Path stability of prices in a nonlinear Leontief model

Annals of Operations Research 37(1992)141-148 141

PATH STABILITY OF PRICES IN A NONLINEAR LEONTIEF MODEL

Ulrich KRAUSE Fachbereich Mathematik/Informatik, Universitiit Bremen, 2800 Bremen, Germany

Abstract

In a nonlinear Leontief model where the input-output coefficients depend on the level of outputs, the dynamics of cost-determined prices is examined. It is shown that under mild assumptions path stability holds in the sense that every path when being disturbed comes finally arbitrarily close to its original form. Path stability may occur even if each single path shows chaotic behaviour. By sharpening the assumptions considerably, convergence of all paths may be achieved.

1. Introduction

Nonlinear Leontief models, that is, Leontief models in which the input-output coefficients are admitted to depend on the quantities of commodities produced, have been studied by various authors [2-4, 9-13]. However, only a few of them were concerned with the question of price dynamics [3, 9, 13]. Already in [12] some kind of nonlinear Perron-Frobenius theorem has been employed which then has been developed further and which proved to be useful in handling nonlinear situations (cf. also [8, 10, 11] and for more recent developments [3, 5, 7, 8]).

The present paper deals with price dynamics in a nonlinear Leontief model and is concerned in particular with a new stability property called path stability. Section 2 of this paper presents a framework for nonlinear Leontief models including the classical rule of prices set according to the production cost (cf. [1]). One arrives at a nonlinear discrete dynamic model which in section 3 is made manageable by the trick of treating the outputs as a parameter. This makes the dynamic model one of iterating different operators instead of iterating just one single operator. Section 4 presents the main result which yields under certain assumptions path stability in the sense that every price path when being disturbed comes finally arbitrarily close to its original form. The proof of this result rests on a theorem on inhomogeneous iterations developed in [6]. Path stability does not by itself imply convergence for every price path. On the contrary, path stability is compatible with such irregular behaviour as chaotic behaviour of each single path. In section 5 it is shown that by sharpening the assumptions considerably, convergence of all the price paths to a unique price equilibrium may be achieved.

As is well-known, nonlinear discrete dynamic systems often exhibit very complex dynamic behaviour called chaotic dynamics. The nonlinear discrete Leontief model examined in the present paper however, shows the additional feature of

© J.C. Baltzer AG, Scientific Publishing Company

142 U. Krause, Path stability of prices

positivity for the relevant variables. This makes it possible to employ particular tools (cf. [5, 6, 8]) which may be viewed to constitute the particular domain of positive discrete dynamical systems.

2. The formulation of a nonlinear Leontief model

Consider n producers who are interdependent in the sense that producer j (j = 1 . . . . . n) produces the specific commodity j by means of the commodities produced by the other producers and by employing (homogeneous) labour. What makes the standard Leontief model nonlinear is the fact that in general the input- output coefficients or, admitting choice of techniques, the technology sets of the producers do depend on the level of outputs. Let Ai(u) denote the non-empty sets of inputs (a, 1) with material input a ~ IR~_ and labour input l ~ IR+ by which producer j is able to produce u units of commodity j. Here, IR~_ = {x = (xl . . . . . xn) [ xi ~ IR, xi > 0 for all i = 1 . . . . . n}. Given prices p e IR~ and wage w ~ IR+the minimum cost for producer j to produce u units of commodity j then is given by

Ci(p, w, u) = inf {pa + wl [ (a, l) ~ Aj(u) } . (1)

In the special case of the standard Leontief model, i.e. a linear time-independent input-output model with input-output matrix A, the minimum cost for j to produce u units is given simply by the j th component of the vector pA + wl multiplied by u. (For the modeling of technical change by time-dependent input-output models see [5].)

For fixed u minimum cost is (jointly) concave and positively homogeneous in (17, w). No such property is available in general with respect to u when p and w are fixed. Rather, one purpose of this paper is to admit a wide range of possibilities for costs to depend on outputs.

