Download A. Additive uncertainty

Mark-up Pricing Under Demand Uncertainty
By
Michael A. Salinger*
March 2009
Abstract: This paper extends the mark-up rule relating a firm’s profit-maximizing price
to its marginal cost and its elasticity of residual demand to the newsvendor problem with
endogenous pricing. Two adjustments are necessary. First, the relevant elasticity of
demand is the elasticity of the average quantity sold with respect to price. Second,
marginal cost is the average cost per unit of an additional unit sold, which is computed as
the marginal cost of an additional output produced divided by the expected fraction of the
marginal unit produced that the firm sells. Applying this rule clarifies why increased
uncertainty about demand can cause an increase in output with additive uncertainty and a
reduction in output with multiplicative uncertainty.
Keywords: newsvendor problem, demand uncertainty, mark-ups, critical loss
Michael A. Salinger
Boston University
School of Management
595 Commonwealth Ave.
Boston, MA 02215
[email protected]
(617) 353-4408
* Professor of Economics, Boston University School of Management. I thank Jim Dana, Dennis Carlton,
and Dan Hosken for helpful comments.
I. Introduction
There is no more fundamental proposition about the application of economic
principles to business than the “mark-up rule” relating price, marginal cost, and the
elasticity of demand. The rule states:
(1)
pc
1

