Download Problems 13 Quantity supplied in monopoly market to maximize

Problems 13
1.
𝐴𝐶 = 𝑀𝐶 = 5
𝑄 = 53 − 𝑃 → 𝑃 = 53 − 𝑄 = 𝐴𝑅
a. 𝑃, 𝑄, 𝜋 monopoly =?
𝑇𝑅 = 𝑃 ∙ 𝑄 = (53 − 𝑄)𝑄 = 53𝑄 − 𝑄 2
𝑑𝑇𝑅
𝑀𝑅 =
= 53 − 2𝑄
𝑑𝑄
Quantity supplied in monopoly market to maximize profit requires: 𝑀𝑅 = 𝑀𝐶
53 − 2𝑄 = 5 → 𝑸 = 𝟐𝟒
Substituting the quantity (Q) into price (P), we have:
𝑃 = 53 − 𝑄 = 53 − 24 = 29
Then the monopolist’s profit becomes:
𝜋 = 𝑇𝑅 − 𝑇𝐶 = 𝑃 ∙ 𝑄 − 𝐴𝐶 ∙ 𝑄 = (𝑃 − 𝐴𝐶)𝑄 = (29 − 5)24 = 576
b. 𝑃, 𝑄 competitive market =?
In a perfect competitive market, we have the following relation: 𝑃 = 𝑀𝐶, thus
𝑷 = 𝟓 and 53 − 𝑄 = 5 → 𝑸 = 𝟒𝟖
c.
P
53
Note:
 Area 1: Consumer Surplus in Monopoly Market
 Area 2: Monopolist’s Profit
 Area 3: Deadweight Loss
1
Eq. in Monopoly Market
29
D = AR
2
3
Eq. in Perfectly Competitive Market
5
AC = MC
24 26.5
48
53
Q
Consumer Surplus in a Monopoly Market:
𝐶𝑆𝑀𝑂𝑁 = 𝐴𝑟𝑒𝑎 1 =
1
× 24 × 24 = 288
2
Monopolist’s Profit:
𝜋𝑀𝑂𝑁 = 𝐴𝑟𝑒𝑎 2 = 24 × 24 = 576
Deadweight Loss:
𝐷𝑊𝐿 = 𝐴𝑟𝑒𝑎 3 =
1
× 24 × 24 = 288
2
Consumer Surplus in a Perfectly Competitive Market:
𝐶𝑆𝑃𝐶𝑀 = 𝐴𝑟𝑒𝑎 1 + 𝐴𝑟𝑒𝑎 2 + 𝐴𝑟𝑒𝑎 3 = 288 + 576 + 288 = 1152
From the calculation above, we can see that the sum of consumer surplus in a monopoly market and the monopolist’s profit is less
than the amount of consumer surplus in a perfectly competitive market.
2.
𝑄 = 70 − 𝑃 → 𝑃 = 70 − 𝑄 = 𝐴𝑅
𝑇𝑅 = 𝑃 ∙ 𝑄 = (70 − 𝑄)𝑄 = 70𝑄 − 𝑄 2
𝑑𝑇𝑅
𝑀𝑅 =
= 70 − 2𝑄
𝑑𝑄
a. 𝐴𝐶 = 𝑀𝐶 = 6
Quantity supplied in monopoly market to maximize profit requires: 𝑀𝑅 = 𝑀𝐶
70 − 2𝑄 = 6 → 𝑸 = 𝟑𝟐
Substituting the quantity (Q) into price (P), we have:
𝑃 = 70 − 𝑄 = 70 − 32 = 38
Then the monopolist’s profit becomes:
𝜋 = 𝑇𝑅 − 𝑇𝐶 = 𝑃 ∙ 𝑄 − 𝐴𝐶 ∙ 𝑄 = (𝑃 − 𝐴𝐶)𝑄 = (38 − 6)32 = 1024
b. 𝐶(𝑄) = 0.25𝑄 2 − 5𝑄 + 300
𝑑𝐶(𝑄)
𝑀𝐶 =
= 0.5𝑄 − 5
𝑑𝑄
Quantity supplied in monopoly market to maximize profit requires: 𝑀𝑅 = 𝑀𝐶
70 − 2𝑄 = 0.5𝑄 − 5
75 = 2.5𝑄 → 𝑸 = 𝟑𝟎
Substituting the quantity (Q) into price (P), we have:
𝑃 = 70 − 𝑄 = 70 − 30 = 40
Then the monopolist’s profit becomes:
𝜋 = 𝑇𝑅 − 𝐶(𝑄) = 𝑃𝑄 − 𝐶(𝑄) = 40𝑄 − (0.25𝑄 2 − 5𝑄 + 300) = 1200 − 225 + 150 − 300 = 825
𝐶(𝑄) = 0.0133𝑄 3 − 5𝑄 + 250
𝑑𝐶(𝑄)
𝑀𝐶 =
= 0.0399𝑄 2 − 5 ≈ 0.04𝑄 2 − 5
𝑑𝑄
Quantity supplied in monopoly market to maximize profit requires: 𝑀𝑅 = 𝑀𝐶
70 − 2𝑄 = 0.04𝑄2 − 5
0.04𝑄 2 + 2𝑄 − 75 = 0
𝑄 2 + 50𝑄 − 1875 = 0
(𝑄 + 75)(𝑄 − 25) = 0
𝑄1 = −75 𝑎𝑛𝑑 𝑄2 = 25 (the negative value of Q is not plausible)
Substituting the positive value of Q into price (P), we have:
𝑃 = 70 − 25 = 70 − 25 = 45
Then the monopolist’s profit becomes:
𝜋 = 𝑇𝑅 − 𝐶(𝑄) = 𝑃𝑄 − 𝐶(𝑄) = 45𝑄 − (0.0133𝑄 3 − 5𝑄 + 250) = 1125 − 207.8125 + 125 − 250 = 792.1875
c.
