Price Effects in Online Product Reviews: An Analytical ... - MIS Quarterly
Li & Hitt/Price Effects in Online Product Reviews–Appendices
RESEARCH ARTICLE
PRICE EFFECTS IN ONLINE PRODUCT REVIEWS: AN ANALYTICAL MODEL AND EMPIRICAL ANALYSIS By: Xinxin Li School of Business University of Connecticut 2100 Hillside Road U1041 Storrs, CT 06279 U.S.A. [email protected]
Lorin M. Hitt The Wharton School University of Pennsylvania 3730 Walnut Street, 500 JMHH Philadelphia, PA 19104-6381 U.S.A. [email protected]
Appendix A Derivation of Optimal Price Functions for the Monopoly Setting We apply backward induction to derive optimal price functions. In the second period, given first period price p1, the firm selects second period price p2 (p2 < Max{q, R}) to maximize its second period profit:
The profit function can be reduced to four possibilities based on the value of p1:
By maximizing profit in each of the four cases, we can derive the optimal second period price p2 as a function of first period price p1:
MIS Quarterly Vol. 34 No. 4,Li & Hitt–Appendices/December 2010
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Li & Hitt/Price Effects in Online Product Reviews–Appendices
The corresponding second period profit as a function of p1 is
Back in the first period, given π*2(p1), the firm selects a first period price p1 (p1 < qe) to maximize its total profit in both periods: p1 (q e − p1 ) + π 2* ( p1 ) . By comparing optimal profit in different ranges of p1, we can derive the optimal first period price for different values t of q:
Combining p*1 and p*2(p1), we can derive that
.
Appendix B Derivation of Optimal Price Functions for the Duopoly Setting We first utilize the case of q1 = 12 to explain in detail how to derive equilibrium prices and then follow the same procedure to solve equilibria for the other two cases (q1 = 1 and q1 = 0). We apply backward induction to derive optimal price functions. In the second period, given the first period prices, p11 and p21 (pj1 < qej = 12),
{
{
3− 4 p
{
}}
{
3q − 2 p
}}
11 2 21 the ratings of the two products are R1 = Min 1, Max 0, 4 and R2 = Min 1, Max 0, 2 . Given a = 1, b = 1, n = 3, t = 16, and qe1 = qe2 = 12, if all second-period consumers purchase from one of the two firms, the second period profits are
)}}) and π = 3( p Min{1, Max{0,3( R − R − p + p + )}}) . If some ( { { second-period consumers expect negative utility from both firms and do not buy from either firm, the profit functions are π = 3( p Min{1, Max{0,6( R − p )}}) and π = 3( p Min{1, Max{0,6( R − p )}}) . Then back in the first period, firms select π 12 = 3 p12 Min 1, Max 0,3( R1 − R2 − p12 + p22 + 12
p11
A2
12
and
p21
1
to
maximize
22
22
12
their
1 6
total
profits
22
in
both
22
2
2
periods:
MIS Quarterly Vol. 34 No. 4Li & Hitt–Appendices/December 2010
1
22
12
1 6
22
{
π 1 = 3 p11 Min 1, Max{0,− p11 + p21 +
1 6
}} + π
* 12
and
Li & Hitt/Price Effects in Online Product Reviews–Appendices
{
π 2 = 3 p21 Min 1, Max{0,− p21 + p11 +
1 6
}} + π
* 22
. It can be proved that in this scenario all of the second-period consumers will purchase
one of the two products in equilibrium. Thus, firms’ second period profit functions are:
{ ( − Min{1, Max{0, }} − p + p + )} , Max{0,3( Min{1, Max{0, }} − − p + p + )} .
π 12 = 3 p12 Min 1,3
π 22 = 3 p22
3− 4 p11 4
3q2 − 2 p21 2
3q2 − 2 p21 2
12
3− 4 p11 2
1 6
22
22
1 6
12
We can then derive the optimal second period prices as functions of the first period prices:
,
The corresponding second period profits as functions of the first period prices thus are:
,
.
