Question

An economy has three firms: X, Y and Z. Every unit of output that X produces creates a benefit of INR 700 for Y and a cost of INR 300 for Z. Firm X’s cost curve is
\(𝐶(𝑄_𝑋) = 2𝑄^2_X + 10\)
where 𝐶 represents cost and 𝑄_𝑋 is the output. The market price for the output of X is INR 1600 per unit. The difference between the socially optimal output and private profit maximizing output of firm X (in INR) is _____ (in integer).

Answer 1

Correct Answer: 100

Given: 
Market price for X's output: \(P=1600\) (INR per unit).
External effect per unit: benefit to Y = 700, cost to Z = 300 → net external effect = \(700-300=400\) (INR per unit).
Cost function of firm X: \(\;C(Q_X)=2Q_X^2+10.\) 
Step 1 — Private profit maximization (price taker)
Marginal cost: \[ MC=\frac{dC}{dQ_X}=4Q_X. \] For a competitive firm \(MR=P\). Set \(MC=MR\): \[ 4Q_p = 1600 \quad\Rightarrow\quad Q_p = \frac{1600}{4}=400. \] 
Step 2 — Socially optimal output (internalize externality)
Social marginal benefit (SMB) = private marginal benefit (price) + net external effect: \[ SMB = P + 400 = 1600 + 400 = 2000. \] Set \(SMB = MC\): \[ 4Q_s = 2000 \quad\Rightarrow\quad Q_s = \frac{2000}{4}=500. \] 
Step 3 — Difference
Difference in output: \[ Q_s - Q_p = 500 - 400 = 100\ \text{units}. \] If you instead interpret the question as the monetary difference at the market price, that would be \[ 100\ \text{units}\times 1600\ \text{INR/unit} = 160000\ \text{INR}. \] 
Final answer (units): \(\boxed{100}\). (If monetary difference asked: \(160000\) INR.)