## Chapter 11Substitutionality 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

$u={\left({x}^{-\rho }+{y}^{-\rho }\right)}^{-\frac{1}{\rho }},$

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 MRTSproduction 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