okishio theorem - kliman

8/13/2019 Okishio Theorem - Kliman

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THE OKISHIO THEOREM: AN OBITUARY

Andrew Kliman

2nd draft. Please obtain author's permission before quoting.

I. INTRODUCTION

Karl Marx (19!"#$% &onsidered the law of the tendential fall in the profit rate ()P*% to be +in e,er-respe&t the most important law in modern politi&al e&onom- and the most essential for understanding the

most diffi&ult relations.+ /owe,er the 0ishio theorem (0ishio 191% is generall- thought to ha,e refuted

this law on logi&al grounds in a manner +so de,astating that it depri,es all arguments (pro and &ontra% ... of

their rele,an&e+ (Pari3s 19$4"9%. 5n other words it is &ommonl- a&&epted that 0ishio's and subsequent

,ersions of the theorem ha,e shown it to be impossible for profit6maximi7ing &apitalists to adopt labor6sa,ing inno,ations that &ause the equilibrium (uniform% rate of profit to fall.

Mu&h of the subsequent literature has shown not surprisingl- that b- altering one of the theorem'spremises one &an deri,e a fall in the profit rate. 5f the theorem were true this would indeed be the onl-

wa- to deri,e a falling profit rate. And be&ause the theorem is widel- a&&epted as true the &ounter6

demonstrations put forward b- proponents of the +temporal single6s-stem+ interpretation of Marx's ,alue

theor- (8rnst 19$2 Kliman 19$$ 199 )reeman 199% are 6 when the- are a&nowledged at all 6

&ommonl- regarded as alternati,e +models+ whi&h lie the rest of the literature alter some premise of the

theorem in order to arri,e at different results.1he tas of the present paper is to show that on the &ontrar-

61 the 0ishio theorem has not been pro,en

62 the :: &ounter6demonstrations are genuine refutations negating the &on&lusions of the theoremwithout altering an- of its stated premises and

6! when the theorem's a&tual results are stated in a ,alid manner the- do no damage to the logi& of Marx's)P*.

M- &laim it should be noted is not that the theorem suffers from a mathemati&al error. he rele,antmatrix algebra theorem of Perron and )robenius upon whi&h the 0ishio theorem is based is true. *ather

the problem is a dis3un&tion between the a&tual impli&ation of the mathemati&s and the e&onomi&

&on&lusions whi&h proponents of the 0ishio theorem draw from it namel- that if a uniform rate of profit

is re6established after a &ost6redu&ing te&hni&al &hange this profit rate &annot be lower than the initial one.

o see this it will help first to re,iew the theorem briefl-.

II. THE OKISHIO THEOREM

;e&ause 5 ha,e &laimed that the :: &ritiques negate the &on&lusion of the 0ishio theorem without

altering an- of its premises it is important to identif- whi&h features of the theorem are premises and whi&hare &on&lusions.


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and the final premise is that

611 the profit rate is re6equali7ed after the adoption of su&h a new te&hnique.

)rom these eight premises proponents of the 0ishio theorem use the Perron6)robenius theorem to draw

the following &on&lusion" the post6inno,ation profit rate &annot be lower than the initial one and must begreater than the initial one if the te&hni&al inno,ation o&&urs in a +basi&+ industr-.

III. THE MISSING PROOF

et us use r to denote the initial profit rate and r@ to denote that profit rate whi&h proponents of the

theorem &all the post6inno,ation profit rate. 5t is indeed true that r@ &annot be lower than r and that it must

be greater than r if the te&hni&al &hange has o&&urred in a +basi&+ industr-. his mu&h is implied b- the

Perron6)robenius theorem. et it is insuffi&ient to support the &on&lusion abo,e. /ow do we now that r@

will be the e&onom-'s a&tual post6inno,ation uniform profit rateB

he t-pi&al answer to this question and the one whi&h presumabl- underlies the belief that the 0ishio

theorem has been pro,en is that te&hni&al and real wage &oeffi&ients together with the stipulation that the

profit rate is equali7ed a&ross se&tors suffi&e to determine a unique rate of return on &apital ad,an&ed.hus r@ is repeatedl- referred to as +the general rate of profit+ b- 0ishio (191" 94 9! 99 emphasis

added% and as +the equilibrium+ profit rate b- *oemer (19$"1=! 1=# emphasis added% impl-ing that ifprofit rates are equali7ed a&ross se&tors the le,el of the uniform profit rate is uniquel- determined b-

ph-si&al &oeffi&ients. And sin&e premise (,iii% stipulates that profit rates are indeed equali7ed after the

te&hni&al inno,ation then supposedl- r@ must be the post6inno,ation profit rate.et &ontrar- to the apparent belief of the proponents of the 0ishio theorem (and most readers of

Cmitrie, and :raffa% it is simpl- not true that the magnitude of the uniform profit rate is uniquel-

determined b- ph-si&al inputDoutput &oeffi&ients. o demonstrate this &ru&ial point let us turn to the

example in whi&h :raffa (194"% supposedl- pro,es that the uniform profit rate is uniquel- determined. /e

posits the following inputDoutput relations"