As to the formation of prices, the paper adopts the view of classical economics according to which prices are in the long run driven by costs (cf. [1]). However, what does that precisely mean in face of costs which vary in an unknown manner on outputs? Prices may be driven by marginal cost or by average cost which both are equal in the case of constant retums by Shephard's lemma, but may be different otherwise. Assuming perfect competition, the maximizing of profits makes marginal cost the relevant concept. However, one may design other mechanisms of price formation, e.g. by average cost to which some kind of mark-up is added. Anyway, in the following we shall deal with a function cj(p, w, u), called the relevant cost function of proclucer j, which may be given by marginal or average cost with respect to (1) or by something else. The classical view of prices driven by costs is now made precise by the assumption that relative costs of one period determine relative prices of the next period, that is,

pi(t + l) ci(p(t) , w(t) , u(t)) - (2)

p j ( t + l) c j (p( t ) , w(t), u(t)) '

U. Krause, Path stability of prices 143

for all i , j ~ {1 . . . . . n}, all t ~ {0, 1 ,2 . . . . }. Equivalently, prices at t + 1 are proportional to costs at t with a factor )t, of proportionality which may be considered to depend upon t alone. Let p = (Pl . . . . . Pn) denote the vector of prices, let Y = (Yl . . . . . Yn) be the vector of outputs of the producers and let c • R 2n÷ 1 ___) Rn denote the function of relevant costs the ith component of which is given by ci( p, w, Yi). Then (2) becomes

p(t + 1) = X(t) . c(p(t), w(t), y(t)). (3)

In general equilibrium theory an equation like (3) is the result of competit ion. However, therein not only the equality of relative prices and relative costs is asserted, but also the equality of absolute prices and costs is claimed. This seems rather strange for a theory which does not specify a theory of money. Formally, in a general equilibrium theory ~,(t) = 1 which means that equilibrium prices p appear as fixed points of (3). In contrast, in this paper equilibrium prices are given by eigenvectors of the nonlinear eigenvalue problem implied by (3) with eigenvalues not necessarily equal to 1.

Of course, eq. (3) is by far too general for obtaining interesting statements concerning price dynamics. To simplify matters we shall not add to (3) an equation concerning wage dynamics but reduce instead wages to prices as follows. The easiest way would be to make the not very reasonable assumption of a given real wage bundle b ~ R n+ by which w = p • b (cf. [3] for an otherwise careful analysis of consumption). Instead, one may conceive of wage eamers as maximizing utility for given wage and prices. That is, a wage earner chooses a bundle b*~ IR+" such that U(b*) = max { U(b) I b ~ IR~, p . b < w}, where U is some common utility function. It follows that w = b*p = rain {p. b l U(b) > U(b*), b ~ IR~}. Fixing a utility level U* = U(b*), the wage is a function of prices and this function is continuous, concave and positively homogeneous (cf. [5]). As in the case of the relevant cost function, what matters is not how the wage function is derived in detail but that the wage function p ~ w(p) exhibits certain properties to be specified later on.

3. Output dependent price dynamics

Inserting the wage function eq. (3) becomes

p(t + 1) = c (p( t ) , w(p(t)), y(t)). (4)

The main problem now is how to treat outputs y(t). This will be handled by treating outputs parametrically, i.e. different vectors of y(t) are conceived of as leading to different cost functions in terms of prices. Thus, for fixed t a mapping T(t) • It~ ~ 1t~ is considered which is defined by

T(t)p = c(p, w(p), y(t)). (5)

Equation (4) which describes price dynamics then becomes


p(t + 1) = ~,(t) T(t) p(t). (6)

If T(t) is positively homogeneous in p, which is the case, e.g,, if relevant costs are given by (1), then £(t) can be eliminated as follows. Prices are normed by

