Suppose that f is a function of n variables defined in a domain D. Suppose also that whenever(x1, x2, . . . , xn)∈Dandt >0, the point(t x1, t x2, . . . , t xn)also lies inD. (A setDwith this property is called acone.) We say thatf ishomogeneous of degreekon Dif
f (t x1, t x2, . . . , t xn)=tkf (x1, x2, . . . , xn) for all t >0 (1) The constantkcan be any number—positive, zero, or negative.
E X A M P L E 1 Test the homogeneity off (x1, x2, x3)= x1+2x2+3x3
x12+x22+x32 .
Solution: Heref is defined on the setDof all points in three-dimensional space excluding the origin. This set is a cone. Also,
f (t x1, t x2, t x3)= t x1+2t x2+3t x3
(t x1)2+(t x2)2+(t x3)2 = t (x1+2x2+3x3)
t2(x12+x22+x32) =t−1f (x1, x2, x3) Hence,f is homogeneous of degree−1.
Euler’s theorem can be generalized to functions ofnvariables:
T H E O R E M 1 2 . 7 . 1 ( E U L E R ’ S T H E O R E M )
Supposefis a differentiable function ofnvariables in an open domainD, where x=(x1, x2, . . . , xn)∈Dandt >0 implytx∈D. Thenf is homogeneous of degreekinDif and only if the following equation holds for allxinD:
n i=1
xifi(x)=kf (x) (2)
Proof: Supposef is homogeneous of degreek, so (1) holds. Differentiating this equa- tion w.r.t. t (withx fixed) yields n
i=1xifi(tx) = ktk−1f (x). Setting t = 1 gives (2) immediately.
To prove the converse, assume that (2) is valid for allxinD. Keepxfixed and define the function gfor allt >0 byg(t )=t−kf (tx)−f (x). Then differentiating w.r.t.tgives
g(t )= −kt−k−1f (tx)+t−k n i=1
xifi(tx) (∗)
Becausetxlies inD, equation (2) must also be valid when eachxiis replaced byt xi. Therefore, we getn
i=1(t xi)fi(tx)=kf (tx). Applying this to the last term of(∗)implies that, for allt >0, one hasg(t )= −kt−k−1f (tx)+t−k−1kf (tx)=0. It follows thatg(t )must be a constantC. Obviously, g(1) = 0, soC = 0, implying thatg(t )≡0. According to the definition ofg, this proves that f (tx)=tkf (x). Thus,f is indeed homogeneous of degreek.
An interesting version of the Euler equation (2) is obtained by dividing each term of the equation by f (x), provided this number is not 0. Recalling the definition of the partial elasticity (Elif (x)=(xi/f (x))fi(x)), we have
El1f (x)+El2f (x)+ ã ã ã +Elnf (x)=k (3) Thus, the sum of the partial elasticities of a function ofnvariables that is homogeneous of degreekmust be equal tok.
The results in (3) to (5) in the previous section can also be generalized to functions of nvariables. The proofs are similar, so they can be left to the interested reader. We simply state the general versions of (3) and (5):
Iff (x)is homogeneous of degreek, then:
fi(x) is homogeneous of degree k−1, i=1,2, . . . , n (4) n
i=1
n j=1
xixjfij(x)=k(k−1)f (x) (5)
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Economic Applications
Let us consider some typical examples of homogeneous functions in economics.
E X A M P L E 2 Letf (v) = f (v1, . . . , vn)denote the output of a production process when the input quantities arev1, . . . , vn. It is often assumed that if all the input quantities are scaled by a factort, thenttimes as much output as before is produced, so that
f (tv)=tf (v) for all t >0 (∗) This implies thatf is homogeneous of degree 1. Production functions with this property are said to exhibitconstant returns to scale.
For any fixed input vectorv, consider the functionϕ(t )=f (tv)/t. This indicates the average returns to scale—i.e. the average output per unit input when all inputs are rescaled together. For example whent = 2, all inputs are doubled. When t = 43, all inputs are reduced proportionally by14.
Now, when(∗)holds, thenϕ(t )=f (v), independent oft. Also, a production function that is homogeneous of degree k < 1 has decreasing returns to scale because ϕ(t ) = tk−1f (v)and soϕ(t ) <0. On the other hand, a production function hasincreasing returns to scaleifk >1 because thenϕ(t ) >0.
