Optimal Solutions for Mixed Bundling of Two Independently-Valued Goods

Abstract

This paper develops analytical results and insights for the mixed bundling problem of pricing a product line consisting of two component goods and a bundle of the two goods. Consumer valuations for the two goods follow a symmetric bivariate uniform distribution, a setting which has previously been considered analytically intractable. Prior analytical work has demonstrated the superiority of mixed bundling, but failed to provide an explicit specification of the optimal solution or prices. Alternately, researchers have employed numerical solution techniques to solve instances of the mixed bundling problem. I reduce the multi-product pricing problem to a univariate nonlinear optimization problem in the bundle price with a global maximum that can be identified uniquely. I derive tight lower and upper bound functions for the optimal bundle price. The average of these two functions can be used as an alternate, and extremely simple, analytical expression for the optimal bundle price. Another alternative formula for the optimal bundle price is obtained by applying statistical estimation over solutions for a large space of problem instances. Both approximations are highly accurate, producing a pro t that is within half percent of the exact optimal pro t. The optimal bundle price has an elegant analytical expression (a linear function of the marginal costs of the two goods) and prices for component goods are an algebraic function of the bundle price. With a bivariate distribution in the unit square and constant unit marginal costs w1 and w2, the optimal bundle price can be written as 0:86032 0:55929(w1 w2)/2.