2$4qr. wheat E 12 t. iron 6F == qr. wheat

124qr. wheat E $ t. iron 6F 24 t. iron

and immediatel- &on&ludes"

he ex&hange6ratio whi&h enables the ad,an&es >on the left6hand sides? to be repla&ed and the profits to

be distributed to both industries in proportion to their ad,an&es >so that the rate of return on &apital

ad,an&ed is equali7ed? is 1= qr. of wheat for 1 t. of iron and the &orresponding rate of profits in ea&h

industr- is 2=G.

o obtain 1="1 as the unique ex&hange ratio :raffa impli&itl- assumes that the ex&hange ratio between

wheat and iron at the time of input is equal to the ex&hange ratio at the time of output. his will not

ne&essaril- be the &ase but e,en if it is the uniform profit rate is not uniquel- determined b- the data

abo,e as we shall see presentl-.

etting Pwt and Pit be the input pri&es and PwtE1 and PitE1 be the output pri&es of wheat and iron

respe&ti,el- :raffa's ex&hange ratio implies that Pit H 1= Pwt and that PitE1 H 1= PwtE1. he rate of return

on &apital ad,an&ed in the wheat se&tor is therefore

(== PwtE1%D(2$4 Pwt E 12 Pit% 6 1 H (== PwtE1%D(2$4 Pwt E 1$4 Pwt% 6 1

H 1.2=(PwtE1DPwt% 6 1

and the rate of return on &apital ad,an&ed in the iron se&tor is

(24 PitE1%D(124 Pwt E $ Pit% 6 1 H (!44 PwtE1%D(124 Pwt E 124 Pwt% 6 1 H 1.2=(PwtE1DPwt% 6 1.

he &onstant ex&hange ratio of 1="1 thus guarantees that the two rates of profit are equal. et the le,el of

the equilibrium profit rate is still not determined. 5t depends as well on the ratio of the output to the input

pri&e of wheat and it &an ,ar- theoreti&all- from 6144G to E (%(%

/en&e if we imagine this to be the post6inno,ation profit rate it &an in prin&iple alwa-s be lower than the

pre6inno,ation rate if PwtE1DPwt is suffi&ientl- small. 5f for instan&e the pre6inno,ation rate was 24G the

post6inno,ation rate will be lower if PwtE1DPwt I 4.9.

5n general then the magnitude of profit rates e,en the magnitude of profit rates that are &onstrained to be

equal depend on the relationship between output and input pri&es. he 0ishio theorem's r@ is therefore

simpl- one parti&ular post6inno,ation profit rate among an infinite number of possible uniform rates a rate

that &orresponds to one parti&ular set of input and output pri&es. :pe&ifi&all- r@ is that parti&ular uniform


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profit rate whi&h &orresponds to the &ase in whi&h post6inno,ation input and output pri&es are equal.


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+pri&e+ rates of profit are equal ,alues &annot be negati,e and profits &an be positi,e onl- if surplus6,alue

is positi,e the rate of profit in luxur- industries affe&ts the general rate and what is most rele,ant to the

present paper labor6sa,ing inno,ation itself &an lead to a lower general (uniform or nonuniform% rate of

profit (see )reeman and Lar&hedi eds. 199%.

he :: refutations of the 0ishio theorem mae use of the fa&t emphasi7ed abo,e that e,en if a

uniform rate of profit is postulated the le,el of the profit rate is not determined b- ph-si&al &oeffi&ients

alone it depends also on the relation between input and output pri&es. 0nl- when input and output pri&eshappen to be equal to ph-si&al &oeffi&ients suffi&e to determine the le,el of the profit rate otherwise one or

another ,alue theor- is needed. he :: refutations of the 0ishio theorem emplo- Marx's ,alue theor- for

this purpose.

:pe&ifi&all- the rate of profit is &omputed in the following wa-. ;oth input and output pri&es are

measured in labor6time per unit of output or equi,alentl- in mone- per unit of output if the monetar-

expression of ,alue is assumed to be &onstant o,er time. 5nput pri&es are taen as gi,en (determined in the

pre,ious period or stipulated as initial &onditions in the first period% and the- together with the means of

produ&tion and subsisten&e ad,an&ed determine the ,alue of the &onstant (L% and ,ariable (% &apital

ad,an&ed.


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hese assumptions satisf- premises (i%6(iii%. Premise (i,% is satisfied b- assuming that the real wage per

unit of li,ing labor is a &onstant 4.2$9! qrs. of wheat. 5nitial input pri&es of 2.!42 and 4.$29 for iron and

wheat respe&ti,el- ensure that initiall- absolute pri&es are stationar- and profit rates are equal satisf-ing

premises (,% and (,i%.


