Problem Set 4: Externalities
Matthew Robson
University of York
Microeconomics 2
1
Question 1
Consider a two-commodity (𝑥 and 𝑦), two-consumer (𝐴 and 𝐵)
pure-exchange economy.
Suppose consumer 𝐴’s utility function is 𝑈𝐴 𝑥𝐴 , 𝑦𝐴 , 𝑥𝐵 =
𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−1 and she is endowed with 100 units of 𝑥 and zero units
of 𝑦.
Suppose consumer B’s utility function is 𝑈𝐵 𝑥𝐵 , 𝑦𝐵 = 𝑥𝐵0.5 𝑦𝐵0.5 and
she is endowed with zero units of x and 100 units of y.
Derive the conditions for Pareto optimality in the above
economy(notice that an externality is present).
2
Question 1
Choose an allocation 𝑥𝐴 , 𝑦𝐴 , 𝑥𝐵 , 𝑦𝐵 to maximise:
𝑥𝐵0.5 𝑦𝐵0.5
subject to:
𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−1 = 𝑈𝐴
𝑥𝐴 + 𝑥𝐵 = 100
𝑦𝐴 + 𝑦𝐵 = 100
Feasible Allocations
3
Question 1
Form the Lagrangian:
ℒ = 𝑥𝐵0.5 𝑦𝐵0.5 + 𝜆 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−1 − 𝑈𝐴 + 𝛼 100 − 𝑥𝐴 − 𝑥𝐵 + 𝛾 100 − 𝑦𝐴 − 𝑦𝐵
Where 𝜆, 𝛼 and 𝛾 are Lagrange multipliers.
Derive the FOC’s:
𝑥𝐴 : 𝜆0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵−1 − 𝛼 = 0,
(1)
𝑦𝐴 : 𝜆0.5𝑥𝐴0.5 𝑦𝐴−0.5 𝑥𝐵−1 − 𝛾 = 0
𝑥𝐵 : 0.5𝑥𝐵−0.5 𝑦𝐵0.5 − 𝜆𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−2 − 𝛼 = 0, (3) 𝑦𝐵 : 0.5𝑥𝐵0.5 𝑦𝐵−0.5 − 𝛾 = 0
𝜆: 𝑥𝐴0.5 𝑦𝐴0.5 − 𝑈𝐴 = 0,
(5) 𝛼: 100 − 𝑥𝐴 − 𝑥𝐵 = 0
𝛾: 100 − 𝑦𝐴 − 𝑦𝐵 = 0
(2)
(4)
(6)
(7)
4
Question 1
(1)=(3)
𝜆0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵−1 = 𝛼 = 0.5𝑥𝐵−0.5 𝑦𝐵0.5 − 𝜆𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−2
0.5𝑥𝐵−0.5 𝑦𝐵0.5
𝜆=
0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵−1 + 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−2
(2)=(4)
𝜆0.5𝑥𝐴0.5 𝑦𝐴−0.5 𝑥𝐵−1 = 𝛾 = 0.5𝑥𝐵0.5 𝑦𝐵−0.5
𝑥𝐵0.5 𝑦𝐵−0.5
𝜆 = 0.5 −0.5 −1
𝑥𝐴 𝑦𝐴 𝑥𝐵
0.5𝑥𝐵−0.5 𝑦𝐵0.5
𝑥𝐵0.5 𝑦𝐵−0.5
−0.5 0.5 −1
0.5 0.5 −2 = 𝜆 = 0.5 −0.5 −1
0.5𝑥𝐴 𝑦𝐴 𝑥𝐵 + 𝑥𝐴 𝑦𝐴 𝑥𝐵
𝑥𝐴 𝑦𝐴 𝑥𝐵
0.5𝑥𝐴0.5 𝑦𝐴−0.5
𝑥𝐵1.5 𝑦𝐵−0.5
−0.5 0.5 −1
0.5 0.5 −2 = −0.5 0.5
0.5𝑥𝐴 𝑦𝐴 𝑥𝐵 + 𝑥𝐴 𝑦𝐴 𝑥𝐵
𝑥𝐵 𝑦𝐵
5
Question 1
0.5𝑥𝐴0.5 𝑦𝐴−0.5
𝑥𝐵2
=
0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵−1 + 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−2 𝑦𝐵
𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵−1 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵−2 𝑦𝐵
0.5 −0.5 +
0.5 −0.5 = 𝑥 2
𝑥𝐴 𝑦𝐴
0.5𝑥𝐴 𝑦𝐴
𝐵
𝑦𝐴
𝑦𝐴
𝑦𝐵
+
2 = 2
0.5𝑥
𝑥𝐵
𝑥𝐴 𝑥𝐵
𝐵
𝑦𝐴
𝑥𝐴
+
2𝑦𝐴
𝑥𝐵
=
𝑦𝐵
𝑥𝐵
Condition required for Pareto Optimality
6
Question 1 (Extension)
Generalise the Utility Function
𝑈𝐴 𝑥𝐴 , 𝑦𝐴 , 𝑥𝐵 = 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟
𝑈𝐵 𝑥𝐵 , 𝑦𝐵 = 𝑥𝐵0.5 𝑦𝐵0.5
• Use 𝑟 as a parameter to enable differences in the preference for 𝑥𝐵 by 𝐴.
