Industry-Level Capital-Labour Isoquants

Transcription

1 Industry-Level Caital-Labour Isoquants Ian Steedman Abstract In both theoretical and alied economics, the long-run conditions of roduction for an industry are often resented in the form of a caitallabour isoquant. This reresentation cannot be derived from the full longrun system of industry-level roduction functions secified in terms of the hysical inuts. Introduction It is not unusual, in either theoretical or alied economics, to find that the conditions of roduction in a articular industry are reresented by a caital-labour isoquant, k (l. This is esecially so, of course, in the context of long-run equilibrium where all inuts are variable. The obect of this short aer is to examine how such a reresentation of technical ossibilities is related to the fuller, hysical roduction function, reresentation.(more exactly, we shall emloy the cost function aroach and, indeed, since we assume constant-return-to-scale, the unit cost function aroach. The Framework Consider an economy with n single-roduct industries, exhibiting constant returns and emloying only circulating caital. Let be the rice

2 of the th commodity (which is uniform throughout the economy, while w and r are the wage rate and the rate of interest (both also uniform throughout. If c ( is the unit cost function for industry then, in longrun equilibrium, c[(,(,...,( n, w] ( if wages are aid ex-ost and all roduced inuts are aid for ex-ante. We suose that system ( can be solved to give (when w 0 by w f ( ( Labour use er unit of outut in industry will of course be given l c ( (3 w where the right- hand side of (3 is homogeneous of degree zero in the inut rices. From (, then, l can be written as a function of ( alone and, indeed, if we take l to be a Hicksian-substitute for every other inut used in industry then l will be a monotonically-increasing function of (. i Similarly, each roduced inut use er unit of outut in industry, a, will be given by the aroriate artial derivative of c (, will be homogeneous of degree zero in the inut rices and can be written as a function of ( alone. The caital-outut ratio in industry will, of course, be given by k a i i ( i / and, taking account of (, we see that k also can be written as a function of ( alone.

3 Consider now all the alternative ossible long-run equilibria corresonding to all the non-negative values of r that ermit a nonnegative real wage. At each such value of r, both l and k are determined and hence as r increases from zero, a k / l locus is traced out. There is no ( doubt, then, that the full system ( does imly a k l relation for each,..., n. ( We cannot lea straight to the conclusion that there is a roer caital-labour isoquant for each industry, however. In order to qualify as such an isoquant, our k l relation must have at least two roerties; it ( must sloe downwards and it must be convex from above. Moreover, if a caital-labour isoquant is to be of use in economic analysis it must have the roerty that it is tangential to the ʻinut iso-cost-line, i.e., that ( dk / dl [ w/( ]. The following sections show that our k l relations need not satisfy these three requirements. ( First and Second Derivatives The k l relation can fail to be downward-sloing and/or everywhere ( convex from above even when the unit cost functions - the c ( - are of the simlest, most stereotyical marginalist kind. Quite secifically, we shall consider a two-industry economy ( n in which the first industry has a constant-elasticity-of-substitution unit cost function (with, while the second industry has a Cobb-Douglas unit cost function. Each industry uses as a roduced inut only the roduct of the other industry. 3

4 To begin, let us set aside consideration of the second industry for the moment and write down for the first industry the rice-cost relation ( ( ( [( ] w (4, where. From (4, and hence a [ ] ( k (5 ( ( ( If we choose our measurement units so that wwhen r 0, we see from (5 that k when r 0. From (5, however, k whenever ( ( / (, i.e., from (4, whenever w ( ( ( [ ] It is imortant to be clear that (6 is not the economy s real wage-interest rate frontier; it is simly a condition the satisfaction of which is equivalent to k ; it deends solely on the cost function for industry. The real wage-interest rate frontier, by contrast, deends on both c ( and c (. It is to be exected, then, that c ( (6 can be so chosen that that frontier intersects (6 not only at r 0, but also at some r 0; follow at once that k ( is non-monotonic and hence that k l ( non-monotonic (given our assumtion that l ( is increasing. it would is also To see how easily that exectation can be fulfilled, let us return to the assumtion that c ( is of the Cobb-Douglas from and write, 4

5 secifically, r / / [( ] w (7 From (4, (5 and (7, k ( k ( (3 ( r (8 Now (8 is clearly satisfied by r 0and k ; but it is also satisfied whenever k and ( ( 3 ( r (9 It is easy to show that (9 has a solution r * 0 if and only if 3 ( (0 (imlying, of course, that. Whenever (0 holds in (4, the k / relation is increasing at r 0, and is as shown in Figure. Note that the ʻroblem with the k relation arises here for low values of r, i.e., for ( l relevant values of the interest rate. ( l Figure Here (Before continuing our discussion of the case n, we may turn aside to suose that (4 still holds in industry but that there are ( n other industries. Equation (6 will still aly and it should be ossible to choose c functions for the ( n ( other industries in such a way that the 5

6 economy s real wage-interest rate frontier intersects (6 many times; that is, so that k for many ositive rates of interest and thus k ( l is decidedly non-monotonic. Leaving the reader to ursue this ossibility, we now return to our simle n examle. If inequality (0 is reversed then k is downward-sloing for all relevant r but this is not, of course, a sufficient condition to make k ( l a conventional caital-labour isoquant. Some tedious differentiation and maniulation shows that ( d k / dl 0 will hold good (at r 0 only if another inequality (of the form f ( holds. When it does not, we ( l 3 have Figure, in which k has the conventional negative sloe but not ( l everywhere the conventional curvature. Note that it is again at low values of r that the convention is violated. Figure Here To conclude this section we resent Figure 3, which shows how different combinations of and yield different signs for ( dk / dl and ( d k / dl, evaluated at r 0 ; note that Figure 3 is only sketched somewhat aroximately and that 0.8 * In region A, ( dk / dl 0 ( d k / dl ; in region B, ( dk / dl 0 ( d k / dl region C, ( dk / dl 0 ( d k /,the conventional case. dl ; while in Figure 3 Here 6

7 The Tangency Condition A downward-sloing, convex from above k l relation is only useful in economic analysis if it has the roerty that its (absolute sloe is equal to the ratio between the real wage rate and (. Now in long-run equilibrium w l ( k ( ( where w is the real wage measured in terms of commodity. On suressing the subscrits we may rewrite ( in differential form: 0 [ wdl ( dk] ( ldw kd ( and we see from ( that ( dk / dl ( w/ if and only if v v ( dw/ d ( k / l. Now it is well known that, at r 0, ( dw/ d ( k / l, v where k and l v are the caital-outut and labour-outut ratios for the vertically integrated sector. Thus, at 0 r, ( dk / dl ( w/ if and v v only if ( k / l ( k / l. This condition will hold for any articular industry only by a comlete fluke. And it could be satisfied for every industry only if all the k / l were exactly equal! Thus outside the fantasy world of an ( ʻas if one-commodity system, in which relative commodity rices never change as r varies, at least two industries must fail to satisfy the ( dk / dl ( w/ condition at 0 r. Even if their k l relations do haen to be downward-sloing and convex from above, they do not constitute useful caital-labour isoquants. ( 7

8 Concluding Remarks When one considers a long-run equilibrium system of n industries, as defined in ( above, one finds that variation of the interest rate does imly the existence of a k l relation for each industry. Such relations ( cannot (generically be interreted as meaningful caital-labour isoquants, however. They can sloe uwards, they need not be globally convex from above and they need not have an absolute sloe equal to the relevant ʻinut-rice-ratio. Industry-level caital-labour isoquants rovide an entirely surious reresentation of long-run roduction ossibilities and they should therefore be cast out of the economist s toolbox. 8