GROWTH WITH OVERLAPPING GENERATIONS, THE DIAMOND MODEL

GROWTH WITH OVERLAPPING GENERATIONS, THE DIAMOND MODEL

The central difference between the Diamond model and the Ramsey–Cass–Koopmans model is that there is turnover in the population: new individuals are continually being born, and old individuals are continually dying. To further simplify the analysis, the model assumes that each individual lives for only two periods.

Assumptions:- ( Please read all t as in subscript)

Lt individuals are born in period t (time is discrete). Population grows at rate n; thus Lt = (1 + n)Lt−1. Since individuals live for two periods, at time t there are Lt individuals in the first period of their lives and Lt−1 = Lt/(1+n) individuals in their second periods. Each individual supplies 1 unit of labour when he or she is young and divides the resulting labour income between first-period consumption and saving. In the second period, the individual simply consumes the saving and any interest he or she earns.

Let C1t and C2t denote the consumption in period t of young and old individuals .

We again assume constant-relative-risk-aversion utility:

Because life- times are finite, we no longer have to assume ρ > n + (1 − θ)g to ensure that lifetime utility does not diverge. If ρ > 0, individuals place greater weight on first-period than second-period consumption; if ρ < 0, the situation is reversed.

The production function is given by

F (•) has constant returns to scale and satisfies the Inada conditions, and A grows at exogenous rate g (so At = [1 + g]At−1). Markets are competitive; thus labour and capital earn their marginal products, and firms earn zero profits. there is no depreciation. The real interest rate and the wage per unit of effective labour are therefore given by

Thus, in period 0 the capital owned by the old and the labour supplied by the young are combined to produce output. Capital and labour are paid their marginal products. The old consume both their capital income and their existing wealth; they then die and exit the model. The young divide their labour income, wtAt, between consumption and saving. They carry their saving for- ward to the next period; thus the capital stock in period t + 1, Kt +1, equals the number of young individuals in period t, Lt, times each of these individuals’ saving, wtAt − C1t. This capital is combined with the labour supplied by the next generation of young individuals, and the process continues.

HOUSEHOLD BEHAVIOUR

The second-period consumption of an individual born at t is

This condition states that the present value of lifetime consumption equals initial wealth (which is zero) plus the present value of lifetime labour income.

To solve this equation we will set up Lagrange as follow

It shows that the interest rate determines the fraction of income the individual consumes in the first period.

If we let s (r ) denote the fraction of income saved equation 11 implies

Thus s is increasing in r if θ is less than 1, and decreasing if θ is greater than 1.

So, we can write equation 11 as

The Dynamics of the Economy

The Equation of Motion of k :- As in the infinite-horizon model, we can aggregate individuals’ behaviour to characterize the dynamics of the economy. The capital stock in period t + 1 is the amount saved by young individuals in period t. Thus,

Note that because saving in period t depends on labour income that period and on the return on capital that savers expect the next period, it is w in period t and r in period t + 1 that enter the expression for the capital stock in period t + 1.

Limitations and Conclusion 

Overlapping generations of the more realistic than infinitely lived representative agents.

Models with overlapping generations fall outside the scope of the First Welfare Theorem: i.e they were partly motivated by the possibility of Pareto suboptimal allocations.

Equilibria may be dynamically inefficient and feature over accumulation: unfunded Social Security can enhance the problem.

Declining path of labor income important for over accumulation, and what matters is not infinite horizons but arrival of new individuals. Over accumulation and Pareto sub-optimality: pecuniary externalities created on individuals that are not yet in the marketplace.

Not overemphasised dynamic inefficiency: major question of economic growth is why so many countries have so little capital.

Although, this model has provided us with new modeling tools and new perspectives on the question of capital accumulation, aggregate saving and economic growth. It has not, however, offered new answers to questions of why countries grow (for example, technological progress) and why some countries are much poorer than others (related to the fundamental cause of income differences). 

References:- Romer, David (2006). Advanced Macroeconomics. McGraw-Hill.