Then back in the first period, firms select the first period prices to maximize their total profits in both periods:
{ Min{1, Max{0,− p
π 1 = 3 p11 Min 1, Max{0,− p11 + p21 +
π 2 = 3 p21
21
+ p11
}} + π ( p , p ) , + }} + π ( p , p ) . * 12
1 6
1 6
11
* 22
21
11
21
By comparing profits in different ranges of p11 and p21, we can derive the optimal first period prices for different values of q2:
,
.
MIS Quarterly Vol. 34 No. 4,Li & Hitt–Appendices/December 2010
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Li & Hitt/Price Effects in Online Product Reviews–Appendices
Combining p*11, p*21, p*12(p11, p21), and p*22(p11, p21), we can derive that:
,
.
Following similar procedure, we can derive the optimal price functions for q1 = 1:
,
.
,
.
Similarly, the optimal price functions for q1 = 0 are
,
.
,
.
In the benchmark scenario, the firms select p11, p21, p12, and p22 to maximize their total profits: 1 6
{
1 6
π 2 = 3 p21 Min 1, Max{0,− p21 + p11 +
A4
}} + 3( p
{
π 1 = 3 p11 Min 1, Max{0,− p11 + p21 +
12
}} + 3( p
22
{
{
{
{
Min 1, Max 0,3(q1 − q 2 − p12 + p22 +
Min 1, Max 0,3(q 2 − q1 = p22 + p12 +
MIS Quarterly Vol. 34 No. 4Li & Hitt–Appendices/December 2010
1 6
)}}) , 1 6
)}}) .
Li & Hitt/Price Effects in Online Product Reviews–Appendices
It can be shown that the optimal first period prices are both 16, the second period prices are: *
16 + 1−3q2 56 − q 2
1− q if 21 < q 2 < 1 * 16 − 3 2 , = p 22 0 if 0 < q 2 < 21
(1)
If q1 = 1, p21 =
(2)
If
(3)
q22 * * q = 0 , p = 0 , p = If 1 21 22 q 2 −
q1 =
1 2
* , p21 =
1 6
+
1 2
− q2 * , p22 = 3
1 6
−
1 2
− q2 . 3
if 0 < q 2 < 1 6
if
1 3
if 21 < q 2 < 1 . if 0 < q 2 < 21
1 3
< q2 < 1
.
MIS Quarterly Vol. 34 No. 4,Li & Hitt–Appendices/December 2010
A5
RESEARCH ARTICLE
PRICE EFFECTS IN ONLINE PRODUCT REVIEWS: AN ANALYTICAL MODEL AND EMPIRICAL ANALYSIS By: Xinxin Li School of Business University of Connecticut 2100 Hillside Road U1041 Storrs, CT 06279 U.S.A. [email protected]
Lorin M. Hitt The Wharton School University of Pennsylvania 3730 Walnut Street, 500 JMHH Philadelphia, PA 19104-6381 U.S.A. [email protected]
Appendix A Derivation of Optimal Price Functions for the Monopoly Setting We apply backward induction to derive optimal price functions. In the second period, given first period price p1, the firm selects second period price p2 (p2 < Max{q, R}) to maximize its second period profit:
The profit function can be reduced to four possibilities based on the value of p1:
By maximizing profit in each of the four cases, we can derive the optimal second period price p2 as a function of first period price p1:
MIS Quarterly Vol. 34 No. 4,Li & Hitt–Appendices/December 2010
A1
Li & Hitt/Price Effects in Online Product Reviews–Appendices
The corresponding second period profit as a function of p1 is
Back in the first period, given π*2(p1), the firm selects a first period price p1 (p1 < qe) to maximize its total profit in both periods: p1 (q e − p1 ) + π 2* ( p1 ) . By comparing optimal profit in different ranges of p1, we can derive the optimal first period price for different values t of q:
Combining p*1 and p*2(p1), we can derive that
.
