ECONOMICS
A monopolist sells good Q and demand is Q = 10 - P, where P is price. The firm must choose to produce Q = 1, 2, 3 or 4 units of output. Assume that MC is $2.95. In this case, the optimal Q* for the monopolist is _____ units, and the resulting area of the triangle for deadweight loss is _____.
(show all work)
a) 3: 4.65
b) 3; 5.55
c) 4; 5.55
d) 4; 4.65
THANK YOU!!
To find the optimal quantity (Q*) for a monopolist, we need to compare marginal cost (MC) with marginal revenue (MR) at each level of output.
Given that the demand function is Q = 10 - P, we can rearrange it to get the price as P = 10 - Q.
To calculate MR, we need to find the derivative of the total revenue function with respect to Q. The total revenue (TR) is obtained by multiplying the price (P) and quantity (Q), so TR = P * Q.
Differentiating TR with respect to Q, we get MR = (dTR/dQ).
Let's calculate the MR for different quantities:
- Q = 1:
P = 10 - 1 = 9
TR = P * Q = 9 * 1 = 9
MR = dTR/dQ = 9
- Q = 2:
P = 10 - 2 = 8
TR = P * Q = 8 * 2 = 16
MR = dTR/dQ = 16 - 9 = 7
- Q = 3:
P = 10 - 3 = 7
TR = P * Q = 7 * 3 = 21
MR = dTR/dQ = 21 - 16 = 5
- Q = 4:
P = 10 - 4 = 6
TR = P * Q = 6 * 4 = 24
MR = dTR/dQ = 24 - 21 = 3
Now, we can compare MC with MR at each level of output:
- Q = 1:
MC = $2.95 < MR = $9
- Q = 2:
MC = $2.95 < MR = $7
- Q = 3:
MC = $2.95 > MR = $5
- Q = 4:
MC = $2.95 > MR = $3
The monopolist will produce the quantity where MC equals MR. From the calculations, we can see that this occurs when Q = 2. Therefore, the optimal Q* for the monopolist is 2 units.
To find the deadweight loss, we need to calculate the consumer surplus (CS) and producer surplus (PS) at the optimal quantity.
Consumer Surplus (CS):
CS is the area between the demand curve and the price paid by consumers for a given quantity (Q).
Given the demand function Q = 10 - P, we can substitute Q = 2 to find the corresponding price:
P = 10 - 2 = 8
To calculate CS, we need to find the area of the triangle above the price line and beneath the demand curve.
The base of the triangle is equal to the quantity (Q* = 2), and the height is the difference between the price (P = 8) and the intercept of the demand curve (P = 10) at Q = 0.
Height = P intercept - P = 10 - 8 = 2
CS = 0.5 * base * height = 0.5 * 2 * 2 = 2
Producer Surplus (PS):
PS is the area between the supply curve (MC curve in this case) and the price received by producers for a given quantity (Q).
To calculate PS, we need to find the area of the triangle below the price line and above the MC curve.
The base of the triangle is equal to the quantity (Q* = 2), and the height is the difference between the MC ($2.95) and the intercept of the MC curve (MC = 0) at Q = 0.
Height = MC - intercept = 2.95 - 0 = 2.95
PS = 0.5 * base * height = 0.5 * 2 * 2.95 = 2.95
Finally, the deadweight loss is given by the difference between the CS and PS:
Deadweight loss = CS - PS = 2 - 2.95 = -0.95
Since the area of a triangle cannot be negative, we can conclude that the deadweight loss is zero (0).
Therefore, the correct answer is:
d) 4; 4.65