p
 ( p)
where p is price, q(p) is the demand relationship, c is the (constant) marginal cost, and
(p) = pq’(p)/q(p) is the elasticity of residual demand facing the firm.1
The rule presumes that firms choose price and output with full knowledge of the
demand curve. Because firms typically make production and pricing decisions in the
presence of uncertainty, the mark-up rule would not be of much value if it only applied in
a world of certainty. It would be surprising if it did. Yet, extensive as the existing
literature on output and pricing under uncertainty is, it does not make clear how the
insight of the mark-up rule extends to uncertainty.
How uncertainty affects pricing depends on the precise sequencing of decisions
and the resolution of uncertainty. One possibility is that a firm must announce a price but
can then adjust output as it observes demand at that price. Another, known as the
newsvendor problem, is that the firm observes the price and then must choose its output
before it knows demand.2 One general theme in the literature is that when they are
feasible, complex pricing strategies yield higher expected profits than the simple strategy
1
For a monopolist, the elasticity of demand facing the firm is also the market elasticity of demand.
Provided one understands the elasticity as the elasticity of residual demand, it extends quite generally
across market structures.
2
See Whitin (1955) and Carlin and Karr (1962).
of charging a constant price per unit.3 Yet, there are situations where only the simple
strategy is feasible, and those are the situations when one might try to apply the intuition
gained from the mark-up rule.
This paper analyzes the case when the firm must choose both its price and its
output before observing demand, the problem known as the newsvendor problem with
endogenous pricing.4 The extensive literature on the problem is highly technical and
difficult to summarize intuitively.5 The typical solution to the problem rests on a set of
first order conditions that might appear to be quite different from the first order condition
for profit-maximization under certainty. Moreover, it has long been known that the
qualitative effect of uncertainty on price and output depends on whether the uncertainty is
additive or multiplicative.6 The sensitivity of the qualitative results to an apparently
technical condition might create the impression that simple intuition might be misleading
in the presence of uncertainty.
Yet, equation (1) does generalize to the newsvendor problem with endogenous
pricing. The only modifications required are that the elasticity of demand is the elasticity
of average demand across states and marginal cost is the marginal cost of an expected
unit sold, rather than a unit produced. In turn, the marginal cost of an expected unit sold
3
For example, Dana (1999) shows that it is better to offer multiple prices with limited quantities available
at the lower prices.This point is analagous to the point that, under certainty, second degree price
discrimination is optimal. See Stole (2007) provides such an interpretation. Dana and Carlton’s (2004)
demonstration that it can be optimal to provide multiple levels of quality can be understood as an extention
of the same point. Another generalization is to allow for the possibility that the good is not completely
perishable (but that there is a cost of carrying over unsold units as inventory). See Kocabıyıkoğlu and
Popescu (2005).
4
According to Petruzzi and Dada (1999), Edgeworth addressed the problem with respect to banks in 1888.
See also Mills (1956), and Carleton (1978).
5
Khouja (1999) and Dana and Petruzzi (2001) provide excellent reviews.
6
Under the assumption of additive uncertainty, Mills (1956) showed that increased uncertainty can cause a
firm to lower price and increase output, which seems counterintuitive. Carlin and Carr (1962) showed that
with mulitplicative uncertainty, increased uncertainty about demand leads to reduced ouptut and a higher
price.
2
is the marginal cost of an extra unit produced divided by the fraction of the marginal unit
produced that is expected to be sold. This characterization of the solution clarifies why
the qualitative effects of demand uncertainty are different for additive and multiplicative
uncertainty.
The remainder of the paper is organized as follows. Section II analyzes the price
and output decision when there are only two states of demand. Assuming discrete states
simplifies the problem and yields insights into the intuition behind the results for the
more general case. Section III extends to the analysis to a continuum of demand states.
Section IV focuses on the special cases of additive and multiplicative uncertainty.
Section V contains concluding comments about the measurement of marginal cost.
II.
Two Demand States
Suppose that demand takes on one of two states. Let q(p) represent demand in the
high demand state, a(p) be the amount by which demand is lower in the low demand
state, and  be the probability with which the high demand state occurs. Assume 0  a(p)
 q(p) for all p.
Additive uncertainty implies a(p) is a constant.7
Multiplicative
uncertainty means that a(p)/q(p) is constant.
We first establish the following lemma.
Lemma: Let π be profit, x be the quantity produced, and (x*,p*) be the production
level and price that maximize expected profits. Then, x* = q(p*) or x* = q(p*) – a(p*).
Proof: The expected profit function is:
(1)
E  p min q p, x 1  p min q p  a  p, x cx
The first order condition with respect to x has three regions to it:
7
The difference would only be constant in the range where q(p) exceeds the constant. For higher prices,
low demand would be 0.
3
(2)
E  
 pc
x
E  
 p  c
x
E  
 c
x
x  q  p a  p
q  p   a  p   x  q p 
q  p  x
If  < 1, then p* > c. Otherwise, expected profits would be negative. The first of the
three derivatives is therefore positive, so x* ≥ q(p*) – a(p*). Since -c < 0, x* ≤ q(p*).8
Within the middle region, the derivative is constant. If it is positive, then x* = q(p*)
maximizes expected profits. If it is negative, x* =q (p*) – a(p*) does.9
The lemma captures a key aspect of the economics of the newsvendor problem.
Under certainty, no distinction exists between the price and the quantity demanded.
Thus, when the firm considers lowering price to increase sales, which of course also
requires an increase in output, the demand curve determines the increase in sales caused
by any given reduction in price. In the newsvendor problem, the firm still makes a pricequantity trade-off, but what determines that rate is more complicated than with certainty.
The lemma implies that with two demand states, the demand curve in one of the two
states determines the price-output trade-off. When it is optimal to produce just for the
low demand, the problem reduces to one of certainty. Thus, the interesting case to
consider is when it is optimal to produce for high demand.
8
These two results meant that the firm never produces less than the minimum demand could be nor more
than the maximum it could be.
9
If, by chance, the derivative is 0 throughout the range, production at an end point of the region maximizes
expected profits, although other outputs would as well.
4
When the firm produces for the high demand state, x = q(p). Substituting for x in
equation (1) implies:
(3) E  pq  p  1  p q p  a  p  cq p
Because equation (3) is a function of only one choice variable, the underlying
logic of the value that maximizes it closely parallels the logic of the solution under
certainty. That is, the solution can be characterized as “price marginal expected revenue
equals price marginal cost” where price expected marginal revenue and price marginal
cost reflect the output adjustment associated with any change in price.
The first order condition for maximizing (3) is:
(4)
dE 
 q p   1   a( p)   p q'  p   1    a'  p   cq'  p   0
dp
The first term in brackets in equation (4) is the average (across states) of the quantity
demanded. The second term is the change in the average quantity demanded multiplied
by the price. Note that unless a’(p) = 0, the change in the average quantity demanded is
different from the change in output. Combined, these first two terms are “price marginal
revenue.” The last term is minus price marginal cost. Dividing the entire equation by the
middle term and rearranging terms yields:
(5)
 q p   1   a( p) 

