MIDTERMM

THE PRODUCTION FUNCTION AND GROWTH

In this note I’ll lay out the analytics of a surplus producing economy. The presentation

relies heavily on the ideas of the classical political economists, specifically the insights of Adam

Smith, David Ricardo, and Karl Marx. To keep things simple we’ll imagine a closed economy,

that is, an economy that does not engage in trade with other economies or involved in an

exploitative or subordinate relation with other systems. It is, in short, a self-contained system.

This assumption is commonly adopted in economics because it helps to focus attention on the

factors that are particular to the system and not the result of outside influence. Once the

characteristics of the system are understood, we can enlarge the analysis to include international

trade, imperialism, or dependency, without destroying the principles captured in the simpler case

of a closed economy.

We’ll start by exploring the relationship between work and gross output. With a given

productive capacity and work environment, that is with a given amount and quality of capital and

land and a given set of work habits and expectations, applying more labor to the production

process will cause output to grow at a diminishing rate. Figure 1A.1 provides a visual

representation of this idea.

Gross output (Q) is measured on the vertical axis, while number of workers (N) is

measured on the horizontal axis. The Q curve, also referred to as a production function, shows

how the existence of diminishing returns causes output to grow at a diminishing rate as a result

of adding more workers to production. But rather than thinking of the production function as a

visual image of a growing economy, the proper interpretation is that it represents the range of

production that’s possible given existing conditions. It provides a visual representation of the

relationship between labor usage and output per time period, given existing conditions of

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production and work. Economic growth, on the other hand, involves additions or improvements

to the system’s productive capacity, causing the system’s productive capacity to grow and the

entire production function to shift over time.

Figure 1A.1

The production function assumes that laborers are working at the same level of intensity.

As a result, the diminishing nature of extra output is not the result of declining effort, but rather

the result of having to interact with increasingly less fertile land. It should also be noted that the

workers measured on the horizontal axis are all assumed to be direct workers; there is no

overhead labor. The graph could be amended to incorporate the role of overhead labor, but it

would complicate the presentation without improving our understanding of basic economic

principles. It should also be noted that all labor is assumed to be productive. There is, in other

words, no unproductive labor.

Given the above production function, let’s say that N’ workers are currently being used

per time period. This would mean that the economy is generating Q’ amounts of gross output per

time period. The worker’s replenishment, socially necessary consumption, is Cn’, depreciation is

Dp (the difference between NP’ and Cn’), and the system’s necessary product is NP’. The

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difference between gross output (Q’) and the necessary product (NP’) represents the system’s

surplus product. The surplus product represents the amount that’s available for surplus

consumption and/or net investment (not shown in the graph). Obviously, the work force must

generate a surplus beyond its own necessary consumption to provide for the consumption

standards of other classes of people, its own possible surplus consumption, depreciation and net

investment. The extent to which this occurs and how the surplus is distributed and used depends

not simply on the productivity of labor but on the political organization of society.

The historical pattern, for surplus producing systems, is to find one class of people

controlling the laboring activities of another class of people – those who do the work. While

there are variations on this theme and classes of people who fall in between these two categories;

it’s nevertheless the case the surplus producing economies have traditionally been class divided

societies with one class of people, the managing or ruling classes, directing or controlling the

laboring activities of the laboring classes. What’s more, the ruling classes oversees the laboring

effort of the workers to ensure that the surplus that’s generated by the workers is used for their

own consumption and/or invested in more productive capacity. They have a direct interest in

ensuring that the workers generate a surplus that’s sufficient to sustain their lifestyle; as a result,

they will generally be intent on having the workers produce as much of a surplus as they possibly

can.

A numerical counterpart to the above graph is presented in Table 1A.1 (the numbers are

provided for heuristic purposes; they are not intended to capture the actual proportions found in

real economies). For the moment, focus on the first six columns of the table. The columns N and

Q are the numerical counterpart to the production function in Figure 1A.1, showing that, with a

given productive capacity and working environment, more workers will bring about greater

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output, but at a diminishing rate. The Cn column represents the socially necessary consumption

of the various workers (represented as line Cn in Figure 1A.1), while the Dp column represents

depreciation (shown in Figure 1A.1 as the difference between the Cn+Dp line and the Cn line).

