Technical progress function

The technical progress function was developed by the economist Nicholas Kaldor . The technological progress is at the growth rate of labor productivity measured. This rate of increase is determined as a function of the rate of increase in capital intensity . Technical progress is therefore no longer exogenously given, as in the Keynesian or neoclassical growth models up to the 1970s, but is determined endogenously depending on the rate of change in capital intensity. In this respect, the technical progress function is a forerunner of the later endogenous growth theories .

The model can be described as a case of a one-good parabola , assuming a produced good that is either used in a certain amount per job (capital intensity) or used to equip additional jobs or even as consumption. Qualitative changes, be it that new and different types of means of production are used or that new and different consumer goods are produced, must be represented by an increase in the quantity of one good, in that more products are produced per worker, more goods are used per worker (capital intensity ) or consumed.

In this way certain simple connections can be represented. From the point of view of the individual company, not every increase in labor productivity is profitable; it has to be bought at the price of an increase in capital intensity. Simplified (neglecting the labor costs) it can be determined that an increase in the capital intensity by a certain percentage is profitable when it an increase in labor productivity of a higher percentage, ie to a higher- proportional leads increase. If both percentages are the same, then the introduction of technical progress is not profitable or just as profitable as you want it to be. The assumption that labor productivity and capital intensity grow at the same rate is a common equilibrium assumption of growth models.

The technical progress function

Kaldor made certain assumptions about the form of the technical progress function (see figure ). Low growth rates in capital intensity lead disproportionately to higher growth rates in labor productivity, but higher growth rates in capital intensity only lead to disproportionately higher growth rates in labor productivity. In between, there must be a growth rate in capital productivity (in the figure 1% growth), which leads to an exactly equally high growth rate in labor productivity.

stability

This point is stable, according to Kaldor, because as long as the rate of growth in capital productivity leads to an even higher rate of growth in labor productivity, firms will try to increase capital intensity even more in the next period. If a growth rate in capital intensity only leads to a disproportionately low growth rate in labor productivity, this will not be profitable and companies will not expand capital intensity as much in the next period.

If the technical progress function has the course claimed by Kaldor, then the equilibrium is that capital intensity and labor productivity grow at the same rate of growth, as is also the case in the Harrod-Domar model or the Solow model . Since this technical progress function contains a balance, it is called "well-behaved", it behaves well.

According to Allen, however, the adjustment process would have to be represented mathematically in an imbalance model in order to be able to decide whether the equilibrium point is actually stable. However, this would not be an easy task.

If any rate of increase in capital intensity would lead to even greater rates of increase in labor productivity, there would be an incentive for companies to invest the entire company profit in the long term not in additional jobs, but in expanding the capital stock per job. If, however, such intensive growth prevails, more and more investments are made for each job, not in new jobs (extensive growth), then there are contradictions on the macroeconomic level, which is the background on a physical or material level to the law of the tendency of Karl's rate of profit to fall Marx educates.

Mathematical properties

In general, the technical progress function cannot be integrated, that is, no production function “suitable” for it can be specified. An exception is the Cobb-Douglas production function with Harrod -neutral, i.e. labor-saving or labor-increasing technical progress with constant economies of scale .

• Y: production amount
• K: Use of capital
• A: Use of work
• c: constant
• t: time
• a: parameters between zero and one.
• m: rate of technical progress

The Cobb-Douglas production function with labor-saving technical progress that grows at the rate m:

${\displaystyle Y=c\cdot K^{a}\cdot (e^{m\cdot t}\cdot A)^{1-a}.}$

Divided with A results in the formulation in per capita variables. The labor productivity is a function of capital intensity :

${\displaystyle {Y \over A}=c\cdot ({K \over A})^{a}\cdot e^{m\cdot t\cdot (1-a)}.}$

So if Y / A = y (labor productivity) and K / A = k (capital intensity)

${\displaystyle y=c\cdot k^{a}\cdot e^{m\cdot t\cdot (1-a)}.}$

The transition to a function that is formulated in the growth rates of labor productivity and capital intensity is made by deriving the logarithm of the function with respect to time : ${\displaystyle y(t)}$ ${\displaystyle t}$

${\displaystyle \ln y=\ln c+a\cdot \ln k+m\cdot t\cdot (1-a),}$

derived from time : ${\displaystyle t}$

${\displaystyle {\frac {1}{y}}\centerdot {\dot {y}}=a\cdot {\frac {1}{k}}\centerdot {\dot {k}}+m\cdot (1-a).}$

On the left is the growth rate of labor productivity as a dependent variable, on the right of the equal sign appears as an explanatory variable the growth rate of capital intensity. Are there

${\displaystyle {\dot {y}}}$and the derivatives with respect to time, and the growth rates are defined as: and ${\displaystyle {\dot {k}}}$ ${\displaystyle {{\dot {y}} \over y}={\hat {y}}}$ ${\displaystyle {{\dot {k}} \over k}={\hat {k}}}$

The result is the growth rate of labor productivity as a linear function of the growth rate of capital intensity with the intercept and the slope with ${\displaystyle m(1-a)}$ ${\displaystyle a}$

${\displaystyle {\hat {y}}=a\cdot {\hat {k}}+m\cdot (1-a).}$

Since , assuming constant returns to scale, the Cobb-Douglas function assumes a value between zero and one, this curve intersects the 45 ° line, so that there is an equilibrium value in which it holds that the growth rate of labor productivity is equal to the growth rate of the Is capital intensity and both are equal to the rate of technical progress : ${\displaystyle a}$ ${\displaystyle m}$

${\displaystyle {\hat {y}}={\hat {k}}=m.}$

Individual evidence

1. ↑ General representation according to Allen 1968. Kaldor himself assumed a linear function with a positive y-axis section and a slope less than 1.
2. ↑ Allen p. 310
3. ↑ Ibid.

literature

• Allen, RGD: Macro-Economic Theory: A Mathematical Treatment . - London, Melbourne, Toronto: Macmillan, 1968.
• Bergheim, Stefan: " Pair-wise cointegration in long-run growth models ". Deutsche Bank Research . Working Papers Series. Research Notes 24 . February 9, 2007. (A technical progress function à la Kaldor is estimated econometrically.)
• Kaldor, Nicholas (1957): "A Model of Economic Growth." The Economic Journal . pp. 591-624.