Constant elasticity of substitution

Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McKenzie, John Hicks and Joan Robinson. The vital economic element of the measure is that it provided the producer a clear picture of how to move between different modes or types of production.

Specifically, it arises in a particular type of aggregator function which combines two or more types of consumption goods, or two or more types of production inputs into an aggregate quantity. This aggregator function exhibits constant elasticity of substitution.

Literature

For every production plant that has business motives, its management has to find ways of ensuring maximum profits through maximization of the available resources. Therefore, direct decision making based on the observable factors can be limiting with their implications unknown to the people as McKenzie states in his article;

[...] elasticity of substitution changes at different points along an isoquant. For example, it may be harder to swap machines for people (low \(\sigma\)) when only a few people are involved in production, whereas it is easier to introduce machines (high \(\sigma\)) while there are still enough people to run the machines.

It is therefore, important for the company to apply a strategy that is well analysed and empirically ascertained that its effects are bearable by the company as far as the vision is concerned. Tom McKenzie explains the same concept as measure that can be used in decision making. In his article about Elasticity of Substitution he states;

Elasticity of substitution measures the ease with which one can switch between factors of production. The concept has a broad range of applications, from comparisons of labor and capital in firms, immigrant versus native workers in the labor market, to assessing ‘clean’ versus ‘dirty’ methods of production for environmental economics.

It is at the point where despite the alteration of products, the production of the company is not affected. This constant level is the point that is desired by many business. Through applying the information in this article and other related topics, a firm can attain the constant Elasticity of Substitution. One of the basic development of the constant is applied in the monopolists revenue function. In this case, a seller whose business is monopoly may have a good selling. Thus the quality of the product demand D(p) is a function of the product of the final price p that he/she sells. The revenue R(p) that the seller gets out of selling the good can be calculated as $R(p)=pD(p)$. It can be noted that when the price increase (decrease), revenue of the seller also increase. In order to realize the expected understanding of the constant of elasticity of substitution, another founding development was done on the measurement of substitutability. Basically, the most common pair of factors are labor and capital. From the derivation of the equation, the substitution is referred to as a measure of the ease with which the varying factor can be substituted for others. This because of its functionality as provided for by an article with its section stating as;

[...] measuring the degree of substitutability between any pair of factors. One of the most famous ones is the elasticity of substitution, introduced independently by John Hicks (1932) and Joan Robinson (1933). Formally, the elasticity of substitution measures the percentage change in factor proportions due to a change in marginal rate of technical substitution.

CES production function

Despite having several factors of production in substitutability, the most common are the forms of elasticity of substitution. On the contrary of restricting direct empirical evaluation, the constant Elasticity of Substitution are simple to use and hence are widely used. McFadden states that;

The constant E.S assumption is a restriction on the form of production possibilities, and one can characterize the class of production functions which have this property. This has been done by Arrow-Chenery-Minhas-Solow for the two-factor production case.

The CES production function is a neoclassical production function that displays constant elasticity of substitution. In other words, the production technology has a constant percentage change in factor (e.g. labour and capital) proportions due to a percentage change in marginal rate of technical substitution. The two factor (capital, labor) CES production function introduced by Solow, and later made popular by Arrow, Chenery, Minhas, and Solow is:

$Q=F\cdot \left(a\cdot K^{\rho }+(1-a)\cdot L^{\rho }\right)^{\frac {\upsilon }{\rho }}$

where

• $Q$ = Quantity of output
• $F$ = Factor productivity
• $a$ = Share parameter
• $K$, $L$ = Quantities of primary production factors (Capital and Labor)
• $\rho$ = ${\frac {\sigma -1}{\sigma }}$ = Substitution parameter
• $\sigma$ = ${\frac {1}{1-\rho }}$ = Elasticity of substitution
• $\upsilon$ = degree of homogeneity of the production function. Where $\upsilon$ = 1 (Constant return to scale), $\upsilon$ < 1 (Decreasing return to scale), $\upsilon$ > 1 (Increasing return to scale).

