Do Buyer-Size Discounts Depend on the Curvature of the Surplus Function?

1. Do Buyer-Size Discounts Depend on the Curvature of the Surplus Function? Experimental Tests of Bargaining Models Hans-Theo Normann University of London Bradley Ruffle Ben-Gurion University Christopher Snyder George Washington University

2. Introduction • Common approach to understanding large-buyer discounts: • buyers of various sizes bargain with a supplier; large buyers • “better bargainers”. • Theoretical bargaining literature suggests nonlinearities in • surplus function determine buyer-size discounts • Horn & Wolinsky (1986) oInderst and Wey (2003) • Stole and Zwiebel (1996a, 1996b) oRaskovich (2003) • Chipty & Snyder (1999) • Concave surplus function  large-buyer discounts • Linear surplus function  no discounts • Convex surplus function  multiple equilibria possible, • including no discounts.

3. A. Concave W(Q) A B Q q q B. Linear W(Q) A B Q q q C. Convex W(Q) A B Q q q Figure 1: Total Surplus Functions with Different Curvatures

4. Experimental Design S S S LB LB LB SB SB SB SB SB SB LB need for two units, value 100 each need for one unit, value 100 SB S various cost functions IMC, CMC, DMC • Bargaining occurs through posted bid market. Buyers post • bids; seller accepts or rejects. • 60 rounds per session. • 12 subjects per session. • 6 sessions. • Shuffle buyers among sellers, sizes, seller cost functions. • Shuffle sellers among buyers, cost functions. • 1,000 points = 1 £.

5. IMC Marginal Cost CMC DMC Quantity Figure 2: Marginal Cost Functions in Different Treatments

6. IMC CMC Total Surplus DMC Quantity Figure 3: Total Surplus Functions in Different Treatments

7. Test theory in experimental laboratory markets. Have seller bargain simultaneously with buyers of different sizes, and vary curvature of the surplus function. • Why might theory be wrong? • Press suggests large-buyer discounts are ubiquitous, unconditional. • Alternative theories with risk aversion (Chae and Heidhues 1999a, 1999b; De Graba 2003) suggest unconditional large-buyer discounts. • Experimental game involves a lot of strategic uncertainty (simultaneous bids of other buyers and seller’s response) • Buyers may not regard themselves as marginal in just the right way.

8. Theoretical Predictions • Simpler bargaining game, with buyers making take-it-or- • leave-it offers, than any of the theoretical papers to simplify • experimental design. • Propositions 1-3 provide general results for this model. • Predictions for parameters used in experiment given in • Figure 4.

9. 450 Small-Buyer Discounts IMC DMC Average Large-Buyer Bid Per Unit CMC Large-Buyer Discounts Average Small-Buyer Bid Per Unit Figure 4: Equilibria for Experimental Parameters

10. Large Buyer Small Buyer A. IMC Treatment Frequency B. CMC Treatment Frequency C. DMC Treatment Frequency Buyer Bid Figure 5: Histograms of Large- and Small-Buyers’ Bids, Last 30 Rounds

11. IMC, Small Buyer IMC, Large Buyer CMC, Small Buyer CMC, Large Buyer DMC, Small Buyer DMC, Large Buyer Probability Seller Accepts Bid Buyer Bid Figure 6: Seller Acceptance Decision from Probit Estimates

12. IMC, Small Buyer IMC, Large Buyer CMC, Small Buyer CMC, Large Buyer DMC, Small Buyer DMC, Large Buyer Expected Buyer Profit Per Unit Buyer Bid Figure 7: Optimal Buyer Bid Given Others’ Experimental Behavior

13. Conclusions • Consistent with theory, significant large-buyer discounts in • IMC treatment: 12 percent for all bids, 14 percent for • accepted bids. • Consistent with theory, no large-buyer discounts in CMC • treatment. • No large-buyer discounts in DMC treatment (consistent with • theory, but theory has multiple equilibria possible). • Consistent with theory, bids higher in IMC than other • treatments. • Seller more likely to reject small-buyer bids in IMC • treatment; no size difference for CMC, DMC treatments. • Small buyer bids in IMC lower than theory predicts; average • bids in CMC and DMC higher than theory predicts (but not • higher than ultimatum experiments typically find).