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Golden Rule Allocation
September 2, 2019
Golden Rule Allocation
Overlapping Generations Economy
In each period t, a new generation with size Nt is born.
Each generation lives for two periods.
There are Nt many young and Nt−1 many old consumers at
date t whereNt
Nt−1= n
is the population growth rate.
At any time t, for each old person there are n many young
persons.
Golden Rule Allocation
Overlapping Generations Economy
In each period t, a new generation with size Nt is born.
Each generation lives for two periods.
There are Nt many young and Nt−1 many old consumers at
date t whereNt
Nt−1= n
is the population growth rate.
At any time t, for each old person there are n many young
persons.
Golden Rule Allocation
Overlapping Generations Economy
In each period t, a new generation with size Nt is born.
Each generation lives for two periods.
There are Nt many young and Nt−1 many old consumers at
date t where
Nt
Nt−1= n
is the population growth rate.
At any time t, for each old person there are n many young
persons.
Golden Rule Allocation
Overlapping Generations Economy
In each period t, a new generation with size Nt is born.
Each generation lives for two periods.
There are Nt many young and Nt−1 many old consumers at
date t whereNt
Nt−1= n
is the population growth rate.
At any time t, for each old person there are n many young
persons.
Golden Rule Allocation
Overlapping Generations Economy
In each period t, a new generation with size Nt is born.
Each generation lives for two periods.
There are Nt many young and Nt−1 many old consumers at
date t whereNt
Nt−1= n
is the population growth rate.
At any time t, for each old person there are n many young
persons.
Golden Rule Allocation
Overlapping Generations Economy
Remember: Each consumer lives for two periods.
Each consumer has an endowment of y units of the
consumption good when young.
When old, a consumer has no endowment.
Consumers want to consume in both periods of their lives.
Golden Rule Allocation
Overlapping Generations Economy
Remember: Each consumer lives for two periods.
Each consumer has an endowment of y units of the
consumption good when young.
When old, a consumer has no endowment.
Consumers want to consume in both periods of their lives.
Golden Rule Allocation
Overlapping Generations Economy
Remember: Each consumer lives for two periods.
Each consumer has an endowment of y units of the
consumption good when young.
When old, a consumer has no endowment.
Consumers want to consume in both periods of their lives.
Golden Rule Allocation
Overlapping Generations Economy
Remember: Each consumer lives for two periods.
Each consumer has an endowment of y units of the
consumption good when young.
When old, a consumer has no endowment.
Consumers want to consume in both periods of their lives.
Golden Rule Allocation
Overlapping Generations Economy
Some notation:
c1,t is young age consumption at time t.
c2,t is old age consumption at time t.
A consumer born at time t has a utility function given by
U(c1,t , c2,t+1)
Golden Rule Allocation
Overlapping Generations Economy
Some notation:
c1,t is young age consumption at time t.
c2,t is old age consumption at time t.
A consumer born at time t has a utility function given by
U(c1,t , c2,t+1)
Golden Rule Allocation
Overlapping Generations Economy
Some notation:
c1,t is young age consumption at time t.
c2,t is old age consumption at time t.
A consumer born at time t has a utility function given by
U(c1,t , c2,t+1)
Golden Rule Allocation
Overlapping Generations Economy
Some notation:
c1,t is young age consumption at time t.
c2,t is old age consumption at time t.
A consumer born at time t has a utility function given by
U(c1,t , c2,t+1)
Golden Rule Allocation
Social Planner
The SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
where
Nt is the number of young at date t and each consume c1,t
Nt−1 is the number of old at date t and each consume c2,t
Nty is the total available endowment at time t.
Golden Rule Allocation
Social Planner
The SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
where
Nt is the number of young at date t and each consume c1,t
Nt−1 is the number of old at date t and each consume c2,t
Nty is the total available endowment at time t.
Golden Rule Allocation
Social Planner
The SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
where
Nt is the number of young at date t and each consume c1,t
Nt−1 is the number of old at date t and each consume c2,t
Nty is the total available endowment at time t.
Golden Rule Allocation
Social Planner
The SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
where
Nt is the number of young at date t and each consume c1,t
Nt−1 is the number of old at date t and each consume c2,t
Nty is the total available endowment at time t.
Golden Rule Allocation
Social Planner
SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
Divide both sides by Nt and assume c1,t = c1 and c2,t = c2
c1 +
(Nt−1
Nt
)c2 ≤ y .
