0.1 Production functions with a single output

0.1.1 Homothetic and Homogeneous Production Functions

Homothetic production functions have the property that f(x) = f(y) impliesf(λx) = f(λy).

Homogeneous production functions have the property that f(λx) = λkf(x)for some k. Homogeneity of degree one is constant returns to scale. Homoge-neous implies homothetic, but not conversely. Example f(x1, x2) = x1x2 + 1 ishomothetic, but not homogeneous. Draw a picture.

Separable production function. Example f(x1, x2, x3) = F (g(x1, x2), x3).Suppose factors 1 and 2 are skilled and unskilled labor and factor 3 is capital.What does separability mean?

Additively separable production function. f(x1, . . . , xn) = F (∑

i g(xi))Abram Bergson’s theorem, A function is additively separable and homothetic

if and only if it is of one of the following two forms: F (∑

i aixρi ) or F (

∑i ai lnxi).

Corollary It is additively separable and homogeneous of degree k iff it is ofthe form (∑

i

aixρi

)k/ρ

or of the form ∏i

xaii

where∑

i ai = k.The first of these cases is known as a constant elasticity of substitution pro-

duction function (ces) and the second as a Cobb-Douglas production function.Why is it called ces? The elasticity of substitution between two factors is

defined to be the absolute value of the percent change in the ratio in which thefactors are used that results from a one percent change in the ratio of the wagesof the two factors. That is, it is the partial derivative of the log of the ratio offactor inputs with respect to the log of the ratio of the factor prices. (elasticityof this with respect to that is always -d log this /d log that.)

For a cost-minimizing firm, the ratio of the marginal products of two factorsthat are both used must equal the ratio of their wages. In this case the marginalrate of substitution between factors i and j equals the wage ratio when

aixρ−1i

ajxρ−1j

=wi

wj.

Take logs of both sides of this equation and you have

lnwi

wj= ln

ai

aj+ (ρ− 1) ln

xi

xj.

Rearrange this equation to write

lnxi

xj=

1(ρ− 1)

lnai

aj− 1

(ρ− 1)ln

wi

wj

1

Therefore the elasticity of substitution between factors i and j is σ = 1/(1− ρ),which is a constant. Hence ces.

For the Cobb-Douglas utility, the elasticity of substitution between any twofactors is 1. (Prove this yourself.) It can be proved that the Cobb-Douglasutility function is the limit as ρ→ 0 of the ces utility functions with parameterρ.

Empirical economists find the ces form especially useful, since if they havedata on changing factor prices and factor utilizations, they can run a log-logregression of factor utilization ratios on factor price ratios to estimate the elas-ticity of substitution as the slope and the ratio a1/a2 as the intercept of theirregression.

If the production function is quasi-concave as well as ces, then it must bethat ρ ≤ 1. Notice that ρ can be negative. Notice also that if ρ > 1, equalizingthe ratio of marginal products to the wage ratio does not minimize cost ofproduction. Look at the example where ρ = 2. How do you minimize productioncosts?

It is instructive to graph the function σ = f(ρ) = 1/(1 − ρ). Notice thatif ρ < 0, then 0 < σ < 1 and that limρ→−∞ f(ρ) = 0. Notice also that if0 < ρ < 1, then σ > 1 and that the limit of f(ρ) as ρ approaches 1 from belowis ∞. Finally, note that if ρ > 1, then σ < 0, with the limit of f(ρ) as ρapproaches 1 from above equal to −∞ and limρ→∞ f(ρ) = 0.

The functional share of a factor is defined as its quantity times its wagedivided by the value of output. Note that in the ces case, an increase in afactor’s wage, holding other wages constant, will increase or decrease the factor’sfunctional share depending on whether the elasticity of substitution is less thanor greater than one.

0.1.2 Cost Function for C.E.S Production Function

It turns out that the cost function for a c.e.s production function is also of thec.e.s. form and if the production function has elasticity of substitution σ, thecorresponding cost function has elasticity of substitution 1/σ.

I leave the Cobb-Douglas case to you.Let’s simplify the ces to the case of two factors, constant returns to scale

and where a1 = a2 = 1. Once we see how to find the cost function for this case,a couple of cheap tricks allow one to handle nonconstant returns differing ai’s.Extending to more than two goods is also easily done and yields no surprises.

Where σ = 11−ρ , we have ρ = σ−1

σ . We will focus on the class of quasi-concave c.e.s. functions, for which ρ ≤ 1 and hence σ ≥ 0. It turns out to beslightly more convenient to write the utility function by substituting σ for ρ.

Let

f(x) =(x

σ−1σ

1 + xσ−1

σ2

) σσ−1

. (1)

Let us find c(w, 1), the cost of producing one unit in the cheapest possible waywhen wages are w = (w1, w2). To do so, we begin by finding the conditionalfactor demand functions x1(w, 1) and x2(w, 1), which are the amounts of the

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two factors used to to produce one unit of product in the cheapest possible waywhen the wage vector is w. Having found the conditional factor demands, wecan then solve for c(w, 1) = w1x1(w, 1) + w2x2(w, 1).

Calculation shows that the ratio of marginal products of factors 1 and 2 is

f1(x1, x2)f2(x1, x2)

=(

x1

x2

)− 1σ

(2)

Cost minimization requires that the ratio of marginal products equals the wageratio. Therefore if x1 = x1(w, 1) and x2 = x2(w, 1), then(

x1

x2

)− 1σ

=w1

w2. (3)

From Equation 3 it follows that

x1wσ1 = x2w

σ = K(w1, w2) (4)

for some function K(w1, w2). Therefore

x1(w1, w2, 1) = K(w1, w2)wσ1 and x2(w1, w2, 1) = K(w1, w2)wσ

2 (5)

It must be that the input vector (x1(w, 1), x2(w, 1)), is sufficient to produceone unit of output. That is

xσ−1

σ1 + x

σ−1σ

2 = 1 (6)

Substituting the expressions for x1 and x2 from Equations 5 into Equation 6and arranging terms, we find that

K(w1, w2) =(w1−σ

1 + w1−σ2

) σ1−σ (7)

Now

c(w, 1) = w1x1(w, 1) + w2x2(w, 1)= w1K(w1, w2)w−σ

1 + w2K(w1, w2)w−σ2

=(w1−σ

1 + w1−σ2

)1+ σ1−σ (8)

=(w1−σ

1 + w1−σ2

) 11−σ (9)

Now we can show the remarkable fact that the elasticity of substitution of thecost function is just the inverse of that of the production function. Rememberthat when a c.e.s. function is of the form

(xρ1 + xρ

2)1ρ

its elasticity of substitution is 1/(1−ρ). In the case of our production function,we set ρ = (σ − 1)/σ and simple algebra shows that σ = 1/(1 − ρ). Our costfunction turns out to be of the form of equation 0.1.2 with ρ = 1 − σ where σis the elasticity of substitution of the production function. Now the elasticityof substitution of this cost function is 1/ (1− (1− σ)), which is just 1/σ.

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