The sum of Cn and Dp is the system’s necessary product (NP) and is represented as the Cn+Dp

line in Figure 1A.1. The column labeled SP represents the system’s surplus product and is the

difference between gross output (Q) and the necessary product (NP).

N Q Cn Dp NP SP ap mp cn NP/N SP/N

0 0 0.00 6.00 6.00 -6.00 8

1 14 8.00 6.00 14.00 0.00 14.00 14.00 8 14.00 0.000

2 27 16.00 6.00 22.00 5.00 13.50 13.00 8 11.00 2.500

3 39 24.00 6.00 30.00 9.00 13.00 12.00 8 10.00 3.000

4 50 32.00 6.00 38.00 12.00 12.50 11.00 8 9.50 3.000

5 60 40.00 6.00 46.00 14.00 12.00 10.00 8 9.20 2.800

6 69 48.00 6.00 54.00 15.00 11.50 9.00 8 9.00 2.500

7 77 56.00 6.00 62.00 15.00 11.00 8.00 8 8.86 2.143

8 84 64.00 6.00 70.00 14.00 10.50 7.00 8 8.75 1.750

9 90 72.00 6.00 78.00 12.00 10.00 6.00 8 8.67 1.333

10 95 80.00 6.00 86.00 9.00 9.50 5.00 8 8.60 0.900

Note that as the employment of labor increases, with a given productive capacity and

work environment, the system’s surplus product grows, reaches a maximum, and then declines.

This can also be seen in Figure 1A.1, by noting that the vertical distance between the production

function (Q) and the necessary product (line Cn+Dp) at first grows, reaches a maximum, then

declines. This behavior is due to the combined effect of both diminishing returns and a growing

necessary product. Adding more workers to the production process, and assuming a stable work

environment, will cause output to grow at a diminishing rate while causing the necessary product

to grow at a constant rate. The interaction of these two trends causes the surplus product to grow,

reach a maximum, and eventually become zero (not shown in either the graph or the table, but

implied in both by the fact that the size of the surplus begins to diminish).

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Another way of thinking about this is that there’s a maximum amount of surplus the labor

force is capable of generating, given productive capacity and work conditions. That is, given the

amount and quality of capital and land, and existing work habits, the system is designed to

generate a maximum surplus with a specific number of workers. In the table that number of

workers is 7; that is, given the system’s productive capacity and work habits, seven workers will

produce the most surplus. Adding more workers beyond that amount will still generate a surplus,

but not as much as the system was designed for. Of course, a smaller number of workers will

also generate smaller surpluses. In Figure 1A.1, the number of workers which will generate a

maximum surplus is N’.

All of this can be interpreted on a per worker basis. Since the workers are the producers

of the gross product, we can interpret the above relationships by comparing the amount that the

average worker produces to the amount the average worker uses up in consumption. Columns

seven through eleven in Table 1A.1 (from column ap to column SP/N), show these relationships.

The column labeled ap represents the average productivity of labor and measures the

amount of gross output produced by the average worker. Thus, when 5 workers are used and

gross output is 60, the productivity of labor is 12 (60 divided by 5). The column labeled mp

represents the marginal productivity of labor and measures the amount of extra output generated

by using one more worker. For example, when the number of workers increases from 2 to 3 per

time period, gross output increases from 27 to 39 per time period; that is, the addition of the third

worker causes output to grow by 12. The marginal product, mp, of the third worker must

therefore be 12. The column labeled cn represents the socially necessary consumption per

worker, which in this simple example is 8 units of gross output per worker. When this level of

consumption per worker is multiplied by the number of workers we arrive at a measure of

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socially necessary consumption (Cn) for the system as a whole. The NP/N column measures the

system’s necessary product per worker (NP divided by N). Note that as the usage of labor

increases necessary product per worker gradually declines because the fixed amount of

depreciation is spread over a larger number of workers. The SP/N column measures the system’s

surplus product per worker (SP divided by N). Figure 1A.2 provides the visual counterpart to

these ideas.

