Determining the model of intermediate surge pricing

Question

Assignment:

Question 1

Consider the model of intermediate surge pricing. Suppose passengers' valuation distribution is uniform, F_b(v) = v, distributed on [0, 1], and the drivers' cost distribution is uniform, F_s(c) = c, distributed on [0, 1]¹ , and the mass of buyers is µL = 1 or µ_H = 2, corresponding to low demand and high demand times respectively. Assume the intermediate firm has no marginal cost. Solve Uber's unconstrained profit maximization problem. What are the optimal passenger fees v_H, v_L and optimal driver compensations cH, c_L?

Question 2

Consider the model of intermediate surge pricing. The distribution functions are as above, but Uber is constrained to maintain a fixed percentage it can take from passenger fees no matter whether the demand is high or low. Suppose half of the time the demand is high and half of the time the demand is low. Solve Uber's constrained profit maximization problem:

max _{qH,qL∈[0,1]} 0.5_qH (F_b^-1  (1 - qH/µH ) - F_s^-1 (qH)
+ 0.5qL (F_b^-1 (1 - qL/µL) - F_s^-1 (qL) )

subject to v_H - c_H/v_H  = v_L - c_L/v_L

where v_H = F_b^-1 (1 - qH/2), c_H = F_s ^-1 (qH), and v_L = F_b^-1 (1 - qL), c_L = F_s^-1 (qL)².