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Sraffa numeraire and reswitching: A note A note
SYED AHMAD McMaster Hamilton,
University Ontario Canada
Sraifa Numeraire and Reswitching: A Note* The paper demonstrates that changing the numeraire from Hicks’ to Sraffa’s does not affect the location and, a fortiori, the number of switch points between two given techniques. This negates some recent claims to the contrary, and supports the earlier general position that the choice of numeraire has no bearing on reswitching, even though it does have a bearing on capital reversal. In the process the paper shows that the rate of growth of the economy and the marginal propensity to save also do not affect the switch points, which may thus be regarded as depending essentially on technical factors alone.
In a recent issue of this Journal (1982), Yi has attempted to show that, given different technical possibilities of producing a consumption good and a capital good, it is possible that we get reswitching if we use the SraEa numeraire when there is no reswitching if we use the Hicks numeraire. SrafFa numeraire equals net output per person (1960); Hicks numeraire equals a unit of the consumption good (1965). This conclusion is not valid and, as we shall see below, has resulted from Yi’s neglect of a basic index number principle in the use of S&a’s numeraire. Since Hicks numeraire consists of one and the same good, its use is not subject to the above criticism. For establishing his proposition, Yi derives the w-r curve of each system’ by using the Sraffa numeraire of that system. He uses the following set of equations for System A’.
PO,= atowa+ ral,Pl,
(14
PI, = aelwa + rallph *Thanks are due to an important point. The author in the note. ‘A “system” is described by a given technique, and ‘The symbols used are has been added, wherever
Journal Copyright
of 0
Macroeconomics, 1986 by Wayne
anonymous referee alone is, of course,
for his suggestion for clarifying an responsible for the views expressed
by a set of input coefficients of the outputs produced the relevant propensities to save. the same as in Yi’s paper except that the subscript ‘a’ necessary, to specify the symbols for System A.
Summer 1986, State University
Vol.
8, No. Press.
3, pp.
381-385
381
Syed Ahmad
1 = atlg,T, + atoc, T, = allgJa
+ al,c,
Taplag, + pooc, = 1
(34 (44 (54
where p,, and p,, are prices of consumption good and capital good respectively, at, and al, are the labour coefficient and capital coefficient of the consumption good sector, and atI and alI of the capital good sector. g, is the growth rate of the capital stock, T, is the (physical) capital-labour ratio, w, the wage rate measured in the units of the chosen numeraire, r the rate of profit, s, is the marginal propensity to save, c, per capita consumption, and an expression to be used later, m,, the relative capital intensity of consumers’ good as compared to capital good: al,ael - allapo. We can first obtain c, and T, from (3a) and (4a). After substituting these in (5a), we can use (la) and (2a) with the new (5a) to obtain w,, and p,,. Thus, Yi obtains:
_ (1- allr)kb+ mad wa- ato+ m,k + f-- allf-g,) ’
(74
and
Pml
= (at, + w-)(aeo+ magal ato + ma (g, + r - all f-g,) .
Note that System A is fully specified by coefficients a’s and s,. Following the same procedure for System B which is analogously specified by b’s and sb, one can get wb and p,b as the counterparts of (7a) and (8a) simply by replacing b’s for a’s and calling them (7b) and (8b) respectively. With the help of (7a) and (7b), Yi goes on to derive the conditions for the equality of the w’s of the two systems corresponding to any given rate of profit, and claims that when these conditions are fulfilled, a switch occurs. Yi’s neglect of the index number principle consists in his weighing the quantities of System A with the prices of A and the quantities of System B with the prices of B, when he obtains their respective numeraires. With different prices as weights, the nu382
Sr-affa Numeraire
and Reswitching:
A Note
meraires are, in general, not comparable. Hence anything measured in terms of one of the numeraires cannot be compared with something measured in terms of the other. In particular, the equality of wages measured in these different units cannot have any significance; it does not represent a switch point. That the prices in the two systems are, in fact, different, can be seen by comparing @a) and (8b); as long as the input coefficients of any one good are different between the two systems, (8a) and (8b) will not be equal. This is true even if the same two goods are produced by the two systems, as they are in Yi’s case. The above procedure of obtaining the numeraires, and comparing values measured in them is similar to obtaining the so-called ‘value index’, which has no economic meaning. For obtaining Laspeyere or Paasche, or any other meaningful quantity index, the prices of one period or situation are used as weights with the quantities of all the periods or situations to be compared. SraEa himself was fully aware of this problem and suggested (1960), without getting into the details of derivation, that for the purposes of comparison between any two systems, the numeraire of any one system be used for both of them3; just as in direct quantity comparison, the prices of one period are used as weights for both the periods to be compared. Following SraEa, let us use the numeraire derived from A for both the Systems A and B. The wr curve of System A can still be represented by (7a); nothing has happened for it to change. For obtaining the w-r curve for System B by using the Sraifa numeraire of System A, however, we shall have to substitute the value of p,, from (8a) for pob in (lb); since the same consumption good valued by the same numeraire will have the same price. This modified (lb) together with (2b) yields the required w-r curve:
(1 - ~~lr)(a~,+ marXat + magJ Wba+ (be0+ mbr)[a&,+ %k + r - allr&)l ’ For obtaining the switch point(s), if any, between tems, we equate (7a) with (7ba). This yields
3To be precise, k&a’s interest here is in the Standard Commodity meraire, but as this is only a special case of Sraffa’s numeraire, the mains unchanged.
