APPENDIX A RETURNS TO SCALE

APPENDIX A RETURNS TO SCALE

The consequences of proportional input scaling on outputs are

(A.1 ) Definition: If '1xER+, A <: 1 and 0 N < Il ::; 1, 1. (a) POx) c ₌ AP(x); (b) IlP(x) c P(llx), 2. P(Yx) YP(x) , Y > 0,

3. (a) AP(x) c POx) ; (b) P(llx) c IlP(x) ,

the output correspondence is called 1. Subhomogeneous, 2. Homogeneous (of degree +1), and 3. Super homogeneous , respectively.

The effects of proportional output scaling on inputs are

(A.2)

(A.3)

Definition: If '1uER+, e <: 1 and M 0 < 0 ::; 1, 1 . (a) L(eu) c eL(u); (b) oL(u) c L(ou), 2. L(Yu) YL( u), Y > 0,

3. (a) eL(u) c L( eu); (b) L(ou) c oL(u),

the input correspondence is called 1. Subhomogeneous, 2. Homogeneous (of degree +1) and 3. Superhomogeneous, respectively.

Proposition: The output correspondence is subhomogeneous, homogeneous or superhomogeneous if and only if the input correspondence is

subhomogeneous, homogeneous or superhomogeneous, respectively.

Proof: Suppose P is subhomogeneous. The input correspondence L(eu) = {x: eu£P(x)}, e;:; 1, ^(2.1.4). Thus, L(eu) = {x: uE1/ep(x)}

S

L is subhomogeneous and e ;:; 1, p(ex) = {u: 6XEL(u)} = {u: xE1/eL(u)} C {u: x£L(u/e)}

ep(x). Similar arguments apply to complete the proof.

Q.E.D.

The homogeneity notions defined above have obvious geometric interpretations. To elucidate these, introduce the following notions.

(A.4)

(A.S)

(A.6)

150

Definition: A set K C ₌ RG is a Cone if vw£K, Yw£K, Y ₊ > O. A set S C RG is ₊ Starred if vw£S, ~U£S, 0 < ~ S 1. 'A set A C R~ is Areoled if vw£A, lwG£A,

l :i: 1.

Proposition: The output correspondence is subhomogeneous, hcmogeneous (of degree +1) or superhomogeneous if and only if the graph (GR) is starred, a cone or areoled, respectively.

Remark: If VX£RN and l ₊ :i: I, P(lx) C ₌ lP(x) (1.(a», then and only then vx£RN and 0 ₊ < ~ S 1, ~P(x) C = P(~x) (1.(b».

Proof: Suppose X£RN, l ₊ :i: 1 and P(lx) C lP(x). Define y ₌ = (lx), and ~ = (Ill), then y£RN, 0 ₊ < ~ S 1 and ~P(y) C ₌ P(~y). The converse is proved analogously.

Q.E.D.

Proof of Proposition (A.S): Suppose the graph is starred. If (x,u)£GR then

(~x,~u)£GR, 0 < ^~ S 1. Thus u£P(x), (2.1.6) and ^~u£P(~x). Therefore, P(x) C

I/~P(~x) and ~P(x) C P(~x). To prove the converse, suppose P is subhomogeneous.

Let UEP(X) and 0 < ^~ S 1. Then (x,u)EGR, (2.1.5), and ~UE~P(X)

S

(~x,~u)EGR. In view of Remark (A.6), the first part is proved. The remaining parts of the proof are left for the reader.

(A.7)

(A.8)

Q.E.D.

Definition: The production technology exhibits Non Increasing Returns to Scale (NIRS), Constant Returns to Scale (CRS) or Non Decreasing Returns to Scale (NDRS) if and only if the output correspondence is subhomogeneous, homogeneous (of degree +1) or superhomogeneous, respectively.

Definition: The production technology exhibits Increasing Returns to scale (IRS) if it exhibits NDRS and not CRS. It exhibits Decreasing Returns to Scale (DRS) if it exhibits NIRS and not CRS.

The above returns to scale definitions are global. However, sometimes local notions are needed, in particular in finding the source of scale inefficiency (Sections 9.3, 9.4 and 9.5) such local notions are required.

Suppose the graph is known, then define the cone

151

(A.9) K(GR) {(x,u): (x,u) (ly,lv), (y,v)EGR, 1 > OJ

(A.10) Definition: (x,u)EGR exhibits IRS, if So(x,u) > 1, (Si(u,x) < 1) and

* *

(x,uWo(x,u»EK(GR), «xWi(u,x),u)EK(GR». It exhibits DRS, if So(x,u) > 1,

* *

(Si(u,x) < 1) and (x,uWo(x,u»iK(GR), «xWi(u,x),u)iK(GR».

