The Canonical Overlapping Generations Model

Even the model with CRRA utility and Cobb-Douglas production function is relatively messy. For this reason, many of the applications of the overlapping generations model use an even more specific utility function, log preferences (or equivalently θ = 1 in terms of the CRRA preferences of the last section).

Since this version of the model is sufficiently common, it may deserve to be called the canonical overlapping generations model and will be the focus of this section. Another inter­esting feature of this model is that the structure of the equilibrium is essentially identical to the basic Solow model we studied in Chapter 2.

Suppose that the utility of the household and generation t is given by where as before β ∈ (0,1) (even though β ≥ 1 could be allowed here without any change in the analysis). The aggregate production technology is again Cobb-Douglas, that is,

The consumption Euler equation now becomes even simpler:

and implies that savings should satisfy the equation

(9.19)

Figure 9.2. Equilibrium dynamics in the canonical overlapping generations model.

which corresponds to a constant saving rate, equal to β/ (1 + β), out of labor income for each individual.

Now combining this with the capital accumulation equation (9.8), we obtain

where the second line uses (9.19) and the last line uses the fact that, given competitive factor markets, the wage rate is equal to w

It is straightforward to verify that there exists a unique steady state with capital-labor ratio given by

Moreover, starting with any k (0) > 0, equilibrium dynamics are identical to those of the basic Solow model and monotonically converge to k*. This is illustrated in Figure 9.2 and stated in the next proposition:

PROPOSITION 9.5. In the canonical overlapping generations model with log preferences and Cobb-Douglas technology, there exists a unique steady state, with capital-labor ratio k* given by (9.20). Starting with any k (0) ∈ (0,k*), equilibrium dynamics are such that k (t) ↑ k*, and starting with any k' (0) > ê*, equilibrium dynamics involve.

Exercise 9.6 asks you to introduce technological progress into this canonical model and to conduct a range of comparative static exercises. Exercise 9.7, on the other hand, asks you to analyze the same economy without the Cobb-Douglas technology assumption.

9.4.