Exercise in Industrial Economics
Consumer Utility
1.1
For consumer i, the utility function is 𝑉𝑉(𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖0 ) = �𝑥𝑥𝑖𝑖 + 𝑥𝑥0𝑖𝑖 . Consumer i has the
endowment 𝑤𝑤𝑖𝑖 , which can be spent for purchase of 𝑥𝑥𝑖𝑖 ≥ 0 goods or 𝑥𝑥0𝑖𝑖 ≥ 0 units of the
numeraire good.
(a) How many units of the numeraire is consumer i willing to pay for 𝑥𝑥𝑖𝑖 = 4, if 𝑤𝑤𝑖𝑖 ≥ 2 ?
Is his willingness to pay influenced by an increase in 𝑤𝑤𝑖𝑖 ?
(b) Suppose consumer i has bought 4 units of good 𝑥𝑥𝑖𝑖 for 1 unit of the numeraire good.
What is the size of his additional utility from this transaction? How many units of the
numeraire good would he be willing to spend for 4 additional units of 𝑥𝑥𝑖𝑖 ?
(c) Let the price of 𝑥𝑥𝑖𝑖 be 𝑝𝑝 = 1. How many units does consumer i want to acquire, if
𝑤𝑤𝑖𝑖 >
1
4
? How many units of 𝑥𝑥𝑖𝑖 will be bought for 𝑤𝑤𝑖𝑖 <
typical for a setting in Industrial Economics?
1.2
1
4
? Which of these cases is more
Producers’ profits, Consumers’ utility
A market for a homogeneous good consists of 𝑖𝑖 = 300 consumers.
Consumers 𝑖𝑖 = 1,2, … 150 have the willingness to pay 𝑈𝑈𝑎𝑎 (𝑥𝑥𝑎𝑎 ) = 2�𝑥𝑥𝑎𝑎 .
Consumers 𝑖𝑖 = 151, … 300 have the willingness to pay 𝑈𝑈𝑏𝑏 (𝑥𝑥𝑏𝑏 ) = 4�𝑥𝑥𝑏𝑏 .
There are two suppliers with the cost functions 𝐶𝐶1 (𝑥𝑥1 ) = 𝑥𝑥12 and 𝐶𝐶2 (𝑥𝑥2 ) = 2𝑥𝑥22 .
(a) Show that the social welfare is maximized by the allocation of goods with 𝑥𝑥𝑎𝑎 =
consumers 𝑖𝑖 = 1,2, … 150 , 𝑥𝑥𝑏𝑏 =
producer 1 and 𝑥𝑥2 =
allocation?
5
2
4
100
1
100
for
for consumers 𝑖𝑖 = 151, … 300 , 𝑥𝑥1 = 5 for
for producer 2. What is the size of the social welfare for this
(b) Show that for the price p, the aggregate demand is represented by 𝐷𝐷(𝑝𝑝) =
the size of the consumer surplus for 𝑝𝑝 = 10 ?
750
𝑥𝑥 2
. What is
(c) What will be the supply function as a function of p of the two producers in a competitive
market? Suppose the price is 𝑝𝑝 = 10 . What is the size of the producer surplus?
(d) Show that for the case of a competitive market, 𝑝𝑝 = 10 is an equilibrium price. Compare
this result to the result of (a)!
1.3
Monopoly and Social Welfare
A monopolistic producer is represented by the cost function 𝐶𝐶(𝑥𝑥) = 0.5𝑐𝑐𝑥𝑥 2 . He is facing a
demand represented by 𝐷𝐷(𝑝𝑝) = 𝑎𝑎 − 𝑝𝑝 .
(a) Calculate the monopoly supply 𝑥𝑥 𝑚𝑚 and the monopoly price 𝑝𝑝𝑚𝑚 . What is the size of the
monopoly profit Π(𝑝𝑝𝑚𝑚 )?
(b) Show that the elasticity of demand 𝜂𝜂𝑝𝑝 is bigger than one!
(c) Calculate the consumer surplus 𝑅𝑅𝐾𝐾 (𝑝𝑝𝑚𝑚 ) for the monopoly price 𝑝𝑝𝑚𝑚 !
(d) What is the amount 𝑞𝑞 ∗ that maximizes social welfare? Calculate the amount of the loss of
social welfare caused by the monopoly!
1.4
Discriminating Monopoly
A monopoly with the cost function 𝐶𝐶(𝑥𝑥) = 𝑥𝑥 2 faces two types of consumers:
-
a fraction 𝜆𝜆 >
1
2
is represented by 𝑈𝑈𝑎𝑎 (𝑥𝑥) = 𝑥𝑥 − 0.5𝑥𝑥 2
a fraction (1 − 𝜆𝜆) is represented by 𝑈𝑈𝑎𝑎 (𝑥𝑥) = 2𝑥𝑥 − 0.5𝑥𝑥 2 .
Suppose 𝜆𝜆 + (1 − 𝜆𝜆) = 1 and that 𝜆𝜆 is located in a range such that the good will be
offered to both types of consumers.
(a) Calculate the demand functions of the two consumer types!
(b) Calculate the optimal two-part tariff for the case of perfect discrimination.
(c) Calculate the optimal two-part tariff for the case where the monopolist can only achieve
discrimination by a price schedule!