1. Pricing with Markups in Competitive Markets with Congestion Nicolás Stier-Moses, Columbia Business School Joint work with José Correa, Universidad Adolfo Ibañez INFORMS Pricing and Revenue Management Montreal, June 2008

2. Pricing with Markups — N. Stier-Moses 2 Motivation ● Since early commitment may limit the possibilities to adjust, in many industries producers or service providers don’t decide price or quantity until time of execution ● Price functions model that producers postpone decisions until demand is realized ● This procedure is more robust to uncertainty ● Examples: electricity, airlines, consulting, …

3. Pricing with Markups — N. Stier-Moses 3 Electricity Markets

4. Pricing with Markups — N. Stier-Moses 4 Overview of Game We consider a pricing game with two stages: 1.Producers learn their cost functions 2.Producers decide their price functions playing a non-cooperative perfect information game 3.Consumers learn price functions of producers 4.Consumers decide from who to buy playing another perfect information game

5. Pricing with Markups — N. Stier-Moses 5 Main Conclusion As long as there is enough competition: Price competition does not lead to big distortion of production costs Hence, consumer-producer assignment is not far from optimal one

6. Pricing with Markups — N. Stier-Moses 6 Questions / Plan for Today ● Does there exist an equilibrium for prices? ● Is it unique? ● How can we compute it? ● What producer charges more at equilibrium? ● How much are costs distorted because of the competition? ● How efficient is the resulting assignment compared to one in which consumers know production costs?

7. Pricing with Markups — N. Stier-Moses 7 Producers: cost side ● Marginally increasing average per-unit production cost that depends on quantity x a ● Different producers have same “shape” but they differ in their efficiency parameter c a ● Total cost of producer is c a x a p(x a ) where p(x)=x for all producers (linear prices) ( but other functions p(x) are possible too )

8. Pricing with Markups — N. Stier-Moses 8 Increasing Marginal Costs ● For electricity markets, it’s been shown empirically that linear production (and price) functions are good approximations (Baldick, Grant & Kahn’04) ● More generally, this model can be used in industries with increasing marginal costs Standard assumption in economics. E.g.: –More demand implies more willingness to pay –Capacities –Labor: additional shifts cost more (regular hours → overtime → temps)

9. Pricing with Markups — N. Stier-Moses 9 Producers: revenue side ● No fixed price or quantity contracts ● Per-unit price of producers depends on total production quantity x a price function = cost function + markup ● We consider that price and production functions have same “shape” (→ constant markups) ● Producers decide markup, then: average per-unit price for x a =  a c a p(x a )

10. Pricing with Markups — N. Stier-Moses 10 Consumers ● In this talk we look at perfect competition (but other market structures are possible too) ● Consumers are price takers and buy at lowest price ● Total demand is inelastic (consumers buy regardless of the price) ● Consumers know producers’ price functions but not their composition

11. Pricing with Markups — N. Stier-Moses 11 Solving the 2 nd Stage of the Game ● An equilibrium satisfies that for all producers i and j such that i is active ● Solving the game, producer a sells ● The profit of producer a is

12. Pricing with Markups — N. Stier-Moses 12 Best Response Function ● Maximizing the profit function leads to an optimal markup  a which can be characterized by the equilibrium condition: ● Equivalently, marginal pricemarginal costmarkup

13. Pricing with Markups — N. Stier-Moses 13 Equilibria of the Game ● Doing iterated best responses produces increasing markups at successive rounds ● If markups are bounded, sequence converges and limit has to be an equilibrium ● Equilibrium is unique when it exists ● It exists iff there are 3 or more producers Intuition: not enough competition with two producers so iterated best responses produces unbounded markups

14. Pricing with Markups — N. Stier-Moses 14 Characterization of Equilibrium ● We want to characterize equilibrium to get algorithm and to understand efficiency ● Observation: scaling costs with constant doesn’t change anything ● Assume w.l.o.g. to simplify best response ● From there one can solve for markup ● Formula implies: more efficiency → larger markup

15. Pricing with Markups — N. Stier-Moses 15 Algorithm to Find Equilibrium ● Condition in previous slide can be written as (  has no solution if 2 producers → no equilibrium) ● Use binary search to find correct  ● Using scaled costs, we can now find markups (formula in previous slide) and assignments with

16. Pricing with Markups — N. Stier-Moses 16 Efficiency of Equilibrium ● Social cost of assignment x measures total production cost: (payments are internal transfers so they vanish) ● Optimal assignment is and its cost is

17. Pricing with Markups — N. Stier-Moses 17 Competitiveness among Producers ● Competitiveness level in a market is (hence, ) ● Equivalently, c a   for all producers a ● When competitiveness is , markups satisfy:

18. Pricing with Markups — N. Stier-Moses 18 Worst-Case Inefficiency of NE The worst instance among  the  -competitive is:

19. Pricing with Markups — N. Stier-Moses 19 Inefficiency as a Function of Competitiveness 

20. Pricing with Markups — N. Stier-Moses 20 Conclusions ● Competition limits markups applied by producers ● Extensions to arbitrary-degree polynomials ● Extensions to more general competitive structures For example, most of this works for vertical and horizontal competition together ● Key question: is this robust when producers can use price functions with other “shapes”?

21. Pricing with Markups — N. Stier-Moses 21 The End

22. Pricing with Markups — N. Stier-Moses 22 Supply Chain Management