Suppose a monopolist faces an inverse demand function P=100-1/2Q, and the monopolist has a fixed marginal cost of $20. How much more would the monopolist make from perfect price discrimination compared to simply producing where marginal revenue equals marginal cost?
Drawing a picture would help.
A normal monopolist would set MC=MR. Under this example, optimal Q=80, thus P=60. Total revenue is 80*60=4800. Total cost (represented by the area under MC between 0 and 80) is 80*20=1600. So, profit = 3200.
For the perfect price discriminator, total profit would be the area under demand, and above MC. In this example it is a simple triangle. Height=100-20=80, width =160. So area=profit=.5*80*160 = 6400
l Accept
To find out how much more the monopolist would make from perfect price discrimination compared to producing where marginal revenue equals marginal cost, we need to calculate the profits under both scenarios.
1. Producing where marginal revenue equals marginal cost:
To determine the monopolist's profit under this scenario, we first need to find the monopolist's marginal revenue function. The marginal revenue (MR) can be calculated by taking the derivative of the inverse demand function, as MR = d(P)/dQ.
Given that the inverse demand function is P = 100 - (1/2)Q, we can find MR by taking the derivative with respect to Q:
MR = d(P)/dQ = -1/2
Since marginal cost (MC) is constant at $20, we set MR equal to MC to find the monopolist's profit-maximizing quantity:
MR = MC
-1/2 = 20
Solving for Q, we get:
Q = -40
However, this is not a meaningful solution in this context, as quantity cannot be negative. Therefore, it does not make sense for the monopolist to produce where marginal revenue equals marginal cost.
2. Perfect price discrimination:
Under perfect price discrimination, the monopolist can charge each customer the maximum price they are willing to pay. This means that the monopolist can extract all consumer surplus and maximize their profit.
To calculate the monopolist's profit under perfect price discrimination, we need to calculate the consumer surplus (CS) and total revenue (TR) first.
The inverse demand function is P = 100 - (1/2)Q. To find the consumer surplus, we need to find the area under the demand curve up to the monopolist's quantity. The formula for consumer surplus (CS) is CS = (1/2)(P)(Q).
Substituting the inverse demand function, we have:
CS = (1/2)(Q)(100 - (1/2)Q)
Expanding and simplifying the equation, we have:
CS = 50Q - (1/4)Q^2
Next, we can find the monopolist's total revenue (TR) by multiplying the monopolist's quantity (Q) by the price (P). Substituting the inverse demand function, we have:
TR = P * Q = (100 - (1/2)Q) * Q
Expanding and simplifying the equation, we have:
TR = 100Q - (1/2)Q^2
To find the monopolist's profit (π), we need to subtract the marginal cost (MC) from the total revenue:
π = TR - MC = 100Q - (1/2)Q^2 - 20Q
Expanding and simplifying the equation, we have:
π = 80Q - (1/2)Q^2
To find the profit-maximizing quantity (Q), we need to set the marginal revenue (MR) equal to the marginal cost (MC):
MR = MC
-1/2 = 20
Solving for Q, we get:
Q = 40
Finally, we can calculate the monopolist's profit under perfect price discrimination:
π = 80Q - (1/2)Q^2
π = 80(40) - (1/2)(40)^2
π = 3,200 - 800
π = $2,400
Therefore, the monopolist would make $2,400 more from perfect price discrimination compared to simply producing where marginal revenue equals marginal cost.
To determine how much more the monopolist would make from perfect price discrimination compared to producing where marginal revenue equals marginal cost, we need to compare the two scenarios:
1. Producing where marginal revenue equals marginal cost:
To find the monopolist's profit-maximizing quantity and price, we first need to find their marginal revenue function. The marginal revenue (MR) is the derivative of the total revenue (TR) function with respect to quantity (Q).
Given the inverse demand function P = 100 - 1/2Q, we can calculate the total revenue function (TR). TR is the product of price (P) and quantity (Q):
TR = P * Q
Differentiating TR with respect to Q will give us the marginal revenue function (MR):
MR = d(TR)/dQ
Using the product rule, we can differentiate the TR function:
MR = d(PQ)/dQ = P + Q*(dP/dQ)
Substituting the inverse demand function P = 100 - 1/2Q into the MR equation, we get:
MR = (100 - 1/2Q) + Q*(-1/2)
Now, set MR equal to marginal cost (MC) to find the profit-maximizing quantity:
MR = MC
(100 - 1/2Q) + Q*(-1/2) = 20
Simplifying the equation, we have:
100 - 1/2Q - 1/2Q = 20
Combining like terms, we find:
- Q + 100 = 20
Solving for Q, we get:
Q = 80
Now, substitute the value of Q into the inverse demand function to find the price (P):
P = 100 - 1/2Q
P = 100 - (1/2)*80
P = 100 - 40
P = 60
So, when producing where marginal revenue equals marginal cost, the profit-maximizing quantity is 80 units, and the price is $60.
2. Perfect price discrimination:
In perfect price discrimination, the monopolist charges each customer the maximum price they are willing to pay. This means that the monopolist can extract all the consumer surplus and maximize their profit.
To calculate the monopolist's profit in this scenario, we need to integrate the demand function from zero to the profit-maximizing quantity.
The total revenue (TR) function, in terms of quantity (Q) for perfect price discrimination, is given by:
TR(Q) = ∫[0 to Q] P(Q) dQ
Substituting the inverse demand function P = 100 - 1/2Q into the TR equation, we get:
TR(Q) = ∫[0 to Q] (100 - 1/2Q) dQ
Integrating the equation, we have:
TR(Q) = [100Q - (1/4)Q^2] evaluated from 0 to Q
Simplifying further, we find:
TR(Q) = 100Q - (1/4)Q^2
Now, find the monopolist's profit by subtracting the total cost (TC) from the total revenue (TR):
Profit = TR(Q) - TC
Given the fixed marginal cost of $20, the total cost function (TC) will be:
TC(Q) = MC * Q
TC(Q) = $20 * Q
Substituting the total revenue (TR) and total cost (TC) functions into the profit equation, we get:
Profit = (100Q - (1/4)Q^2) - ($20Q)
Simplifying the equation, we have:
Profit = 100Q - (1/4)Q^2 - $20Q
Profit = 80Q - (1/4)Q^2
To find the profit-maximizing quantity, we set the derivative of the profit function equal to zero:
d(Profit)/dQ = 80 - (1/2)Q = 0
Solving for Q, we get:
(1/2)Q = 80
Q = 160
Now, substitute the value of Q into the profit function to find the monopolist's profit:
Profit = 80Q - (1/4)Q^2
Profit = 80*160 - (1/4)*160^2
Profit = $12,800 - $6,400
Profit = $6,400
Therefore, the monopolist would make $6,400 from perfect price discrimination.
To calculate the difference in profit between perfect price discrimination and producing where marginal revenue equals marginal cost, subtract the profit under marginal revenue equals marginal cost ($6,400) from the profit under perfect price discrimination ($6,400 - $6,400 = $0).
So, the monopolist would not make any additional profit from perfect price discrimination compared to producing where marginal revenue equals marginal cost.