Introduction to Advanced
Microeconomics
Seminar 1
Q1. If the supply curve is
perfectly elastic and the
demand curve is linear and
downward sloping.
a.
What is the effect of a £1
specific tax collected from
producers on equilibrium
price and quantity?
The tax shifts the supply curve
upwards.
A specific tax of £1 shifts
the
1
pre-tax supply curve,
𝑆 ,
2
upward by £1 to 𝑆 at 𝑝1 + 1.
At the after tax equilibrium,
𝑒2 the price consumers pay is
𝑝2 = 𝑝1 + 1, the price firms
receive is 𝑝2 − 1 = 𝑝1 , and the
quantity is 𝑄2 .
Q1. If the supply curve is
perfectly elastic and the
demand curve is linear and
downward sloping.
b. What is the incidence on
consumers? Why?
The equilibrium quantity falls from 𝑄1
to 𝑄2 , the price firms receive remains
at 𝑃1, and the equilibrium price
consumers pay rises from 𝑝1 to 𝑝2 =
𝑝1 + 1.
The entire incidence of the tax falls
on the consumer:
Δ𝑝 𝑝2 − 𝑝1
𝑝1 + 1 − 𝑝1
£1
=
=
=
Δ𝑡
Δ𝑡
£1
£1
The consumer absorbs the entire tax
because firms will not supply the
good at a price that is any lower than
𝑝1 .
Thus, the price must rise enough that
the price suppliers receive after tax is
unchanged. As consumers do not
want to consume as much at a higher
price, the equilibrium quantity falls.
Q2 A monopolist sells in two markets. The inverse demand curve in market 1 is
𝑝1 = 200 − 𝑞1 while the inverse demand curve in market 2 is 𝑝2 = 300 − 𝑞2
The firm’s total cost function is 𝑐 𝑞1 + 𝑞2 = 𝑞1 + 𝑞2 2
a. If the firm can price discriminate, what price will it set in each market
and how much will be sold?
Note that the total cost function is a quadratic which implies increasing
marginals cost. Will have to consider both markets at the same time
since an increase in the output sold in one market increases the
common marginal cost relevant to solving the optimal output in the
other market.
Marginal cost 𝑀𝐶 =
𝑑𝑐(.)
𝑑𝑞
= 2 𝑞1 + 𝑞2
Next compute the revenue and the marginal revenue from each
market.
Q2 A monopolist sells in two markets. The inverse demand curve in market 1 is
𝑝1 = 200 − 𝑞1 while the inverse demand curve in market 2 is 𝑝2 = 300 − 𝑞2
The firm’s total cost function is 𝑐 𝑞1 + 𝑞2 = 𝑞1 + 𝑞2 2
a. If the firm can price discriminate, what price will it set in each market
and how much will be sold?
Marginal cost 𝑀𝐶 =
𝑑𝑐(.)
𝑑𝑞
= 2(𝑞1 + 𝑞2 )
Total revenue: TR1= 200𝑞1 − 𝑞1 2 , TR2= 300𝑞2 − 𝑞2 2
Marginal Revenue: MR1= 200 − 2𝑞1 , MR2= 300 − 2𝑞2
Both these marginal revenues must be equal to marginal cost. Gives
two simultaneous equations:
2(𝑞1 + 𝑞2 ) = 200 − 2𝑞1
2(𝑞1 + 𝑞2 ) = 300 − 2𝑞2
Q2 A monopolist sells in two markets. The inverse demand curve in market 1 is
𝑝1 = 200 − 𝑞1 while the inverse demand curve in market 2 is 𝑝2 = 300 − 𝑞2
The firm’s total cost function is 𝑐(𝑞1 + 𝑞2 ) = 𝑞1 + 𝑞2 2
a. If the firm can price discriminate, what price will it set in each market and
how much will be sold?
2(𝑞1 + 𝑞2 ) = 200 − 2𝑞1  4𝑞1 + 2𝑞2 = 200
2(𝑞1 + 𝑞2 ) = 300 − 2𝑞2  2𝑞1 + 4𝑞2 = 300
Write first equation as 8𝑞1 + 4𝑞2 = 400
{1}
Second one unchanged 2𝑞1 + 4𝑞2 = 300 {2}
Take {2} from {1} => 6𝑞1 = 100 =>𝑞1 = 16.7
Sub back into either equation for 𝑞2 :
Equation {2) => 𝑞2 = (300 − 2𝑞1 )/4 =
300−33.33
4
= 66.7
Q2 A monopolist sells in two markets. The inverse demand curve in market 1 is
𝑝1 = 200 − 𝑞1 while the inverse demand curve in market 2 is 𝑝2 = 300 − 𝑞2
The firm’s total cost function is 𝑐(𝑞1 + 𝑞2 ) = 𝑞1 + 𝑞2 2
b. If it is unable to price discriminate, what price will it set.
Combine the two markets : 𝑞1 = 200 − 𝑝, 𝑞2 = 300 − 𝑝
𝑄
=> 𝑄 = 500 − 2𝑝 or 𝑝 = 250 −
𝑄2
2
Total revenue 𝑇𝑅 = 250𝑄 − => 𝑀𝑅 = 250 − 𝑄
2
2
Total Cost TC = 𝑄 and MC = 2Q
Set MR = MC => 250 − 𝑄 = 2𝑄
So 3𝑄 = 250
Therefore 𝑄 = 83.3 is the profit maximising output
and 𝑝 = 250 −
𝑄
2
1
= 250 −
833
2
= 208.333 is the profit maximising price.
