WEB APPENDIX
Proof of Corollary 1
We use the proof by contradiction method. Suppose d L  d H . Since
 MD
Price to L-Types
d L
 MD
Price to H-Types
d L
 0 and
 0 , the highest value of  MD   ND in the region where d L  d H is obtained when
d L  d H . Substituting d L  d H , we arrive at:
 MD   ND  1  Max  0,d H  d H vLF   vLF   vLF  d H vLF 
  2vLF  1 d H  1  0
MD
ND
Thus, a necessary condition for     0 is
dL  dH .
Proof of Corollary 2
We use the proof by contradiction method. Suppose
vHU  1  vLF  vLU . Using equations (1) and
  PS   ND 
1  vHU  vLF  vLU  vLF  vLU
(7), we find:  PS   ND  Min 
. Since 
,1


2
v
 0 , in the
LF


2
2
vHU


vHU  1  vLF  vLU , the highest value of  PS   ND is obtained when
 1  vLF  vLU . Substituting vHU  1  vLF  vLU , we arrive at:
region where
vHU
1  (1  vLF  vLU )  vLF  vLU  vLF  vLU
 PS   ND  Min 
,1 
 2vLF
2
2


3v
 2v
 v
 Min  LF ,1  LU  LF
2
 2
 2
v v
 LU LF  0
2
PS
ND
Thus, a necessary condition for     0 is
vHU  1  vLF  vLU , or rearranging,
vLF  vLU  1  vHU .
Proof of Proposition 1
Part (a) asserts that there is a subset of parameters such that MD  ND  PS . Using equations (1),
(4), and (7), these inequalities will hold if:
1  vHU  vLF  vLU  vLF  vLU
1  Max  dH  dL vLF ,0  dL vLF  2vLF  Min 
,1 
2
2


(A1)
The regions in Figures 1 a) and b) denoted as “Only PS is advantageous to ND” show that the subset denoted
by (A1) is non-empty.
Part (b) is a direct calculation of  PS   MD using equations (4) and (7).
Part (c) is based on the following calculations:
  PS   MD 
vLU
1
 if vLU  1  vHU  vLF
 2
1 if vLU  1  vHU  vLF
1
(A2)
d

vLF if d L  H
PS
MD



     
vLF

d L
2v if d  d H
L
 LF
vLF
  PS   MD 
vHU
  PS   MD 
d H
(A3)
 1
 if vHU  vLU  vLF  1
 2
0 if vHU  vLU  vLF  1
(A4)
1 if d H  d L vLF

0 if d H  d L vLF
(A5)
Proof of Lemma 2
In this proposed equilibrium, the firm select prices such that H-type consumers purchase their
favored product and L-type consumers purchase the probabilistic good. To induce such segmentation, prices
much be set according to equation (6). Substituting in vLU  vHU  0 , we have P1PS  1  vLF and P0PS  vLF . At
2
2
2  1
these prices, total demand for the popular good is
and the total demand for the unpopular good is
2
3  2 . To meet this demand, the seller must purchase 2  1 units of each product. Such an inventory
2
2
1

v
v
2


1
1

PS
,
DU
LF
LF
decision results in a profit of 


 2c 
  vLF    2  1 c .
2
2
2
2


It is obvious that ordering more that 2  1 units of each product cannot be optimal since this would
2
increase costs without impacting revenue. Thus, the key to proving Lemma 2 is showing that it is not optimal
for the seller to reduce its inventory orders below 2  1 . It is important to note that, to induce price
2
discrimination under PS, prices must be set according to equation (6) regardless of the inventory order (and
without price discrimination, PS is equivalent to ND). Let K be the number of units ordered for each product
and X be the number of units held back of each good to use for buyers of the probabilistic good. There are
three parameter regions of K to consider:
 


1. If K  1   , the firm sells out of both products at the regular price P1PS . Profit is 2 P1PS  c K ,
which is increasing in K. Thus, there is no interior solution for K  1   .
 
