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Talking about Dynamic Pricing in Brno.
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Lectures – Dynamic PricingPart I – Background Material
BRNO – 2009
By
Kjetil K. Haugen
The classic linear Monopoly model• Assumptions: (linear
demand and linear and proportional costs)c(Q)=cQ
P=a-bQ
• a,b > 0 Normal good
• Profit function:(Q)=R(Q)-C(Q)
=PQ-cQ
=(a-bQ)Q -cQ
Demand curve plot
Q
P
a
ab
Monopoly solution
• Analytically: (Max )
• Graphically:
b
caQ
bQca
Q
2
02)(
0)('
*
*
MCMR
QMCQMR
Q
QcQRQ
0)()(
0)('
)()()(
cQcMC
bQaQRMR
bQaQQbQa
QQPQR
)('
2)('
)(
)()(
2
• MR and MC:
Q
P
a
ab
a2b
MR
Demand
c
MC
QM
PM
Perfect Competition
• Monopoly:– 1 producer
– consumers
• Perfect competition– producers
– consumers
– Q give no P
MCP
MCPQ
cQQPQ
0)('
)(
Q
P
a
ab
a2b
MR
Demand
=c
MC
QM
PM
QPC
PPC
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
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?
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
A Mathematical Programming formulation (NLP)
0,,
)(
.
)(
1
1
ttt
tt
ttttt
tttt
T
t
ttt
pIx
tRx
tpxII
ts
IhxcppZMax
f
f
Xt : Amount produced in period tIt : Amount stored between periods t-1, tpt : Price set in period t
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)
A Mathematical Programming formulation (Case 1)
0,
.
1
1
tt
tt
tttt
tttt
T
t
Ix
tRx
tdxII
ts
IhxcZMin
A given constant price pt=p0 t leads to a constant term in the objec-tive. Maximising a constant minus ”something” is the same as minimis-ing ”something”. Then defining ft(p
0)=dt yields the above LP-model.
A Mathematical Programming formulation (Case 2)
0,,
.
1
1
2
pIx
tRx
tpbaxII
ts
IhxcBpApZMax
tt
tt
ttttt
tttt
T
t
T
t
T
t
tttt
t
BpApbpapppba
tpp
1 1
22T
1t
)(
then if
A Mathematical Programming formulation (Case 3)
Discrete prices implies a given set of possible prices in all periods to choose from.
I
1i
1
1
:price picked oneonly secure toand
:by expressed becan revenue then the
otherwise0
periodin chosen is price1
periodin ,1, ealternativ Price :
t
t
pd
pbad
ti
tIiip
it
it
I
i
it
ititit
itttit
it
it
A Mathematical Programming formulation (Case 3)
0,1
0,
1
.
1
1
1
1 1
it
tt
tt
I
i
it
itit
I
i
ttt
tttt
T
t
I
i
itit
Ix
tRx
t
tdxII
ts
IhxcZMax
A Mathematical Programming formulation (Case 4)
0,,
)(
.
)(
1
1
ttt
tt
tttttt
tttt
T
t
tttt
pIx
tRx
tpbaxII
ts
IhxcppbaZMax
Solving some cases:
• Case data:
– T=4
– Rt=20 t
– ct=20 t
– ht=2 t
– Demand functions (see individual case presen-tations)
Case 1 (Cost minimization)
• Only one given price (p* = 95)
• Constant capacity constraint (Rt = 20 t)
* = 4294,5
Case 2: (max – single price)
• Only one (but variable) price (p* = 88)
• Constant capacity constraint (Rt = 20 t)
* = 5658
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
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
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%
0
1000
2000
3000
4000
5000
6000
7000
Case 1 Case 2 Case 3 Case 4
Series1
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!
Some extensions:
• Multiple products
• Set-up costs (and times)
• Marketing
• Uncertainty
• Gaming (competition among Dynamic Pricer’s)
Lectures – Dynamic PricingPart II – Research Material
BRNO – 2008
By
Kjetil K. Haugen
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