d. P
MCc
45
40
38
MCb
c
b
a
6
10
25 30 32 MR
MCa
D = AR
Q
9.
a.
The undesirable effect of a monopoly is a higher price and a lower quantity of goods if it is compared to a price competitive market.
To overcome this situation, the government tries to reduce the price and increase the quantity supplied in the monopoly market. A
lump-sum subsidy will not make the government’s goal achieved due to the following reasons:
The total cost of a firm is given by 𝑇𝐶 = 𝐹𝐶 + 𝐴𝐶(𝑞), and its Marginal Cost is given by 𝑀𝐶 =
b.
𝑑𝑇𝐶
𝑑𝑞
= 𝐴𝐶 ′ (𝑞)
If the government gives a lump-sum subsidy to the firm with an amount of S, then its total cost becomes
𝑇𝐶′ = 𝐹𝐶 + 𝐴𝐶(𝑞) − 𝑆
Therefore, its Marginal Cost function is given by
𝑑𝑇𝐶
𝑀𝐶′ =
= 𝐴𝐶 ′ (𝑞)
𝑑𝑞
We can see that MC’ = MC. Thus, when the government gives a lump-sum subsidy into a firm, it would not change the Marginal Cost
function of the firm. As a result, the equilibrium price and quantity in the monopoly market would not change because to find the
optimal quantity supplied in the monopoly market, we need to set MR=MC.
If the government give a per-unit-of-output subsidy to the firm, then the total cost of the firm becomes
𝑇𝐶′ = 𝐹𝐶 + 𝐴𝐶(𝑞) − 𝑆(𝑞)
Therefore, its Marginal Cost function is given by
𝑑𝑇𝐶
𝑀𝐶 ′ =
= 𝐴𝐶 ′ (𝑞) − 𝑆 ′ (𝑞)
𝑑𝑞
We can see that the Marginal Cost of the firm after subsidy (MC’) is smaller than its initial Marginal Cost (MC).
The graphical proof is shown in the following picture:
P
MC
MC’
b
c
a
MR
c.
D=AR
Q
In a perfectly competitive market, the equilibrium occurs at point a, in the intersection of supply curve (MC) and demand curve (AR).
In a monopoly market, equilibrium occurs at point b, where MC=MR. Consequently, the price is higher and the quantity supplied in
the market decreases. A per-unit-of-output subsidy by the government will reduce the total cost of the firm and shift the marginal
cost curve down. As a result, a new equilibrium with lower price and higher quantity than the condition in monopoly market is
created.
In a perfectly competitive market, the equilibrium occurs at P=MC. In a monopoly market, maximum profit is achieved when
MR=MC. To achieve both condition, the government should allocate a per-unit subsidy t such that MR=MC-t. Therefore, we have the
following relation:
1
𝑀𝑅 = 𝑀𝐶 − 𝑡 = 𝑃 (1 +
)
𝑒𝑄,𝑃
Since P=MC, then
1
𝑃 − 𝑡 = 𝑃 (1 +
)
𝑒𝑄,𝑃
Divide both side by P, we have:
𝑡
1
1− =1+
𝑃
𝑒𝑄,𝑃
Thus, we have:
𝒕
𝟏
=−
𝑷
𝒆𝑸,𝑷
A monopoly creates inefficiency in the market; as a result, the required condition P=MC is not achieved, because the price in a
monopoly market is higher than MC. Consequently, a subsidy with an amount that fulfilled the condition above is required to
overcome the gap between P and MC.
Problems 20
P
1.
a.
b.
3.
