Economics

Econ 441 Homework 5 (for grad students)

1 Question 1

Consider the model of intermediate surge pricing. Suppose passengers’ valuation distribution is uniform,

Fb(v) = v, distributed on [0, 1], and the drivers’ cost distribution is uniform, Fs(c) = c, distributed on

[0, 1]1, and the mass of buyers is µL = 1 or µH = 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 vH,vL and optimal driver compensations

cH,cL?

2 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]

0.5qH ( F−1b (1 −

qH µH

) −F−1s (qH ) )

+ 0.5qL ( F−1b (1 −

qL µL

) −F−1s (qL) )

subject to vH − cH vH

= vL − cL vL

where vH = F −1 b (1 − qH/2),cH = F

−1 s (qH ), and vL = F

−1 b (1 − qL),cL = F

−1 s (qL).

2

1The distribution of passengers’ valuation gives you, for each v ∈ [0, 1], the proportion of passengers whose valuation of a ride is below v. For example, if the fee is 0.7, then the proportion of passengers whose valuation of a ride is below 0.7

is 70 percent since Fb(0.7) = 0.7. In other words, 30% of the passengers will be willing to pay the fee 0.7 to get a ride.

Likewise, if the driver’s compensation is 0.4, 40% of the drivers will find it worthwhile to drive for Uber since Fs(0.4) = 0.4 2Hint: solutions of both questions can be found in Example 1 and 2 of the paper Intermediated Surge Pric-

ing by S. Bikhchandani, but the paper skips some intermediate steps. Fill in the details. After simplify-

ing the KT conditions, you may want to use an equation solver such as https://www.symbolab.com/solver/

non-linear-system-of-equations-calculator to compute the answer.

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