 
2. If 1    K   , profit equals 1    K  X  P1PS   2 X  PoPS  2cK s.t. 0  X  K    1 . Since
profit is linear in X, only corner solutions can be optimal. At X=0, profit is linear in K, so we have a
corner solution: K  1   or K   . If X  K    1 , profit is linear in K, which implies the
maximum on this range would be achieved at either K  1   or K   . Note that if K  1   ,
then X  K    1  0 . Thus, there are three relevant corner solutions: (1) X=0& K  1   , which
yields a profit of (1)  1   1  vLF  2c  ; (2) X=0& K   , which yields a profit of
1  vLF
(3)
 2 c ; and (3) X  2  1 & K   , which yields   1    vLF  1  2c  .
2
2  1
If   K 
, all H-types can purchase their favored product, and thus the amount of the popular
2
 (2) 
3.
good remaining is X  K   . This can be used to create 2 X units of the probabilistic good. Profit
 
is P1PS  2  K    PoPS  2cK . This function is linear in K. The profit at the corner solution with
2
K   is (2) . The profit at the corner solution with K  2  1 is given by  PS , DU as defined in part
2
(b) of Lemma 2.
Comparing the four candidate solutions, we find
v  2c
 PS , DU   (2)  LF
 0 and (3)   (1)   2  1 vLF  2c   0 , where these differences are positive since
2
v  2c   2  11  vLF 
v
v
c  LF . Furthermore,  PS , DU   (3)  LF
 0 since c  LF and vLF  1 .
2
2
2
PS , DU
Thus, 
is the maximum profit available under PS when there is demand uncertainty.
Proof of Proposition 2
For the parameter region given in the text, in the absence of demand uncertainty (as analyzed in
Section 3.2), under PS, we have P1PS  1  vLF , P0PS  vLF , and  PS  vLF  1  2c . Lemma 2 gives the profit
2
2
2
under PS when demand uncertainty is present. Thus, the magnitude of the profit decrease caused by demand
uncertainty is:
(A6)
 PS , DU   PS   PS , DU   2  1 c  0
Profits decrease because revenue is unchanged, but inventory costs rise with demand uncertainty. In
particular, without demand uncertainty, the firm is able to match inventory orders to sales. But under demand
uncertainty, 2  1  3  2  2  1 units of the ex post unpopular good will go unsold. Finally, with and
2
2
without demand uncertainty, all consumers purchase, and thus total sales equal 2.
In the absence of demand uncertainty, equations (3) and (4) give the prices and profit under MD. For
the parameter region under consideration, we have: P1MD  1 , P2MD  dL vLF , and MD  1  dL vLF  2c . With
demand uncertainty, the retailer can either mimic the prices and allocation that were used when there was no
demand uncertainty or it can sell fewer units. First, suppose the firm retains the same prices and allocation.
Since the H-type consumers purchase their favored product in Period 1 and the L-type consumers purchase
their favored product in Period 2, total demand for the popular good equals 2 and total demand for the
unpopular good equals 2 1    . With demand uncertainty, in order to meet this demand, the seller must
purchase 2 units of each product (which will result in 2(2α - 1) units going unsold). The profit to the seller
is:
(A7)
 aMD , DU  1  d L vLF  4 c
Second, suppose the retailer reduces its inventory orders so that fewer units go unsold. Let
MD , DU
K
 2 be the number of units ordered of each product. Note that, since the value of the L-type’s less
favored good is zero, the seller cannot benefit from inducing the L-types to purchase their less-preferred
product.1 Thus, as long as it is profitable to make second period sales, the prices charged will be the same as
in the case without demand uncertainty. The seller earns a profit of:
(A8)
bMD, DU  Min  , K MD, DU   1     1    Max  K MD, DU   , 0 dL vLF  2 K MD, DU c


