Economics
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A monopolist produces widgets and faces the following demand curve:
,
where is the quantity demanded and is the price. The monopolist\'s total cost function is given by .
Tasks:
- Derive the monopolist\'s total revenue (TR), marginal revenue (MR), and marginal cost (MC) functions.
- Determine the profit-maximizing output and price by setting .
- Calculate the monopolist\'s total profit at the profit-maximizing output.
- Analyze the impact on the monopolist\'s price and output if the government imposes a per-unit tax of $20.
- Discuss the welfare implications of the monopolist\'s pricing strategy, including deadweight loss.
Image Requirement: Provide a graph showing the demand curve, marginal cost (MC), marginal revenue (MR), and the profit-maximizing point. Also, indicate the deadweight loss area on the graph.
'Answer
Answer
1.
Q = 500 - 2P
P = 250 - Q/2
TR = Q*P = Q*(250 - Q/2) = 250Q – Q2/2
MR
= d(TR)/dQ = d(250Q - Q2/2)/dQ
Differentiating
TR with respect to Q, we get
MR
= 250 – Q
MC
= d(TC)/dQ = d(1000 + 50Q)/dQ = 50
MC
= 50
Answer
2.
At
profit maximizing output Q,
MR
= MC
250
– Q = 50
Q
= 200
Profit
maximizing output is 200 units
Putting
Q = 200 in demand function
P
= 250 – 200/2 = 150
Profit
maximizing Price is $150
Answer
3.
Total
profit = TR – TC
TR
= 250Q - Q2/2 = 250*200 - 2002/2 = 50000 – 40000/2= 500000
– 20000 = 30000
TR
= 30000
TC
= 1000 + 50Q = 1000 + 50*200 = 1000 + 10000 = 11000
Total
Profit = 30000 – 11000 = 19000
Total
profit is $19000
Answer
3.
The
$20 tax raises the cost per unit for the monopolist by $20
MC
= 50 + 20 = 70
Now
MR = MC(New)
250
– Q = 70
Q
= 180
After
the tax, the monopolist will reduce the output by 180 units
New
Price = 250 – 180/2
New
Price = 160
So, new price will be $160 after the implementation of tax
A monopolist's pricing policy has important welfare ramifications in terms of efficiency and equity, particularly when a per-unit tax is in place. The impact of a monopolist's pricing approach on social welfare is broken down below, with an emphasis on deadweight loss (DWL).
When a monopolist produces less than the quantity that is socially ideal, where P=MC, deadweight loss occurs. A deadweight loss results from this decreased output since some customers who would have been prepared to pay more than the marginal cost are unable to obtain the good.
Optimal Quantity (Q) in society: This would happen at the intersection of P and MC.
Quantity of Monopoly (Qm): Because the monopolist limits output in order to raise prices, this is less than Q∗.
Deadweight Loss: The triangle formed by the supply and demand curves, extending from Qm to Q∗. It stands for the loss of overall excess as a result of lower producer and consumer surplus.
In our example (before the tax):
- The monopolist’s output (Q) is 200 units, while the price (P) is $150.
- If the market were competitive, the quantity would likely be higher and the price lower, maximizing total surplus.
- By producing less than the competitive quantity, the monopolist creates a deadweight loss as fewer consumers can access the good at a price they are willing to pay.
When the government imposes a $20 per-unit tax:
- The monopolist's marginal cost increases, leading to a further reduction in output from 200 units to 180 units.
- The price rises from $150 to $160, exacerbating the welfare loss for consumers who now face an even higher price.