4 January 2018

2 hour
Full points: 40
ECN 201E Final Examination Solutions
Answer all 4 questions below. I suggest you spend 25 minutes on each question. If you are
stuck on a question, I recommend moving on to the next question and returning to the
problem after you have finished answering all the other questions.
Make sure to show all your working and calculations for significant partial credit. You must
show all the steps that you did to get to your answer. If you just give me the answer, you
will get 0 even if it is correct. I will also give you 0 if all you give me is a formula without
any working.
Good luck!
Please write your name and ITU ID number on your answer sheet.
Question 1
The market demand and market supply for a good that has price 𝑝 are 𝐷(𝑝) = 210 − 𝑝 and
𝑆(𝑝) = 𝑝 respectively.
a) What is the equilibrium price 𝒑∗ and equilibrium quantity 𝒒∗ ? [4 points]

b) Suppose an ad valorem sales tax of 10% is levied by the government. Find the after-tax
equilibrium price paid by the demander 𝑝𝐷∗ and the after-tax equilibrium price received by
the supplier 𝑝𝑆∗ . [4 points]

c) What is the deadweight loss (numerical value) associated with the ad valorem tax? [2
points]
Question 2
Suppose a firm has the Cobb-Douglas the production function 𝑓(𝑥1 , 𝑥2 ) = 𝑥10.3 𝑥20.3 where 𝑥1
is the amount of factor 1 that the firm uses in the production process and 𝑥2 is the
corresponding amount of factor 2. Let the price of the firm’s output be 𝑝 and the factor
prices of good 1 and good 2 be 𝑤1 and 𝑤2 respectively. Find the long-run supply function
of the firm by solving the profit-maximization problem of the firm. [10 points]
Question 3
a) A firm in a perfectly competitive market has a cost function given by 𝑐(𝑦) = 10𝑦 2 +
1000 where 𝑦 is the output level of the firm.
i) Is 𝑐(𝑦) = 10𝑦 2 + 1000 the long-run or short-run cost function of the firm? Give a
reason for your answer. [1 point]
ii) What is the minimum average cost of the firm? [2 points]
iii) The equilibrium market price is $50. Will the firm produce any positive output at this
price? Explain your answer. [2 points]
b) Find the firm’s cost function 𝑐(𝑤1 , 𝑤2 , 𝑦) if the two factors of production are perfect
complements i.e. 𝑓(𝑥1 , 𝑥2 ) = min⁡{𝑥1 , 𝑥2 }. [5 points]
Question 4
a) A monopolist faces a linear inverse market demand curve 𝑃(𝑦) = 𝑎 − 𝑏𝑦 where 𝑦
represents quantity, and 𝑎 and 𝑏 are positive constants. The cost function of the
monopolist is 𝐶(𝑦) = 𝑐𝑦 + 𝑑 where 𝑐 and 𝑑 are positive constants. The government
levies a per unit tax of 𝑡. Prove that the rate of change of the after-tax demand price with
respect to the per unit tax is 0.5. [7 points]
𝑀𝐶(𝑦 ∗ )
b) Using the mark-up pricing formula 𝑝(𝑦) = 1 , argue that a monopolist who faces a
1−|𝜖(𝑦)|

constant-elasticity demand curve and has a constant marginal cost 𝑐 will increase the
after-tax demand price more than the per unit tax 𝑡 once it is levied by the government. [3
units]
Answer to Question 1
a) The equilibrium price 𝑝∗ can be found by solving the equation below:
𝐷(𝑝∗ ) = 210 − 𝑝∗ = 𝑝∗ = 𝑆(𝑝∗ ).

2𝑝∗ = 210,
∴ 𝒑∗ = $𝟏𝟎𝟓.
To find the equilibrium quantity 𝑞 ∗ we can substitute 𝑝∗ into either the demand function
or supply function. Below we substitute into the supply function:
𝑞 ∗ = 𝑆(105) = 105,
∴ 𝒒∗ = 𝟏𝟎𝟓⁡𝒖𝒏𝒊𝒕𝒔.

b) The after-tax equilibrium is determined by the equations below:

