# Consider a market with 190 consumers. Of these, 90 of them have individual (inverse) demands given by: PM(Q)=10−Q, while each of the other 100 has an individual (inverse) demand of PS(Q)=10−10Q. The cost function of the monopolist serving this market is C(Q) = 6Q - Q^2/400 . (a) Find the aggregate demand. Analyze the cost function and find what kind of returns to scale it exhibits. Compute the efficient total output (ignoring break-even constraints).(b) Compute the optimal linear price (and quantity) for this monopolist, and the deadweight loss.

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1. Consider a market with 190 consumers. Of these, 90 of them have individual (inverse) demands given by: PM(Q)=10−Q, while each of the other 100 has an individual (inverse) demand of PS(Q)=10−10Q. The cost function of the monopolist serving this market is
C(Q) = 6Q - Q^2/400 .
(a) Find the aggregate demand. Analyze the cost function and find what kind of returns to scale it exhibits. Compute the efficient total output (ignoring break-even constraints).
(b) Compute the optimal linear price (and quantity) for this monopolist, and the deadweight loss.
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Step 1

(a) Aggregate demand refers to the sum total of demand of all individual. It is provided that an individual inverse demand curve of 90 individual is PM(Q) = 10-Q and individual inverse demand curve of 100 individual is PS(Q)=10-10Q. Thus aggregate demand curve  is Q=1000-100P

Step 2

Returns to scale refers to the change in average total cost with an increase in output. total cost function of the monopolist firm is C(Q)=6Q-Q^2/400. By dividing the total cost function with Q, average total cost function can be achieved. since the first derivative of ATC is negative, this can be concluded that monopolist firm experiences increasing returns to scale.

Step 3

(b)  Optimal linear price and quantity can be obtained at a point where marginal revenue is exactly equal to marginal cost where MR is  ...

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