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In macroeconomics, the Inada conditions, named after Japanese economist Ken-Ichi Inada, 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.
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. 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: ,
- the limit of the first derivative is zero as approaches positive infinity:
- 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.
- Uzawa, H. (1963). "On a Two-Sector Model of Economic Growth II". The Review of Economic Studies. 30 (2): 105–118. JSTOR 2295808.
- Takayama, Akira (1985). Mathematical Economics (2nd ed.). New York: Cambridge University Press. pp. 125–126. ISBN 0-521-31498-4.
- 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.