x) = pt(z(T,c)) 0 ≤ t ≤ T
5. that react to changes in dt(p) in t but not to changes in the
marginal value ∆V (t,x). Typically
z(T,c) is chosen to capture the marginal value of capacity by
solving the following fluid
program:
V
̄ (T,c) = min
z≥0
[cz +
∫ T
0
rt(z)dt]. (3)
Program (3) arises in at least three different ways: 1) By using
Approximate Dynamic
Programming (ADP) with affine functions, 2) By using a fluid
limit approximation and du-
alizing the capacity constraint and 3) By modifying the
differential equation (2) by replacing
2
∆V (t,x) with the partial derivative Vx(t,x) of V (t,x) with
respect to x. We will later show
that (3) is a convex minimization program. We refer the reader
to Gallego and van Ryzin
[5] for a proof that V
̄ (T,c) is an upper bound on V (T,c) and for
a discussion of the asymp-
totic optimality properties of the quasi-static pricing policy for
the case d(p) = dt(p) for all
t ∈ [0,T].
6. Quasi-static pricing policies are responsive to changes in
willingness to pay but not re-
sponsive to changes in state dynamics. It can be shown that
quasi-static pricing policies are
asymptotically optimal, see Gallego [7], and they are a natural
extension of the fixed priced
policies in [5]. The fact that quasi-static pricing policies ignore
state dynamics is materially
detrimental only when both capacity and aggregate demand are
relatively small. On the posi-
tive side, quasi-static pricing policies do not suffer from the
nervousness of full dynamic pricing
policies that react instantaneously to state dynamics, e.g.,
decreasing prices between sales and
increasing them after each sale. This is an important advantage
in practice as quasi-static
policies are easier to implement. Limits are often are imposed
on prices, so the optimization
is restricted to p ∈ Xt where Xt may be a finite price menu. The
design of the price menu is
considered part of the problem. For example, if the cardinality
of the set of different prices
utilized by the static pricing heuristic {pt(z(T,c)) : 0 ≤ t ≤ T} is
M and M is considered too
large, then the task may be to select a pricing menu with at most
J < M different prices,
to prevent the pricing policy from being too nervous. We study
a variant of this problem in
Section 6.1.
The quasi-static heuristic is often made more dynamic by
frequently resolving (3), ob-
taining an updated value of the marginal value of capacity each
time (3) is resolved. More
precisely, if the realization of demand deviates significantly
from its deterministic path, then
7. the value of z can be updated at state (s,y) to z(s,y) where z(s,y)
is the minimizer of
[yz +
∫s
0 rt(z)dt]. Prices are then updated to pt(z(s,y)) for t ∈ [0,s] or
until the deterministic
problem (3) is solved again. If the system is updated
continuously, we get a feedback policy
Ph(t,x) = pt(z(t,x)), which tends to perform better than the
quasi-static policy but is also
requires more computations and results in more nervous prices.
The reader is referred to
Maglaras and Meissner [13] who show that the feedback policy
is also asymptotically optimal,
and to Cooper [2] who presents and example that shows that
updating z when the inventory
and the time-to-go are small can hurt rather than help
performance.
The near optimality of quasi-static pricing policies motivates
the study of static optimiza-
tion problem (1). Although this is a special case of the basic
pricing problem where marginal
costs are constant there are some subtle issues regarding
existence and uniqueness. In addi-
tion, there are a number of variants of the problem that are of
interest in their own right. Our
aim on this Chapter is to present the reader with a unified and
comprehensive analysis of the
problem.
In Section 2 of this Chapter we present basic properties and
existence of finite maximizers.
We first show that r(z) is decreasing convex in z and present
8. conditions on d(p) that guarantee
the existence of a finite price p(z) increasing in z such that r(z)
= (p(z)−z)d(p(z)). In Section
3 we present sufficient conditions for the uni-modality of r(p,z)
in p and for the uniqueness
of p(z). We analyze the case of bounded capacity and lower
bounds on sales in Section 4.
Multiple market segments are treated in Section 6. We first look
into the question of existence
3
and uniqueness when the demands of two or more market
segments are aggregated. We show
that existence conditions for the individual market segments are
inherited by the aggregate
demand. This is not so for uniqueness conditions. We then
explore heuristics to price M
market segments with at most J < M different prices. This
problem may arise either because
only a few prices are allowed or because detailed demand
information from the different market
segments is not enough to support using more prices. We show
that it is often possible to
design near optimal price menus for values of J that are small
relative to M. The welfare
problem is discussed in Section 5 where call options on capacity
are presented as a viable
solution when booking and consumption are separated by time
and customers learn their
valuations between booking and the time of consumption.
2 Basic Properties and Existence of Finite Maximizers
9. Let d(p) : X ⊂ [0,∞) → [0,∞] be a function representing the
demand for a product at
price p ∈ X. For any z ≥ 0 let r(p,z) = (p − z)d(p) be the profit
function for any p ∈ X.
We treat z as an exogenous unit cost and r(p,z) as the profit
function. For z ≥ 0, we define
r(z) = supp∈ X r(p,z), the optimal profit as a function of the unit
cost. We write sup instead of
max in the definition of r(z) because the maximum may not be
attained. To see this consider
the demand function d(p) = 1 for p ∈ [0, 10) and d(p) = 0 for p
≥ 10 then r(z) = (10−z)+ but
the maximum is not attained. As an example where a finite
maximizer fails to exist, consider
the demand function d(p) = p−b,p ≥ 0 for b ∈ (0, 1). Then r(p,
0) = p1−b so r(0) = ∞ and
there is no finite maximizer. Later we will present sufficient
conditions for the existence of a
finite maximizer p(z) < ∞ such that r(z) = r(p(z),z). However,
even if the supremum is not
attained we can show that r(z) is decreasing1 convex in z.
Theorem 1 r(z) is decreasing convex in z.
Proof: Notice that for any z < z′, r(p,z) = (z′ − z)d(p) + r(p,z′) ≥
r(p,z′). Therefore
r(z) = supp∈ X r(p,z) ≥ supp∈ X r(p,z′) = r(z′). To verify
convexity, let α ∈ (0, 1), and let
z(α) = αz + (1 −α)z′. Then
r(z(α)) = sup
p∈ X
r(p,z(α))
= sup
10. p∈ X
[αr(p,z) + (1 −α)r(p,z′)]
≤ α sup
p∈ X
r(p,z) + (1 −α) sup
p∈ X
r(p,z′)
= αr(z) + (1 −α)r(z′).
Remark 1: The convexity of r(z) implies that cz +
∫T
0 rt(z)dt is a convex problem in z, so to
obtain z(T,c), the marginal cost to be used for quasi-static
pricing, all we need to do is to
1We use the term increasing and decreasing in the weak sense
unless stated otherwise.
4
find the unconstrained minimizer of the convex function cz +
∫T
0 rt(z)dt and take its positive
part.
Remark 2: Jensen’s inequality implies that Er(Z) ≥ r(EZ). This
means that a retailer
11. prefers a random unit cost Z than unit cost EZ, provided that he
can charge random prices
p(Z). This also explains why dynamic pricing reacts to state
dynamics, ∆V (t,x), even when
demand dt(p) = d(p) is time invariant.
Remark 3: If r is twice differentiable then r(Z) ' r(E[Z]) + (Z −
E[Z])r′(E[Z]) + 0.5(Z −
E[Z])2r′′(E[Z]). Taking expectations yields E[r(Z)] − r(E[Z]) '
0.5Var[Z]r′′(E[Z]). Conse-
quently, the benefits of responding to cost changes is large
when Z has a large variance and
r has large curvature at E[Z].
Example 1 If d(p) = 1 − p over p ∈ [0, 1] then for z ∈ [0, 1],
p(z) = (1 + z)/2 maximizes
r(p,z) = (p−z)(1−p) resulting in r(z) = (1−z)2/4. If z = 1/2 then
r(1/2) = 1/16. Notice a
retailer with demand d(p) prefers a wholesaler with unit cost z1
= 1/3 with probability 1/2 and
unit cost z2 = 2/3 with probability 1/2 since this leads to more
than a 10% increase in expected
profits from 1/16 to 5/72. However, this does not mean that the
retailer prefers to randomize
prices if his true cost is z = 1/2 for any deviation from p(1/2)
leads to lower profits.
The following Corollary pushes the idea a bit further. The proof
is provided in the Ap-
pendix.
Corollary 1 If g(y) : <m → <+ is increasing in y then r(g(y)) is
decreasing in y. If g(y)
is concave, then r(g(y)) is convex. Moreover, if Y ∈ <m is
random, then Er(g(Y )) ≥
r(Eg(Y )) ≥ r(g(EY )).
12. We can interpret z = g(y) as the unit cost where y is the vector
of component costs. As an
example, g(y) = f ′y where f ∈ <m+ is the vector of resource
requirements. This shows, again,
that the retailer is better off with random cost Y than with
deterministic costs EY .
We have not assumed that d(p) is decreasing in p to allow for
prestige goods whose demand
may increase in price over a certain range. We will now show
that we can construct a decreasing
function d̄ (p), based on d(p), such that under mild conditions we
can find a maximizer p(z) of
r(p,z) by finding a maximizer, say p̄ (z), of r̄ (p,z) = (p−z)d̄ (p).
Indeed, let d̄ (p) = supp′≥p d(p′)
for all p ≥ 0, and assume that d(p) is upper-semi-continuous
(USC). Recall that a function
d(p) : X ⊂ [0,∞) → [0,∞] is USC at po ∈ X if lim supp→po
d(p) ≤ d(po) and d(p) is USC
in p ∈ X if it is USC at every point p ∈ X. Clearly d(p) USC
implies that d̄ (p) is USC.
Moreover any decreasing USC function is left-continuous with
right limits (LCRL), so d̄ (p) is
LCRL. Let r̄ (z) = supp∈X r̄ (p,z). It is easy to construct
examples where r̄ (z) > r(z). The
next Lemma shows that this is not possible if d(p) is upper-
semi-continuous (USC).
Lemma 1 If d(p) is USC and p̄ (z) is a finite maximizer of r̄ (p,z),
then p(z) = p̄ (z) is a
maximizer of r(p,z) and r(z) = r̄ (z).
5
13. The proof of Lemma 1 can be found in the Appendix.
Our next task is to find conditions that guarantee the existence
of a finite price that
attains the maximum of r(p,z) and for this we need a few
definitions from convex analysis,
see Rockefeller [15]. A function d(p) is said to be proper if d(p)
< ∞ for all p ∈ [0,∞).
The product of two non-negative, proper USC functions is also
USC. The product of two
non-negative USC, proper or not, is also USC provided we treat
0 ×∞ = ∞, and we will
agree to this convention to develop a unified theory for both
proper and improper functions.
Let s̄ (z) =
∫∞
z d̄ (y)dy be the area under the function d̄ (y) to the right of z,
and notice that
r̄ (z) ≤ s̄ (z) ≤ s̄ (0) for all z ≥ 0.
The following result presents conditions that guarantees the
existence of a finite maximizer.
The proof of the result is somewhat technical and can be found
in the Appendix.
Theorem 2 If d(p) is USC and s̄ (0) < ∞ then for every z ≥ 0
there exist a finite price
p(z) ∈ [z,∞) such that r(z) = r(p(z),z). Moreover, p(z) can be
selected so that it is increasing
in z.
Remark 1: If we want to guarantee the existence of a finite price
for a given z, rather than for
14. all z ≥ 0, then it is enough to require d to be USC on X ∩{p ≥
z} and to require s̄ (z) < ∞.
Remark 2: Notice that the condition s̄ (0) < ∞ is sufficient but
not strictly necessary. To see
this notice that d̄ (p) = 1/p results in s̄ (0) = ∞ yet r̄ (p, 0) = 1 for
all p > 0, so p̄ (0) = 1 is
optimal. However, p̄ (z) = ∞ for all z > 0 since r̄ (p,z) = 1 −z/p is
increasing in p.
