PRODUCTION-POSSIBILITY FRONTIER WITH VARIABLE RETURNS TO SCALE. A GENERAL RESULT ∗ By Jules GAZON † February 28, 2003 Abstract Even if, for over four decades, substantial progress has been made for determining the Production-Possibility Frontier (PPF ) shape, when returns to scale are increasing or variable, the interest of the PPF cur- vature and its calculation have been intermittent and limited to ho- mogeneous or homothetic production functions: Herberg (1969 and 2000), Herberg and Kemp (1969). We decompose the local curvature into the effect of returns to scale and the effect of inputs intensities from production functions, which are not necessarily homothetic or homogeneous, but which exhibit variable returns to scale. The effect of returns operates in the sense of previous results. But, while the economic literature generally waits for input intensities difference to make PPF concave to origin returns effect apart, we show that this is not necessarily the case if the relative variation of partial returns is different for at least one production unit. Keywords: Production-Possibility Frontier, Returns to scale, Effi- cient Allocation of Resources. Increasing returns are an essential characteristic of the economic world. The present technological revolution based on data processing and telecom- munications increasingly implies internal and external economies of scales in ∗ I am grateful to Jacques Bair, professor at the University of Liège (Belgium), Horst Herberg, professor at Christian Albrechts University at Kiel (Germany) and Murray C. Kemp, professor at the University of New South Wales (Australia) for their very helpful comments that enable me to improve a first version of this article. The usual caveat applies. † Université de Liège, Economie Internationale, Boulevard du Rectorat, 7 (B31) - B - 4000 — LIEGE (Belgium) - Tél. + 32(0)43663121 - Fax : + 32(0)43663120 - e-mail: [email protected], Université Paris 1 (Sorbonne), Centre de Recherche Crifes-Matisse. 1
Transcript
PRODUCTION-POSSIBILITY FRONTIERWITH VARIABLE
RETURNS TO SCALE. A GENERAL RESULT∗
By Jules GAZON†
February 28, 2003
AbstractEven if, for over four decades, substantial progress has been made
for determining the Production-Possibility Frontier (PPF ) shape, whenreturns to scale are increasing or variable, the interest of the PPF cur-vature and its calculation have been intermittent and limited to ho-mogeneous or homothetic production functions: Herberg (1969 and2000), Herberg and Kemp (1969). We decompose the local curvatureinto the effect of returns to scale and the effect of inputs intensitiesfrom production functions, which are not necessarily homothetic orhomogeneous, but which exhibit variable returns to scale. The effectof returns operates in the sense of previous results. But, while theeconomic literature generally waits for input intensities difference tomake PPF concave to origin returns effect apart, we show that thisis not necessarily the case if the relative variation of partial returns isdifferent for at least one production unit.
Keywords: Production-Possibility Frontier, Returns to scale, Effi-cient Allocation of Resources.
Increasing returns are an essential characteristic of the economic world.The present technological revolution based on data processing and telecom-munications increasingly implies internal and external economies of scales in
∗I am grateful to Jacques Bair, professor at the University of Liège (Belgium), HorstHerberg, professor at Christian Albrechts University at Kiel (Germany) and Murray C.Kemp, professor at the University of New South Wales (Australia) for their very helpfulcomments that enable me to improve a first version of this article. The usual caveatapplies.
†Université de Liège, Economie Internationale, Boulevard du Rectorat, 7 (B31) - B- 4000 — LIEGE (Belgium) - Tél. + 32(0)43663121 - Fax : + 32(0)43663120 - e-mail:[email protected], Université Paris 1 (Sorbonne), Centre de Recherche Crifes-Matisse.
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both production process and production organization. Traditional use andpowerful results of convex analysis in economic theory show that our under-standing of increasing returns to scale is far from complete. The problemcomes from production side, and especially from the Production-PossibilityFrontier (PPF ) shape. As PPF is the frontier of the output set, it wouldhave to concern production theory. But this latter generally limits its analy-sis to returns to scale as the output set variation due to equiproportionalinputs variation, when PPF results from a Pareto-efficient sharing of lim-ited inputs between all production units. Therefore it concerns all problemsdealing with an efficient allocation of limited resources as the productionside of general equilibrium theory and its implications, international tradetheory, having the most studied, but intermittently, PPF shape with re-gard to returns to scale. Given the possible non-existence of a competitiveequilibrium when there are increasing returns, the theorists of general equi-librium are worried about this situation. As a result (see Heal, 1999), theyhave made important theoretical progress in the three last decades in deal-ing with increasing returns to scale in order to find equilibria, which mightbe efficient for types of pricing. It is not our scope to discuss the existenceor uniqueness of a general equilibrium, but rather to how important it is toknow PPF shape for determining these results. Seeking a direction processto reach a Pareto-efficient allocation of resources, the non market approachthough a central planning board as proposed in the ”tâtonnement process”(Arrow and Hurwicz, 1960), or in the algorithm of Cremer (1977) as in theresearch of a saddle point of the Langangian by Heals (1984) based on theresult of Arrow, Gould and Howe (1977) directly concerns PPF shape. Thisis also the case of the marginal pricing rule starting with Guesnerie, (1975),and developed by Beato (1982) who have exhibited several equilibria onPPF under the assumption that this later is smooth, while Cornet (1982)supposed that it has no ”inward kinks”. But, as shown in Beato and Mas-Colell (1985) if there is more than one firm producing the same commodityand some have increasing returns, PPF exhibits ”inward kinks”. Further-more, Bonnisseau and Cornet (1990) treated the existence of marginal costpricing equilibria in an economy with several nonconvex firms. A third ap-proach, initiated by Dieker, Guesnerie and Neuefeind (1985) and developedin Brown, Heal, Khan and Voha (1986), where non-convex producers fulfilonly the first order Kuhn-Tucker necessary conditions, adopts price settingbehaviors to follow different rules like, profit maximizing, average cost pric-ing, bounded losses; for this later pricing, it is necessary and sufficient thatthe production set be star-shaped (Bonnisseau and Cornet, 1988).
It appears that, with the useful and realistic assumption of convex con-
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sumer behaviors, non convexity of output set caused by increasing returnsis always at the origin of problems to define an efficient general equilib-rium, which must be a tangency point between the pricing hyperplane andthe PPF , even if certain pricing rules proposed equilibria where produc-tion efficiency fails. This paper doesn’t treat general equilibrium, but webelieve that our results, linking local curvature of PPF, notably to returnsto scale, should help to improve the knowledge of the existence, uniquenessand stability of equilibrium.
International trade theorists, often in a different way than general equi-librium theorists, have shown that the PPF shape resulting from increasingreturns might imply the non-uniqueness of equilibrium for international com-petitive economies as demonstrated, for imperfect competition and economiesof scale, by Venables (1984), Kehoe (1985) and Kemp and Schweinberger(1991), confirmed by several applications as this of Mercenier (1995). Obvi-ously, the multiplicity and the stability of the equilibria always depend onconvexity, concavity and the existence of saddle points of PPF.
