4. Overlapping-Generations Model of Economic Growth

Definitions of physical capital will vary depending on the purpose at hand. In some cases physical capital is defined to include inventories, software, land, and other inputs that extend beyond plant and equipment.

You can think of the value of r_t as actually determined in period t−1. In that period households make their saving decision based on the firms’ commitments to rent capital in period t and pay the rental rate r_t. In other words, r_t is determined in period t−1 based on the savings behavior of households and the planned investment demands of firms.

The economy never literally reaches the steady state, although it will get arbitrarily close.

The weakening effect of the capital-labor ratio on wages, stems from the diminishing marginal product of capital. As capital accumulates relative to labor, the effect of further capital accumulation on output and wages gets smaller. Formally, note that the effect of an increase in k on the marginal product of labor is (1 − α)αAk^α − 1 = (1 − α)×marginal product of capital.

There are differences of opinion about what qualifies as an appropriate target. Some believe that calibration should not involve previous econometric estimation. According to this view, all parameters within a model should be set to match particular data points or statistical moments of a data set (sample means, variances, and covariances), but not to match econometric estimates found in the literature. Others broaden the targets to include previous statistical estimates of the model’s parameters and behavioral responses, even if the model used in the estimation is not the same as the one used in the calibration. We are comfortable with either approach. The important point from our perspective is that all quantitative models, however calibrated, should be tested by comparing their predictions against observations or statistics not used in the calibration process. The fact that these “tests” or comparisons are not as formal and refined as traditional hypothesis testing in statistics does not particularly concern us. At this stage in the profession’s understanding of macroeconomics, models that even roughly approximate reality are difficult to find. Hopefully, as our approximations become more refined, we will need to worry about more formal testing procedure.

For a further discussion of the issues associated with quantifying overlapping generations models see Appendix B of Chap. 2 from Das et al. (2018).

Estimates of the rate of depreciation in academic studies tend to be smaller than the estimates used by national income accounts, where the rate of depreciation is typically in the 10–15 percent rage. The higher the rate of depreciation, the more accurate is the simplification used in future chapters where we assume the capital stock completely depreciates after a single period of the model (associated with continual use for 30 actual years).

See Das et al. (2018, Chaps. 2 and 4) for an extension of the quantitative theory to include human capital accumulation through public schooling.