VW
the two sys-
as the argument
nure-
383
Syed Ahmad 1 - allr
1 - b,,r
a,+m,r=b,,+mbt-’
(9)
That the choice of numeraire has no effect on the switch points becomes obvious when we find that Hicks’s numeraire used with (la) and (2a) of System A and (lb) and (2b) of System B yields exactly the same condition, equation (9), for switching. If we now write a,, = b,, and ael = bel, the case Yi considers through his numerical example, (9) gives only one value for r:
1-= -(at0 - ho,) ma- mb ’
(10)
Thus there is no reswitching if the capital goods sectors in the two systems have the same input coefficients. This result is, of course, also independent of the choice of the numeraire, contrary to Yi’s conclusion, since (9) is independent of the numeraire. The above is simply an illustration of the general principle already noted in the literature that the switch points remain unchanged with the change of numeraire. An important corollary of the above argument is that since (9) does not contain g or s, switch points are also independent of the rate of growth, or the propensity to save. Finally, the above conclusions do not invalidate Yi’s and Akyuz’s (1972) claim that the choice of the numeraire may not be neutral regarding capital reversal within a technique since no comparison between techniques is required for this result. Received: July Final version:
1984 February
1986
References Akyuz, A. “Income Distribution, Value of Capital, and Two Notions of the Wage-Profit Trade-Off.” Oxford Economic Papers 24 (July 1972): 156-165. Hahn, F. “The Neo-Ricardians.” Cambridge Economic Journal 6 (December 1982): 353-74. Hicks, J.R. Capital and Growth. Oxford: Clarendon Press, 1965. Pasinetti, L. Lectures in the Theory of Production. New York: Columbia University Press, 1977. 384
Sr-affa Numeraire
and Reswitching:
A Note
SraEa, P. Production of Commodities by Means of Commodities. Cambridge: Cambridge University Press, 1960. Yi, G. J.L. “The Choice of Numeraire, Capital Intensity Uniqueness, and the Reswitching of Techniques.” Journal of Macroeconomics 4 (Winter, 1982): 89-100.
385
University Ontario Canada
Sraifa Numeraire and Reswitching: A Note* The paper demonstrates that changing the numeraire from Hicks’ to Sraffa’s does not affect the location and, a fortiori, the number of switch points between two given techniques. This negates some recent claims to the contrary, and supports the earlier general position that the choice of numeraire has no bearing on reswitching, even though it does have a bearing on capital reversal. In the process the paper shows that the rate of growth of the economy and the marginal propensity to save also do not affect the switch points, which may thus be regarded as depending essentially on technical factors alone.
In a recent issue of this Journal (1982), Yi has attempted to show that, given different technical possibilities of producing a consumption good and a capital good, it is possible that we get reswitching if we use the SraEa numeraire when there is no reswitching if we use the Hicks numeraire. SrafFa numeraire equals net output per person (1960); Hicks numeraire equals a unit of the consumption good (1965). This conclusion is not valid and, as we shall see below, has resulted from Yi’s neglect of a basic index number principle in the use of S&a’s numeraire. Since Hicks numeraire consists of one and the same good, its use is not subject to the above criticism. For establishing his proposition, Yi derives the w-r curve of each system’ by using the Sraffa numeraire of that system. He uses the following set of equations for System A’.
PO,= atowa+ ral,Pl,
(14
PI, = aelwa + rallph *Thanks are due to an important point. The author in the note. ‘A “system” is described by a given technique, and ‘The symbols used are has been added, wherever
Journal Copyright
of 0
Macroeconomics, 1986 by Wayne
anonymous referee alone is, of course,
for his suggestion for clarifying an responsible for the views expressed
by a set of input coefficients of the outputs produced the relevant propensities to save. the same as in Yi’s paper except that the subscript ‘a’ necessary, to specify the symbols for System A.
Summer 1986, State University
Vol.
8, No. Press.
3, pp.
381-385
381
Syed Ahmad
1 = atlg,T, + atoc, T, = allgJa
+ al,c,
Taplag, + pooc, = 1
(34 (44 (54
where p,, and p,, are prices of consumption good and capital good respectively, at, and al, are the labour coefficient and capital coefficient of the consumption good sector, and atI and alI of the capital good sector. g, is the growth rate of the capital stock, T, is the (physical) capital-labour ratio, w, the wage rate measured in the units of the chosen numeraire, r the rate of profit, s, is the marginal propensity to save, c, per capita consumption, and an expression to be used later, m,, the relative capital intensity of consumers’ good as compared to capital good: al,ael - allapo. We can first obtain c, and T, from (3a) and (4a). After substituting these in (5a), we can use (la) and (2a) with the new (5a) to obtain w,, and p,,. Thus, Yi obtains:
_ (1- allr)kb+ mad wa- ato+ m,k + f-- allf-g,) ’
(74
and
Pml
= (at, + w-)(aeo+ magal ato + ma (g, + r - all f-g,) .