Remark 3: In many cases d(p) is eventually decreasing, i.e.,
there is a p′ such that d(p) is
decreasing on p ≥ p′. However, Theorem 2 does not require this.
For example, the demand
function d(p) = a exp(−bp) sin2(p) is not eventually decreasing
yet d̄ (p) ≤ a exp(−bp) so
s̄ (0) ≤ a
b
.
The following result shows that if the demand comes from a
maximum willingness to pay
function with finite mean then the conditions of Theorem 2
apply.
Corollary 2 If d(p) = λP(W ≥ p) for some random variable W
with E[W] < ∞, then there
exist a finite maximizer p(z) such that r(z) = r(p(z),z).
Under the first two conditions of Corollary 2, the actual
demand, say D(p) is random with
d(p) = E[D(p)]. As an example, the potential demand may be
Poisson with parameter λ and
demand at price p may be a thinned Poisson with parameter
λH(p). Notice that by defining
15. H(p) = P(W ≥ p) instead of H(p) = P(W > p) we are able to
claim that H(p) is LCRL.
This is an innocuous assumption if the distribution of W is
continuous, or if W is discrete and
pricing is, as it is in practice, restricted to discrete values, e.g.,
dollars and cents. However, the
case where W is discrete and prices are allowed to be
continuous leads to technical problems.
Thus, if a customer is willing to pay any price lower than $10,
then there is no finite price
that maximizes the revenue that we can generate from such a
customer, but things are fine if
he is willing to pay up to and including $10. For this reason it is
convenient to think of W as
the maximum willingness to pay when H(p) is defined as P(W ≥
p).
6
As an example, if W is exponential with mean θ, then d(p) =
λe−p/θ, p(z) = z + θ and
r(z) = θe−(z+θ)/θ = θe−1e−z. In this case the demand function
d(p) has two parameters, λ
representing the expected market size and θ representing the
mean willingness to pay. There
are, however, examples where d(p) = λH(p) with H(p)
decreasing in p, where H(p) is not of
the form P(W ≥ p) for some random variable W. To see this
consider the demand function
d(p) = λp−β for some β > 1. Then s(z) < ∞ and p(z) = βz/(β − 1)
for all z > 0, yet
there is no random variable W such that λP(W ≥ p) = d(p), as
p−β > 1 for p ∈ (0, 1).
Often d(p) = λH(p), with H decreasing can be written as d(p) =
16. λf(αp + β ln p), where f is
a decreasing function and α and β are non-negative parameters.
For example, f(x) = e−x,
α = 1/θ and β = 0 yields d(p) = λe−p/θ, while α = 0 and β > 1
yields d(p) = λp−β.
2.1 Demand Estimation
Suppose that time is rescaled into tiny intervals so that the
demand Dt = D(pt) at price pt in
period t is a Bernoulli random variable with expected value
d(pt) = λf(αpt + β ln(pt)) << 1,
for some positive, decreasing function f, e.g., f(x) = e−x or f(x)
= e−x/(1 + e−x). Then
Dt = 1 with probability d(pt) and Dt = 0 with probability 1
−d(pt). Suppose we have data
(ps,ds) : s = 1, . . . , t, where ds is the realized value of Ds in
period s. The likelihood function
up to time t is given by
Lt(λ,α,β) = Π
t
s=1d(ps)
d
s(1 −d(ps))
1−ds.
The log-likelihood function is given by
lt(λ,α,β) =
s∑
s=1
[ds ln(d(ps)/(1 −d(ps)) + ln(1 −d(ps))].
17. The score equations are obtain by setting the derivatives
lt(λ,α,β) with respect to λ,α and
β equal to zero. The solution to the score equations are the
maximum likelihood estimators
λ̂t, α̂t, β̂t. One important concern is whether the sequence of
estimators λ̂t, α̂t, β̂t converges to
the true parameter values λ,α and β. An interesting finding is
that to guarantee convergence
there needs to be enough variability in the prices. Without
enough variability, it is possible
for the estimates to converge to incorrect values of the
parameters.
3 Unimodality of r(p,z) and uniqueness of p(z)
We now turn to conditions on the demand function d(p) that
guarantee that r(p,z) does not
have local, non-global, maximizers or more succinctly that
r(p,z) is uni-modal in p ≥ z. This
equivalent to r(p,z) being quasi-concave in p ≥ z and to r(p,z)
having convex upper level
sets: {p ≥ z : r(p,z) ≥ α} for all α. If d(p) is continuous and
differentiable, we define the
hazard rate at p to be h(p) = −d′(p)/d(p) where d′(p) is the
derivative of d at p. The hazard
rate function h(p) is defined for all p < p∞ = sup{p : d(p) > 0}.
Notice that p∞ may be ∞.
p∞ is the null price as d(p) > 0 for all p < p∞ and d(p) = 0 for
all p ≥ p∞. We say that a
function f(p) has a unique sign change from + to − over p ≥ z if
the function starts positive,
7
18. becomes non-positive and stays non-positive once it becomes
non-positive for the first time.
Notice that we are not requiring f(p) to be decreasing, nor for a
root of f(p) = 0 to exist.
The following Theorem provides sufficient conditions for the
existence of a finite maximizer.
The proof of the Theorem is in the Appendix.
Theorem 3 If d(p) is differentiable and
f(p) = 1 − (p−z)h(p) (4)
has a unique sign change from + to − on p ≥ z, then r(p,z) is
unimodal and
p(z) = sup{p : 1 − (p−z)h(p) ≥ 0} (5)
is a global maximizer of r(p,z).
Proof: The derivative of r(p,z) with respect to p can be written
as
∂r(p,z)
∂p
= d(p) + d′(p)(p−z)
= d(p) [1 − (p−z)h(p)] (6)
for all p < p∞. As a result r(p,z) is increasing in p for all p <
p(z) and decreasing for all
p ≥ p(z). Moreover, r(p,z) = 0 for all p ≥ p∞, proving that p(z)
is a global maximizer.
Notice that we cannot guarantee the existence of a root to
19. 1−(p−z)h(p). This is because
d′(p) and therefore f(p) need not be continuous. While Theorem
3 rules out the existence of
local, non-global, maximizers, there may be multiple global
maximizers, i.e., multiple roots of
f(p) = 0, if there is an interval over which h(p) = 1/(p− z). The
following corollary provide
stronger conditions for the existence and uniqueness of a finite
maximizer and also provides
bound on p(z).
Proposition 1 a) If h(p) is continuous and increasing in p and
h(z) > 0, then there is a
unique optimal price satisfying z ≤ p(z) ≤ z + 1/h(z).
b) If ph(p) is continuous and increasing in p and there exists a
finite z′ ≥ z such that
1 < z′h(z′), then there is a unique optimal price satisfying z ≤
p(z) ≤ z/(1−1/z′h(z′)).
c) If d̃(p) is a demand function with hazard rate h̃(p) such that
h̃(p) ≥ h(p) or ph̃(p) ≥ ph(p)
for all p, then p̃(z) ≤ p(z) where p̃(z) is a maximizer of r ̃(p,z) =
(p−z)d̃(p).
Proof: Part a) If h(p) is continuous and increasing in p then f(p)
is continuous and strictly
decreasing in p ≥ z. Moroever, f(z + 1/h(z)) = 1 − h(z +
1/h(z))/h(z) ≤ 0 < 1 = f(z),
on account of h(z + 1/h(z)) ≥ h(z) > 0. Therefore there exist a
unique p(z) satisfying (5)
that is bounded below by z and above by z + 1/h(z). Part b) If
ph(p) is increasing in p and
z′h(′z) > 1 then f(p) is continuous in p > z and the equation f(p)
= 0 can be written as
20. 8
ph(p) = p/(p−z) with the left hand side increasing in p and the
right hand side decreasing to
one for p > z. Since zh(z) < ∞ it follows that p(z) ≥ z. Notice
that z/(1−z′h(z′)) is the root
of z′h(z′) = p/(p−z). Since ph(p) ≥ z′h(z′) ≥ p/(p−z) for all p ≥
z/(1 −z′h(z′)) it follows
that p(z) is unique and bounded above by z/(1 − 1/z′h(z′)). Part
c) Clearly f ̃(p) ≤ f(p) so
p̃(z) ≤ p(z).
The reader may wonder whether there are demand functions that
achieve the bounds in
part a) and b) of Proposition 1. Part c) suggest that the bounds
may be attained when
h(p) or ph(p) increase the least, e.g., when they are constant.
For part a) this suggest the
hazard rate h(p) = 1/θ that corresponds to the exponential
demand function d(p) = λe−p/θ,
resulting in p(z) = z + θ = z + 1/h(z). For part b) we try ph(p) =
b > 1, corresponding
to d(p) = λp−b, which is known as the constant price elasticity
demand model. In this case
p(z) = bz/(b−1) = z/(1−1/b) = z/(1−1/z′h(z′)). Notice that the
condition b > 1 is crucial
as there is no finite root p(z) if b < 1, or if b = 1 and z > 0.
The reader is directed to van den Berg [17] and references
therein for earlier efforts to
characterize the existence or uniqueness of global maximizers.
In particular van den Berg
assumes that H exist, is continuous and E[V ] < ∞ to show
existence. He assumes that ph(p)
21. is strictly increasing to show uniqueness. He calls this condition
increasing proportional failure
rate condition (IPFR) and gives a large list of distribution
functions that satisfy the IPFR
condition. Economists frequently write the first order condition
f(p) = 0 as
p−z
p
=
1
ph(p)
=
1
|e(p)|
where e(p) = −ph(p) = pd′(p)/d(p) is the elasticity of demand.
Since ph(p) is the (absolute)
elasticity of demand at price p, the IPFR condition is equivalent
to assuming an increasing
(absolute) demand elasticity. The reader is also referred to
Lariviere and Porteus [10] for an
equivalent assumption where ph(p) is called the generalized
hazard rate.
The problem of maximizing r(p,z) can sometimes be
transformed so that demand rather
than price is the decision variable. This can be done if there is
an inverse demand function,
say p(d), that yields demand d at price p(d). This results in the
problem of maximizing
(p(d) −z)d over d. It is sometimes advantageous to use this
22. formulation as there are demand
functions for which (p(d) −z)d is concave in d while r(p,z) is
not concave in p. While we are
cognizant of this advantage, and have used it in some of our
research, it is interesting to note
that there are also demand functions for which r(p,z) is concave
in p without (p(d)−z)d being
concave in d. We refer the reader to Ziya et al. [19] for an
interesting analysis that shows
that non-equivalence of the following assumptions (i) concavity
of pd(p) in p, (ii) concavity of
dp(d) in d and (iii) ph(p) increasing in p.
9
4 Bounded Capacity and Sales Constraints
Consider pricing a product where up to c units can be procured
at marginal cost z. At
price p we can sell at most dc(p) = min(d(p),c) units. It is
possible to sell up to dc(p)
units assuming that customers are willing to take partial orders
or that demand comes from
many customers with small demands. In this case the pricing
problem can be formulated as
rc(z) = supp∈ X rc(p,z) where rc(p,z) = (p−z)dc(p).
Proposition 2 If d(p) satisfies the conditions of Theorem 2, then
so does dc(p) = min(d(p),c),
and as a result there exists a finite maximizer pc(z), increasing
in z, of rc(p,z) such that
rc(z) = rc(pc(z),z) is decreasing convex in z.
Proof: Since dc(p) ≤ d(p) it follows that x̄ c(z) ≤ s̄ (z) for all z
23. and consequently x̄ c(0) ≤
s̄ (0) < ∞. If d(p) is USC then so is dc(p) because the minimum
of USC functions is USC.