Traditionally, returns to scale is defined in terms of scale elasticity. Suppose a production function F is known, and suppose it is differentiable. The Elasticity of Scale is defined as

(A.11) E(x,u)

To generalize (A.11) note that Do(x,u) ~ u/F(x). Then

DO(X'U) ^N -aDo(x,u) x

(A.12) E(X,U) 1: n

u _n~1 Ox n (Do(X'U» 2

N aDo(x,u) x

1: n

ax Do(X'U)

n~1 n

In light of (A.12), the Primal Output-Based measure of scale elasticity is given by

(A.13) -XVXDO(X'U)

Do(x,u)

The Primal Input-Based measure of scale elasticity is

(A. 14)

(A.15 )

-uV D. (u,x)

( u 1 )-1

D. (u,x)

1

Proposition: If inputs and outputs are proportionally scaled (dxn/xn ) (prop x), vn and (dum/um) ^~ (prop u), vm, then EO(U,X) ~ Ei(u,x).

Proof: The output distance function is homogeneous of degree +1 in output, thus x(prop x)VxDo(x,u) + (prop u)Do(x,u) ~ O. The input distance function is homogeneous in inputs, thus u(prop u)VuDi(u,x) + (prop x)Di(u,x) ^~ O. The two expressions yield

(prop x) (prop u)

DO(X,U) xV'xDo(x,u)

152

uV' uDi (u,x) Di(u,x)

To characterize the input-based measure of scale elasticity, recall the input conjugate duality

(A.16) px C(U,p)Di(u,x).

where x is chosen optimality.

Differentiation of (A.16) with respect to u yields

Multiply by u to obtain

(A.17) uV'UC(u,p) C(u,p)

UV'UDi (u,x) Di(u,x)

The left hand side is thus a dual, cost representation of Ei(u,x).

From the output conjugate duality (6.6.9) we obtain,

(A.18) xV' xR(x,r) R(x,r)

xV'xDo(x,u) Do(X'U)

as the dual, revenue representation of EO(U,X).

Q.E.D.

(A.19) Definition: (x,u) exhibits IRS if EO(U,X) > 1, CRS if EO(U,X) if EO(U,X) < 1.

1 and DRS

APPENDIX B

STANDARD NOTATIONS AND MATHEMATICAL APPENDIX

Let A and B be two sets, we mean by

£ a£A a is an element in A;

i aU a is not an element in A;

C A ^C B A is a subset of B;

C A C B A is a proper subset of B;

eJ A III A is an empty set;

{a£A: *} the subset of A formed by the elements satisfying property *;

n A n B {x: x£A and x£B};

u AU B {x: x£A or x£B};

\ A\B {x: x£A, xiB};

Compl. Complement;

+ A ⁺ B A ⁺ B (z: a£A, b£B, Z a + b);

RN Euclidean space of dimension N;

~ x,y£RN, x ~ Y if and only if xn ~ Yn' n 1,2, ... ,N;

> ^{x ~} Y if and only if x ~ y and x ~ y;

> ^x > y if and only if xn > Yn' n 1,2, ••• ,N;

*

*

> ^{x >} y if and only if xn > Yn or xn Yn ^{0, n} 1,2, ••• ,N;

154 RN + RN + {x: xe:RN, x ii: O};

RN ++ RN

++ {x: xe:RN, x > 0);

RN RN {x: xe:RN, x ~ 0);

R+ R+ R+ U {+"');

2 RN

2 RN

{A: A C RN);

[, ] [a,b] {x: a ^~ x ^:;: b};

[a,b) [a,b) {x: a ~ x < b);

A is convex if for all 0 ~ A ~ 1, x,ye:A, AX + (l-A)ye:A;

3 there exists;

for all;

sum sign;

x product sign;

the sequence x converges to xo; ~

tends to +"';

closure of it;

=> xe:A => xe:B, x belongs to A only if x belongs to B;

<=> if and only if;

gradient of F(x);

s.t. subject to;

positive integers;

px

B.1

B.2 B.3 B.4

B.5

B.6 B.7 B.8

B.9

B.10

B.11

B.12 B.13

px

155

N L P x , p and xeRN, the inner product.

n=1 n n

A set A c RN is bounded <=> sup{lIx-yll! x,yeA}

A set F c RN is closed <=> 'ts 9- ^{-+ so,} s9- eF ,'t9-,

< +"'.

sOeF.

A set Q c RN is compact <=> it is closed and bounded.

A set Q in a topological space is compact <=> every cover of Q has a finite subcover.

