Only Exceptional Proff

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The Firm: Capital Accumulation and Labor Extraction

I. Introductory

In this chapter we’ll explore the use of capital and labor by firms and industries. Until

now we’ve been assuming that the capital structure of the firms is given and that production

levels are changed by employing more or less labor (with, of course, the appropriate complement

of needed raw materials). That is, we have been analyzing the behavior of the firm in the short

run, where the existing capital structure is a result of past decisions informed by the technology

in use at that time. In our discussion of the firm all of this has been summarized by what we have

been calling the capital-labor ratio, k. What we now need to do is explain how the capital-labor

ratio is determined in the first place, and explore the relationship between capital, labor, and

production, as well as the relationship between the capital-labor ratio and the relative price of

capital and labor.

II. Capital labor ratios

We begin by reviewing the role of capital-labor ratios in the cost of production.

Remember that the unit cost of a firm (or industry) can be represented in the following terms

!" = !!" + !!! ·! !" 5.1

where the first part of the expression (w/ap) represents the firm’s average variable cost (which, in

this case is also the firm’s unit labor cost), and the second part [(i+d)·k/ap] represents the firm’s

average fixed cost. Keep in mind that, to keep our attention focused on the key economic

relationship between capital and labor, we have simplified the unit cost of a firm by imagining

that the only two inputs are direct labor and capital. A more realistic, and complicated, equation

would incorporate the role of raw materials in the firm’s variable cost and the role of overhead

labor in the firm’s fixed cost. This would make the argument a bit more realistic but it wouldn’t

add to our understanding of the underlying relationship between capital and labor, which is the

point of theoretical economics.

Now, in focusing on average fixed cost, it should be apparent that it’s behavior will

depend on the capital-labor ratio (k) and the productivity of labor (ap), since the rate of interest

(i), and the depreciation rate (d) are thought of as given and beyond the firm’s control; that is,

afc=f(k/ap). In the short run, when the firm’s capital structure is given, output can only be

changed through changes in the usage of labor. But, this in turn means that the measured capital-

labor ratio must be changing with production levels. Specifically, the measured capital-labor

ratio must be decreasing as output increases (with a given capital structure, increasing the hours

of work or the number of workers, against a given capital structure, implies that the measured

capital-labor ratio is getting smaller; as the denominator increases against a fixed numerator, the

ratio k must fall).

But, what of the productivity of labor? In the case of firms exhibiting a fixed proportions

technology, the productivity of labor remains stable so long as the technology of production and

labor relations remains stable. This will ensure that the firms average fixed cost will gradually

decline as output (and the usage of labor) increases. The capital-labor ratio gradually falls while

the productivity of labor remains constant, ensuring a gradually declining average fixed cost with

growing output levels.

In the case of firms exhibiting diminishing returns, the productivity of labor gradually

falls, even if technology and labor relations remain stable. At first, this would seem to suggest

that at some point, as output grows, average fixed cost might begin to increase as the decline in

labor productivity begins to outweigh the decline in the measured capital-labor ratio. But, it turns

out, this does not occur. While both the productivity of labor and the measured capital-labor ratio

will be declining as output increases, the decline in the productivity of labor will always be less

than the decline in the capital-labor ratio; ensuring, once again, a gradually declining average

fixed cost with growing output levels.

The reason for this last result is not readily apparent, but the intuition behind it involves

remembering the distinction between the marginal product of labor and the average product of

labor. As output increases, in the context of diminishing returns, the usage of labor also increases

but at increasingly greater amounts. This means that as output grows one unit at a time, the

capital-labor ratio will have to be falling at increasingly faster rates. At the same time, an

increasingly greater usage of labor will be required in the production of each extra unit of output.

The marginal product of labor will be falling, but not as rapidly as the decline in the capital-labor

ratio. And since the marginal product of labor must fall faster than the average product of labor,

in the context of diminishing returns, this will guarantee that the average product of labor will

decline at a slower pace than the capital-labor ratio.

In the construction of an establishment (a firm), the capitalist invests in an amount of

capital that he/she assumes is the minimally necessary amount to produce, with the usage of

labor, a volume of output that will meet expected demand. That is, the capitalist has an idea of

the optimal relationship between the amount of capital and labor needed to produce the normal

level of output. This optimal or normal capital-labor ratio, kn, represents the combination of

capital and labor the firm is assuming will be in effect when producing the expected or normal

level of output. One would expect a profit maximizing capitalist to choose a capital-labor ratio

that would minimize its cost of producing the expected level of demand. But once the

establishment (firm) has been created with its full complement of buildings, machines and

inventory, then it’s quite possible that the demand for the product, and consequently the

employment of labor, will differ from the expected amount. And this, in turn, will have the effect

of causing the measured capital-labor ratio (the capital-labor ratio that is actually being used) to

differ from the normal capital-labor ratio (the capital-labor ratio for which the firm was

designed). If the demand for the product happens to be less than anticipated, then the firm has no

option but to use less labor than normal, causing the measured capital-labor ratio to be higher

than the normal capital-labor ratio. Likewise, if the demand for the product happens to be greater

than anticipated, then the firm – assuming it still has some leeway – will have to use more labor

than normally expected, causing the measured capital-labor ratio to be lower than the normal

capital-labor ratio.

