Inada conditions

A Cobb-Douglas-type function satisfies the Inada conditions when used as a utility or production function.

In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada,^[1] are assumptions about the shape of a function, usually applied to a production function or a utility function. When the production function of a neoclassical growth model satisfies the Inada conditions, then it guarantees the stability of an economic growth path. The conditions as such had been introduced by Hirofumi Uzawa.^[2]

Statement

Given a continuously differentiable function [math]\displaystyle{ f \colon X \to Y }[/math], where [math]\displaystyle{ X = \left\{ x \colon \, x \in \mathbb{R}_{+}^{n} \right\} }[/math] and [math]\displaystyle{ Y = \left\{ y \colon \, y \in \mathbb{R}_{+} \right\} }[/math], the conditions are:

1. the value of the function [math]\displaystyle{ f(\mathbf{x}) }[/math] at [math]\displaystyle{ \mathbf{x} = \mathbf{0} }[/math] is 0: [math]\displaystyle{ f(\mathbf{0})=0 }[/math]
2. the function is concave on [math]\displaystyle{ X }[/math], i.e. the Hessian matrix [math]\displaystyle{ \mathbf{H}_{i,j} = \left( \frac{\partial^2 f}{\partial x_i \partial x_j} \right) }[/math] needs to be negative-semidefinite.^[3] Economically this implies that the marginal returns for input [math]\displaystyle{ x_{i} }[/math] are positive, i.e. [math]\displaystyle{ \partial f(\mathbf{x})/\partial x_{i}\gt 0 }[/math], but decreasing, i.e. [math]\displaystyle{ \partial^{2} f(\mathbf{x})/ \partial x_{i}^{2}\lt 0 }[/math]
3. the limit of the first derivative is positive infinity as [math]\displaystyle{ x_{i} }[/math] approaches 0: [math]\displaystyle{ \lim_{x_{i} \to 0} \partial f(\mathbf{x})/\partial x_i =+\infty }[/math], meaning that the effect of the first unit of input [math]\displaystyle{ x_{i} }[/math] has the largest effect
4. the limit of the first derivative is zero as [math]\displaystyle{ x_{i} }[/math] approaches positive infinity: [math]\displaystyle{ \lim_{x_{i} \to +\infty} \partial f(\mathbf{x})/\partial x_i =0 }[/math], meaning that the effect of one additional unit of input [math]\displaystyle{ x_{i} }[/math] is 0 when approaching the use of infinite units of [math]\displaystyle{ x_{i} }[/math]

Consequences

The elasticity of substitution between goods is defined for the production function [math]\displaystyle{ f(\mathbf{x}), \mathbf{x} \in \mathbb{R}^n }[/math] as [math]\displaystyle{ \sigma_{ij} =\frac{\partial \log (x_i/x_j) }{\partial \log MRTS_{ji}} }[/math], where [math]\displaystyle{ MRTS_{ji}(\bar{z}) = \frac{\partial f(\bar{z})/\partial z_j}{\partial f(\bar{z})/\partial z_i} }[/math] is the marginal rate of technical substitution. It can be shown that the Inada conditions imply that the elasticity of substitution between components is asymptotically equal to one (although the production function is not necessarily asymptotically Cobb–Douglas, a commonplace production function for which this condition holds).^[4][5]

In stochastic neoclassical growth model, if the production function does not satisfy the Inada condition at zero, any feasible path converges to zero with probability one provided that the shocks are sufficiently volatile.^[6]

References

Further reading

• Barro, Robert J.; Sala-i-Martin, Xavier (2004). Economic Growth (Second ed.). London: MIT Press. pp. 26–30. ISBN 0-262-02553-1. https://books.google.com/books?id=jD3ASoSQJ-AC&pg=PA26. 
• Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 176–178. ISBN 3-540-60988-1. https://books.google.com/books?id=ouC6AAAAIAAJ&pg=PA176. 
• Romer, David (2011). "The Solow Growth Model". Advanced Macroeconomics (Fourth ed.). New York: McGraw-Hill. pp. 6–48. ISBN 978-0-07-351137-5. 

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