If

n

11 P II = ~ Pi, P ~ IR~. i=I

p(t) = p ( t ) and T p - Tp Up(t) fl IITpll

denote the normalizations of prices and cost operator, respectively, then from (6) it follows that p(t + 1) = T(t) p(t). Rewriting p as p, the following system of price dynamics is obtained:

p(t + 1) = T(t)p(t) if

where T(t)p = T(t)p with IlT(t)Pll

p(t)~s = {p ~ R + [ l l p l l = 1},]

7](t)p = ci(p,w(p) , yi(t)). (7)

In normalizing prices one has to be careful. The processes of price normalization and price formation do not commute in general. If T(t) = T for all t, to approach eigenvectors of T, the relevant iterates are that of the normalized cost operator, i.e. the ~-k, but not the nomlalizations of the iterated cost operator, i.e. the ~.k. This difference may be relevant if mark-ups are involved. The difference vanishes if T is positively homogeneous. To simplify matters, we make this assumption. Since the operator T(t) in (7) depends on t, not simply the iterations T k of one single operator T have to be analyzed but so-called inhomogeneous iterations T(k) o T(k-1) o . . . o T(0) (cf. [6]). This will be done in the next section.

4. Path stability

To prove something definite about the asymptotic behaviour of the prices, assumptions have to be made, some of which have already been discussed. Concerning c(-), w(.), y(.) the following will be assumed.

ASSUMPTIONS

(1) Costs are (jointly) positively homogeneous and concave in prices and wage, i.e. the function (p, w) ~-~ cj (p, w, yj ) is positively homogeneous and concave for each j and each yj ~ ~+.

(2) The wage is positively homogeneous and concave in prices, i.e. the function p ~-~ w(p) has these properties.


(3) Costs are positive if the price of some particular commodity is positive and costs are irreducible, i.e.

(a) there exists some i such that cj(e i, w(ei), yj) > 0 for all j for which Yi > 0 (ei = the ith unit vector in lR");

(b) for any subset ~ c I c { 1 .. . . . n} there exist i ~ I a n d j ~ I such that c)(e i, w(ei), y)) > 0, provided yj > O.

(4) Costs are mono tone in outputs , i.e. for f ixed p and w the funct ion y) ~ c ) ( p , w , y j ) is always non-decreasing or is always non-increasing.

(5) Outputs remain finally bounded and positive, i.e. there exist 0 < u/, vj ~ IR+ and t o ~ R+ such that uj < yj(t) < vj for all j and all t > t o.

The following lemma draws some useful conclusion from assumption 3. For elements x, y e R~_, x _< y means xi < Yi for all i, x < y means x i < Yi for all i and x < y m e a n s x < y b u t x ~:y .

LEMMA

Let f " ~ . ---) IR~ be positively homogeneous, non-decreasing and such that

(i) f(e i) > 0;

(ii) for any subset O ~ / ~ { 1 ... . . n} there exist i ~ I and j ~ I with fj(e i) > O.

Then for the mth iterate f " of f there holds fro(x) > 0 for all x ~ 0 and m > n.

Proof

From ( i ) f ( e 0 > 0 by an appropriate numbering of coordinates. It will be shown by induction, that for any I < k < n fk(e i) > 0 for all 1 0 for 1 0 such that u > Zel .Because f is positively homogeneous and monotone it follows that f ( u ) > ~, f (e])> O. Therefore, f k + l(ei) =f( fk(ei) ) =f (u ) > 0 for 1 0. From f(ei) >fj(ei)ej it follows that f k ( f ( e l ) ) > f j ( e l ) f k ( e j ) . Since j < k, f k ( e j ) > 0 by the induction hypothesis. Thus, f k+ 1~ i) > 0 for some i > k + 1. By appropriate numbering i = k + 1 may be assumed. This completes the induction. Now, let m > n and x ¢ 0. Then x > xi el with x i > 0 for some 1 xi f"(e i) > 0. Applying f (m - n) times to fn(x) > 0, it follows fro(x)> 0 because for any y > 0 and appropriate Z > 0 there holds

Y ->- ~el. []


Remarks The lemma generalizes an argument in [3, proposition 3.1(b)] to nonlinear

mappings. The lemma yields that the mapping f is primitive. Hence, the mapping f considered in [3] is primitive too, contrary to what is stated there.