q'  p 
 p q'  p   1    a'  p   1 p  c q'  p   1    a'  p  


The fraction in brackets on the left-hand side is the inverse of the elasticity of average
demand, which we can denote as A. Thus, we can rewrite (9) as:
5
(6)
 1 
c
p 1   
  A  q'  p   1    a'  p 
q'  p 
The left-hand side of (6) is quantity marginal revenue, i.e. the marginal revenue
from an additional expected unit sold. When a company sells an extra unit of output, it
gets the price it charges for the marginal unit, but the fraction of the price that it realizes
as marginal revenue is determined by the elasticity of demand. Demand uncertainty does
not change the way in which the ratio of the marginal revenue to price from an additional
unit sold depends on the elasticity of demand.
The right hand side is marginal cost of an expected unit sold. When producing for
the high demand state, the price reduction associated with a one unit increase in output
increases demand in the high demand state by 1. It also increases demand in the low
demand state but, except in the special case where a’(p) = 0 (i.e., additive uncertainty),
one extra unit of output leads to something other than one additional unit sold in the low
demand state. With multiplicative uncertainty, a’(p) is positive, so the denominator of
the right hand side is a fraction less than 1 (because q’(p) < 0).
A simple numerical example illustrates the point. Suppose low demand is always
half high demand, that the two states occur with equal probability, and that the marginal
cost of a unit of output is $3. If the firm expands output by 1, it lowers its price to
increase demand in the high demand state by 1. Because demand in the low demand state
is half the demand in the high demand state, the price reduction increases demand in the
low demand state by only 0.5. Thus, even though output increases by 1, expected sales
increase by only 0.75. Yet, the firm incurs the full production cost of $3. Per expected
6
unit sold, which is the appropriate basis of comparison for marginal revenue, marginal
cost is $3/0.75 = $4.
Rearranging (6) gives the extension of (1) to two-state uncertainty:
p
(7)
c
q'  p   1    a '  p 
1
q'  p 

p
A
Equation (7) clarifies why additive and multiplicative uncertainty have different
qualitative effects on price and the quantity produced.
As noted above, additive
uncertainty means that a(p) is constant and a’(p) = 0. Since the price reduction needed to
increase demand by one unit in the high demand state also increases demand by 1 in the
low demand state, the marginal cost of an additional unit sold equals the marginal cost of
a unit produced. In addition, the fact that the slope of the demand relationship is constant
across states has implications for how increased uncertainty affects the average elasticity
of demand. If one thinks of increased uncertainty as either a bigger difference between
high and low demand or as a higher probability of the low demand state, increased
uncertainty lowers average demand while the slope of the demand relationship remains
constant. As a result, increased uncertainty increases the elasticity of average demand.
This qualitative effect of increased uncertainty on price only applies in the range
where it is optimal to produce for the high demand states. Figure 1 depicts how optimal
prices vary with demand uncertainty for additive uncertainty, linear demand, and constant
unit costs. It shows the optimal price as a function of  for three different levels of a.
Starting from the point where demand is high with probability 1, decreases in  - that is,
increases in the probability of low demand – cause a linear reduction in price. In this
7
region, a firm produces for the high demand state, so a price reduction implies an
increase in output. The rate of decrease is greater for larger values of a. Thus, within
this region, the greater the amount of output that goes unsold in the low-demand state, the
greater the increase in output as the probability of low demand increases. For all three
values of a, however, there is a critical value of  beyond which the firm produces for the
low demand state and charges the price that would maximize profits if that state occurred
with certainty. Once a firm reaches that break, it simply ignores the opportunity to sell
extra in the high demand state. As a result, variations in the probability of demand do not
cause it to change its price or output.
With multiplicative uncertainty, a(p) = k q(p). This has two implications that
help simplify the pricing rule. First, the elasticity is constant across states. As a result,
increased uncertainty has no effect on the average elasticity of demand. Second, the
marginal changes in output in different states are proportional to output. Qualitatively,
increased uncertainty means that a greater portion of marginal output goes unsold, which
in turn increases the scaling factor from the marginal cost of a unit produced to marginal
cost of a unit sold. Quantitatively, it means that the average rate of unused output is also
the fraction of marginal output that goes unused. As a result, (7) reduces to:
p
(8)
c
1 k
1