It is also these economists who have studied the influence of returns toscale nature on the PPF shape the most. Savosnick’s technique (1958) isan important reference for deriving PPF with two production units, whichuses two inputs for producing two outputs. The use by Savosnick of theEdgeworth box showing that not only returns to scale are determinant forPPF shape, but also the difference in input intensities, interested theo-reticians of international trade. But the limitation of geometric technique,especially when Savosnik introduces increasing returns, needed a more rigor-ous demonstration. This was done by Johnson (1966), Quirk and Saposnik(1966), Lancaster (1968, pp 127-134), Melvin (1968) and Kelly (1969). Butabove all, is the article of Herberg (1969), which has established defini-tive results in the case 2x2 for homogeneous production functions, showingthat the PPF shape interpretation of this time was often false in economicliterature. Herberg (2000) generalizes these results for any finite numberof production units and any number of production factors in the case ofhomothetic production functions. Mayer (1974) provided sufficient condi-tions for the local concavity to origin where production functions are onlyquasi-concave but with other hypotheses. Furthermore Herberg and Kemp(1969) have demonstrated a decisive result for external variable returns tothe firm but internal to industry through homothetic production functions,each firm producing one commodity with constant internal returns. Theydetermine PPF shape in the neighborhood of the axis, when the produc-tion of one production unit tends to zero, the resources being nearly fullallowed to other production units and conclude that, if for output value
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near zero, one good is produced under condition of increasing (decreasing)returns to scale, then in the neighborhood of zero output, PPF is strictlyconvex (concave) to origin and this, whatever the nature of returns to scalein the other industry. Answering to Markusen and Melvin (1984) who esti-mate that this result is inapplicable in context of specific factors, Herbergand Kemp (1991) show that if there are two or more mobile factors, theirproposition is valid whether or not there are industry-specific factors. ButWong (1996) showed that the Herberg-Kemp’s proposition does not dependon the number of mobile factors and is more general than they have re-alized. Panagariya (1981) demonstrated that, with one unique productionfactor, two commodities and homogeneous production functions, PPF hasonly one inflection point. When there are several factors, there may haveto be more than one point of inflection. During the seventies, for homo-geneous and linear production functions, several trade theorists completedthe task of specifying necessary and sufficient conditions for a single-countryand world PPF to be locally of any assigned degree of flatness. First in thecase of non-joint-product technology, see: Khang and Uekawa (1973) withthe presence of intermediary goods and the generalisation of Kemp, Khangand Uekawa (1978) who demonstrate that the PPF , with endowment of nprimary-factor inputs, contains an (n − r) dimensional flat if and only ifthere are r linearly independent vectors of primary-factor inputs. Second,Kemp, Manning, Nishimura and Tawada (1980) derive the result under con-ditions of joint production. Furthermore, still in trade theory, it has longbeen recognized that when the number of commodities exceeds the numberof production factors, PPF is not strictly convex to origin. Kemp, Manning,et al. (1980), using a result of Melvin (1968) based on the degree of linearlyindependent vectors of primary factors, generalize this result to show thatPPF contains ”flats” when there are more commodities than primary fac-tors in a world of constant returns to scale. Finally, Manning and Melvin(1992) have shown that any arbitrary concave, negatively sloped PPF canbe generated by a pair of linearly homogeneous quasi-concave productionsfunctions given fixed total endowment of two factors.
Even if the previous results are not always based on the same assump-tions, they give a global description of PPF shape with regard to returns toscale showing that it depends on two forces: in this literature the non lin-earity of input vectors plays in favor of a PPF concave to origin, while thereturns to scale operates in the same direction if they are decreasing and tothe opposite direction if they are increasing; with constant returns to scalePPF is flat if and only if input vectors are linearly dependant which meansthat, with two primary inputs, PPF is a straight line because of identical
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input intensities.This literature limits the analysis to homogeneous or homothetic pro-
duction functions, which is very restrictive when one wants to take intoaccount the variation of returns. The purpose of the present article is to fillthis gap through a general result. We think that the sporadic interest forPPF in economic literature, which has implied no decisive references duringthe recent years, except Herberg (2000) and Wong (1996), doesn’t justifythe abandoning of research, given its importance for efficient allocation ofresources.
In section 1 we define concepts, assumptions and notation for produc-tion technology and production functions with only primary inputs in orderto induce PPF from the efficient allocation of inputs. A production-unitproduces only one output, the multi-input - multi-output relation with fixedendowment of inputs is only based on technology without consideration ofmarket pricing. The production functions follow the usual basic hypothesis;they are quasi-concave, twice differentiable but neither necessarily homoge-neous nor homothetic. The general curvature problem of PPF is posed insection 2 through the Hessian matrix integrating the second order condi-tion of optimisation. Section 3, using vectorial analysis for two productionunits with two inputs, gives the general result of the curvature of PPF atthe neighborhood of axis (Theorem 1) and for any locus of the definition do-main (Theorem 2). It not only confirms the actions of both above-mentionedforces, but also gives a general formulation that specifies calculation of PPFlocal curvature. The local effects of variable returns to scale are clearlyisolated, while the input intensities effect is decomposed: first, into the in-put intensities difference effect,which jointly acts with the marginal rateof substitution relative variation on input frontier (IF ) and the secondly,partial returns ratios of inputs relative variation effect. This general resultclarifies the Hessian matrix elements, which allows the generalisation formulti-outputs, multi-inputs production economy.
Section 4 confronts our general result to main results on PPF in economicliterature, (i) the situation when inputs tend to be devoted to only oneactivity, which generalises the result of Herberg and Kemp (Theorem 1), (ii)the non variable returns case, which corresponds to homogeneous productionfunction and the case where productions functions are homothetic confirmthe Herberg (1969 and 2000)’s statements (Theorem 3).
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1 Production technology andProduction-Possibility Frontier
1.1 Production technology
Let producers use inputs {1, ...,N} in quantities such as x = (x1, ..., xN ) ∈RN
+ to produce different outputs y = (y1, ..., yM ) ∈ RM+ , R+ = [0,∞[ and
where {1, ...,M} are the M production-units.The technology set consists of all feasible input-output vectors: T =
{(x,y) : x can produce y} . The input correspondence is: y −→ L (y) ⊆RN
set P (x) is called the output set: it contains all output vectors which canbe produced by input vectors x. As a consequence, we have the followingequivalent relations: (x,y) ∈ T ⇐⇒ y ∈ P (x)⇐⇒ x ∈ L (y).
The input and output correspondences satisfy the following assumptions:A.1. P (0) = {0} (inactivity),A.2. If y ≥ 0 (y 6= 0), y 6= P (0), (no free lunch),A.3. P (x) is bounded for each x ∈ RN
+ (scarcity),A.4. P (x) is closed,A.5. L (y) is closed and convex,A.6. P (resp. L) is continuous at x (resp. y),A.7. If y ∈ P (x), x ≤ x0 ⇐⇒ y ∈ P (x0), (strong disposability of
inputs),A.8. If y ∈ P (x), y ≥ y0 ⇐⇒∈ y0 ∈ P (x), (strong disposability of
outputs).A.9. There are M production-units, one for each output {1, ...,M} .This assumption allows avoiding ”inward kinks”on PPF which could
appear when several units produce the same output as shown in Beato andMas-Colell (1985).
Let’s define input-isoquant by:
IsoqL (y) = {x : x ∈ L (y.) , λ ∈ [0, 1[ =⇒ λx /∈ L (y)} ,y ≥ (0) , y 6= (0) .A.10. If x ∈ IsoqL (y) then x0≤ x (x0 6= x) =⇒ x0 /∈ L(y),y ≥ 0(y 6=
0): input vector x corresponds to a Pareto-efficiency allocation of inputs.
This means that a reduction in any input requires an increase in at leastone input to hold the same output y.
6
Furthermore, the output-isoquant is defined by:IsoqP (x) = {y : y ∈ P (x) , θ ∈ ]1,∞[ =⇒ θy /∈ P (x)} ,x ≥ (0) ,x 6= 0.
A.11. If y ∈ P (x), then y0≥ y =⇒ y0 /∈ P (x),x ≥ 0,x 6= 0 : outputvector y corresponds to a Pareto-efficiency distribution of outputs.
A.12. On an input-isoquant, one input might be continuously substitutedfor another.
A.13. On an output-isoquant, one output might be continuously substi-tuted for another.