Q4. Jackie has a Cobb-Douglas utility function 𝑈 = 𝑞1𝛼 𝑞21−𝛼 where 𝑞1 is the number
of tracks of recorded music she buys a year, and 𝑞2 is the number of live music
events she attends. What is her marginal rate of substitution?
First work out the marginal utilities:
The marginal utility from extra tracks is:
𝛼 1−𝛼
𝛿𝑈
𝑞
𝑞2
𝑈
1
𝛼−1 1−𝛼
𝑈1 =
= 𝛼𝑞1
𝑞2 = 𝛼
=𝛼
𝛿𝑞1
𝑞1
𝑞1
The marginal utility form extra live music is:
𝛼 1−𝛼
𝛿𝑈
𝑞
1 𝑞2
𝛼 −𝛼
𝑈2 =
= 1 − 𝛼 𝑞1 𝑞2 = 1 − 𝛼
= 1−𝛼
𝛿𝑞2
𝑞2
Express this as the marginal rate of substitution
𝛿𝑈
𝑈
𝛼
𝑑𝑞2
𝛼
𝑞2
𝑞1
𝛿𝑞1
𝑀𝑅𝑆 =
=−
=−
=−
∗
𝑈
𝛿𝑈
𝑑𝑞1
1
−
𝛼
𝑞1
1−𝛼
𝑞2
𝛿𝑞2
𝑞2
So for example if 𝛼 = 0.5 then 𝑀𝑅𝑆 = −
𝑞1
Q5 Show the substitution and income effects for the following price change (the price
of 𝑥1 has fallen). Assume that you have a budget of £100, the price of 𝑋1 is £10 and the
price of 𝑋2 is £10 and the drop in price for 𝑥1 reduces it to £5.
Also assume that the indifference curves you have are of the form: 𝑈 = 𝑥10.5 𝑥20.5 .
a. Find the initial optimal allocation.
Budget constraint is 100 = 10𝑋1 + 10𝑋2
Re-write this as 𝑋2 = 10 − 𝑋1
=> The slope of the budget line is -1
Know from Q4 that minus the slope of the indifference
curve is the MRS where
𝛼
𝑋
0.5
𝑋
𝑋
𝑀𝑅𝑆 =
∗ 2=
∗ 2= 2
1−𝛼
𝑋1
1−0.5
𝑋1
𝑋1
-MRS equals the slope of the budget line at the optimal
𝑋
point. ie −1 = − 2
𝑋1
=> 𝑋1 = 𝑋2
Budget line of 100 = 10𝑋1 + 10𝑋2
So 100 = 10𝑋1 + 10𝑋1 = 20𝑋1
𝑋1 = 5 and 𝑋2 = 5
Q5 Show the substitution and income effects for the following price change (the price
of 𝑥1 has fallen). Assume that you have a budget of £100, the price of 𝑋1 is £10 and the
price of 𝑋2 is £10 and the drop in price for 𝑥1 reduces it to £5.
Also assume that the indifference curves you have are of the form: 𝑈 = 𝑥10.5 𝑥20.5 .
b. Find the intermediate allocation showing the substitution
effect.
Substitution effect => looking for the same level of utility but
with the new slope of the budget line.
New budget line is 100 = 5𝑋1 + 10𝑋2
𝑋2 = 10 − 0.5𝑋1 and the slope is −0.5
𝑋
The MRS must be 0.5 ie −0.5 = − 𝑋2
1
So 𝑋2 = 0.5𝑋1
We need to find a allocation that gives a utility of 5 but which
contains twice as much X1 as X2
𝑈 = 0.5𝑋1 0.5 ∗ 𝑋10.5 = 5
Q5 Show the substitution and income effects for the following price change (the price
of 𝑥1 has fallen). Assume that you have a budget of £100, the price of 𝑋1 is £10 and the
price of 𝑋2 is £10 and the drop in price for 𝑥1 reduces it to £5.
Also assume that the indifference curves you have are of the form: 𝑈 = 𝑥10.5 𝑥20.5 .
b. Find the intermediate allocation showing the substitution
effect.
𝑈 = 0.5𝑋1
0.5
∗ 𝑋10.5 = 5
Square both sides
0.5𝑋1 ∗ 𝑋1 = 25
0.5𝑋12 = 25
𝑋12 = 50
𝑋1 = 7.07
And therefore
𝑋2 = 3.54
So a fall in the price of 𝑋1 leads to an increase of 𝑋1 and a
reduction of 𝑋2 as we would expect.
Q5 Show the substitution and income effects for the following price change (the price
of 𝑥1 has fallen). Assume that you have a budget of £100, the price of 𝑋1 is £10 and the
price of 𝑋2 is £10 and the drop in price for 𝑥1 reduces it to £5.
Also assume that the indifference curves you have are of the form: 𝑈 = 𝑥10.5 𝑥20.5 .
c. The new optimal allocation showing the income effect.
Know due to the substitution effect that we are looking for
𝑋2 = 0.5𝑋1 and we know that our new budget line is 100 =
5𝑋1 + 10𝑋2
Putting this together we get: 100 = 5𝑋1 + 10 0.5𝑋1 = 10𝑋1
Therefore 𝑋1 = 10
And 𝑋2 = 5
So the overall effect has been that we buy the same amount of
𝑋2 but more 𝑋1 the reason for this is that while we substitute
out some of 𝑋2 as 𝑋1 is cheaper we also have more real
income so while the ratio is different we still want both.