Talking about Dynamic Pricing in Brno.

1. Lectures – Dynamic Pricing
Part I – Background Material
BRNO – 2009
By
Kjetil K. Haugen

2. The classic linear Monopoly model
• Assumptions: (linear
demand and linear and
proportional costs)
c(Q)=cQ
P=a-bQ
Demand curve plot
P
a
• a,b > 0  Normal good
• Profit function:
(Q)=R(Q)-C(Q)
=PQ-cQ
=(a-bQ)Q -cQ
a
b
Q

3. Monopoly solution
• Analytically: (Max )
• MR and MC:
 ' (Q)  0
R(Q)  P(Q)Q
 (a  c)  2bQ*  0
ac
 Q* 
2b
 (a  bQ)Q  aQ  bQ 2
 MR  R' (Q)  a  2bQ
MC  c' (Q)  c
• Graphically:
 (Q)  R(Q)  c(Q)
 ' (Q)  0
 MR(Q)  MC (Q)  0
 MR  MC
P
a
P
MR
Demand
M
MC
c
QM a
2b
a
b
Q

4. Perfect Competition
• Monopoly:
– 1 producer
–  consumers
• Perfect competition
–  producers
–  consumers
– Q give no P
 (Q)  P  Q  cQ
 ' (Q)  P  MC  0
 P  MC
P
a
P
M
MR
Demand
MC
PPC=c
QM a
2b
QPC
a
b
Q

5. A practical example – Dynamic
Monopoly (1)
• Look at a grocery store selling milk.
• Shop owner has experienced a customer daily
demand pattern like:
0800
1200
1600

6. A practical example – Dynamic
Monopoly (2)
• Suppose that in order to serve such a daily
demand, the shop needs 4 employees for the
tops , 1 for the bottoms and 2 for the average
demand.
• Could you device a simple dynamic pricing
strategy for improving profits comparded to a
fixed price strategy?
• What determines if a dynamic pricing strategy is
better than a fixed price strategy?
• What about existence of other milk sellers?

7. A simple lot-size Dynamic Pricing
Problem (Dynamic Monopoly)
• Assume:
– A producer produce one product in T timeperiods
– In each of these T timeperiods, a demand function
ft(pt) exist. These functions are given and
represent consumer (steady) preferences.
– The producer can store finished products and the
products are assumed non-perishable
– Product storage leads to inventory costs - ht, also
production costs - ct are present. We assume
linearity and proportionality for these costs.
– Production capacity is limited by Rt

8. A Mathematical Programming
formulation (NLP)
T
Max Z   ft ( pt ) pt  ct xt  ht I t
t 1
s.t
I t  I t 1  xt  ft ( pt ) t
xt  Rt t
xt , I t , pt  0
Xt :
It :
pt :
Amount produced in period t
Amount stored between periods t-1, t
Price set in period t

9. Comparing various versions of this
model
• Assumptions:
– Linear demand: ft(pt)=at-btpt
• Case 1:
– Fixed given price (LP – cost minimization)
• Case 2:
– Constant variable price (QP – profit maximisation)
• Case 3:
– Discrete price choice (MILP – profit maximization)
• Case 4:
– Dynamic continuous price (QP – profit maximisation)

10. A Mathematical Programming
formulation (Case 1)
T
Min Z   ct xt  ht I t
t 1
s.t
I t  I t 1  xt  d t t
xt  Rt t
xt , I t  0
A given constant price pt=p0 t leads to a constant term in the objective. Maximising a constant minus ”something” is the same as minimising ”something”. Then defining ft(p0)=dt yields the above LP-model.

11. A Mathematical Programming
formulation (Case 2)
T

2
Max Z  Ap  Bp   ct xt  ht I t 
 t 1

s.t
I t  I t 1  xt  at  bt p t
xt  Rt t
xt , I t , p  0
if pt  pt then
T
 (a
t 1
t
T
 bt p) p  p  at  p
t 1
T
2
b
t 1
t
 Ap  Bp 2

12. A Mathematical Programming
formulation (Case 3)
Discrete prices
implies a given
set of possible
prices in all
periods to
choose from.
pit : Price alternative i, i   ,  I in period t
1
1 price i is chosen in period t
 it  
0 otherwise
d it  at  bt pit
 it  d it pit
then the revenue can be expressed by :
I
 
i 1
it it
t
and to secure only one picked price :
I

i 1
it
 1 t

13. A Mathematical Programming
formulation (Case 3)
 I

Max Z     it  it   ct xt  ht I t
t 1  i 1

s.t
T
I
I t  I t 1  xt   d it  it t
i 1
I

i 1
it
 1 t
xt  Rt t
xt , I t  0
 it   ,0
1

14. A Mathematical Programming
formulation (Case 4)
T
Max Z   (at  bt pt ) pt  ct xt  ht I t
t 1
s.t
I t  I t 1  xt  (at  bt pt ) t
xt  Rt t
xt , I t , pt  0

15. Solving some cases:
• Case data:
– T=4
– Rt=20  t
– ct=20  t
– ht=2  t
– Demand functions (see individual case presentations)

16. Case 1 (Cost minimization)
• Only one given price (p* = 95)‫‏‬
• Constant capacity constraint (Rt = 20 t)‫‏‬
* = 4294,5

17. Case 2: (max  – single price)
• Only one (but variable) price (p* = 88)‫‏‬
• Constant capacity constraint (Rt = 20 t)‫‏‬
* = 5658

18. Case 3: (Max  discrete prices - 3 options 90, 80, 70)‫‏‬
• Optimal discrete prices - p*=[90, 70, 80, 80]‫‏‬
• Constant capacity constraint (Rt = 20 t)‫‏‬
* =5973,33

19. Case 4: (Max  - full continuous dynamic pricing)
• Optimal prices - p*=[88, 66.29, 77.79, 92.74]‫‏‬
• Constant capacity constraint (Rt = 20 t)‫‏‬
* =6154,33

20. Summing up
Case 1
Case 2
Case 3
Case 4
*
4294.5
5658
5973.33
6154.33
%
-
31.75%
5.57%
3.03%
% - Total
-
-
-
43.31%
7000
6000
5000
4000
Series1
3000
2000
1000
0
Case 1
Case 2
Case 3
Case 4

21. Some discussion topics
• How does the effects of Dynamic pricing relate
to:
– Demand time variability?
– Size of storage costs?
– Size of Capacity Constraints?
– What risks will a manufacturer face by applying
Dynamic Pricing in a non-monopolistic
(oligopolistic) environment?
– Does Dynamic Pricing in a Perfectly Competitive
market make sense?
Use the DPD (Dynamic Pricing Demonstrator) to discuss these questions!

22. Some extensions:
•
•
•
•
•
Multiple products
Set-up costs (and times)
Marketing
Uncertainty
Gaming (competition among Dynamic Pricer’s)

23. Lectures – Dynamic Pricing
Part II – Research Material
BRNO – 2008
By
Kjetil K. Haugen