Inada conditions

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

In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada, ^[1] are assumptions about the shape of a function, usually applied to a production function or a utility function. When the production function of a neoclassical growth model satisfies the Inada conditions, then it guarantees the stability of an economic growth path. The conditions as such had been introduced by Hirofumi Uzawa. ^[2]

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]

References

1. Inada, Ken-Ichi (1963). "On a Two-Sector Model of Economic Growth: Comments and a Generalization". The Review of Economic Studies . 30 (2): 119–127. doi:10.2307/2295809. JSTOR   2295809.
2. Uzawa, H. (1963). "On a Two-Sector Model of Economic Growth II". The Review of Economic Studies. 30 (2): 105–118. doi:10.2307/2295808. JSTOR   2295808.
3. Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. pp.  125–126. ISBN   0-521-31498-4.
4. Barelli, Paulo; Pessoa, Samuel de Abreu (2003). "Inada Conditions Imply That Production Function Must Be Asymptotically Cobb–Douglas". Economics Letters . 81 (3): 361–363. doi:10.1016/S0165-1765(03)00218-0. hdl: 10438/1012 .
5. Litina, Anastasia; Palivos, Theodore (2008). "Do Inada conditions imply that production function must be asymptotically Cobb–Douglas? A comment". Economics Letters . 99 (3): 498–499. doi:10.1016/j.econlet.2007.09.035.
6. Kamihigashi, Takashi (2006). "Almost sure convergence to zero in stochastic growth models" (PDF). Economic Theory . 29 (1): 231–237. doi:10.1007/s00199-005-0006-1. S2CID   30466341.

Further reading

• Barro, Robert J.; Sala-i-Martin, Xavier (2004). Economic Growth (Second ed.). London: MIT Press. pp. 26–30. ISBN   0-262-02553-1.
• Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 176–178. ISBN   3-540-60988-1.
• Romer, David (2011). "The Solow Growth Model". Advanced Macroeconomics (Fourth ed.). New York: McGraw-Hill. pp. 6–48. ISBN   978-0-07-351137-5.