Substitution and size effect for factor demand revisited

Bröcker, Johannes; Requate, Till Article — Published Version Substitution and size effect for factor demand revisited Economic Theory Bulletin Provided in Cooperation with: Springer Nature Suggested Citation: Bröcker, Johannes; Requate, Till (2022) : Substitution and size effect for factor demand revisited, Economic Theory Bulletin, ISSN 2196-1093, Springer International Publishing, Cham, Vol. 10, Iss. 2, pp. 251-265, https://doi.org/10.1007/s40505-022-00229-z This Version is available at: https://hdl.handle.net/10419/311270 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Economic Theory Bulletin (2022) 10:251–265 https://doi.org/10.1007/s40505-022-00229-z RESEARCH ARTICLE Substitution and size effect for factor demand revisited Johannes Bröcker1·Till Requate1 Received: 21 January 2022 / Accepted: 8 July 2022 / Published online: 24 August 2022 © The Author(s) 2022, corrected publication 2022 Abstract We reconsider the decomposition of the comparative statics effect of a factor price increase on (unconditional) factor demand into a substitution and a size (or level) effect. While for the own price effect the substitution effect and the size effect go into the same negative direction, the cross price effect cannot be signed unambiguously, in general. But for two cases of regularity, homotheticity and complementarity, the size effect is negative. We furthermore show by example that in both special cases the cross substitution effects are still ambiguous, but that under complementarity the unconditional effect is always negative. Hence, if the cross substitution effect happens to be positive, it is dominated by the negative size effect in this case. We further show by example that without such regularity assumptions the cross price effect is ambiguous. Finally, we study the impact of a factor price increase on the cost, which can also increase or decrease. Keywords Slutsky equation ·(Conditional)Factor demand ·Comparative statics · Substitution effect ·Size effect ·Level effect JEL Classification B16 ·B21 ·D11 ·D21 ·D41 1 Introduction In a seminal but initially unnoticed paper, Slutsky (1915,1952) suggested the decomposition of individual consumer demand change responding to a price increase into the substitution and the income effect.1A major conclusion from the Slutsky equa1Allen (1936) popularized and further developed Slutsy’s work. See Dooley (1983), Weber (1999), and above all Chipman and Lenfant (2002) for a detailed history on Slutsky’s and related work. For further applications of consumer demand for goods and financial assets see Bierwag and Grove (1968), Mundlak (1968), Fischer (1972), Kalman et al. (1974), Ellis (1976), Laitinen and Theil (1979), Sedaghat (1996), and Menezes and Wang (2005). For empirical evidence see Barten (1967). Johannes Bröcker (1950–2021): Deceased. BTill Requate [email protected] 1Department of Economics, Kiel University, Kiel, Germany 123 252 J. Bröcker, T. Requate tion is the individual consumer’s law of demand, stating that for normal goods the own price effect is always negative since substitution and income effect work into the same direction. By contrast, nothing interesting can be said about cross price effects. Typically, these are ambiguous or zero for special cases. Similar decompositions have been suggested for factor demand, the earliest by Puu (1966) and replicated by Bertoletti (2005). These authors show that when a firm’s technology is given by a strictly convex cost function, the unconditional factor demand can be decomposed into a factor substitution effect and a “size” (or “level”) effect that determines how a factor price increase impacts on the production level. In this article, we dive a little deeper into that decomposition by studying under what circumstances the substitution, the size (or level), and the total effect can be signed. In case of the own price increase, similarly to the Slutsky equation for consumer demand of a normal good, the substitution effect and the level effect always work into the same direction (confirming the well known result that there is nothing like a Giffen factor). Regarding the cross price effect we show that, in general, both effects, and thus the total effect, cannot be signed unambiguously. But