A.14.: Technology is characterized by a fixed input endowment, x̄ =(x̄1, ..., x̄N ) ⊂ RN
+ fully employed where ∀j ∈ {1, ...,N} , xij ∈ [0, x̄j ] ⊂ R1+, is
the quantity of input j used to produce output i withXM
i=1xij = x̄j (scarcity
constraint).Input endowment could be considered as the set of full employment in-
put vectors: x̄ =
½x : x =
·XM
i=1xi1 = x̄1, ...,
XM
i=1xjN = x̄N
¸¾and the
corresponding full employment technology is−T =
n³−x,y
´: y ∈ P
³−x´o.
Scarcity constraint A.14. refers to the Edgeworth box for the input setand implies that the inputs are mobile between production units but theirsupply is completely inelastic.
Assumption A.14 doesn’t suppose that the inputs are necessarily essen-tial. They are usable but it is possible that one production unit doesn’t useall inputs. In this case, the inputs that are not used by some productionunits are specific to the others. This is less restrictive than Herberg’s (1969,2000) assumption. However, Herberg (2000) shows that his results don’tneed to have all input indispensable. His demonstration holds if there is atleast one input that is essential and there is at least one output for which allinputs are essential. In the case 2x2 treated in section 3, assumption A.14implies this situation.
Note that, even if x ∈ −x is a full employment input vector, the entire
map of P (x) is not necessarily technologically efficient, as full employment ofinputs is not sufficient to reach an efficient output. Given input endowmentx̄, Production Possibility Frontier (PPF), which corresponds to the Pareto-
efficiency output set, is defined by: PPF =n
y : y ∈ IsoqP (x) ,x ∈ −xo.
Similarly, Input Frontier (IF ), which corresponds to Pareto-efficiency allo-cation of inputs and which is the contract curve in Edgeworth box, is defined
as: IF =n
x : : x ∈ IsoqL (P (x)) ,x ∈ −xo. Because the correspondences P
and L are continuous (A.6.) with strong disposability ( A.7, A.8), we have,
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as demonstrated in R.W. Shephard (1970, pp 212-222):
(1) L (IsoqP (x)) = IsoqL (P (x)) .
Consequently as it is well known, PPF is mapping Input frontier (IF ) andreciprocally: T = {(x,y),x ∈IF ,y ∈PPF} is an efficient technology. Inother words, to have an output vector included in PPF, it is necessary andsufficient that the corresponding input vectors be included in IF.
Now let define technology i, also called activity i, as an input-output vec-tor subset
¡xi,yi
¢= Ti, where the input vector is defined by
xi =¡xi1, ..., x
iN
¢,xi ≤ x̄ (< or =) and the output vector by: yi =
(0, ..., 0, yi, 0, ..., 0) ∈ P¡xi¢.
If yi ∈ Isoq (P (xi)), technology Ti is efficient and from (1) one has:L¡IsoqP
¡xi¢¢= IsoqL
¡P¡xi¢¢
1.2 Production functions
For the efficient technology Ti, component yi of the output vector yi ∈IsoqP (xi) may be considered as the production function yi
knowing that due to optimisation one has: ∇yi(xi) =
XM
k 6=iµk∇yk(x
k)
where∇yi(xi),∇yk(xk) are the gradient vectors of yi(xi) and yk(x
k) in RN+
and µk ∈ R.Therefore PPF might be defined as implicit joint production function in
RM+ : F
¡y1, ..., yM
¢= 0 or by any explicit production function yi : RM−1
+ −→R1
+:
(3) yi = yi¡y1, ...,
£yi¤, ..., yM
¢.
Due to assumptions A.13, A.15. and relation (1), the reciprocal function(yi)−1 of yi : yi −→ xi exits and is twice continuously differentiable.
B A
21
3
IF
01
02 y1
y2
PPF
A’
B’
0 02
0)1( y=
(x1)01yIsoqmax
01yIsoq (x1)
y
( )( )101 xyIsoqL:1
( )( )202 xyIsoqL:2
( )( ) 21101 xxxwithxyIsoqmaxL:3 ==
( ) ( )xPIsoqBP'B ∉=
( ) ( )xPIsoqAP'A ∈=
B A
21
3
IF
01
02 y1
y2
PPF
A’
B’
0 02
0)1( y=
(x1)01yIsoqmax (x1)01yIsoqmax
01yIsoq (x1)01yIsoq (x1)
y
( )( )101 xyIsoqL:1
( )( )202 xyIsoqL:2
( )( ) 21101 xxxwithxyIsoqmaxL:3 ==
( ) ( )xPIsoqBP'B ∉=
( ) ( )xPIsoqAP'A ∈=
Figure 1Efficiency correspondence between Edgeworth box and output set
2 General curvature ofProduction-Possibility Frontier
We will determine PPF shape on the basis of the explicit equation 3 con-sidering, without losing generality, that y1 describes PPF on y(1) ⊂ RM−1
+ ,equation 3 is rewritten by: y1 = y1
¡y2, ..., yM
¢.
Studies on function concavity or convexity usually postulate that the de-finition domain is convex. This is inappropriate for studying concavity of y1
10
because to maintain the generality of analysis, supposing that the definitiondomain y(1) ⊂ RM−1
+ of y1 is convex could imply concavity properties ofproduction functions: y2, ..., yM , and consequently would destroy the mainaim of our analysis because for its generality, it is necessary that the func-tions y2, ..., yM have no more restrictions than y1 on the nature of theircurvature. They only respect, as y1, the property of quasi-concavity. Butthen, from our assumptions and properties of production functions, we haveno guarantee on the y(1)convexity.
We need a criterion of PPF concavity or convexity defined by y1 onthe set y(1) ⊂ RM−1
+ , which is not necessarily convex. Let us define localconcavity:
Definition 1: y1 defined on y(1) ⊂ RM−1+ is locally concave at y ∈
PPF ⊂ RM+ where L (y) ⊂ IF if and only if, the subset in the neighborhood
of y : N (y) = {y0 ∈ P (x0) : ky0 − yk < ε} is convex.In fact N (y) is the set under the PPF in the neighborhood of y.With the above definition, local concavity may be approached by gener-
alization of second derivative test of the y1 through the Hessian H:
H =
22y1 · · · 2My
1
.... . .
...M2y
1 · · · MMy1
where
ijy1 = ∂2y1/∂yi∂yj (i, j = 2, ...,M) .
Instead of using the borded Hessian built from the production function yi(xi)as is usual for an optimisation problem under constraints, in order to de-termine local concavity or convexity of PPF, defined by y1, we propose tointegrate in each second derivative of yi(xi), i = 1, ...,M , the results com-ing from the fixed endowment of inputs and from the efficient constraint.Then it is possible to determine each ijy
1, , i, j = 2, ...,M . As said before,∂yi(xi)/∂xij > 0 if input xij is essential to activity i. If input x
ij is not
essential to i , it is possible to have ∂yi(xi)/∂xij = 0.Hence, name the essential input set ( the Edgeworth box except the
sides):ixh ⊂ x as: ixh =n
x ∈x : xij > 0,∀i = 1, ...,M ;∀j = 1, ...,N,o.
Then ∀x ∈ ix̄h, the constraints are:
(i) ∀i = 1, ...,M ; ∀j = 1, ...,N : xij = xj −XM
k=2xik
(ii) ∇y1(xi) =XM
k=2x∈ixh
µk ∇yk(xk)
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The constraint (ii) of efficient allocation of inputs means that the gradi-ents of each production function are co-linear ∀x ∈ ixh. In other words, theyare perpendicular to the hyperplane that is tangent at x ∈ IF,∀y ∈ PPFto all IsoqL
¡yi¢. If x /∈ ixh , it is possible that constraint (ii) doesn’t hold.
We will treat this case in the next section.Consequently, the function y1 defined on y(1) is locally concave (convex)
on P (x) ∈ PPF if and only if the HessianH is semi-negative (semi-positive)defined. It has a saddle point if H is undefined.