The normal capital-labor ratio should thus be thought of as that combination of capital

and labor that allows the firm to produce its normal output at the lowest unit cost. It represents

the optimal size of the establishment (or firm) for the production of the expected, normal, level

of output. But once that capital structure is in place, the firm’s output can only be changed by

changing the usage of labor. In the context of a firm experiencing diminishing returns, what this

means is that the optimal size of the firm, the point at which the measured capital-labor ratio is

equal to the normal capital-labor ratio, occurs when unit cost is at it’s minimum – the minimum

point on the average cost curve. Production levels beyond that point involve increasingly greater

amounts of labor, in relation to capital, causing unit cost to rise. But production levels below that

point involve more capital than is needed with the declining amounts of labor, causing unit cost

to be greater than at the normal level of production.

In the case of firms experiencing a fixed proportions technology the optimal size of the

firm, the point at which the measured capital-labor ratio equals the normal capital-labor ratio,

occurs when unit cost is at a minimum; and this corresponds to the maximum amount of output

the firm can produce with that combination of capital and labor. But, the problem with this is that

it leaves the firm with no way of meeting a sudden increase in demand beyond the normal level.

As a result, it’s common for firms with a fixed proportions technology to choose a capital-labor

ratio that will allow them to meet the usual, normal, level of demand, but with enough excess

capacity to meet the upswings in demand that occur over the course of a fiscal year.

At any moment in time it’s common to find within an industry a range of firms operating

with differing capital-labor ratios. The firms are producing the same, or similar, output, but the

technologies used by the firms might differ from each other because of the time at which the

different technologies were introduced into the businesses. The newer or more aggressive firms

will generally be operating with the most efficient technologies and, as such, operating with the

higher and more productive capital-labor ratios. The older and/or less aggressive firms will be

operating with older technologies, lower capital-labor ratios, and consequently less productive

technologies. Firms that are operating with higher capital-labor ratios often have a cost

advantage, lower unit cost, brought on by the increased productivity made possible by the higher

capital-labor ratio and improved technology. Firms that are operating with a lower capital-labor

ratio will tend to have a higher unit cost as a result of the lower productivity associated with the

lower capital-labor ratio and less efficient technology. While these differences may persist for

quite some time, in the context of perfect competition these costs differences would vanish over

time. Any cost advantage that a firm might have, as a result of using a highly productive

technology (a high capital-labor ratio), would soon be eliminated as all the other firms in that

same industry rush to adopt the same technology in an effort to remain competitive. In the

extreme, all the firms in the industry would end up using the exact same technology, the same

capital-labor ratio, and – as a result – incurring the same unit cost. Competition, therefore, tends

to bring about conformity in the capital-labor ratios employed by the firms in an industry.

A similar process is operating within any one firm. That is, it’s not uncommon to find

within firms a range of technologies being used in the production of output. The newer machines

may have higher capital-labor ratios and, consequently, a greater level of productivity, while the

older machines have lower capital-labor ratios and, as a result, a lower level of productivity. The

firm will generally be operating with the more productive machinery but as the demand for the

product begins to exceed the normal level, the firm will bring out the older technologies so as to

meet the greater than normal demand. The growing reliance upon older technologies will have

the effect of increasing the unit cost of the product.

III. Choice of technique

The following graph shows two possible capital/labor ratios. The lines k1 and k2 represent

two possible capital/labor ratios, that is two possible techniques of production. As will soon be

shown, moving along any one technique, from the origin to the northeast corner of the graph,

means that more is being produced – as should be expected since both more capital and labor are

being employed. Yet, if the technique of production remains unchanged, then inputs must grow

at the same rate as output because of the fixed capital-labor ratio. Of course, technology can

change and when it does it usually involves not only a change in the capital-labor ratio it also

involves a change in the productivity of labor, a phenomenon not depicted in this graph. Higher

capital-labor ratios are associated with higher levels of labor productivity, while lower capital-

labor ratios are associated with lower levels of labor productivity. That is, firms and/or industries

invest in relatively capital-intensive (i.e. high capital-labor ratios) techniques only if the extra

productivity made possible by the increased capital counterbalances or outweighs the rising cost

associated with higher capital-labor ratios, allowing unit cost to remain unchanged or fall.

In general, there are two huge traditions in the discussion of productive techniques: one

tradition (usually associated with the Classical or Marxian tradition, but also reflected in the

contemporary engineer’s view of the world) argues that the technique of production, the capital-

labor ratio that a firm or industry chooses is driven almost exclusively by the knowledge

embodied in existing technology. That is, the way in which any one product is produced, the

combination of capital and labor that’s used to produce the product, is a function of the

technology that exists. From this perspective it doesn’t really matter what happens to the relative

price of capital (in terms of labor), when figuring out how to produce the product. Capital can be

relatively expensive or relatively cheap, but since there’s only one way of producing the good

then the capital/labor ratio that’s embodied in existing technology will be the one that’s used.

From this perspective, the ratio of capital to labor is determined by the technology of production.

The other tradition (associated with the Neoclassical tradition) argues that the technique

of production, the capital-labor ratio, employed by a firm depends on the relative price of capital

and labor. If, for example, the relative price of capital (in terms of labor) is high, then firms will

employ a technique of production that economizes on capital and uses more labor, that is, firms

will be motivated to pick a low capital/labor ratio technique. On the other hand, if the relative

price of capital (in terms of labor) is low, then the firms will employ a technique of production

that economizes on labor and uses more capital, that is, they will pick a high capital/labor ratio

technology.