for two prominent special cases, homotheticity and complementarity, we show that the size (or level) effect is negative. As this is what is typically assumed in applications, we call this the regular case. Furthermore, we show by examples that in both special cases the cross substitution effects are still ambiguous, but that under complementarity for differentiable and strictly concave production functions the total effect is always strictly negative for an arbitrary number of input factors. While non-positivity has been shown using the tools of super-modularity (Topkis 1995,1998), negativity follows from the Frobenius matrix algebra (see Takayama 1985, and more recently Amir 1996). Hence, if the cross substitution effect happens to be positive, it is dominated by the negative size effect in this case. While it is easy to see that for homothetic production functions total output goes down when some factor price increases, it can also increase for sufficiently asymmetric isoquants. By contrast, maybe surprisingly, production costs may increase or decrease through a factor price increase. In our analysis we start with a dual approach in Sect. 2where we present our decomposition in terms of observable effects. In Sect. 3, we provide an overview under what conditions own and cross price effects can be signed and provide a graphical illustration. In Sect. 4, we study how factor price increases impact on costs. In Sect. 5, we wrap up and draw some policy conclusions. 2 Model and decomposition of factor demand Consider a firm producing one output by means of ninputs subject to a production function y=f(x), where x=(x1,...,xn). About fwe make the following assumption. 123 Substitution and size effect for factor demand... 253 Assumption 1 fis strictly monotonic and unbounded in x, twice continuously differentiable, it has positive marginal products, i.e. fi≡∂f/∂xi>0, and is strictly concave. For output price pand input prices wi,i=1,...,n,letG(p,w) := supx{pf(x)− w·x}be the profit function, and let C(w, y):= infx{w·x|f(x)≥y}be the cost function. We denote by x(p,w) the factor demand and by ˜x(w, y)the conditional factor demand, respectively. Moreover, let y(p,w) := f(x(p,w))denote the firm’s supply. Since fis differentiable and strictly concave, G(p,w)is strictly convex and differentiable. By Hotelling’s Lemma we get Gp=y,Gw=−x(denoting partials by subscripts), and the Hessian of Gis symmetrical positive-definite (S-PD), while C is differentiable and strictly concave in wwith Cw=˜x, and its Hessian Cw,w is S-ND almost everywhere.2 We thus get yp>0, ∂xi ∂wi<0 and pyp= i wi ∂xi ∂p.(1) This follows from differentiating py − i wixi=G w.r.t. pand using Gp=y(suppressing the function arguments, for short). Partials of xw.r.t. pmay be negative, but they cannot all be negative. In particular, if fis homothetic, they are all positive, because they respond to pin fixed proportions. Similarly, we get ∂˜xi ∂wi<0 and wi ∂˜xi ∂wi=− j=i wj ∂˜xj ∂wi .(2) Partial cross derivatives of the conditional factor demand ˜xmay be negative, but they cannot all be negative. In particular, for only two inputs, this implies ∂˜xj ∂wi>0, i= j. Define the inverse function of yw.r.t. pby π(w, ¯y):= {p|y(p,w)=¯y}.3Since y is monotonically increasing in p,π(w, ¯y)is a singleton. The price function π(w, ¯y)can be considered as a compensating price for changes in factor prices wto sustain a fixed output level ¯y. Then by taking the total differential of π, keeping output constant, we obtain πw=−yw yp .(3) 2I.e. negative-definiteness may fail only on a measure-zero subset of the domain. 3To see that {p|y(p,w)=¯y}is not empty, observe that x(p,w)=argmax{pf(x)−wx},andy(p,w)= f(x(p,w)).Ifpgoes to infinity, y(p,w)goes to infinity, and if pbecomes sufficiently small, y(p,w)goes to zero. Since y(p,w)is a function with R0=f[R0],{p|y(p,w)=¯y}is non-empty. 123 254 J. Bröcker, T. Requate By definition of π(·), we can write ˜x(w, y)=x(w, π(w, y)). Differentiating both sides with respect to the factor prices we get ˜xw=xw+xpπw, or by rearranging, we obtain the decomposition for factor demand into a substitution effect and size (or level) effect: xw=˜xw−xpπw.(4) 3 Own and cross price effects For the own price effect we have 0>∂xi ∂wi=∂˜xi ∂wi+−∂xi ∂p ∂π ∂wi(5) Both RHS terms are non-positive. See above for the first term. The proof for the second term goes as follows. From the symmetry of G’s Hessian it follows that Gp,w =yw=Gw,p=−xp.