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