p
A
As with linear uncertainty, the effect of increased uncertainty on price with
multiplicative demand modeled here depends on being in the range where it is optimal to
produce for the high demand state. Figure 2 depicts the optimal price as a function of 
for three different values of . As in Figure 1, each value of the demand shift parameter
8
has two ranges, one where the firm produces for high state and one where it produces
only for low demand.
In contrast to Figure 1, however, the prices increase as 
decreases, implying that output also decreases. The bigger the demand reduction in the
low demand state, the more price goes up as the probability of the low demand state
increases.
III.
Continuous Distribution of Demand
The two key aspects of the solution from the two-demand state extend more
generally to a continuum of states. Equation (1) applies with the elasticity of demand
being the elasticity of average demand and marginal cost being the marginal cost of a unit
sold, computed as the marginal cost of a unit produced divided by a scaling factor that
reflects the expected fraction of the marginal unit produced that the firm ends up selling.
Let Q(ε;p) be the inverse cumulative distribution function for the quantity
demanded conditional on p, where p is price and ε is a random variable uniformly
distributed between 0 and 1 with Q/ > 0 and Q/p < 0. As with just two states of
demand, the firm chooses output x which it produces at constant marginal cost c. It sells
x if Q(ε,p) ≥ x and Q(ε,p) otherwise. Let (q,p) = Q-1(q,p) be the probability that at a
price p, the quantity demanded is less than Q. The firm maximizes expected profits, Π,
given by:
  x, p 
(9)
E  px 1  x, p   p  Q ; p  d  cx
0
The two (standard)10 first order conditions for the maximum are:
(10)
10
E  
 p 1  x, p  c  0
x
See the discussion below.
9
and
(11)
E
 x 1  x, p 
p
 x, p 
 x, p 
0
0
 Q ; p d  p

Q ; p 
d  0
p
Both equations (10) and (11) have natural interpretations. Equation (10) says that,
holding price constant, the optimal output is where the price multiplied by the probability
that the marginal output is sold equals the marginal cost of a unit of output. Equation
(11) says that, holding the quantity produced (and, therefore, costs) constant, the optimal
price is where the increased revenue on inframarginal sales from a price increase is offset
by the lost revenues on the marginal sales. Because production is being held constant,
the lost revenues are the entire purchase price, not just the margin.
These equations appear quite different from the first order condition that arises
under certainty. Equation (10) appears to imply choosing output where price equals the
marginal cost of an additional expected unit sold.
Equation (11) appears to imply
choosing a price where price marginal revenue is 0. Neither seems to imply an output
where marginal revenue, taking account of the necessary price reduction to sell an extra
unit, equals marginal cost. Yet, such a condition is an implication of the simultaneous
solution of (10) and (11) even if it is not transparent from either one separately.
In solving (9), there are an infinite number of partial derivatives that hold at the
maximum. One can, as in (11), take the derivative of expected profits with respect to
price holding output constant; but one can just as well take the derivative of expected
profits with respect to price, increasing the output by 2 for every one unit increase in
price. The key to deriving an analog to equation (1) for the case of continuous states is to
look at a first order condition of the form ∂E[Π]/∂p + b ∂E[Π]/∂q = 0, where b is a
constant chosen to reflect the change in output associated with a one-unit change in price.
10
In choosing b, one might consider using the implicit function theorem on equation
(10) to find the change in output that is optimal for any given price. While this approach
yields a condition that is valid at the optimum, it turns out not to be the most useful for
isolating the effect of the “pure” elasticity of demand. To understand why, consider how
the continuous-state analysis differs from the two-state analysis of section II. With twostate uncertainty, the firm produces either for the high demand or low demand state.
More generally with discrete states, the company must decide for any given price which
state to produce for. The same point applies with continuous demand. The random
variable ε represents the state of demand. An interpretation of equation (10) is that, for
any price, it gives the optimal state of demand for which to produce. With a continuum
of states, the state of demand that determines the quantity produced varies continuously
with price. Thus, the derivative of optimal output with respect to price reflects two
factors. One is the change in demand due to a price change holding constant the state for
which the firm produces (i.e., holding constant the probability that the firm sells all its
output). The other is a change in output due to a change in the marginal state for which
the firm produces. Only the first of these reflects a pure effect of demand on price. Thus,
to generate a condition that reflects the pure effect of price on demand, the appropriate
value for b is ∂Q[Φ(x,p),p]/∂p, i.e., the partial derivative of demand with respect to price
evaluated at the marginal state for which demand equals output:
11
(13)
E   Qx, p , p  E 

