OLG Model

#economics #macro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Kang, Minwook.

Over-Lapping-Generation, i.e. Diamond Model

population growth rate determined by birth rate
finite lifetime of household

Assumptions

Model Setting

discrete time: .
2-period lifetime: the young born at , the old die at
Labor supplied by the young, Capital owned by the old
the young divide their labor income into consumption and savings, which equals to capital stock in the next period
Constant population & technology growth rate: , .

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Behaviors of Firms

Same as Solow Model, we have a production function

: CRS function
, , where are exogenous.
: capital stock at is owned by old.
: depreciation rate, : saving rate

Profit Maximization Conditions Profit Maximization Conditions:

Behaviors of Households

CRRA Utility Functions:

: constant relative-risk-aversion coefficient
: discount rate, : discount factor
: consumption of young() at time
: consumption of old() at time

Budget Constraint

Young divides their labor income into consumption and savings. While old consumes their savings when they were young. Thus the composite constraint is:

Optimization Problems

Problem: Lagrangian: F.O.C. The Euler equation is: thus the consumption growth rate increases when

interest rate increases; then and .
discount rate decreases; then and .
relative-risk-aversive rate decreases; then the consumer's response on and decreases.

Saving Rate

From the budget constraint, where the individual saving rate is

Saving rate and Interest rate

By taking derivatives with , we have

If (Logarithmic utility), then is independent of , thus
- substitution effect of saving equals to its wealth effect.
- change in the interest rate does not affect the saving rate.
- thus the consumption is not affected by the interest rate.
If , then
- substitution effect is greater than wealth effect.
- the saving rate increases when the interest rate goes up.
If , then
- substitution effect is smaller than wealth effect.
- the saving rate decreases when the interest rate goes up.

Dynamics

Dynamics of

Since the capital stock is determined by the savings by young, as , where the third equality holds by Profit maximization conditions Profit maximization conditions.

Steady State

Cobb-Douglas Function

Suppose and . then we have thus the dynamic equation of is at Steady-State, since , thus the S.S. capital stock per effective labor increases when

: labor share increases
, : growth rate of technology or population decreases
: discount rate decreases

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Note that in Diamond model, the Steady-State analysis is 1-dimensional problem.

General Function

S.S. exists if: is concave increasing function, , .
S.S. is stable if: : if not, then is not stable.

From the dynamic equation:

if increases, then decreases by diminishing marginal productivity, thus the interest rate decreases.
if , then since substitution effect greater than the wealth effect, decreasing in will decrease .
thus if labor share does not increases constantly, we have .

Dynamic Inefficiency

Since the optimization problem in Diamond model is only considered in each generation separately, the equilibrium may not be dynamically efficient:

Their decision making is only for their own lifetime, not for the social welfare as a whole.
Diamond model does not assume any trade between young and old generation.
If inter-generational redistribution is possible, then the utility of total society may bettter off.

Golden Rule

Similar to Solow Model Solow Model, consider a budget constraint as a whole economy: Assume , , and . then we have
therefore, if the production function is Cobb-Douglas form, then

Dynamic Inefficiency

The equilibrium equations are: since the dynamic inefficiency occurs when i.e. .

labor's share is big ( is small); or is small
over-savings decrease in interest rate inefficiency
if the social planner makes a decision, there is another solution that can maximize everyone's consumption.
thus OLG solution is not pareto optimal pareto optimal.