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L162 Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/alternativetheor401brem FACULTY WORKING PAPERS College of Commerce and Business Administration University of Illinois at Urbana-Champaign May 20, 1977 ALTERNATIVE THEORIES OF PRICING, DISTRIBUTION, SAVING, AND INVESTMENT Hans Br ems #401 ;;.-.i-;!:-| S-:;i(;f •:;• May 17, 1977 ALTERNATIVE THEORIES OF PRICING, DISTRIBUTION, SAVING, AND INVESTMENT By Hans Brems University of Illinois at Urbana-Champaign 61801 100-word summary : In order to identify similarities and dissimilarities between neo- classical and post-Keynesian distribution models, the present paper specifies both models mathematically, in their stripped form, and in the same notation. Four questions are then asked and answered. First, does saving or investment adjust to a higher propensity to save? Second, within that adjustment what are the roles played by the real wage rate and mark-up pricing? Third, within that adjust- ment does a Wicksell Effect emerge? Fourth, are the models examined open or closed ones? In conclusion, the relative analytical appeal of the two models is appraised. V'i; , I'.' ■• •.."t- '.•^Y'V-'-'V-^'-' ' ■■ ;V!.; .'!T:*"iri:^':iyr ■ ■ •?■•:. ,v;-iv.,'Vjr. ;• '-ij.-fs VJ'-.ar'"'vrij -J t;.j. fjri ; -1; il-i rb-i .-oJ- Ti^.b -•(*:/?..•■ >.vs ■'■ "i' ■ ' .-pj li-'off ■ ''lO-o ::rt. . I-* ■ ALTERNATIVE DISTRIBUTION, May 17, 1977 THEORIES OF PRICING, SAVING, AND INVESTMENT By Hans Brems University of Illinois at Urbana-Champaign 61801 "The whole dispute between Keynesian and non-Keynesian theories is whether investment determines savings, or vice versa," Kaldor [3], 301 The purpose of the present paper is to identify rigorously some similarities and dissimilarities between neoclassical [7] and post -Keynesian [2], [3], [5], [6] distribution models. Four interrelated questions will be asked and answered. First, does saving or invest- ment adjust to a higher propensity to save? Second, within that adjust- ment what are the roles played by the real wage rate and mark-up pric- ing? Third, within that adjustment does a Wicksell Effect emerge? Fourth, are tha models examined open or closed and if closed, how? Ansv.'ers will be facilitated by specifying both models mathematic- ally, in their stripped form, and in the same notation: Variables C = consumption g = proportionate rate of growth of variable v = S and X I = investment K = physical marginal productivity of capital stock L = labor employed m = mark-up pricing factor •■-•:^V/■ :ii; Tv^i. ri;ic.;; . -"^iv i:->J y '■• fJ i J A ■;:i>.;^ ■{.:.;■ C^ r*.."*-. ■ : T'"- Vu-p iv J I -J .. - 2 - P = price of good S = physical capital stock W = wage bill X = physical output y = national money income Z = profits bill Parameters a, b = input-output coefficients a, 6 = exponents of Cobb-Douglas production function c = propensity to consume F = available labor force g_ = proportionate rate of growth of parameter F M = multiplicative factor of Cobb-Douglas production function w = money wage rate I. EQUATIONS COMMON TO BOTH MODELS We confine ourselves to the stripped form of either model having Vy-v-y :: >a- = ' -J - . ' .~- *t :-iv! .■/•": 'i\; ; .'■".i. iP"" ■';:i :•; ■■■•:.-, . T ■■. r ' ■■ '^v :..;;; •>~i. •: •.••!:> o.\ - 3 - one good, an immortal capital stock of that good, and no technological progress in it. Four definitions and one equilibrium condition are common to both models. Define the proportionate rate of growth dv 1 (1) g^ = dt V Define investment as the time derivative of capital stock: dS (2) I = — dt Define the wage bill as the money wage rate times employment: (3) W = wL Define national money income as the sum of wage and profits bills (4) Y = W + Z Equilibrium requires output to equal demand for it (5) X = C + I n- :.;i;t., ■| '''•,( '.