The Encyclopedia of Economic Principals

Chapter 97

Marxian, Sraffian, and Class-Based Distribution

Surplus, frontiers and prices of production in small value accounts.

Four of the chapter's worked examples, made interactive: necessary and surplus hours, the one-good wage-profit frontier, prices of production with a wage gap, and composition against exploitation. All numbers are the chapter's hypothetical value accounting.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Necessary labor and the rate of surplus value

How do a longer day and a cheaper wage bundle change the rate of surplus value?

The wage is recovered in the necessary hours; every further hour creates surplus. Lengthening the day (absolute surplus) or cutting the value of the wage bundle (relative surplus) both raise s / v.

Equation, written in LaTeX: v=mh, s=m(L-h), \frac{s}{v}=\frac{L-h}{h}.

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m = $30 is new value per socially necessary hour, L the length of the working day, h the hours needed to reproduce the wage v, and s surplus value. Constant capital c = $300 transfers to the product.

Predict first. Lengthen the day from 8 to 10 hours with the $120 wage. What is s / v?

Your prediction

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Figure: Necessary labor and the rate of surplus value. An 8 hour day bar: 4 necessary hours and 4 surplus hours, rate of surplus value 100.0%.
Working day (hours): 8, Daily wage value ($): 120
Constructed example: the chapter's hypothetical values are m = 30, days of 8 and 10 hours, wages of 120 and 90 and c = 300; a 12 hour day is added.

Calculated values

Necessary hours h
4
Wage v ($)
120
Surplus s ($)
120
Rate s / v
100.0%
Batch value c + v + s ($)
540

Hypothetical value accounting, not estimates. At $30 of new value an hour, the 120 dollar wage is reproduced in 4 hours. The remaining 4 hours create surplus of 120, so s / v = 120 / 120 = 100.0%. A longer day raises absolute surplus; a cheaper wage bundle raises relative surplus.

Worked steps

  1. h = 120 / 30 = 4 hours
  2. s = 30 x (8 - 4) = 120
  3. s / v = 120 / 120 = 100.0%
  4. Batch value = 300 + 240 = 540

Use the idea

Use the split of the day into necessary and surplus hours to read what a change in hours or in the cost of wage goods does to the Marxian exploitation rate.

Where the conclusion applies

Labor-value accounting with a fixed value per hour, as in the chapter's hypothetical example; none of the amounts is an estimate.

Check your understanding: With a 12 hour day and a wage value of $90, what is s / v?
h = 90 / 30 = 3, s = 30 x 9 = 270, so s / v = 300%.

Chapter 97 source: section "Surplus value and exploitation".

Demonstration 2 of 4

The one-good wage-profit frontier

How does a higher real wage trade off against the profit rate, and what shifts the frontier?

With technique fixed, wages and profits divide a given surplus, so the frontier slopes down. A technique that uses less input moves the frontier out rather than along it.

Equation, written in LaTeX: p=(1+r)pA+wl.

Equation, written in LaTeX: 100=(1+r)50+100w_g.

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100 units of grain are produced from an advance of 50 (40 after technical change) and 100 labor hours. w is the real wage per hour in grain and r the uniform profit rate.

Predict first. Raise the wage from 0.20 to 0.35 with input 50. What is r?

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Figure: The one-good wage-profit frontier. Two downward lines of profit rate against wage for inputs 50 and 40; the point at w = 0.20 on the input 50 line shows r = 60.0%.
Real wage per hour (grain): 0.2, Grain input per 100 output: 50
Constructed example: all values (output 100, inputs 50 and 40, 100 hours, wages 0, 0.20, 0.35 and 0.50) are the chapter's hypothetical numbers; some combinations are added.

Calculated values

Wage bill
20.00
Profit
30.00
Profit rate r
60.0%
Maximum rate R
100.0%

Hypothetical value accounting, not estimates. With 50 grain advanced and a wage of 0.20, profit is 100 - 50 - 20.00 = 30.00 and r = 60.0%. A higher wage moves along the frontier; a smaller input shifts the whole frontier out.

Worked steps

  1. Wages = 100 x 0.20 = 20.00
  2. Profit = 100 - 50 - 20.00 = 30.00
  3. r = 30.00 / 50 = 60.0%

Use the idea

Separate a distributive change (moving along the frontier) from a change in technique (shifting the frontier) before reading wage and profit data.

