Consider a competition of quantities between two producers, such that the marginal cost of producer 1 is equal to 10$ and the marginal cost of producer 2 is equal to 40$. If the aggregated demand is equal to: QD(P)=40-0.25P{"version":"1.1","math":"QD(P)=40-0.25P"}, what is the welfare inequilibrium?单项选择题
A
2137.5
B
1012.5
C
900
D
1125
E
None of the other answers.

Question

Consider a competition of quantities between two producers, such that the marginal cost of producer 1 is equal to 10$ and the marginal cost of producer 2 is equal to 40$. If the aggregated demand is equal to: QD(P)=40-0.25P{"version":"1.1","math":"QD(P)=40-0.25P"}, what is the welfare inequilibrium?单项选择题

A

2137.5

B

1012.5

C

900

D

1125

E

None of the other answers.

BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: two producers compete in quantities (Cournot-like) with constant marginal costs MC1 = 10 and MC2 = 40. The aggregated demand is QD(P) = 40 − 0.25P, which implies the inverse demand is P(Q) = 160 − 4Q. The question asks for the welfare at the equilibrium (welfare inequilibrium).

First, consider the best responses given the other producer's quantity. If firm 2 produces q2, firm 1 chooses q1 to maximize its profit π1 = (P − MC1) q1 = (160 − 4(q1 + q2) − 10) q1 = (150 − 4q1 − 4q2) q1. The FOC for firm 1 is: 150 − 8q1 − 4q2 = 0, yielding q1 = (150 − 4q2)/8 = 18.75 − 0.5q2.

For firm 2, profit is π2 = (P − MC2) q2 = (160 − 4(q1 + q2) − 40) q2 = (120 − 4q1 − 4q2) q2. The FOC is: 120 − 4q1 − 8q2 = 0, giving q2 = (120 − 4q1)/8 = 15 − 0.5q1.

Solving the two best-response equations simultaneously:
- q1 = 18.75 − 0.5 q2
- q2 = 15 − 0.5 q1
Substitute the second into the first: q1 = 18.75 − 0.5(15 − 0.5 q1) = 18.75 − 7.5 + 0.25 q1 = 11.25 + 0.25 q1. This yields 0.75 q1 = 11.25, so q1 = 15. Then q2 = 15 − 0.5(15) = 15 − 7.5 = 7.5.

However, recall that MC2 = 40 is quite high; it can be the case that the second firm produces zero in equilibrium if its profit is nonpositive at any positive output. Let’s check the profitability of firm 2 when firm 1 produces q1 = 15: P = 160 − 4(15 + q2) = 160 − 60 − 4q2 = 100 − 4q2. Profit for firm 2 is π2 = (P − 40) q2 = (60 − 4q2) q2. This is maximized at dπ2/dq2 = 60 − 8q2 = 0 → q2 = 7.5, with π2 = (60 − 30) × 7.5 = 30 × 7.5 = 225. But we must also ensure consistency with firm 1’s best response. If q2 = 0, firm 1’s best response becomes q1 = 18.75. If q1 = 18.75, P = 160 − 4(18.75) = 160 − 75 = 85, and firm 1’s profit would be (85 − 10) × 18.75 > 0. Thus the joint interior solution above may not be consistent with the condition that both firms have nonnegative profits; but given MC2 is high, the second firm may still participate depending on the resulting price.

An alternative, simpler approach is to test the natural candidate: the lower-cost firm (MC1 = 10) tends to produce as much as needed up to the point where price falls to its marginal cost, since the other firm has a much higher marginal cost. If firm 2 produces nothing (q2 = 0), firm 1 maximizes π1 = (P − 10) q1 with P = 160 − 4q1. The FOC is: dπ1/dq1 = (dP/dq1) q1 + (P − 10) = (−4) q1 + (150 − 4q1) = 150 − 8q1 = 0, so q1 = 18.75. Then P = 160 − 4(18.75) = 160 − 75 = 85. In this outcome, firm 1 earns π1 = (85 − 10) × 18.75 = 75 × 18.75 = 1406.25, and firm 2 earns zero (since producing would require price to drop below its marginal cost). Total welfare equals CS plus PS. Consumer surplus with P = 85 and choke price P0 = 160 is CS = 0.5 × (160 − 85) × Q, where Q = 18.75, so CS = 0.5 × 75 × 18.75 = 0.5 × 1406.25 = 703.125. Total welfare = CS + PS = 703.125 + 1406.25 = 2109.375. Note that different modeling choices (e.g., allowing the high-cost firm to participate under certain q allocations) can yield different welfare levels, but under the standard interpretation with the high MC firm not producing when P < MC2, this result is plausible.

Now compare the computed welfare values to the provided numerical options: 2137.5, 1012.5, 900, 1125, None of the other answers. The welfare value derived in this analysis (approximately 2109.38) does not match any of the listed numbers exactly. Among the given options, none aligns perfectly with the computed welfare figure, and several numbers are notably different from the 2100s range. Therefore, based on this calculation, none of the listed numeric options exactly matches the welfare value under the assumed equilibrium path.

Consequently, when assessing which of the listed options matches the welfare outcome under the considered equilibrium framework, you would observe a mismatch with the concrete welfare value computed here. This leads to the conclusion that the listed numeric answers do not coincide with the welfare figure derived from the standard Cournot-with-different-MCs calculation, leaving the option that denotes a mismatch as the appropriate choice among the given set.