Question at position 8 Section 2: (Hard) Suppose that on weekends 50% of consumers have a demand curve for rides of P = 6 - 0.25Q (“high demand consumers”), and the remaining 50% of consumers have a demand curve for rides of P = 5 – 0.25Q (“low demand consumers”). Because we cannot distinguish these low demand consumers and high demand consumers at the entrance gate, we will be charging the same entry fee (T) and price per unit (P) for both types of consumers. What is the best mix of entry fee (T) and price per ride (P) to maximize profits on weekends?P = MC, T = all CS of high demand customers at that price(none of the choices listed here)P = Monopoly price for low demand customers, T = all CS of of low demand customers at that priceP MC, T = all CS of high demand customers at that priceP = MC, T = all CS of low demand customers at that priceP MC, T = all CS of low demand customers at that priceP = Monopoly price for high demand customers, T = all CS of high demand customers at that price

Question

Question at position 8 Section 2: (Hard) Suppose that on weekends 50% of consumers have a demand curve for rides of P = 6 - 0.25Q (“high demand consumers”), and the remaining 50% of consumers have a demand curve for rides of P = 5 – 0.25Q (“low demand consumers”). Because we cannot distinguish these low demand consumers and high demand consumers at the entrance gate, we will be charging the same entry fee (T) and price per unit (P) for both types of consumers. What is the best mix of entry fee (T) and price per ride (P) to maximize profits on weekends?P = MC, T = all CS of high demand customers at that price(none of the choices listed here)P = Monopoly price for low demand customers, T = all CS of of low demand customers at that priceP > MC, T = all CS of high demand customers at that priceP = MC, T = all CS of low demand customers at that priceP > MC, T = all CS of low demand customers at that priceP = Monopoly price for high demand customers, T = all CS of high demand customers at that price

BlackTom AI · Accepted Answer

We start by restating the scenario and all answer choices clearly so we can evaluate each option.

Question setup:
- Weekend demand curves: high-demand consumers: P = 6 − 0.25Q; low-demand consumers: P = 5 − 0.25Q.
- 50% of customers are high-demand, 50% low-demand. We cannot distinguish at entry, so everyone faces the same entry fee T and the same ride price P.
- Profit equals revenue from rides (P times total rides) plus entry fees collected, minus any costs (MC). In these options, some choices set P equal to MC and set T to capture consumer surplus (CS) of one or both groups.
- For a given P, quantities demanded are:
  Q_high = (6 − P)/0.25 = 24 − 4P
  Q_low  = (5 − P)/0.25 = 20 − 4P
  Total Q = Q_high + Q_low = 44 − 8P.
- Consumer surplus for a linear demand curve with intercept A and slope −0.25 can be computed as CS = 0.5 × (A − P) × Q, where Q is the quantity demanded at price P.
  Hence:
  CS_high(P) = 0.5 × (6 − P) × (24 − 4P)
  CS_low(P)  = 0.5 × (5 − P) × (20 − 4P)

Now, we evaluate each option in turn, explaining why it is or isn’t consistent with profit maximization given the inability to discriminate at entry.

Option 1: P = MC, T = all CS of high-demand customers at that price
- If P = MC, ride output is chosen to maximize profit from rides for the combined demand, but we are also collecting T as the entire CS of high-demand customers. This leaves high-demand consumers paying the ride price plus T, while low-demand consumers pay only T and price P. The effect on profit depends on how much CS we can extract while not drastically reducing total participation. Since high-demand CS is typically larger at a given P than low-demand CS (because high-demand willing to pay more for each quantity), extracting all CS from high-demand customers could reduce their participation if T is too large, potentially causing volume losses. Meanwhile, low-demand CS is left entirely unrecovered by this option, meaning some potential extraction is foregone. Conceptually, this option could be suboptimal because it targets the group with greater willingness to pay for CS extraction, which may depress overall participation more than extracting CS from the lower-paying group.

Option 2: (none of the choices listed here)
- This option claims none of the above are correct. To assess it, we would need to determine whether any listed option is a plausible profit-maximizing mechanism. Since at least one of the others is designed to reflect standard insights from market design with equal entry and posted price, it’s unlikely that none are sensible without rigorous counterexample.