𝐷(𝑝𝐷 ) = 210 − 𝑝𝐷 = 𝑝𝑆 = 𝑆(𝑝𝑠 ), (1)
𝑝𝐷 = (1 + 𝜏)𝑝𝑆 , (2)
where 𝜏 is the ad valorem tax and is equal to 0.1 if the ad valorem tax is 10%.
Hence equation (2) becomes:
𝑝𝐷 = 1.1𝑝𝑆 . (3)
Substituting 𝑝𝐷 in equation (1) from (3):
210 − 1.1𝑝𝑆 = 𝑝𝑆 ,
⇒ 2.1𝑝𝑆 = 210,
∴ 𝒑∗𝑺 = $𝟏𝟎𝟎.
We can find after-tax equilibrium price paid by the demander 𝑝𝐷∗ by substituting 𝑝𝑆∗ in
equation (3):
 𝑝𝐷∗ = 1.1𝑝𝑆∗ ,
⇒ 𝑝𝐷∗ = 1.1(100),
∴ 𝒑∗𝑫 = $𝟏𝟏𝟎.
c) The after-tax equilibrium quantity 𝑞𝑇 is (you will need to find it to calculate the
deadweight loss):
𝑞𝑇 = 𝑆(𝑝𝑆∗ ) = 100⁡𝑢𝑛𝑖𝑡𝑠.
[It is not necessary to draw the figure to get full points but I would advise you to draw the
figure whenever possible]

From the figure above we see that the deadweight loss is the area of the triangle whose
area equals C and D.
1 1
𝐷𝑒𝑎𝑑𝑤𝑒𝑖𝑔ℎ𝑡⁡𝑙𝑜𝑠𝑠 = 2 × (110 − 100) × (105 − 100) = 2 × 10 × 5,

∴ 𝑫𝒆𝒂𝒅𝒘𝒆𝒊𝒈𝒉𝒕⁡𝒍𝒐𝒔𝒔 = $𝟐𝟓.
Answer to Question 2
The firm’s profit-maximization problem is:
max 𝑝𝑥10.3 𝑥20.3 − 𝑤1 𝑥1 − 𝑤2 𝑥2 . (1)
𝑥1 ,𝑥2

The first-order conditions are:

0.3𝑝𝑥1−0.7 𝑥20.3 − 𝑤1 = 0, (2a)

0.3𝑝𝑥10.3 𝑥2−0.7 − 𝑤2 = 0. (2b)

Multiply both sides of equation (2a) by 𝑥1 , and similarly, both sides of equation (2b) by 𝑥2 :

0.3𝑝𝑥10.3 𝑥20.3 − 𝑤1 𝑥1 = 0, (3a)

0.3𝑝𝑥10.3 𝑥20.3 − 𝑤2 𝑥2 = 0. (3b)

Let 𝑦 = 𝑥10.3 𝑥20.3 denote the output level of the firm. Substituting 𝑦 into equations (3a) and
(3b) and re-arranging:
0.3𝑝𝑦 = 𝑤1 𝑥1 , (4a)
0.3𝑝𝑦 = 𝑤2 𝑥2 . (4b)
Solving for 𝑥1 and 𝑥2 we have:
0.3𝑝𝑦
𝑥1∗ = , (5a)
𝑤1

0.3𝑝𝑦
𝑥2∗ = . (5b)
𝑤2

We can substitute 𝑥1∗ and 𝑥2∗ back into the Cobb-Douglas production function and solve for 𝑦:
0.3𝑝𝑦 0.3 0.3𝑝𝑦 0.3
( ) ( ) = 𝑦,
𝑤1 𝑤2

0.3𝑝 0.3 0.3𝑝 0.3

⇒(𝑤 ) (𝑤 ) 𝑦 0.6 = 𝑦,
1 2

0.3 0.3 0.3 0.3

⇒ (𝑤 ) (𝑤 ) 𝑝0.6 = 𝑦 0.4,
1 2

0.3 0.75 0.3 0.75

⇒ 𝑦 = (𝑤 ) (𝑤 ) 𝑝1.5,
1 2

𝟎.𝟏𝟔𝟒
∴ 𝒚 = 𝒘𝟎.𝟕𝟓 𝒘𝟎.𝟕𝟓 𝒑𝟏.𝟓 . (6)
𝟏 𝟐

Equation (6) is the long-run supply function of the firm.

