Inada conditions

In the neoclassical theory of production and growth, Inada conditions are several conditions that are usually applied to the production functions used. The name goes back to an article by the Japanese economist Ken-Ichi Inada from 1963, in which he explicitly formulated it for a growth model.

The term “Inada conditions” is used vaguely in the literature; The majority of the authors confine themselves to the requirements below, others also attribute other classically assumed (and also adopted by Inada) conditions to the Inada conditions, such as the assumption of decreasing marginal productivity (see also the following section).

Explanation

Example of a production function that meets the Inada conditions

Let it be a production function, where stands for the capital input and the labor input. Then the Inada conditions say (in the narrower sense) that the marginal product of every factor of production converges to infinity, if only one lets the respective factor input tend towards zero; On the other hand, if the respective factor input is allowed to tend towards infinity, the marginal product of the factor converges towards zero. So formally applies ${\displaystyle F(K,L)}$ ${\displaystyle K}$ ${\displaystyle L}$

${\displaystyle \lim _{K\to 0}{\frac {\partial F(K,L)}{\partial K}}=\infty \quad {\textrm {und}}\quad \lim _{K\to \infty }{\frac {\partial F(K,L)}{\partial K}}=0}$

respectively

${\displaystyle \lim _{L\to 0}{\frac {\partial F(K,L)}{\partial L}}=\infty \quad {\textrm {und}}\quad \lim _{L\to \infty }{\frac {\partial F(K,L)}{\partial L}}=0}$.

A typical reading of these conditions, which is helpful for technical purposes, is, for example, that given the technology in an economy, the output cannot be increased at will by increasing the labor input.

In a broader sense the Inada conditions denote the following 6 properties in accordance with the formulation of Hirofumi Uzawa : for a function applies ${\displaystyle f(x)}$

1. the value of the function at position 0 is 0: ${\displaystyle f(x)}$ ${\displaystyle f(0)=0}$
2. the function is twice continuously differentiable ,
3. the function is strictly increasing in : , ${\displaystyle x_{i}}$ ${\displaystyle \partial f(x)/\partial x_{i}>0}$
4. the second derivative of the function is negative (therefore it is a concave function ) , ${\displaystyle x_{i}}$ ${\displaystyle \partial ^{2}f(x)/\partial x_{i}^{2}<0}$
5. the limit value of the first derivative is positive infinity for up to 0: , ${\displaystyle x_{i}}$ ${\displaystyle \lim _{x_{i}\to 0}\partial f(x)/\partial x_{i}=+\infty }$
6. and the limit value of the first derivative is zero for infinite against: . ${\displaystyle x_{i}}$ ${\displaystyle \lim _{x_{i}\to +\infty }\partial f(x)/\partial x_{i}=0}$

Implications

If one assumes, as is typically assumed for production functions, that both input factors show a positive but decreasing marginal productivity , so that the following applies:

${\displaystyle {\frac {\partial F(K,L)}{\partial K}}>0\quad {\textrm {sowie}}\quad {\frac {\partial ^{2}F(K,L)}{\partial K^{2}}}<0}$

respectively

${\displaystyle {\frac {\partial F(K,L)}{\partial L}}>0\quad {\textrm {sowie}}\quad {\frac {\partial ^{2}F(K,L)}{\partial L^{2}}}<0}$,

and that the production function has constant returns to scale (= homogeneous of grade one):

${\displaystyle F(\alpha K,\alpha L)=\alpha F(K,L)\quad (\alpha \in \mathbb {R} ^{+})}$,

then it follows from the above Inada conditions that every factor used is essential (also: essential ). This means that an economy in a state in which there is either no capital or no work cannot generate any output. Formally:

${\displaystyle F(K,0)=0\quad {\textrm {und}}\quad F(0,L)=0}$.

If a production function satisfies the Inada conditions, marginal solutions are therefore excluded in which a factor input in the profit maximum disappears or increases without restriction.

It was assumed that the Inada conditions imply that the production function must be asymptotically of the Cobb-Douglas type , since they assumed that all functions which asymptotically have a substitution elasticity of one belong to the class of Cobb-Douglas functions. However, it turned out that the Inada conditions imply that the production function does not necessarily have to be of the Cobb-Douglas type for this property .

literature

• Rolf Färe and Daniel Primont: Inada Conditions and the Law of Diminishing Returns. In: International Journal of Business and Economics. 1, No. 1, 2002, pp. 1–8 ( online free of charge ; PDF; 166 kB).
• Ken-Ichi Inada: On a Two-Sector Model of Economic Growth: Comments and a Generalization. In: The Review of Economic Studies. 30, No. 2, 1963, pp. 119-127 ( JSTOR 2295809 ).

Remarks

1. ↑ Inada 1963.
2. ↑ As here for example Färe / Primont 2002; Stefan Baumgärtner: The Inada Conditions for Material Resource Inputs Reconsidered. In: Environmental & Resource Economics. 29, No. 3, 2004, pp. 307-322, doi : 10.1007 / s10604-003-5267-5 ; Knut Sydsæter u. a .: Further Mathematics for Economic Analysis. 2nd Edition. Pearson 2008, p. 214; More broadly, however, for example, Thomas Wagner and Elke J. Jahn: Neue Arbeitsmarkttheorien. 2nd Edition. Lucius & Lucius (UTB), Stuttgart 2004, ISBN 3828202535 .
3. ↑ Wagner / Jahn 2004 provide a practical example for the use of the condition : Applies to company profit ( : real wages). If, with a given capital stock, every worker in the national economy had a marginal productivity that is above the real wage that forms on the market, that would be for every worker ${\displaystyle \pi (L,w)=F(K,L)-wL}$ ${\displaystyle w}$ ${\displaystyle K={\overline {K}}}$ ${\displaystyle i}$

   ${\displaystyle {\frac {\partial F({\overline {K}},L_{i})}{\partial L_{i}}}>w\quad (\forall i)}$,

   then generally for the partial derivation of the profit function according to , that is ${\displaystyle L}$

   ${\displaystyle {\frac {\partial \pi (L,w)}{\partial L}}={\frac {\partial F({\overline {K}},L)}{\partial L}}-w}$

   hold that this is always positive. With this, however, the company could theoretically generate an infinitely high profit without the Inada conditions by asking for an ever greater amount of work. Cf. Thomas Wagner and Elke J. Jahn: New labor market theories. 2nd Edition. Lucius & Lucius (UTB), Stuttgart 2004, ISBN 3828202535 , p. 29.
4. ↑ Uzawa, Hirofumi. "On a two-sector model of economic growth II." The Review of Economic Studies (1963): 105-118. P. 108.
5. ↑ A proof can be found, for example, in Färe / Primont 2002, p. 3 f.
6. ↑ See Paulo Barelli and Samuel de Abreu Pessôa: Inada conditions imply that production function must be asymptotically Cobb – Douglas. In: Economics Letters. 81, No. 3, 2003, pp. 361-363, doi : 10.1016 / S0165-1765 (03) 00218-0 .
7. ↑ litina, Anastasia, and Theodore Palivos. "Do Inada conditions imply that production function must be asymptotically Cobb – Douglas? A comment." Economics Letters 99.3 (2008): 498-499.