SP’s budget set is thus given by (see Figure 1)
c1 +c2n≤ y
Golden Rule Allocation
Social Planner
SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
Divide both sides by Nt and assume c1,t = c1 and c2,t = c2
c1 +
(Nt−1
Nt
)c2 ≤ y .
SP’s budget set is thus given by (see Figure 1)
c1 +c2n≤ y
Golden Rule Allocation
Social Planner
SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
Divide both sides by Nt and assume c1,t = c1 and c2,t = c2
c1 +
(Nt−1
Nt
)c2 ≤ y .
SP’s budget set is thus given by (see Figure 1)
c1 +c2n≤ y
Golden Rule Allocation
Social Planner
SP’s budget set is
Ntc1,t + Nt−1c2,t ≤ Nty
Divide both sides by Nt and assume c1,t = c1 and c2,t = c2
c1 +
(Nt−1
Nt
)c2 ≤ y .
SP’s budget set is thus given by (see Figure 1)
c1 +c2n≤ y
Golden Rule Allocation
Social Planner
SP’s budget set is
c1 +c2n≤ y
Key observation: By reducing young age consumption by
one unit, the SP can increase old age consumption by n units.
Reason: At any time t, for each old person there are n many
young persons.
Golden Rule Allocation
Example 1
When young, a consumer has y = 400, and has no
endowment when old.
Population grows at a rate n = 1.25.
The slope of a consumer’s indifference curve is
− c23c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
Example 1
When young, a consumer has y = 400, and has no
endowment when old.
Population grows at a rate n = 1.25.
The slope of a consumer’s indifference curve is
− c23c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
Example 1
When young, a consumer has y = 400, and has no
endowment when old.
Population grows at a rate n = 1.25.
The slope of a consumer’s indifference curve is
− c23c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
Example 1
When young, a consumer has y = 400, and has no
endowment when old.
Population grows at a rate n = 1.25.
The slope of a consumer’s indifference curve is
− c23c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 2)
c1 +1
nc2 ≤ 400
c1 +1
1.25c2 ≤ 400.
1.25c1 + c2 ≤ 500.
The slope of SP’s budget line is −1.25.
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 2)
c1 +1
nc2 ≤ 400
c1 +1
1.25c2 ≤ 400.
1.25c1 + c2 ≤ 500.
The slope of SP’s budget line is −1.25.
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 2)
c1 +1
nc2 ≤ 400
c1 +1
1.25c2 ≤ 400.
1.25c1 + c2 ≤ 500.
The slope of SP’s budget line is −1.25.
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 2)
c1 +1
nc2 ≤ 400
c1 +1
1.25c2 ≤ 400.
1.25c1 + c2 ≤ 500.
The slope of SP’s budget line is −1.25.
Golden Rule Allocation
An Example
The golden rule allocation (cGR1 ,cGR2 ) must satisfy two
equations.
First, it must be on the budget line of SP
1.25cGR1 + cGR2 = 500. (1)
Second, the slope of SP’s budget line must be equal to slope
of the consumer’s indifference curve at the golden rule
−1.25 = − cGR2
3cGR1
(2)
Golden Rule Allocation
An Example
The golden rule allocation (cGR1 ,cGR2 ) must satisfy two
equations.
First, it must be on the budget line of SP
1.25cGR1 + cGR2 = 500. (1)
Second, the slope of SP’s budget line must be equal to slope
of the consumer’s indifference curve at the golden rule
−1.25 = − cGR2
3cGR1
(2)
Golden Rule Allocation
An Example
The golden rule allocation (cGR1 ,cGR2 ) must satisfy two
equations.
First, it must be on the budget line of SP
1.25cGR1 + cGR2 = 500. (1)
Second, the slope of SP’s budget line must be equal to slope
of the consumer’s indifference curve at the golden rule
−1.25 = − cGR2
3cGR1
(2)
Golden Rule Allocation
An Example
From (2), we write
− cGR2
3cGR1
= −1.25⇒ cGR2 = 3.75cGR1 (3)
We substitute (3) into (1)
1.25cGR1 + cGR2 = 500
⇒ 1.25cGR1 + 3.75cGR1 = 500
⇒ 5cGR1 = 500
⇒ cGR1 = 100 and cGR2 = 375.
Golden Rule Allocation
An Example
From (2), we write
− cGR2
3cGR1
= −1.25⇒ cGR2 = 3.75cGR1 (3)
We substitute (3) into (1)
1.25cGR1 + cGR2 = 500
⇒ 1.25cGR1 + 3.75cGR1 = 500
⇒ 5cGR1 = 500
⇒ cGR1 = 100 and cGR2 = 375.
