Assume that the extraction of water from an aquifer by a coal mining company imposes a cost on farmers that grow citrus crops. Assume that the supply curve for coal is given by the following: p = 240 +2Q. Further, assume that the demand curve for coal is given by the following: p = 900 – 3Q. If the marginal external cost (MEC) of coal mining is equal to 0.5Q, which of the following is true?

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BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: a coal mining operation imposes an external cost on citrus farmers, with the coal supply curve given by p = 240 + 2Q and demand for coal given by p = 900 – 3Q. The marginal external cost (MEC) of mining is MEC = 0.5Q. The question asks which of the given statements is true.

First, compute the competitive (private) equilibrium without accounting for external costs. Set supply equal to demand:
240 + 2Q = 900 – 3Q
5Q = 660
Qc = 132
Price at this quantity: p = 240 + 2(132) = 504

Next, determine the social optimum by internalizing the external cost. The social marginal cost (SMC) is the private marginal cost (the supply inverse) plus MEC: SMC = (private MC) + MEC = (240 + 2Q) + 0.5Q = 240 + 2.5Q.
Set the demand price equal to the social marginal cost to find the socially efficient quantity Qs:
900 – 3Q = 240 + 2.5Q
660 = 4.5Q
Qs = 660 / 4.5 ≈ 146.666...

Now, determine the Pigouvian tax that could align private incentives with the social optimum. The usual approach is to set a per-unit tax equal to MEC at the socially efficient quantity, i.e., t ≈ MEC(Qs) = 0.5 * Qs ≈ 0.5 * 146.666... ≈ 73.333...
(Note: some treatments use the marginal external cost evaluated at the social optimum; others may consider a tax that equates private marginal cost with social marginal cost across quantities, which also leads to a similar magnitude but depends on the exact approach. Importantly, the tax is not exactly 60 under these standard calculations.)

Now evaluate the statement in the answer choice: “At the competitive equilibrium a deadweight loss of 396 occurs and a Pigouvian tax of 60 will lead to an efficient outcome.”
- Deadweight loss (DWL) from the external cost arises because the private equilibrium Qc = 132 is below the social optimum Qs ≈ 146.67, and the externality causes welfare loss over the range from 132 to 146.67. The DWL for a linear demand and linear private MC with MEC increasing with Q can be computed as the integral of MEC from Qc to Qs: DWL = ∫_{Qc}^{Qs} MEC dQ = ∫_{132}^{146.666...} 0.5Q dQ = 0.25 [Qs^2 − Qc^2]. Plugging in numbers gives a DWL on the order of thousands (approximately 1000 in the rough calculation shown here), not 396. Therefore, the claimed DWL of 396 is inconsistent with these curves and MEC.
- The Pigouvian tax argument: if one were to implement a tax to internalize the externality, the tax should reflect MEC at the relevant quantity. If we evaluate MEC at the social optimum Qs, t = 0.5 * Qs ≈ 73.33, not 60. If, instead, the tax were chosen to shift the private marginal cost up so that private equilibrium equals the social optimum, the required tax would still be around a value that aligns with MEC growth and the slope of the curves; 60 is noticeably lower than the approximate 73 range derived from MEC at the social optimum.

So, breaking down the two components of the statement: (1) the DWL of 396 at the private equilibrium does not align with the calculated DWL given MEC = 0.5Q and the linear curves; (2) a Pigouvian tax of 60 is not the standard correction for this setup, since the MEC at the socially efficient quantity is closer to 73.3, and the resulting tax should reflect MEC to achieve efficiency.

In light of this, the assertion described in the option is not consistent with the model and calculations above, given the standard interpretation of MEC and Pigouvian taxation in this context.

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