The general relation between variable returns to scale and output shapeis therefore determined from Hessian H, where second derivatives must beexpressed as functions of returns to scale. This general relation could bedifficult to interpret. This is why we will precise the analysis for two inputs(N = 2) and two outputs (M = 2), but without added restrictions on input-output correspondence and productions functions. Consequently, HessianH will be replaced by a second derivative. But this latter will have the same”mathematical structure” as second derivatives of Hessian. Therefore themain properties linking PPF and returns to scale will appear with an easierinterpretation.
3 Production-Possibility Frontiershape as a result of variable returns to scale
3.1 Definition of the problem
For M = 2 and N = 2, if we take input endowment as unit of measure, wehave x̄ = (1, 1); then, with xij/xj = λij ∈ [0, 1] ,∀i, j = 1, 2, λij is the shareof input x̄j used in activity i. For production functions y1 = y1
¡λ1
1, λ12
¢and
y2 = y2
¡λ2
1, λ22
¢, optimisation problem becomes:
objectif : max y1 = y1
¡λ1
1, λ12
¢product. constr. : y2 = y2
¡λ2
1, λ22
¢full/empl. constr. : λ1
1 + λ21 = 1 and λ
12 + λ
22 = 1.
Consequently, with the same notation convention as in preceding sec-tion, PPF corresponds to production function y1 defined on y(1) =
as solution of optimisation (see Production-Possibility Frontier, Figure 1).Then: y1 = y1
¡y2¢respects all assumptions defined above.
Furthermore,due to the full employment constraint, posing¡λ1
1, λ12
¢=
(λ1, λ2), we have¡λ2
1, λ22
¢= (1− λ1, 1− λ2). One notes the input intensi-
ties k1 = λ1/λ2; k2 = (1− λ1) / (1− λ2). which are positive and finite if
12
and only if both inputs are essential for both production units. The mar-
ginal rates of substitution along the isoquants, τ s1 = −∂y1
∂λ2/∂y1
∂λ1|dy1=0 and
τ s2 = − ∂y2
∂(1− λ2)/
∂y2
∂(1− λ1)|dy2=0, are continuously differentiable over
]0,∞[. because of the assumption A.15. For ki and τ si (i = 1, 2) beingwell-defined functions, all inputs must be essential for the production units.Then ∂yi/∂λj > 0,∀i, j = 1, 2. In the essential input set (the Edgeworth boxexcept the sides)
®λ=nλ ∈λ : λ ∈ ]0, 1[2
o⊂ λ, the efficiency condition
(ii), induces, in the 2X2 case, the well known equality between the marginalrates of substitution on IF (contact curve). Then
∀λ ∈ ®λ ∈ and λ ∈IF : ∂y1
∂λ2/∂y1
∂λ1=
∂y2
∂(1− λ2)/
∂y2
∂(1− λ1)= −τ s
where τ s = dλ1/dλ2 | λ∈IFdy1=dy2=0
, is the identical marginal rate of substitution
between inputs for both production functions, which ensures an efficientallocation of inputs. We will introduce the case where ∀λ /∈ ®
λ, after
having established the general result for the essential input set.Now let define returns to scale. For α > 0, activity 1 exhibits constant
returns to scale (CRS1) if αy1 = y1 (αλ1, αλ2), and for α > 1, activity 1exhibits decreasing returns to scale (DRS1) if αy1 > y1 (αλ1, αλ2) and hasincreasing returns to scale (IRS1) if αy1 < y1 (αλ1, αλ2).Mutatis, mutandisfor activity 2. In addition, the local measure of returns to scale of yi isdefined on the basis of equi-proportional variation of inputs and is notedρ1 = dy1/γ1y1, ρ2 = dy2/γ2y2 where γ1 = dλ1/λ1 = dλ2/λ2 and γ2 =d(1− λ1)/ (1− λ1) = d(1− λ2)/ (1− λ2) .
Activity i(i = 1, 2) has locally increasing (decreasing or constant) returnsto scale if and only if at this level of activity i, one has: ρi > 1 (ρi < 1 orρi = 1). Returns to scale of activity i are variable if and only if ρi varies onthe definition interval domain of yi.
So, ρ1 is defined from the derivative of production function y1(λ1, λ2) inthe direction of the vector (λ1, λ2) and ρ2 from the derivative ofy2(1− λ1, 1− λ2) in the direction of the vector (1− λ1, 1− λ2).
3.2 Vectorial analysis of return to scale in Edgeworth box
Because integrating returns to scale in the derivatives of production func-tions implies different derivative directions, vectorial analysis stands out asa good method for our problem.
Local returns to scale of each production function may be defined throughthe scalar product of two vectors divided by the level of production: for ρ1,the scalar product of the gradient vector∇y1(λ) = (∂y1(λ)/∂λ1, ∂y1(λ)/∂λ2)and λ divided by y1, and for ρ2, the scalar product of ∇y2(1−λ) =(∂y2(1− λ)∂(1− λ1), ∂y2(1− λ)∂(1− λ2)) and 1− λ divided by y2.
2
The scalar product being the product of the magnitudes of both vectorsand the cosine of the angle between vectors one has:
(4) ρ1 =∇y1(λ)λ
y1=k∇y1(λ)k kλk cos θ1
y1;∀λ ∈ ]0, 1[2
(5) ρ2 =∇y2(1−λ) (1− λ)
y2=k∇y2(1− λ)k k1− λk cos θ2
y2; ∀λ ∈ ]0, 1[2
For λ ∈ ®λ ,if λ ∈IF and 1− λ ∈IF, the angles θ1 ∈£0, π2
£and θ2 ∈
£0, π2
£,
because they are acute as a result of P4, P5, P6 and A.15, as shown in theEdgeworth box of Figure 2. In addition, in this case, because the gradientvectors are calculated for an efficient allocation of inputs they are bothperpendicular to the common tangent to isoquants of each production unit,which has the direction of vector (1, τ s) Therefore gradient vectors havethe same direction equal to vector (1,−1/τ s). As a result, they are notperpendicular to vectors λ and 1− λ for all that IF is include in the essentialsubset.3
Furthermore, as the derivative direction of production functions at apoint y0 ∈ PPF is the tangent to PPF if, without loosing generality, oneconsiders for our problem that dy1 > 0 and dy2 < 0, one has:
his the acute angle between the gradient direction and the
tangent to IF which are not perpendicular due to P4, P5, P6 and A.15.
2 In Edgeworth box, as shown in Figure 2, the vectors are reported to axis having thelow-left point 01 as origin when activity 1 is concerned and the high righ point 02 foractitivity2
3We will suppose λ ∈ ®λ , except if we mention the contrary14
02
01λ 2
∇∇∇∇ y¹ =∇∇∇∇ y²(dλ 1, dλ 2)
(1,k1)
(1,k2)
(τ s ,1) θ2θ1 ϕ
A
02
01λ 2
∇∇∇∇ y¹ =∇∇∇∇ y²(dλ 1, dλ 2)
(1,k1)
(1,k2)
(τ s ,1) θ2θ1 ϕ
A
(1,k2)(τs,1)
(1,k1)
(dλ1, dλ2)
∇∇∇∇y¹ =∇∇∇∇y²
θ2ϕ
θ1
C1
C2
DB
A
(1,k2)(τs,1)
(1,k1)
(dλ1, dλ2)
∇∇∇∇y¹ =∇∇∇∇y²
θ2ϕ
θ1
C1
C2
DB
A
(τs,1)
(1,k1)
(dλ1, dλ2)
∇∇∇∇y¹ =∇∇∇∇y²
θ2ϕ
θ1
C1
C2
DB
A
Figure 2Vectorial analysis of returns to scale in Edgeworth box
15
As functions yi (i = 1, 2) appear as a solution of the optimisation prob-lem, hence (y1, y2) ∈ PPF and from 6 and 7, integrating the efficiencyconstraint (ii) , one deduces that dy1/dy2 = dy1(λ)/dy2 (1− λ) | λ∈IF
1−λ∈IF.