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/en&e no firm is te&hnologi&all- ba&ward or te&hnologi&all- ad,an&ed relati,e to the others in the

industr-. his is true not onl- of existing firms but of new entrants. An- new entrant into either se&tor in

an- period emplo-s the exa&t same te&hniques as the existing firms sin&e better te&hniques are not -et

a,ailable.

herefore super6profit (surplus profit% does not exist here. hat is no firm e,en new entrants &an obtain

a higher rate of return than the a,erage b- ha,ing produ&ed at a lower &ost per unit. he- ha,e all

in&luding new entrants produ&ed at the exa&t same &ost per unit and therefore the- all in&luding newentrants obtain the exa&t same rate of return as the others. /en&e the fall in the a&tual profit rate despite a

&onstant real wage and a &ontinuing series of 0ishio6,iable te&hni&al &hanges simpl- cannotbe attributed

to the existen&e of te&hnologi&all- ba&ward firms. he a&tual rate of profit is not the rate of profit obtained

b- the te&hnologi&all- ba&ward firms onl-.


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)P* and to anal-7e &urrent realities on its basis will wrongl- &ontinue to be regarded as ignorant and as

dogmati& apologists.

Jnfortunatel- nearl- fifteen -ears ha,e elapsed sin&e the publi&ation of the first :: refutation of the

theorem (8rnst 19$2% and the re&ord has -et to be &orre&ted.

NOTES

1. his interpretation whi&h will be dis&ussed below is also nown as +sequential non6dualist.+ he term +temporal single6s-stem+

was first used in :illman (199=%. *e&entl- *amos (199% has also &hallenged the 0ishio theorem from a :: perspe&ti,e fo&usingon the problem of &hanges in the mone-Dlabor6time relation brought about b- te&hni&al &hange.

2. *oemer (19$1% generali7ed 0ishio's theorem to in&lude fixed &apital. )or simpli&it- 5 will abstra&t from fixed &apital below

although the ensuing dis&ussion applies equall- to the fixed &apital ,ersion of the theorem. he refutations of the theorem b- Kliman

(19$$ 199% and )reeman (199% appl- to both the &ir&ulating &apital and fixed &apital &ases.

!. Apparentl- the rationale for this restri&tion is a passage in Lapital 555 in whi&h Marx dis&usses wh- a profit6maximi7ing &apitalist

would ,oluntaril- adopt inno,ations that lower the general profit rate i.e. pre&isel- the argument that the 0ishio theorem re6

examines. Marx (19$1"!!6#% argues that the inno,ating &apitalist at first reaps superprofits (and thus raises hisDher own profit rate%

but that +&ompetition maes the new pro&edure uni,ersal and sub3e&ts it to the general law >of ,alue?. A fall in the profit rate then

ensues 6 firstl- perhaps in this sphere of produ&tion and subsequentl- equali7ed with the rest 6 a fall that is &ompletel- independent ofthe &apitalists' will.+ Oi,en that the theorem is intended as a disproof of Marx's )P* and not a &laim &on&erning a&tual e&onomies

it is reasonable for its s&ope to be restri&ted to the &ase of uniform profitabilit-. Marx's referen&e here to the equali7ation of the rate of

profit howe,er is best understood as a referen&e to a tenden&- not a stati& out&ome.

#. he simulation models that in,estigate the d-nami& stabilit- of the :raffian model do not address this issue at all sin&e the- worwith relati,e pri&es onl-. o guarantee that pri&es will be stationar- throughout all time neo&lassi&al intertemporal equilibrium models

need to assume not onl- perfe&t mobilit- of &apital but also perfe&t nowledge of the future perfe&t futures' marets for all

&ommodities and no time preferen&es. :u&h postulates are of &ourse absent from Marx's )P* and from all ,ersions of the 0ishiotheorem.

=. 5n addition to the :: wors alread- &ited see for instan&e Oiussani (1991692% Maldonado )ilho (199=% and the &ontributions b-

Lar&hedi de /aan )reeman and M&Olone and Kliman in )reeman and Lar&hedi eds. (199%.

. he importan&e of the s-stemati& di,ergen&e between these two profit rates and the &on&omitant failure of the a&tual input andoutput pri&es to &on,erge &annot be o,eremphasi7ed. 5t is &ommonl- belie,ed that the ,alue pri&e and profit magnitudes of the ::

interpretation differ from those deri,ed from simultaneous equation models onl- during a +disequilibrium+ d-nami& ad3ustment

pro&ess so that the +true+ magnitudes are those of the simultaneous equation models whi&h a&t as +&enters of gra,it-+ to whi&h the

a&tual magnitudes &on,erge or around whi&h the- flu&tuate. *esults su&h as those abo,e show the utter falsit- of this belief.

. hese ,iews ha,e been put forward on the 0utline of Politi&al 8&onom- ist (0P86% a &losed e6mail dis&ussion list.

REFERENCES

Cumnil O. and C. i,-. 199!. he 8&onomi&s of the Profit *ate" Lompetition Lrises and /istori&al enden&ies in Lapitalism.(Aldershot JK" 8dward 8lgar%.

8rnst .*. 19$2. :imultaneous aluation 8xtirpated" a &ontribution to the &ritique of the neo6*i&ardian &on&ept of ,alue *e,iew of

*adi&al Politi&al 8&onomi&s 1#"2.)reeman A. 199. Pri&e alue and Profit66a &ontinuous general treatment in A. )reeman and O. Lar&hedi (eds.% Marx and

okishio theorem - kliman

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