As an example, suppose that z = 0, d(p) = 3 for p ≤ 10 and d(p)
= 0 for p > 10. If
c = 2 then d2(p) = 2 if p ≤ 10 and d2(p) = 0 for p > 10. Then
r2(p, 0) is maximized at
p2(0) = 10 resulting in r2(0) = 20. Notice that at this price three
units are demanded but
only two units are sold. If demand comes from three different
customers each requesting one
unit this is not a problem, but if it comes from a single customer
that wishes to fulfill all
of his demand or none at all then the formulation proposed here
would be inappropriately
optimistic. Indeed, if customers are not willing to take partial
orders we can use a more
conservative formulation: supp≥z r(p,z) subject to d(p) ≤ c. The
set of feasible prices for
the current example is {p : p > 10} and over this range r(p, 0) =
0, so the profit under this
formulation is zero. This would be the correct profit if demand
comes from a single customer
unwilling to take partial orders but the formulation would be
excessively pessimistic if the
demand came from three different customers each demanding
one unit at any price p ≤ 10.
We now turn to the questions of unimodality of rc(p,z) and
uniqueness of pc(z).
Proposition 3 If the hazard rate h(p) of d(p) satisfies the
conditions of Theorem 3 for a fixed
z then so does the hazard rate hc(p) of dc(p) and as a result
rc(p,z) is unimodal in p.
24. Proof: Suppose that 1 − (p− z)h(p) has a unique sign change
from + to −. Let dc(p) =
min(d(p),c). If d(0) < c then dc(p) = d(p) for all p ≥ 0 and there
is nothing to show.
Otherwise the hazard rate, say hc(p), of dc(p) is zero when d(p)
> c and is equal to h(p)
otherwise. Thus, if 1 − (p − z)h(p) has a unique sign change
then so does 1 − (p − z)hc(p),
showing that rc(p,z) is unimodal in p.
Let pmin(c) = sup{p ≥ 0 : d(p) ≥ c}. It is useful to think of
pmin(c) as the market
clearing price as demand exceeds supply for all p < pmin(c) and
supply exceeds demand for
all p > pmin(c). The following result links pc(z) to p(z) via the
market clearing price pmin(c).
Corollary 3 The price
pc(z) = sup{p ≥ pmin(c) : 1 − (p−z)h(p) ≥ 0} =
max(p(z),pmin(c))
10
is a global maximizer of rc(p,z). Moreover, if either hc(p) or
phc(p) are strictly increasing or
the equation 1 − (p−z)hc(p) has a unique root, then pc(z) is
unique.
If d(p) is continuous then the formulation maxp≥z rc(p,z) is
equivalent to the formulation
maxp≥z r(p,z) subject to d(p) ≤ c and we can bring in the
machinery of Lagrangian Relaxation.
25. The idea is to impose a penalty γ(d(p) − c) for violations of the
capacity constraint where γ
is a non-negative Lagrange multiplier. Subtracting the penalty
results in the Lagrangian:
L(p,γ) = r(p,z) −γ(d(p) − c) = r(p,z + γ) + γc.
The agenda is to find minγ≥0 maxp≥z L(p,γ). The inner
optimization is solved by p(z + γ)
and the outer optimization is equivalent to minγ≥0[r(z + γ) + γc]
which is a convex program
in γ. Notice that γ ≥ 0 increases the marginal cost of capacity.
Let γc be any unconstrained
minimizer of r(z + γ) + γc. Then the outer optimization is
solved by γ∗ c = max(γc, 0). If
d(p(z)) ≤ c, then γc ≤ 0 and consequently γ∗ c = 0. In other
words, p(z) is an optimal solution
if capacity is ample.2 On the other hand, if d(p(z)) > c, then
capacity is scarce and γc is the
root of d(p(z + γ)) = c. This corresponds to using the market
clearing price pmin(c) discussed
before. In summary, an optimal price is given by
max(p(z),pmin(c) and if pmin(c) > p(z) then
there exists a γ∗ c > 0 such that pmin(c) = p(z + γ
∗
c ). As an example, consider the problem with
d(p) = λe−p/µ then p(z) = µ+z and pmin(c) = µ ln(c/λ) so pc(z)
= max(µ+z,µ ln(c/λ)) solves
the pricing problem and the problem is capacity constrained
whenever c < d(p(z)) = e−1d(z).
Also, γ∗ c = max(0,µ[ln(c/λ) − 1] −z).
4.1 Sales Constraints
26. Management may be interested in achieving a certain sales
volume and impose the constraint
d(p) ≥ c on sales. This is the opposite of a capacity constraint
and if d(p) is continuous the
constraint can be handled by imposing a penalty γ(c−d(p)) on
violations of the constraint.
Subtracting the penalty results in the Lagrangian L(p,γ) = r(p,z)
− γ(c − d(p)) = r(p,z −
γ) −γc. The program is to maximize r(z −γ) −γc over γ ≥ 0.
Notice that now γ ≥ 0 acts
as a subsidy to the unit cost z. This is a convex program in γ.
Let γc be the unconstrained
optimizer of r(z−γ)−γc. Then γ∗ c = max(γc, 0). If d(p(z)) ≥ c
then γc ≤ 0 and consequently
p(z) is an optimal solution. In this case, the target sales c is
overshot. On the other hand,
if d(p(z)) < c, then γc is the root of d(p(z − γ)) = c. This
corresponds to using the market
clearing price pmin(c) discussed before. In summary, the
optimal price is given by pc(z) =
min(p(z),pmin(c)).
5 Call Options and Social Welfare
Assume that demand is d(p) = λH(p) where H(p) = P(W ≥ p).
While the seller is naturally
interested in maximizing r(p,z) = (p − z)d(p), a social planner
may be more interested
2In this case c−d(p(z)) units will go unsold. Any attempt to
reduce the price to sell these additional units
will result in lower profits.
11
27. in maximizing the sum of the seller’s profit r(p,z) plus the
consumer’s surplus s(p) where
s(p) =
∫∞
p d(y)dy = λE[(W −p)
+] =. The social welfare problem is to maximize
w(p,z) = s(p) + r(p,z) = λ[E[(W −p)+] + (p−z)H(p)].
Let w(z) = maxp≥z w(p,z). It is easy to see, by just drawing a
graph of E[(W − p)+] +
(p − z)H(p), that an optimal solution to the welfare problem is
to set p = z, so w(z) =
s(z) + r(z,z) = s(z). Unfortunately, this solution reduces the
profit of the seller to zero, as
r(z,z) = 0, while giving all of the surplus s(z) to the customers.
Welfare planners call dead-weight loss the difference
w(z)−w(p(z),z) between the opti-
mal social welfare and the social welfare that results when the
seller maximizes his profits. We
now explore a situation where the dead-weight loss can be
eliminated. The situation requires
the use of call options on capacity when booking and
consumption are separated by time and
customers have homogeneous ex-ante valuations at the time of
booking. Examples include a
group of homogeneous customers booking air transportation a
month in advance of traveling
or a single customer buying a service contract for services over
a certain period of time.
Suppose there is a time separation between booking and
28. consuming a service and that each
customer has random valuation, say W, for the service at the
time of consumption. We assume
that customers know the distribution of W at the time of
booking and learn the realization
of W at the time of consumption. We assume that the
distribution H(p) = P(W ≥ p) is
known by the seller. Under these conditions, the seller can
benefit from offering call options
to consumers. A call option requires an upfront non-refundable
payment x that gives the
customer the non-transferable right to buy one unit of the
service at price p at the time of
consumption; see Gallego and Sahin [6], Png [14], Shugan and
Xie [16], Xie and Shugan [18].
The special case where p = 0, is called advanced selling.
Customers evaluate call options by the surplus they provide. A
customer who buys an (x,p)
option will exercise his right to purchase one unit of the service
at the time of consumption if
and only if W ≥ p. By doing this, an individual customer obtains
expected surplus E[(W −
p)+]. Since the consumer needs to pay x for this right, the
consumer receives surplus E[(W −
p)+]−x. We will impose a participation constraint λ[E(W −p)+
−x] = s(p)−λx ≥ s̃, where
s̃ ≥ 0 is a lower bound on the aggregate consumer surplus.
If all customers buy the call option then the seller’s profit is
given by
λ[x + (p−z)H(p)]. (7)
This consists of the revenue from the non-refundable deposit x
plus the profit p−z from those
29. customers who exercise their options.
Consider now the problem of maximizing the expression in
equation (7) with respect to
(x,p) subject to the surplus constraint s(p) −λx ≥ s̃. Notice that
the seller may set s̃ = 0 to
extract as much surplus from consumers. Here we will analyze
the problem for other values
of s̃ to show that it is possible to eliminate the dead-weight loss
and use s̃ as a mechanism to
distribute profits and surplus between the seller and the
consumers.
Since the objective function (7) is increasing in x, it is optimal
to set λx = s(p) − s̃, so
12
the problem reduces to that of maximizing s(p) + r(p,z) − s̃ =
w(z,p) − s̃ with repect to p.
We already know that w(p,z) is maximized at p = z. Thus, the
solution to the provider’s
problem is to set p = z and x = [s(z)− s̃]/λ, so the provider
obtains profits equal to s(z)− s̃,
while consumers receive surplus s̃. We now explore the range of
values of s̃ that guarantees
that both the seller and the consumers are at least as well off as
the solution (x,p) = (0,p(z)),
where price p(z) is offered to consumers after they know their
valuations. At price p(z), the
provider makes profit r(z), while purchasing customers obtain
aggregate surplus s(p(z)). As
a result, consumers are better off whenever s̃ ≥ s(p(z)), while
the seller is better off whenever
30. s(z)−s̃ ≥ r(z), so a win-win is achieved for any value of s̃ such
that s(p(z)) ≤ s̃ ≤ s(z)−r(z).
Since the solution eliminates dead-weight loss, s(z) ≥ r(z) +
s(p(z)), and consequently the
win-win interval is non-empty. Absent competition or an
external regulator, the provider may
simply select s̃ = 0, to improve his profits from r(z) to w(z)
extracting all consumer surplus
while also capturing the dead-weight loss.
The idea of using call options can be extended to the case where
the variable cost Z
of providing the service at the time of consumption is random.
In this case, the option be
designed by setting λx = Es(Z) − s̃ and p = Z, so that by paying
x in advance the option
bearer has the right to purchase one unit of the service at the
random marginal cost Z.
It is interesting to measure the benefits to the provider of
offering call options on capacity
instead of selling at p(Z) when customers already know their
valuations. In essence we want
to compare Es(Z) − s̃ to Er(Z). To make this a fair comparison
we will set s̃ = Es(p(Z)),
so that both (x,p) with λx = Es(Z) −Es(p(Z)) and p = Z, and
(x,p) = (0,p(Z)) result in
the same consumer surplus. However, the benefits of offering
call options may be larger as
a monopolist need not compete against himself and can in fact
extract all surplus by setting
s̃ = 0. Our next result is for exponentially distributed W with
mean θ. For convenience, we
will let θ∗ = θ/e.
Proposition 4 If W is exponentially distributed with mean θ and
the moment generating
31. function MZ(−1/θ) = E[e−Z/θ] < ∞, then the lift in expected
profits from offering call option
(Es(Z) −Es(p(Z)),Z) relative to offering call option (0,p(Z)) is
72%. Moreover, the lift in
profits for a monopolist who sets s̃ = 0 is 172%.
Proof: If W is exponential with mean θ. Then p(Z) = Z + θ and
r(Z) = λθ∗ e−Z/θ.
Consequently, the expected profit from (0,p(Z)) is E[r(Z)] =
λθ∗ MZ(−1/θ). Since s(Z) =
λθe−Z/θ and s(p(Z)) = r(Z)/, it follows that the expected profits
from the call option is given
by λ(θ−θ∗ )MZ(−1/θ), and the relative lift in profits is equal to
(θ−2θ∗ )/θ∗ = (e−2) = 72%.
