Question

An upstream paper mill dumps effluents in a river. The total benefit and total cost to the mill are $TB = 120Q - Q^2$ and $TC = 20Q$, respectively, where $Q$ is the amount of output it produces. The environmental cost due to the negative externality is $EC = Q^2$. The government wants to impose a production tax of $t$ per unit of output on the mill. The value of $t$ to achieve the socially optimal level of production is

• A. 6
• B. 25
• C. 50
• D. 70

Hint:

At the socially optimal output, Pigouvian tax equals the marginal external cost — internalizing the negative externality.

Answer 1

The Correct Option is B

Step 1: Private equilibrium.
Private firm maximizes profit where marginal benefit (MB) = marginal cost (MC). \[ MB = \frac{dTB}{dQ} = 120 - 2Q, \quad MC = \frac{dTC}{dQ} = 20. \] At private optimum: \[ 120 - 2Q = 20 \Rightarrow Q_p = 50. \]
Step 2: Socially optimal condition.
Social planner includes external cost: \[ MSC = MC + MEC, \quad \text{where } MEC = \frac{dEC}{dQ} = 2Q. \] Set $MB = MSC$: \[ 120 - 2Q = 20 + 2Q \Rightarrow 4Q = 100 \Rightarrow Q_s = 25. \]
Step 3: Find the optimal tax $t$.
Tax should equal marginal external cost at $Q_s$: \[ t = MEC(Q_s) = 2(25) = 50. \]
Step 4: Conclusion.
The socially optimal tax per unit of output is Rs. 50.