To aid in the description and estimation of the tremendous recent growth in the collaborative economy, we provide a model for the dynamic sharing, subject to fixed costs. The sharing economy comprises a set of infinitely lived, heterogeneous suppliers, who take recurring decisions about entering or leaving the market. We provide a closed-form solution for the nonlinear evolution of the equilibrium in the sharing economy, typically resulting in an S-curve diffusion pattern. In general, the sharing economy can evolve in a nonmonotonic way, with downward adjustments followed either by a steady state or by a positive diffusion towards a steady state. The conversion costs produce a decision hysteresis for potential sharers. Unless these costs are so large that the economy rests in steady state (almost) immediately, the adjustment process does not converge in finite time.