Optimal Growth in Two-Sided Markets

We develop a theoretical model of optimal growth in two-sided markets. The model posits that market output (number of transactions) is a function of the stock of supply and demand. This market output is modeled using a homogeneous production function, which can have increasing or decreasing returns to scale. The supply and demand stock levels follow a growth model in which the rate of growth at each point in time is a function of both the surplus each side of the market receives and the attrition of supply and demand (supply and demand lifetimes). The surplus can be apportioned between the two sides of the market by changing the price paid to sellers and the price charged to buyers, which we assume the platform controls. Through these price levers, the platform can pay subsidies to one or both sides of the market. We investigate the behavior of optimal market growth, including the point at which the market becomes self-sustaining and the long-run optimal size of the market. We characterize the optimal balance between supply and demand as the market size grows and determine optimal subsidy policies that maximize discounted total profit. For the case of both increasing and decreasing returns without price constraints, we show the optimal policy is to grow using an impulse of subsidy spending (a subsidy shock) to move the market immediately to its optimal long-run size. This result is consistent with the race to growth observed in many two-sided markets like ride-sharing.