The first derivative 2y1 = dy1/dy2 determines the slope of PPF 4. One has:
(8) 2y1 =
dy1
dy2= −
°°∇y1°°°°∇y2°° = −� t < 0
where � t=− dy1/d�2
dy2/d(1− �2)= − dy1/d�1
dy2/d(1− �1)represents the absolute value
of the marginal transformation rate on PPF and may also be interpreted asthe absolute value of the marginal input transfer rate equal for each input.
Furthermore, the equiproportional variation of inputs �i, i = 1, 2, usedfor the calculation of returns to scale on IF when dy1 > 0 and dy2 < 0 asone supposes, are from 4, 6, for �1 and from 5, 7 for �2:
(9) �1 =kdλk cos�kλk cos �1
> 0; �2 =− kdλk cos�k1− λk cos �2
< 0
They express the equiproportional variation of inputs taking in considerationthat they are not necessarily used in the same proportion by productionunits. The respect of efficiency when productions vary, implies that inputsallocation vary along IF . But in order to make appear returns to scale inthe formalization, the equiproportional variation must be linked to efficiencyallocation of inputs. It is what 9 does. Zoom on Figure 2 clarifies thisquestion. For an infinitesimal variation of inputs from A ∈ IF , the directionwill be the tangent direction dλ. The gradient vectors ∇y1 = ∇y2 =(1,−1/� s) having A as origin and being perpendicular to the direction ofthe vector (1, � s), � s being the marginal rate of substitution at A, one hasAB = AC1 cos �1 = AC2 cos �2 = AD cos� or:
(10) d kλk cos �1 = d k1− λk cos �2 = kdλk cos�.Consequently from 9 and 10 �1 = d kλk / kλk and �2 = d k1−λk / k1−λk .
At once, let us calculate first derivative 2y1 on¤0,max y2
£, from 4, 5 and
8, each production function being derivable on input interval ]0, 1[:
(11) 2y1 = −y
1�1 k1− λk cos �2
y2�2 kλk cos �1
4Because y1 and y2 are the optimisation solutions, dy1 and dy2 replace dy1(λ)and dy2 (1− λ) , like this for ∇y1 and ∇y2 which take place for ∇y1(λ) and for∇y2 (1− λ) . This means that the calculation from initial production functions integratesthe efficiency constraint.
16
Considering 9 and 11:
(12) 2y1 =
y1�1�1
y2�2�2
Finally, for considering variable returns to scale, we propose to integrate thevariation through the local elasticity of return to scale �1 and �2 as follow:
(13) �1 =d�1
�1�1
; �2 =d�2
�2�2
.
A.16. The derivative of returns to scale of each activity to the scale ofproduction (d�i/ ||dλ|| ,∀i = 1, 2) is finite for �1, �2 ∈ ]0, 1[.
To characterize the local effect of variable returns to scale note the defi-nition 2.
Definition 2: A production unit i has locally increasing (constant anddecreasing) variable returns to scale if and only if, at this level of production,the sum of returns to scale and their elasticity is superior to unity: �i+�i > 1(equal to unity: �i + �i = 1 and inferior to unity: �i + �i < 1).
3.3 General result
3.3.1 The case of essential inputs for each production unit
We need the second derivative 22y1 to infer the curvature properties of PPF .
It is given by 22y1 =
d
µ−y1�1k1−λk cos �2
y2�2kλk cos �1
¶dy2 .
First calculate, considering equations 9: d³−y1�1k1−λk cos �2
with cos �i = 1/p1 + tg2�i and cos� defined by 10. 6
5Knowing that �i ∈£0, π
2
£if ki > −1/�s, one has �i = arctgki + arctg1/�s =⇒
tg�i = (1 + ki�s) / (� s − ki) > 0 and d�i = dki/¡1 + k2
i
¢ − d�s/¡1 + �2
s
¢; if ki < −1/� s,
one has �i = −arctgki − arctg1/�s =⇒ tg�i = − (1 + ki�)s/ (�s − ki) > 0 and d�i =
d�s/¡1 + �2
s
¢− dki/¡1 + k2
i
¢; if ki = −1/�s, one has tg�i = 0. Then ∀i = 1, 2 : tg�id�i =
(1 + ki�s) / (�s − ki)£dki/
¡1 + k2
i
¢− d�s/¡1 + �2
s
¢¤.
6 It s also possible to calculate tg�id�i,from cos �i defined by 6 and 7
17
Therefore because dy2 = y2�2�2 = −y2�2 ||dλ|| cos�||1− λ|| cos �2
, dividing equation
14 by dy2, one deduces the general result of PPF local curvaturethrough the second derivative 22y
1 for λ ⊂ ®λ:22y
1 = y1�1k1−λk cos �2
(y2�2kλk cos �1)2 [k1− λk cos �2 (�1 + �1 − 1)(15)
+kλk cos �1 (�2 + �2 − 1)+ kλk k1−λk cos �1 cos �2
cos�
³tg�1
d�1kd�k − tg�2
d�2kd�k
´], IF ⊂ ®λ
This general result may also be expressed by:
(16)
22y1 = y1�1k1−λk cos �2
(y2�2kλk cos �1)2 {k1− λk cos �2 (�1 + �1 − 1) + kλk cos �1 (�2 + �2 − 1)+ kλk k1− λk cos �1 cos �2
cos�
h1+k1�s�s−k1
³dk1/kdλk
1+k21− d�s/kdλk
1+�2s
´−1+k2�s
�s−k2
³dk2/kdλk
1+k22− d�s/kdλk
1+�2s
´io,λ ⊂ ®λ .
3.3.2 The case with specific inputs
What happens if the optimisation induces a solution whereby the inputs arenot essential for the production units: λ /∈ ®λ̄? For the case 2X2, we havetwo possibilities.
First, both activities use only one input which is necessarily differentbetween activities due to the full employment assumption (A.14.). In thiscase, PPF is reduced to one isolated point. The production units are totallyindependent because they don’t use the same input. This case is outside oursubject.
Second, one activity k limits itself to one input while the other i usesboth. In the Edgeworth box, L(PPF ) is represented by one of the sides cor-responding to the input used by both activities. Then for λ = L(y),y = (y1,y2),one has: L(PPF ) = IF =
ªwhere input j is specific to activity 1. If we call � the input which is usedby both activity when their production is positive, because each produc-tion functions may be treated as one variable function, equation 11 becomes2y1 = −y1�1(1− �)/y2�2� although equations 15 or 16 become:
(17) 22y1 =
y1�1(1− �)(y2�2�)
2 [(1− �) (�1 + �1 − 1) + � (�2 + �2 − 1)]
18
As far as curvature nature of PPF is concerned, the sign of 22y1 is
determinant. We first propose to analyse this problem, at the neigborhoodof the axis and secondly at any other locus.
3.4 PPF shape at the neighborhood of axis
Before giving the general proposition describing PPF shape, it is useful toknow how it is at the neighborhood of axis.
Theorem 1: From assumptions A1 to A15 applied to the case with twoproduction units with two usable inputs and variable returns to scale, oneconcludes:
(i) PPF is strictly concave (convex) to origin at the neighborhood ofproduction axis of the production unit for which all inputs tend to be devotedif for this allocation of inputs the returns to scale of the other productionunit are decreasing (increasing).
(ii) If the returns to scale of the activity whose production tends to zeroare constant, PPF at the neighborhood of production axis of the unit forwhich all inputs tend to be devoted is strictly concave (convex) to origin ifand only if this production unit has decreasing (increasing) variable returnsto scale.