• We can then see no externalities, positive externalities and negative externalities.
7
Question 1
Form the Lagrangian:
ℒ = 𝑥𝐵0.5 𝑦𝐵0.5 + 𝜆 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟 − 𝑈𝐴 + 𝛼 100 − 𝑥𝐴 − 𝑥𝐵 + 𝛾 100 − 𝑦𝐴 − 𝑦𝐵
Where 𝜆, 𝛼 and 𝛾 are Lagrange multipliers.
Derive the FOC’s:
𝑥𝐴 : 𝜆0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵𝑟 − 𝛼 = 0,
(1)
𝑥𝐵 : 0.5𝑥𝐵−0.5 𝑦𝐵0.5 + 𝜆𝑟𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟−1 − 𝛼 = 0, (3)
𝜆: 𝑥𝐴0.5 𝑦𝐴0.5 − 𝑈𝐴 = 0,
𝑦𝐴 : 𝜆0.5𝑥𝐴0.5 𝑦𝐴−0.5 𝑥𝐵𝑟 − 𝛾 = 0 (2)
𝑦𝐵 : 0.5𝑥𝐵0.5 𝑦𝐵−0.5 − 𝛾 = 0
(5) 𝛼: 100 − 𝑥𝐴 − 𝑥𝐵 = 0
𝛾: 100 − 𝑦𝐴 − 𝑦𝐵 = 0 (7)
(4)
(6)
8
Question 1
(1)=(3)
𝜆0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵𝑟 = 𝛼 = 0.5𝑥𝐵−0.5 𝑦𝐵0.5 + 𝜆𝑟𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟−1
0.5𝑥𝐵−0.5 𝑦𝐵0.5
𝜆=
0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵𝑟 − 𝑟𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟−1
(2)=(4)
𝜆0.5𝑥𝐴0.5 𝑦𝐴−0.5 𝑥𝐵𝑟 = 𝛾 = 0.5𝑥𝐵0.5 𝑦𝐵−0.5
𝑥𝐵0.5 𝑦𝐵−0.5
𝜆 = 0.5 −0.5 𝑟
𝑥𝐴 𝑦𝐴 𝑥𝐵
0.5𝑥𝐵−0.5 𝑦𝐵0.5
𝑥𝐵0.5 𝑦𝐵−0.5
−0.5 0.5 𝑟
0.5 0.5 𝑟−1 = 𝜆 = 0.5 −0.5 𝑟
0.5𝑥𝐴 𝑦𝐴 𝑥𝐵 − 𝑟𝑥𝐴 𝑦𝐴 𝑥𝐵
𝑥𝐴 𝑦𝐴 𝑥𝐵
0.5𝑥𝐴0.5 𝑦𝐴−0.5
𝑥𝐵0.5−𝑟 𝑦𝐵−0.5
−0.5 0.5 𝑟
0.5 0.5 𝑟−1 =
0.5𝑥𝐴 𝑦𝐴 𝑥𝐵 − 𝑟𝑥𝐴 𝑦𝐴 𝑥𝐵
𝑥𝐵−0.5 𝑦𝐵0.5
9
Question 1
0.5𝑥𝐴0.5 𝑦𝐴−0.5
𝑥𝐵1−𝑟
−0.5 0.5 𝑟
0.5 0.5 𝑟−1 =
0.5𝑥𝐴 𝑦𝐴 𝑥𝐵 − 𝑟𝑥𝐴 𝑦𝐴 𝑥𝐵
𝑦𝐵
𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵𝑟 𝑟𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟−1
𝑦𝐵
−
=
𝑥𝐵1−𝑟
𝑥𝐴0.5 𝑦𝐴−0.5
0.5𝑥𝐴0.5 𝑦𝐴−0.5
𝑦𝐴 𝑥𝐵𝑟
𝑥𝐴
𝑟𝑦𝐴 𝑟−1
𝑦𝐵
−
𝑥𝐵 = 1−𝑟
0.5
𝑥𝐵
𝑦𝐴
𝑟𝑦𝐴 −1
𝑦𝐵
−
𝑥𝐵 = 1−𝑟 𝑟
0.5
𝑥𝐵 𝑥𝐵
𝑥𝐴
𝑦𝐵
=
𝑦𝐴
−
2𝑟𝑦𝐴
𝑥𝐵
𝑥𝐴
𝑥𝐵
Condition required for Pareto Optimality (take this further to get 𝑦𝐴 = ⋯)
10
Question 1
From (6):
100 − 𝑥𝐴 − 𝑥𝐵 = 0
𝑥𝐵 = 100 − xA
From (7):
100 − 𝑦𝐴 − 𝑦𝐵 = 0
𝑦𝐵 = 100 − yA
Plug into:
𝑦𝐵
𝑥𝐵
=
𝑦𝐴
𝑥𝐴
100 − yA 𝑦𝐴
2𝑟𝑦𝐴
=
−
100 − xA 𝑥𝐴
100 − xA
100
𝑦𝐴
−1=
100
𝑥𝐴
−
2𝑟𝑦𝐴
𝑥𝐵
100 − yA
∴
− 1 − 2𝑟
1
𝑦𝐴
∴
2𝑟
𝑦𝐴 =
−
100
𝑥𝐴
1
𝑦𝐴
=
100 − xA
𝑥𝐴
− 2𝑟
1
2𝑟
=
−
100
𝑥𝐴
−1
11
Question 2
For the economy in Question 1, derive the conditions that prevail in
a general competitive equilibrium, and thus demonstrate that the
general competitive equilibrium allocation is not Pareto optimal.