Too be sure, there’s an element of truth in both traditions. Obviously, at any moment in

time there’s a technology that exists in the production of things that firms must use and relative

prices have little to do with the choice. But it’s also true that at any moment in time there’s an

Labor

C ap

it al

k1 > k2

existing range of technologies, though usually somewhat narrow, that are employed in the

production of a good, permitting a firm to choose within that set. The technology chosen from

that set would more than likely depend on the relative prices of the inputs. Thus, while it makes

sense to imagine that capitalists change their technique of production in response to a change in

the relative prices of capital and labor, it does not make sense to imagine that the range of input

substitutability is as extreme as neoclassical theory suggests; there’s a way in which the range of

technologies available for the production of almost anything is constrained within a limited

range, with the extreme version of this being that there exists only one technology.

We’ll explore these issues by first explaining the neoclassical view and then pointing out

the difficulties confronting that perspective.

Since this whole discussion involves a discussion of choice of technique, that is of the

possible capital-labor ratios a firm might use in the production of a good, it should be apparent

that we’re in the long run, a situation where all inputs can be changed and there is no one fixed

input. Of course, the one “input” that could be thought of as fixed is technology itself, namely

the knowledge that currently exists regarding the various possible ways in which a product can

be produced. But the problem with this interpretation is that technology isn’t an input as such;

rather it’s the knowledge that’s available to organize inputs in specific ways. A change in

technology is reflected in a change in the use of inputs as well as a change in the volume or type

of output.

Neoclassical theory starts out by imagining that capital and labor can be used in any

combination to produce any level of output, in short, the neoclassical theory of production

assumes extreme substitutability among the inputs to production. In addition, neoclassical theory

typically assumes that all inputs experience diminishing returns. If we conceive of production in

this fashion then the total production function can be depicted as shown in the following graph.

Neoclassicals refer to this as a well-behaved production function. What they mean by this is that

all the inputs experience diminishing returns and that the inputs can be substituted for one

another in an infinite variety of ways to produce the product.

Now, if we imagine cutting the above production function at the point depicted by the

dotted curve, then we could imagine this “mountain” sliced off with a flat surface at that level.

Every point on that surface would represent the same level of output, so any combination of

labor and capital on that surface could produce that same level of output. Note, however, that it’s

only the combinations of capital and labor that are on the edge of the mountain, the rim of the

flat surface closest to the origin, where one finds the most efficient possible combinations of

capital and labor to produce that same level of output. If we repeated this exercise for each level

of output we’d end up with a very large set of rims showing the most efficient combinations of

capital and labor in the production of the corresponding level of output. Rotating the above graph

so that the output axis is facing us, jutting straight out from the paper, then what we’d see is a

large number of theses rims showing the various combinations of capital and labor that produce

various levels of output. The following graph depicts this idea.

The three curves show three possible “rims” of the huge number that could conceivably

exist. These “rims” are called isoquants to underscore the idea that all the possible combinations

of capital and labor depicted on any one “rim” generates the same level of output. So, for

example, in looking at isoquant Q1, all the possible combinations of capital and labor that lie on

that isoquant generate the same level of output. The same interpretation applies to isoquant Q2

and isoquant Q3, with the understanding that the further the isoquant is from the origin, the

greater the output, and vice versa.

The rate at which labor can be substituted for capital in the production of the same level

of output is called the rate of technical substitution, i.e. RTS(L.K). This rate of technical

substitution can be thought of as the negative of the slope of the isoquant at any given point. This

slope shows the extra amount of capital that would have to be given up for the extra amount of

labor that would be needed to produce the same level of output. It turns out that the rate of

technical substitution can also be depicted as the ratio of the marginal product of labor to the

marginal product of capital. That is,

RTS(L,K) = - ΔK/ΔL = MPL/MPK

Note that if isoquants have the strictly convex shape, as shown in the graph, then the

RTS(L,K) would be diminishing as the usage of labor increases (and the usage of capital

decreases) in the production of any one level of output. What this means is that as more labor is

substituted for capital, in the production of any one level of output, the amount of capital the firm

(or industry) would be willing to give up, and still produce the same volume of output, would get

increasingly smaller. This implies that there’s some minimal level of capital the firm thinks it

must have, as it’s usage of labor increases, but the volume of output remains unchanged.

Labor

C ap

it al

∆K

∆L

Slope = -∆K/∆L = RTS(L,K)

Of course, there’s no reason why the isoquants must have these shapes, the idea of a

strictly convex isoquant is a theoretical creation generated by neoclassical theorists intent on

showing that the choice of technique is driven exclusively by relative prices; a proposition which

requires that substitutability be a feature of the system. In short, strictly convex isoquants were

not generated as a result of observing real production processes; they were instead generated by

the need to demonstrate that the choice of technique is driven by relative prices. If inputs are not

easily substitutable, then the isoquant map ends up looking like the following graph, which

depicts a fixed proportions technology.

