The Overlapping Generations Model (Romer chapter 2, Part B) By Ole Hagen Jørgensen,
OJ@cebr.dk4/10 2006
The Overlapping Generations Model (Romer chapter 2, Part B) By Ole Hagen Jørgensen,
OJ@cebr.dk
4/10 2006
Introduction
I will teach three lectures:
One lecture: on the basic OLG model in Romer (2001), chapter 2, part B
One lecture: on a recently developed solution method for the OLG model:
One lecture: on the Real Business Cycle literature (RBC)
My own research actually applies RBC-techniques for solving OLG models!
There will be a presentation in PowerPoint on each subject - Therefore, you will receive 3 handouts (or download from Blackboard or www.cebr.dk/oj)
Intergenerational issues
Motivation for life-cycle approach to economic dynamics
Life-cycle aspects of human behavior are important to study…
We model explicitly the different periods of life
Distribution of welfare over generations
How the choices of one generation can affect the succeeding generation
How different exogenous shocks to the economy affects different generations (demographic shocks, productivity shocks)
Intergenerational transfers
Purpose: If the market equilibrium allocates consumption unevenly across generations there may be a scope for redistribution.
How: Taxes and benefits
Case: Pensions, education, health
The model
The OLG model will be presented according to the following outline:
Description of the economy
Basic assumptions
Demographics
Household utility
Life-cycle consumption
Firms
Resources
Dynamics of the economy
Household utility maximization
Capital accumulation and Steady State
Case study
Efficiency and welfare
Dynamic efficiency
Government policy
Overall assumptions about the economy:
Time is discrete
There is one good, to be consumed or saved/invested
The economy “lives” on forever (no last generation)
Individuals have finite lifetime (finite horizon)
Infinite number of agents
Closed economy
Perfect competition
Absence of externalities
No government sector (could be included easily)
No uncertainty (perfect foresight)
Of course the economy has more detailed characteristics – we turn to
those when relevant…
The life-cycle of generations
The lifetime is divided into two periods: young and old
When individuals are young they work
When individuals are old they are retired
One period therefore amounts to a half lifetime
Who are alive at the same time? (vertical box)
We want to inspect the behavior of one specific generation (horizontal box)
We trace generation “0” that is born at time t=0 and is old in t+1
Demographics
Assumptions on the demographic structure of the economy
Could be modeled in great detail
Different sexes
Survival probabilities
Different skills by different people
Very simple assumption in this model
Fixed growth rate of the population over generational periods:
(Equivalent to the continuous time variant )
where Lt is the number of individuals born at time t,
where n is the growth rate of the population
Individuals derive utility only from consumption in their two periods of life
Two factors determine how individuals decide to divide consumption over time in a risk-free (certain/perfect foresight) environment
The consumption “smoothing” motive, captured by the term ρ
The consumption “fluctuation” motive, captured by the term θ
We discuss each in turn…
Household utility
The consumption “smoothing” motive
Individuals generally like to smooth (evenly divide) their consumption over periods
The degree of impatience towards consuming today is captured by the discount rate, ρ
The discount rate of future consumption is generally 1/(1+ρ) so that household utility can be represented in present value terms as:
Household utility
Consumption “fluctuation” motive
Uncertain environment
In an uncertain environment you might be risk averse, and might not be willing to shift consumption very freely over time.
Say, if you decide to smooth consumption 50/50 over your two periods, and if you are uncertain about how your consumption will vary your tend to stick to the safer level in each period.
If you expect the interest rate to increase in the next period, you would get a higher lifetime consumption if you shift some units of consumption. If you are risk averse you would rather stick to the safer levels of consumption – you then miss out on the extra consumption
θ measures the degree of consumption risk aversion
Certain environment
In this case there is no risk (perfect foresight)
You can still appreciate stable consumption levels in each period, so the parameter θ then measures the degree to which you like stable and consumption
Again, if you expect the interest rate to increase in the future period, and you prefer consumption not to fluctuate – you will then not take advantage of the potentially higher lifetime consumption.
θ measures the degree of consumption fluctuation aversion
Household utility
The utility function:
utility from consumption
features intertemporal consumption smoothing motive, ρ
features consumption fluctuation motive, θ
We divide by (1-θ) to ensure positive marginal utility in case θ>1
Note: for ρ>0, second period utility is valued less than first period utility
We assume that ρ>-1: weight on second period consumption>0.
Also,
People live for two periods, as adults and as old, and they need to consume in each period
Adults (workers)
The adults work, consume, and save:
where
Old (retirees)
The elderly are retired, and consume (they do not work, but live of their savings and interest earnings)
Intertemporal budget constraint (IBC)
Firms use two factors in production: labor and capital
Firms pay the wage rate, wt , to the labor, Lt, supplied by workers
Firms rent capital, Kt, from retirees at a rental price of rt
The production function is generally:
Due to CRS we can restate the capital in efficiency units, where:
The wage rate is the marginal product of labor in production:
Return to capital is defined by marginal product of capital in production (assume no capital depreciation, δ=0)
Total Returns:
We have seen the consumption budget constraint for the household (IBC)
There is also a constraint on consumption for the economy as a whole: society cannot prioritize over more than is actually produced (closed economy)
In each period people save and consume (to save is to invest):
Resource constraint (RC):
In efficiency units:
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