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| In [[macroeconomics]], the '''Inada conditions''', named after Japanese economist [[Ken-Ichi Inada]],<ref>{{cite journal |last=Inada |first=Ken-Ichi |year=1963 |title=On a Two-Sector Model of Economic Growth: Comments and a Generalization |journal=[[The Review of Economic Studies]] |volume=30 |issue=2 |pages=119–127 |jstor=2295809 }}</ref> 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]].<ref>{{cite journal |last=Uzawa |first=H. |title=On a Two-Sector Model of Economic Growth II |journal=The Review of Economic Studies |volume=30 |issue=2 |year=1963 |pages=105–118 |jstor=2295808 }}</ref>
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| The six conditions for a given function <math>f(x)</math> are:
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| #the value of the function <math>f(x)</math> at 0 is 0: <math>f(0)=0</math>
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| #the function is [[continuously differentiable]],
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| #the function is strictly increasing in <math>x_{i}</math>: <math>\partial f(x)/\partial x_{i}>0</math>,
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| #the second [[derivative]] of the function is decreasing in <math>x_{i}</math> (thus the function is [[concave function|concave]]): <math>\partial^{2} f(x)/\partial x_{i}^{2}<0</math>,
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| #the [[limit of a function|limit]] of the first derivative is positive infinity as <math>x_{i}</math> approaches 0: <math>\lim_{x_{i} \to 0} \partial f(x)/\partial x_i =+\infty</math>,
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| #the [[limit of a function|limit]] of the first derivative is zero as <math>x_{i}</math> approaches positive infinity: <math>\lim_{x_{i} \to +\infty} \partial f(x)/\partial x_i =0</math>
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| It can be shown that the Inada conditions imply that the elasticity of substitution is asymptotically equal to one (although the production function is ''not'' necessarily asymptotically [[Cobb–Douglas]]).<ref>{{cite journal |last=Barelli |first=Paulo |first2=Samuel de Abreu |last2=Pessoa |year=2003 |title=Inada Conditions Imply That Production Function Must Be Asymptotically Cobb–Douglas |journal=[[Economics Letters]] |volume=81 |issue=3 |pages=361–363 |doi=10.1016/S0165-1765(03)00218-0 }}</ref><ref>{{cite journal |last=Litina |first=Anastasia |first2=Theodore |last2=Palivos |year=2008 |title=Do Inada conditions imply that production function must be asymptotically Cobb–Douglas? A comment |journal=[[Economics Letters]] |volume=99 |issue=3 |pages=498–499 |doi=10.1016/j.econlet.2007.09.035 }}</ref>
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| 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.<ref>{{cite journal |first=Takashi |last=Kamihigashi |year=2006 |title=Almost sure convergence to zero in stochastic growth models |journal=[[Economic Theory (journal)|Economic Theory]] |volume=29 |issue=1 |pages=231–237 |doi=10.1007/s00199-005-0006-1 }}</ref>
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| == References ==
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| {{Reflist}}
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| == Further reading ==
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| *{{cite book |last=Gandolfo |first=Giancarlo |year=1996 |title=Economic Dynamics |chapter=The Neoclassical Growth Model |edition=Third |location=Berlin |publisher=Springer |pages=175–189 |isbn=3-540-60988-1 }}
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| *{{cite book |first=David |last=Romer |authorlink=David Romer |chapter=The Solow Growth Model |title=Advanced Macroeconomics |edition=Fourth |location=New York |publisher=McGraw-Hill |year=2011 |pages=6–48 |isbn=978-0-07-351137-5 }}
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| {{DEFAULTSORT:Inada Conditions}}
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| [[Category:Economic growth]]
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