Inada conditions
This article may be too technical for most readers to understand.(August 2017) |
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.
Statement
[edit]Given a continuously differentiable function , where and , the conditions are:
- the value of the function at is 0:
- the function is concave on , i.e. the Hessian matrix needs to be negative-semidefinite.[3] Economically this implies that the marginal returns for input are positive, i.e. , but decreasing, i.e.
- the limit of the first derivative is positive infinity as approaches 0: , meaning that the effect of the first unit of input has the largest effect
- the limit of the first derivative is zero as approaches positive infinity: , meaning that the effect of one additional unit of input is 0 when approaching the use of infinite units of
Consequences
[edit]The elasticity of substitution between goods is defined for the production function as , where 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
[edit]- ^ 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.
- ^ 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.
- ^ Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. pp. 125–126. ISBN 0-521-31498-4.
- ^ 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.
- ^ 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.
- ^ 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
[edit]- 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.