>r l\ -r. '! ■ ■^{ ■ ■•T/"- , -.?.•=•..' - u - II. EQUATIONS PECULIAR TO THE NEOCLASSICAL MODEL 1 . Production Let entrepreneurs apply a Cobb-Douglas production function (6) X = Ml"s^ where 0 V:' ■' > -^ .; > « t ;•. . I: ....M -J .■ • K. K".: - 5 - Assume full employment: (10) F = L 2 . Distributive Shares Insert (7) into (3) and find the wage bill W = aPX. Insert that and (9) into (4) and find Y = PX, so the distributive shares are W/Y = a and Z/Y = 3. Notice that we derived this familiar result before spec- ifying our consumption function. For that result, then, the form of the consumption function is immaterial and could easily be post-Keynesian : C = c^W/P + CgZ/P But if W = aPX and Z = BPX we may define c = ac^ + 0c_ and write it (11) C = cX where < c < 1, which is the customary neoclassical form. Equipped with a Cobb-Douglas production function with a + B = 1, then, our neoclassical model is rich enough to accommodate a consumption ; f • ... ; ■.' .V ! .1 '- ••■ J> .- V ■) 'i !'■ ■• '■■£<:i '. it V JL .' :, '. .( :■' >!' ■■t,o . n ■} ':■ ,: •!■ ^"i;. ;■; -j;^- :i'' 'i'- ?! I. " nci ■; ^Ji l\c: '■■ I ' '..etJ >,' u ■' •O.'li . •.:ffv./ t i I ri :t '. ,-/ iro i , ..-;)) \ ;;>• <•'. li-i'-! 1 . - 6 - function with different propensities to consume real wages and real profits. But such a consumption function is a nice luxury, not a ne- cessity. Considering marginal productivity a meningful concept, hence capable of deriving distributive shares from it, neoclassicists don't need such a consumption function for that derivation. 3 . Does Saving or Investment Adjust to a Higher Propensity to Save? In the neoclassical model the overall propensity to save 1 - c is a parameter or may be written as one, as we just saw. If that para- meter were twice as high, how would the model adjust? The neoclassical clue lies on the investment side and begins with a typical neoclassical response of factor proportion to relative factor price. Divide (6) by L and (7) by a and set the resulting expressions for X/L equal. Find (12) S/L = (aM)"^/^(w/P)^/^ So a higher real wage rate w/P will induce a higher capital in- tensity S/L, and the elasticity of S/L with respect to w/P is 1/8. Un- der pure competition individual entrepreneurs take the real wage rate ■' h-: < .-. ,. ^s. f. .-•. I- •:jL;:;i/ .-,:..;., ., • ^;^ ;. V f :V J «..' >*- bs-i.fi , -"t:! ; ^ v^ fj j.,£,. c '.■:■■■:■: V--'. . •».l: ru ■ B -!. '■:> '■! f^}\ ■,1.- .-. •. 'n' for granted. But a macroeconomist cannot take it for granted; he must consider it a variable to be solved for before he can see the whole picture. 4. The Real Wage Rate To solve for the real wage rate divide (6) by S, raise both sides to the power -1, and find the capital coefficient (13) S/X = (1/M)(S/L) a Next use (1) and (2) to write I = ggS. insert that and (11) into (5), and find another expression for the capital coefficient S/X = (1 - c)/gg Finally set the right-hand sides of the two expressions for S/X equal, insert the result into (12) and find the real wage rate (!»*) w/P = aM^/"[(l - c)/gg3P^^ .'■' •"' irt •? •■ :;.~!. d c< ■■' • ■ !l <. ■ .^ ) \C. 't; ■; ) ■■>:. i: '.■ ■■ ■ •H a, > .'.Ji V - • 51 (f -^ - 8 - 5 . Closing the Neoclassical Model Eq. (m) is not a solution yet. So far it merely expresses one unknown, the real wage rate w/P, in terms of another, the rate of growth of capital stock g . But neoclass icists do close their models. In the absence of technological progress they find proportionate rates of growth to be converging to the steady-state solutions (15) ^S ~ ^X " ^F Neoclassicists modify their solutions by allowing for technolog- ical progress, ignored here. Insert (15) into (14), and you have a neoclassical solution for the real wage rate w/P. Now we can see the whole picture: According to (14) with (15) inserted, an economy with twice the overall propensity to save will 6/ot have a 2 times higher real wage rate. According to (12) such a 1 /ot real wage rate will induce a 2 times higher capital intensity. According to (13) such a capital intensity means a twice as high capital coefficient. Summing up: The economy with twice the overall propensity to save will have a capital coefficient twice as high. In this sense the give of the neoclassical model lies on the investment side . rj-. ■■ '^ •;■ c, . !" i: ■. . ■ ■? !. -.