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JjXl + fix 2 +

----1- fnXn - kf(x\,x2, ■ . -Vj,)

Proof

Since / is homogeneous of degree k, we have f (,VA"1. SX2 SXn ) = SkJ (a-1. ,V2 xn)

Writing each .va, as z, and differentiating both sides with respect to s gives 3/ , 'àf dz2 , , 3/ for.

which implies that

/iJft + fjx2 H-----1- fnx„ = ksk"lf{x\,x2 x„)

Since this condition holds for any .s > 0. it holds for î = I which implies that f\x\ + f7X2 + ••••+■ f„x„ = kf(xi, a2. .. -,x„) (11.12)

Euler's theorem slates that if / is homogeneous of degree k. then multiplying the marginal product of each input /' by the level of that input and summing ./¡am gives a value equal to k times the value of output. For the specific case of k = 1, we have that J\ a, equals the value of output.

Example 11.39 Show that for y = /(a, . x2) = jfj/4a,1/4 it follows that /, v, + f2x2 = /(.v,, .r:). Solution r _ ir- V4_J/1 r _ \ |/4 -

Homotheticity

The concept of returns to scalc is not a meaningful one for utility functions. However. some of the properties of the level curves corresponding to homogeneous functions are useful in the context of consumer theory. In figure 11.29 we illustrate some level curves for some function y - /(x), X e R2. which are radial expansions and contractions of each other, as for a homogeneous function. However. from the function values attached to the level curves, it is clear that this

Figure 11.29 Level curves of a function thai is homothetic but not homogeneous

function is not homogeneous (i.e.. /(2x°) = 2/ix"), implies homogeneity of degree I, while /( 3x°) = 2.4/(x°) implies homogeneity of degree less than I). It follows that the class of functions that satisfies the property that level curves are radial expansions and contractions of each other is larger than the set of functions that is homogeneous. It includes all homogeneous functions as well as all mono-tonic transformations of homogeneous functions. The latter are called homothetic functions.

Definition 11.8

A function is homothetic if it is a monotonie transformation of some homogeneous function.

Theorem 11.20

Let / be a function defined on IP;'.. The function f is homothetic if and only if along any ray from the origin the slope of each level curve (i.e.. the value of /1//2) is constant.

This theorem extends to functions defined on IR" in the same way as does the equivalent result for homogeneous functions. That is. if a function / defined on is homothetic. then the value of dxt _ A

Example 11.40 The function fix.r:> = I -Fx,1'2*-!' defined on E; is not homogeneous but is homolhetic.

Solution

We know already thai g(x\.X2) = is homogeneous, and since / is a monotonic transformation of it follows diat J is homolhetic. It is straightforward to show that / is not homogeneous. For instance, beginning with (x®, .v?) = (I, I) we can see by comparing the values /(1, 1) = 2, /(2. 2) = 3. and ,/'(3.3) = 4 that there is no pattern f(sx\\ sxf) = skfix\\ jtV). ■

Since we are only interested in the shape of the indifference curves implied by a utility function, and a utility function is unique only up to a positive, monotonic transformation, then only the requirement of homotheticity is relevant and not the additional properties implied by homogeneity.

Elasticity of Substitution

The elasticity of substitution between inputs, a, is defined as proportionate rate of change of the input ratio n - -

proportionate rate of change of the MRTS

when MRT'S, the marginal rate of technical substitution, is the slope of the isoquant. In figure 11.30 we have drawn two level curves (isoquants) for two different production functions / and g. The proportionate rate of change in the input ratio

Figure 11.30 Production function associated with level curve jl' showing a greater elasticity of substitution than lhat associated with level curve f

(i.e.. the numerator of a) in moving from point.v° to x is the same for either isoquant / or,?. However, the proportionate rate of change of the slopes of the isoquants— the MRTS (i.e., the denominator of <r)—is less for # than /. Thus g displays greater elasticity of substitution. To make such comparisons mathematically, we use the following definition:

Definition 11.9

The elasticity of substitution between inputs for a production function v = /(x), x e K2 which has continuous marginal product functions is defined as

The constant elasticity of substitution (CES) production function has the property, as its name suggests, that the value of a is the same at any point on any isoquant. The Cohb-Douglas production function has this property, with a = I.

Example 11.41 Show that the Cobb-Douglas production function y - Ax"x^. A. a, fi > 0 defined on + has a = I everywhere.

Solution

First we find the partial derivatives f\ and f-i'■

and so

This implies that

0 0

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