Dynamic Pricing with Risk Analysis and Target Revenues

Baichun Xiao Long Island University

Outline

Risk-neutral- a basic assumption of most RM models; its inability to deal with short-term behavior;

Literature review; What affects short-term risk? A RM model with a target revenue and

penalty function; Concluding remarks.

Risk Neutral Assumption of RM Models

Decision makers are risk neutral; i.e., all models attempt to maximize the expected revenues at the end of the disposal period;

Optimal in the long run (the law of large numbers): no single realization has the potential for severe revenue impacts on the company;

Not necessarily the best option in the short run.

Risk-Neutral May Not Apply to Short-Term

Compelling reasons for concerning short-

term revenues: Financial constraints; Uncompromised revenue goals or

minimum probability of achieving these goals;

Shareholders’ requirements; All of the above are escalated by the

perishability of products.

Risk with Short-term Revenues

Short-term revenues can swing drastically from their long-term estimates because:• Uncertainty of demand;• Forecast errors;• Speculations;• Unexpected capacity changes.

A single poor performance can be very damaging and may not be compromised by the long-term average.

Example One

20, =3, =20, =5.

(0, ) 295.75

Target revenue = 280

( 280) 53.4%

p M T

V M

P X

Practitioners’ Solutions

Sacrifice expected revenues in return for higher

probability of achieving a revenue goal. Liquidations; Clearances; Negotiated discounts; Favorable price for large-volume

demand.

Example Two

15, =5

306.6

, =20, =5.

(0, )

Target revenue = 280

(

6

9 %280) 0

M T

V M

P X

p

Theoretical Framework?

Current RM models are unable to explain why a dynamic control

policy leads to a steep dive of prices during liquidation periods;

unable to explain discount policies for large-volume demand.

Related Research

McGill and van Ryzin (1999): addressed importance and challenge of pricing group demand.

Bitran and Caldentey (2002): “essentially all the models that we have discussed assume that the seller is risk neutral.”

Feng and Xiao (1999): a risk-sensitive pricing model to maximize sales revenue of perishable products.

Related Research

Kleywegt and Papastavrou (1997), Slyke and Young (2000), Brumell and Walczak (2003): pricing group demand, but not from the perspective of risk.

Lim and Shanthikumar (2004): (i) a model built upon erroneous forecast parameters may perform badly and present a risk; (ii) robust dynamic pricing; (iii) equivalent to single product dynamic RM with exponential utility function without parameter uncertainty;

Related Research

Levin, McGill, and Nediak (2005): (i) motivated by inventory clearance of high-value items (automobiles, electronic equipment, appliances, etc.); (ii) permit control of the probability that total revenues fall below a minimum acceptable level; (iii) augment the expected revenue objective with a penalty term for the probability that revenues drop below a desirable level.

Related Research

Feng and Xiao (2005): (i) Maximize the risk-averse utility function instead of risk-neutral revenue; (ii) The risk-averse utility model retains monotone properties of the optimal policy; (iii) Risk-neutral models are special cases of risk-averse models; (iv) The risk-averse model explains behaviors that cannot be rationalized by the risk-neutral assumption; e.g., group discount policy.

How is Risk Measured?

Risk is normally measured by variance and standard deviation;

Variance and standard deviation are policy dependent;

Variance and standard deviation are affected by the remaining inventory and time-to-go;

Penalty function using the standard deviation is not a proper choice when the time-to-go is diminishing.

Risk is Affected by Policy

20 15

3 5

20 20

5 5

295.75 306.66

69.3 21.8

p

M

T

(0, )V M

Variance vs. Remaining Inventory

5, 15, 5, ,

5, 15, 5,

10 0.65

20 21.,

5,

76

4015, 74.775, ,

T p M

T p M

T p M

Variance increases with the remaining Inventory.

Variance vs. Remaining Inventory

A single-policy model ( ):

2 2

1 1

2

1

( ) ( )

! !

( )

: variance with items;

: expected revenue.

k T k Tn

nk n

n

nk

n

n

k

T e T eVAR k n

k k

P

VAR n

V

V k

1p

Variance vs. Remaining Inventory

11

2 21

11 1

1

1

11 1

( )(2 1)

!