(iii) PPF is a straight line at the neighborhood of one axis if and onlyif the two production units have constant variable returns to scale. Thisimplies that a straight line at the neighborhood of one axis implies a straightline at the neighborhood of the other axis.
Proof: Take the limit of 15 when y2 −→ 0 or �j −→ 1, ∀j = 1, 2 (thedemonstration is mutatis mutandis if y1 −→ 0 or �j −→ 0).
and k1 −→ 1 and from 15, calculate the limit of the third term in brackets.Consider that kλk k1− λk cos �1 cos �2/ cos� (tg�1d�1/ kdλk− tg�2d�2/ kdλk)= kλk k1− λk 1/ cos�(sin �1 cos �2
d�1kdλk−sin �2 cos �1
d�2kdλk),where d�i/ kdλk
(i = 1, 2) is finite. One has cos� 6= 0 because when all inputs are essential,the vector dλ might not be perpendicular to gradient vectors and havingthe same direction as the common tangent to isoquant, because dyi 6= 0 forthe definition of dλ.
Then kλk k1− λk cos �1 cos �2cos�
³tg�1
d�1kdλk − tg�2
d�2kdλk
´→ 0.
When inputs are not essential, if one input is not used by a productionunit, one refers to 17 where the above expression is equal to zero.
Therefore from 4 and 5, one concludes that:
19
limy2−→0
22y1 = lim
y2−→0
°°∇y1°°°°∇y2°°2
µ�1 + �1 − 1√2 cos �1
+�2 + �2 − 1cos �2 k1− �k
¶. From A.16,
d�2
d k1− �k is finite which implies limy2−→0
�2 = 0, and from P3, ∇y1 and ∇y2
are also finite. Consequently, one concludes that: limy2−→0
22y1
¯̄̄̄+∞−∞
if and only
if �2¯̄>1
<1what established (i).
Furthermore, in this case, if �2 = 1, it is �1+�1−1 which determines thesign of second derivative because, applying Hospital theorem on the basis of
distance k1− λk → 0, one has�2 − 1
cos �2 k1− λk → 0 what demonstrates (ii)
and induces (iii). Q.E.D.This theorem generalizes the Herberg and Kemp’s (1969) theorem.
3.5 Local PPF shape on the whole definition domain
´ioIf one poses:A = �1 + �1 − 1; signA = sign k1− λk cos �2 (�1 + �1 − 1) in 15 and
16 or sign(1− �) (�1 + �1 − 1) in 17, if one input is not productive for oneproduction unit,B = �2 + �2 − 1; signB = sign kλk cos �1 (�2 + �2 − 1) in 15 and 16 or
sign� (�2 + �2 − 1) in 17, if one input is not productive for one productionunit. Then one has: A T 0 if and only if �1 + �1 T 1 and B T 0 if and only
if �2 + �2 T 1. Pose also
∆ =1 + k1� s� s − k1
µdk1/ kdλk1 + k2
1
− d� s/ kdλk1 + �2
s
¶−1 + k2� s� s − k2
µdk2/ kdλk1 + k2
2
− d� s/ kdλk1 + �2
s
¶= tg�1
d�1
kdλk − tg�2d�2
kdλk .Even when A and B constitute the variable returns to scale effect of each
activity on PPF curvature, ∆ corresponds to the input intensities effect.
As shown in 15 and 16, ∆ additionally operates with returns to scale todescribe the PPF shape. A deep analysis of sign∆ justifies itself. We just
20
propose to express ∆ in a way, which permits to understand how the mainvariables play and to collate our result with the main results in economicliterature.
3.6 Analysis of input intensities effect
3.6.1 The case of one specific input
Lemma 1: From assumption A1 to A16 applied to the case with two produc-tion units with two usable inputs and variable returns to scale, if one inputis locally specific to one production unit, there is no local input intensitieseffect on PPF .
Proof: In the case 2x2, if one input is specific to one production unit,one has ∆ = 0 as shown in 17. Q.E.D.
If the inputs are only usuable but not essential, with one production unitusing only one input, one has:
(18)
22y1 = y1�1k1−λk cos �2
(y2�2kλk cos �1)2 [k1− λk cos �2 (�1 + �1 − 1) + kλk cos �1 (�2 + �2 − 1)]
3.6.2 The case of equal input intensities
A particular case, but of interest for many applications is this when alldifferences in input intensities are disregarded in order to have equal inputintensities for each activity on the whole definition domain. Because offull employment assumption, passage from one efficient input-output vectorto another implies transfer of freed inputs from the first activity to thesecond, without changing input intensities, as inputs are freed in the sameproportion. In fact the above hypothesis implies that for each activity, eachinput is used in the same proportion of its endowment. This last proportiondetermines the scale of activities which is kλk for activity 1 and k1− λk foractivity 2.
Lemma 2: From assumption A1 to A16 applied to the case with twoproduction units with two usable inputs and variable returns to scale, wheninputs are both locally essential for the production units, there is no localinput intensities effect on PPF if the input intensities of both productionsare equal.
Proof: If k1 = k2 = 1 at the neighborhood of any PPF locus, one hasfrom note 5: �1 = �2, and d�1 = d�2. Therefore tg�1d�1 = tg�2d�2 = 0what implies ∆ = 0. Q.E.D.
21
One deduces equation 18 for ∆ = 0.Furthermore first derivative 11 may be written:
(19) 2y1 = −y
1�1 k1− λky2�2 kλk
Note also that ∀i = 1, 2 : kλk = √2�i and k1− λk =
√2 (1− �i),
because �1 = �2.
3.6.3 The case of different input intensities
According to economic literature often based on linear homogeneous pro-ductions functions, the waited effect when input intensities are different orlinearly independent when they are more than two, is ∆ < 0, which meansthat difference of input intensities, would operate, additionally to returnsto scale, to make PPF more concave to origin or less convex to origin de-pending on the nature of returns to scale. But for production functionsrespecting our assumptions, this proposition is false because one may have∆ > 0, as we are going to show.
In order to understand how the marginal rates of substitution and inputintensities operate, let notice the following property which establishes theirinterdependence through the partial returns to scale.
Lemma 3: From assumptions A1 to A16 applied to the case with twoproduction units with two essential input and variable returns to scale, therelative variations on PPF of the marginal rate of substitution is equal foreach production unit to the sum of the relative variation of input intensitiesand the relative variation of partial returns to scale ratio.
Proof: Actually, �i =P
j=1,2 �j yij/y
i(i = 1, 2) where �j yij/yi =
j�i represents the partial returns of production unit i to input j. Dueto efficiency condition (a common marginal rate of substitution) one has:1 − � s/ki = �i/1�i(i = 1, 2). Therefore � s = −ki (2�i/1�i) . With ri =2�i/1�i (i = 1, 2) , we conclude that:
(20) � s = −kiri(i = 1, 2)Consequently:
(21)d� s� s
=dkiki+driri(i = 1, 2)
Q.E.D.
22
We must express equation 21 under the variation of productions deter-mined by kdλk .
Interdependence of the three variables in equation 21 implies several ex-pressions of ∆ depending on what variables are retained. It is interesting
to hold in position first,d� s/ kdλk� s
because the constraint of efficiency im-
plies the equality of the marginal rates of substitution and second,dri/ kdλkri
because the variation of partial returns to scale constitutes a technical char-acteristic of production functions. As a consequence, one has Lemma 4.
Lemma 4: From assumptions A1 to A16 applied to the case with twoproduction units, two essential inputs and variable returns to scale, the inputintensities local effect (∆) on PPF may be decomposed by addition intofirst, the input intensities difference effect ( IIDE) depending on the inputsintensities difference which jointly acts, by multiplication, with the relativevariation of the marginal rate of substitution on input frontier (IF ) andsecond, the partial returns to scale ratios relative variation effect (PRSV E)which operates through the difference of partial returns to scale ratio relativevariation of both activities each of them being weighted by the tangent of theacute angle between the gradient vector and the input vector multiplied by apositive function of input intensity.
