To explain why some countries grow more rapidly than others or why a country may grow more rapidly during one period of history than another, economists have found it convenient to think in terms of a “production function.” This is a mathematical way of relating some measure of output, such as GNP, to the inputs required to produce it. For example, it is possible to relate GNP to the size of the labour force measured in man-hours, to capital stock measured in dollars, and to various other inputs that are considered important. An equation can be written that states that the rate of growth of GNP depends upon the rates of growth of the labour force, the capital stock, and other variables. A common procedure is to assume that the influence of the separate inputs is additive—i.e., that the increase in the growth of output caused by increasing the rate of growth of, say, capital is independent of the rate of growth of the labour force. This is the starting point of a great deal of current empirical work that attempts to quantify the importance of different inputs.

Under certain assumptions, some reasonable and some patently false, it is possible to conclude that what labour and capital receive in the form of wages, profits, and interest is a fair measure of what they contribute to the productive process. Thus in the United States in the period following World War II the share of output going to labour was approximately 79 percent, while the share of output distributed as “profits” was 21 percent. If we assume that these proportions determine how much we should weight the rate of growth of the labour force and of capital respectively in determining their contribution to the rate of growth of output, we must conclude that the relative contribution of capital is slight. Alternatively, we may say that some given percentage increase in the rate of growth of the labour force will have a much larger influence on the rate of growth of output than the same percentage increase in the rate of growth of capital. This is a puzzling result and can be traced to the assumption that the influence of separate inputs is additive.

Much work has been done in an effort to measure the inputs in the productive process more accurately by taking account of improvements in the quality of both labour and capital over time. For example, it has been argued that the amount of a worker’s time spent on his formal education is positively related to the income he receives and to his productive contribution. Measuring the number of man-hours worked from one period to the next will not give a true picture of the increase in labour input if the average amount of education received by workers is changing. Man-hour units must be converted to “efficiency” units. Thus if a labour force of 100 workers in the first year all had an eighth-grade education, while 20 years later each member had a 10th-grade education, then measured in efficiency units the labour force had grown. If the length of time spent on formal education increases over time, then the growth of the labour input will be larger if measured in efficiency units. There is, thus, an element of capital in the labour force.

Examples of investment in human capital are expenditures on health and on all types of education, including on-the-job training. Expenditures of this sort increase the quality of the labour force and its ability to perform productive tasks. Many economists have argued that technological progress is really nothing but quality improvements in human beings. Some economists take an even broader view and speak of the “production of knowledge” as the clue to technological progress. The production of knowledge is a broad category including outlays on all forms of education, on basic research, and on the more applied type of research associated especially with industry. It is argued that fast-growing industries tend to be those having a high research and development component in their total costs. In addition, firms within an industry that have large research and development budgets tend to experience the most rapid technological progress. The argument is that technical change and improvements must originate in inventions that lead to innovations in the products produced or in the processes whereby existing products are manufactured.

A similar argument applies to the size of the capital stock. It can be maintained that design improvements increase the efficiency of capital goods so that a dollar’s worth of machinery purchased today may be much more efficient than a dollar’s worth of depreciated machinery purchased yesterday. The rate of growth of the capital stock measured so as to take account of quality improvements will be greater than the rate of growth of the capital stock measured in a way that neglects the differences between “vintages.”

Some economists have stressed “economies of scale.” For example, if an increase in the use of capital and labour leads to a greater than proportionate increase in output, this is said to result from economies of scale. Economies of scale may arise because an expansion of the market justifies a radical change in productive techniques. These new techniques may be so much more efficient that the returns in the way of increased output are much greater proportionately than the increase in inputs.

Another source of growth and of technical progress in particular has been seen in shifts of demand from low productivity sectors to high productivity sectors, thus causing resources to be reallocated. The most notable movement has been the shift of resources, especially labour, out of agriculture—a traditionally low-productivity sector. Such shifts act to increase the rate of growth of output in ways that cannot be accounted for by simply measuring growth in total inputs. Historically, the allocation of both capital and labour have shifted during the growth process from low productivity sectors to high ones, causing the rate of growth of output to exceed the weighted average of the rates of growth of total inputs.