The way to interpret these fixed- proportions isoquants is that, for example in the

production of any one level of output, say Q1, there’s a specific combination of capital and labor

that must be used to produce that output, namely the combination K1 and L1 corresponding to the

point where the isoquant kinks along the capital labor technique k. Any usage of labor that

exceeds this amount, on the horizontal portion of the Q1 isoquant, would still require the same

amount of capital, namely K1, and involve an unnecessary use of labor (labor would be

redundant). The same holds true for any usage of capital that lies along the vertical portion of the

Q1 isoquant. Any usage of capital beyond K1 that amount would be redundant. Thus the only

combination of capital and labor that must be used to produce Q1 units of output would be K1,

L1.

Let’s continue with the neoclassical assumption that isoquants are strictly convex, i.e.

extreme substitutability and no fixed proportions. What combination of inputs would the firm use

Labor

C ap

it al

L 1

K 1

in the production of any one given level of output? To answer this question we must remember

that cost can be represented as

TC = w·L+v·K

We can rewrite this equation to make capital a function of everything else. If we assume that TC

is fixed, then we can call this equation an isocost since it would show all the combinations of

capital and labor, given wages and rental rates, which generate the same cost.

The following graph shows one way of thinking about the way in which a firm goes

about minimizing its cost of producing any one level of output. The graph shows one isoquant,

Q1, and two isocost lines. The capital intercept of any one isocost will always be TC/v, while the

labor intercept of any one isocost will always be TC/w. The slope of any one isocost will always

be –w/v, that is, the relative price of labor in terms of capital (the ratio of the wage rate to the

rental rate). Note that, since the wage rate and the rental rate are assumed fixed, then the cost of

production will depend on the amounts of labor and capital the firm employs in the production of

Q1.

€

K = TC v − w v ⋅ L

Labor

C a p it

a l

TC/v

TC/w TC'/w

TC'/v

RTS(L,K) > w/v

RTS(L,K) = w/v

RTS(L,K) < w/v

The firm could start out by trying to use the combination of capital and labor represented

by point a. But at that point, the rate of technical substitution of labor for capital exceeds the

relative price of labor in terms of capital. Another way of saying this is

That is, at point “a” the firm gets a little more output from a dollar spent on labor than it does

from a dollar spent on capital. As a result, a firm intent on using the least cost combination of

inputs would be induced to use a bit more labor and a bit less capital. That is, the firm would

move down the isoquant. But as it does so, the isocost line starts shifting down. This process will

continue until the firm finds that one combination of capital and labor that minimizes it’s cost.

That point will occur when the rate of technical substitution of labor for capital just equals the

relative price of labor in terms of capital. Or stated differently, when

The same logic applies if instead we had started at point b. Under these circumstances the

rate of technical substitution of labor for capital would fall short of the relative price of labor in

terms of capital. That is

implying that the firm gets a little more output from a dollar spent on capital than it does from a

dollar spent on labor. This would induce the firm to move up the isoquant in its search for more

capital and less labor. But as it moves up the isoquant the isocost line start shifting closer to the

origin. Eventually, once again, the firm will find itself moving to that combination of capital and

labor that minimizes its cost and this will occur where the slope of the isocost is just equal to the

slope of the isoquant, that is, the rate of technical substitution of labor for capital just equals the

relative price of labor in terms of capital (or stated differently, the marginal product generated by

a dollar spent on labor equals the marginal product of a dollar spent on capital).

Now, let’s imagine that the firm has chosen its capital labor ratio, i.e. it has chosen it’s

technique of production. Once the capital is put in place then there isn’t much the firm can do, if

output levels change, other than to alter the usage of labor with the existing amount of capital.

What this means is that, in the short run, where the capital structure of the firm is given, changes

in production levels can be met by using more or less labor, but in ways that inevitable involve

€

MPL w

> MPK v

€

MPL w

= MPK v

€

MPL w

< MPK v

some inefficiency. The following graph provides one way of thinking about this. Imagine that the

firm is using a capital labor ratio depicted by point b, which, as can be seen, involves the least

cost combination of capital and labor, since the slope of the isocost is equal to the slope of the

isoquant in the production of Q1 level of output. Now, what if sales, and consequently output,

either increase to Q2 or decrease to Q0? Note that in both cases, the firm has no option but to

either increase or decrease the usage of labor with the fixed amount of capital. If the firm is

induced to produce Q2 units of output then it will be using a combination of labor and capital that

is not cost minimizing. At that point the rate of technical substitution of labor for capital would

be less than the relative price of labor in terms of capital, clearly implying that cost minimization

is not taking place.

If instead the firm is induced to produce Q0 units of output, then it will be using a

combination of labor and capital that, once again, is not cost minimizing. In this case the rate of

technical substitution of labor for capital would be greater than the relative price of labor in

terms of capital, implying once again that cost minimization is not taking place.

What if, instead, the relative price of labor in terms of capital changes? That is, assume

that the firm has chosen the cost minimizing combination of capital and labor, but sometime after

having constructed the buildings and installed the machinery, the relative price of labor in terms

of capital changes. What happens then? Well, then this would have the effect of inducing the

firm to search for another combination of capital and labor that would, once again, minimize its

cost in the production of output. In other words, the search for the right cost minimizing

combination of capital and labor. The search for another capital to labor ratio still has the same

0 L0

Labor

C ap

it al

b c

RTS(L,K) > w/v

RTS(L,K) = w/v

RTS(L,K) < w/v

K fixed

L1 L2

properties that we’ve outlined above but with an added twist. The following graph provides a

visual image of the issues involved.

