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

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. u i (x) = w i 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.

Problem Formulation System Problem: total user utility maximization subject to. User’s Problem: surplus maximization subject to. Provider’s Problem: revenue max. subject to.

Optimal Pricing: Logarithmic Utility Logarithmic utility function: Single-bottleneck case: Multi-bottleneck case:

Elasticity Demand-price elasticity: Utility-bandwidth elasticity: For a surplus-maximizing user:

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.

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

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