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.

Question
Asked Sep 17, 2019
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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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Expert Answer

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

m+O90(10-P)+100 10-P
10
Q 1000-100P
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m+O90(10-P)+100 10-P 10 Q 1000-100P

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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.

60-
400
АТC -
о
о
АТС 3 6-
400
ДАТС
1
до
400
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60- 400 АТC - о о АТС 3 6- 400 ДАТС 1 до 400

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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  ...

aTR
MR =
до
1000-Q
xо
100
MR
MR = 1000-20
100
атс
MС-
до
Q
MC 6
200
MC = MR
Q1000-20
6-
200
100
1200 0 2000-4Q
Q 266.66
1000-0
P =
100
1000-266.66
P =
100
P $7.33
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aTR MR = до 1000-Q xо 100 MR MR = 1000-20 100 атс MС- до Q MC 6 200 MC = MR Q1000-20 6- 200 100 1200 0 2000-4Q Q 266.66 1000-0 P = 100 1000-266.66 P = 100 P $7.33

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