Inada conditions

Given a continuously differentiable function f \colon X \to Y, where X = \left\{ x \colon \, x \in \mathbb{R}_{+}^{n} \right\} and Y = \left\{ y \colon \, y \in \mathbb{R}_{+} \right\}, the conditions are: • the value of the function f(\mathbf{x}) at \mathbf{x} = \mathbf{0} is 0: f(\mathbf{0})=0 • the function is concave on X, i.e. the Hessian matrix \mathbf{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(\mathbf{x})/\partial x_{i}>0, but decreasing, i.e. \partial^{2} f(\mathbf{x})/ \partial x_{i}^{2} • the limit of the first derivative is positive infinity as x_{i} approaches 0: \lim_{x_{i} \to 0} \partial f(\mathbf{x})/\partial x_i =+\infty, meaning that the effect of the first unit of input x_{i} has the largest effect • the limit of the first derivative is zero as x_{i} approaches positive infinity: \lim_{x_{i} \to +\infty} \partial f(\mathbf{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(\mathbf{x}), \mathbf{x} \in \mathbb{R}^n as \sigma_{ij} =\frac{\partial \log (x_i/x_j) }{\partial \log MRTS_{ji}}, where 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). 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. == References ==