This graph assumes that the firm is first producing Q1 units of output and, given the

relative prices in effect at the time, has chosen the cost minimizing combination of capital and

labor depicted by point a, that is, K1 units of capital and L1 units of labor. Now, if the relative

price of labor in terms of capital were to decrease, either because the wage rate falls or the rental

rate increases, or some combination of the two (so long as the percentage decline in the wage

rate exceeds the percentage increase in the rental rate), then the slope of the isocost line would

flatten out and the firm would be induced to search for a technique (a capital to labor ratio) that

involves more labor and less capital. The new cost minimizing point that would be chosen, in the

production of the same volume of output, is shown as point b. The firm would now be employing

K3 units of capital and L2 units of labor.

The movement from a to b is referred to as the substitution effect. It underscores the idea

that it makes sense to imagine that profit-seeking capitalists would use less capital and more

labor if the relative price of labor in terms of capital were to fall; that is techniques of production

would move from being relatively capital intensive to being relatively labor intensive. It’s

important to note, however, that the movement from a to b would be time consuming and involve

a total abandonment of technique a, liquidating the assets associated with the method of

production, and building an entirely different structure corresponding to the more labor intensive

technique b. Perhaps another way of thinking of this is that a firm that has multiple plants

(establishments), might very well have two plants operating at the same time: the plant

Labor

C a p it

a l

L1 L3L2

TC/v

TC/w

represented by technique “a” can be thought of as having been constructed when the relative

price of labor in terms of capital was high and the plant represented by technique “b” can be

thought of as having been constructed when the relative price of labor in terms of capital was

low. Regardless of the interpretation, neoclassical theory calls the shift to an alternative

technology in the face of a change in relative prices, while holding levels of output constant, as

the substitution effect.

However, the story doesn’t end there. Indeed it gets more complicated. If the relative

price of labor in terms of capital diminishes, say as a result of a reduction in the wage rate, then it

must be that the short run cost of production will be falling. The impact this reduction in short

run cost might have on the price of the product and the consequent demand and thus production

levels is complicated and depends on the kind of output market the firm is operating in

(competitive or monopolistic) and the price elasticity of demand for the output of the firm. The

simplest case (though still complex) is that of perfect competition. In a perfectly competitive

output market, a reduction in the firms’ unit cost of production would mean (since all firms are

using the same technology) that the total volume of output offered on the market would increase

causing the market price to fall (a growing supply against a given demand causes the price to

fall). As the price falls this induces an increase in quantity demanded (consumers purchase more

as the price decreases). However, the amount by which quantity demanded increases as a result

of the reduction in market price depends on the price elasticity of demand. In some cases the

quantity demanded might be significant, in other cases it might be relatively trivial. There is no

way of knowing a-priori, it would depend on the nature of the market being studied.

What if, instead, the market were monopolized, that is, the firm is the single seller of the

product. Then under these circumstances, a reduction in unit cost, brought on by the reduction in

the wage rate, might induce the firm to charge a lower price as a way of enticing more customers

to purchase the product. The amount by which quantity demanded would increase as a result of a

reduction in unit price would, once again, depend on the price elasticity of demand. What’s more

all of this assumes that the firm is single-mindedly focused on maximizing profits. If instead the

firm is a satisficer, the monopolist might be quite content to simply allow unit cost to fall while

holding onto the existing output price, which in turn would not bring about a change in demand

and consequently output levels, but it would bring about an increase in profits.

The point of the above two paragraphs is that what happens to quantity demanded as a

result of a reduction in the wage rate depends on a number of particular circumstances that

cannot be universalized to all firms, it can only be considered on a case by case basis. But,

however it works out, it clearly makes sense to imagine that a change in the wage rate will bring

about a change in unit cost which might also affect output price and consequently quantity

demanded and, ultimately, levels of production. And, of course, a change in production levels

(different isoquants) will also bring about a change in productive techniques.

The chain of events that starts with a change in the wage rate to a change in output, and

consequently to a change in the combination of capital and labor employed in the production of

the product is called the output effect. This is depicted in the above graph as the move from point

“b” to point “c”. The graph is drawn on the assumption that the reduction in the relative price of

labor to capital has brought about an increase in production levels as a result of the growth in

quantity demanded brought on by the lower price which in turn was motivated by the lower unit

cost. It’s important to note that this need not always be the case. The output effect might be

positive, as shown in the above graph, or it might be zero or even negative; it would all depend

on the structure of the output market being considered. For pedagogical purposes we will assume

that the output effect is positive, in the sense that the wage reduction eventually leads to a greater

level of production and, consequently a different combination of capital and labor.

We close this section by noting that, if substitutability is not a feature of a productive

technology, then a change in relative prices will have absolutely no impact on the capital labor

ratio. This is shown in the graph below, which depicts a fixed capital labor ratio in the

production of a given level of output. Note that it doesn’t really matter what the relative price of

labor in terms of capital might, relative prices might be (w/v)1 or (w/v)2, yet the amount of

capital and labor that will be used to produce Q1 units of output will remain K1 units of capital

and L1 units of labor.

IV. Elasticity of Substitution

The ease with which labor can be substituted for capital in the production of a given level

of output is called the elasticity of substitution, which we’ll denote by the symbol !!. It provides a way of measuring the degree to which the capital-labor ratio, k, changes as a result of a change

in the rate of technical substitution, RTS(l,K). Formally, the elasticity of substitution can be

expressed as

!! = %∆!

