Mixed Bundling of Two Independently Valued Goods

This paper develops analytical results and insights for the mixed bundling problem of pricing a product line consisting of two component goods (with valuations distributed uniformly and independently in [0, a(1)] and [0, a(2)], respectively, with a(1) < a(2)) and a bundle of the two goods. This setting has previously been considered analytically intractable. By deriving component good prices as exact algebraic functions of the optimal bundle price, I reduce the multiproduct pricing problem to a univariate nonlinear optimization problem in the bundle price. When the two component goods are sufficiently asymmetric in valuations, the optimal solution is partial mixed bundling (one component good is not sold separately), for which I derive exact conditions and optimal prices. An exact analytical solution is also given for mixed bundling of two information goods (component goods with zero marginal costs). For the general case, I derive a closed-form approximation of the optimal bundle price: P-B = 0.5724w(1) + 0.5695w(2) + 0.3516a(1) + 0.4889a(2) + 0.0054(a(2)/a(1))w(1)-0.0201(a(2)/a(1))w(2). The approximation is highly 'accurate (the resulting profit is within a percent of the exact optimal profit), efficient (instant solutions for specific problem instances), and useful in other applications where mixed bundling is a subproblem. I demonstrate this by applying the solution to the mixed bundle problem in a vertical channel where a retailer must bundle and price component goods from multiple independent manufacturers.