12
Question 2
General Competitive Equilibrium
• Both consumers are maximising utility subject to their budget
constraints
• max 𝑈𝐴 (𝑥𝐴 , 𝑦𝐴 ) subject to 𝑝𝑥 𝑥𝐴 + 𝑝𝑦 𝑦𝐴 = 𝑚𝐴 = 𝑝𝑥 𝜔𝑥𝐴 + 𝑝𝑦 𝜔𝑦𝐴
• max 𝑈𝐵 (𝑥𝐵 , 𝑦𝐵 ) subject to 𝑝𝑥 𝑥𝐵 + 𝑝𝑦 𝑦𝐵 = 𝑚𝐵 = 𝑝𝑥 𝜔𝑥𝐵 + 𝑝𝑦 𝜔𝑦𝐵
• Market Clear (Demand = Supply)
• 𝑥𝐴 + 𝑥𝐵 = 𝜔𝑥𝐴 + 𝜔𝑥𝐵
• 𝑦𝐴 + 𝑦𝐵 = 𝜔𝑦𝐴 + 𝜔𝑦𝐵
13
Question 2
For Consumer A:
ℒ = 𝑥𝐴0.5 𝑦𝐴0.5 𝑥𝐵𝑟 + 𝜆 100𝑝𝑥 − 𝑝𝑥 𝑥𝐴 − 𝑝𝑦 𝑦𝐴
FOC:
(1)=(2)
𝑥𝐴 : 0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵𝑟 − 𝑝𝑥 λ = 0
(1)
𝑦𝐴 : 0.5𝑦𝐴−0.5 𝑥𝐴0.5 𝑥𝐵𝑟 − 𝑝𝑦 λ = 0
(2)
𝜆: 100𝑝𝑥 − 𝑝𝑥 𝑥𝐴 − 𝑝𝑦 𝑦𝐴 = 0
(3)
0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑥𝐵𝑟
0.5𝑦𝐴−0.5 𝑥𝐴0.5 𝑥𝐵𝑟
=𝜆=
𝑝𝑥
𝑝𝑦
𝑝𝑥
𝑦𝐴 = 𝑥𝐴 ,
𝑝𝑦
𝑝𝑦
𝑥𝐴 = 𝑦𝐴 ,
𝑝𝑥
∴
𝑦𝐴 𝑝𝑥
=
𝑥𝐴 𝑝𝑦
14
Question 2
For Consumer B:
ℒ = 𝑥𝐵0.5 𝑦𝐵0.5 + 𝜆 100𝑝𝑦 − 𝑝𝑥 𝑥𝐵 − 𝑝𝑦 𝑦𝐵
FOC:
(1)=(2)
𝑥𝐵 : 0.5𝑥𝐵−0.5 𝑦𝐵0.5 − 𝑝𝑥 λ = 0
(1)
𝑦𝐵 : 0.5𝑦𝐵−0.5 𝑥𝐵0.5 − 𝑝𝑦 λ = 0
(2)
𝜆: 100𝑝𝑦 − 𝑝𝑥 𝑥𝐵 − 𝑝𝑦 𝑦𝐵 = 0
(3)
0.5𝑥𝐵−0.5 𝑦𝐵0.5
0.5𝑦𝐵−0.5 𝑥𝐵0.5
=𝜆=
𝑝𝑥
𝑝𝑦
𝑝𝑥
𝑦𝐵 = 𝑥𝐵 ,
𝑝𝑦
𝑝𝑦
𝑥𝐵 = 𝑦𝐵 ,
𝑝𝑥
∴
𝑦𝐵 𝑝𝑥
=
𝑥𝐵 𝑝𝑦
15
Question 2
• Therefore, in General Competitive Equilibrium, we have:
𝑦𝐴 𝑝𝑥 𝑦𝐵
=
=
𝑥𝐴 𝑝𝑦 𝑥𝐵
• However, for Pareto Optimality our (generalised) condition is:
𝑦𝐵
𝑥𝐵
=
𝑦𝐴
𝑥𝐴
−
2𝑟𝑦𝐴
𝑥𝐵
• Then, GE = PO when 𝑟 = 0 (i.e. no externalities)
16