If the seller extracts all the surplus then the relative lift in
profits is (e− 1) = 172%.
The lift in expected profits from the exponential distribution is
quite large and one may
wonder whether large lifts are also possible for other
distributions. It is possible to show
that if d(p) = λp−b, then for z > 0 and b > 1, the lift in profits is
at least as large as that
for the exponential demand model, with the benefits converging
to those of the exponential
distribution as b → ∞. Consequently, the benefits are at least as
large under the constant
price elasticity model than under the exponential demand
model. Here we show that if W
has a uniform distribution, then the lift in expected profits can
be up to 50%. Readers not
interested in the details of the analysis can skip to the next
section.
13
32. Example 2 : If W is uniformly distributed over the interval [a,b]
then s(p) = E[W] −p for
p < a, s(p) = 0.5(b−p)2/(b−a) for p ∈ [a,b] and s(p) = 0 for p >
b. The revenue maximizing
price is p(z) = max(a, (b+z)/2) for 0 < z ≤ b. For z > b there is
no demand so we will confine
our analysis for z < b. Then r(z) = a−z for 0 ≤ z < (2a−b)+ and
r(z) = 0.25(b−z)2/(b−a)
for (2a − b)+ ≤ z ≤ b. The expected surplus from offering price
p(z) is s(p(z)) = s(a) =
E[W]−a = 0.5(b−a) = 2r(a) for 0 ≤ z < (2a−b)+, s(p(z)) =
0.125(b−z)2/(b−a) = 0.5r(z)
for 2a − b ≤ z ≤ b. An (x,p) option with p = z results in surplus
−x + s(z) and for this
to be more attractive we need x ≤ s(z) − s(p(z)). The contract
(s(z) − s(p(z)),z) results in
profits s(z) − s(p(z)) = E[W] − z − E[W] + a = a − z = r(z) for z
< (2a − b)+ so there
is no benefit in offering contracts when z < (2a − b)+. For (2a −
b)+ ≤ z ≤ a we have
s(z) −s(p(z)) = E[W] −z − 0.5r(z) > r(z) on account of θ(z) =
E[W] −z − 1.5r(z) ≥ 0 on
(2a− b)+ ≤ z ≤ a. This can be verified by checking that θ(2a− b)
= 0 and θ′(z) > 0 on the
interval (2a − b)+ ≤ z ≤ a. In fact at z = a we have θ(a) = E[W]
− a − 1.5r(a) = 0.5r(a)
so the lift from contracts is between (0, 0.5] over the interval
((2a− b)+,a). Finally, over the
interval a ≤ z ≤ b we have s(z) −s(p(z)) = 2r(z) − 0.5r(z) =
1.5r(z) so there is a 50% lift.
5.1 Call Options and Service Contracts
As mentioned earlier, the idea of a call option may also apply to
33. an individual customer
buying a service contract for services over a certain period of
time. The contract allows the
customer to pay x in advance for the right to pay the marginal
cost z each time the service
need arises over a certain pre-specified horizon. If the expected
number of services during this
period of time is λ, and each service need has random value W,
then a contract of the form
(x,p) = (λ(s(z) − s̃),z) may be designed, by selecting s̃, to be as
attractive as offering á la
carte services at p(z). In this case, obtaining the surplus from á
la carte services is a bit trickier
because the decision of whether or not to buy a service at price
p(z) for a current service of
value W may influence the need for future services. As an
example, consider the problem of
repair services for a certain product. If the customer declines
the service at price p(z) because
W < p(z), then the customer forgoes the future utility associated
with this product while
the service provider forgoes the opportunity to continue
servicing the product. This situation
forces the customer to think carefully about whether or not to
pay for the service at p(z) and
forces the service provider to carefully design the contracts so
they are win-win.
5.2 Call Options with Bounded Capacity
Assume there is a bounded capacity c. We will assume that each
will buy at most one call
option. We will formulate the problem with the unconstrained
demand function and impose
a condition on the number of customers that exercise the (x,p)
option at the exercise price
34. p. Under this formulation the seller’s profit is [s(p) − s̃] + r(p,z)
subject to the constraint
d(p) = λH(p) ≤ c. The constraint is equivalent to p ≥ pmin(c) so
an optimal solution is to set
the exercise price at max(z,pmin(c)) and the option price at
s(max(z,pmin(c)))− s̃. This leads
to profit [s(max(z,pmin(c))) − s̃] + r(max(z,pmin(c)),z) for the
seller and aggregate consumer
surplus s̃. It is instructive to compare the two cases: pmin(c) ≤ z
and pmin(c) > z. In the
first case the capacity constraint is not relevant as d(z) = λH(z)
≤ c, so the optimal option is
14
p = z and λx = s(z)−s̃, the profit to the seller is s(z)−s̃, and the
aggregate consumer surplus
is s̃. On the other hand, if pmin(c) > z then λx = s(pmin(c)) − s̃
and p = pmin(c) resulting in
seller’s profit equal to s(pmin(c))− s̃ + r(pmin(c),z) =
s(pmin(c))− s̃ + c(pmin(c)−z). It is also
possible to work directly with the truncated demand function
dc(p). This leads to essentially
the same result but it is a bit more subtle to interpret.
6 Multiple Market Segments
Suppose we have multiple market segments with demands
dm(p),m ∈ M = {1, . . . ,M}.
For any subset S ⊂ M, let dS(p) =
∑
m∈ S dm(p) denote the aggregate demand over market
35. segments in S and let rS(p,z) = (p−z)dS(p) denote the profit
function for market segments in
S when the variable cost is z, and a common price p is offered
to all market segments in S. We
will first deal with questions related to the existence and
uniqueness of finite maximizers of
rS(p,z) before exploring using a finite price menu of J different
prices to price the M market
segments.
The following result shows that dS(p) inherits some desirable
properties from the individual
market demand functions dm(p),m ∈ S.
Proposition 5 If dm(p) satisfies the conditions of Theorem 2 for
every m ∈ S ⊂ M, then
so does dS(p). Moreover, there exists a finite price pS(z),
increasing in z, such that rS(z) =
rS(pS(z),z) is decreasing convex in z.
Proof: Since the sum of USC is USC it follows that dS(p) is
USC. Moreover x̄ m(0) < ∞ for
all m ∈ M implies that x̄ S(0) =
∑
m∈S x̄ m(0) < ∞. As a result dS(p) satisfies the conditions
of Theorem 2 so there exists a finite price pS(z), increasing in
z, such that rS(z) = rS(pS(z),z)
is decreasing convex in z.
It may be tempting to conclude that under the conditions of
Proposition 5 pS(z) would
lie in the convex hull of {pm(z),m ∈ S}, i.e., in the interval
[minm∈ S pm(z), maxm∈ S pm(z)].
However, Example 3 shows that this is not true.
36. Example 3 Suppose that d1(p) = 1 for p ≤ 10 and d1(p) = 0 for p
> 10. Then r1(p, 0) is
maximized at p1(0) = 10 and r1(0) = 10. Suppose that d2(p) = 1
for p ≤ 9, d2(p) = .1 for
9 < p ≤ 99 and d2(p) = 0 for p > 99. Then r2(p, 0) is maximized
at p2(0) = 99 resulting
in r2(0) = 9.9 and total profit equal to 19.9 if each is allowed to
be priced separately. Let
S = {1, 2}, then rS(p, 0) = r1(p, 0) + r2(p, 0) is maximized at
pS(0) = 9 < mini∈ S pi(0)
resulting in rS(0) = 18.
Since the sum of quasi-concave functions is not, in general,
quasi-concave, it should not
be surprising that properties of dm(p) that imply quasi-
concavity of rm(p,z), for each m ∈ M
are not, in general, inherited by dS(p) =
∑
m∈ S dm(p). Example 4 illustrates this.
15
Example 4 a) Suppose that dm(p) = exp(−p/bm) for m = 1, 2
with b1 < b2. Then the
hazard rate hm(p) = 1/bm, is constant, and there is a unique
price pm(z) = z + bm that
maximizes rm(p,z). Let S = {1, 2} and notice that the hazard
rate hS(p) of dS(p) is
decreasing in p.
b) Suppose that dm(p) = 1/p
bm for some bm > 1, then phm(p) = bm and there is a unique
37. price pm(z) = bmz/(bm − 1) that maximizes rm(p,z). However,
the proportional hazard
rate phS(p) of dS(p) is decreasing in p.
This state of affairs is very unsatisfying because in both cases
in Example 4 the profit
function rS(p,z) is actually quasi-concave, even if the aggregate
demand function dS(p) has
decreasing hazard rate (part a) or decreasing proportional
hazard rate (part b). Some level of
satisfaction may be restored if sufficient conditions can be
founds so that rS(p,z) has a finite
bounded maximizer. Here we present such conditions.
Theorem 4 Assume that the hazard rate hm(p) is continuous in p
and there is a finite root
pm(z) of fm(p) = 1 − (p − z)hm(p) = 0 for each m ∈ S. Assume
further that phm(p)
is increasing for each m ∈ S. Then rS(p,z) has a finite
maximizer in the convex-hull of
{pm(z),m ∈ S}.
Proof: It is easy to see that pm(z) > z is the root of
p
p−z = phm(p). Since the left hand
side is decreasing in p and phm(p) is increasing in p, it follows
that there is a unique root
p > z. This implies that fm(p) > 0 on p < pm(z) and fm(p) < 0 on
p > pm(z). Let
fS(p) = 1 − (p − z)hS(p) where hS(p) is the hazard rate of dS(p).
Since fS(p) is a convex
combination of fm(p) = 1 − (p − z)hm(p) with weights θm(p) =
dm(p)/dS(p), it follows that
fS(p) > 0 for all p < minm∈ S pm(z) because over that interval
38. fm(p) > 0 for all m ∈ S. Also
fS(p) < 0 for all p > maxm∈ S pm(z) because over that interval
fm(p) < 0 for all m ∈ S.
Since the derivative of rS(p,z) is proportional to fS(p) it follows
that rS(p,z) is increasing
over p < minm∈ S pm(z) and decreasing over p > maxm∈ S
pm(z). Moreover, since rS(p,z) is
continuous over the closed and bounded interval [minm∈ S
pm(z), maxm∈ S pm(z)] and appeal to
the EVT yields the existence of a global maximizer pS(z) of
rS(p,z).
Corollary 4 Theorem 4 holds if hm(p) is increasing in p for all
m ∈ S
The Corollary follows since then phm(p) is increasing in p for
all m ∈ S.
6.1 Pricing with Finite Price Menus
Consider now the situation where it is possible to use third
degree price discrimination so
that a different price can be used for each market segment m ∈
M without worrying about
incentive compatibility. This situation arises when it is possible
to vary price by time, location
16
or customer attributes without cannibalizing demand from other
market segments. We will
embed this problem as part of a more general problem where we
are allowed a price menu that
consist of at most J ≤ M different prices. The use of a finite
39. price menu J < M may result
from constraints in pricing flexibility or because the demand
functions of some of the market
segments is not know with sufficient accuracy. We will assume
that the demand functions
dm(p),m ∈ M belong to the same family. By this we mean that
dm(p) = λmHm(p),m ∈ M
and the tail distributions Hm(p) = P(Vm ≥ p),m ∈ M differ only
on their parameters.
Examples of families of demand functions include linear, log-
linear, CES, Logit, among others.
We will assume that the profit function rm(p,z) = (p−z)dm(p) is
quasi-concave for each m and
that there is a unique finite maximizer pm(z) for each m ∈ M.
We will assume that the market
segments are ordered so that p1(z) ≤ . . . ≤ pM (z). Finally, we
will assume that for any S ⊂M,
the profit function rS(p,z) has a finite maximizer in the interval
[minm∈ S pm(z), maxm∈ S pm(z)],
as guaranteed under the conditions of Theorem 4.
