Substitutionality of production factors

Test: In the above graph the isoquants of a CES- production function are
shown. The output quantity was normalized to have the isoquants always pass
through the point x=5, y=5 to visualize the change in substitution elasticity.
Thus, all points on the graph indicate factor combinations that provide the same
output quantity. So you can see, to what extent you can replace the use of
$\mathit{Factor}2$ with
more $\mathit{Factor}1$
and vice versa (substitution).

The degree of substitutability can be measured in two different ways. The
marginal rate of technical substitution (MRTS) indicates how many units of
$\mathit{Factor}1$ must be used to
replace one unit of $\mathit{Factor}2$.
The MRTS thus represents the slope of the isoquant. In most cases (exception: linear
production function and Leontief- production function) the MRTS is decreasing in
$\mathit{Factor}1$, i.e. if already
much of $\mathit{Factor}1$ is used
in relation to $\mathit{Factor}2$ an
additional unit of $\mathit{Factor}1$
increases the output only a little, while reducing the use of
$\mathit{Factor}2$ by
one unit decreases the output relatively much.

Alternatively, the substitutability can be measured via the substitution elasticity
of the production function. The elasticity of the relative factor input is a function
of the relative marginal products and can be interpreted as the curvature of the
isoquant.

The CES- production function shown here has a constant elasticity of
substitution. This property is advantageous in many economic applications. Here,
we illustrate the function

where the elasticity of substitution is
$\sigma =\frac{1}{1+\rho}$. By
variation of $\rho $
the type of utility function can be changed from Leontief to Cobb-Douglas to
perfect substitution (linear utility) function.

$\rho $ | Type of production function | substitutability | comments |

$\rho =-1$ $\sigma =\infty $ | linear | complete substitution | constant MRTS production possible with only one factor |

$-1<\rho $ | production possible with only one factor | ||

$\rho =0$ $\sigma =1$ | Cobb Douglas | partial substitutability | both factors necessary for production |

$\rho =\infty $ $\sigma =0$ | Leontief | no substitution | MRTS=0 or $\infty $ Example: right and left shoes |

(c) by Christian Bauer

Prof. Dr. Christian Bauer

Chair of monetary economics

Trier University

D-54296 Trier

Tel.: +49 (0)651/201-2743

E-mail: Bauer@uni-trier.de

URL: https://www.cbauer.de