Proof: Taking into consideration the expression of ∆ from 16 and theLemma 3 (equation 21), after calculations, one has:
∆ =d� s/ kdλk� s
k21 − k2
2
(1 + k21)(1 + k
22)
+1 + k2� s� s − k2
k2
1 + k22
dr2/ kdλkr2
− 1 + k1� s� s − k1
k1
1 + k21
dr1/ kdλkr1
(22)
Then ∆ = IIDE+PRSV E where IIDE =d� s/ kdλk� s
k21 − k2
2
(1 + k21)(1 + k
22)
and PRSV E =1 + k2� s� s − k2
k2
1 + k22
dr2/ kdλkr2
− 1 + k1� s� s − k1
k1
1 + k21
dr1/ kdλkr1
.7
Note that tg�i = |(1 + ki� s) / (� s − ki)| . Q.E.D.
Furthermore, signIIDE = sign·(d� s/ kdλk� s
(k1 − k2)
¸.
If we suppose k1 > k2, without loosing generality, from our assumptions
signIIDE = sign(d� s/ kdλk� s
) is undetermined because on IF for dλ, one
7Note that ki/¡1 + k2
i
¢= sin�i cos�i, with �i = arctgki (i = 1, 2) .
23
may have d� s positive, negative or null. However notice from equation 22,
that one has d� s > 0 (which means thatd� s/ kdλk� s
< 0 and IIDE < 0 for
k1 > k2) if and only ifdri/ kdλkri
< −dki/ kdλkki
(i = 1, 2). Consequently,
for having IIDE < 0, the difference between partial returns to scale rela-tive variation must be inferior to the opposite of input intensities relativevariation. SignPRSV E is also undetermined.8
Finally, from equation 15 or 16 and the above discussion on ∆, theLemma 1, 2, 3, 4 and theorem 1, one deduce the following theorem 2.
Theorem 2: From assumptions A1 to A16 applied to the case with twoproduction units with two essential inputs and variable returns to scale, oneconcludes:
(i) PPF is strictly locally concave (convex) to origin if and only if R =k1− λkA cos �2+ kλkB cos �1+ kλk k1− λk∆cos �1 cos �2
cos� < 0 (> 0). It is alocal straight line if and only if R = 0 at the neigborhood of the consideredlocus and there is an inflection point if and only if R = 0 at this isolatedlocus with change of sign.
(ii) When the input intensities local effect is negative ∆ < 0 (positive:∆ > 0), PPF is strictly locally concave (convex) to origin if no productionunit produces with increasing (decreasing) variable returns.
(iii) When the input intensities local effect is negative: ∆ < 0 (posi-tive: ∆ > 0), PPF may be strictly locally convex (concave) to origin if atleast one production unit exhibits sufficiently strongly increasing (decreas-ing) variable returns. The amount by which these variable returns must besuperior (inferior) to unity, is all the greater that the variable returns of theother production unit and/or the absolute input intensities effect (∆) areweak.
(iv) When production units use input in the same proportion or if oneinput is specific to a production unit, PPF is locally concave (convex ora straight line) to origin if each production unit, exhibits local decreasing(increasing or constant at the neighborhood) variable returns.
Proof: Proposition (i) results from local strictly concavity (convexity)definition through second derivative by considering sign22y
1.
8But, taking into consideration equation 20 which implies that r1/r2 = k2/k1, and theequation 21, one deduces 1 + r1 = 2 + r2 = 1, where (i = 1, 2) i = dki
ki
/ d�s
�s|IF is
the elasticity of substitution of inputs along IF, and ri= dri
ri/d�s
�s|IF is the elasticity
of substitution of partial returns to scale along IF . From this property one may induceseveral sufficient conditions for having PRSV E < 0.
24
Proposition (ii) is obvious considering the definition 2 of local variablereturns which determines the sign of A and B.
Proposition (iii) results from (i) and (ii).Proposition (iv) considers the Lemma 2 and 1 which show that for the
considered cases one has ∆ = 0; the application of (i) for ∆ = 0, established(iv). Q.E.D.
Our priority in this paper, after having established the general result 15or 16, is now to bring it face-to-face with the main statements in economicliterature.
4 The general result face to face with anteriorstatements
Because we have determined the curvature of PPF with the possibility tocalculate it through the second derivative, we refer to economic literatureinterested by this problem. We must notice that, until now, only the ho-mogeneous or homothetic production functions have been studied in detail,which is much more restrictive than production functions submitted to ourassumptions. In fact the study of homogenous production functions con-cerns the non-variable returns, when homothetic functions refer to variablereturns in a very restrictive sense as we will see. It is also the reason whywe use a very different methodology. Even though with homogeneous or ho-mothetic functions, the calculation of second derivative on PFF is not toodifficult, when they are neither homogeneous nor homothetic, it is very dif-ficult to exploit the same calculation. This is the reason we have worked oninput frontier (IF ) to deduce the second derivative on PPF.
4.1 PPF shape at the neighborhood of axis
Determining PPF shape when all disposable inputs tend to be devoted toonly one production unit might largely contribute to characterize it. Herberg(1969) was the first to establish the result for the transformation curve inthe case of homogeneous production functions. Afterwards, Herberg andKemp (1969) established the nature of PPF curvature at the neighborhoodof the axis when returns to scale are external to the production unit butinternal to industry. It was a seminal article which especially improved thetheory of international trade when national economies are characterized bythis type of returns to scale. They demonstrated their results through anhomothetic production function with external returns to scale depending
25
on the scale of industry and constant internal returns for each firm whoseproduction functions are homogeneous of the first degree. It is because theauthors referred to markets prices and were anxious to include the firmsin competitive markets that they maintain constant the internal returns toscale. Our proposition implies a similar conclusion, but for any returnsto scale, internal or external, analysing the PPF only on the basis of thetechnology of each production unit, with respect of our assumptions, muchless restrictive than for production functions used by Herberg and Kempbecause the production functions are neither homogeneous nor homothetic.
Furthermore, in our general result, proposition 1 holds even if only oneinput is mobile between production units as demonstrated for the Herberg-Kemp case by Wong (1996).
4.2 PPF shape on the whole definition domain
Let first notice that when production functions exhibit non variable returnsto scale, they are homogeneous.
The decomposition of returns to scale into partial returns to scale inducesfor the production functions 1 and 2:�1y1 = �1∂y1/∂�1 + �2∂y1/∂�2 and�2y2 = (1− �1)∂y2/∂(1− �1) + (1− �2)∂y2/∂(1− �2).If �1 and �2 are constant they are the degree of homogeneity of the
production functions y1 and y2 because of the reciprocal of Euler’s theorem.
4.2.1 The case of equal input intensities
If returns to scale are not variable as for homogeneous functions and if inputintensities of the two production units are equal, ∆ = 0 and equation 15 or16 becomes:
(23) 22y1 =
k1− λk �1y1 [kλk (�2 − 1) + k1− λk (�1 − 1)]
(kλk�2y2)2
As a result it is possible to define PPF shape, not only locally, but ∀ kλk ∈£0,√2¤. Applying theorem 1 for calculating lim
y2−→022y
1 of 23, in this case
of identical input intensity, the result is obvious if returns of both activ-ities are increasing (resp. decreasing) or constant. If one activity has in-creasing returns while the other one has decreasing returns, on λ ∈ ]0, 1[2means that sign of 22y1 changes, and PPF has an inflection point for�i = (�1 − 1) / (�1 − �2), ∀i = 1, 2. It is the only inflection point, whatalso confirms the result of Herberg (1969) and Panagariya (1981).
