Problem Set 3: General Equilibrium
Matthew Robson
University of York
Microeconomics 2
1
Question 1
Producer Theory
• The firm chooses 𝑦, 𝐿 and 𝐾 to maximise profit , 𝜋, subject to its
production function, 𝑦
Or:
• The firm chooses 𝐿 and 𝐾 to minimise cost, 𝐶, subject to a given
level of output, 𝑦
2
Question 1 i)
Consider the profit function 𝜋 = 𝑝2 /𝑤, where 𝑝 is the price of
output and 𝑤 is the price of labour. Derive the producer’s outputsupply and input-demand functions.
3
Question 1 i)
𝑝2
𝜋=
𝑤
Hotelling’s lemma:
𝜕𝜋
𝑦=
,
𝜕𝑝
Output-supply function:
𝜕𝜋
𝐿=−
𝜕𝑤
2𝑝
𝑦=
𝑤
Input-demand function:
𝑝
𝐿=
𝑤
2
4
Question 1 ii)
1 1
4 4
Consider the cost function 𝐶 = 4𝑦𝑤 𝑟 , where 𝑦 is output, 𝑤 is the
wage rate, and 𝑟 is the rental rate. Derive the producer’s conditional
input-demand functions for labour and capital.
5
Question 1 ii)
𝐶=
1 1
4𝑦𝑤 4 𝑟 4
Shepard’s Lemma:
𝜕𝐶
𝐿=
,
𝜕𝑤
𝜕𝐶
𝐾=
𝜕𝑟
Conditional Input-Demand Functions:
𝐿=
𝐾=
1
𝑦𝑟 4
3
𝑤4
1
𝑦𝑤 4
3
𝑟4
6
Question 2
Consider a two-commodity (𝑥 and 𝑦), two-consumer (𝐴 and 𝐵) pureexchange economy.
Suppose consumer 𝐴’s utility function is 𝑈𝐴 𝑥𝐴 , 𝑦𝐴 = 𝑥𝐴0.5 𝑦𝐴0.5 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 equation that describes the economy’s contract curve.
7
Edgeworth Box
8
Question 2
Choose an allocation 𝑥𝐴 , 𝑦𝐴 , 𝑥𝐵 , 𝑦𝐵 to maximise:
𝑥𝐵0.5 𝑦𝐵0.5
subject to:
𝑥𝐴0.5 𝑦𝐴0.5 = 𝑈𝐴
𝑥𝐴 + 𝑥𝐵 = 100
𝑦𝐴 + 𝑦𝐵 = 100
Feasible Allocations
9
Question 2
Form the Lagrangian:
ℒ = 𝑥𝐵0.5 𝑦𝐵0.5 + 𝜆 𝑥𝐴0.5 𝑦𝐴0.5 − 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 − 𝛼 = 0, (1) 𝑦𝐴 : 𝜆0.5𝑥𝐴0.5 𝑦𝐴−0.5 − 𝛾 = 0 (2)
𝑥𝐵 : 0.5𝑥𝐵−0.5 𝑦𝐵0.5 − 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
(4)
(6)
𝛾: 100 − 𝑦𝐴 − 𝑦𝐵 = 0 (7)
10
Question 2
(1)=(3)
𝜆0.5𝑥𝐴−0.5 𝑦𝐴0.5 = 𝛼 = 0.5𝑥𝐵−0.5 𝑦𝐵0.5
0.5𝑥𝐵−0.5 𝑦𝐵0.5
𝜆=
0.5𝑥𝐴−0.5 𝑦𝐴0.5
(2)=(4)
𝜆0.5𝑥𝐴0.5 𝑦𝐴−0.5 = 𝛾 = 0.5𝑥𝐵0.5 𝑦𝐵−0.5
0.5𝑥𝐵0.5 𝑦𝐵−0.5
𝜆=
0.5𝑥𝐴0.5 𝑦𝐴−0.5
𝑥𝐴0.5 𝑦𝐵0.5
𝑥𝐵0.5 𝑦𝐴0.5
0.5 0.5 = 𝜆 = 0.5 0.5
𝑥𝐵 𝑦𝐴
𝑥𝐴 𝑦𝐵
𝑥𝐴 𝑦𝐵 = 𝑥𝐵 𝑦𝐴
11
Question 2
From (6):
100 − 𝑥𝐴 − 𝑥𝐵 = 0
𝑥𝐵 = 100 − xA
From (7):
100 − 𝑦𝐴 − 𝑦𝐵 = 0
𝑦𝐵 = 100 − yA
Plug into:
𝑥𝐴 𝑦𝐵 = 𝑥𝐵 𝑦𝐴
𝑥𝐴 (100 − yA ) = (100 − xA )𝑦𝐴
100
100
−1=
−1
𝑦𝐴
𝑥𝐴
100𝑥𝐴 = 100𝑦𝐴
𝑥𝐴 = 𝑦𝐴
12
Contract Curve
13
Question 3
For the economy in question 2, derive the general competitive
equilibrium prices and quantities.
