The Search for Liquidity

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We formulate and solve a dynamic programming exercise that extends search theory to perfectly divisible assets, leading us to a theory of search at the margin. Our showcase application is the liquidation sale of a divisible asset --- such as the time of an accountant, or a physical asset like recently inherited land. A seller faces a stream of buyers periodically arriving with random limit orders. His trading behavior optimally adjusts as the asset position falls, reflecting the endogenous time-varying value of the asset position. Our search theory story adds a different dimension to liquidity: the waiting time between trades.We shift the focus from Bellman values to Bellman value functions in dynamic search. We recursively characterize three derivative properties of the Bellman value. First, even though dividend payoffs rise linearly in the position, so long as selling opportunities are individually bounded in size, there is “diminishing returns to optionality”: The value function is strictly concave and 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. Second, if the density of purchase caps is falling, the marginal value of assets is strictly convex. Hence, the seller's supply response is less elastic at higher prices, and that the depth of the induced market rises in the asset position. The asset value function therefore shares standard properties of the utility for money: it is increasing, strictly concave, and the marginal value is strictly convex in the asset position.Our model is amenable to price-quantity bargaining. We show that greater buyer bargaining power is tantamount to greater search frictions. Supply is higher and the negotiated prices lower with bargaining, and thus the induced market is more liquid.