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?