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United States and German real capital formation and social investment in the sciences and humanities.


This brief essay links together two distinct rules for optimizing economic growth. One is the golden rule for aggregate saving, and the other is the optimal rule for maximizing a particular kind of social investment, that in the state of the arts. The latter requires application of Euler's equation from the calculus of variations. The former means basically choosing the proper discount rate between current and future consumption of the work force, measured in efficiency units. Two countries, the United States and Germany, are tested for their performance. Previous work has taken two forms: One is to define an optimal level of investment in progress, as in Barro (2001) or Gehrels (2010); the other is to compare countries at different levels of development and to seek the causes for their differences. Prominent here are Hall and Jones (1999), Hanushek and Woessmann (2008), and Viane and Zilcha (2009).

Testing against the Golden Rule

This first issue examined here is whether at near full employment the United States economy succeeded in providing the amount of real capital formation which comes near that required to satisfy the Golden Rule of Investment, first put forward by Joan Robinson and exposited by a number of other authors. The accepted finding is that in an economy driven by inputs of labor and capital, and subject to constant returns to scale in the aggregate, but diminishing returns to proportion, an amount of net investment is required such that the marginal product of capital just matches the growth rate of the work force, as measured in efficiency units. This in turn requires that an amount of investment matching the share of capital in the social product be invested continuously over time. For that purpose, one can take either the net share in the net national product, or the gross share in the gross product. Strictly speaking, the observer would want a number of successive years of full employment. Lacking these in recent decades, we concentrate on one year, 2012, which comes relatively close, and for which the U.S. Bureau of Economic Analysis provides data. The finding from Tables 1 and 2 is that net capital formation is well below the share of capital in the net national product. Hence the United States economy, even if it maintained itself on a full-employment path, would not be on its highest steady-state consumption path--one where it is not possible to increase consumption per head at one point in time without reducing it at another.

A brief exposition of the Golden Rule of Accumulation now follows, as this is at the foundation of the rule applied just above. Economic policy makers are assumed to value consumption per unit at different future dates equally with present consumption. So, for example, if a worker 10 years hence is twice as effective as a present worker, he counts as two units of present labor. Now, one maximizes f(k)--nk with respect to k, and this yields f'(k)=n. Then multiply both sides by k, and this gives kf'(k)=nk. The left-hand side is defined as the share of the national product going to "capital", and the right-hand side is the amount invested. The conclusion is that an amount equal to the capital share of the national product should be invested, in order to yield a Pareto optimum for the entire time interval--where an increase of consumption at one date can only occur through a decrease at another date. This is a necessary but not yet a sufficient condition for maximal consumption per unit of labor.

Figure 1 below, first used to my knowledge by Matthews and Hahn (2005), can be used to illustrate this point. At the golden age, when the marginal return to capital is zero, net social output per head is maximal. This is measured either by cg or by 0f. The entire social product goes to the work force (who may through taxes and transfer payments share this with the non-working part of the population). When capital per head is 0b, the share to capital is de, and the rest, 0d, goes to labor. If more than ed is invested, the production point, h, moves upward toward g, the golden age point. This occurs at a decelerating rate, as the share of capital shrinks toward zero. Once more, the golden rule, if applied at every date preceding the golden age, maximizes the difference between net output and investment, which is consumption per head. But more saving than this is needed to bring society toward the golden age. Once arriving at the golden age, society needs to continue saving enough to match the growth of the population, but at a zero rate of return.

Comparable data for Germany show a similar picture, that net and gross investment as reported in the Statistical Yearbook of 2012 are much less than the gross and net profit shares. In the German case, however, the growth in the labor force has been slower. If indeed this condition continues and the rise in productivity is Hicks-neutral, then one can argue that net investment as measured is adequate. The capital stock augments itself through technical improvement and not much through additional saving (Table 3).

Can one draw any policy conclusions from this, to the present author, surprising evidence? If in recent years, for which 2012 serves as a sample, productivity growth can be described as Harrod-neutral (labor-augmenting) then the amount of new real capital needs to match the growth of labor in efficiency units. If, on the other hand, progress can be described as Hicks neutral (equally labor-and capital-augmenting) then the amount of new investment needs only to match the growth of labor in natural units. This has been slow in the German case, faster in the United States. There is of course an infinity of possibilities in between. If indeed progress has augmented both capital and labor, as seems plausible, then there is less cause for alarm, especially in the German case. *

* Note: The assumption of Harrod-neutral progress seems overly well-imbedded in the macroeconomic literature. A representative example of this is the textbook of D. Romer, Advanced Macroeconomics (2001).

