Long run input use-input price relations and the cost function Hessian. Ian Steedman Manchester Metropolitan University

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1 Long run input use-input price relations and the cost function Hessian Ian Steedman Manchester Metropolitan University Abstract By definition, to compare alternative long run equilibria is to compare alternative points on the real input price frontier. It follos at once that one can never move beteen long run equilibria by changing just one input price; one must change at least to. And in some cases, indeed, such as the Wicksellian one, to change one price is ipso facto to change all the others in a determinate manner. Hence the Hessian of the cost function can quite obviously never represent the long run comparative statics of input price-input quantity relations ith accuracy. More detailed investigation in fact shos the Hessian to be a hopeless guide to [dl i /d j ], both qualitatively and quantitatively. I thank Christian Bidard (Paris) and Arrigo Opocher (Padova) for their encouraging comments on an earlier version. I.Steedman@mmu.ac.uk Department of Economics Manchester Metropolitan University Mabel Tylecote Building Cavendish Street Manchester M5 6BG April 005

2 Long run input use-input price relations and the cost function Hessian Ian Steedman The comparative statics analysis of input use and input price must, at a minimum, be able to include the comparison of alternative long run equilibria, since it ould otherise be incomplete. It might ell be thought, moreover, that for policy purposes, it is indeed such long run comparisons that are of the greatest use. The purpose of this simple paper is to emphasize that the Hessian matrix of the cost function by no means provides the long run comparative statics required and, more positively, to sho just ho it is related to the relevant results. Let represent a vector of positive input prices and c() be a cost function relating to a single-product process. The output level does not appear in c() if e interpret the cost function in either of to ays. First, e may take c() to be a unit cost function for a constant-returns-to-scale production process. Alternatively, e may take it to be an indirect average cost function (Silberberg, 974) shoing the minimum average cost at hich output can be produced. In either case, the use of the ith input per unit of output, l i, is given by l i = ( c/ i ) and thus dl = Cd (), here C is the Hessian matrix of c( ). It is of course a symmetric, negative semidefinite matrix. (On the indirect average cost function, see Silberberg, 974) Here, e shall also assume that every off-diagonal element of C is positive, i.e., that all pairs of inputs are Hicksian substitutes; e make this (arbitrary) assumption merely in order to stress that nothing to be said belo ill presuppose the presence of Hicksian complementarity and the difficulties to hich it can give rise.

3 From (), then, and ( l i / i ) = c ii < 0 () ( l i / j ) = c ij = ( l j / i ) > 0 () Do the familiar results () and () constitute comparisons beteen alternative long run equilibria? Certainly not. By definition, any long run equilibrium involves that c() is equal to the product price, or taking that price to be unity, so that the are no real input prices that c() = (4) From (4), one can never change just one input price at a time hen comparing long run equilibria. In (), then, d is constrained by l T d = 0 (5) and any study of the long run comparative statics of input use and input price must take account of both () and (5). We shall begin sloly, by considering our old friend the to-input production process but later sections ill of course generalize the argument. To inputs In this basic case () and (5) simplify to and dl dl c = c c c d d (*) l d + l d = 0 (5*) We also kno, of course, that c + c 0 c + c 0 (6)

4 because c(, ) is homogeneous of the first degree. Consider, for example, (dl /d ). From (*) and (5*). and hence, from (6), (dl /d ) = c (l /l ) c (dl /d ) = (c / l ) (7), since ( l + l ) = Result (7) shos that (dl /d ) < ( l / ) < 0 and, indeed, that if the share of the second input is small, (dl /d ) ill be much less than ( l / ). Repeating the derivation of (7) for the other (dl i /d j ) dl/d dl/d dl /d dl /d l/ l = l / l l/ l l / l (8) We see from (8) that the Hessian matrix C does give the correct sign pattern for the matrix [dl i /d j ]; beyond that, hoever, the former matrix is a very poor guide to the latter. Every element of C underestimates the absolute magnitude of the corresponding (dl i /d j ) and may do so grossly. Moreover hilst C is necessarily symmetric, matrix [dl i /d j ] ill be so only in the marginal case l = l = 0.5. And then, of course, [dl i /d j ] = [ l i / j ] so that the Hessian is a hopeless approximation to the true matrix of long run comparative statics effects. (Note that l = l is equivalent to (d / ) = - (d / ) and that there ill be at most one such point on c(, ) =. Note too that if either l or l becomes very small, one column of the Hessian becomes a very good and the other column a