 x 1  x, p 
p
p
x

  x, p 
 x, p 
0
0
 Q ; p  d  p 
Q ; p 
d
p
Qx, p , p 
 p 1  x, p   c 0
p
Rearranging terms reveals even more clearly the parallels with the certainty case:

Qx, p , p 
p  1  x, p 

p

(14)

  x1  x, p 

  x, p 

0
Q ; p  
d 
p

 Qx, p , p 
c 0

p

 x, p 
 Q ; p  d p  
0
The first term on the left-hand side of (14) is price multiplied by the change in the
quantity sold when the firm changes price and the quantity produced to hold constant the
probability of selling out. When the firm sells out, which occurs with probability 1(x,p), the increase in the quantity sold is the increase in output. In all other states, the
change in demand determines the change in the quantity sold. The next set of brackets is
average demand. The last term is the increase in cost from increasing output enough to
hold constant the probability of selling out. Rearranging terms implies:
(15)
p c/S
1

p
A
where:
12
Q x, p ; p 
1  x, p   Q ; p  d
dp
p
0
S
Qx, p ; p 
dp
  x, p 
and
 x, p 
 Qx, p ; p 

1 x, p   Q ; p  d 
p
dp
p


0
A  
 x, p 


 x1  x, p   Q ; p  d p 
0


While the expressions for S and A are relatively complicated, the interpretation of
equation (15) is essentially the same as the interpretation of equation (7). S is a scaling
factor for marginal cost that reflects the ratio of the increase in demand in the marginal
state in which demand just equals supply to the average across states of the increase in
the quantity sold in response to a price increase. As in (7), η A is the elasticity of average
demand with respect to price.
IV.
Special Cases
The two special cases that have garnered interest in the past are additive and
multiplicative uncertainty. Both simplify the general case considerably.
A. Additive uncertainty
With additive uncertainty,
(16)
Q( ; p) q( p)  r ( )
13
where q(p) is the deterministic component of demand.
As before,  is distributed
uniformly from 0 to 1, so r() is an inverse cumulative distribution function for the
additive random component. Let r(1) = 0 and r’() > 0, which implies that q(p) is the
maximum possible demand at any given price.
Expected profits are given by:
(17)
E   px Pr obr ( ) x  q( p) p
r 11  x  q ( p ) 
 q p    d  cx
0
The (standard) first order conditions are:
(18)
E  
 p Pr obr ( ) x  q( p)  c  0
x
and
E 
x Pr obr ( ) x  q( p)
p
r 11  x  f ( p ) 
r 11  x  f ( p ) 
(19)
p

0
  f  p   r ( )d
0
df ( p)
d  0
dp
With additive uncertainty, one need not resort to the “trick” of using an
appropriately weighted sum of the two “standard” first order conditions to derive a markup rule.
Since (18) implies that the probability that demand exceeds the quantity
produced is c/p, the probability that demand is less than the quantity produced is (p-c)/p.
Direct substitution into the last term of (19) implies:
r 11  x q ( p ) 
pc

