a monopolist has the following demand and Tc functions:
Q=2,500-25P
TC= 100,000 + 20Q+ 0.05Q^2
How much consumer surplus ($) will a monopolist transfer to itself
Take a shot, what do you think?
Hint: On a graph, Consumer surplus is the area below the demand curve but above price. Compare the area using the Price the Monopolist would change to maximize his profits vs the price where P=MC. You will need to use some geometry.
doing this exercise
To determine the consumer surplus transferred to the monopolist, we first need to find the monopolist's optimal price and quantity, and then calculate the consumer surplus.
Step 1: Find the monopolist's optimal price and quantity
The monopolist maximizes its profit by setting marginal cost equal to marginal revenue. To find the monopolist's optimal quantity, we equate the marginal cost (MC) and marginal revenue (MR) functions:
MR = MC
The marginal revenue is the derivative of the monopolist's demand function:
MR = d(Q)/dP
The marginal cost is the derivative of the monopolist's total cost function:
MC = d(TC)/dQ
Step 2: Calculate the consumer surplus
Consumer surplus is the difference between what consumers are willing to pay for a good and what they actually pay. It can be calculated by finding the area of the triangle under the demand curve and above the price line.
Consumer Surplus = (0.5) * (Q_max - Q)^2 * (P_max - P)
Where Q_max and P_max represent the quantity and price at the demand curve's intercept with the quantity and price axes.
Let's calculate the monopolist's optimal price and quantity, and then the consumer surplus:
1. Find the monopolist's optimal quantity:
Set MR equal to MC:
d(Q)/dP = d(TC)/dQ
Differentiate the demand equation to get MR:
-25 = d(TC)/dQ
Differentiate the total cost equation to get MC:
20 + 0.1Q = -25
Solve for Q:
0.1Q = -45
Q = 450
2. Find the monopolist's optimal price:
Substitute the value of Q into the demand equation:
Q = 2,500 - 25P
450 = 2,500 - 25P
Solve for P:
25P = 2,500 - 450
25P = 2,050
P = 82
3. Calculate the consumer surplus:
Consumer Surplus = (0.5) * (Q_max - Q)^2 * (P_max - P)
Q_max = 2,500 - 25(0) = 2,500
P_max = $0
Consumer Surplus = (0.5) * (2,500 - 450)^2 * (0 - 82)
= (0.5) * (2,050)^2 * (82)
= $85,457,500
Therefore, the monopolist transfers $85,457,500 of consumer surplus to itself.