Golden Rule Allocation
An Example
From (2), we write
− cGR2
3cGR1
= −1.25⇒ cGR2 = 3.75cGR1 (3)
We substitute (3) into (1)
1.25cGR1 + cGR2 = 500
⇒ 1.25cGR1 + 3.75cGR1 = 500
⇒ 5cGR1 = 500
⇒ cGR1 = 100 and cGR2 = 375.
Golden Rule Allocation
Example 2
When young, a consumer has y = 600, and has no
endowment when old.
Population is constant, that is, n = 1.
The slope of a consumer’s indifference curve is
− c24c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
Example 2
When young, a consumer has y = 600, and has no
endowment when old.
Population is constant, that is, n = 1.
The slope of a consumer’s indifference curve is
− c24c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
Example 2
When young, a consumer has y = 600, and has no
endowment when old.
Population is constant, that is, n = 1.
The slope of a consumer’s indifference curve is
− c24c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
Example 2
When young, a consumer has y = 600, and has no
endowment when old.
Population is constant, that is, n = 1.
The slope of a consumer’s indifference curve is
− c24c1
Find the golden rule allocation (cGR1 ,cGR2 ).
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 3)
c1 +1
nc2 ≤ 600
c1 + c2 ≤ 600.
The slope of SP’s budget line is −1.
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 3)
c1 +1
nc2 ≤ 600
c1 + c2 ≤ 600.
The slope of SP’s budget line is −1.
Golden Rule Allocation
An Example
The SP’s budget set is (see Figure 3)
c1 +1
nc2 ≤ 600
c1 + c2 ≤ 600.
The slope of SP’s budget line is −1.
Golden Rule Allocation
An Example
The golden rule allocation (cGR1 ,cGR2 ) must satisfy two
equations.
First, it must be on the budget line of SP
cGR1 + cGR2 ≤ 600. (4)
Second, the slope of SP’s budget line must be equal to slope
of the consumer’s indifference curve at the golden rule
−1 = − cGR2
4cGR1
(5)
Golden Rule Allocation
An Example
The golden rule allocation (cGR1 ,cGR2 ) must satisfy two
equations.
First, it must be on the budget line of SP
cGR1 + cGR2 ≤ 600. (4)
Second, the slope of SP’s budget line must be equal to slope
of the consumer’s indifference curve at the golden rule
−1 = − cGR2
4cGR1
(5)
Golden Rule Allocation
An Example
The golden rule allocation (cGR1 ,cGR2 ) must satisfy two
equations.
First, it must be on the budget line of SP
cGR1 + cGR2 ≤ 600. (4)
Second, the slope of SP’s budget line must be equal to slope
of the consumer’s indifference curve at the golden rule
−1 = − cGR2
4cGR1
(5)
Golden Rule Allocation
An Example
From (5), we write
− cGR2
4cGR1
= −1⇒ cGR2 = 4cGR1 (6)
We substitute (6) into (4)
cGR1 + cGR2 ≤ 600.
⇒ cGR1 + 4cGR1 = 600
⇒ 5cGR1 = 600
⇒ cGR1 = 120 and cGR2 = 480.
Golden Rule Allocation
An Example
From (5), we write
− cGR2
4cGR1
= −1⇒ cGR2 = 4cGR1 (6)
We substitute (6) into (4)
cGR1 + cGR2 ≤ 600.
⇒ cGR1 + 4cGR1 = 600
⇒ 5cGR1 = 600
⇒ cGR1 = 120 and cGR2 = 480.
Golden Rule Allocation
An Example
From (5), we write
− cGR2
4cGR1
= −1⇒ cGR2 = 4cGR1 (6)
We substitute (6) into (4)
cGR1 + cGR2 ≤ 600.
⇒ cGR1 + 4cGR1 = 600
⇒ 5cGR1 = 600
⇒ cGR1 = 120 and cGR2 = 480.
Golden Rule Allocation
c₁
c₂
c₁ + c₂𝑛𝑛
= 𝑦𝑦
𝑦𝑦
𝑛𝑛𝑦𝑦
Figure 1: Social Planner’s Budget Set
slope = −𝑛𝑛
c₁
c₂c₁ + c₂
1.25= 400
400
Figure 2: SP Budget Set (Example 1)
500
slope = −1.25
c₁
c₂c₁ +c₂ = 600
600Figure 3: SP Budget Set (Example 2)
600slope = −1