- : . ; i-, .'.) ■i ■)• .■= c ■• - 9 - 6. The Wicksell Effect Raising the proportion of one factor to another raises the mar- ginal productivity of the latter. In our neoclassical model such rais- ing manifests itself as a Wicksell Effect: According to (lU) the real wage rate w/P is the higher the higher the propensity to save 1 - c or, as Wicksell [8], 164, put it: "The capitalist saver is thus, funda- mentally, the friend of labour." III. EQUATIONS PECULIAR TO POST-KEYNESIAN MODELS 1 . Production We confine ourselves to the simplest form of a post-Keynesian mo- del having fixed input-output coefficients. Two simultaneous equations will then take the place of a neoclassical production function: (16) L = aX (17) S = bX i '■■)!■:.: ■•'•-/ >.r;; ■J 'i ^1 I V .. -T O ,,^ ' - 10 - Two conclusions follow at once. First, there can be no response of factor proportion to anything, relative factor price or otherwise, for according to (16) and (17) the factor proportion S/L = b/a is a parameter. Second, because the system (16) and (17) is a simultaneous one, any variation of X implies simultaneous variation of L and S. Consequently partial derivatives of X with respect to L or S are mean- ingless. Marginal productivity is defined as such a partial derivative. Considering marginal productivity a meaningless concept, hence incap- able of deriving distributive shares from it, post-Keynesians need something else, i. e., different propensities to consume wages and profits , deployed as follows . 2 . Distributive Shares We confine ourselves to the simplest form of a post-Keynes ian model having a propensity to consume wages equalling one. In that case the consumption function is (18) C = W/P + c„Z/P Li With immortal capital stock the entire value of output represents ■:\':- .i ■" { i ., t 's 1 1 i u - 11 value added, i. e., money national income (19) PX = Y Insert (4) into (19), divide by P, and write (20) X = W/P + Z/P Subtract (18) from (20) and insert (5). Use (1) and (2) to write I = g„S, insert (17) into that, divide by X, and use (19) to express the profits share in terms of the capital coefficient, the proportion- ate rate of growth of physical capital stock, and the propensity to save real profits: (21) Z/Y = bgg/(l - c^) 3 . Closing the Post-Keynesian Model Eq. (21) is not a solution yet. So far it merely expresses one unknown, the profits share Z/Y, in terms of another, the rate of growth of capital stock g . How do post-Keynesians close their system? ^-•l^-Ci] ifil c '■ >i-n : •■? .; 'T I ■ I. 'i •• 1- > ;'■'■«■ ;■■■. ■ I .-.t.r-- 'I •^ n> ; .» + . • •■-irr^t; - 12 - At this point Kaldorian and Robinsonian ways are parting. Kaldor does consider the rate of growth of capital stock g_ a va- riable and solves for it by assuming full employment as neoclassic- ists do. Insert (10) into (16), take the derivatives of (16) and (17) with respect to time, use (1), and find that in the absence of tech- nological progress steady-state proportionate rates of growth are (15) 'X " ^F Like neoclassicists , Kaldor modifies his solution by offering a "technical-progress function," ignored here. Insert (15) into (21), and you have a Kaldorian solution for the profits share Z/Y. Joan Robinson doesn't consider the rate of growth of capital stock g- a variable. To her g„ is autonomously given by the "animal spirits" of non-profit-maximizing and otherwise nonrational entrepren- eurs. So (21) is already a Robinsonian solution for the profits share Z/Y. •+ . Does Saving or Investment Adjust to a Higher Propensity to Save? In the post-Keynesian model the overall propensity to save is not ■< >>■ ■•! xc* .A rir !..