This historical fact points to an element that has received little attention so far: the influence of entrepreneurship. If the allocation of resources changes during the course of growth and development, it does so under the leadership of an entrepreneurial class. The quality of entrepreneurship is seen by many economists as an important explanation of differences in the rate of technical progress between countries. Decisions must be made somewhere along the line as to whether a new product or process will be introduced. It has been argued that two countries undertaking similar amounts of investment leading to more or less identical rates of growth in the capital stock will not necessarily show the same rate of technical progress. In one country entrepreneurs may be undertaking enterprise investment that has as its aim the introduction of the most advanced types of production techniques, those that will lead to a rapid growth of labour productivity. In the other, because of hesitation or ignorance, the investment program may lead only to marginal changes in productive processes; the resulting growth in labour productivity and GNP will be small. For example, much has been said since World War II about the more aggressive nature of German businessmen as compared to their English counterparts. The emphasis on the role of the entrepreneur in economic growth stems from the theoretical work of the economist Joseph A. Schumpeter, but many others have echoed it.

Much thinking assumes, then, that contributions to output from growth of individual inputs are independent of one another. This assumption allows many growth theorists to conclude that capital investment is relatively unimportant as a growth factor. If there is interaction between the rates of growth of the different inputs, however, then it is possible to draw different conclusions. For example, over time there are likely to be improvements in the quality of capital goods. A machine that requires so much steel and so much labour to manufacture may be twice as productive as an older machine that required the same amount of raw materials and labour in its manufacture. Thus the rate of growth of technical progress and the rate of growth of the capital stock measured in natural units interact. Furthermore, the interaction between technical progress and capital formation is not necessarily in one direction. New knowledge opens up new production possibilities and gives rise to potential increases in technical progress and profits. Or the better educated the labour force, the more adaptable it is likely to be and therefore the better able to cope with new production techniques. At the same time, the higher the rate of growth of capital, the higher will be the growth of incomes and therefore the demand for education. The fact that much of the overall growth of technical progress stems from the transfer of resources and the positive association between the rate of transfer of resources and the rate of growth of the capital stock is another example of interdependence or complementarity between the growth of the inputs. But, again, capital investment undertaken to develop new lines of production will also be dependent on technological progress going on in those areas.

Conventional marginal productivity doctrine argues that as an input such as capital rises relative to labour, the additional output or marginal product that can be attributed to this extra amount of capital will be less than what a unit of capital on the average had been producing before. Marginal productivity doctrine also assumes that each unit of capital is identical with the next. This assumption is the basis for the argument that as more units of capital are utilized in production with a given amount of labour, it will push down the former’s marginal product. There is the possibility, however, that additional units of capital may enhance the productivity of existing units: for example, an increase in the amount of capital resources devoted to the development of transportation and distribution may raise the productivity of capital employed, say, in manufacturing. The development of this kind of social overhead capital is certainly a prerequisite for a high return to capital in manufacturing, wholesaling, and retailing.

The analysis can be carried back one more step, to the basic determinants of growth. Economists ask why it is that capital, labour, or technical progress has grown more rapidly in one economy than in another or at one time than at another. Historically, the transition from a subsistence-level, underdeveloped state to a higher-level, developed one has been accompanied by a decline in the death rate followed by a decline in the birth rate. This has the effect of first speeding up the rate of growth of the population and labour force and then reducing it as birth rates fall. Migration can alter this picture, often unpredictably. In the United States, for example, the rate of growth of the population and labour force during the 19th and early 20th centuries was higher than in most other developed countries, mainly because of high rates of immigration. From 1840 to 1930, the native-born U.S. population increased about 600 percent, while the number of those of foreign birth increased 1,300 percent.