Note that System A is fully specified by coefficients a’s and s,. Following the same procedure for System B which is analogously specified by b’s and sb, one can get wb and p,b as the counterparts of (7a) and (8a) simply by replacing b’s for a’s and calling them (7b) and (8b) respectively. With the help of (7a) and (7b), Yi goes on to derive the conditions for the equality of the w’s of the two systems corresponding to any given rate of profit, and claims that when these conditions are fulfilled, a switch occurs. Yi’s neglect of the index number principle consists in his weighing the quantities of System A with the prices of A and the quantities of System B with the prices of B, when he obtains their respective numeraires. With different prices as weights, the nu382
Sr-affa Numeraire
and Reswitching:
A Note
meraires are, in general, not comparable. Hence anything measured in terms of one of the numeraires cannot be compared with something measured in terms of the other. In particular, the equality of wages measured in these different units cannot have any significance; it does not represent a switch point. That the prices in the two systems are, in fact, different, can be seen by comparing @a) and (8b); as long as the input coefficients of any one good are different between the two systems, (8a) and (8b) will not be equal. This is true even if the same two goods are produced by the two systems, as they are in Yi’s case. The above procedure of obtaining the numeraires, and comparing values measured in them is similar to obtaining the so-called ‘value index’, which has no economic meaning. For obtaining Laspeyere or Paasche, or any other meaningful quantity index, the prices of one period or situation are used as weights with the quantities of all the periods or situations to be compared. SraEa himself was fully aware of this problem and suggested (1960), without getting into the details of derivation, that for the purposes of comparison between any two systems, the numeraire of any one system be used for both of them3; just as in direct quantity comparison, the prices of one period are used as weights for both the periods to be compared. Following SraEa, let us use the numeraire derived from A for both the Systems A and B. The wr curve of System A can still be represented by (7a); nothing has happened for it to change. For obtaining the w-r curve for System B by using the Sraifa numeraire of System A, however, we shall have to substitute the value of p,, from (8a) for pob in (lb); since the same consumption good valued by the same numeraire will have the same price. This modified (lb) together with (2b) yields the required w-r curve:
(1 - ~~lr)(a~,+ marXat + magJ Wba+ (be0+ mbr)[a&,+ %k + r - allr&)l ’ For obtaining the switch point(s), if any, between tems, we equate (7a) with (7ba). This yields
3To be precise, k&a’s interest here is in the Standard Commodity meraire, but as this is only a special case of Sraffa’s numeraire, the mains unchanged.
VW
the two sys-
as the argument
nure-
383
Syed Ahmad 1 - allr
1 - b,,r
a,+m,r=b,,+mbt-’
(9)
That the choice of numeraire has no effect on the switch points becomes obvious when we find that Hicks’s numeraire used with (la) and (2a) of System A and (lb) and (2b) of System B yields exactly the same condition, equation (9), for switching. If we now write a,, = b,, and ael = bel, the case Yi considers through his numerical example, (9) gives only one value for r:
1-= -(at0 - ho,) ma- mb ’
(10)
Thus there is no reswitching if the capital goods sectors in the two systems have the same input coefficients. This result is, of course, also independent of the choice of the numeraire, contrary to Yi’s conclusion, since (9) is independent of the numeraire. The above is simply an illustration of the general principle already noted in the literature that the switch points remain unchanged with the change of numeraire. An important corollary of the above argument is that since (9) does not contain g or s, switch points are also independent of the rate of growth, or the propensity to save. Finally, the above conclusions do not invalidate Yi’s and Akyuz’s (1972) claim that the choice of the numeraire may not be neutral regarding capital reversal within a technique since no comparison between techniques is required for this result. Received: July Final version:
1984 February
1986
References Akyuz, A. “Income Distribution, Value of Capital, and Two Notions of the Wage-Profit Trade-Off.” Oxford Economic Papers 24 (July 1972): 156-165. Hahn, F. “The Neo-Ricardians.” Cambridge Economic Journal 6 (December 1982): 353-74. Hicks, J.R. Capital and Growth. Oxford: Clarendon Press, 1965. Pasinetti, L. Lectures in the Theory of Production. New York: Columbia University Press, 1977. 384
Sr-affa Numeraire
and Reswitching:
A Note
SraEa, P. Production of Commodities by Means of Commodities. Cambridge: Cambridge University Press, 1960. Yi, G. J.L. “The Choice of Numeraire, Capital Intensity Uniqueness, and the Reswitching of Techniques.” Journal of Macroeconomics 4 (Winter, 1982): 89-100.
385