E X A M P L E 3 The general Cobb–Douglas functionF (v1, . . . , vn)=Av1a1ã ã ãvann is often used as an example of a production function. Prove that it is homogeneous, and examine when it has constant/decreasing/increasing returns to scale. Also show that formula (3) is confirmed.
Solution: Here
F (tv)=A(t v1)a1. . . (t vn)an =Ata1v1a1. . . tanvnan =ta1+ããã+anF (v)
SoFis homogeneous of degreea1+ã ã ã+an. Thus it has constant, decreasing, or increasing returns to scale according asa1+ ã ã ã +an is= 1,< 1, or> 1. Because EliF = ai, i=1, . . . , n, we getn
i=1EliF =n
i=1ai, which confirms (3) in this case.
E X A M P L E 4 Consider a market with three commodities with quantities denoted byx,y, andz, whose prices per unit are respectivelyp,q, andr. Suppose that the demand for one of the com- modities by a consumer with incomemis given byD(p, q, r, m). Suppose that the three prices and incomemare all multiplied by somet >0. (Imagine, for example, that the prices of all commodities rise by 10%. Or that all prices and incomes have been converted into euros from, say, German marks.) Then the consumer’s budget constraintpx+qy+rz≤m becomestpx+t qy+t rz≤t m, which is exactly the same constraint. The multiplicative constanttis irrelevant to the consumer. It is therefore natural to assume that the consumer’s demand remains unchanged, with
D(tp, t q, t r, t m)=D(p, q, r, m)
Requiring this equation to be valid for allt > 0 means that the demand function D is homogeneous of degree 0. In this case, it is often said that demand is not influenced by
“money illusion”; a consumer with 10% more money to spend should realize that nothing has really changed if all prices have also risen by 10%.
As a specific example of a function that is common in demand analysis, consider D(p, q, r, m)= mpb
pb+1+qb+1+rb+1 (bis a constant) Here
D(tp, t q, t r, t m)= (t m)(tp)b
(tp)b+1+(t q)b+1+(t r)b+1 =D(p, q, r, m) sincetcancels out. So the functionishomogeneous of degree 0.
Sometimes we encounter nonhomogeneous functions of several variables that are, however, homogeneous when regarded as functions of some of the variables only, with the other variables fixed. For instance, the (minimum) cost of producingyunits of a single output is often expressed as a functionC(w, y)ofyand the vectorw=(w1, . . . , wn)of prices ofn different input factors. Then, if all input prices double, one expects cost to double. So econo- mists usually assume thatC(tw, y)=t C(w, y)for allt >0—i.e. that the cost function is homogeneous of degree 1 inw(for each fixedy). See Problem 6 for a prominent example.
Homothetic Functions
Letf be a function ofnvariablesx=(x1, . . . , xn)defined in a coneK. Thenf is called homotheticprovided that
x,y∈K, f (x)=f (y), t >0 ⇒ f (tx)=f (ty) (6) For instance, iff is some consumer’s utility function, (6) requires that whenever there is indifference between the two commodity bundlesxandy, then there is also indifference after they have both been magnified or shrunk by the same proportion. (If this consumer is indifferent between 2 litres of soda and 3 litres of juice, she is also indifferent between 4 litres of soda and 6 litres of juice.)
A homogeneous functionf of any degreekis homothetic. In fact, it is easy to prove a more general result.
Hstrictly increasing and
f is homogeneous of degreek ⇒ F (x)=H (f (x))is homothetic (7) Indeed, suppose thatF (x)=F (y), or equivalently, thatH (f (x))=H (f (y)). BecauseH is strictly increasing, this implies thatf (x)=f (y). Becausefis homogeneous of degreek, it follows that ift >0, then
F (tx)=H (f (tx))=H (tkf (x))=H (tkf (y))=H (f (ty))=F (ty)
This proves thatF (x)is homothetic. Hence, any strictly increasing function of a homo- geneous function is homothetic. It is actually quite common to take this property as the definition of a homothetic function, usually withk=1.2
2 Suppose thatF (x)is any continuous homothetic function defined on the coneK of vectorsx satisfyingxi≥0, i=1, . . . , n. Suppose too thatF (tx0)is a strictly increasing function oftfor each fixedx0=0inK. Then one can prove that there exists a strictly increasing functionHsuch thatF (x)=H (f (x)), where the functionf (x)is homogeneous of degree 1.
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The next example shows that not all homothetic functions are homogeneous.