The extreme cases are J = 1 where a single price is used for all
market segments and
J = M where each market segment can be individually priced. In
practice, one seldom has
the freedom or sufficiently detailed knowledge to use J = M
prices, particularly if M is large.
In this section we solve to optimality the case J = M assuming
detailed knowledge of the
demand functions. In addition, we develop heuristics for J ∈ {1,
. . . ,M − 1} that are robust
to possible misspecification of demand functions dm(p),m ∈ M.
If J = M the problem is to
separately select prices pm,m ∈ M to maximize
∑
40. m∈ M rm(pm,z). This problem has a trivial
solution, namely to price market segment at pm(z),m ∈ M, so
the optimal profit is given by
RM (z) =
∑
m∈ M
rm(z).
Since each rm(z) is decreasing convex in z it follows that RM
(z) is decreasing convex in z.
RM (z) will serve as a benchmark upper bound against which
we will measure heuristics when
the price menu allows only J < M prices.
Since we will be using heuristic prices, it is convenient to have
a measure of how efficient it
is to use price p instead of pm(z) for market segment m. This
motivates defining the relative
efficiency of price p instead of pm(z) for market segment m
when the unit cost is z as
em(p,pm(z),z) =
rm(p,z)
rm(z)
(8)
Notice that em(p,pm(z),z) ≤ 1, em(p,pm(z),z) reaches maximum
efficiency at p = pm(z),
and decays on both directions as a result of our quasi-concavity
assumption. It is possible to
find closed form formulas for em(p,pm(z),z) for many families
of demand functions including
41. linear, log-linear and CES. However, there are distributions that
do not admit closed form
expressions for em(p,pm(z),z) but the results that we will derive
here can also be applied,
numerically, for distributions that do not admit closed form
expressions. The relative efficien-
cies of prices will help us deal with situations where we may
not know the exact parameters
of some of the market segments.
We will be particularly interested in families of demands for
which em(p,pm(z),z) is inde-
17
pendent of m, i.e, that the functional form of e does not depend
on the market segment. The
following result confirms that em is independent of m for the
linear, for the log-linear and for
the logit demand functions.
Lemma 2 For the linear demand function dm(p) = am − bmp
e(p,pm(z),z) =
p−z
pm(z) −z
(
2 −
p−z
pm(z) −z
42. )
, (9)
for the log-linear demand function dm(p) = am exp(−p/bm)
e(p,pm(z),z) =
p−z
pm(z) −z
exp
(
1 −
p−z
pm(z) −z
)
, (10)
and for the logit demand function dm(p) = λme
am−p/(1 + eam−p),
e(p,pm(z),z) =
p−z
pm(z) −z + (ep−pm(z) − 1)
. (11)
Proof: For the linear demand function d(p) = a − bp, p(z) − z =
(a − bz)/2b. Since
a− bp(z) = b(p(z) −z) it follows that r(z) = b(p(z) −z)2.
Therefore
e(p,p(z),z) =
(a− bp)(p−z)
43. b(p(z) −z)2
.
Then (9) follows from (a− bp) = 2b(p(z) −z) − b(p−z) since
2b(p(z) −z) = a− bz.
For the log-linear demand function d(p) = ae−p/b, p(z) = z + b,
so d(p(z)) = e−1d(z) and
r(z) = be−1d(z). On the other hand, r(p,z) = (p−z)e(p−z)/bd(z).
As a result,
e(p,p(z),z) =
p−z
p(z) −z
exp{(p(z) −z)/b− (p−z)/b}.
The result (10) follows since b = p(z) −z.
For the logit demand function ea−p/(1 + ea−p), p(z) is the root
of the equation p − z =
1 + ea−p, so r(z) = ea−p(z) = p(z) −z − 1. Consequently, the
ratio r(p,z)/r(z) can be written
as (p−z)/[(p(z)−z−1)/d(p)] and the result follows if we can show
that (p(z)−z−1)/d(p) =
p(z) − z − 1 + ep−p(z). But this is equivalent to showing that
(p(z) − z − 1)/ea−p = ep−p(z)
or equivalently p(z) − z − 1 = ea−p(z). But we know this to be
true since r(z) = ea−p(z) =
p(z) −z − 1.
Notice that in the first two cases what is important is the
markup ratio (p−z)/(p(z)−z).
On occasions we will write e(p,q,z) and this should be
interpreted as the efficiency of using
44. price p when q is optimal, so for example, e(p,q,z) = (p−z)/(q
−z)[2 − (p−z)/(q −z)] for
the linear demand model.
18
The following result will be helpful in establishing our results.
Lemma 3 Suppose q1 < q2 and q ∈ (q1,q2) is selected so that
e(q,q1,z) = e(q,q2,z), then
e(q,p,z) ≥ e(q,q1,z) = e(q,q2,z) for all p ∈ (q1,q2).
Proof: Recall that e(q,p,z) deteriorates as q gets further from p
in either direction. If p ∈
(q1,q) then e(q,p,z) > e(q,q1,z) as q is closer to p than to q1. On
the other hand, if p ∈ (q,q2)
then e(q,p,z) > e(q,q2,z) as q is closer to p than to q2.
We will now provide a bound when only one price is allowed
for all of the market segments.
We will make use of Lemma 6.1 to lower bound the ratio
R1(z)/RM (z) where for J = 1 we
write R1(z) = rM(z) as the maximum profit when all market
segments are priced at pM(z).
Theorem 5 Assume that the functions rm(p,z) are quasi-concave
and each has a unique finite
maximizer pm(z). Suppose that the market segments are indexed
so that pm(z) is increasing in
m ∈ M. Assume that em(αpm(z),pm(z),z),m ∈ M is independent
of m ∈ M for all α > 0.
Let q1 be the root of
e(q,p1(z),z) = e(q,pM (z),z) (12)
45. and let γ1(z) = e(q1,p1(z),z) = e(q1,pM (z),z) be the loss of
efficiency of using q1 for market
segments 1 and M. Then
R1(z)
RM (z)
≥
rM(q1,z)
RM (z)
≥ γ1(z),
Proof: Assume p1(z) and pM (z) are respectively the smallest
and the largest optimal prices
for the M market segments. Let q1 be the root of e(q,p1(z),z) =
e(q,pM (z),z). Then, by
Lemma 6.1 we know that e(q1,pm(z),z) ≥ γ1(z) for all m = 2, . .
. ,M−1. From this it follows
that
R1(z)
RM (z)
≥
rM(q1,z)
RM (z)
=
∑
m∈ M
e(q1,pm(z),z)
46. rm(z)
RM (z)
≥
∑
m∈ M
γ1(z)
rm(z)
RM (z)
= γ1(z).
Notice that Theorem 5 does not require precise knowledge of
the demand functions dm(p)
other than knowing that pm(z) ∈ [p1(z),pM (z)]. Without detail
knowledge of the demand
functions dm(p),m ∈ {2, . . . ,M −1} it is not possible to find
RM (z) or even R1(z). However,
it is possible to find q1, the root of equation (12). Theorem 5
guarantees that pricing all
19
segments at q1 is not too far from optimal when p1(z) and pM
(z) are not too far apart.
Moreover, the actual performance R1(q1,z)/RM (z) can be
significantly better than the lower
bound γ1(z). Closed form expressions for γ1(z) will be
presented shortly for the linear and
log-linear demand functions after we generalize Theorem 5 to J
> 1.
47. We will now define RJ(z) the maximum expected revenue that
we can obtain if we are
allowed to use up to J different prices for J ∈ {2, . . . ,M − 1}.
Fix 1 < J < M and consider
any partition S1, . . . ,SJ of M such that ∪ Jj=1Sj = M and Si
∩Sj = ∅ for i 6= j. Let
rSj (z) = sup
p≥z
∑
m∈ Sj
rm(p,z)
and let
RJ(z|S1, . . . ,SJ) =
J∑
j=1
rSj (z).
Optimizing over the partitions we obtain
RJ(z) = max
S1,...,SJ
RJ(z|S1, . . . ,SJ)
where the maximum is taken over all mutually exclusive and
collectively exhaustive partitions
of M into J subsets. Notice that finding RJ(z) can be a difficult
as there are a combinatorial
number of possible partitions of M. Moreover, solving for RJ(z)
48. requires precise knowledge
of all of the demand functions dm(p),m ∈ M.
To extend the heuristic for J > 1 we proceed as follows: Select
break-points p1(z) = s0 <
s1 < s2 . . . < sJ−1 < sJ = pM (z) and prices qj ∈ (sj−1,sj) such
that e(qj,sj−1,z) = e(qj,sj,z)
for each j and the efficiencies e(qj,sj,z) are independent of j.
More precisely, the sjs and qjs
are selected so that
e(qj,sj−1,z) = e(qj,sj,z) for all j = 1, . . . ,J (13)
and
e(q1,s1,z) = e(q2,s2,z) = . . . = e(qJ,sJ,z). (14)
Let γJ(z) = e(q1,s1,z) and define the sets Mj = {m : pm(z) ∈
[sj−1,sj)} for j = 1, . . . ,J − 1
and MJ = {m : pm(z) ∈ [sJ−1,sJ]}. Notice that the qjs and sjs
are independent of the
precise specification of dm(p),m = 2, . . . ,M −1 and
consequently γJ(z) is also independent of
the intermediate demands. However, identifying the sets Mj,j =
1, . . . ,J does require some
knowledge of the intermediate demand functions in the sense
that we need to identify the
subset Mj to which each pm(z) belongs.
Theorem 6 Under the assumptions of Theorem 5, offering price
qj to all segment in Sj for
j = 1, . . . ,J results in
RJ(z)
RM (z)
≥ γJ(z).
57. The results for the linear demand function for z = 0 and J = 1
first appeared in Gallego
and Queyranne [4].
One may wonder how large J needs to be to achieve γJ(z) ≥ 1−α
for some pre-specified α
and given ∆1(z), ∆M (z). The following corollary answers this
question and Table 1 illustrates
the results for a range of values of α and of the ratio ∆M
(z)/∆1(z).
Corollary 5 Let a(z) = ∆M (z)/∆1(z) and w(α) = (1 +
√
α)2/(1 − α). If J is an integer
greater or equal to ln(a(z))/ ln(w(α)), then γJ(z) ≥ 1 −α.
Proof: Let aJ(z) = a(z)
1/J. Then γJ(z) = 4aJ(z)/(1 + aJ(z))
2. Notice that w(α) is a
solution to the equation 4w/(1 + w)2 = 1 −α. Thus γJ(z) ≤ 1 −α
whenever aJ(z) ≤ w(α),
or equivalently whenever a(z) ≤ w(α)J. Solving for J gives the
result.
∆M (z)/∆1(z)
1 −α w(α) 2 5 10 25
90% 1.92 2 3 4 5
93% 1.75 2 3 5 6
95% 1.58 2 4 6 8
98% 1.38 3 6 8 11
99% 1.22 4 9 12 17
Table 1: Smallest J such that γJ(z) ≥ 1 −α
58. From Table 1 we see that if the markup ratio ∆M (z)/∆1(z) =
(pM (z) −z)/p1(z) −z) = 2
we need only J = 2 to achieve an effectiveness of 95%
regardless of the number of products
M. If the markup ratio is 5 then J = 6 is enough to guarantee an
effectiveness of 98%. The
following example illustrates the lower bounds for a set of 10
products with linear demands
as well as the actual performance of the heuristic for J = 1.