Appendix B Derivation of Optimal Price Functions for the Duopoly Setting We first utilize the case of q1 = 12 to explain in detail how to derive equilibrium prices and then follow the same procedure to solve equilibria for the other two cases (q1 = 1 and q1 = 0). We apply backward induction to derive optimal price functions. In the second period, given the first period prices, p11 and p21 (pj1 < qej = 12),
{
{
3− 4 p
{
}}
{
3q − 2 p
}}
11 2 21 the ratings of the two products are R1 = Min 1, Max 0, 4 and R2 = Min 1, Max 0, 2 . Given a = 1, b = 1, n = 3, t = 16, and qe1 = qe2 = 12, if all second-period consumers purchase from one of the two firms, the second period profits are
)}}) and π = 3( p Min{1, Max{0,3( R − R − p + p + )}}) . If some ( { { second-period consumers expect negative utility from both firms and do not buy from either firm, the profit functions are π = 3( p Min{1, Max{0,6( R − p )}}) and π = 3( p Min{1, Max{0,6( R − p )}}) . Then back in the first period, firms select π 12 = 3 p12 Min 1, Max 0,3( R1 − R2 − p12 + p22 + 12
p11
A2
12
and
p21
1
to
maximize
22
22
12
their
1 6
total
profits
22
in
both
22
2
2
periods:
MIS Quarterly Vol. 34 No. 4Li & Hitt–Appendices/December 2010
1
22
12
1 6
22
{
π 1 = 3 p11 Min 1, Max{0,− p11 + p21 +
1 6
}} + π
* 12
and
Li & Hitt/Price Effects in Online Product Reviews–Appendices
{
π 2 = 3 p21 Min 1, Max{0,− p21 + p11 +
1 6
}} + π
* 22
. It can be proved that in this scenario all of the second-period consumers will purchase
one of the two products in equilibrium. Thus, firms’ second period profit functions are:
{ ( − Min{1, Max{0, }} − p + p + )} , Max{0,3( Min{1, Max{0, }} − − p + p + )} .
π 12 = 3 p12 Min 1,3
π 22 = 3 p22
3− 4 p11 4
3q2 − 2 p21 2
3q2 − 2 p21 2
12
3− 4 p11 2
1 6
22
22
1 6
12
We can then derive the optimal second period prices as functions of the first period prices:
,
The corresponding second period profits as functions of the first period prices thus are:
,
.
Then back in the first period, firms select the first period prices to maximize their total profits in both periods:
{ Min{1, Max{0,− p
π 1 = 3 p11 Min 1, Max{0,− p11 + p21 +
π 2 = 3 p21
21
+ p11
}} + π ( p , p ) , + }} + π ( p , p ) . * 12
1 6
1 6
11
* 22
21
11
21
By comparing profits in different ranges of p11 and p21, we can derive the optimal first period prices for different values of q2:
,
.
MIS Quarterly Vol. 34 No. 4,Li & Hitt–Appendices/December 2010
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Li & Hitt/Price Effects in Online Product Reviews–Appendices
Combining p*11, p*21, p*12(p11, p21), and p*22(p11, p21), we can derive that:
,
.
Following similar procedure, we can derive the optimal price functions for q1 = 1:
,
.
,
.
Similarly, the optimal price functions for q1 = 0 are
,
.
,
.
In the benchmark scenario, the firms select p11, p21, p12, and p22 to maximize their total profits: 1 6
{
1 6
π 2 = 3 p21 Min 1, Max{0,− p21 + p11 +
A4
}} + 3( p
{
π 1 = 3 p11 Min 1, Max{0,− p11 + p21 +
12
}} + 3( p
22
{
{
{
{
Min 1, Max 0,3(q1 − q 2 − p12 + p22 +
Min 1, Max 0,3(q 2 − q1 = p22 + p12 +
MIS Quarterly Vol. 34 No. 4Li & Hitt–Appendices/December 2010
1 6
)}}) , 1 6
)}}) .
Li & Hitt/Price Effects in Online Product Reviews–Appendices
It can be shown that the optimal first period prices are both 16, the second period prices are: *
16 + 1−3q2 56 − q 2
1− q if 21 < q 2 < 1 * 16 − 3 2 , = p 22 0 if 0 < q 2 < 21
(1)
If q1 = 1, p21 =
(2)
If
(3)
q22 * * q = 0 , p = 0 , p = If 1 21 22 q 2 −
q1 =
1 2
* , p21 =
1 6
+
1 2
− q2 * , p22 = 3
1 6
−
1 2
− q2 . 3
if 0 < q 2 < 1 6
if
1 3
if 21 < q 2 < 1 . if 0 < q 2 < 21
1 3
< q2 < 1
.
MIS Quarterly Vol. 34 No. 4,Li & Hitt–Appendices/December 2010
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