In
macroeconomics, the Inada conditions, named after Japanese economist
Ken-Ichi Inada
was a Japanese economist.
Beginning in the 1950s, Inada wrote a number of important papers on welfare economics, economic growth and international trade. His contributions include an early extension of Kenneth Arrow's impossibility theorem o ...
, are assumptions about the shape of a function, usually applied to a
production function
In economics, a production function gives the technological relation between quantities of physical inputs and quantities of output of goods. The production function is one of the key concepts of mainstream neoclassical theories, used to define ...
or a
utility function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
. When the production function of a
neoclassical growth model
Neoclassical or neo-classical may refer to:
* Neoclassicism or New Classicism, any of a number of movements in the fine arts, literature, theatre, music, language, and architecture beginning in the 17th century
** Neoclassical architecture, an a ...
satisfies the Inada conditions, then it guarantees the stability of an
economic growth path. The conditions as such had been introduced by
Hirofumi Uzawa
was a Japanese economist.
Biography
Uzawa was born on July 21, 1928 in Yonago, Tottori to a farming family.
He attended the Tokyo First Middle School (currently the Hibiya High School ) and the First Higher School, Japan (now the University o ...
.
Statement
Given a
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
function
, where
and
, the conditions are:
#the value of the function
at
is 0:
#the function is
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset ...
on
, i.e. the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
needs to be
negative-semidefinite. Economically this implies that the
marginal return
Marginal Return is the rate of return for a marginal increase in investment; roughly, this is the additional output resulting from a one-unit increase in the use of a variable input, while other inputs are constant.
See also
*Diminishing returns ...
s for input
are positive, i.e.
, but decreasing, i.e.
#the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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
The
elasticity of substitution Elasticity of substitution is the ratio of percentage change in capital-labour ratio with the percentage change in Marginal Rate of Technical Substitution. In a competitive market, it measures the percentage change in the two inputs used in respons ...
between goods is defined for the production function
as
, where
is the
marginal rate of technical substitution
In microeconomic theory, the marginal rate of technical substitution (MRTS)—or technical rate of substitution (TRS)—is the amount by which the quantity of one input has to be reduced (-\Delta x_2) when one extra unit of another input is used ( ...
.
It can be shown that the Inada conditions imply that the
elasticity of substitution Elasticity of substitution is the ratio of percentage change in capital-labour ratio with the percentage change in Marginal Rate of Technical Substitution. In a competitive market, it measures the percentage change in the two inputs used in respons ...
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).
In stochastic
neoclassical growth model
Neoclassical or neo-classical may refer to:
* Neoclassicism or New Classicism, any of a number of movements in the fine arts, literature, theatre, music, language, and architecture beginning in the 17th century
** Neoclassical architecture, an a ...
, 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.
References
Further reading
*
*
*
{{DEFAULTSORT:Inada Conditions
Economic growth