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# Golden Rule in Growth Models

BIBLIOGRAPHY

The Nobel laureate Robert M. Solow (b. 1924) famously argued that a steady state growth will involve a higher savings and a higher capital-to-labor ratio to achieve a higher per capita income. Higher per capita income, however, does not automatically imply higher per capita consumption. Therefore this is a typical case of optimization in which we have to find the level of capital-to-labor ratio that provides at the same time the maximum possible per capita consumption. This problem was posed initially by Edmond Phelps (1961), and the solution provided was called the golden rule of accumulation (growth). According to this rule, an economy reaches the optimum growth rate at the point where each generation saves and invests at the level that it wishes the previous generation would have invested. That is, each generation saves (on behalf of future generations, as it were) that fraction of income that it would have liked past generations to have saved, subject to the constraint that all generations past and present are to save the same fraction of income. This condition requires that the rate of profit (interest) equals the rate of growth of an economy.

Let us define the per capita consumption (c ) as the difference between the per capita income (f (k )) and the per capita savings/investment (sf (k )), that is: c = f (k ) sf (k ). Equilibrium is attained at the point where the per capita saving (investment) is equal to the amount needed to keep the increasing labor force (n ) with the same capital equipment (growth rate of capital labor ratio equals zero). Formally, sf (k ) = nk. If we now revise the previous equation accordingly, we have c = f (k ) nk. Because we are not interested in any level of consumption, but rather the maximum possible, we get the first derivative of the above equation and we set it equal to zero: dc/dk = f' (k ) n = 0 and f' (k ) = n. Thus the marginal product of capital (read rate of profit) equals the growth rate of the economy, because at the steady state growth rate, the growth rate of output equals the growth rate of labor force.

It is clear that public policy and private propensities can be designed to achieve this golden rule. That is, for example, even though a relatively low capital-poor economy will pursue optimum growth by steadily increasing its capital-to-labor ratio, it may accomplish this by saving relatively more in the future than in the present. However, major concerns about the existence of the golden rule of growth have been raised that refer mainly to the conditions for its existence. Some economists suggest that the rule exists only in neoclassical models in which capital and labor are continuously substitutable, there is no technical progress, unlimited influence on subsequent generations choices is present, proper population policies exist, and so on. Phelps showed that the golden rule always exists in the neoclassical and Harrod-Domar models if the labor force increases at a constant rate, the depreciation rate is constant, technical progress, if any, is purely labor-augmenting, labor augmentation occurs at a constant rate, and positive labor is required for positive output (Phelps 1965, p. 812). Otherwise, the growth paths are dynamically inefficient and the golden growth rate of accumulation is unattainable.

SEE ALSO Economic Growth; Neoclassical Growth Model; Optimal Growth; Solow, Robert M.

## BIBLIOGRAPHY

Phelps, Edmond. 1961. The Golden Rule of Accumulation: A Fable for Growthmen. American Economic Review 51 (4): 638643.

Phelps, Edmond. 1965. Second Essay on the Golden Rule of Accumulation. American Economic Review 55 (4): 793814.

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