A function f: RN ₊ ~ R is upper semi-continuous <=> 'tx9-₊ ~ xO, lim sup f(xi ) ~ f(xO) <=) (x: f(x) ~ ~} is closed, 't~eR+ <=>

i~+'"

GR = ((x,~): f(x) ~ ~, 't~eR+} is closed.

f is lower semi-continuous <=> -f is upper semi-continuous.

f is upper semi-continuous, Q compact, max f(x) is achieved.

xeQ

f: R~ ~ R+ is quasi-concave <=> 'tA, 0 ~ A ~ 1, X,yeR~, f(Ax + (1-Aly) ~ min{f(x), fey)}

f is quasi-convex <=> -f is quasi-concave.

f is quasi-concave <=> (x: f(x) ^{~ ~}} is convex ^'t~eR+.

f is continuous <=> f is upper and lower semi-continuous.

f: R ~ R is strictly increasing if x > u => f(x) > fey).

A continuous strictly increasing function f has an inverse

C 1 .

REFERENCES

Aczel, J. (1966) Lectures on Functional Equations and Their Applications, Academic Press, New York.

Afriat, S. N. (1972) "Efficiency Estimation of Production Functions," International Economic Review, 13:3, pp. 568-598.

AI-Ayat, R. and R. Fare (1979) "On the Existence of Joint Production Functions,"

Naval Research Logistics Quarterly, 26:4, pp. 627-630.

Arrow, K., E. Barankin and D. Blackwell (1953) "Admissable Points of Convex Sets,"

in Contributions to Theory of Games. H.W. Kuhn and A.W. Tucker, eds., Princeton University Press, Princeton.

Blackorby, C., D. Primont and R.R. Russell (1978) Duality Separability, and Functional Structure: Theory and Economic Applications, North-Holland, New York.

Carlson, S. (1939) A Study on the Pure Theory of Production, King, London.

Diewert, E. (1982) "Duality Approaches to Microeconomic Theory," in Handbook of Mathematical Economics, Vol. II, K. Arrow and D. Intriligator, eds., North-Holland, Amsterdam.

Eichhorn, W. (1978) Functional Equations in Economics, Addison-Wesley, Reading Mass.

Eichhorn, W. (1987) "Vorlesung Producktionstheorie," Memo.

Fare, R. (1980) Laws of Diminishing Returns, Springer-Verlag, Berlin.

Fare, R. (1986) "A Dynamic Non-Parametric Measure of Output Efficiency,"

Operations Research Letters, 5:2, pp. 83-85.

157

Fll.re, R. and S. Grosskopf (1985) "A Nonparametric Cost Approach to Scale Efficiency," The Scandinavian Journal of Economics, 87:4, pp. 594-604.

Fll.re, R., S. Grosskopf and E. Kokkelenberg (1987) "Measuring Plant Capacity, Utilization and Technical Change: A Nonparametric Approach," Memo.

Fll.re, R., S. Grosskopf and C.A.K. Lovell (1985) The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston.

Fll.re, R., S. Grosskopf and C.A.K. Lovell (1986) "Scale Economies and Duality,"

Zeitschrift fur Nationalokonomie,. 46:2, pp. 175-182.

Fll.re, R., S. Grosskopf and C.A.K. Lovell (1987) "An Indirect Efficiency Approach to the Evaluation of Producer Performance," Memo.

Fll.re, R. and L. Jansson (1975) "On VES and WDI Production Functions,"

International Economic Review, 16:3, pp. 745-750.

Fll.re, R. and J. Logan (1983:a) "The Rate of Return Regulated Version of Shephard's Lemma," Economics Letters, 13, pp. 297-302.

Fll.re, R. and J. Logan (1983:b) "The Rate-of-Return Regulated Firm: Cost and Production Duality," The Bell Journal of Economics, 14:2, pp. 405-414.

Fll.re, R. and T. Mitchell (1987) "The Family Tree of Production Functions," Memo.

Fll.re, R. and R.W. Shephard (1977) "Ray-Homothetic Production Functions,"

Econometrica, 45:1, pp. 133-146.

Farrell, M.J. (1957) "The Measurement of Productive Efficiency," Journal of the Royal Statistical Society, Series A, General, 120, Part 3, pp. 253-281.

F0rsund, F. and L. Hjalmarsson (1987) Analyses of Industrial Structure: A Putty-Clay Approach, Almquist and Wiksell, Stockholm.

Fukuyama, H. (1987) "Alternative Notions and Measures of Returns to Scale: A Multi-Output Duality Approach," Ph.D. TheSiS, Southern Illinois University, Carbondale.

Fuss, M. and D. McFadden (1978) Production Economics, A Dual Approach to Theory and Applications, Vol. I and II, North-Holland, Amsterdam.

158

Kothe, G. (1969) Topolotical Vector Spaces I, Springer-Verlag, Berlin.