%∆!"#(!,!) ,

where the numerator, %∆!, represents the percentage change in the capital-labor ratio, and the denominator, %∆!"#(!,!), represents the percentage change in the rate of technical substitution of labor for capital.

The elasticity of substitution can vary between zero and infinity. If the production

function involves a fixed-proportions technology, then there is no substitutability between labor

and capital and !! = 0. This is the case of the isoquant depicted in the above graph, where there’s only one capital-labor ratio and a change in the rate of technical substitution will not

affect the choice of technology; that is, the capital-labor ratio remains fixed.

If instead the production function involves an infinite range of possible technologies, i.e.

infinite substitutability, then the elasticity of substitution between labor and capital is infinite and

!! = ∞. The isoquants reflecting this theoretical possibility would be straight, negatively sloped,

Labor

C a p it

a l

(w/v)1 (w/v)2

lines. In this case the rate of technical possibility is constant (i.e., the rate at which labor is

substituted for capital remains fixed along the isoquant) and, as a result, an infinite number of

capital-labor ratios is possible along that isoquant. The following graph depicts this unreal

theoretical possibility. It shows two possible capital-labor ratios, out of the infinite amount that’s

possible on the given isoquant.

V. The Demand for Labor (a Neoclassical Perspective)

We’ll start by exploring the demand for labor under conditions of extreme competition;

assuming that the firm is a price taker in both the output and input market. After we’ve explored

that case, we’ll study the demand for labor on the assumption that the firm is a price setter, with

a certain amount of monopoly power, in both the output and input markets. The intermediate

possibilities (such as when the firm is price setter in the input market but a price taker in the

output market, or vice versa) will be left unexplored since they do not significantly change the

conclusions arrived at in the first two cases.

The Price-Taking Firm (Diminishing Returns)

We’ve already seen that the price-taking firm will maximize profits when marginal

revenue is equal to marginal cost. Since, for price taking firms, marginal revenue is the price of

the product, while marginal cost (in a context where labor is the only variable input) is equal to

Labor

C ap

it al

k 1

k 2

the wage rate divided by the productivity of labor, the profit-maximizing rule can be restated in

the following terms:

Rearranging the last expression provides us with the following:

The expression on the left side of this equality measures the marginal revenue generated by one

extra hour of labor, it is called the marginal value product of labor. Given this definition, it

should be clear that what this equality is saying is that the profit maximizing firm will hire labor

up to the point at which the marginal value added by one extra hour of labor is equal to the wage

that must be paid for that extra hour of labor. This is the core principle underscoring the

neoclassical theory of the demand for labor (or any input for that matter).

A graphic version of this idea is presented below. This graph is assuming that the firm is

experiencing diminishing marginal returns. The value of the marginal product of labor is

diminishing because, even though the product price remains unchanged (since the firm is a price

taker), the marginal product of labor declines due to the presence of diminishing returns.

The profit-maximizing firm will hire L1 units of labor when the wage rate is w1, because

at that point, the marginal value product of labor just equals the wage rate. If, for whatever

€

Πmax ⇒ mr = mc ⇒ p = w mpL

€

p⋅ mpL = w

Labor

w ,

M V

MVPL=p*mpL

reason, the firm hired an amount of labor that falls short of L1, the firm would quickly discover

that the marginal value product of labor is greater than the wage rate, inducing the firm to hire

more labor in an effort to capture more profits. On the other hand, if the firm hired more than L1

amount of labor, then it would realize that the marginal value product of labor is less than the

wage rate, inducing the firm to cute back on the use of labor in an effort to capture more profits.

L1 is thus the profit-maximizing amount of labor, given the firm’s capital/labor ratio, the price of

the product, and the wages of labor. It’s important to underscore that this is an equilibrium

position; that is, it represents a position the firm would move toward if everything else remained

the same. Another way of thinking about this is that, so long as conditions are fairly stable, then

real firms would be gravitating toward that position, in the vicinity of the profit-maximizing

usage of labor. It’s also important to note that all of this assumes the firm can hire more or less

labor, and consequently produce more or less output, without affecting output price or the wages

of labor; the context, in short, is one of extreme competition where the firm is a price taker in

both the output market and the labor market.

Another way of viewing this relationship is to introduce the value of the average

productivity of labor. Doing so helps to bring out the difference between the marginal product of

labor (or the marginal value product) and the average product of labor (or the average value

product). The following graph introduces the average value product of labor. Note that it’s the

same graph is the previous one, with the exception that the average value product is now shown

explicitly.

Labor

w ,

M V

P , A

V P

MVPL=p*mpL

p*apL

AVPL=p*apL

This graph makes it easier to see that the firm’s profit-maximizing choice of labor will

ensure that the value generated by the average worker will be greater than the average wage rate.

This, of course, is what one would expect of a capitalist enterprise. The whole idea of a business

is to generate a profit, and this in turn requires that the value that’s generated by the labor force

exceed the cost of hiring that labor force. The difference between the average value product (i.e.

the average product of labor multiplied by the price of the output) and the wage rate must be

sufficient to cover the firm’s overhead and target profits, it’s a measure of the surplus value

that’s generated by labor.

