Table 18-2
The information in the following table shows the total demand for internet radio subscriptions in a small urban market. Assume that each company that provides these subscriptions incurs an annual fixed cost of $20,000 (per year) and that the marginal cost of providing an additional subscription is always $16.

Quantity Demanded

(Internet radio subscriptions)

Price

(Dollars per subscription per year)

0 64
500 60
1,000 56
1,500 52
2,000 48
2,500 44
3,000 40
3,500 36
4,000 32
4,500 28
5,000 24
5,500 20
6,000 16
6,500 12
7,000 8
7,500 4
8,000 0

Refer to Table 18-2. Assume that there are two profit-maximizing internet radio providers operating in this market. Further assume that they are not able to collude on the price and quantity of subscriptions to sell. How much profit will each firm earn when this market reaches a Nash equilibrium?

Question

Table 18-2
The information in the following table shows the total demand for internet radio subscriptions in a small urban market. Assume that each company that provides these subscriptions incurs an annual fixed cost of $20,000 (per year) and that the marginal cost of providing an additional subscription is always $16.

​

Quantity Demanded

(Internet radio subscriptions)

Price

(Dollars per subscription per year)

0	64
500	60
1,000	56
1,500	52
2,000	48
2,500	44
3,000	40
3,500	36
4,000	32
4,500	28
5,000	24
5,500	20
6,000	16
6,500	12
7,000	8
7,500	4
8,000	0

​

Refer to Table 18-2. Assume that there are two profit-maximizing internet radio providers operating in this market. Further assume that they are not able to collude on the price and quantity of subscriptions to sell. How much profit will each firm earn when this market reaches a Nash equilibrium?

BlackTom AI · Accepted Answer

We start by laying out the key setup from Table 18-2: there are two internet radio providers, each facing a fixed annual cost of 20,000 and a constant marginal cost of 16 per subscription. The market demand at each price P (in dollars per subscription per year) is given as Q_total.

Option by option analysis:
- $52,000: At price 52, the table shows Q_total = 1,500 subscriptions. If the two firms share this evenly, each firm would sell Q_i = 750. The margin per subscription is (P − MC) = 52 − 16 = 36. So per-firm profit before fixed costs is 36 × 750 = 27,000. Subtracting the fixed cost of 20,000 yields a net profit of 7,000, not 52,000. This option miscalculates both revenue potential and the effect of fixed costs.

- $16,000: We need to check whether this aligns with any price point in the table. If P = 40, Q_total = 3,000. Each firm sells Q_i = 1,500. Marginal margin is P − MC = 40 − 16 = 24. Profit before fixed costs is 24 × 1,500 = 36,000. After subtracting the 20,000 fixed cost, net profit is 16,000. This matches a plausible Nash outcome under noncollusive Bertrand competition with identical firms, where the price is driven down to where marginal cost plus the fixed-cost constraint balances profit against entry. Other prices will yield different profits, as seen below.

- $12,000: To get 12,000 per firm, consider the price 48 where Q_total = 2,000, so Q_i = 1,000. Margin is 32, so profit before fixed costs is 32 × 1,000 = 32,000, and after fixed costs it’s 12,000. While this gives 12,000, it is not the maximum possible profit among price outcomes; higher profits occur at P = 40, which yields 16,000, and at P = 44 with 15,000, etc. Thus 12,000 is feasible but not optimal.

- $64,000: This option would imply exceptionally high profitability, which is inconsistent with the given data. At the highest price in the table (P = 64), Q_total = 0, so there is no quantity to sell; profit would be negative due to fixed costs, not positive, making this option clearly incorrect.

To determine the Nash equilibrium profit, compare profits across the feasible price points where there is positive demand and compute each firm’s profit as: Profit per firm = (P − 16) × (Q_total/2) − 20,000.
- At P = 60, Q_total = 500, Profit per firm = (60 − 16) × 250 − 20,000 = 44 × 250 − 20,000 = 11,000 − 20,000 = −9,000.
- At P = 56, Q_total = 1,000, Profit per firm = (56 − 16) × 500 − 20,000 = 40 × 500 − 20,000 = 20,000 − 20,000 = 0.
- At P = 52, Q_total = 1,500, Profit per firm = (52 − 16) × 750 − 20,000 = 36 × 750 − 20,000 = 27,000 − 20,000 = 7,000.
- At P = 48, Q_total = 2,000, Profit per firm = (48 − 16) × 1,000 − 20,000 = 32 × 1,000 − 20,000 = 12,000.
- At P = 44, Q_total = 2,500, Profit per firm = (44 − 16) × 1,250 − 20,000 = 28 × 1,250 − 20,000 = 35,000 − 20,000 = 15,000.
- At P = 40, Q_total = 3,000, Profit per firm = (40 − 16) × 1,500 − 20,000 = 24 × 1,500 − 20,000 = 36,000 − 20,000 = 16,000.
- At P = 36, Q_total = 3,500, Profit per firm = (36 − 16) × 1,750 − 20,000 = 20 × 1,750 − 20,000 = 35,000 − 20,000 = 15,000.
- At P = 32, Q_total = 4,000, Profit per firm = (32 − 16) × 2,000 − 20,000 = 16 × 2,000 − 20,000 = 32,000 − 20,000 = 12,000.
- At P = 28, Q_total = 4,500, Profit per firm = (28 − 16) × 2,250 − 20,000 = 12 × 2,250 − 20,000 = 27,000 − 20,000 = 7,000.
- At P = 24, Q_total = 5,000, Profit per firm = (24 − 16) × 2,500 − 20,000 = 8 × 2,500 − 20,000 = 20,000 − 20,000 = 0.
- At P = 20, Q_total = 5,500, Profit per firm = (20 − 16) × 2,750 − 20,000 = 4 × 2,750 − 20,000 = 11,000 − 20,000 = −9,000.
- At P = 16, Q_total = 6,000, Profit per firm = (16 − 16) × 3,000 − 20,000 = 0 − 20,000 = −20,000.

From these calculations, the highest per-firm profit occurs at P = 40, giving 16,000, which is larger than the profits at all other price points. In a Nash equilibrium with two noncolluding, symmetric firms facing the same demand, the price tends toward the level that maximizes each firm’s profit given the competitor’s similar pricing, and here that peak profit point is 16,000 per firm.

Therefore, the option that matches the Nash-equilibrium profit per firm is $16,000. The other options either overstate profit, understate it, or rely on price-quantity outcomes that do not maximize profit under the given fixed and marginal costs.

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