Figure 1A.2

The ap line represents the productivity of labor. Note that, as in the table, it shows the

productivity of labor declining as more labor is applied to existing productive capacity. Once

again, this is not due to a failing on the part of the workers but rather to the existence of

diminishing returns. The mp line represents the marginal productivity of labor. Note that the

marginal productivity of labor is declining at a faster pace than the average productivity of labor.

Socially necessary consumption per worker is shown as the straight line labeled cn. The

necessary product per worker is shown as a gradually declining curve (cn+Dp/N) that lies above

the cn line; this is the visual counterpart to the NP/N column in Table 1A.1.

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The marginal productivity of labor has a special role to play in our understanding of the

surplus. In looking over Table 1A.1 it should be apparent that the surplus product, SP, reaches a

maximum when 7 workers are employed. Note that it’s also at that point that the marginal

productivity of labor (mp) is equal to socially necessary consumption per worker (cn). This is an

important principle of economics. The basic idea is that the surplus product will reach a

maximum when the extra output generated by one more worker (mp) is just equal to (or greater

than, but close to) the extra cost of using one more worker (cn). Note that in Figure 1A.2, the

point at which the marginal product of labor line intersects the consumption per worker line is

comparable to using 7 workers in Table 1A.1. In Figure 1A.2, that point occurs when N’ workers

are used; the surplus product is at a maximum at that point.

When N’ workers are used, average productivity is ap’, socially necessary consumption

per worker is cn’ and necessary product per worker is NP/N’. Note that the difference between

the productivity of labor (ap) and socially necessary consumption per worker (cn) represents the

amount in excess of the average worker’s consumption that is used by the rest of society for

depreciation, surplus consumption, or net investment. The proportions in which the surplus is

used for these various purposes depends on the nature of society’s technology and political

economic institutions. But it should be obvious from the graph that, beyond some point, the

excess that’s generated by the average worker will start to diminish and eventually be just

enough to cover depreciation, leaving nothing for surplus consumption or net investment. Of

course, one would not expect society to consciously reach such a state of affairs. Those who live

off the surplus, the ruling classes, would begin to search for new technologies and/or new

productive capacity in the hope of improving the productivity of labor and increasing the size of

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the surplus. Indeed, the pressure to search for alternative methods of generating more surpluses

will begin to occur when more than N’ workers are employed.

Figure 1A.3 provides a visual image of the impact of capital accumulation on economic

growth. The figure assumes that there are only two eras or phases. The first one, corresponding

to production function Q, shows the relationship between number of workers and gross output

with the initial productive capacity. The second one, corresponding to production function Q’,

shows the relationship between number of workers and gross output with an enhanced

productive capacity due to capital accumulation.

Figure 1A.3

The figure shows depreciation increasing with the process of capital accumulation, both

in absolute terms and as a fraction of gross output. The existence of more capital has had the

effect of increasing the amount of gross product that has to be allocated to capital replenishment.

However, the extent to which this occurs depends on the nature of capital accumulation. We will

consider the types of capital accumulation later on, but for the moment it’s enough to know that

the rate and level of depreciation can vary and need not take the form suggested in Figure 1A.3,

even though it is not uncommon.

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Figure 1A.4 provides the per worker counterpart to Figure 1A.3. Note that, when seen

from this perspective, the process of capital accumulation has the effect of improving the

productivity of labor; the line labeled ap’ represents the greater range of productivity brought on

by the accumulation of capital. The process of capital accumulation would also affect the

marginal productivity of labor. In general, and once again depending on the nature of

technological change, the marginal productivity of labor would increase in tandem with the

average productivity of labor. However, for ease of exposition (to keep the figure relatively

uncluttered), the marginal productivity of labor (mp) is not shown.

Figure 1A.4 also shows, consistent with Figure 1A.3 a greater level of depreciation, and

thus a greater necessary product. But the improvement in labor productivity and thus gross

output is, at a minimum, equal to the growth in necessary product.

Figure 1A.4