Econ 441 Homework 5 For Grad Students Question 1 ✓ Solved

Econ 441 Homework 5 For Grad Students1 Question 1consider The

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], 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?

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−1b (1 − qH/2), cH = F−1s (qH), and vL = F−1b (1 − qL), cL = F−1s (qL).

Paper For Above Instructions

The concepts of surge pricing and dynamic pricing models represent a significant evolution in the supply of services, particularly in sectors such as transportation, hospitality, and entertainment. This paper will focus on the model of intermediate surge pricing used by firms such as Uber and how they can optimize profits under various conditions. Specifically, we will break down the unconstrained and constrained profit maximization problems Uber faces in order to derive optimal passenger fees and driver compensations.

Unconstrained Profit Maximization

In the unconstrained profit maximization scenario, we consider a simple model where the distribution of passenger valuations and driver costs are uniformly distributed over the interval [0, 1]. The valuation distribution for passengers is given by F_b(v) = v, meaning that if a passenger is charged a fee of v, the proportion of passengers who value the ride is exactly that percentage. Therefore, if a fee of 0.7 is charged, then 70% of passengers would be willing to pay that amount for a ride.

On the driver's side, the cost of the drivers is also uniformly distributed, described by F_s(c) = c. Hence, if a driver is compensated at 0.4, then only 40% of drivers will choose to work for Uber, as they find it worthwhile given their cost structure.

For the purpose of this model, we define two demand scenarios: low demand (µ_L = 1) and high demand (µ_H = 2). The firm will maximize profit (π) by setting appropriate passenger fees (v_L, v_H) and driver compensations (c_L, c_H). The total profit can be expressed as follows:

π = (v_L - c_L) µ_L + (v_H - c_H) µ_H

The firm’s optimization leads us to establish the first-order conditions that need to be solved to find the optimal values for v_L, v_H, c_L, and c_H.

Optimal Fees and Compensations

To find the optimal values, we can analyze the constraints and parameters based on demand levels. For the high-demand scenario:

The optimal passenger fee (v_H) can be expressed as follows:

v_H = F_b(1 - q_H/2)

And the optimal driver compensation for high demand is:

c_H = F_s(q_H)

Similarly, for low demand, we have:

v_L = F_b(1 - q_L)

c_L = F_s(q_L)

By applying uniform distributions and solving through these expressions, we determine the optimal solutions for v and c under both demand conditions. These calculations involve algebraic simplifications and solvers for non-linear systems as hinted in the prompt.

Constrained Profit Maximization

In the constrained profit maximization case, Uber is restricted in the percentage it can take from passenger fees. This adds a layer of complexity to the optimization problem as it must maintain a fixed relationship between the fees and the compensations over varying demand situations.

The optimization function is set up as follows:

max q_H, q_L ∈ [0, 1] 0.5q_H (F_b(1 - q_H µ_H) - F_s(q_H)) + 0.5q_L (F_b(1 - q_L µ_L) - F_s(q_L))

This is subject to the constraint: v_H − c_H / v_H = v_L − c_L / v_L.

To solve this, we need to analyze how the fixed percentage influences both passenger fees and driver compensations across the two demand scenarios. The given conditions imply a relationship between the demand responses and how Uber can price its services accordingly.

Conclusion

Through the examination of both unconstrained and constrained frameworks of surge pricing in the Uber model, we can derive substantial insights into profit optimization strategies. By understanding the interplay between passenger fees, driver compensations, and demand fluctuations, Uber can more effectively manage its pricing mechanism to enhance profitability while considering market dynamics. Future developments in this realm will focus on either improvements in modeling consumer behavior or technological enhancements to optimize pricing algorithms in real-time scenarios.

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