5 very poor approximation to the comparative statics matrix. Note finally that in the Cobb-Douglas case the ( j l j ) terms become constants and that (generically) e never have (dl /d ) = (dl /d ); in the fluke case, c( ) = equal.), hoever, they are alays Even in the basic, to-input case, then, the [ l i / j ] matrix is a poor guide to the [dl i /d j ] matrix of long run comparative statics effects. We no ask ho matters stand in the more general case. More inputs To have more than to inputs is not, of course, going to restore symmetry of [dl i /d j ]; that can be taken for granted. But e do need to ask hether (ith Hicksian substitution throughout) e can at least continue to say that (dl i /d i ) < ( l i / i ) < 0 and that (dl i /d j ) > ( l i / j ) for i j. It ill suffice to consider the case of three inputs, since the interested reader can readily generalize further. The relations and dl = Cd () C 0 (6) no hold, C being the x Hessian of c(,, ); relation (5) becomes l d + l d + l d = 0 (5**) The crucial difference beteen the present case and the above to-input case is that (5**) unlike (5*) no longer allos us to express the vector d, in (), in terms of a single (scalar) d j. Consequently, although the Hessian matrix C is fully defined at every point on the input price frontier c(,, ) =, the matrix [dl i /d j ] is not. The latter matrix depends on ho vector moves on the frontier, on ho relative input prices change. (With only to inputs this as completely determined.) 4

6 Not all is lost, hoever. Consider for example, (dl /d ); e have dl d = c + c d d + c d d d d here l l = l d + d If either of (d /d ) and (d /d ) is zero, or if they are both negative, then certainly (dl /d ) < ( l / ) < 0 Hoever, if (d /d ) and (d /d ) are of opposite signs e can no longer be sure, a priori, that (dl /d ) < ( l / ); or even that (dl /d ) < 0! A more geometric ay of seeing this is to note that c(,, ) = and l (,, ) = some constant, together define a curve on the input price frontier; by construction, along the curve dl = 0 alays. Starting from any point on that curve e may change a little and find corresponding changes in (, ) that make dl either positive, or zero or negative. Intuition might suggest that if, say, and both change by the same percentage then some definite results should emerge for the (dl i /d ) since e are almost back to the to-input case. And it can indeed be shon that, no, dl l (9) ( ) = l d (9) is a nice enough result to be sure but, of course, the analogous result for (dl/d ) ould require that and both change by the same percentage, hich is completely inconsistent ith the assumption underlying (9). Thus no general nice relation beteen [dl i /d j ] and C is going to be reachable by this route. 5

7 What if, say, d = 0? In this case, e find that dl d dl d has the sign pattern + + ( )( ) + / / +, here e note that (dl /d ) (dl /d ) is naturally negative. The complete sign pattern is again not predicted. On a more constructive note, it is easy to sho that at least one of dl d dl c and c d must be strictly positive (and similarly for the other to d j ). Yet neither this result, nor the fact that T [dl i /d j ] = 0, can give one great cheer. Definite results have been sparse in this section and ill not be made abundant by generalization to n > inputs. In order to reach more specific conclusions, e shall need to supplement restriction (5) [l T d = 0] ith some further limitation on d. Fortunately, this can arise quite naturally, as in the folloing section. The Wicksellian case Although the argument of this section can readily be extended to the general n-input case, it may be helpful to concentrate again on the case n =. Suppose no that the cost function c(,, ) refers to a constant returns, three period Wicksellian process of production. If ages are paid in advance, e have j = ( + r) j, here is the real (product) age rate and r is the period rate of interest. On defining ρ ( + r), e see that in long run equilibrium 6

8 c(, ρ, ρ ) = c(ρ -,, ρ) = (0) c(ρ -, ρ -, ) = From the first and third elements of (0) e see that is monotonically decreasing and monotonically increasing in ρ; ho varies ith ρ ill depend on the exact nature of the c( ) function. Alternatively, on noting that, e may rite c (,, ) = and deduce immediately that is monotonically decreasing in but that ho varies ith ill depend on c( ). Whether e think of ρ varying parametrically or, more directly, of varying parametrically, the upshot is the same. Although there is a complete, smooth input price surface c(, ) =, only the vectors lying on a particular curve across that surface are potential long run equilibrium. Hence only vectors on that curve need ever be considered in calculating [dl i /d j ] and it can almost alays be calculated exactly, because each j and therefore each l i is a determinate function of. The indeterminacy hich plagued us in the previous section has been removed by our Wicksellian interpretation of c ( ); note that this ould continue to be true for an arbitrary number of dated labour inputs. To keep our eye firmly fixed on (,, ), let us use the identity and treat (rather than ρ) as a parameter. Since and d - d + d 0 l d + l d + l d = 0 e can alays determine any to of the d j in terms of the third one. Using the relation dl = Cd e can thus determine all nine (dl i /d j ). What can e say about them? Bearing in mind that all three first partial derivatives of c( ) are homogeneous of degree zero, e see that e can rite 7