14
Question 3
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
15
Question 3
For Consumer A:
ℒ = 𝑥𝐴0.5 𝑦𝐴0.5 + 𝜆 100𝑝𝑥 − 𝑝𝑥 𝑥𝐴 − 𝑝𝑦 𝑦𝐴
FOC:
(1)=(2)
𝑥𝐴 : 0.5𝑥𝐴−0.5 𝑦𝐴0.5 − 𝑝𝑥 λ = 0
(1)
𝑦𝐴 : 0.5𝑦𝐴−0.5 𝑥𝐴0.5 − 𝑝𝑦 λ = 0
(2)
𝜆: 100𝑝𝑥 − 𝑝𝑥 𝑥𝐴 − 𝑝𝑦 𝑦𝐴 = 0
(3)
0.5𝑥𝐴−0.5 𝑦𝐴0.5
0.5𝑦𝐴−0.5 𝑥𝐴0.5
=𝜆=
𝑝𝑥
𝑝𝑦
𝑝𝑥
𝑦𝐴 = 𝑥𝐴 ,
𝑝𝑦
𝑝𝑦
𝑥𝐴 = 𝑦𝐴
𝑝𝑥
16
Question 3
Plug 𝑦𝐴 into (3)
Plug 𝑥𝐴 into (3)
𝑝𝑦
100𝑝𝑥 − 𝑝𝑥 𝑦𝐴 − 𝑝𝑦 𝑦𝐴 = 0
𝑝𝑥
100𝑝𝑥 = 2𝑝𝑦 𝑦𝐴
50𝑝𝑥
= 𝑦𝐴
𝑝𝑦
𝑝𝑥
100𝑝𝑥 − 𝑝𝑥 𝑥𝐴 − 𝑝𝑦 𝑥𝐴 = 0
𝑝𝑦
100𝑝𝑥 = 2𝑝𝑥 𝑥𝐴
50 = 𝑥𝐴
Similarly for Consumer B we obtain:
50𝑝𝑦
𝑥𝐵 =
𝑝𝑥
𝑎𝑛𝑑
𝑦𝐵 = 50
Market clearing requires that 𝑥𝐴 + 𝑥𝐵 = 100 and 𝑦𝐴 + 𝑦𝐵 = 100. Therefore:
𝑥𝐴 = 50 ⇒ 𝑥𝐵 = 50
and
𝑦𝐵 = 50 ⇒ 𝑦𝐴 = 50
17
Question 3
Finally, from utility maximisation we have:
0.5𝑥𝐴−0.5 𝑦𝐴0.5 𝑦𝐴 𝑝𝑥
𝑀𝑅𝑆𝐴 =
=
=
0.5𝑥𝐴0.5 𝑦𝐴−0.5 𝑥𝐴 𝑝𝑦
Because
𝑦𝐴
𝑥𝐴
= 1, general competitive equilibrium prices are
𝑝𝑥
𝑝𝑦
= 1.
18
Equilibrium Prices and Quantities
19
Question 4
Consider a Robinson Crusoe economy in which the utility function is
𝑢 𝑐, 𝐿 = 2 ln 𝑐 + ln 1 − 𝐿 and the production function is 𝑐 =
1
2
2𝐿 , where 𝑐 is consumption and 𝐿 is labour. Find the Pareto optimal
allocation for this economy.
20
Question 4
Robinson Crusoe Economy
• One person is both a consumer and a producer
• Two commodities, 𝑐 the consumption good and 𝐿 is the production input
labour.
• One consumer with the utility function: 𝑢 𝑐, 𝐿 = 2 𝑙𝑛 𝑐 + 𝑙𝑛 1 − 𝐿
1
2
• One producer with the production function 𝑐 = 2𝐿
21
Question 4
1
2
Choose 𝑐 and 𝐿 to maximise 2 ln 𝑐 + ln 1 − 𝐿 subject to 𝑐 = 2𝐿 . We can turn
this constrained maximisation problem into the unconstrained maximisation
problem:
1
2
Choose 𝐿 to maximise 2 ln 2𝐿
+ ln 1 − 𝐿 .
Which is equivalent to:
2
2
Choose 𝐿 to maximise 2 ln 2 + ln 𝐿 + ln 1 − 𝐿 .
FOC:
𝐿:
1
2
1
1
−
=0
𝐿 1−𝐿
1
2
Which implies that 𝐿 = , and therefore: 𝑐 = 2𝐿 = 2
1
2
1
2
=2
1
2
= 2
22