22
Example 5 Suppose that M = 10 with market sizes 100, 200,
300, 400, 500, 500, 400, 300,
200, 100, each with uniform willingness to pay functions
U[Am,Am + 100] with Am = 100 +
5(m−1),m = 1, . . . , 10. Table 2 reports q1, γ1(z) and the actual
performance rM(q1,z)/RM (z)
of the heuristic. Table 3 reports the improvements on the
efficiency lower bound as we enlarge
the menu J. Recall that the results from the table are lower
bounds on performance whereas the
actual realization from a limited price menu can be significantly
higher than the lower bound.
z q1(z) γ1(z) rM(q1,z)/RM (z)
0 $110.11 99% 100%
50 $134.78 98% 100%
100 $159.18 97% 99%
120 $168.78 95% 99%
140 $178.18 93% 98%
160 $187.20 87% 95%
180 $195.29 72% 86%
59. Table 2: Prices, Lower Bounds and Actual Performance for
Example 5.
z γ1(z) γ2(z) γ3(z) γ4(z) γ5(z)
0 98% 100% 100% 100% 100%
50 99% 100% 100% 100% 100%
100 97% 99% 100% 100% 100%
120 95% 99% 99% 100% 100%
140 93% 98% 99% 100% 100%
160 87% 97% 98% 99% 99%
180 72% 92% 96% 98% 99%
Table 3: Efficiency Lower Bounds: J ∈ {1, . . . , 5}, Example 5
Notice that the lower-bound γJ(z) deteriorates as z increases and
improves as J increases,
and even for fairly high values of z, it is possible to obtain
reasonably high lower bounds with
J = 3 or J = 4. For most demand models the contribution
margins (pm(z) − z)/pm(z) go
down as the unit cost z increases. The behavior of the lower
bound indicates that as margins
become thinner it becomes more important to have more pricing
flexibility. In other words,
higher marginal costs require a higher J to achieve near
optimality. In the context of Revenue
Management this suggest that a rich fare menu is more
important when capacity is scarce
than when it is ample.
Remark: Sometimes it is possible to improve on the
performance of a limited price menu by
giving up on the lower market segments. For example, for J = 1
and z = 180 the profit from
market segment 1 is less than 1% of the total. This suggest we
60. can do better by dropping the
effort to keep the relative efficiency of market segment 1 high.
If we use the single price
q′1(z) = z + 2
∆2(z)∆M (z)
∆2(z) + ∆M (z)
= $198.06
to control the efficiency of markets 2 through 10, the
performance for J = 1 improves from
86% to 91.5% even though the efficiency of market segment 1
drops significantly.
23
6.3 Log-Linear Demand Functions
The family of log-linear, or exponential, demand functions is of
the form dm(p) = am exp(−p/bm).
The maximizer of rm(p,z) is given by pm(z) = z + bm and the
efficiency function is given by
e(p,pm(z),z) =
p−z
pm(z) −z
exp
{
1 −
p−z
61. pm(z) −z
}
.
Let
sj = z + b1u
j/J, j = 0, 1, . . . ,J (18)
and prices
qj = z + b1u
j/JUJ j = 1, . . . ,J. (19)
where u = bM/b1 and UJ =
ln(u)
J(u1/J−1) .
Proposition 7 Equations (18,19) are roots of equations (13, 14).
Moreover,
γJ = UJe
1−UJ. (20)
Proof:
To show that e(qj,sj,z) =
qj−z
sj−z
exp(1 − qj−z
sj−z
) = γJ(z) for each j = 1, . . . ,J, first notice
62. that
qj −z
sj −z
= UJ,
so e(qj,sj,z) = UJe
1−UJ = γJ(z).
Notice that
qj −z
sj−1 −z
= u1/JUJ,
so to show e(qj,sj−1,z) = e(qj,sj,z) it is enough to show that u
1/JUJe
1−u1/JUJ = UJe
1−UJ but
this is equivalent to showing that ln(u1/J) = UJ(u
1/J−1) but this is true because UJ(u1/J−1) =
ln(u)/J = ln(u1/J).
Notice that unlike the linear demand function, for log-linear
demand functions γJ is inde-
pendent of z. However, just like the linear demand function the
lower bound improves with
J. On the other hand, γJ(z) deteriorates as u = bM/b1 increases.
One may wonder how large J needs to be to achieve γJ ≥ 1 −α
for some pre-specified α
and given ∆1(z), ∆M (z). The following corollary answers this
63. question and Table 4 illustrates
the results for a range of values of α and of the ratio ∆M
(z)/∆1(z).
Corollary 6 Let w(α) be the root of ln(a)/(a − 1) = 1 − α and let
u = bm/b1. If J is an
integer greater or equal to ln(u)/ ln(w(α)), then γJ ≥ 1 −α.
24
Proof: Let bJ = b
1/J. Then γJ = ln(bJ)/(bJ − 1), so setting bJ = b1/J = w(α),
solving for
J and rounding up achieves γJ ≥ α.
∆M (z)/∆1(z)
1 −α w(α) 2 5 10 25
90% 1.92 2 4 6 8
93% 1.75 2 5 7 9
95% 1.58 2 6 8 11
98% 1.38 4 9 12 17
99% 1.22 6 12 17 24
Table 4: Smallest J such that γJ(z) ≥ 1 −α
Example 6 Suppose that M = 10 with log-linear demand
functions with parameters a1, . . . ,am
given by 100, 200, 300,400, 500,500, 400,300,200,100,
respectively and with parameters bm =
50+10(m−1),m = 1, . . . , 10. Table 5 reports q1, γ1(z) and the
actual performance R(q1,z)/R(z)
of a common pricing policy for a range of values of z. Notice
that in this case γ1(z) is inde-
64. pendent of z and that the actual performance is significantly
better than the lower bound but
does deteriorate slowly with z.
Table 6 reports the improvements on the efficiency lower bound
as we enlarge the menu J
for different values of u. The key observation is that for log-
linear demand functions pricing
flexibility is important when u is large, but just a little
flexibility can result in a fairly high
lower bound on efficiency, with the true performance of the
system likely to be significantly
better.
z q1(z) γ1 rM(q1,z)/RM (z)
$0.00 $80.08 88% 96%
$50.00 $130.08 88% 96%
$100.00 $180.08 88% 96%
$150.00 $230.08 88% 95%
$200.00 $280.08 88% 95%
$250.00 $330.08 88% 95%
Table 5: Efficiency Lower Bounds and Actual Performance J =
1
u γ1 γ2 γ3 γ4 γ5
1 100% 100% 100% 100% 100%
2 94% 99% 99% 100% 100%
3 86% 96% 98% 99% 99%
4 79% 94% 97% 99% 99%
5 73% 92% 96% 98% 99%
Table 6: Efficiency Lower Bounds: J ∈ {1, . . . , 5}
25
65. 6.4 Logit Demand Model
Consider the logit demand functions dm(p) = λmHm(z) where
Hm(z) = e
αm−p/(1 + eαm−p)
denotes the probability that a customer will select a product of
quality αm at price p over
a no purchase alternative under the logit model. This model
arises when the utlity of the
product is αm−p+� and the the no purchase alternative has
utility �′ where � and �′ are both
standard Gumbel random variables. It is easy to see that pm(z),
the maximizer of rm(p,z) is
the unique root of p = z + 1 + eαm−p and the efficiency
function is given by
e(p,pm(z),z) =
p−z
pm(z) −z + (ep−pm(z) − 1)
.
The key to showing the form of e(p,pm(z),z) is that rm(z) =
pm(z) − z − 1 = eαm−pm(z) =
Hm(pm(z))/(1−Hm(pm(z)), so the optimal profit per customer is
the purchase to no-purchase
odds ratio. The highest efficiency common markup, say ∆ = p−
z, for two market segments
with optimal markups ∆i(z) = pi(z) −z,i = 1, 2 is given by
∆ = ln
(
66. ∆2 − ∆1
e−∆1 −e−∆2
)
where ∆ is the root of the equation e(z + ∆,p1(z),z) = e(z +
∆,p2(z),z). This result in the
lower bound
R1(z)
R2(z)
≥ γ1(z) = e(z + ∆,p1(z),z) =
∆(e−∆1 −e−∆2 )
(∆2 − 1)e−∆1 − (∆1 − 1)e−∆2
.
Finding a closed form solution to γJ(z) is quite involved, but
γJ(z) can be computed
numerically by finding breakpoints p1(z) − s0 < s1 < .. . < SJ =
pM (z) and prices qj ∈
(sj−1,sj) such that e(qj,sj−1,z) = e(qj,sj,z) and e(qi,si,z) =
e(q1,s1,z) for all j = 1, . . . ,J.
The following example illustrates the behavior of the heuristic
q1 for the case of J = 1 and
the performance of γJ(z) for several values of J and z.
Example 7 Suppose that M = 10 with market sizes λm =
220−20m and quality parameters
am = m,m = 1, . . . , 10. Table 7 reports q1, γ1(z) and the actual
performance rM(q1,z)/RM (z)
of the heuristic that prices all market segments at q1 where q1
is the root of the equation
e(q1,p1(z),z) = e(q1,pM (z),z). Table 8 provides values of γJ(z)
for J = 1, . . . , 5 and z =
67. 2k,k = 0, . . . , 5. In sharp contrast to the linear demand
function, where γj(z) decreases with
z, here γj(z) increases with z. The reason for this is that for the
logit function the difference
pM (z) − p1(z) is decreasing in z, implying that restricting the
price menu works better as z
increases.
7 Multiple Products
So far we have explored how to price a single product in one or
more markets. In this section
we explore the problem of maximizing r(p,z) = (p − z)′d(p)
where z ∈ <n is the vector of
26
z q1(z) γ1(z) rM(q1,z)/RM (z)
$0.00 $3.44 49% 77%
$2.00 $4.78 52% 82%
$4.00 $6.35 62% 85%
$6.00 $7.91 77% 89%
$8.00 $9.46 92% 95%
$10.00 $11.14 99% 99%
Table 7: Prices, Lower Bounds and Actual Performance for
Example 7.
z γ1(z) γ2(z) γ3(z) γ4(z) γ5(z)
$0.00 49% 77% 88% 93% 95%
$2.00 52% 80% 90% 94% 96%
$4.00 62% 86% 93% 96% 97%
$6.00 77% 93% 97% 98% 99%
$8.00 92% 98% 99% 99% 100%
68. $10.00 99% 100% 100% 100% 100%
Table 8: Efficiency Lower Bounds: J ∈ {1, . . . , 5}, Example 7
unit costs, p ∈ <n+ is the price vector and d(p) ∈ <n is the
demand function. As before, it is
easy to see that r(z) = supp r(p,z) is decreasing convex in z. The
problem of existence of a
finite maximizer p(z) and conditions for the uniqueness of p(z)
have attracted the attention
of several researchers, but most of the work is for specific
demand functions. Here we present
some results for the linear demand function and demands driven
by the nested logic model.
7.1 Linear Demand Function
Let d(p) = a − Bp where a and p are n-dimensional vectors and
B is an n × n matrix. We
are interested in finding conditions on a and B that guarantee
the existence of a unique, non-
negative, profit maximizing price vector p(z) such that r(z) =
(p(z)−z)′d(p(z)) for all z ≥ 0.
Maximizing r(p,z) with respect to p is equivalent to minimizing
1
2
p′(B+B′)p−(a+B′z)′p+a′z
which is quadratic function. A sufficient condition for this
function to be convex is that
S = B + B′ is positive definitive. Recall that a matrix S is
positive definitive if and only if
p′Sp ≥ 0 for all p 6= 0. It is known that S is positive definitive,
if and only if B is, see [9]. If
B is positive definitive then S is invertible and since S is
69. symmetric, so is its inverse S−1. If
B is positive definitive then the maximizer of r(p,z) is given by
p(z) = S−1(a + B′z). (21)
We will impose conditions on a and B so that p(0) = S−1a ≥ 0.