Where the conclusion applies

A one-good economy with wages paid at harvest, as in the chapter's hypothetical case.

Check your understanding: With input 40 and wage 0.50, what is r?
Profit = 100 - 40 - 50 = 10, so r = 10 / 40 = 25%.

Chapter 97 source: section "Sraffian wage-profit frontier".

Demonstration 3 of 4

Production prices and unequal exchange

With equal labor content, why does the low-wage sector's good sell for less?

A common profit rate spreads the total surplus in proportion to capital advanced. The low-wage sector advances less capital, so its price falls below its value and the high-wage sector's price rises above.

Equation, written in LaTeX: p_i=c_i+v_i+rK_i,

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Each sector uses constant capital c = 40 and adds 50 units of new value. v is the wage bill (30 in H), s = 50 - v surplus, K = c + v capital advanced and r the common profit rate.

Predict first. Raise L's wage bill to 30, equal to H's. What happens to the exchange ratio?

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Figure: Production prices and unequal exchange. Paired bars: both sectors have value 90; production prices are 75.00 for L and 105.00 for H.
Low-wage sector wage bill: 10
Constructed example: the chapter's hypothetical values are c = 40, new value 50 and wage bills 10, 30 (L) and 30 (H); a wage bill of 20 is added.

Calculated values

Common profit rate r
50.0%
Price p_L
75.00
Price p_H
105.00
Exchange ratio p_H / p_L
1.400

Hypothetical value accounting, not estimates. With L's wage bill at 10, the common rate is 60 / 120 = 50.0%. Prices are 75.00 and 105.00: L sells 15.00 below its value and H 15.00 above, so one unit of H buys 1.400 units of L.

Worked steps

  1. s_L = 50 - 10 = 40; s_H = 50 - 30 = 20
  2. r = (40 + 20) / (50 + 70) = 60 / 120 = 0.5000
  3. p_L = 50 x 1.5000 = 75.00; p_H = 70 x 1.5000 = 105.00
  4. Ratio = 105.00 / 75.00 = 1.400

Use the idea

Check for wage gaps, mobile capital and equal profit rates before reading a price gap as a transfer of value.

Where the conclusion applies

Two sectors, equal turnover and the contested labor-value categories, as the chapter notes.

Check your understanding: With L's wage bill at 20, what are r and the two prices?
r = 50 / 130 = 38.46%; p_L = 60 x 1.3846 = 83.08 and p_H = 70 x 1.3846 = 96.92.

Chapter 97 source: section "Unequal exchange".

Demonstration 4 of 4

Composition, exploitation, and the profit rate

Does automation that raises capital per worker lower the profit rate?

A rising composition lowers r at a given exploitation rate, but higher exploitation or cheaper machinery can offset it. The sign depends on which moves faster.

Equation, written in LaTeX: r=\frac{s}{c+v}=\frac{s/v}{c/v+1}.

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c is constant capital, v = 15 variable capital after automation and s surplus value; k = c / v is composition and e = s / v exploitation. The baseline is c = 80, v = 20, s = 20.

Predict first. After automation surplus reaches only 24 (c = 135). Does the rate fall below 20%?

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Figure: Composition, exploitation, and the profit rate. Profit rate curves against composition for e = 1 and e = 2.00; the automated point at k = 9.00 gives r = 20.0%, against the baseline 20%.
Constant capital: 135, Surplus value: 30
Constructed example: the chapter's hypothetical values are c = 80, 135 and 105, v = 20 and 15, s = 20, 30 and 24; c = 165 is added.

Calculated values

Composition k
9.00
Exploitation e
2.00
Profit rate r
20.0%

Hypothetical value accounting, not estimates. After automation, k = 9.00 and e = 2.00, so r = 30 / 150 = 20.0%, equal to the 20% baseline. The rate falls only if composition rises faster than exploitation.

Worked steps

  1. k = 135 / 15 = 9.00; e = 30 / 15 = 2.00
  2. r = 30 / (135 + 15) = 30 / 150 = 20.0%
  3. Check: e / (k + 1) = 2.00 / 10.00 = 20.0%

Use the idea

Track composition and exploitation together before claiming a falling or rising profit rate.

Where the conclusion applies

Value accounting for a single hypothetical firm, as in the chapter.

Check your understanding: With c = 165 and s = 30, what is r?
30 / (165 + 15) = 30 / 180 = 16.7%.

Chapter 97 source: section "Tendency of the rate of profit to fall".