Assignment Presentation 8: Monopolist Analysis and Cost Functions | Microeconomics | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

Group assignm ent presenta tion 8
Problem 1. The following table shows the demand & cost data for a monopolist: Quantity Price ($) Total Marginal Total cost Average Marginal revenue revenue ($) total cost cost ($) ($) ($) ($) 0 8.5 5 1 8.0 9 2 7.5 11.5 3 7.0 12.5 4 6.5 13.5 5 6.0 14.0 6 5.5 16.0 7 5.0 20.0 8 4.5 25.0 9 4.0 32.0 10 3.5 40.0
a. Complete the table We have: TR= P x Q. Δ TR MR= . Δ Q TC ATC = . Q Δ TC MC = . Δ Q
After calculating, we have the result in the table below: Quantity Price ($) Total Marginal Total cost Average Marginal revenue revenue ($) total cost cost ($) ($) ($) ($) 0 8.5 0 - 5 - - 1 8.0 8.0 8 9 9 4 2 7.5 15 7 11.5 5.75 2.5 3 7.0 21 6 12.5 4.167 1 4 6.5 26 5 13.5 3.375 1 5 6.0 30 4 14.0 2.8 0.5 6 5.5 33 3 16.0 2.67 2 7 5.0 35 2 20.0 2.86 4 8 4.5 36 1 25.0 3.125 5 9 4.0 36 0 32.0 3.56 7 10 3.5 35 -1 40.0 4 8
b. What quantity will the monopolist produce?
The monopolist will produce the quantity at maximum profit. At maximum profit: MR = MC
According to the table, we have: Q* = 6
c. What price will the monopolist charge?
To maximize profit, the monopolist will charge the price at Q*=6 ⇒ P* = $5,5
d. What will the profit be at this price?
At P*=$5,5, we have the profit:
Π = TR- TC = Q*. (P* – ATC*) = 6. (5,5 – 2,67) = $17. Problem 2:
A firm has a demand function of P=100-Q ($) and a total cost function of TC=500+ 4Q+Q2 ($).
a) Is this firm perfectly competitive? Why? Method 1: We have:
TR= P.Q = Q. (100-Q) = 100Q – 2 Q Δ TR MR = = (TR)’ = 100 – 2Q Δ Q
⇒ MR < P ⇒ imperfectly competitive. Method 2:
If the firm is perfectly competitive, we have a perfectly elastic Demand curve (Horizontal).
But we have D: P = 100 – Q ⇒Downslope D ⇒ Imperfect competitive.
b) What are the price and quantity to maximize total revenue? What is the
maximum total revenue? We have:
𝑇𝑅 = 𝑃. 𝑄 = (100 − 𝑄). 𝑄 = 100𝑄 − 𝑄2
= −(𝑄2 − 2.50. 𝑄 + 502) + 2500
= −(𝑄 − 50)2 + 2500 ≤ 2500. So, 𝑇𝑅 = 2500 when Q -50 = 0 𝑚𝑎𝑥 ⇒ Q = 50. P = 100 -Q = 100 -50 = 50.
c) What are the price and optimal quantity to maximize profit? What is the
maximum total profit? Δ TR MR = = (TR)’ = 100 – 2Q Δ Q MC = (TC)’ = 2Q +4.
To maximize profit, we have MR = MC. ⇒ 100 - 2Q = 2Q +4 ⟺ 4Q = 96 ⟺ Q = 24.
At Q* = 24, we have P* = 100 - Q* = 76 Π max = TR -TC
= 100𝑄 − 𝑄2 − (500 + 4𝑄 + 𝑄2)
= −2𝑄2 + 96𝑄 – 500 = 652 ($)
d) Assume government imposes a tax of 8 $ per unit of goods sold, what are
the price and optimal quantity that gives the firm maximum profit? What is this maximum profit?
As the government imposes a tax of $8, so: 𝑇𝐶 2
1 = 500 + 4𝑄 + 𝑄 + 8𝑄 = 500 + 12𝑄 + 𝑄2  𝑀𝐶1 = 2𝑄 + 12. At maximum profit: MR = MC  100 − 2𝑄 = 2𝑄 + 12  𝑄 = 22 → 𝑃 = 78 TC 1 2 2 500 500 624 ATC = +12 Q+Q +12.22 22 + = = = Q Q 22 11 624
Π = Q. (P - ATC) = 22. (78 - ) = 468. 11
e) Assume government imposes a fixed tax of 100 $, what is the price and the
optimal quantity that gives the firm maximum profit.
As the government imposes a fixed tax of $100, so: 𝑇𝐶 2 2
2 = 500 + 4𝑄 + 𝑄 + 100 = 600 + 4𝑄 + 𝑄 → 𝑀𝐶 = 2𝑄 + 4 2
To maximize profit: 𝑀𝑅 = 𝑀𝐶  100 − 2𝑄 = 2𝑄 + 4  𝑄 = 24 → 𝑃 = 76