profit, surplus product, exploitation and less than maximized utility: a new equivalence proposition...

Profit, Surplus Product, Exploitation and Less than

Maximized Utility: A New Equivalence

Proposition on the Fundamental Marxian Theorem

Tadasu Matsuo

Metroeconomica, 59-3, 2008

The Problem

• The Fundamental Marxian Theorem Okishio(1955)The positive profit exists if and only if there is the exploitation of labor.

From my brain cache

The Problem

• Petri’s (1980) criticism “Profit without exploitation”

Commodity 2

Commodity 1

ly

lR

p

p

RR2

yy2

y1

y : net products per unit labor

R : real wage rate bundle

A New Exploitation Concept

• Matsuo’s(1994) solution

Commodity 2

Commodity 1

y

R

y2

R2

y1

z

Introducing worker’s indifference curve.

Workers could work less to maintain their welfare than they actually do work.

Exploitation!

Washida (1988) Kawaguchi’s (1994) System of Exploitation Theory

• The Profit Warranty Condition

• The Surplus Condition

• Exploitation Defined with Effective Value Vector (EVV)

• Exploitation Defined with Minimized Labor

The 4 conditions below are equivalent.

The Strong System of Exploitation Theory

The Profit Warranty Condition

• There exist production processes that yield a positive profit under any semi-positive price vector.

　　 ¬∃p≥0, p(B−A−Rτ)≦0

This is equivalent with “warranted rate of profit is positive.” (Kawaguchi, 1994)

The Surplus Condition

• There exist semi-positive activity vectors, that yield positive surplus products for all commodity types.

　　　∃x≥0, (B−A−Rτ)x≥0

• The value of the real wage basket for unit labor, evaluated by any semi-positive labor value vector that is calculated from the most labor-productive combinations of the available techniques, is less than unity.

∀t∈{t | t(B−A)≦τ, t ≥ 0, t(B−A)<τ}, 1-tR>0 　　

Exploitation Defined with EVV

Exploitation Defined with Minimized Labor

• The minimum labor necessary to produce the real wage basket for unit labor is less than unity

　　 1>minτx s.t. (B−A)x≧R, x≥0

By Figure

The Surplus Condition

• ∃x≥0, (B−A−Rτ)x>0Commodity 2

Commodity 1

R

y

The Profit Warranty Condition

• 　 ¬∃p≥0, p(B−A−Rτ)≦0

Commodity 2

Commodity 1

R

y

ly

lR

p

p

Before showing the Exploitation Defined with EVV

• We must interpret this type of labor value vector.

t∈{t | t(B−A)≦τ, t ≥ 0, t(B−A)<τ}

Suppose there are many processes.

Commodity 2

Commodity 1

net products per unit labor

Compare A and B.

Commodity 2

Commodity 1

A is inferior to B for the labor productivity.

A

BThus we vanish A.

All these are vanished.

Commodity 2

Commodity 1

These remain.

How about this.

Commodity 2

Commodity 1

Combine these two processes.

Net products on this segment can be produced.

This is inferior, thus vanished

Net Production Possibility Frontier per Unit Labor

Commodity 2

Commodity 1

Net Production Possibility Frontier per Unit Labor

Commodity 2

Commodity 1

If there are sufficiently many processes.

We can approximate this as a curve.

Then we can show this labor value vector as…

Commodity 2

Commodity 1

t∈{t | t(B−A)≦τ, t ≥ 0, t(B−A)<τ}

1/t2

1/t1

• ∀t∈{t | t(B−A)≦τ, t ≥ 0, t(B−A)<τ}, 1-tR>0 　　

Now,Exploitation Defined with EVV

Commodity 2

Commodity 1

1/t2

1/t1

R

tR/t1

tR/t2

Exploitation Defined with Minimized Labor

• 　 1>minτx s.t. (B−A)x≧R, x≥0Commodity 2

Commodity 1

R

y

This equivalence system does not encompass the type of

positive profit expressed in the Petri’s (1980) criticism.

The Surplus Condition does not hold here.