(20)
p
x Pr obr ( ) x  q( p)
 q p   r ( )d
0
p
dq( p)
dp
14

1
A
The numerator on the right hand side of (20) is the average quantity sold so the entire
right-hand side is the negative of one divided by the elasticity of the average quantity
sold.
The key feature that distinguishes equation (20) from equation (15) is that there is
no scaling factor for marginal cost (or, more accurately, the scaling factor is 1). As with
two-state uncertainty, a price reduction with additive uncertainty causes the same
increase in the quantity sold in all states, so there is no distinction between the marginal
cost of unit produced and the marginal cost of a unit sold.
B. Multiplicative Uncertainty
With multiplicative uncertainty,
(21)
Q( ; p) q( p)m( )
where q(p) is again the maximum possible demand conditional on price. As before,  is
distributed uniformly from 0 to 1, so m() is an inverse cumulative distribution function
for the multiplicative random factor, m(1) = 1 and m’() > 0.
Substituting (21) into the expressions for s in (15) yields:
S 
(22)

dq
m x, p 
dp
dq
dq
mx, p 1   x, p 
dp
dp
m x, p 
mx, p 1  x, p 
  x, p 
 m( ) d
0
  x, p 
 m( ) d
0
As a general proposition, the scaling factor is one divided by the fraction of the marginal
unit of output that is sold. Equation (22) indicates that with multiplicative uncertainty,
15
one can use the fraction of total output sold to measure the fraction of the marginal output
that is sold.
Substituting (21) into the expression for A implies:
A
(23)
 x, p 
 dq

dq
p  mx, p 1  x, p  
m( ) d 
dp
 dp

0
 
 x, p 










q
(
p
)
m

x
,
p
1


x
,
p

q( p)m  d 




0
dq p

dp q
With multiplicative uncertainty, the elasticity of demand at a given price is the same in all
states. Thus, the elasticity in any one state is the average elasticity over all states.
As with additive uncertainty, the insights gained from the analysis of the twodemand state with multiplicative uncertainty apply to continuous demand states as well.
V. Conclusion
The main contribution of this paper is to suggest a way of characterizing the solution
to the newsvendor problem with endogenous pricing. The approach clarifies how
uncertainty affects optimal mark-ups.
The characterization clarifies why the form of uncertainty can have a qualitative
effect on how uncertainty affects prices and output. It also suggests that the quantitative
effect of the form of uncertainty on a firm’s optimal price and output can be large. If
uncertainty is additive, one can compute marginal cost directly from the production
function. For all other forms of uncertainty, marginal cost depends on demand uncertainty
as well. With multiplicative uncertainty, which is arguably a natural assumption absent
evidence to the contrary, the marginal cost of a unit produced must be converted to a
16
marginal cost of a unit sold by dividing by the fraction of output sold. For industries in
which firms end up discarding a substantial fraction of output, the scaling factor can be
large.
In deciding what to assume about the nature of uncertainty, one would ideally be
guided by econometric estimates of demand. One factor to consider is the statistical
distribution of the residuals in the demand relationship. In modeling demand, however, one
might control for factors that firms consider uncertain at the time they make their price and
output decisions. To infer the statistical distribution of demand from the perspective of the
firm making the price and output decision, one would also need to consider the (joint)
distribution of these factors.11
There are many extensions of the newsvendor problem with endogenous pricing.
One can allow for a depreciated value of unsold output and a reputation cost for unmet
demand. While the problem is posed as a monopoly problem, it applies across market
structures. In applying the insights from monopoly to more competitive market structures,
one would need to understand how the form of market demand uncertainty affects the form
of uncertainty regarding residual demand. There is potentially a great deal of insight to be
gained by applying the approach for characterizing the solution developed here to more
general forms of the problem.
11
One practical application where increased care in the measurment of marginal cost can be important is
critical loss analysis in antitrust. For a discussion of the controversy surronding this issue, see Katz and
Shapiro (2003) and O’Brien and Wickelgren (2003).
17
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19
Figure 1
Additive Uncertainty
1.6
1.5
a = .15
1.4
Optimal Price
a = .30
1.3
a = .45
1.2
Assumptions
1.1
High Demand Curve:
P
Costs:
Q=2c=1
1
1
0.95
0.9
0.85
0.8
0.75
0.7
Probability of High Demand
20
0.65
0.6
0.55
0.5
Figure 2
Multiplicative Uncertainty
1.60
k = .55
1.55
Optimal Price
k = .70
k = .85
1.50
Assumptions
High Demand Curve:
P
Costs:
Q=2c=1
1.45
1
0.95
0.9
0.85
0.8
0.75
21
0.7
0.65
0.6
0.55
0.5