Overlapping generations model

For the topic in population genetics, see Overlapping generations.

An overlapping generations model, abbreviated to OLG model, is a type of economic model in which agents live a finite length of time and live long enough to endure into at least one period of the next generation's lives.

All OLG models share several key elements:

• Individuals receive an endowment of goods at birth.
• Goods cannot endure for more than one period.
• Money endures for multiple periods.
• Individuals must consume in all periods, and their lifetime utility is a function of consumption in all periods.

The concept of an OLG model was inspired by Irving Fisher's monograph The Theory of Interest. Notable improvements were published by Maurice Allais in 1947, Paul Samuelson in 1958, and Peter Diamond in 1965.

Basic OLG model

Generational Shifts in OLG Model

The most basic OLG model has the following characteristics :

• Individuals live for two periods; in the first period of life, they are referred to as the Young. In the second period of life, they are referred to as the Old.
• A number of individuals is born in every period.N^t_t denotes individuals born in period t.
• N^t-1_t denotes number of old people in period t. Since the economy begins in period 1. In period 1, there is a group of people who are already old. They are referred to as the initial old. They can be denoted as N₀ .
• The size of the initial old generation is normalized to 1 i.e. N ⁰₀ = 1.
• People do not die early, N ^t_t = N ^t_t+1.
• Population grows at a constant rate n:

             Ntt = (1+n)t N00 = (1+n)t

• There is only one good in this economy, and it cannot endure for more than one period.
• Each individual receives a fixed endowment of this good at birth. This endowment is denoted as y. This endowment of goods can also be thought of as an endowment of labor that the individual uses to work and create a real income equal to the value of good y produced. Under this framework, individuals only work during the young phase of their life.
• Preferences over consumption streams is given by:

                  u(ctt,ctt+1) =  U(ctt) + βU(ctt+1)

'

First Welfare Theorem

First Welfare Theorem states that it is not possible to increase the aggregate demand or the aggregate utility of the society more than the aggregate supply available in the society.So, the total utility derived by a society must be equal to the total supply in the society. It states that as the resources in the economy are limited, the pareto optimal outcome is achieved by the invisible hands of the market forces and it is not possible to live beyond the means to achieve a pareto-improving situation.Incase every agent has a positive quantity of every good and also has a utility function that is convex, continuous and strictly increasing then the First Welfare Theorem holds.The first welfare theorem states that a Walrasian equilibrium is weakly pareto optimal.This theorem is true in a large number of general static equilibrium models.

Pure-Exchange OLG model

Two important aspects of the OLG model are that the steady state equilibrium need not be unique nor efficient. Essentially, because there is an infinite number of agents in the economy (over time) there is no prior restriction on the differential equation that relates the capital stock to investment (note that the First welfare theorem requires that there be a finite number of consumers in an economy). Hence, multiple equilibria, even a continuum of them, are possible. The Cass Criterion gives necessary and sufficient conditions for when an OLG competitive equilibrium allocation is inefficient.

Furthermore, it is possible that 'over saving' can occur - a situation which could be improved upon by a social planner. Since there is an infinite number of generations, a social planner could transfer some consumption from one generation to the previous one, compensate the first generation with a transfer from the next and so on, into infinity. However, certain restrictions on the underlying technology of production and consumer tastes can ensure that the steady state level of saving corresponds to the Golden Rule savings rate of the Solow growth model and thus guarantee intertemporal efficiency. Along the same lines, most empirical research on the subject has noted that oversaving does not seem to be a major problem in the real world.

A third fundamental contribution of OLG models is that they justify existence of money as a medium of exchange. In an OLG model, money is a welfare-improving "innovation."

Comparison between the First Welfare Theorem and Pure-Exchange OLG model

With respect to the overlapping generations model, the first welfare theorem breaks down because of the simple fact that what is impossible in the first welfare theorem premise becomes possible because of the ability to borrow from the future generations and thus in theory, there is an infinite amount of resources available at the disposal of the society as long as it keeps borrowing for the future. It must be understood that as you stretch your future leverage more and more the borrowing will get more and more expensive but the fundamental difference is that what is impossible in the First Welfare theorem becomes possible, albeit expensive in the Overlapping Generations model.Samuelson’s model while holding the premise and the theory of the First Welfare Theorem manages to breach it.Indeed in Samuelson’s model it is possible to improve on the resources by redistribution from old to young forever.

OLG models with Production

A OLG model with an aggregate neoclassical production was constructed by Peter Diamond. A two-sector OLG model was developed by Oded Galor.

Unlike the Ramsey growth model the steady state level of capital need not be unique. Moreover, as demonstrated by Diamond (1965), the steady-state level of the capital labor ratio need not be efficient which is termed as "dynamic inefficiency".

The Diamond OLG Model

Convergence of OLG Economy to Steady State

The economy has the following characteristics :

• 1. Two generations are alive at any point in time, the young (age 1) and old (age 2).
• 2. The size of the young generation in period t is given by N_t = N₀ E^t.
• 3. Households work only in the first period of their life and earn Y_1,t income. They earn no income in the second period of their life (Y_2,t+1 = 0)
• 4. They consume part of their first period income and save the rest to finance their consumption when old.
• 5. At the end of period t, the assets of the young are the source of the capital used for aggregate production in period t+1.So K_t+1 = N_t,a_1,t where a_1,t is the assets per young household after their consumption in period 1.In addition to this there is no depreciation.
• 6. The old in period t own the entire capital stock and consume it entirely, so dissaving by the old in period t is given by N_t-1,a_1,t-1 = K_t.
• 7. Labor and capital markets are perfectly competitive and the aggregate production technology is CRS, Y = F(K,L).

The article so far has considered The Diamond OLG Model and if the competitive equilibrium of the economy features dynamic inefficiency, its citizens save more than what is socially optimal. Hence Government programs are needed that reduce national saving wherein arises the need for economic applications of OLG model.

Economic Applications

Now that we have considered the theoretical understanding of the model, listed below are the practical applications of the model:

• Pay-As-you-go Social Security Schemes'

One of the best examples of the use of Overlapping generations model is the Social Security net that countries provides to its citizens when the current young provides for pensions and other social security measures for the old.

• OLG and Public Debt'

Overlapping Generations model also provides the basis for Public Financing of government debt because the future generations of the economy are expected to pay at least a part of the debt undertaken by the nation.The level of sustainability of Public debt is variable and depends on the demographics, current and future spending and other factors and most importantly the growth prospects.The reality of the OLG model (new generations come along always) allows the economies to borrow from future generations and thus provide for higher wealth for the current generations and also provide the basic model for deficit financing for the government with their ability to borrow at a low rate of interest.However, if the population and the economic growth rate don’t match, then it creates an imbalance and leads to budgetary deficit problems for the economy.

See also

• Peter A. Diamond
• Karl Shell
• Macroeconomic model
• First Welfare Theorem
• Walrasian equilibrium
• static