Commodity 2

Commodity 1

R

y

Then

Commodity 2

Commodity 1

R

y

ly

lR

p

p

Then

Commodity 2

Commodity 1

R

y

ly

lR

p

p

Then

Commodity 2

Commodity 1

R

y

ly

lR

p

p

Then

Commodity 2

Commodity 1

R

y

lylR

p

p

There is a case of no profit

Commodity 2

Commodity 1

R

y

lylR

p

p

p2=0

AndCommodity 2

Commodity 1

1/t2

1/t1

R

tR/t1

tR/t2

AndCommodity 2

Commodity 1

1/t2

1/t1

R

tR/t1

tR/t2

AndCommodity 2

Commodity 1

1/t1

R

tR/t1

There is a case of no Exploitation Defined with EVV

Commodity 2

Commodity 1

=1/t1

R

tR/t1

t2=0

And there is no Exploitation Defined with Minimized Labor.

Commodity 2

Commodity 1

R

y

Here I proposed an alternative system

• The Weak Profit Warranty Condition

• The Weak Surplus Condition

• Exploitation Defined with Narrow Effective Value Vector (NEVV)

• Exploitation Defined with Minimized Labor for Equal Utility (MLEU)

The 4 conditions below are equivalent.

The Weak System of Exploitation Theory

The Weak Profit Warranty Condition

• There exist production processes that yield a positive profit under any positive price vector.

　　 ¬∃p>0, p(B−A−Rτ)≦0

The Profit Warranty Condition

• There exist production processes that yield a positive profit under any semi-positive price vector.

　　 ¬∃p≥0, p(B−A−Rτ)≦0

The Weak Profit Warranty Condition

• There exist production processes that yield a positive profit under any positive price vector.

　　 ¬∃p>0, p(B−A−Rτ)≦0

The Weak Profit Warranty Condition is satisfied here.

• 　 ¬∃p>0, p(B−A−Rτ)≦0Commodity 2

Commodity 1

ly

lR

p

p

RR2

yy2

y1

These lines must be sloped

The Weak Surplus Condition

• There exist semi-positive activity vectors, that yield positive surplus products for at least one commodity types.

　　　∃x≥0, (B−A−Rτ)x≥0

The Surplus Condition

• There exist semi-positive activity vectors, that yield positive surplus products for all commodity types.

　　　∃x≥0, (B−A−Rτ)x>0

The Weak Surplus Condition

• There exist semi-positive activity vectors, that yield positive surplus products for at least one commodity types.

　　　∃x≥0, (B−A−Rτ)x≥0

This satisfies the Weak Surplus Condition

Commodity 2

Commodity 1

R

y

Surplus product

• The value of the real wage basket for unit labor, evaluated by any positive labor value vector that is calculated from the most labor-productive combinations of the available techniques, is less than unity.

∀t∈{t | t(B−A)≦τ, t > 0, t(B−A)<τ}, 1-tR>0 　　

Exploitation Defined with NEVV

• The value of the real wage basket for unit labor, evaluated by any semi-positive labor value vector that is calculated from the most labor-productive combinations of the available techniques, is less than unity.

∀t∈{t | t(B−A)≦τ, t ≥ 0, t(B−A)<τ}, 1-tR>0 　　

Exploitation Defined with EVV

• The value of the real wage basket for unit labor, evaluated by any positive labor value vector that is calculated from the most labor-productive combinations of the available techniques, is less than unity.

∀t∈{t | t(B−A)≦τ, t > 0, t(B−A)<τ}, 1-tR>0 　　

Exploitation Defined with NEVV

Exploitation Defined with NEVV is satisfied here.

Commodity 2

Commodity 1

1/t2

1/t1

R

tR/t1

tR/t2

These lines must be sloped

Exploitation Defined with MLEU• The minimum labor necessary to produce the

commodity bundle which does not decrease worker’s utility of any worker’s utility function than real wage basket for unit labor, is less than unity.

