Inada conditions


In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada, are assumptions about the shape of a production function that guarantee the stability of an economic growth path in a neoclassical growth model. The conditions as such had been introduced by Hirofumi Uzawa.
Given a continuously differentiable function, where and, the conditions are:
  1. the value of the function at is 0:
  2. the function is concave on, i.e. the Hessian matrix needs to be negative-semidefinite. Economically this implies that the marginal returns for input are positive, i.e. , but decreasing, i.e.
  3. the limit of the first derivative is positive infinity as approaches 0:,
  4. the limit of the first derivative is zero as approaches positive infinity:
It can be shown that the Inada conditions imply that the elasticity of substitution is asymptotically equal to one.
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.