THEOREM 1

Under assumptions (1) to (5), any two price paths t ~ p(t) and t ~ q(t) finally approach each other, i.e. d(p(t), q(O) -~ 0 for t -~ oo. (d is the Euclidean distance on IR".)

Proof

It suffices to consider the case where yj ~ c j (p ,w,y j ) is non-decreasing. Then, using assumptions (4) and (5), it follows that

cj(p, w(p), uj) < Tj(t)p < cj(p, w(p), vj) for all t > t o.

Let f : IR~ ---) IR n + be defined byf jp = cj(p, w(p), u), and let g" IR]. ---) IR] be defined by gjp = ci( p, w(p), vj). The above inequalities become

f ( p ) < T(t)p < g(p) for all t > t 0, all p ~ IR~. (8)

From assumptions (1) and (2) it follows that the function p ~-> f jp is concave. This implies that f / i s non-decreasing in prices. Namely, if 0 - ( 1 - 1 ) f j ( P ) + l f j ( p + n ( q - p ) ) >- ( 1 - 1 ) f j ( p ) .

Since n is arbitrary it follows that f/(q)_>f/(p). Similarly, gj(q)>_gj(p). Using induction one obtains

fro(p)< T ( k + m - 1 ) o...o T ( k + l ) o T ( k ) p < gin(p) (9)

for all p ~ P,~. and all k > to. Obviously, inequalities (9) hold for m = 1 because of (8). Suppose (9) holds for some m > 1. Applying f to the left-hand inequality of (9) with x = T ( k + m - 1 ) o . . , o T(k+ 1) oT(k)p yields fro(p) < f ( x ) < T(k + m)o x by taking into account that f is non-decreasing and by applying (8), Similarly, the right-hand inequality of (9) yields T(k + m) o x <= g(x) < g,,,+1 (p). Putting together, it follows

fm+l(p)< T ( k + m ) o x < gm+l(p),


which is (9) for m + 1 instead of m. This completes the proof of (9) by induction. Furthermore, by assumptions (1), (2) and (3) the func t ionf satisfies the assumptions of the lemma. Hence, by the l emmafn(p ) > 0 for p rt 0. This together with (9) means that the sequence (Sk)k~ given by the lumped operators SkP = T(k + n - 1) o . . . o T(k + 1) o T(k)p is uniformly pointwise bounded (cf. [6, p. 846] for the definition). Obviously, the operators T(t) are concave and positively homogeneous by assumptions (1) and (2). Furthermore, the operators T(t) are proper, i.e. T(t)p = 0 i f f p = 0. Namely, T(t)0 = 0 because (p ,w) F-~ c j (p ,w ,y j ) as well as p ~ w(p) is positively homogeneous. If T(t)p = 0, then the lemma implies that p = 0. Thus all assumptions of the concave version of the Coale-Lopez theorem proved in [6, p. 848] are satisfied and this theorem yields theorem 1. []

Theorem 1 ensures, under the assumptions made, path stability in the sense that each path when being distorted exogeneously comes back after a while to its original form. Of course, such a path stability holds if each path would converge to the same position. The point of theorem 1 however, is to yield path stability although each path may diverge. (There are simple examples where this actually happens.) One might also say that according to theorem 1 all starting points lead to the same limit set, which however, need not be a single point but could be a cycle. Even more, the limit set might be a fractal set on which the dynamics is chaotic. The path stability of theorem 1 holds without specifying in detail the time path of outputs y(t) or the dependence of costs on outputs.

5. Convergence to a single point

By making stronger assumptions than those in the previous section, for output- dependent prices even the convergence of all paths to a single point can be shown.

