Inada conditions

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In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada,[1] 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.[2]

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.[3] 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:

In the class of CES production functions only the Cobb–Douglas production function meets all of these conditions.


  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. JSTOR 2295809. 
  2. ^ Uzawa, H. (1963). "On a Two-Sector Model of Economic Growth II". The Review of Economic Studies. 30 (2): 105–118. JSTOR 2295808. 
  3. ^ Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. pp. 125–126. ISBN 0-521-31498-4. 

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