( ) ( )

( )(2 1)

!

( ) ( ) ( 1)

0.

k T

n nk n

n n

n nk k

k T

k n

n n

n n nk k

T eVAR VAR n

k

P V k P V k

T en

k

P V k P V k P V n

Variance vs. Remaining Inventory

( )2

2( )

2

( , ) ( ) ( , 1)

( ) ( , 1)

s

it

i

s

it

i

dT

i iti

dT

i iti

VAR t n s V s n p e ds

s V s n p e ds

General cases (inventory control)

Variance and Time Remaining

Standard Deviation of Revenue (p=1,lambda=3, n=5)

0

0.5

1

1.5

2

1 5 9 13 17 21 25 29 33 37 41

Time Remaining

ST

D

Coefficient of Variation

Coefficient of Variation

0

2

4

6

8

10

12

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Time Remaining

ST

D/E

(X)

Risk of Selling Perishable Products

For a given policy, risk increases if the remaining inventory increases;

For a given policy, risk increases if the time-to-go diminishes;

Risk can be controlled by policy.

Alternatives for Reducing Risk

Expected revenue with a penalty function when the target revenue is not met;

Expected revenue with a penalty function when the probability of not achieving the target revenue is above a threshold;

Risk-averse expected utility function;

Distribution of Revenues

1

3

10

5

p

M

T

Distribution of Revenues

1

3

10

1

p

M

T

Distribution of Revenues

1

3

20

5

p

M

T

A Continuous-Time Pricing Model with Target Revenue

Assumptions: Management has a target revenue in the

short run; If the target is not met, a penalty is

incurred; The penalty is proportional to the deficit

of revenue; Management makes price decisions to

maximize the expected revenue with a penalty of not meeting the target.

Notations

1

0

= { , }, price set, for

( ) = demand intensity at given

= target revenue

( ) = remaining inventory at

( ) = revenue collected up to

= penalty coeffici

m i j

i i

p p p p i j

t t p

r

n t t

r t t

c

ent

= initial inventory

= end of sales period

M

T

Objective Function

optimal expected revenue at T given the remaining inventory and realized revenue at t be n(t) and r(t), respectively;

The revenue function has three parameters t, n, and r.

( , ( ), ( )) :V t n t r t

Boundary Conditions when t = T

0

0

0 0

( , , ) ( )

(1 )

V T n r r c r r

r r r

c r cr r r

Note: (i) Penalty is incurred if r < r0; (ii) Thepenalty function is piecewise linear.

Boundary Conditions when t = T

0

0 0

( , 1, ) ( , , )

(1 ) , ;

( ) ,

0.

i

i i

i i

V T n r p V T n r

c p r p r

p c r r r p r

Implication: sell remaining inventory in theneighborhood of T for any price.

Boundary Conditions when n = 0

0

0 0

, ;( ,0, )

(1 ) , .

r r rV t r

c r cr r r

Note: V(t, 0, r) is independent of t.

Boundary Conditions when n = 0

1 2

1 2 1 2 0

1 2 0 2 1 0 2

1 2 0 1 2

( ,0, ) ( ,0, )

, ;

( ) ( ), ;

(1 )( ), .

V t r V t r

r r r r r

r r c r r r r r

c r r r r r

For r1 > r2,

Optimality Condition

Only one price is accepted at any given time;

Optimal price is chosen from the price set P (may not be the highest price).

( , , )max ( )[ ( , 1, ) ( , , )] 0.i i i

V t n rt V t n r p V t n r

t

Optimality Condition

If pi is the optimal solution at t with realized revenue r, then

( ) ( , 1, ) ( ) ( , 1, )( , , ), ,

( ) ( )

( ) ( , 1, ) ( ) ( , 1, )( , , ), .

