Search at the Margin

Abstract

We formulate and solve dynamic programming models extending search theory to (1) multiple indivisible units and (2) perfectly divisible assets. Buyers arrive seeking randomly many units at a random price in (1), or with a random limit order in (2). The seller can partially exercise orders - hence, search at the margin.

The optimal selling strategy adjusts as the asset position falls, reflecting the endogenous holding costs: i.e. the opportunity cost of delaying optionality for the inframarginal units. This depresses reservation prices at the margin. For instance, model (1) subsumes the wage search model, but we inductively argue that the seller is willing to accept less and less money for each additional unit he sells.

While we use induction for the indivisible units model (1), our analysis of continuously divisible assets in (2) exploits recursion and contraction methods. We characterize three derivatives of the Bellman value function. It is firstly increasing and strictly concave. Thus, the seller takes greater advantage of more generous offers, and his marginal value shifts up as he unwinds his position, making him less willing to trade. Next, with a falling purchase cap density, the marginal value is strictly convex, and so the seller's supply response is less elastic at higher prices.

Our model is readily amenable to price-quantity bargaining. We show that greater buyer bargaining power has the same effect as greater search frictions.

Keywords: Search Theory, Dynamic Programming

JEL Classification: D83, G14