Option 3: P = Monopoly price for low-demand customers, T = all CS of low-demand customers at that price
- Setting P to the monopoly price for the low-demand segment means choosing P to maximize revenue from low-demand buyers alone, ignoring high-demand optimizers, while still charging the same price to high-demand buyers. Then T captures all CS of low-demand buyers at that price, extracting their entire surplus. This leaves high-demand buyers paying the monopoly price (which is below their own stand-alone monopoly price but possibly still above MC) plus T. Because high-demand demand is stronger, the total quantity and revenue could be decent, but the extraction is only from low-demand CS; high-demand CS remains with them plus maybe some entry fee. Whether this is profit-maximizing depends on how much total CS you can capture from high-demand buyers without destroying their participation; it’s plausible but not evidently optimal over the other options.

Option 4: P > MC, T = all CS of high-demand customers at that price
- If P exceeds MC, rides generate more gross revenue, and extracting all CS of high-demand buyers via T captures a large share of the surplus from the group that values the ride most. Since high-demand customers have higher willingness to pay, extracting their CS can be highly profitable if it does not push them out of the market. However, charging P > MC with T capturing all CS of high-demand buyers may still leave some high-demand CS untapped (the portion of CS not captured by T if their CS exceeds T). The key consideration is that high-demand customers’ participation must remain positive for the pricing plan to be feasible; otherwise, you could lose volume. This option is consistent with a high-extraction regime aimed at the group with stronger demand, but you must ensure total surplus extraction does not deter purchases.

Option 5: P = MC, T = all CS of low-demand customers at that price
- Here, price equals marginal cost, ensuring the per-ride operation is efficient, and the ride price does not distort allocative efficiency. All low-demand CS is captured through the entry fee T, so low-demand customers pay their full surplus upfront. High-demand customers pay T plus MC-equivalent ride price. Since high-demand consumers typically have higher CS, extracting only low-demand CS while charging P = MC can be advantageous because you secure a large portion of total surplus from the less price-elastic group without sacrificing too much volume from high-demand buyers. This combination often aligns with maximizing profits under a single price and non-distinct entry, because it balances extracting surplus from the group with the smaller or more elastic willingness to pay (low-demand) while keeping P at the efficient level and not overburdening high-demand buyers.

Option 6: P > MC, T = all CS of low-demand customers at that price
- This choice sets a price premium above marginal cost and extracts all CS from low-demand customers. Because high-demand customers are more willing to pay, they may still participate and pay the higher P plus T. Extracting low-demand CS entirely while charging P > MC could be profitable if demand from both groups remains positive and the high-demand group’s participation is preserved. However, pushing P above MC could shrink ride demand, and extracting all CS from low-demand buyers might be suboptimal if you could instead extract more total CS from high-demand buyers or balance both groups more evenly.

Option 7: P = Monopoly price for high-demand customers, T = all CS of high-demand customers at that price
- This would maximize extraction from the high-demand group at their own monopoly price, but because you cannot distinguish at entry, you must also set T that could deter low-demand buyers or leave their CS unextracted. The risk is that charging the high-demand monopoly price to everyone may reduce total demand enough to reduce overall profit, especially since the low-demand group would be paying the same high price for rides they value less highly. While this captures high-demand CS, it ignores the potential value (and risk) of cross-subsidization effects on the low-demand group.

Synthesis and takeaway:
- In a setting with a single posted price and a single entry fee, the standard intuition is to set P where marginal revenue equals marginal cost (P = MC) to maximize ride-profit under the constraints, and choose T to extract some or all of the CS from one group while leaving the other group to participate.
- If the goal is to concentrate extraction on the group with the greatest willingness to pay without driving away participation, it is common to set P at MC and capture the CS of the low-demand group via the entry fee, leaving the high-demand group to contribute ride revenue while still paying T. This aligns with P = MC and T capturing all CS of the low-demand customers at that price, which matches the typical structure of the optimal screening under a single price and single entry fee when you cannot differentiate at entry but can extract CS through T for one segment.

Conclusion note:
- Among the listed options, the configuration that aligns with extracting full low-demand CS at the efficient ride price P = MC (i.e., P = MC, T = all CS of low-demand customers at that price) is the one that matches the standard approach to maximize profits when you cannot distinguish customer types at entry and you want to allocate surplus extraction to the lower-willingness-to-pay segment while keeping rides efficiently priced. The other options either misallocate extraction across groups, rely on price points (monopoly prices) that misalign with the inability to distinguish at entry, or introduce conditions that could reduce overall participation or fail to capture as much total surplus as possible.

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