A sufficient condition for
this is that a ≥ 0 and S is a Stieltjes matrix or s-matrix. An s-
matrix is a real symmetric,
positive definitive matrix with non-positve off-diagonal
elements. Since we have already as-
sumed that S is positive definitive the only additional
requirement is that Bij + Bji ≤ 0 for
all i 6= j, which is something we expect from the economics of
the linear demand model. An
27
important consequence is that an s-matrix has a non-negative
inverse implying that p(0) ≥ 0
whenever a ≥ 0. Since p(z) is non-decreasing in z it follows that
p(z) ≥ p(0) ≥ 0 for all z ≥ 0.
By adding and subtracting Bz to the expression in parenthesis
on the righthand side of
(21) we can write
p(z) = z + S−1d(z) (22)
where d(z) is the demand at p = z. It is also possible to write
d(p(z)) = a − Bp(z) =
a − B(p(z) ± z) = a − Bz − B(p(z) − z) = (I − BS−1)d(z) and
then use the fact that
I −BS−1 = B′S−1 to obtain
70. d(p(z)) = B′S−1d(z). (23)
This allow us to write
r(z) = (p(z) −z)′d(p(z)) = d(z)′S−1B′S−1d(z) = d(z)′Nd(z) (24)
where N = S−1BS−1.
7.1.1 Random Potential Demand
A natural extension to the linear demand model is to have
random potential demand d(0) = a.
We will assume that a is a non-negative, random vector, with
mean µ. Are we better off if a is
random? The answer is yes if we can observe a before deciding
the price p(z) = z +S−1d(z) to
offer. From equation (24) we can write the optimal profit
function as r(z) = (a−Bz)′N(a−Bz)
which is a convex function of a given that N is positive-
definitive. By Jensen’s inequality
Ea(a − Bz)′N(a − Bz) ≥ (µ − Bz)′N(µ − Bz) = d̄ (z)′Nd̄ (z) = r̄ (z),
where d̄ (z) = µ − Bz
is the expected demand at z and r̄ (z) is the optimal profit
corresponding to demand d̄ (z).
Suppose now the decision maker has to price before observing
a. Is he worse off because
of randomness? The answer is no, since if he prices in
anticipation of average demand, his
optimal price is p̄ (z) = z + S−1d̄ (z), resulting in expected
profits r̄ (z), which are equal to the
profits that the decision maker would have made if a = µ, i.e., if
demand were deterministic.
The implication here is that dynamic pricing can also be driven
by randomness in the potential
demand even if the marginal value of capacity is unchanged.
71. 7.1.2 Linear Component Costs
Suppose that the n products are built from m components
according to the recipe matrix
A = (Aij) where Aij is the number of units of component j used
by product i. Suppose
further that component can be procured at a linear cost y, where
y is the vector of unit
component costs. How many units of each component should the
firm buy? And, at what
price should the products be sold if the demand function is d(p)
= a−Bp? Since the demand
is deterministic we can solve this problem by using the fact that
the unit cost vector z is
given by z = Ay. Therefore, it is optimal for the firm to price at
p(Ay) and to sell d(p(Ay))
resulting in profits r(Ay) = (p − Ay)′d(p(Ay)). The problem
becomes more interesting if
a is non-negative random vector and there is a need to procure q
units before observing the
realization of a. Committing to q before observing a hurts, and
if the decision maker has to set
28
prices before observing a then randomness in a is detrimental to
profits. On the other hand,
if a can be observed before setting prices, the revenue
advantage from Jensen’s inequality can
in some cases overcome the disadvantage of having to commit
to q before observing a.
7.2 Log-Linear Demand
72. The log-linear demand function for multiple products can be
written as d(p) = exp(a−Bp).
Unfortunately this demand function is not very amenable to
analysis as attempts to maximize
r(p,z) = (p−z)′d(p) leads to unbounded solutions whenever the
non-diagonal elements of B
are negative. Indeed, if bij < 0 for some i then there is an
incentive to make pj very large which
has the negative effect of bringing demand and revenues from
product j to near zero, but also
the positive effect of artificially increasing demand and revenue
for product i. If bij > 0 then
increasing the price of product j decreases the demand of
product i which is what we would
expect if the products are complements (e.g., a shirt and a tie)
instead of substitutes. This
leaves the case bii = 0 which reduces to independent demands.
This analysis shows that the
log-linear demand function has limited applications to
independent demands and the pricing
of complementary products.
7.3 The Nested Logit Model
In this section we consider pricing under the Nested Logit (NL)
model, which is a popular
generalization of the standard MNL model. Under the NL
model, customers make product
selection decisions sequentially: at the upper level, they first
select a branch, called a “nest”
that includes multiple similar products; at the lower level, their
subsequent selection is within
that chosen nest (see McFadden [12], Carrasco and Ortuzar [1]
and Green [3]). Suppose
that the substitutable products constitute n nests and nest i has
73. mi products. Let pi =
(pi1,pi2, . . . ,pimi) be the price vector corresponding to nest i =
1, . . . ,n, and let (p1, . . . ,pn)
be the price vector for all the products in all the nests. Let
Qi(p1, . . . ,pn) be the probability
that a customer selects nest i at the upper level; and let qk|i(pi)
denote the probability that
product k of nest i is selected at the lower level, given that the
customer selects nest i where
pi is the price vector for all the products in nest i. Qi(p1, . . .
,pn) and qk|i(pi) are defined as
follows:
Qi(p1, . . . ,pn) =
eγiIi
1 +
∑n
l=1 e
γlIl
, (25)
qj|i(pi) =
eαij−βijpij∑mi
s=1 e
αis−βispis
, (26)
where αis can be interpreted as the “quality” of product s in nest
i, βis ≥ 0 is the product-
specified price sensitivity for that product, Il = log
∑ml
s=1 e
74. αls−βlspls represents the attractiveness
of nest l, which is the expected value of the maximum of the
utilities of all the products in
nest l, and nest coefficient γi can be viewed as the degree of
inter-nest heterogeneity. When
0 < γi < 1, products are more similar within nest i than across
nests; when γi = 1, products
in nest i have the same degree of similarity as products in other
nests, and the NL model
29
reduces to the standard MNL model; when γi > 1, products are
more similar to the ones in
other nests.
The probability that a customer will select product k of nest i,
which can also be considered
the market share of that product, is
πij(p1, . . . ,pn) = Qi(p1, . . . ,pn)qj|i(pi). (27)
The monopolist’s problem is to determine the price vectors (p1,
. . . ,pn) to maximize the total
expected profit
R(p1, . . . ,pn) =
n∑
i=1
mi∑
j=1
75. (pij −zij)πij(p1, . . . ,pn), (28)
where zij is the unit cost of product j in nest i. The objective
function R(p1, . . . ,pn) fails
to be quasi-concave in prices. When the objective function is
rewritten with market shares
as decision variables then the objective function can be shown
to be concave if the price
sensitivity parameters βij = βi are product independent in each
nest and γi ≤ 1 for all i as
shown in Li and Huh [11]. However, the objective function fails
to be concave in the market
shares in the more general case where the price sensitivities are
product dependent.
Let pij(z) denote the optimal price for product j in nest i as a
function of the vector of unit
costs z. Gallego and Wang [8], show that the optimal price
pij(z) adds to the unit cost zij two
components. The first component is the reciprocal of the price
sensitivity βij and the second
one is a nest dependent constant θi, so that pij(z) = zij + 1/βij +
θi. They also show that the
nest dependent constants θi, i = 1, . . . ,n are linked as explained
in the following Theorem.
Theorem 7 If γi ≥ 1 or maxs βismins βis ≤
1
1−γi
, then there exist a unique constant φ such that
θi + (1 −
1
γi
76. )wi(θi) = φ,
and
pij(zij) = zij +
1
βij
+ θ∗ i ,
where wi(θ) =
∑mi
k=1
1
βik
· qk|i(θi) and qk|i(θi) = e
α̃ik−βikθi∑mi
s=1
eα̃is−βisθi
.
Theorem 7 is interesting because a non-concave optimization
problem over
∑n
i=1 mi vari-
ables can be reduced, under mild conditions, to a root finding
problem over a single variable.
Even in the mild condition γi ≥ 1 or maxs βismins βis ≤
1
1−γi
77. fails to hold, Gallego and Wang [8] show
that the problem reduces to a single variable maximization
problem of a continuous function
over a bounded interval, so the problem can be easily solved
numerically. Gallego and Wang
[8] also show that if different firms control different nests the
pricing problem under com-
petition is strictly log-supermodular in the nest markup
constants, so the equilibrium set is
nonempty with the largest equilibrium preferred by all the
firms.
30
8 Acknowledgments
I acknowledge the feedback from my students and collaborators.
In particular, I would like to
recognize the contributions and feedback from Anran Li and
Richard Ratliff.
9 Appendix
Proof of Corollary 1 Proof: Clearly y′ ≥ y implies g(y′) ≥ g(y)
and r decreasing implies that
r(g(y′)) ≤ r(g(y)), showing that r(g(y)) is decreasing in y. Tho
show that r(g(y)) is convex,
notice that from the concavity of h it follows that g(αy +
(1−α)y′) ≥ αg(y) + (1−α)g(y′) for
all α ∈ [0, 1]. Then, since r is decreasing, it follows that r(g(αy
+ (1−α)y′) ≤ r(αg(y) + (1−
α)g(y′)). Finally, from the convexity of r we see that r(αg(y) +
(1 − α)g(y′)) ≤ αr(g(y)) +
(1−α)r(g(y′)). Consequently, r(g(αy + (1−α)y′) ≤ αr(g(y)) +
78. (1−α)r(g(y′)), showing that
r(g(y)) is convex.
Since r is convex, by Jensen’s inequality Er(Z) ≥ r(EZ), In
particular, Er(g(Y )) ≥
r(Eg(Y )), By the concavity of g and Jensen’s inequality we
have Eg(Y ) ≤ g(EY ). Since r is
decreasing, it follows that r(Eg(Y )) ≥ r(g(EY )).
Proof of Lemma 1 Proof: If d(p̄ (z)) = d̄ (p̄ (z)) then r(z) ≤ r̄ (z) =
r̄ (p̄ (z),z) = r(p̄ (z),z) ≤
r(z) implying that r(z) = r̄ (z) and that p(z) = p̄ (z) is a finite
maximizer of r(p,z). Next,
we will show that d(p̄ (z)) < d̄ (p̄ (z)) leads to a contradiction. To
see this, first notice that
d(p) ≤ d̄ (p) < d̄ (p̄ (z)) for all p > p̄ (z),p ∈ X for otherwise there
is a p ∈ X, p > p̄ (z)
such that r̄ (p,z) > r̄ (z) contradicting the optimality of p̄ (z). But
then, d(p(z)) < d̄ (p(z)) =
supp≥p(z) d(p) together with d(p) < d̄ (p) < d̄ (p(z)) for all p >
p(z) contradicts the fact that d
is upper-semicontinuous at p(z) since
d(p̄ (z)) < d̄ (p̄ (z)) = lim
�↓0
sup
p̄ (z)≤p≤p̄ (z)+�
d(p) ≤ lim sup
p→p̄ (z)
d(p) ≤ d(p̄ (z)),
where the last inequality follows from the USC of d(p) at p̄ (z).
79. Proof of Theorem 2. Proof: Since d(p) is USC and the product
of non-negative USC
functions is also USC, it follows that r(p,z) and r̄ (p,z) are USC
in p ∈ [z,∞). If d(p) = 0
for all p ≥ z, then p(z) = z and r(z) = r(z,z) = 0 and there is
nothing to prove. Otherwise
there exists a p′ > z such that 0 < d(p′) < ∞ for if not then s̄ (z) =
∞. We will show that
there is a q > p′ such that r̄ (p,z) ≤ r(p′,z) for all p > q. This will
allow us to restrict the
optimization of r̄ (p,z) to p ∈ [z,q]. Since r̄ (p,z) is USC and the
supremum is now taken over
a closed and bounded set, the Extreme Value Theorem (EVT)
guarantees the existence of a
finite price, say p̄ (z) ∈ [z,q] such that r̄ (z) = r̄ (p̄ (z),z) = maxp≥z
r̄ (p,z). Then, by Lemma 1,
p(z) = p̄ (z) is a finite maximizer of r(p,z).
