ANNEX: Derivation of Criteria for Optimal Capital Accumulation
The theory of optimal capital accumulation was initiated by Phelps (1961) and Diamond (1965) and uses the neoclassical Solow-Swan model of economic growth. Its main contents can be summarized as follows. Let output be given by a linear homogenous production function Y=F(K,L) with inputs capital K and employment in efficiency terms L. Employment in efficiency terms is employment adjusted for labor augmenting technical progress, which is assumed to take place at rate g. In a closed economy output is either consumed or invested I in capital accumulation K’:
If the employment ratio is fixed, employment grows proportional to population at rate n. Normalization in per-worker terms (lower-case letters) gives the behavior of the per-worker capital stock k=K/L, which is proportional to the per-capita capital stock as
and in the steady state it holds that
In the steady state, investment per worker, i=f(k)-c, must equal the capital stock times the sum of the growth rate of population and the rate of technical progress so as to ensure that the capital stock grows in line with population (employment) growth and technical progress. This is true for any steady state and does not yet determine the optimal level of the capital stock. The desired level of capital stock is the level that maximizes the level of consumption. From c=f(k)-(n+g)k this is obviously the one for which holds:
To derive the implications of this rule for the level of savings, express total savings S as a fraction σ of the level of profits, which are given by the return to capital r times the capital stock:
The question then is: how high should total savings be in relation to profits, what is the optimal value of σ? Since savings equals investment and hence the change in the capital stock K’, the growth rate of the capital stock will be equal to σr=K’/K. In order to be in a steady state the capital stock has to grow at a K’/K=n+g. Combining these two conditions yields σr=n+g, from which, together with (4), it follows that σ=1 in the optimal, consumption maximizing steady state. This provides the second criterion for the golden rule: savings and hence investment should equal profits.
The criterion in section 4 is derived as follows. The return to capital r is given by the total return to capital (capital income over capital employed) net of depreciation d: r = αY/K - d. Since, under the modified golden rule, the net return equals the sum of population growth n, technological progress g and time preference rate p, it follows that K/Y = α/(n+g+p+d).
References Chapter II
Abel, A., N. Mankiw, L. Summers and R. Zeckhauser, “Assessing Dynamic Efficiency: Theory and Evidence,” Review of Economic Studies (London), Vol. 55 (1989) pp. 1–19.
Evans, O., “National Savings and Targets for the Federal Budget Balance,” Chap. 4 in The United States Economy: Performance and Issues. Y. Horiguchi et al. (eds.), (Washington: International Monetary Fund, 1992).
Miranda, K., “Does Japan Save Too Much?,” Chap. 2 in: Saving Behavior and the Asset Price “Bubble” in Japan - Analytical Studies. U. Baumgartner and G. Meredith (eds.), IMF Occasional Paper no. 124 (Washington: International Monetary Fund, April 1995)
Phelps, E., “The Golden Rule of Capital Accumulation: A Fable for Growthmen,” American Economic Review (Nashville) Vol. 51 (1961) pp.638–643.
This chapter was prepared by C. Thimann.
See, for example, IMF, World Economic Outlook, May 1995, Chapter V.
The choice of social discount rates is often regarded as an ethical question. One of the earliest and best known proponents of a zero social discount rate, on the grounds that present generations have no right to discount future generations’ welfare, was Ramsey (1928).
IMF, World Economic Outlook, May 1995, p. 83.
A modified golden rule, which attaches greater weight to the present generation by discounting future generations’ consumption, is discussed in section 4.
For simplicity the rate of technical progress and depreciation are assumed to be identical across countries and are set at average levels derived in other studies. Since we are only interested in an international comparison of rates of time preference and not their levels themselves, the levels of technical progress and depreciation are not important as such.