Karl Marx’s notion of the rate of surplus value, also known as the rate of exploitation,

measures the amount of surplus value generated per unit of variable capital and can be expressed

€

s'= s v

where s represent the surplus value generated by labor and v represents variable capital, the value

of labor power. If we think of the value of labor power as the going wage rate and surplus value

as the difference between the average value product of labor and the wage rate, then the rate of

exploitation can also be restated as

€

s'= s v

= p⋅ ap − w

Thus the profit maximizing choice of the firm affects not only the volume of output the

firm will produce, but the amount of labor it will hire as well as the rate of exploitation. Since

we’re assuming extreme competition, it must be the case that the productivity of labor is as high

as it can possibly be. Thus, profit maximization under conditions of extreme competition, where

the firms are price takers, also implies that the rate of exploitation is as high as it can be, given

existing technology, the price of output, and the wage rate.

The Price-Setting Firm (Diminishing Returns)

In the case of price setting firms, the logic is the same; the only difference is that the firm

must now pay attention to the marginal revenue it brings in from selling the extra output

generated by the marginal worker. Remember that in the case of price-setting firms, marginal

revenue is always lower than price. Given this, the profit-maximizing choice of labor for a price-

setting firm experiencing diminishing returns, can now be restated as

€

mr⋅ mpL = w

The expression on the left is called the marginal revenue product of labor. It measures the

marginal revenue generated by the marginal product of labor. While the equation looks different,

the underlying logic is the same: the profit maximizing firm will hire labor up to the point at

which the marginal revenue product of labor just equals the wage rate. However, it’s important

to note that this firm is moving the output price up and down to arrive at the profit maximizing

level of output which, in turn, impacts the usage of labor. So, at the same time the firm is finding

the appropriate profit-maximizing price in the output market, it’s also finding the profit-

maximizing usage of labor. The two decisions are occurring simultaneously. The following

graph provides a visual image of this logic, but focusing exclusively on the input market, in this

case, the demand for labor.

It’s important to note that while the logic is similar to the case of a price-taking firm, the

difference is that the price setter will be moving the price up and down until the profit-

maximizing level of output (and consequently usage of labor) is found. In the above diagram, the

firm’s price choice is built into the average value product of labor curve. However, this average

value product of labor curve is falling at a steeper rate than the average value product of labor

curve for the price-taking firm. The reason for this is that both the price and the average product

of labor are falling with increased production (sales) levels. In the case of the price-taking firm,

the price remains constant while the productivity of labor declines due to diminishing returns.

Labor

w ,M R P, A V P

MRPL=mr*mpL

p*apL

AVPL=p*apL

But in the case of the price-setting firm, the price falls (due to the downward sloping demand

curve) at the same time that the productivity of labor is also declining due to diminishing returns.

Price-Taking Firm (Fixed Proportions)

The logic underlying the firm’s demand for labor remains the same in the case of firms

experiencing fixed proportions technology; that is the firm will still hire that number of workers

that will maximize it’s profits, and that occurs when the marginal value product of labor is

greater than or equal to the wage rate. So long as the marginal value product of labor is greater

than or equal to the wage rate, the firm will have an incentive to produce at capacity. The

following graph depicts this situation. It’s assumed, for the sake of simplicity, that the marginal

value product is greater than the wage rate. But, the outcome shown in the graph would still be

the same, namely that the firm will hire the maximum amount it can, given its capacity (capital

structure), even if the marginal value product happened to coincide with the wage rate. However,

in this case the choice would have to be a short-term decision, since the firm would not be

covering any of its overhead or profits.

Price-Setting Firm (Fixed Proportions)

In the case of price setting firms, the profit-maximizing demand for labor will be

determined by the point at which the marginal revenue product of labor just matches the wage

Labor

w ,

M V

P, A

V P

p*mpL = p*apL

rate. This remains true regardless of whether the firm experiences diminishing returns or fixed

proportions. The following graph illustrates one possible scenario, where the marginal revenue

product of labor meets the wage rate at some level of production that falls short of the firm’s

maximum level of ouptut. Of course, it’s also possible for the marginal revenue product of labor

to never match the wage rate, as would be the case if the demand for the product were very

strong, exceeding the firm’s capacity. In this case, the outcome would be the same as in the

previous case (the price-taking firm with a fixed proportions productive technology); that is the

firm would be induced to hire the maximum amount of labor, consistent with the firm’s capacity

(capital). Note that regardless of whether the marginal revenue product of labor matches the

wage rate (as in the graph) or remains above it, the average value product will always exceed the

wage rate.

Output and Substitution Effects

What if the wage rate were to change? In particular, what would happen to the quantity of

labor demanded, if the wage rate were to decrease? The neoclassical argument is that the firm

would hire more workers and begin to change its technology of production by employing more

labor-intensive techniques. Since the price of labor has fallen relative to capital, the firm would

eventually use a lower capital/labor ratio in the production of the output. The extent to which this

might be true depends on the extent to which labor can be substituted for capital. If the

technology of production is narrowly constrained so that only a specific amount of labor can be

Labor

w ,

M V

P, A

V P

mr*mpL = mr*apL

p*mpL = p*apL

p*apL

employed with a specific amount of capital, then a reduction in the wage rate may not cause the

firm to substitute more labor-intensive techniques for the older capital-intensive ones. But if the

technology of production is quite flexible then it’s reasonable to imagine that the firm would

eventually substitute more labor for capital.