••• v; V. •; I •••r -. -ay.'/ -f:;-' v:-:;(-jr; .IT i/ ' iS ^^ r. ,-; -, ? - 1 • •■ •.M. J a •-. : n •■■ .■ ■> ■vrtu '^i^' ■ I .-.■■' A, [a Y 7 • -i i i. < . • "• r. (•'*•■ ' ,-i ^ l^ .1 ' : : I ■;• .' -■ '■• ■•■ ".^ vo:^- r - 13 - a parameter but a variable depending on income distribution. The para- metric propensity to save is the propensity to save real profits 1 - c„. If that parameter were twice as high, how would the model adjust? A post-Keynesian clue cannot lie in a response of factor proportion to relative factor price, for according to (16) and (17) the factor pro- portion S/L = b/a is a parameter. The clue must lie elsewhere, and it lies in (21) which we read as follows: If two economies have the same capital coefficient b and are growing at the same proportionate rate g„, but one economy has a propensity to save real profits 1 - c„ twice as high as that of the other economy, then the former economy will have a profits share Z/Y half that of the latter. That allows the overall propensities to save, and with them the capital coefficients S/X, to stay equal and they had better, for the two economies were said to have the same capital coefficient S/X = b. In this sense the give of post-Keynesian models lies on the savings side. 5 . The Real V:a go R"-::o r;-cl :: "■:-r::^ Pricing In Sec. II, 3 above we found a typical neoclassical response of factor proportion to relative factor price. In the post-Keynesian model the factor proportion S/L = b/a is a parameter unresponsive to the real wage rate w/P. But the real wage rate is still there. It V • >.^: '■■, iljy*'..'., -iJii .V <■■ ?:-•.:. :v i^.;-v. <». • I- '- :-iii/ i- ^*.^•. i - 14 - hides behind the price formula that "in modern manufacturing industry ...prices are formed by adding a margin to prime cost" [6], 179. Now in our one-good version of the post-Keynesian model, "prime cost" is labor cost only, and according to (16) per-unit labor cost is aw. Con- sequently the formula for price is P = amw or for the real wage rate (22) w/P = l/(am) where m is the mark-up factor, and m > 1. Mark-up pricing may be a deviation from neoclassical language but not from neoclassical substance. Under neoclassical pure competition, too, there are overhead costs to be covered, and freedom of entry and exit will see to it that they are, so neoclassical price, too, will exceed "prime cost". The proportion in which it does is easily found: Write (7) as (7) P = w L a S or, in English: Price P exceeds per-unit labor cost wL/X in the pro- r. • -■ i'i .' - 15 - portion 1/a. Since we assumed < a < 1, 1/a > 1, as it should. That proportion could well be labelled a "mark-up factor". So far, post -Keynesian substance doesn't seem to deviate from a neoclassical one. But a deviation will emerge once we ask whether the mark-up factor is a parameter or not. The neoclassical one 1/a clearly is. Joan Robinson's less than rigorous style may give the impression that so is the post-Keynesian one m. Could m perhaps be an interesting structural parameter reflecting such things as Kalecki's "degree of monopoly" or at least business convention? It cannot. If it were, Joan Robinson's system would be overdetermined : Divide (^) by Y, in- sert (3), (15), (19), and (22), and find another expression for the profits share: (23) Z/Y = 1 - 1/m Consider our two expressions for the profits share (21) if Kaldorian, with (15) inserted and (23). If m were a parameter those expressions would be two equations in one unknown Z/Y, hence would be an overdetermined system. If m were a variable those expressions would be two equations in the two unknowns Z/Y and m. Set the right -hand sides of (21) and (23) equal and solve for m: ■ >i T^.