Let � > 0. We claim there exists a p1 > p
′ such that s̄ (p) < � for all p > p1. This follows
because s̄ (0) − s̄ (p) is increasing and converges to s̄ (0) as p →
∞. Consequently, there exist
a p1 > p
′ such that s̄ (0) − s̄ (p) > s̄ (0) − ∈ for all p > p1, or equivalently
s̄ (p) < � for all
p > p1. We claim there exist a price q ≥ p1 such that r̄ (q,z) < �.
If q does not exist, then
31
r̄ (p,z) > ∈ for all p > p1, implying that d̄ (p) > �/(p − z) for all
p > p1 and consequently
80. s̄ (0) ≥ s̄ (p1) = ∞, contradicting the finiteness of s̄ (0).
Therefore for all p > q,
r̄ (p,z) = (q −z)d̄ (p) + (p− q)d̄ (p)
≤ (q −z)d̄ (q) + s̄ (q)
= r̄ (q,z) + s̄ (q)
≤ 2�.
By taking ∈ ∈ (0, 0.5r(p′,z)) we guarantee that r̄ (p,z) ≤ r̄ (p′,z)
for all p ≥ q, so we can
limit the optimization to the closed and bounded set [0,q],
enabling us to call on the EVT to
show the existence of p̄ (z) ≤ q such that r̄ (z) = r̄ (p̄ (z),z).
We now turn to the monotonicity of the largest maximizer, say
p(z), of r(p,z). Suppose
that z ≤ z′. If p(z) ≤ z′ then there is nothing to show as then p(z)
≤ z′ ≤ p(z′). On the
other hand, if z′ < p(z) we will show that r(p′,z′) < r(p(z),z′) for
all prices p′ ∈ [z′,p(z)] so
then r(z′) = maxp≥p(z) r(p,z
′) and therefore p(z′) ≥ p(z). To see this notice that r(p′,z) =
(p′ − z)d(p′) ≤ (p(z) − z)d(p(z)), so (p(z) − p′)d(p(z)) ≥ (p′ −
z)(d(p′) − d(p(z)) > (p′ −
z′)(d(p′) −d(p(z)), and this implies that r(p′,z′) < r(p(z),z′),
showing p(z′) ≥ p(z).
Proof of Corollary 2 Proof: If d(p) is proper and decreasing, and
λ = d(0) < ∞, then
H(p) = d(p)/λ ∈ [0, 1] is also decreasing and therefore it is the
complement of the cumulative
distribution function (CCDF) of a non-negative random
variable, say W. If H(p) = P(W ≥
81. p), then H(p) is left continuous with right limits (LCRL). Since
a decreasing LCRL function
is USC it follows that H(p) and therefore d̄ (p) = d(p) is USC. In
addition, E[W] < ∞ implies
that s̄ (0) = s(0) = λE[W] < ∞. As a result the conditions of
Theorem 2 are satisfied.
References
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33
“A Rose for Emily”
by William Faulkner (1930)
I
WHEN Miss Emily Grierson died, our whole town went to her
funeral: the
men through a sort of respectful affection for a fallen
84. monument, the
women mostly out of curiosity to see the inside of her house,
which no
one save an old man-servant--a combined gardener and cook--
had seen
in at least ten years.
It was a big, squarish frame house that had once been white,
decorated
with cupolas and spires and scrolled balconies in the heavily
lightsome
style of the seventies, set on what had once been our most select
street.
But garages and cotton gins had encroached and obliterated
even the
august names of that neighborhood; only Miss Emily's house
was left,
lifting its stubborn and coquettish decay above the cotton
wagons and
the gasoline pumps-an eyesore among eyesores. And now Miss
Emily had
gone to join the representatives of those august names where
they lay in
the cedar-bemused cemetery among the ranked and anonymous
graves
of Union and Confederate soldiers who fell at the battle of
Jefferson.
Alive, Miss Emily had been a tradition, a duty, and a care; a
sort of
hereditary obligation upon the town, dating from that day in
1894 when
Colonel Sartoris, the mayor--he who fathered the edict that no
Negro
woman should appear on the streets without an apron-remitted
her
85. taxes, the dispensation dating from the death of her father on
into
perpetuity. Not that Miss Emily would have accepted charity.
Colonel
Sartoris invented an involved tale to the effect that Miss Emily's
father had
loaned money to the town, which the town, as a matter of
business,
preferred this way of repaying. Only a man of Colonel Sartoris'
generation
and thought could have invented it, and only a woman could
have
believed it.
When the next generation, with its more modern ideas, became
mayors
and aldermen, this arrangement created some little
dissatisfaction. On
the first of the year they mailed her a tax notice. February came,
and
there was no reply. They wrote her a formal letter, asking her to
call at
the sheriff's office at her convenience. A week later the mayor
wrote her
himself, offering to call or to send his car for her, and received
in reply a
note on paper of an archaic shape, in a thin, flowing calligraphy
in faded
ink, to the effect that she no longer went out at all. The tax
notice was
also enclosed, without comment.
They called a special meeting of the Board of Aldermen. A
86. deputation
waited upon her, knocked at the door through which no visitor
had
passed since she ceased giving china-painting lessons eight or
ten years
earlier. They were admitted by the old Negro into a dim hall
from which a
stairway mounted into still more shadow. It smelled of dust and
disuse--
a close, dank smell. The Negro led them into the parlor. It was
furnished
in heavy, leather-covered furniture. When the Negro opened the
blinds of
one window, they could see that the leather was cracked; and
when they
sat down, a faint dust rose sluggishly about their thighs,
spinning with
slow motes in the single sun-ray. On a tarnished gilt easel
before the
fireplace stood a crayon portrait of Miss Emily's father.
They rose when she entered--a small, fat woman in black, with a
thin
gold chain descending to her waist and vanishing into her belt,
leaning
on an ebony cane with a tarnished gold head. Her skeleton was
small and
spare; perhaps that was why what would have been merely
plumpness in
another was obesity in her. She looked bloated, like a body long
submerged in motionless water, and of that pallid hue. Her eyes,
lost in
the fatty ridges of her face, looked like two small pieces of coal
pressed
into a lump of dough as they moved from one face to another
while the
87. visitors stated their errand.
She did not ask them to sit. She just stood in the door and
listened
quietly until the spokesman came to a stumbling halt. Then they
could
hear the invisible watch ticking at the end of the gold chain.
Her voice was dry and cold. "I have no taxes in Jefferson.
Colonel Sartoris
explained it to me. Perhaps one of you can gain access to the
city records
and satisfy yourselves."
"But we have. We are the city authorities, Miss Emily. Didn't
you get a
notice from the sheriff, signed by him?"
"I received a paper, yes," Miss Emily said. "Perhaps he
considers himself
the sheriff . . . I have no taxes in Jefferson."
"But there is nothing on the books to show that, you see We
must go by
the--"
"See Colonel Sartoris. I have no taxes in Jefferson."
"But, Miss Emily--"
"See Colonel Sartoris." (Colonel Sartoris had been dead almost
ten years.)
"I have no taxes in Jefferson. Tobe!" The Negro appeared.
88. "Show these
gentlemen out."
II
So SHE vanquished them, horse and foot, just as she had
vanquished
their fathers thirty years before about the smell.
That was two years after her father's death and a short time after
her
sweetheart--the one we believed would marry her --had deserted
her.
After her father's death she went out very little; after her
sweetheart went
away, people hardly saw her at all. A few of the ladies had the
temerity to
call, but were not received, and the only sign of life about the
place was
the Negro man--a young man then--going in and out with a
market
basket.
"Just as if a man--any man--could keep a kitchen properly, "the
ladies
said; so they were not surprised when the smell developed. It
was another
link between the gross, teeming world and the high and mighty
Griersons.
A neighbor, a woman, complained to the mayor, Judge Stevens,
eighty
years old.
"But what will you have me do about it, madam?" he said.
89. "Why, send her word to stop it," the woman said. "Isn't there a
law? "
"I'm sure that won't be necessary," Judge Stevens said. "It's
probably just a
snake or a rat that nigger of hers killed in the yard. I'll speak to
him about
it."
The next day he received two more complaints, one from a man
who
came in diffident deprecation. "We really must do something
about it,
Judge. I'd be the last one in the world to bother Miss Emily, but
we've got
to do something." That night the Board of Aldermen met--three
graybeards and one younger man, a member of the rising
generation.
"It's simple enough," he said. "Send her word to have her place
cleaned
up. Give her a certain time to do it in, and if she don't. .."
"Dammit, sir," Judge Stevens said, "will you accuse a lady to
her face of
smelling bad?"
So the next night, after midnight, four men crossed Miss
Emily's lawn and
slunk about the house like burglars, sniffing along the base of
the
brickwork and at the cellar openings while one of them
performed a
90. regular sowing motion with his hand out of a sack slung from
his
shoulder. They broke open the cellar door and sprinkled lime
there, and
in all the outbuildings. As they recrossed the lawn, a window
that had
been dark was lighted and Miss Emily sat in it, the light behind
her, and
her upright torso motionless as that of an idol. They crept
quietly across
the lawn and into the shadow of the locusts that lined the street.
After a
week or two the smell went away.
That was when people had begun to feel really sorry for her.
People in
our town, remembering how old lady Wyatt, her great-aunt, had
gone
completely crazy at last, believed that the Griersons held
themselves a
little too high for what they really were. None of the young men
were
quite good enough for Miss Emily and such. We had long
thought of them
as a tableau, Miss Emily a slender figure in white in the
background, her
father a spraddled silhouette in the foreground, his back to her
and
clutching a horsewhip, the two of them framed by the back-
flung front
door. So when she got to be thirty and was still single, we were
not
pleased exactly, but vindicated; even with insanity in the family
she
wouldn't have turned down all of her chances if they had really
materialized.
91. When her father died, it got about that the house was all that
was left to
her; and in a way, people were glad. At last they could pity
Miss Emily.
Being left alone, and a pauper, she had become humanized. Now
she too
would know the old thrill and the old despair of a penny more
or less.
The day after his death all the ladies prepared to call at the
house and
offer condolence and aid, as is our custom Miss Emily met them
at the
door, dressed as usual and with no trace of grief on her face.
She told
them that her father was not dead. She did that for three days,
with the
ministers calling on her, and the doctors, trying to persuade her
to let
them dispose of the body. Just as they were about to resort to
law and
force, she broke down, and they buried her father quickly.
We did not say she was crazy then. We believed she had to do
that. We
remembered all the young men her father had driven away, and
we knew
that with nothing left, she would have to cling to that which had
robbed
her, as people will.
92. III
SHE WAS SICK for a long time. When we saw her again, her
hair was cut
short, making her look like a girl, with a vague resemblance to
those
angels in colored church windows--sort of tragic and serene.
The town had just let the contracts for paving the sidewalks,
and in the
summer after her father's death they began the work. The
construction
company came with riggers and mules and machinery, and a
foreman
named Homer Barron, a Yankee--a big, dark, ready man, with a
big voice
and eyes lighter than his face. The little boys would follow in
groups to
hear him cuss the riggers, and the riggers singing in time to the
rise and
fall of picks. Pretty soon he knew everybody in town. Whenever
you heard
a lot of laughing anywhere about the square, Homer Barron
would be in
the center of the group. Presently we began to see him and Miss
Emily on
Sunday afternoons driving in the yellow-wheeled buggy and the
matched
team of bays from the livery stable.
At first we were glad that Miss Emily would have an interest,
because the
ladies all said, "Of course a Grierson would not think seriously