( ) ( )

i i j j

i j

i i j j

i j

t V t n r p t V t n r pV t n r i j

t t

t V t n r p t V t n r pV t n r i j

t t

Optimality Condition

Result: If then the optimal price in the neighborhood of T is the lowest price ;

The lowest price has the highest revenue rate.

mp

( ) ( ) ,i i j jt p t p i j

Solve V(t, 1, r)

In the neighborhood of T, ( , , )

( )[ ( , 1, ) ( , , )] 0m m

V t n rt V t n r p V t n r

t

leads to

( )( ,1, ) ( ) ( ,0, ) .

s

mt

T d

m mtV t r s V s r p e ds

Solve V(t, 1, r)

( )

0

( )

0 0

( ,1, )

( ) ( ) , ;

[(1 )( ) ] ( ) , .

s

mt

s

mt

T d

m m mt

T d

m m mt

V t r

r p s e ds r p r

c r p cr s e ds r p r

Solve V(t, 1, r)

Let

In the left neighborhood of , the optimal price becomes

Other thresholds are similarly defined.

,1

1 1

1

max{ | 0 } such that

( )( ) ( )( )( ,1, )

( ) ( )

m

m m m m

m m

z t t T

t r p t r pV t r

t t

,1mz

1mp

Solve V(t, n, r)

Assume has been obtained;

In the neighborhood of T, is the optimal price, V(t, n, r) is given by

( , 1, )V t n r

( )( , , ) ( ) ( , 1, ) .

s

mt

T d

m mtV t n r s V s n r p e ds

mp

Solve V(t, n, r)

Let

In the left neighborhood of , the optimal price becomes

Other thresholds are similarly defined.

,

1 1

1

max{ | 0 } such that

( ) ( , 1, ) ( )( , 1, )( , , )

( ) ( )

m n

m m m m

m m

z t t T

t V t n r p t t n r pV t n r

t t

,m nz

1mp

Numerical Experiment

Data:

0

P={10, 9, 8, 7, 6, 5}; 1

{0.05,0.07,0.09,0.11,0.13,0.14}, 1/ 2;

{0.06,0.08,0.09,0.10,0.11,0.12}, 1/ 2;

100, 2, 700.

T

t

t

M c r

Expected revenue with the target r0

r0 V(0, M, 0)500 738.93550 738.88600 738.24650 734.16700 718.27750 678.39800 609.03850 516.27900 416.81

Expected Revenue with the Target r0

Expected Revenue

400

450

500

550

600

650

700

750

800

400 500 600 700 800 900 1000r0

V(0

,M,0

)

V(t,n,r) is a decreasing function of r0.

Expected revenue as Function of c

c V(0, M, 0)1.0 728.531.2 726.471.4 724.411.6 722.361.8 720.312.0 718.272.2 716.232.4 714.192.6 712.15

Note: r0=700

Expected revenue as Function of c

Expected Revenue

705

710

715

720

725

730

1 1.5 2 2.5 3

c

V(0

,M,0

)

V(t,n,r) is a decreasing and linear function of c.

Expected revenue as Function of n

-600

-400

-200

0

200

400

600

0 20 40 60 80 100

V(600,n,300)

V(700,n,300)

V(t,n,r) is increasing and concave in n forfixed t and r. (r0=700)

Expected Revenue as Function of r

-800

-400

0

400

800

1200

0 100 200 300 400 500

V(600,50,r)

V(700,50,r)

V(t, n, r) is an increasing function of r for fixed t and n (r0=700).

Expected Revenue as Function of t

-600

-400

-200

0

200

400

600

800

1000

0 200 400 600 800 1000

0

1

2

3

4

5

6

7

8

9

10

V(t,50,300)

V(t,49,300)

Delta

V(t,n,r) is a decreasing and concave function of t for given n and r; but V(t,n,r)-V(t,n-1,r) may not decrease in t (r0=700).

Property of V(t,n,r+s) -V(t,n,r)

V(t,50,300+s)-V(t,50,300)

048

121620242832

0 200 400 600 800 1000t

s=4

s=6

s=8

s=10

( , , ) ( , , )V t n r s V t n r s

Concluding Remarks

Decision makers are risk-averse in financial market and many other areas, revenue management should not be an exception;

The proposed pricing model handles risk with a target revenue and a penalty function;

Many properties of risk-neutral models seem to hold except the marginal expected revenue;

Concluding Remarks

More structural properties of the value function need to be uncovered;

Whether pricing policy for group demand can be dealt with by the proposed model need to be explored.