Answer to Question 3
ai) Short-run cost function. Because the cost function has a fixed cost (1000).
aii) The average cost function 𝐴𝐶(𝑦) can be fund by dividing the cost function by output 𝑦:
1000
𝐴𝐶(𝑦) = 10𝑦 + . (1)
𝑦

To find the minimum we differentiate 𝐴𝐶(𝑦) with respect to 𝑦 and set equal to 0:
𝑑𝐴𝐶 1000
= 10 − = 0. (2)
𝑑𝑦 𝑦2

Solving (2) we get 𝑦𝑚𝑖𝑛 = 10. Substituting 𝑦𝑚𝑖𝑛 into 𝐴𝐶(𝑦) to get the minimum average
cost:
𝐴𝐶(𝑦𝑚𝑖𝑛 ) = 100 + 100 = 200.
[An alternative way to do it is to use the fact that MC and AC are equal at minimum average
cost. I will accept bot methods.]
aiii) The firm will produce positive output as long as price is more than minimum average
variable cost. We see from the cost function that the average variable cost 𝐴𝑉𝐶(𝑦) is:
𝐴𝑉𝐶(𝑦) = 10𝑦. (3)
By inspection, the minimum average variable cost is 0. Therefore, the firm will produce
positive output when the price is $50.
b) When 𝑓(𝑥1 , 𝑥2 ) = min⁡{𝑥1 , 𝑥2 } we need at least 𝑦 amount of factor 1 and 𝑦 amount of
factor 2 to produce 𝑦 amount of output. Thus the minimal cost of production will be:
𝒄(𝒘𝟏 , 𝒘𝟐 , 𝒚) = 𝒘𝟏 𝒚 + 𝒘𝟐 𝒚 = (𝒘𝟏 + 𝒘𝟐 )𝒚.
Answer to Question 4
a) By inspection, we see that the marginal cost of the monopolist, 𝑀𝐶, after the tax is levied
is 𝑐 + 𝑡.
The next step in the analysis is to find the marginal revenue function, 𝑀𝑅(𝑦). The revenue of
the monopolist is:
𝑅 = 𝑦𝑃(𝑦)
⇒ 𝑅 = 𝑦(𝑎 − 𝑏𝑦)
⇒ 𝑅 = 𝑎𝑦 − 𝑏𝑦 2 . (1)
Differentiating revenue in (1) with respect to 𝑦 to obtain the marginal revenue function,
𝑀𝑅(𝑦):
𝑀𝑅(𝑦) = 𝑎 − 2𝑏𝑦. (2)
We can find the after-tax equilibrium quantity 𝑦 ∗ by equating 𝑀𝑅(𝑦) and 𝑀𝐶:
𝑎 − 2𝑏𝑦 ∗ = 𝑐 + 𝑡
𝑎−𝑐−𝑡
⇒ 𝑦∗ = . (3)
2𝑏

The after-tax demand price 𝑝∗ is given by the inverse market demand curve when 𝑦 = 𝑦 ∗ :
𝑝∗ = 𝑎 − 𝑏𝑦 ∗ . (4)
𝑑𝑝∗
We can find the rate of change of the after-tax demand price with the tax, , by the Chain
𝑑𝑡
Rule:
𝑑𝑝∗ 𝑑𝑝∗ 𝑑𝑦 ∗
= 𝑑𝑦 ∗ .
𝑑𝑡 𝑑𝑡

𝑑𝑝∗ 1
⇒ = −𝑏. − 2𝑏
𝑑𝑡

𝑑𝑝∗ 1
⇒ =2. (5)
𝑑𝑡

Equation (5) shows that the rate of change of the after-tax demand price with respect to the
per unit tax is 0.5. This completes the proof.
b) Using the mark-up pricing formula, the after-tax demand price 𝑝∗ is:
𝑐+𝑡
𝑝∗ = 1 . (1)
1−|𝜖|

Differentiating (1) with respect to 𝑡:

𝑑𝑝∗ 1
= 1 . (2)
𝑑𝑡 1−|𝜖|

We know that at an optimal price |𝜖| > 1 (a profit-maximizing producer will never price in
𝑑𝑝∗
the inelastic part of his demand curve), therefore it must be > 1. This proves for a
𝑑𝑡
monopolist facing a constant elasticity demand curve the increase in demand price after a tax
𝑡 is levied will be more than the tax.