On Influencing the Rate of Progress

It is useful and important to consider ways to improve the rate of technological progress, and one way we consider here is public support of basic research and its twin, higher education. We take as given that the social product is homogeneous of first degree in the two factors, labor(L) and capital(K), and that we can therefore write:

Y(t) - A(t)[L(f - kf'(k)) + Kf'(k)]

L is here standardized labor in natural units, K is malleable capital. A is state of the arts, which changes over time in a partly autonomous way. But this can be influenced by government support, via its own institutions, and grants to various nongovernmental organizations. Generally, governments are averse to risk of failure, and a proxy for this risk aversion can be taken to be the estimated variance of A, times some coefficient. Maximizing a social utility function which takes account of both expected return and degree of risk is then a fairly straightforward matter.

Comparing the two problems of policy makers, one can say that in the one case a government can through tax, transfer, and enforcement measures affect within limits the share of the national product which is saved. For example, removing foreign tax havens would increase the amount of household and business saving available for domestic capital formation. Somewhat paradoxically, increasing the share of income tax placed on higher incomes tends to reduce the rate of household saving. On the second problem, the evidence is incomplete. It is one thing to compare advanced countries with developing and threshold countries, and to show that the levels of education differ between them, and correspondingly the levels of per capita income; and it is another to test the effectiveness of education and research in generating progress.

In order to make the problem and its difficulties in application clearer we write Euler's equation, the necessary condition for the maximization of an integral over a given finite time span.


The components here deserve some comment: The marginal utility of consumption per head of labor in efficiency units is taken to be a constant (a standard assumption); hence (1-s) is also a constant, dEA(t)/DR(t) is the expected marginal value of the return (say, in constant dollars). The second expression is the relative departure of future value from its starting value, as a consequence of the investment. The whole expression is remotely analogous to comparing the marginal efficiency of investment with the rate of interest, but with the difference that here we take explicit account of an entire time span. In any event we have a formal condition which needs to be satisfied in order to obtain an optimal investment of present resources for a stream of future returns. The expression can also be expanded to take explicit account of risk aversion; but for this see Gehrels (2010).

Concluding Remark

The two tasks of government should be viewed as simultaneous. Analytically it is more demanding to optimize the path of income per head with respect to research support than it is to put through the golden rule, but politically the latter may be the more difficult task. The latter implies a zero discount rate for social planners, which rate is also applied to investment in technical and scientific progress.

DOI 10.1007/s11293-013-9378-y

Published online: 4 August 2013


Barro, R. J. (2001). Human capital and growth. American Economic Review, 90, 12-17.

Gehrels, F. (2010). On optimal social investment in the sciences and humanities. Atlantic Economic Journal, 38, 325-330.

Hall, R. E., & Jones, C. J. (1999). Why do some countries produce more output per worker than others. Quarterly Journal of Economics, 114(1), 83-116.

Hanushek, E. A., & Wocesmann, L. (2008). The role of cognitive skills in economic development. Journal of Economic Literature, 46, 607-668.

Matthews, R. C. O., & Hahn, F. H. (2005). Two-sector models. Capital Theory, 2, 113-123.

Romer, D. (2001). Advanced macroeconomics (4th ed.). New York: McGraw-Hill.

Statistisches Bundesamt. Statistisches Jahrbuch Deutschland und Internationales (2012).

U.S. Department of Commerce, Bureau of Economic Analysis. (2012).

Viane, J. V., & Zilcha, I. (2009). Human capital and inequality dynamics; the role of education technology. Economica, 76, 760-778.

F. Gehrels ([mail])

University of Munich, 83737 Irschenberg, Oberbayem, Germany


Table 1 Net capital share in United States National Income, 2012
(billions of dollars)

Proprietors' net income     $1,809.3
Rental net income           $1,500.6
Corporate profits           $7,800.6

Total                      $14,462.5

U.S. Department of Commerce, Bureau of Economic Analysis 2012

Table 2 Gross and net investment in United States National Income,
2012 (billions of dollars)

Gross domestic investment      $10,138.5
Consumption of fixed capital    $8,046.9
Net domestic investment         $2,081.6

U.S. Department of Commerce, Bureau of Economic Analysis 2012

Table 3 Gross and net investment in the German National Product, 2012
(in euro)

Gross national product            2,295.5 [euro]
Gross investment                    468.7 [euro]
Depreciation                        383.7 [euro]
Net investment                       85.0 [euro]
Business and proprietary income     651.9 [euro]

Statistisches Bundcsamt, Statistisches Jahrbuch Deutschland und
Internationales 2012
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Author:Gehrels, Franz
Publication:Atlantic Economic Journal
Geographic Code:1USA
Date:Sep 1, 2013
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