All of this can be explained by exploring the output and substitution effects of using labor

as a result of a reduction in the wages of labor. The following three graphs help explain this

process.

Labor

w ,

M V

MVP1

L2 L3

MVP2

Labor

C ap it al

Q1 Q2

Kfiexed

w1/v

w2/v

L1 L2 L3

Much of this has already been explained in a previous section that explained the output

and substitution effects of a change in the relative price of labor with respect to capital. The basic

idea is that a reduction in the wages of labor will have two effects. The first thing that would

occur is the output effect. But over the long run, the substitution effect could also take hold.

The output effect refers to the idea that a change in the wage rate would have the effect of

changing the marginal cost of production that in turn can affect the amount produced and

consequently the amount of labor needed to produce that greater output. Let’s consider the case

of a wage reduction to see how this works. The wage reduction would have the effect of reducing

the marginal cost of production. In the case of price taking firms, the reduction in marginal cost

would have the effect of inducing the firm to produce a larger volume of output. This is depicted

in the third graph, showing the firm producing a larger volume of output as a result of the

reduction in marginal cost. It’s also depicted in the first graph as the movement from L1 to L2,

and in the second graph as the shift from Q1 to Q2 and consequently the growth in the use of

labor (with a fixed capital structure) from L1 to L2.

It should be noted that the output effect gets more complicated than suggested above. To

keep things simple we’re imagining that all that happens is that more is produced as a result of a

reduction in the wage rate. However, the extent to which this might occur, or indeed the

possibility that output could actually decrease, depends on a host of other factors we’ve left

untouched. A glimpse of this can be captured by noting that an increase in output (and

consequently supply), brought on by a reduction in the marginal cost of production, would be

expected to bring about a reduction in the price of the product. The extent of that price reduction

would depend on the price elasticity of market demand and supply. But regardless of the extent

of the price reduction, it should also be obvious that the price reduction would, in turn, have the

Output

D ol la rs

MC1

MC2

Q2Q1

effect of inducing the firm to produce a smaller volume of output, something less than Q2. We

have, of course, no way of knowing the extent of that quantity reduction; it would depend on the

magnitude of the price reduction. It’s possible that the quantity reduction forces the firm to

produce less than Q2 but still more than Q1. But it’s also possible that quantity is forced to fall

back to Q1 or even below that amount. To keep things simple we’ll assume that the output effect

is negative, meaning that a reduction in the wage rate brings about an increase in output and

consequently an increase in the usage of labor, as depicted in the above diagrams.

Now, in addition to the output effect, a change in the wage rate will also have the effect

of changing the relative price of labor in terms of capital and induce the firm to start searching

for productive techniques that use a different capital/labor ratio. This is called the substitution

effect. In this case, a reduction in the wage rate will have the effect of inducing the firm to pick a

lower capital/labor ratio, one that relies on more labor (since it’s now cheaper), and less capital

(since it’s now relatively more expensive). The shift to a lower capital to labor ratio is depicted

in the second diagram as the movement from L2 to L3 units of labor in the production of output

Q2. Note that the firm is still producing the same volume of output (namely Q2), but with a more

efficient combination of capital and labor. This same effect is depicted in the first diagram as a

movement from L2 to L3. In this case, the substitution effect is shown as a shift in the marginal

value product of labor.

It should be noted that an increase in the marginal value product of labor might seem

counterintuitive. Since the firm is now relying on less capital, why would the marginal product

of labor increase? It’s important to review the fact that a reduction in the relative price of labor in

terms of capital (i.e. the w/v ratio) will induce the firm to search for a lower capital/labor ratio or,

stated differently, a productive technique where the ratio of the marginal product of labor to the

marginal product of capital is lower. This last requirement would suggest that the marginal

product of labor should fall; yet as noted in the above graphs, the marginal product of labor is

shown to be increasing. The way to think about this is to keep in mind that it’s the ratio of the

marginal products that’s diminishing, not simply the marginal product of labor. Thus, this ratio

can be falling as a result of an increase in the marginal product of capital combined with an

increase in the marginal product of labor (but one where the increase in the marginal product of

capital is greater than the increase in the marginal product of labor). Both capital and labor are

now more productive at the margin, but capital is now slightly more productive than labor.

Monopsony

Up to now we’ve been assuming competitive labor markets. But we should explore the

more realistic case where firms have some price-setting ability within the labor market. The

extreme version of this is found in the case of a monopsonist, that is a firm that has monopoly

power in the purchasing of inputs, in this case in the hiring of labor. An understanding of the

case of monopsony provides us with insights into the more general case where firms are not

necessarily formal monosponists but nevertheless have varying degrees of power within the labor

market.

This graph depicts a monopsonist intent on hiring the profit-maximizing amount of labor.

The key to understanding this behavior is that the firm is forced to increase the wages it pays its

labor force every time it wishes to hire a bit more labor. This has the effect of causing the

marginal expense of hiring one more worker to increase at a faster pace than the wages that must

be paid to attract one more worker. The firm will pick that usage of labor that will insure that the

marginal revenue product of labor just matches marginal expense of labor. The firm will pick

that wage rate that brings this about.

The thing to note about this choice is that the monopsonist will always pay its labor force

less than what would be the case in a competitive market while, at the same time, hiring less

labor than would be the case in a competitive market.

Labor

S L ,M E L ,M R P L

MEL

MRPL