*- . rr.:f ■- , • -'io : I'd Iv ■■^">^:;-I. : .'-■ I'i V -• ;■! -! -a .•■'■ IJ. ■) .Ji'.*: J; •1 r : .••. f - t I ■ { i • ^ .1 'w :' '•:» (24) 16 m = 1/[1 - bgg/(l - c^)] Joan Robinson's rate of growth of capital stock g was given exo- genously by the "animal spirits" of entrepreneurs. Thus (21) was a Ro- binsonian solution for the profits share Z/Y, and (22) and (24) are ac- companying Robinsonian solutions for the real wage rate w/P and the mark-up factor m, respectively. Entrepreneurs may take their pick: Choosing a lower g. would reduce the profits share (21), would lower the mark-up factor (24) and thereby raise the real wage rate (22). All in one scoop! What would make entrepreneurs choose to do such things? Or refrain from doing them? Since her entrepreneurs are neither profit -maximizing nor otherwise rational, Joan Robinson cannot say. 6. A Wicksell Effect? In a neoclassical world, raising the proportion of one factor to another raises the marginal productivity of the latter. But in post -Keynesian models, as we saw, in the first place the factor proportion S/L = b/a is a parameter, hence cannot vary. In the second place, marginal productivity is a meaningless concept. Under such circumstan- ces we should hardly expect a Wicksell Effect in post-Keynesian ': •: !■■■ ■ n J ii ■■•• ■ rr .' '■11 -f-'^ ! ;«•>. .!- ■ -f ■ ! :'. .-.: '.::. ■■ r- 4 ^y. .n- (:.?'• ■.> k. :s- - 17 - models. But accepting (21), (22), and (24) as equations accompanying one another, we do find a Wicksell Effect just the same: Like a low- er rate of growth g , a higher propensity to save real profits 1 - c would reduce the profits share (21), would lower the mark-up Z factor (21) and thereby raise the real wage rate (22). Again in one scoop. In other words, as in neoclassical models the real wage rate is the higher the higher the propensity to save. Even a Robinsonian capitalist is "fundamentally the friend of labour"! IV. CONCLUSIONS The present paper has confronted neoclassical and post-Keynes- ian distribution models and has tried to answer four questions. First, is Kaldor correct in saying that in neoclassical models savings determine investment whereas in post-Keynesian models invest- ment determines saving? Taken literally, he is wrong: Savings and investment are both variables and as such both determined by the parameters of the system. A correctly asked question would be: How sensitive are they to those parameters? What Kaldor really means is ■,..,i.Vi:: / . .' HI -•J-' ' It'L/W -iP . » o fi I •'. c- \ „ ■- .• - f- . ■•'M • r. n fc •^-..1; N»;".f ;■ ; s:;V«iJ .i". ,iiv,- -'jj? */;;;< - 18 - that neoclassical and post-Keynesian models have very different sensiti- vities to the parametric propensity to save. In the neoclassical model, doubling that propensity was found to double the capital coefficient, and in that sense the give of the neoclassical model lies on the in- vestment side. In the post-Keynesian model, doubling that propensity was found to halve the profits share, and in that sense the give of the post-Keynesian model lies on the savings side. Second, is the post-Keynesian mark-up factor an interesting new structural parameter reflecting "the degree of monopoly" or business convention? Or, less dramatically, is it merely a variable inherent in the savings-investment adjustment? The latter interpretation was found necessary to avoid overdetermination of the post-Keynesian model. Third, does a Wicksell Effect emerge? As expected, it does in the neoclassical model. But accepting the interpretation of the mark-up factor as merely a variable inherent in the savings-investment adjust- ment, we found the post-Keynesian model to have a Wicksell Effect of its own. Fourth, are the models examined open or closed ones? Yes and No! Kaldor's profit share as well as his mark-up factor are determined by a full-employment assumption requiring capital stock, output, and •.,: -I-; I a-'^rTt .h. i '.'