Why are Limit Orders Always Linear?

Abstract

A number of results in liquidity asset pricing and market microstructure depend on the assumption that, when investors trade strategically, recognizing limited market participation (i.e. thin markets), trade volume impacts prices linearity with respect to order size. The assumption is usually made explicitly or implicitly in the traders' linear investment demand schedules or limit order books. Yet, in the reviewed literature, this assumption on investment demand is always imposed, rather than derived from investors' endowments and preferences for risk. In many cases, the result is that the lack of market depth has little or no impact on asset prices. We first show that the validity of these results depends strongly on the linearity assumption, as we demonstrate that investment demand linearity is equivalent to investors being homogenously price-sensitive regardless of endowment heterogeneity. We then eliminate the need for the assumption, endogenizing investment demand schedules from risk preferences instead, and show that, in the case of CARA investors facing a stream of normally distributed instantaneous dividends, the equilibrium demand is necessarily linear on prices, and thus that investors, regardless of differences in endowments, are all equally price-sensitive. This result reconciles the theory with both economic intuition and with empirical studies on price impact functions.