Inada conditions

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.^[1][2]

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

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 ${\displaystyle f\colon X\to Y}$, where ${\displaystyle X=\left\{x\colon \,x\in \mathbb {R} _{+}^{n}\right\}}$ and ${\displaystyle Y=\left\{y\colon \,y\in \mathbb {R} _{+}\right\}}$, the conditions are:

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

Consequences

The elasticity of substitution between goods is defined for the production function ${\displaystyle f(\mathbf {x} ),\mathbf {x} \in \mathbb {R} ^{n}}$ as ${\displaystyle \sigma _{ij}={\frac {\partial \log(x_{i}/x_{j})}{\partial \log MRTS_{ji}}}}$, where ${\displaystyle MRTS_{ji}({\bar {z}})={\frac {\partial f({\bar {z}})/\partial z_{j}}{\partial f({\bar {z}})/\partial z_{i}}}}$ 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).^[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]