Monopoly Cable Operator in a Hill Panchayat
A hill panchayat in Idukki has only one cable television operator, serving households that have no other multi-channel option. The operator's monthly demand is estimated as P = 120 − 2Q, where P is the monthly subscription in rupees and Q is the number of connections in hundreds. Its total cost is TC = 20Q + 100, measured in hundreds of rupees, so fixed cost is ₹100 hundred and marginal cost is constant at ₹20 hundred per hundred connections. The panchayat is concerned about high subscription charges and is considering a ceiling of ₹50 per month. The operator says the ceiling will make service unviable, while consumer groups say it will improve welfare. The panchayat secretary needs a monopoly equilibrium, a comparison with the competitive benchmark, and an evaluation of the proposed ceiling.
1. Calculate the profit-maximising quantity, price and profit for the cable operator. Show the MR and MC conditions.
Read the worked answer
Total revenue is TR = P × Q = (120 − 2Q)Q = 120Q − 2Q².
Marginal revenue is MR = dTR/dQ = 120 − 4Q.
Marginal cost is MC = 20. Set MR = MC:
120 − 4Q = 20 → 4Q = 100 → Q = 25 hundred connections.
The monopoly price comes from the demand curve: P = 120 − 2(25) = ₹70 per month.
Profit = TR − TC = (70 × 25) − (20 × 25 + 100) = 1,750 − 600 = 1,150 hundred rupees, i.e. ₹1,15,000. The operator restricts output to 25 hundred connections and charges ₹70, well above marginal cost of ₹20.
2. Find the perfectly competitive price and quantity, and calculate the deadweight loss of monopoly.
Read the worked answer
A competitive market would set P = MC: 120 − 2Q = 20 → 2Q = 100 → Q = 50 hundred connections and P = ₹20.
The monopoly produces 25 hundred and charges ₹70. The deadweight loss is the triangle between the demand curve and the MC line from Q = 25 to Q = 50:
DWL = ½ × (P_monopoly − MC) × (Q_competitive − Q_monopoly)
DWL = ½ × (70 − 20) × (50 − 25) = ½ × 50 × 25 = 625 hundred rupees (₹62,500).
This measures the value of the extra connections that would have been worth more to consumers than their marginal cost but are not supplied. The monopoly is allocatively inefficient: it restricts quantity to keep price high, and consumer surplus is lower than under competition.
3. Evaluate the proposed price ceiling of ₹50. Will the operator supply the quantity demanded? What happens to profit and deadweight loss?
Read the worked answer
At P = ₹50, quantity demanded is 50 = 120 − 2Q → 2Q = 70 → Q = 35 hundred connections. The ceiling is below the monopoly price of ₹70 but above MC of ₹20, so the firm still covers variable cost.
With a binding ceiling, the firm's marginal revenue is ₹50 for each additional connection up to the quantity demanded, because it cannot charge more. Since ₹50 > MC = ₹20, the operator is willing to supply the 35 hundred connections demanded. Profit becomes:
TR − TC = (50 × 35) − (20 × 35 + 100) = 1,750 − 800 = 950 hundred rupees.
Profit falls from 1,150 to 950 hundred, but remains positive. New DWL = ½ × (50 − 20) × (50 − 35) = ½ × 30 × 15 = 225 hundred rupees. The ceiling reduces deadweight loss from 625 to 225 hundred and transfers surplus from producer to consumers, so welfare improves but is not first-best. A ceiling at ₹20 would achieve the competitive quantity, but the firm's revenue would be 1,000 hundred against TC of 1,100 hundred, a loss, so it would need a subsidy to stay in business.