9 l = c (, /, / ) l = c ( /,, / ) l = c ( /, /,) () If all the c ij > 0 (i j), e see from the first and last elements of () that l is increasing in and l decreasing. (The movement of l is less obvious.) Hence (dl /d ) and (dl /d ) are certainly both negative, hile (dl /d ) and (dl /d ) are certainly both positive (although there is no reason to expect them to be equal). It may be noted that dl d dl d dl = d dl d d d d d hich is negative, being the product of three negative terms and one positive one; even hen ( l / ) and ( l / ) are both positive, (dl /d ) and (dl /d ) must be of opposite signs. It may be noted also that if d = 0 at some point on the (,, ) curve then, at that point, (dl i /d i ) < ( l i / i ), for i =,, and that dl/d dl/d dl/d dl /d dl dl /d /d l/ = l / l/ l / l / + l / c c c [, ] But, of course, e cannot evaluate the dl i /d magnitudes at this particular point We kno that (dl /d ) and (dl /d ) are alays negative in the Wicksellian model but does (dl /d ) share this property? From (0) and () ( /,, / ) c = and l = ( /,, ) c / 8

10 so that each of and l is a function of the single variable /. As this variable increases, say, it exerts conflicting pressures on and, if c > 0 < c, on l. It is thus plausible to suppose that one or both of and l may have a turning point ith respect to ( / ). And if that is so for ( / ) > (i.e., for ρ > ) then the l ( ) relation ill be non-monotonic for economically relevant values of (,, ). Consider the cost function c = λ + λ + λ + λ + λ + λ for hich l = λ + λ + λ and thus (l λ ) = λ λ + λ + λ l has a minimum value at = (λ /λ ) and hence at a positive rate of interest if λ > λ. Also, on setting c = and defining x 4 ( / ) e find that [λ x - + λ x - + (λ + λ ) + λ x + λ x ] = Thus has a maximum value at λ x + λ x 4 = λ + λ x and the solution ill be greater than one hen λ + λ > λ + λ. It is thus certainly possible that, as increases, both a minimum l and a maximum ill occur at (different) positive rates of interest. Then a graph of the l ( ) relation ill take the form of a (non-closed) loop, ith initial and final donard sloping sections and an intermediate upard sloping section. (The direction of movement around the loop as and ρ increase ill depend on hich of 9

11 the l ( ) and ( ) turning points occurs first.) The l ( ) loop should not be called a demand curve, of course, since by construction both and are alays changing along the curve. But it is the relationship obtaining beteen l and hen only long run equilibria are compared, i.e. it is relevant to long run comparative statics, hereas demand curves (ith, constant) are not. It ould naturally be possible to calculate the complete matrix [dl i /d j ] for our quadratic-square-root cost function, noting for example that both (dl /d ) and (dl /d ) change sign along the above-mentioned loop, even hen ( l / ) and ( l / ) are both positive throughout, and that both (dl /d ) and (dl /d ) do likeise. But the reader is probably tired by no and the general thrust of our argument ill already be clear. Neither the symmetric nature, nor the sign pattern, nor again the absolute magnitudes displayed by the Hessian of the cost function give any useful guide to the corresponding properties of [dl i /d j ], the matrix that does actually display the magnitudes relevant to the comparison of long run equilibria. And the underlying reason for this is very simple; input prices cannot be changed one at a time hen such equilibria are being compared. (It is simple to extend the Wicksellian case to n dated labour inputs. If j = (+r) j (n-), as before, then i = (n-i) (i-) n, for i (n-). It follos easily both that (, n ; l, l n ) behave just like (, ; l, l ) above and that every intermediate ( i, l i ) for i (n-) is just as akard a customer as (, l ) above; the reader ho so desires can spend many a happy hour explicating all of this.) Concluding remarks By definition, to compare alternative long run equilibria is to compare alternative points on the real input price frontier. It follos at once that one can never 0

12 move beteen long run equilibria by changing just one input price; one must change at least to. And in some cases, indeed, such as the Wicksellian one, to change one price can be ipso facto to change all the others in a determinate manner. Hence the Hessian of the cost function can quite obviously never represent the long run comparative statics of input price input quantity relations ith accuracy. More detailed investigation in fact shos the Hessian to be a hopeless guide to [dl i /d j ], both qualitatively and quantitatively. Reference Silberberg, E., 974. The theory of the firm in long-run equilibrium. American Economic Revie, LXIV, pp H:\files from admin acc\steedman\005\long run input use-input relations.doc