As its name suggests, the CES production function exhibits constant elasticity of substitution between capital and labor. Leontief, linear and Cobb–Douglas functions are special cases of the CES production function. That is,

• If $\rho$ approaches 1, we have a linear or perfect substitutes function;
• If $\rho$ approaches zero in the limit, we get the Cobb–Douglas production function;
• If $\rho$ approaches negative infinity we get the Leontief or perfect complements production function.

The general form of the CES production function, with n inputs, is:

$Q=F\cdot \left[\sum _{i=1}^{n}a_{i}X_{i}^{r}\ \right]^{\frac {1}{r}}$

where

• $Q$ = Quantity of output
• $F$ = Factor productivity
• $a_{i}$ = Share parameter of input i, $\sum _{i=1}^{n}a_{i}=1$
• $X_{i}$ = Quantities of factors of production (i = 1,2...n)
• $s={\frac {1}{1-r}}$ = Elasticity of substitution.

Extending the CES (Solow) functional form to accommodate multiple factors of production creates some problems. However, there is no completely general way to do this. Uzawa showed the only possible n-factor production functions (n>2) with constant partial elasticities of substitution require either that all elasticities between pairs of factors be identical, or if any differ, these all must equal each other and all remaining elasticities must be unity. This is true for any production function. This means the use of the CES functional form for more than 2 factors will generally mean that there is not constant elasticity of substitution among all factors.

Nested CES functions are commonly found in partial equilibrium and general equilibrium models. Different nests (levels) allow for the introduction of the appropriate elasticity of substitution.

CES utility function

The same CES functional form arises as a utility function in consumer theory. For example, if there exist $n$ types of consumption goods $x_{i}$, then aggregate consumption $X$ could be defined using the CES aggregator:

$X=\left[\sum _{i=1}^{n}a_{i}^{\frac {1}{s}}x_{i}^{\frac {s-1}{s}}\ \right]^{\frac {s}{s-1}}.$

Here again, the coefficients $a_{i}$ are share parameters, and $s$ is the elasticity of substitution. Therefore, the consumption goods $x_{i}$ are perfect substitutes when $s$ approaches infinity and perfect complements when $s$ approaches zero. The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969).

CES utility functions are a special case of homothetic preferences.

The following is an example of a CES utility function for two goods, $x$ and $y$, with equal shares:

$u(x,y)=(x^{r}+y^{r})^{1/r}.$

The expenditure function in this case is:

$e(p_{x},p_{y},u)=(p_{x}^{r/(r-1)}+p_{y}^{r/(r-1)})^{(r-1)/r}\cdot u.$

The indirect utility function is its inverse:

$v(p_{x},p_{y},I)=(p_{x}^{r/(r-1)}+p_{y}^{r/(r-1)})^{(1-r)/r}\cdot I.$

The demand functions are:

$x(p_{x},p_{y},I)={\frac {p_{x}^{1/(r-1)}}{p_{x}^{r/(r-1)}+p_{y}^{r/(r-1)}}}\cdot I,$

$y(p_{x},p_{y},I)={\frac {p_{y}^{1/(r-1)}}{p_{x}^{r/(r-1)}+p_{y}^{r/(r-1)}}}\cdot I.$

A CES utility function is one of the cases considered by Dixit and Stiglitz (1977) in their study of optimal product diversity in a context of monopolistic competition.

Note the difference between CES utility and isoelastic utility: the CES utility function is an ordinal utility function that represents preferences on sure consumption commodity bundles, while the isoelastic utility function is a cardinal utility function that represents preferences on lotteries. A CES indirect (dual) utility function has been used to derive utility-consistent brand demand systems where category demands are determined endogenously by a multi-category, CES indirect (dual) utility function. It has also been shown that CES preferences are self-dual and that both primal and dual CES preferences yield systems of indifference curves that may exhibit any degree of convexity.

External links

• Anatomy of CES Type Production Functions in 3D
• Closed form solution for a firm with an N-dimensional CES technology
• Monopolists revenue function