Elasticity Considerations for Optimal Pricing of Networks Murat Yüksel and Shivkumar Kalyanaraman Rensselaer Polytechnic Institute, Troy, NY {yuksem, shivkuma}

Elasticity Considerations for Optimal Pricing of Networks

Murat Yüksel and Shivkumar KalyanaramanRensselaer Polytechnic Institute, Troy, NY

{yuksem, shivkuma} @ecse.rpi.edu

Outline

Literature Problem formulation Optimal pricing: logarithmic utility Elasticity:

utility-bandwidth elasticity demand-price elasticity

Optimal pricing: non-logarithmic utility Summary

Literature Network optimization by pricing: The problem: maximization of total user utility Kelly et al. divided the problem into two sub-

problems: User’s surplus maximization Provider’s revenue maximization

For logarithmic user utilities (i.e. ui(x) = wi log x), Kelly showed that the system will reach an equilibrium by using prices as Lagrange multipliers.

Then, Low et al. generalized the concepts to users with concave (but not necessarily logarithmic) utility.

We investigate effect of user’s elasticity on optimal pricing strategies.

f

ffU )(max

Problem Formulation

System Problem: total user utility maximization subject to .

User’s Problem: surplus maximization subject to .

Provider’s Problem: revenue max. subject to .

llFff c

)(

llFff c

)(

Optimal Pricing: Logarithmic Utility

Logarithmic utility function: Single-bottleneck case:

Multi-bottleneck case:

)log()( xwxu ii

c

wp Ff f

)( )(

)(

lFf fLk k

lFf f

l c

wp

Elasticity Elasticity

Demand-price elasticity:

Utility-bandwidth elasticity:

For a surplus-maximizing user:

.1 where,1

1

.1,0 where,1

1

pxuxpxux

)('})({max

dp

pdX

pX

p

pdp

pXpdX )(

)(/

)(/)(

dx

xdu

xu

x )(

)(

appX )(

bxxu )(

Elasticity (cont’d)

Optimal Pricing: Non-logarithmic Utility

Non-Logarithmic utility function: Since , .

Single-bottleneck case:

Multi-bottleneck case:

Simply estimate and calculate prices accordingly..

Be more conservative in capacity, if more elasticity.

10 where

,)(

xwxu ii

|/1|||

c

wp Ff f

|/1|

)( )(

)(

||

lFf fLk k

lFf f

l c

wp

10 1

Summary We investigated effect of user’s

elasticity in pricing. Also, we identified demand-price and

utility-bandwidth elasticity. We addressed how should user’s

elasticity to price and bandwidth effect pricing strategy.

We observed that pricing strategy should be more conservative on capacity if user’s elasticity is higher.

Future work: Development of a distributed pricing

algorithm

Questions, Ideas?

THANK YOU!