THEOREM 2

Suppose assumptions (1)-(3) of the previous section and suppose in addition that

(6) cj(p, w(p) , yj) is continuous in p and uniformly continuous in Yi on S = { p ~ ~ " + I Ilpll = 1 },

(7) Outputs converge, yj(t) ---> y~> 0 for t ---> ~ and all j.

Let T denote the operator defined by T:p = cj(p, w(p), y)). Then the nonlinear eigen equation Tp = &p has a umque solution ( p , Z ) under the constraints p ~ S, )~ > 0 and for each price path t ~ p( t ) ,p( t ) converges to p* if t approaches oo.

Proof

By assumptions (6) and (7), to e > 0 there exists t o such that ITy(t)p -Tip I = I ci(P, w(p), yj(t)) - ci( p, w(p), y~) I < e for all p ~ S. Hence, the operators


T(t) converge for t ~ oo uniformly on S to the operator T with components T/. By assumptions (1) and (2) the operators T(t) and T are concave and positively homogeneous. As in the proof of theorem 1, the operators T(t) with t > t o for some t o and T are proper. Finally, let p, q ~ S with ~,p < q for some/~ ~ IR÷, i.e. q = $p + r with r > 0. By assumptions (1) - (3) the lemma yields Tmr ~ 0 for some m. Since T is concave and positively homogeneous it follows that Tmq >$Tmp + Tr% > $Tmp. Thus, all the assumptions of theorem 6 in [6, p.850] are satisfied and this theorem implies theorem 2. []

For the linear Leontief model the unique solvability of the eigen equation Tp = )~p, i.e. c(p, w(p))= 3.p, may be viewed as a "dynamic" non-substitution theorem."Dynamic" because & ~ 1 is admitted. For this view of the non-substitution theorem, in case of ~ = 1, see [1]. A similar interpretation is possible by theorem 2 for nonlinear Leontief models. However, whereas in the linear case the unique (normalized) price p* does not depend on output at all, in the nonlinear case it can only be said that p* does not depend on how outputs behave in time but may depend on the limit position y* for outputs.

Acknowledgement

I would like to thank three anonymous referees for helpful comments on a previous draft.

References

[1] K. Arrow and D. Starrett, Cost-theoretical and demand-theoretical approaches to the theory of price determination, in: Carl Menger and the Austrian School of Economics, ed. J.R. Hicks and W. Weber (Oxford Clarendon Press, Oxford, 1973) pp. 129-148.

[2] P. Chander, The nonlinear input--output model, J. Econ. Theory 30(1983)219-229. [3] R.A. Dana, M. Florenzano, C. Le Van and D. Levy, Production prices and general equilibrium prices,

J. Math. Econ. 18(1989)263-280. [4] T. Fujimoto, Non-linear Leontief models in abstract spaces, J. Math. Econ. 15(1986)151-156. [5] T. Fujimoto and U. Krause, Ergodic price setting with technical progress, in: Competition, Instability

and Nonlinear Cycles, ed, W. Semmler (Springer, Berlin, 1986). [6] T. Fujimoto and U. Krause, Asymptotic properties for inhomogeneous iterations of nonlinear operators,

SIAM J. Math. Anal. 19(1988)841-853. [7] E. Kohlberg, The Perron-Frobenius theorem without additivity, J. Math. Econ. 10(1982)299-303. [8] U. Krause, Perron's stability theorem for nonlinear mappings, J. Math. Econ. 15(1986)275-282. [9] M. Morishima, Equilibrium, Stability and Growth (Clarendon Press, Oxford, 1964). [10] H. Nikaido, Convex Structures and Economic Theory (Academic Press, New York, 1968). [11] T.T. Read, Balanced growth without constant returns to scale, J. Math. Econ. 15(1986)171-178. [12] R.M. Solow and P.A. Samuelson, Balanced growth under constant retums to scale, Econometrica

21(1953)412-424. [13] C.C. yon Weizs~icker, Steady State Capital Theory (Springer, Berlin, 1971).

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Path stability of prices in a nonlinear Leontief model