Assignment 9: Monopoly Profit Maximization & DWL Analysis | Microeconomics | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

A monopoly has a demand function of P=15-Q ($) and total cost function of TC= 7Q ($)
a. What is the price and optimal quantity that gives the firm maximum profit? Using Lerner indicator
(L) to identify the market power of this firm? We have:
TR = P.Q =(15-Q). Q = 15Q - Q^2.
MR = TR/QQ = (TR)’ = 15 - 2Q. MC = TC/QQ = (TC)’ = 7.
The firm gets maximized profit when: MR = MC → 15 -2Q = 7 → Q = 4
With Q* = 4 → P* = 15 - Q* = 15-4 = 11.
L = (P -MC) / P =(11-7)/11=4/11
So, with P = 11 and Q =4, the firm gets maximized profit.
The market power of this firm is 4/1111.
b. What is society’s price and optimal quantity (for a perfectly competitive market)? Identify dead-
weight loss (DL) created by this firm
We have competitive market point is the intersection of the demand curve and the marginal cost curve . So: MC=7=15-Q => Q=8 at P=7
The DWL equals to area of ABE so DWL = SABE=1/2.AE.BE= 1/2.4.4=8$
So, the price and optimal quantity for society are P=7, Q=8 and the dead-weight lost is 8$.
Problem 2: A monopolist has a demand function of P= 100-Q and a cost function of AVC= Q+4. FC=200
a. What is the optimal output level that maximizes profit? What is that maximum profit? We have:
TR = P.Q = (100-Q). Q = 100Q - Q^2. MR =TR/Q= (TR)’ = 100 - 2Q.
TC = VC + FC = Q.AVC + FC = Q. (Q+4) + 200 = Q^22 + 4Q + 200. MC =TC/QQ = (TC)’ = 2Q + 4.
The firm gets maximized profit when: MR = MC. →100 - 2Q = 2Q + 4. →Q = 24. P = 76.
With Q* = 24 = TR - TC = 100Q - Q^2 - (Q^22 + 4Q + 200) = 952.
So, the optimal output level to maximize profit is 24. The maximum profit is $952.
b. What are consumer surplus (CS) and deadweight loss (DWL) created by this firm
+ The price at 24 units is P(24) = 100- 24 = 76
CS (the red right triangle) = 1/2 × (100- 76) × (24- 0) = 288 + P = MC 100- Q = 2Q +4 Q = 32
DL (the black triangle) = 1/2 × (76- 52) × (32- 24) = 96.
c. Assume this firm applies perfect price discrimination, what is the quantity and variable profit of the firm?
This firm applies perfect price discrimination, the monopolist set all of the price level on the demand curve.
That expands the output from Q* to Q1.
So, by perfect price discrimination, firm increases profit by taking all CS and DWL.
=> The variable profit of the firm (PS) is IBC instead of P*AEC like before.
d. Identify the relationship between variable profit (PS) that the firm gets before and after applying
perfect price discrimination and consumer surplus and deadweight loss?
Because the price discrimination is the practice of selling a good at different prices to different consumers
(or different quantity) to take consumer surplus and DWL.
We also can see on the graph that: PSa = PSb + CS + DWL