9.1 Homogeneous and Homothetic Production Functions
A production function is homogenous of degree r if when all inputs are decreased/increased by the same proportion, output decreases/increases by the rth power of that increase. Formally, if \(f(x_1 ...., x_n)\) is homogenous to the degree r,
\(f(tx_1 ...., tx_n) \equiv t^r f(x_1..., x_n)\)
Several properties of homogeneous functions in general were noted in an earlier chapter, especially Euler's theorem. They also have a geometric property such that
\(\frac{f(tx_1 ...., tx_n)}{f(tx_1 ...., tx_n)} \equiv \frac{f(x_1..., x_n)}{f(x_1..., x_n)}\)
i.e. the slopes of the level curves are the same along every point of a given ray out of the origin. However, homogeneous functions are not the only functions with this geometric property. Consider any transformation of a homogeneous production function, where the slope of the level curve in the plane is
\(\frac{H_i}{H_j} = \frac{F'(z)f_i}{F'(z)f_j} = \frac{f_i}{f_j}\)
The class of functions is called the homothetic functions.
Example
Consider a production function, \(y = H(x_1, x_2) = x_1 x_2 + x_1 ^2 x_2 ^2\). This function is not homogeneous but it is homothetic, however, since \(H(x_1, x_2) \equiv z + z^2\), where \(z = x_1 x_2\). That is, \(H(x_1, x_2) \equiv F(f(x_1, x_2))\), where \(F'(z) = 1 + 2z \neq 0\) since production is presumed to be nonnegative. The slope of a level curve is
\(- \frac{H_1}{H_2} = - \frac{x_2 + 2x_1 x_2 ^2}{x_1 + 2 x_1 ^2 x_2}\)
\(= \frac{x_2 (1 + 2x_1 x_2)}{x_1(1 + 2x_1 x_2)} = - \frac{x_2}{x_1}\)
The function is homogeneous of degree 0. Thus the level curves of the equation have the same slope at all points along any given ray out of the origin.
Still another way to express homotheticity is to state that the output elasticities for all factors are equal at any given point. Any increase, say, in output from y to y' will result in a new tangency point B along a straight line through the origin and the former tangency point A. Hence for homothetic production functions, output elasticities are equal in all factors. See below:
The result is that the slope of the ray is the same for any output level as long as factor prices are held constant. That is,
\(\frac{\partial (x_j^*/ x_i ^*)}{\partial y} \equiv 0\)
Using the quotient rule and multiplying through yields
\(x_i^* \frac{\partial (x_j^*)}{\partial y} - \frac{\partial (x_i ^*)}{\partial y} \equiv 0\)
\(\frac{y}{x_j^*} \frac{\partial x_j^*}{\partial y} \equiv \frac{y}{x_i ^*} \frac{\partial x_i ^*}{\partial y}\) or \(\epsilon _{jy} = \epsilon _{iy}\)
Thus using the value of common output elasticity,
\(\epsilon _y = \frac{MC}{AC}\)
Thus the common value of output elasticity, for homothetic functions, is the ratio of marginal to average cost. For all firms with increasing average costs, the factors are all output-elastic and vice versa for firms with declining average cost. Also, if the firm is at the minimum point of its AC curve, the output elasticities of its factors are all unity if the production function is homothetic.
9.2 The Cost Function: Further Properties
If a cost function is homogenous to the degree r > 0, the cost function can be partitioned into
\(C*(w_1..., w_n, y) \equiv y^{1/r} A(w_1..., w_n)\)
When r = 1, this case exhibits constant returns to scale. This would mean that AC = MC = constant. Below is a production function homogeneous of the degree 1/2: