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].
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.
Given a continuously differentiable function
=
, the conditions are: The elasticity of substitution between goods is defined for the production function
σ
∂ log (
∂ log
{\displaystyle \sigma _{ij}={\frac {\partial \log(x_{i}/x_{j})}{\partial \log MRTS_{ji}}}}
{\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.