1
Therefore, prices in the second period are the same for both the popular good and the unpopular one. Symmetric prices
are consistent with much of current retail practice. For example, major department stores often offer the same
markdown percentage on all color-patterns of a particular product even when demand differs across these color patterns.
3
 2 1    so that the seller can meet the entire demand for the
Equation (A8) assumes that K
unpopular good. Since demand for the popular good is strictly higher than the demand for the unpopular
good, it cannot be optimal for the firm to produce such limited units that it sells out of the unpopular good.
MD , DU
Notice that  bMD , DU is linear in K
. Thus, there is no interior solution and we must have a corner
MD, DU
MD, DU
 2 1    , K MD , DU   , or K MD , DU  2 . The profit when K MD , DU  2 is given by
solution at K
 aMD , DU as defined by (A7). For the other two corner solutions, we consider two parameter regions separately.
2
, then 2 1      . In this range, both additional corner solutions are feasible.
3
MD , DU
, DU
 2 1    , then bMD
If K
 3 1     1    dL vLF  4 1    c . Here, there are just enough units
,2
If  
of the unpopular product to meet the demand from both the H-types and the L-types. However, the popular
MD , DU
  , then
product sells out in the first period before meeting the entire demand from the H-types. If K
, DU
bMD
 1  1    dL vLF  2 c . Here, there are just enough units of the popular good to meet the demand
,3
in the first period from the H-types. Thus, no L-types who favor the popular good are able to purchase in the
second period. Furthermore, after both first period and second period sales of the unpopular product, units of
, DU
, DU
inventory will remain unsold. Furthermore, we calculate bMD
 bMD
  3  21  2c   0 . Thus, it is
,3
,2
never optimal to choose K
MD, DU
 2 1    when  
2
.
3
2
MD, DU
 2 1    ,
, then 2 1      . In this range, the only feasible corner solution is K
3
, DU
which yields a profit of bMD
 1   3  4  dL vLF  4 1    c . Here, all H-types are able to purchase their
,4
favored product in the first period, all L-types who favor the unpopular product can purchase in the second
period, but the popular good stocks out during the second period. All units of both goods will be sold.
The seller chooses the corner solution which yields the highest profit:
If  
 MD , DU

1  d L vLF  4 c


 1   3  4  d L vLF  4 1    c


1  1    d L vLF  2 c

d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
c
(A9)
The profit decrease caused by demand uncertainty when the seller employs the MD strategy is:
 MD , DU   MD   MD , DU

2c  2  1


 2  2  1 d L vLF  c 


 d L vLF  2c 1   

c
d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
(A10)
Without demand uncertainty, under MD, the firm is able to match inventory orders to sales. But, when there
is demand uncertainty, some units of the unpopular good go unsold:
Unsold MD , DU

2  2  1


 0


3 - 2

c
d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
4
(A11)
In the absence of demand uncertainty, all consumers purchase under MD, and thus total sales equal 2.
However, when there is demand uncertainty, the market may not be fully covered. In particular, sales are:
Units Sold MD , DU

2


 4 1   


2  

c
d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
(A12)
Using (A6) and (A10), we can compare the profit loss caused by demand uncertainty under MD versus PS:
 MD , DU   PS , DU

c  2  1  0


  2  1 2d L vLF  3c 


 d L vLF  c

c
d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
(A13)
The difference in profit decrease is positive unless costs are too large. In particular, the first line of (A13) is
positive. The second line is positive if c < 2(dL vLF)/3. The third line is positive if c < α dL vLF . Thus, a
sufficient condition for ΔMD,DU > ΔPS,DU is c < 2(dL vLF)/3.
Comparing the number of unsold inventory under MD versus PS:
Unsold MD , DU  Unsold PS , DU

2  1  0


 1- 2  0


 -1  0

d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
c
(A14)
Comparing total sales under MD versus PS:
Units Sold MD , DU  Units Sold PS , DU

0


 2  2  1  0


  0

5
d L vLF
2
d L vLF
 c  d L vLF &   2
3
2
d L vLF
 c  d L vLF &   2
3
2
c
(A15)