.r. V : J x: -'iar \. . . ^ •.'o:> :■/■»■■' M -^ p. :r s :' n ",> '^ '!■ ,;r> ■• ; ;•: r «. '•'ii;? ; r :; I ■! ' r v.;.- r; v .1. ;,' or. v.,'--!".^ •fj ■<=i n'-f n' r" ;!fT ■j '.> n ^ . _ 1. J - •- . ( b.V...,T.- .^ r..v. f; ■PS;'.: - 19 - available labor force to be growing at the same rate (15). The Kal- dorian system may be unnecessarily rigid. The burden of adjustment it imposes upon the distributive shares may be unnecessarily heavy. The distributive shares may be unlikely to be actually carrying that burden. But at least the Kaldorian system is a closed one. By contrast, Joan Robinson's system is an open one. Her profits share as well as her mark-up factor are determined by letting non- profit-maximizing and otherwise nonrational entrepreneurs fix an arbitrary growth rate of capital stock. One fails to see the analytical appeal of post-Keynesian distrib- ution theory. Neoclassical analysis seems more flexible and has less to fear from confrontation with the real world [1]. Perhaps the very openness of Joan Robinson's system appeals to interventionists: Con- trol investment, and you control income distribution! Perhaps the post-Keynesian appeal is ideological rather than analytical, as Krelle and Gabisch 1^1, 203, suggest: Warum aber nun gerade der Widerstand gegen die neoklassische Wachs- tumstheorie, die doch aus wenigen, plausiblen Voraussetzungen sehr viel mehr erklaren kann als die meisten anderen Wachstumstheorien und ' 3 ! :? ">. *V ^'"-'i '• ' ' 1 •,:; .:■>. !.;<• -c: r. j,;f: 'tofc V , :^ n.- , ■r fi 'X r^ - 20 - die Konfrontierung mit der Wirklichkeit auf ihren Abstraktionsgrad nicht zu scheuen braucht? In einer solchen Situation wird man nach ideologischen GrQnden und nicht nach logischen suchen. Sie sind auch nicht schwer zu finden. Der Stein des AnstoBes ist im Grunde die Grenzproduktivitatstheorie der Verteilung. . . Wenn der Lohn gleich dem Grenzprodukt der Arbeit und der Zins gleich dem Grenzprodukt des Kapitals ist, so ist bei Vollbeschaftigung von Arbeit und Kapital die Verteilung des Sozialprodukts sozusagen naturgesetzlich festge- legt (die Produktionsfunktion ist ja so definiert, da3 sie unabhangig von der Gesellschaf tsstruktur ist und allein die technischen und or- ganisatorischen Kenntnisse der Wirtschaftenden wiederspiegelt) . • '. V. r - 21 - REFERENCES Cl] Brems, H., "Reality and Neoclassical Theory," Journal of Economic Literature 15, March 1977, 72-83. [2] Kaldor, N., "A Model of Economic Growth," Economic Journal 67, Dec. 1957, 591-624. [3] Kaldor, N., "Marginal Productivity and Macroeconomic Theories of Distribution," Review of Economic Studies 33 , 1966, 309-19; reprint- ed in G. C. Harcourt and N. F. Laing (eds.). Capital and Growth , Baltimore, Md., 1971, 295-313. [4] Krelle, W., and G. Gabisch, Wachstumstheorie , Berlin, Heidelberg, and New York, 1972. [5] Robinson, J., The Accumulation of Capital , London, 1956. [6] Robinson, J., "Solow on the Rate of Return," Economic Journal 74 , June 1964, 410-417; reprinted in G. C. Harcourt and N. F. Laing (eds.). Capital and Growth , Baltimore, Md., 1971, 168-179. ;: ;; li:. j-'f .!.!■. n'i:. c r.-y~:'r 1 j;- ■ , n-r;: ■ j :> - ;J :.' i ■ . ■ f.j .1 [r- CO - 22 - [7] Solow, R. M., "A Contribution to the Theory of Economic Growth," Quarterly Journal of Economics 70 , Feb. 1956, 65-94-. [8] Wicksell, K. , Forelasningar i nationalekonomi , I , Lund 1901; English translation. 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