𝑃 = $20, 𝑀𝐶 = 0.4 𝑞
In a perfectly competitive market, the condition for optimal production is:
𝑃 = 𝑀𝐶 → 20 = 0.4 𝑞 → 𝒒 = 𝟓𝟎
Thus, the optimal production of output for the firm is 50 widgets.
𝑃 = $20, 𝑆𝑀𝐶 = 0.5 𝑞
20
The socially optimal condition requires:
Tax = $4
16
𝑃 = 𝑆𝑀𝐶 → 20 = 0.5 𝑞 → 𝒒 = 𝟒𝟎
Thus, the socially optimal production of output for the firm is 40 widgets.
At 𝑞 = 40, the marginal cost of the firm is 𝑀𝐶 = 0.4 𝑞 = 0.4 ∙ 40 = $16
Therefore, the government should impose a tax with an amount of:
𝑡 = 𝑃 − 𝑀𝐶 = $20 − $16 = $4
c.
P = $10/barrel
MC = AC = 1000
𝑄 = 500𝑛 − 𝑛2
𝑄
𝑞 = = 500 − 𝑛
𝑛
SMC =0.5 q
MC =0.4 q
40
50
q
a.
𝑇𝑅 = 𝑃 ∙ 𝑞 = 10(500 − 𝑛) = 5000 − 10𝑛
Equilibrium condition requires:
𝑇𝑅 = 𝑀𝐶
5000 − 10𝑛 = 1000
10𝑛 = 4000 → 𝑛 = 400
The total output of the well is:
𝑄 = 500𝑛 − 𝑛2 = 500(400) − 4002 = 40,000
Marginal Productivity of each well is:
𝑑𝑞
𝑑
(500 − 𝑛) = −1
=
𝑑𝑛 𝑑𝑛
Q
62,500
Diminishing Marginal
Productivity of Q shows the
existence of externalities
0
b.
c.
7.
𝑞𝑎
𝑞𝑏
a.
b.
c.
250
500
n
We can see that the marginal productivity of each well diminish if there is additional wells, therefore, we can conclude that there is
an externality in the production of the oil, because additional wells reduces the output of each well. From the graph above, we can
also observe that the total output also experiences diminishing marginal productivity.
Total Revenue from all wells is:
𝑇𝑅 = 𝑃𝑄 = 10(500𝑛 − 𝑛2 ) = 5000𝑛 − 10𝑛2
𝑑𝑇𝑅
𝑀𝑅 =
= 5000 − 20𝑛
𝑑𝑛
An optimal output requires 𝑀𝑅 = 𝑀𝐶, hence
5000 − 20𝑛 = 1000 → 𝒏 = 𝟐𝟎𝟎
Then, the total output is: 𝑄 = 500𝑛 − 𝑛2 = 500(200) − (200)2 = 60,000
And, the output per well is: 𝑞 = 500 − 𝑛 = 500 − 200 = 300
The optimal number of well is 200 wells, therefore, the government should limit the number of well to 200 wells.
The marginal cost of each well is 1000. The total revenue of each well when n = 200 is:
𝑇𝑅 = 5000 − 10𝑛 = 5000 − 10(200) = 3000
Therefore, the license fee for the firm is:
𝒕 = 𝑨𝑹 − 𝑴𝑪 = 𝟑𝟎𝟎𝟎 − 𝟏𝟎𝟎𝟎 = 𝟐𝟎𝟎𝟎
= 100 − 𝑃 → 𝑃 = 100 − 𝑞𝑎
= 200 − 𝑃 → 𝑃 = 200 − 𝑞𝑏
𝑀𝐶 = $120
Total Willingness to Pay of the society is:
𝑇𝑊𝑃 = 𝑃𝑎 + 𝑃𝑏 = 300 − 2𝑄
The optimal condition occurs when TWP=MC
300 − 2𝑄 = 120
𝑄 ∗= 90
In a private market we have P = MC
Demand of mosquito control for individual a:
120 = 100 − 𝑞𝑎 → 𝑞𝑎 = −20 (thus, demand of individual a for mosquito control is 0)
Demand of mosquito control for individual b:
120 = 200 − 𝑞𝑏 → 𝑞𝑏 = 80
Total demand of the society is 80, which is less than the optimal demand when mosquito control is treated as public goods.
The cost for producing mosquito control:
𝑇𝐶 = 𝑃 ∙ 𝑄 = $120 ∙ 90 = $10,800
Willingness to pay of individual a is: 𝑃 = 100 − 𝑄 ∗= 100 − 90 = 10. Hence, the proportion of individual a is $10 ∙ 90 = $900
Willingness to pay of individual b is: 𝑃 = 200 − 𝑄 ∗= 200 − 90 = 110. Hence, the proportion of individual b is $110 ∙ 90 = $9900