Inada conditions

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

In macroeconomics, the Inada conditions are assumptions about the shape of a function that ensure well-behaved properties in economic models, such as diminishing marginal returns and proper boundary behavior, which are essential for the stability and convergence of several macroeconomic models. The conditions are named after Ken-Ichi Inada, who introduced them in 1963.

The Inada conditions are commonly associated with ensuring the existence of a unique steady state and preventing pathological behaviors in production functions, such as infinite or zero capital accumulation.

Statement

Given a continuously differentiable function $f:X\to Y$, where $X=\{x:x\in {ℝ}_{+}^n\}$ and $Y=\{y:y\in {ℝ}_{+}\}$, the conditions are:

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

Consequences

The elasticity of substitution between goods is defined for the production function $f(𝐱),𝐱\in {ℝ}^n$ as ${\sigma }_{ij}=\frac{\partial \log (x_i/x_j)}{\partial \log MRT{S}_{ji}}$, where $MRT{S}_{ji}(\overline{z})=\frac{\partial f(\overline{z})/\partial z_j}{\partial f(\overline{z})/\partial z_i}$ 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).

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.

Further reading

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