Tadasu Matsuo Metroeconomica , 59-3, 2008

1. Profit, Surplus Product, Exploitation and Less than Maximized Utility: A New Equivalence Proposition on the Fundamental Marxian Theorem Tadasu Matsuo Metroeconomica, 59-3, 2008

2. 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

3. p y2 y p R2 ly R lR y1 The Problem • Petri’s (1980) criticism “Profit without exploitation” Commodity 2 y : net products per unit labor R : real wage rate bundle Commodity 1

4. z A New Exploitation Concept • Matsuo’s(1994) solution Commodity 2 Introducing worker’s indifference curve. y y2 Workers could work less to maintain their welfare than they actually do work. R R2 Exploitation! Commodity 1 y1

5. Washida (1988) Kawaguchi’s (1994) System of Exploitation Theory The 4 conditions below are equivalent. • The Profit Warranty Condition • The Surplus Condition • Exploitation Defined with Effective Value Vector (EVV) • Exploitation Defined with Minimized Labor The Strong System of Exploitation Theory

6. 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)

7. 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

8. Exploitation Defined with EVV • 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

9. Exploitation Defined with Minimized Labor • The minimum labor necessary to produce the real wage basket for unit labor is less than unity1>minτx s.t. (B−A)x≧R, x≥0

10. By Figure

11. The Surplus Condition • ∃x≥0, (B−A−Rτ)x>0 Commodity 2 y R Commodity 1

12. The Profit Warranty Condition • ¬∃p≥0, p(B−A−Rτ)≦0 Commodity 2 p y p ly R lR Commodity 1

13. 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}

14. Suppose there are many processes. Commodity 2 net products per unit labor Commodity 1

15. A Compare A and B. Commodity 2 A is inferior to B for the labor productivity. Thus we vanish A. B Commodity 1

16. All these are vanished. Commodity 2 These remain. Commodity 1

17. Combine these two processes. Net products on this segment can be produced. How about this. Commodity 2 This is inferior, thus vanished Commodity 1

18. Net Production Possibility Frontier per Unit Labor Commodity 2 Commodity 1

19. Net Production Possibility Frontier per Unit Labor If there are sufficiently many processes. Commodity 2 We can approximate this as a curve. Commodity 1

20. Then we can show this labor value vector as… t∈{t| t(B−A)≦τ, t ≥ 0, t(B−A)<τ} Commodity 2 1/t2 Commodity 1 1/t1

21. Now,Exploitation Defined with EVV • ∀t∈{t| t(B−A)≦τ, t ≥ 0, t(B−A)<τ}, 1-tR>0 Commodity 2 1/t2 tR/t2 R Commodity 1 tR/t1 1/t1

22. Exploitation Defined with Minimized Labor • 1>minτx s.t. (B−A)x≧R, x≥0 Commodity 2 y R Commodity 1

23. This equivalence system does not encompass the type of positive profit expressed in the Petri’s (1980) criticism.

24. The Surplus Condition does not hold here. Commodity 2 y R Commodity 1

25. p p ly lR Then Commodity 2 y R Commodity 1

26. Then Commodity 2 p y p R ly lR Commodity 1

27. Then Commodity 2 p y p R ly lR Commodity 1

28. Then Commodity 2 p y p R ly lR Commodity 1

29. There is a case of no profit Commodity 2 p p2=0 y p R ly lR Commodity 1

30. And Commodity 2 1/t2 tR/t2 R Commodity 1 tR/t1 1/t1

31. And Commodity 2 1/t2 tR/t2 R Commodity 1 tR/t1 1/t1

32. And Commodity 2 R Commodity 1 tR/t1 1/t1

33. There is a case of no Exploitation Defined with EVV Commodity 2 t2=0 R Commodity 1 tR/t1 =1/t1

34. And there is no Exploitation Defined with Minimized Labor. Commodity 2 y R Commodity 1

35. Here I proposed an alternative system The 4 conditions below are equivalent. • 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 Weak System of Exploitation Theory

36. 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

37. 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

38. 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

39. p y2 y p R2 ly R lR y1 The Weak Profit Warranty Condition is satisfied here. • ¬∃p>0, p(B−A−Rτ)≦0 Commodity 2 These lines must be sloped Commodity 1

40. 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

41. 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

42. 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

43. Surplus product This satisfies the Weak Surplus Condition Commodity 2 y R Commodity 1

44. Exploitation Defined with NEVV • 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

45. Exploitation Defined with EVV • 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

46. Exploitation Defined with NEVV • 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

47. Exploitation Defined with NEVV is satisfied here. These lines must be sloped Commodity 2 1/t2 tR/t2 R Commodity 1 tR/t1 1/t1

48. 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∈U1>minτx s.t. (B−A)x≧z, x≥0, u(z)≧u(R)Uis a set of the functions which are strictly increasing and continuous.

49. z Exploitation Defined with MLEU exists here. • This is Matsuo’s(1994) solution. Commodity 2 y y2 R R2 Commodity 1 y1

50. I proved the equivalence between these four conditions. The Weak Profit Warranty Condition The Weak Surplus Condition Exploitation Defined with MLEU Exploitation Defined with NEVV