Koopmans, T.C. (1951) Activity Analysis of Production and Allocation, John Wiley and Sons, New York.

Lee, H. and R. Chambers (1986) "Expenditure Constraints and Profit Maximization in u.S. Agriculture," American Journal of Agricultural Economics, pp. 857-865.

Lehmijoki, U. (1984) "Notes on Quasi-Convexity of the Cost Function," The Scandinavian Journal of Economics, 86:3, pp. 385-389.

Malmquist, S. (1953) "Index Numbers and Indifference Surfaces," Trabajos de Estatistica, 4, pp. 209-242.

Menger, K. (1936) "Bemerkungen zu den Ertragsgesetzen," Zeitschrift fUr Nationalokonomie, 7, pp. ~5-56.

Rockafellar, R. (1970) Convex Analysis, Princeton University Press, Princeton.

Sato, R. (1981) Theory of TeChnical Change and Economic Invariance: Application of Lie Groups, Academic Press, New York.

Schumpeter, J.A. (1954) History of Economic Analysis, Oxford University Press, N.Y.

Shephard, R.W. (1953) Cost and Production Functions, Princeton University Press, Princeton.

Shephard, R.W. (1967) "The Notion of a Production Function," Unternehmensforschung, 11, pp. 209-232.

Shephard, R.W. (1970) Theory of Cost and Production Functions, Princeton University Press, Princeton.

Shephard, R.W. (1974) Indirect Production Functions, Verlag Anton Hain, Meisenheim am Glan.

Shephard, R.W. and R. Fare (1980) Dynamic Theory of Production Correspondence, Oelgeschlager, Gunn and Hain, Cambridge.

159

Teusch,

w.

Turgot, A.R.J. (1767) See Schumpeter (1954).

von Neumann, J. (1938, 1945) "A Model of General Economic Equilibrium," Review of Economic Studies, 13:1, pp. 1-9.

Attainability, ~, 21, 44

Congestion, 14, 18, 135 OP-Congestion, 14 OL-Congestion, 14

Cost Function, 83 derivatives, 91-93 duality, 84

properties of, 83, 81

rate of return regulated, 101 ray-homothetic, 88

revenue indirect, 113 short run, 99

Cost Minimization Set, 89

Disposability

strong input,

2,

weak input,

2,

Distance Function

input, 35, 36, 40, 91, 128 duality, 85

properties of, 31-38 ray-homothetic, 56 revenue indireot, 114 output, 29, 36, 31, 40, 41, 128

cos~indirect, 110 duality. 94

properties of, 30-34 ray-homothetic, 52

INDEX

Duality

ccst indirect, 110, 111 ccst-prcducticn, 84, 92 revenue indirect, 114, 115 revenue-prcduction, 94, 197

Efficiency Measures, 127-148 Farrell decompcsiticn, 129-133 input, 128, 129-131

output, 128, 131-133

Efficient Subsets

input,

11,

11,

Essential, 65, 70 ccngestion, 75 cost, 86

not essential, 66

Graph, ~, 119

Indirect Production ccst indirect, 106

properties of, 106-111 efficiency, 143-144

revenue indirect, ^~ prcperties cf, 112-115

Input Correspondence, ~, 36, 38, 39 piecewise linear, 44-49 prcperties of, 8-10 ray-homothetic, 54-56

Input Efficiency Theorem, 89

Input Set, ~, 36, 38, 39 properties of, 14-17

161

Isoquant

input,

.!..l,

.!..l,

Joint Produotion Funotion, 38 properties of, 39-40

Law of Variable Proportions, 79

Limitational strong, 68, 70 weak, 64, 67

Linear Expansion Paths input, 87, 90 output, 96

Maintained Axioms, lQ, 24 independenoe of, 18-20

Null Joint, 77 oongestion, 78

Output Correspondenoe, ^~, 23, 25, 31, 39 piecewise linear, 44-49

properties of, 6-8 ray-homothetic, 49-51, 56

Output Set, ~, 17, 31, 39

Plant Capacity,

II

efficiency, 145-146 properties of, 71-73

Product Technology, 118 properties of, 118-122

162

Produotion Funotion, 23 dynami c, 1 22 examples of, 26-29 indirect, 108 properti es of, 23-29 ray-homothetic, 59-60, 69

Profit Function, 100

Pr ices, 82-86 shadow, 97-98

Quantity Index, 41

Returns to Scale, 147-152

efficiency, 134, 137, 140-143

Revenue Function, 93 cost indirect, 109 duali ty of, 94 properties of, 93-95 ray-homothetic, 95

Shephard's Lemma, 92, 102

Time

substitution, 124-126 support, 121

163