(6) Thus, using (3), the second term in Eq. (5) becomes −∂xi ∂p ∂π ∂wi=−∂xi ∂p ∂xi ∂p/yp<0.(7) Even though well known, we restate this result as follows: Proposition 1 Under Assumption 1the own price effect of a partial factor price increase is negative, and both the substitution and the size (or level) effect move into the same direction. One might suspect that ∂π ∂wiis always non-negative. But we will show in Sect. 3.4 that this need not be the case. Recall also that the last result stands in contrast to consumer demand, where income and substitution effect in the Slutsky equation can go into opposite directions. Next we take a look at the cross price effect. Here we obtain ∂xi ∂wj=∂˜xi ∂wj−∂xi ∂p ∂π ∂wj (8) =∂˜xi ∂wj+∂xi ∂p ∂y ∂wj ∂y ∂p (9) where in the last equation we make use of (3). In general, neither term can be signed unambiguously for i= j, except that the first term (i.e. the substitution term) must be positive for at least one i.Tobemore conclusive, we thus study two prominent cases in the following, homotheticity and 123 Substitution and size effect for factor demand... 255 complementarity. Both imply a negative size (or level) effect. As this is the typical assumption in applied work, we coin this the regular case. Despite regularity, the total effect remains ambiguous under homotheticity but can be shown to be negative under complementarity. 3.1 Homothetic production functions Intuitively, output seems to be decreasing in input prices, inputs seem to be increasing in the output price, and the price compensation function seems to be increasing in input prices. We call these responses regular. The following result shows regularity to hold for homothetic production functions. But as shown later, it does not hold in general. Proposition 2 If f is homothetic, we get ∂y/∂wi<0,∂xi/∂ p>0, and ∂π/∂wi> 0, for all i =1,...,n, while both, the cross price substitution effect ∂˜xi ∂wjand the unconditional cross price effect ∂xi ∂wjare ambiguous. Proof Note that a production function fis homothetic iff the cost function can be written as C(w, y)=h(y)c(w) with convex hand concave c. It is homogeneous, if, in addition, his a power function. Profit maximization implies h(y)c(w) =p(10) Thesolutioninydefines an implicit supply function y(p,w). Differentiating this equation totally, we obtain h(y)c(w) ∂y ∂wi+h(y)∂c ∂wi=0 (11) and thus ∂y ∂wi=−h(y) h(y) ∂c ∂wi c(w) <0(12) as ∂c/∂wi>0. To see ∂xi/∂ p>0, we write (using Shepard’s Lemma) xi(w, p)=˜xi(w, y(p,w)) =∂c ∂wih(y(p,w)). Thus: ∂xi ∂p=∂c ∂wi h(y)yp>0 (13) Finally, ∂π/∂wi>0 follows from (7). The ambiguity of ∂˜xi ∂wjand ∂xi ∂wjis shown in the appendix by using the following Example 1. q.e.d. 123 256 J. Bröcker, T. Requate Example 1 Let the production function be nested CD-CES, f(x1,x2,x3)=(xρ 1+ xρ 2)αβ/ρ x(1−α)β 3, with CES composite zρ=xρ 1+xρ 2that x1and x2symmetrically enter into and with parameters 0 <α<1, 0 <β<1, and ρ<1. Note that Proposition 2immediately implies that also the size (or level) effect −∂xi ∂p ∂π ∂wjis negative. 3.2 Complements In this section we provide a condition under which cross price effects are negative. This holds even if the substitution effect is positive, meaning that in this case a negative size effect dominates the positive substitution effect. The respective condition is that all inputs are complements, i.e. an increase of the input amount of one factor increases the productivity of all other factors. Definition 1 The inputs of a differentiable production function are called pairwise strong complements if fij >0 for all i,j=1,...,n,i= j. From the Frobenius matrix algebra (see Takayama 1985, also Amir 1996) we can derive the following result. Proposition 3 (Takayama) If all inputs are strong complements, then ∂xi ∂wj <0. The proof follows from Proposition 4.D.3 in Takayama (1985) and the equivalence of part (IV”), which is satisfied under strict concavity of the production function, and (VIII”), which is our conclusion. It can also be derived from the strict version of Topkis’s monotonocity proved by Amir (1996) (see also Theorem 2.8.5, Topkis 1998). The result implies that if the substitution effect is positive, the size effect must be negative and must dominate the substitution effect. In fact, we can show that the size effect is always negative. Proposition 4 If all inputs of a differentiable, strictly concave production function are pairwise strong complements, then ∂xi/∂ p>0and ∂π/∂wi>0for all i =1,...,n. The proof is given in the appendix. With this result we can immediately obtain: Corollary 1 If all inputs are pairwise strong complements, then the size (or level) effect is always negative. Proof By the decomposition (8) the level effect is −(∂xi/∂ p)·(∂π/∂wj)<0.  