Laura runs a nightclub, ‘Two Standard Drinks’. Given the popularity and cache of the club, she has a monopoly position in the market. The market demand curve is given by P = 192–q. Laura has a marginal cost per drink of MC = 2q and a fixed cost FC = $142. If Laura charges the same price to all customers, what is the resulting DWL? [Round your final answer to 2 decimal points when necessary]
Question
Laura runs a nightclub, ‘Two Standard Drinks’. Given the popularity and cache of the club, she has a monopoly position in the market. The market demand curve is given by P = 192–q. Laura has a marginal cost per drink of MC = 2q and a fixed cost FC = $142. If Laura charges the same price to all customers, what is the resulting DWL? [Round your final answer to 2 decimal points when necessary]
Solution
To calculate the Deadweight Loss (DWL), we first need to find the quantity and price at the socially optimal point (where Marginal Cost (MC) equals Demand (D)) and the quantity and price at the monopoly point (where Marginal Revenue (MR) equals MC).
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Set MC = D to find the socially optimal quantity (q) and price (P): MC = D 2q = 192 - q 3q = 192 q = 192 / 3 q = 64
Substitute q = 64 into the demand equation to find P: P = 192 - 64 P = 128
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To find the monopoly quantity and price, we first need to find the MR. The MR is the derivative of the total revenue (TR), which is P*q. But since P = 192 - q, TR = (192 - q) * q. The derivative of TR is MR = 192 - 2q.
Set MR = MC to find the monopoly quantity: MR = MC 192 - 2q = 2q 192 = 4q q = 192 / 4 q = 48
Substitute q = 48 into the demand equation to find P: P = 192 - 48 P = 144
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Now we can calculate the DWL, which is the area of the triangle formed by the socially optimal and monopoly quantities and prices. The formula for the area of a triangle is 0.5 * base * height.
The base of the triangle is the difference in quantities: 64 - 48 = 16. The height of the triangle is the difference in prices: 144 - 128 = 16.
DWL = 0.5 * 16 * 16 DWL = 128
So, the resulting Deadweight Loss (DWL) is $128.
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