A Monopoly faces market demand given by Q = 30 – P. where Q stands for quantity and
P for price. Total cost function is given by C(Q) = 20°. Find the profit maximising price
and quantity and the resulting profit to the monopoly. Compare your results with the
equilibrium quantity and price of that of a perfect competitive industry.
Answers
Answer:
The socially optimal price is attained when the product is provided to the consumer at a price that is equal to the marginal cost(MC).
Explanation:
The total cost(TC) of producing the good is:
() = 1 2 . 2
The cost function is incorrect and does not make sense so the correct TC function is assumed to be:
() = (½)^2
The MC of the function is derived from taking the derivative of TC with respect to Q.
MC=Q
The demand function is given as:
= 30 − P
Or
P=30-Q
So, the socially optimal price will be:
P=MC
30-Q=Q
30=2Q
30/2=Q
Q=15
The price will be P=MC=Q
Q*=15
P*=15
Thus, the socially optimal price is $15.
Due to monopoly, the quantity and price will be attained by equating MR (marginal revenue) and MC.
MR is calculated by taking the derivative of TR(total revenue).
TR= P*Q
P=30-Q
TR=(30-Q)*Q
TR=30Q-Q^2
MR=30-2Q
MR=MC
30-2Q=Q
30=3Q
10=Q
P=30-Q
P=30-10
P=20
The DWL(dead-weight loss) due to monopoly can be calculated as:
DWL=½*(P-MC)(Q-Q)
DWL=½*(20-15)*(15-10)
DWL=½*(5)*(5)
DWL=25/2
DWL=12.5
The CS(consumer surplus) in case of monopoly can be calculated as:
CS=½*(max P-P in monopoly)*(Q at monopoly-0)
Maximum P is the price when quantity is zero.
P=30-Q
At Q=0
P=30
CS=½*(30-20)*(10-0)
CS=½*(10)*(10)
CS=50
The CS in case of social optimal level will be:
CS=½*(max P-P*)(Q-0)
CS=½*(30-15)*(15-0)
CS=½*(15)*(15)
CS=112.5
The PS(producer surplus) in case of monopoly will be:
PS=area of rectangle + area of the triangle
PS=(P in monopoly-MR at Q in monopoly)(Q in monopoly)+½(MR at Q in monopoly-P, when Q supplied, is 0)*(Q in monopoly-0)
MR at Q in monopoly
MR=30-2Q
MR=30-2(10)
MR=30-20
MR=10
P, when Q supplied, is 0
MC=Q
Q=0
MC=0
PS=(20-10)(10)+½(10-0)*(10-0)
PS=1010+½(10)*(10)
PS=100+50
PS=150
PS in the case of social optimal level will be:
PS=½*(P*-P, when Q supplied, is 0)(Q-0)
PS=½*(15-0)*(15-0)
PS=½*(15)*(15)
PS=112.5
The graph can be shown by drawing a price and quantity graph, where the x-axis shows quantity and the y-axis represents/depicts price, MC, and MR. The demand curve can be drawn by marking the x-intercept at 30 (when Q is 0) and the demand equation. The MR curve can be drawn by marking the x-intercept at 30 and the MR equation. The MR curve will be below the demand curve.
The MC will be an upward sloping curve starting from the origin.
In perfect competition(social optimal level):
The social optimal price will be at the intersection of MC and the demand curve, that is 15 and the quantity at the corresponding level is also 15.
The CS in this is marked as a triangle that is below the demand(D) curve and above the socially optimal price.
The PS is marked as a triangle below the optimal price and above the MC curve.
There will be no DWL at the socially optimal level.
In the case of monopoly,
The equilibrium quantity is attained where MR intersects MC and the price is attained at the corresponding demand curve. The CS is marked as the triangle below the demand curve and above the monopoly price. The PS is marked as the area above the MC and below the monopoly price.
The DWL is marked as the triangle that is not covered in CS or PS.
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