Thus, if the substitution effect is negative, the substitution and the size effect enforce each other, as is the case for the own price effect. In Fig. 1, we have, for the case of a Cobb-Douglas production function, visualized the decomposition of the comparative statics effect of a partial factor price increase of factor 2. Note that, contrasting from the case of a price increase in the consumer’s choice problem, the iso-cost curve does not tilt around a fixed point on one axis, but also shifts. 123 Substitution and size effect for factor demand... 257 Fig. 1 Decomposition of factor demand change into substitution and size effect: the regular case with homogenous production function. We denote: (x∗ 1,x∗ 2)= original factor demand, (xc 1,xc 2) = compensated factor demand, (x∗∗ 1,x∗∗ 2)= new factor demand, convex curves (red) = isoquants, C1C1= original iso-cost line, C2C2= new iso-cost line, dashed line = price compensated iso-cost line 3.3 Decomposition formulas for 2 and 3 inputs We can further substantiate the decomposition for the cases of two and three inputs, assuming fij >0 for all i= jthroughout. From micro textbooks (e.g. Varian 1992) it is well known that the comparative statics expressions for the unconditional factor demand, in the case of two factors, are given by: ∂xi ∂wi=fjj p|H|<0(14) ∂xi ∂wj=− fij p|H|<0 (15) ∂xi ∂p=fjfij −fifjj p|H|>0 (16) where |H|=f11 f22 −f2 12 >0 is the determinant of the Hessian of f. By contrast, the comparative statics expressions for the conditional factor demand are given by ∂˜xi ∂wi=f2 j λ|˜ H|<0 (17) ∂˜xi ∂wj=−f1f2 λ|˜ H|>0 (18) where |˜ H|=f2 1f22 +f2 2f11 −2f1f2f12 <0 is the determinant of the bordered Hessian, which by the second-order condition needs to be negative, and λis the Langrange multiplier for the output constraint of the cost minimization problem. By definition of π(w, y)we have f(x(w, π(w, y))) =y(19) 123 258 J. Bröcker, T. Requate Differentiating this with respect to wiwe obtain: ∂π ∂wi=−n j=1fj ∂xj ∂wi n j=1fj ∂xj ∂p (20) Using the last equation we obtain ∂π ∂wi=fifjj −fjfij |˜ H|>0 (21) as the denominator is exactly the determinant of the bordered Hessian. Thus, the total size effect for the own price change can be expressed as −∂xi ∂p·∂π ∂wi=−fjfij −fifjj p|H|·fifjj −fjfij |˜ H|=(fifjj −fjfij)2 p|H|| ˜ H|<0(22) while for the cross-price effect it is given by −∂xi ∂p·∂π ∂wj=−fjfij −fifjj p|H|·fjfii −fifij |˜ H|<0 (23) Now, exploiting that due to (19)wehaveλ=p, and inserting (16), (17), and (22) into (5) and rearranging we can easily verify that ∂xi ∂wi=fjj p|H|=f2 j |˜ H|−fjfij −fifjj p|H|·fifjj −fjfij |˜ H|=∂˜xi ∂wi−∂xi ∂p ∂π ∂wi (24) holds, similarly for the cross price effect. For n≥3 one can derive similar expressions. For instance for n=3 the total cross price effect is given by ∂xi ∂wj=−fij fkk +fik fjk p|H|<0 (25) where again |H|<0 is the determinant of the Hessian, while the substitution effect is now given by ∂˜xi ∂wj=fk[−fifjk −fjfik +fkfij]+fifjfkk p2|˜ H|(26) where the determinant of the bordered Hessian, |˜ H|, is negative. Note that in the numerator all terms except f2 kfij are negative. Therefore, the substitution effect cannot be signed unambiguously (see Example 1). We know, however, that not all substitution effects can be negative. 123 Substitution and size effect for factor demand... 265 Sedaghat, H.: A variant of the Slutsky equation in a dynamical account based model. Econ. Lett. 50, 367–371 (1996) Slutsky, E.E.: Sulla teoria del bilancio del consumatore. Giornale degli Economisti 51, 1–26 (1915) Slutsky, E.E.: On the theory of the budget of the consumer. Translated by Olga Ragusa. In: Stigler, G.J., Boulding, K.E. (eds.) Readings in Price Theory, pp. 27–56. 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