Embed Size (px)
Citation preview
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!