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4.2.2 The case of different input intensities
The first result determining PPF shape is this of Savosnick (1958) who hasshowed that PPF, for linearly homogeneous production functions, is strictlyconcave to origin if both production functions use input in different inten-sities. Manning R. and J.R. Melvin (1992) reverse Savosnick’s technique inorder to establish that any negatively sloped, concave PPF , for an economywith fixed endowment of inputs, can be derived from some pair of linearlyhomogeneous, quasi-concave production functions.
But the main results determining the curvature of PPF were estab-lished by Herberg (1969) for any homogeneous production functions. Afterhaving shown that when the literature on international trade at this timereferred to increasing returns, the statements on PPF shape were generallyfalse, Herberg (1969) proposed a demonstration consisting in transformingthe homogeneous production function into linearly homogeneous functionsbecause the contract curve (IF ) of two production functions depends onlyon the shape but not on the output indices of their isoquants. This permitshim, first to treat the problem with constant returns to scale for the trans-formed functions, second, to come back to the initial production functionswhich are simply the degree of homogeneity power of transformed functions.Herberg (2000) has generalised the results for the n commodities m factorscase using the same idea as in Herberg and Kemp (1969) by consideringhomothetic production functions resulting from a monotonously increasingtransformation of linear homogeneous functions.
In order to show the generality of our result, it is necessary to establishthat it integrates the Herberg’s statements for homogeneous production andhomothetic functions.
Lemma 5: From assumptions A1 et A16 applied to the case with twoproduction units with essential inputs and for homogeneous or homotheticproduction functions, the relative variations of partial returns to scale areequal.
Proof: If returns to scale �1 and �2 are not variable they represent thehomogeneity degrees of y1 and y2. Therefore the ratios of partial returnsto scale ri = 2�i
1�i(i = 1, 2) are homogeneous of zero degree. As a conse-
quence for i = 1, for instance, one has: �1dr1/d�1 + �2dr1/d�2 = 0, Then(�1/d�1+�2/d�2)dr1 = 0. The problem being treated along the input fron-tier (IF ), d�1 > 0 and d�2 > 0 if the two inputs are essential. Consequentlydr1/r1 = 0 which implies d1�1/1�1 = d2�1/2�1 mutatis mutandis dr2/r2 = 0.
Furthermore if production functions are homothetic, the returns to scale
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may be variable but in a much more restrictive sense than ours, because theirmarginal rate of substitution, like for homogeneous production functions, arehomogeneous of zero-degree which implies also that dri/ri = 0 (i = 1, 2) .Q.E.D.
In other words, when the production functions are homogeneous or ho-mothetic, from equation 21 the relative variation of the common marginalrate of substitution on input frontier (IF ) is equal to the relative variationof the input intensities.
Therefore the relation 16 becomes, when production functions are ho-mogeneous or homothetic with IF ⊂ ®λ:
22y1 = y1�1k1−λk cos �2
(y2�2kλk cos �1)2 [k1−λk cos �2 (�1 + �1 − 1)(24)
+kλk cos �1 (�2 + �2 − 1)+kλk k1− λk cos �1 cos �2
cos� (d�s/kdλk�s
k21−k2
2
(1+k21)(1+k2
2))],
Note that, if the production functions are homogeneous: �1 = �2 = 0 in 24.With dy1 > 0 and dy2 < 0, one has d� s/� s < 0 if and only if k1 > k2
and d� s/� s > 0 if and only if k1 < k2. Consequently ∆ < 0 if and onlyif k1 6= k2. This property is true not only locally but also on the wholedefinition domain �1, �2 ∈ ]0, 1[.
Consequently taking in consideration theorem 1, lemma 1, 2, 4, 5 andequations 24, 18, one deduces the theorem which confirms the Herberg (1969,2000)’s conclusions for homogeneous and homothetic production functions.
Theorem 3: From assumptions A1 to A16 applied to the case with twoproduction units and two usable inputs, which exhibit non variable returns toscale which means that production functions are homogenous one concludes:
(i) PPF is strictly concave (convex) to origin at the neighborhood ofthe axis of one output if the other output is produced under decreasing (in-creasing) returns. Therefore PPF has at least one inflection point if thereare increasing returns in one production unit and decreasing returns in theother.
(ii) For a given relative variation of the marginal rate of substitutionalong input frontier ( IF ),on the whole domain of definition, if input inten-sities of the two production units are different, the greater this difference is,the more concave to origin PPF is, if returns of both production units aredecreasing. If returns of both production units are increasing, the greater thedifference of input intensities is, the less convex to origin PPF is. For suf-ficiently strong difference of input intensities, PPF may be locally concave
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to origin even if the two production units exhibit increasing returns, exceptat the neighborhood of the axis in application of (i). Consequently, to have aPPF locally convex to origin, the smaller the increasing returns of the twoproduction unit are, the less the input intensities of productions must differ.
(iii) If the input intensities of both production units are different, PPFis strictly concave to origin on the whole domain of definition if and only ifno production unit exhibits increasing returns.
(iv) If one production unit uses only one input, PPF is strictly con-cave (convex) to origin on the whole domain of definition if and only if noproduction unit exhibits increasing (decreasing) returns.
(v) When production functions use inputs in the same proportion, PPFis strictly concave (convex) to origin, on the whole definition domain, if foreach production function, returns to scale are decreasing (increasing) or ifone production unit produces with decreasing (increasing) returns while theother produces with constant returns. PPF is a straight line on the wholedefinition domain, if and only if each production function exhibits constantreturns to scale. If returns to scale of one production unit are increasingand decreasing for the other, PPF has an inflection point.
(vi) If production functions are homothetic, the propositions (i), (ii),(iii), (iv), (v) hold, except that, depending on the case, must be taken inconsideration increasing variable, decreasing variable, constant variable re-turns (�i + �i, i = 1, 2) instead of increasing, decreasing constant returns.
5 Concluding remarks
Our general result determines the PPF curvature through the second deriv-ative, for the case 2x2. Equation 15 or 16 decomposes it into two maineffects. On one hand, the returns to scale of each production unit effectwhich shows that the greater (lower) than one is the sum of local returnsand their local elasticity, the more one would expect to have PPF locallyconvex (concave) to origin. On the other hand, the input intensities effectwhich appears as a result of two effects, first, the input intensities differenceeffect and the partial returns to scale effect. When the inputs are essential,the latter only operates if the relative variation of partial returns is differentfor at least one production unit or, consequently if the relative variation ofthe common marginal rate of substitution is different from the relative vari-ation of input intensities for at least one production unit (Lemma 3). Thisestablishes a main difference with the earlier statements of Herberg (1969,2000) and Herberg and Kemp (1969) who limited their analysis to homoge-
29
neous or homothetic production functions for which the relative variation ofthe common marginal rate of substitution is equal to the relative variationof input intensities of each production unit (Lemma 3 and 5). Actually,when the production functions have non variable returns to scale as for ho-mogeneous functions, or variable returns, but submitted to the hypothesisof homotheticity, the PPF shape is even, while it may be very jagged if therelative variation of partial returns to scale is different for at least one pro-duction unit. In fact, this situation of jagged shape, constitutes the generalcase corresponding to our assumptions A.1. to A.16. Therefore the inputintensities effect may act to make PPF concave as well as convex to origin,while the economic literature, based on homogeneous or homothetic pro-duction functions, expects, returns effect apart, that input intensities effectinfluences PPF will be concave to origin because, in this case, the inputintensities effect is reduced to input intensities difference effects.
Finally, new progress in the knowledge of PPF shapes implies a deeperanalysis of the input intensities effects, especially of the partial returns toscale ratio relative variation effect (PRSVE). Sufficient conditions shoulddetermine for jagged shapes if this latter effect operates in favor of a PPFconcave or convex to origin.
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