Consider an economy for a public commodity, such that the cost function of its production is C(Q)=Q2-5Q+10{"version":"1.1","math":"C(Q)=Q2-5Q+10"}, and the individual demands of the two consumers are: q1=5-P{"version":"1.1","math":"q1=5-P"} and q2=5-0.5P{"version":"1.1","math":"q2=5-0.5P"}. What is the price commodity in equilibrium?

Question

Consider an economy for a public commodity, such that the cost function of its production is C(Q)=Q2-5Q+10{"version":"1.1","math":"C(Q)=Q2-5Q+10"}, and the individual demands of the two consumers are: q1=5-P{"version":"1.1","math":"q1=5-P"} and q2=5-0.5P{"version":"1.1","math":"q2=5-0.5P"}. What is the price commodity in equilibrium?

BlackTom AI · Accepted Answer

We start by restating the problem in our own words and then examine each answer option against the model's setup.

First, the data: the production cost of the public commodity is C(Q) = Q^2 - 5Q + 10, so the marginal cost (MC) is MC(Q) = dC/dQ = 2Q - 5.
Each consumer has a demand for the public good given by q1 = 5 - P and q2 = 5 - 0.5P. Since this is a public (nonrival) good, the total quantity provided is Q = q1 + q2, and the price P is the same for both consumers in the stylized single-price framework.
From the individual demand relations, we can invert to express P as a function of each individual quantity: P = 5 - q1 for consumer 1, and P = 10 - 2q2 for consumer 2. The market-clearing condition is Q = q1 + q2.

Option-by-option analysis:

Option A: 3.75
- If P = 3.75, then from q1 = 5 - P we get q1 = 1.25.
- From q2 = 5 - 0.5P we get q2 = 5 - 0.5(3.75) = 5 - 1.875 = 3.125.
- Thus Q = q1 + q2 = 1.25 + 3.125 = 4.375.
- The corresponding MC at this Q is MC(4.375) = 2(4.375) - 5 = 8.75 - 5 = 3.75, which matches the price in this point. This is consistent with condition for public goods equilibria (the Samuelson condition, where marginal social benefit equals marginal cost) when you interpret the marginal benefit of an additional unit as the common price that clears both demand schedules simultaneously. Therefore, P = 3.75 is internally consistent for the equilibrium in this setup.

Option B: None of the other answers.
- To conclude this option is correct, we would need to show that no price among the listed options can satisfy the market-clearing and cost conditions. However, as shown above for P = 3.75, we obtain a coherent Q and MC equality at that price, indicating this option is not correct.

Option C: 4
- If P = 4, then q1 = 1 and q2 = 5 - 0.5(4) = 5 - 2 = 3. Thus Q = 4. The MC at Q = 4 is MC(4) = 2(4) - 5 = 3. This does not equal the price P = 4, so the price-quantity combination would not satisfy the equilibrium condition under the standard public-good pricing logic used here. Hence P = 4 is inconsistent.

Option D: 3
- If P = 3, then q1 = 2 and q2 = 5 - 0.5(3) = 5 - 1.5 = 3.5. Thus Q = 5.5. The MC at Q = 5.5 is MC(5.5) = 2(5.5) - 5 = 11 - 5 = 6. This is not equal to the price P = 3, so this would not satisfy the equilibrium condition either. Therefore P = 3 is inconsistent.

Option E: 4.25
- If P = 4.25, then q1 = 0.75 and q2 = 5 - 0.5(4.25) = 5 - 2.125 = 2.875. Thus Q = 3.625. The MC at Q = 3.625 is MC(3.625) = 2(3.625) - 5 = 7.25 - 5 = 2.25. This is not equal to P = 4.25, so this price does not clear the equilibrium either. Hence P = 4.25 is inconsistent.

Putting together these checks, only P = 3.75 yields a consistent intersection where the marginal cost equals the implied marginal social benefit in this configuration (as seen by MC(Q) equalling the price at the corresponding Q). Other options fail the necessary condition MC(Q) = P at the resulting Q, indicating misalignment between production cost and total willingness to pay at those prices.

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