∀u∈U　　 1>minτx s.t. (B−A)x≧z, x≥0, u(z)≧u(R)

U is a set of the functions which are strictly increasing and continuous.

Exploitation Defined with MLEU exists here.

• This is Matsuo’s(1994) solution.Commodity 2

Commodity 1

y

R

y2

R2

y1

z

I proved the equivalence between these four conditions.

The Weak Surplus Condition

Exploitation Defined with MLEU

Exploitation Defined with NEVV

The Weak Profit Warranty Condition

And then I showed an another condition equivalent with

these.

Consider the workers’ utility for the working hours.

Assumptions

• A worker’s utility function u is an element of the function set U, which is given as follows:

U={u : Rm+×R1

+ (∋(y, T))→R1 | (i) y1≥y2⇒u(y1, T)>u(y2, T), (ii) u is continuous}

• Each worker’s labor time T has the upper limit T. That is, T≦T.

Definition

• A worker’s maximum utility umax is defined as:umax=max u(y,τx), s.t. (B−A)x≧y, x≥0 y, x

Since 0≦τx≦T and hence y is bounded by the constraint, then u is a continuous function from a compact set to a one-dimension real number. Then, from The Weierstrass Theorem, there is a maximum value of u.

Proposition

• ∃x≥0, (B−A−Rτ)x≥0　　 ⇔ u(RT, T)<umax, for any u∈U

That is; The conditions of the Weak System of the Exploitation Theory are equivalent to the situation in which the utility for any worker from the present real wage bundle and working hours is less than the maximum utility, which could be obtained from free access to all the processes of the entire economy.

A new controversy after submitting the paper.

• Yoshihara’s(2005) criticism against FMT.A case of positive profit without exploitation under different consumption bundles of workers.

Commodity-1

Commodity-2

Ｒ＝ (0.75,0.75)ｙ＝ (1,1)

1.51

1.5

1

Ｒ 2

Ｒ 1

Matsuo’s(2007) refutation.• If 2 workers of both types work 0.75 each,

1.5 of each commodity are produced and both workers can get their reward of 1 labor.

Commodity-1

Commodity-2

Ｒ＝ (0.75,0.75)ｙ＝ (1,1)

1.51

1.5

1

Ｒ 2

Ｒ 1

Exploitation

0.75<1

Then I must change The Weak System of the Exploit

ation Theory to extend to the situation of the

workers’ different consumption bundles.

• Matsuo’s(2007) proof supposes homo-thetic utility functions. → Generalize!

Perhaps the last condition of The Weak System will be;

• for any i, ui(Ci, Ti)<uimax, pCi≦wTi

That is; Non-exploitation situation is that each worker can achieve an optimal reproduction without pre-constraint of the means of production.

A case of non-exploitation

YUA

UB

EA＊

EB＊

LALB

Net products

Labor

Roemer’s non-exploitation is;

Y

P

LZ

UA

UB

Distributing equal means of production represented by point P.

But production at P is not optimal for each worker.

Net products

Labor

Roemer’s non-exploitation is;

Y

E

P

LZ

UA

UB UA′UB′

EAEB

LALB

Π

YA

YB

ΔY

ΔY

ΔLΔL

Net products

Labor

B rents A some amounts of means of production which requires ΔL labor, and gets ΔY interest.

→Better off.

If initial endowments increase,

Y

E

P

LZ

UA′UB′

EAEB

LALB

Π

YA

YB

ΔY

ΔY

ΔLΔL

Net products

Labor

If initial endowments increase,

YE

P

LZ

UA′

UB′

EA

EB

LALB

Π

YA

YB

ΔY

ΔY

ΔLΔL

Net products

Labor

Interest rate decreases.

If initial endowments increase,

YE

P

LZ

UA′

UB′

EA

EB

LALB

Π

YA

YB

ΔY

ΔY

ΔLΔL

Net products

Labor

Interest rate decreases.

Converge to my non-exploitation situation.

Y

P

LZ

UA′

UB′

EA

EB

LALB ΔLΔL

Net products

Labor

To prove this is my goal.

Interest rate converges to zero.