Uber posts a price ${p}$ per ride and keeps a commission ${\alpha}$ on the price. Suppose Uber is the only ride matching service in town. If ${D(p)}$ is the demand function for rides at per ride price ${p}$ and ${S(w)}$ is the supply curve for drivers at wage ${w}$ per ride, Uber must choose ${\alpha}$ and ${p}$ to solve the following:

$\displaystyle \max_{\alpha, p} \alpha p D(p)$

subject to

$\displaystyle D(p) \leq S((1-\alpha)p)$

The last constraint comes from the assumption that Uber is committed to ensuring that every rider seeking a ride at the posted price gets one.

Suppose, Uber did not link the payment to driver to the price charged to rider in this particular way. Then, Uber would solve

$\displaystyle \max_{p,w} pD(p) - wS(w)$

subject to

$\displaystyle D(p) \leq S(w)$

The first optimization problem is clearly more restrictive than the second. Hence, the claim that Uber is not profit maximizing. Which raises the obvious puzzle, why is Uber using a revenue sharing scheme?