E X A M P L E 5 Show that the function F (x, y) = xy +1, which is obviously not homogeneous, is nevertheless homothetic.
Solution: DefineH (u)=u+1. ThenHis strictly increasing. The functionf (x, y)=xy is homogeneous of degree 2, andF (x, y)=xy+1=H (f (x, y)). According to (7),F is homothetic. (You can also use definition (6) to show directly thatF is homothetic.) Suppose thatF (x)=F (x1, x2, . . . , xn)is a differentiable production function, defined for all(x1, . . . , xn)satisfyingxi ≥ 0, i = 1, . . . , n. Themarginal rate of substitutionof factorj for factoriis defined by
hj i(x)=∂F (x)
∂xi
∂F (x)
∂xj , i, j=1,2, . . . , n (8) We claim that ifF is a strictlyincreasing transformation of a homogeneous function (as in (7)), these marginal rates of substitution are homogeneous of degree0.3To prove this, suppose thatF (x)=H (f (x)), whereHis strictly increasing andf (x)is homogeneous of degreek. Then∂F (x)/∂xi =H(f (x))(∂f (x)/∂xi), implying that
∂F (x)
∂xi
∂F (x)
∂xj =∂f (x)
∂xi
∂f (x)
∂xj , i, j=1,2, . . . , n
because the factorHcan be cancelled. Butf is homogeneous of degreek, so we can use (4) to show that, for allt >0,
hj i(tx)= ∂f (tx)
∂xi
∂f (tx)
∂xj =tk−1∂f (x)
∂xi tk−1∂f (x)
∂xj =hj i(x), i, j =1,2, . . . , n (9) Formula (9) shows precisely that the marginal rates of substitution are homogeneous of degree 0. This generalizes the result for two variables mentioned at the end of Section 12.6.
P R O B L E M S F O R S E C T I O N 1 2 . 7
⊂SM⊃1. Examine which of the following functions are homogeneous, and find the degree of homogeneity for those that are:
(a) f (x, y, z)=3x+4y−3z (b) g(x, y, z)=3x+4y−2z−2 (c) h(x, y, z)=
√x+ √y+√z
x+y+z (d) G(x, y)=√xy lnx2+y2 xy (e) H (x, y)=lnx+lny (f) p(x1, . . . , xn)=n
i=1xin
⊂SM⊃2. Examine the homogeneity of the following functions:
(a) f (x1, x2, x3)= (x1x2x3)2 x41+x24+x34
1 x1+1
x2+1 x3 (b) x(v1, v2, . . . , vn)=A
δ1v1−+δ2v2−+ ã ã ã +δnv−n
−μ/
(The CES function.)
3 Because of footnote 2, the same must be true ifF is any homothetic function with the property thatF (tx)is an increasing function of the scalartfor each fixed vectorx.
3. Examine the homogeneity of the three meansx¯A,x¯G, andx¯H, as defined in Example 11.5.2.
4. D. W. Katzner has considered a utility functionu(x)=u(x1, . . . , xn)with continuous partial derivatives that for some constantasatisfy
n i=1
xi∂u
∂xi =a (for allx1>0,. . .,xn>0)
Show that the functionv(x)=u(x)−aln(x1+ ã ã ã +xn)is homogeneous of degree 0. (Hint:
Use Euler’s theorem.)
⊂SM⊃5. Which of the following functionsf (x, y)are homothetic?
(a) (xy)2+1 (b) 2(xy)2
(xy)2+1 (c) x2+y3 (d) ex2y HARDER PROBLEMS
6. Suppose thatf (x)andg(x)are homogeneous of degreerands, respectively. Examine whether the following functionshare homogeneous. Determine the degree of homogeneity in each case.
(a) h(x)=f (x1m, x2m, . . . , xnm) (b) h(x)=(g(x))p (c) h=f+g
(d) h=f g (e) h=f/g
⊂SM⊃7. Thetranslog cost functionC(w, y), wherewis the vector of factor prices andyis the level of output, is defined implicitly by
lnC(w, y)=a0+c1lny+ n i=1
ailnwi+1 2
n i,j=1
aijlnwilnwj+lny n i=1
bilnwi
Prove that this translog (short for “transcendental logarithmic”) function is homogeneous of degree 1 inw, for each fixedy, provided that all the following conditions are met:n
i=1ai=1;
n
i=1bi=0;n
j=1aij=0 for alli; andn
i=1aij=0 for allj.