Golden Rule Allocation - Southern Methodist Universityfaculty.smu.edu/ozerturk/Golden-Rule-lecture.pdf · Golden Rule Allocation. Overlapping Generations Economy In each period t,

Golden Rule Allocation

September 2, 2019


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.






date t whereNt

Nt−1= n



persons.






date t where

Nt

Nt−1= n



persons.






date t whereNt

Nt−1= n



persons.






date t whereNt

Nt−1= n



persons.



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.
























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)



Some notation:




U(c1,t , c2,t+1)



Some notation:




U(c1,t , c2,t+1)



Some notation:




U(c1,t , c2,t+1)


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.


Social Planner



where





Social Planner



where





Social Planner



where





Social Planner

SP’s budget set is


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


Social Planner




c1 +

(Nt−1

Nt

)c2 ≤ y .


c1 +c2n≤ y


Social Planner




c1 +

(Nt−1

Nt

)c2 ≤ y .


c1 +c2n≤ y


Social Planner




c1 +

(Nt−1

Nt

)c2 ≤ y .


c1 +c2n≤ y


Social Planner


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.


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 ).


Example 1


endowment when old.



− c23c1



Example 1


endowment when old.



− c23c1



Example 1


endowment when old.



− c23c1



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.


An Example


c1 +1

nc2 ≤ 400

c1 +1

1.25c2 ≤ 400.

1.25c1 + c2 ≤ 500.



An Example


c1 +1

nc2 ≤ 400

c1 +1

1.25c2 ≤ 400.

1.25c1 + c2 ≤ 500.



An Example


c1 +1

nc2 ≤ 400

c1 +1

1.25c2 ≤ 400.

1.25c1 + c2 ≤ 500.



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)


An Example


equations.


1.25cGR1 + cGR2 = 500. (1)



−1.25 = − cGR2

3cGR1

(2)


An Example


equations.


1.25cGR1 + cGR2 = 500. (1)



−1.25 = − cGR2

3cGR1

(2)


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.


An Example

From (2), we write

− cGR2

3cGR1

= −1.25⇒ cGR2 = 3.75cGR1 (3)


1.25cGR1 + cGR2 = 500

⇒ 1.25cGR1 + 3.75cGR1 = 500

⇒ 5cGR1 = 500

⇒ cGR1 = 100 and cGR2 = 375.


An Example

From (2), we write

− cGR2

3cGR1

= −1.25⇒ cGR2 = 3.75cGR1 (3)


1.25cGR1 + cGR2 = 500

⇒ 1.25cGR1 + 3.75cGR1 = 500

⇒ 5cGR1 = 500

⇒ cGR1 = 100 and cGR2 = 375.


Example 2


endowment when old.

Population is constant, that is, n = 1.


− c24c1



Example 2


endowment when old.



− c24c1



Example 2


endowment when old.



− c24c1



Example 2


endowment when old.



− c24c1



An Example


c1 +1

nc2 ≤ 600

c1 + c2 ≤ 600.

The slope of SP’s budget line is −1.


An Example


c1 +1

nc2 ≤ 600

c1 + c2 ≤ 600.



An Example


c1 +1

nc2 ≤ 600

c1 + c2 ≤ 600.



An Example


equations.


cGR1 + cGR2 ≤ 600. (4)



−1 = − cGR2

4cGR1

(5)


An Example


equations.


cGR1 + cGR2 ≤ 600. (4)



−1 = − cGR2

4cGR1

(5)


An Example


equations.


cGR1 + cGR2 ≤ 600. (4)



−1 = − cGR2

4cGR1

(5)


An Example

From (5), we write

− cGR2

4cGR1

= −1⇒ cGR2 = 4cGR1 (6)


cGR1 + cGR2 ≤ 600.

⇒ cGR1 + 4cGR1 = 600

⇒ 5cGR1 = 600

⇒ cGR1 = 120 and cGR2 = 480.


An Example

From (5), we write

− cGR2

4cGR1

= −1⇒ cGR2 = 4cGR1 (6)


cGR1 + cGR2 ≤ 600.

⇒ cGR1 + 4cGR1 = 600

⇒ 5cGR1 = 600

⇒ cGR1 = 120 and cGR2 = 480.


An Example

From (5), we write

− cGR2

4cGR1

= −1⇒ cGR2 = 4cGR1 (6)


cGR1 + cGR2 ≤ 600.

⇒ cGR1 + 4cGR1 = 600

⇒ 5cGR1 = 600

⇒ cGR1 = 120 and cGR2 = 480.


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

Golden Rule Allocation - Southern Methodist Universityfaculty.smu.edu/ozerturk/Golden-Rule-lecture.pdf · Golden Rule Allocation. Overlapping Generations Economy In each period t,

Documents