By Sakshi Bhardawaj
Theory of production, in
economics, an effort to explain the principles by which a business firm decides
how much of each commodity that it sells (its “outputs” or “products”) it will
produce, and how much of each kind of labour, raw material, fixed capital good, etc.,
that it employs (its “inputs” or “factors of production”) it will use. The
theory involves some of the most fundamental principles of economics. These
include the relationship between the prices of commodities and the prices (or
wages or rents) of the productive factors used to produce them and also the
relationships between the prices of commodities and productive factors, on the
one hand, and the quantities of these commodities and productive factors that
are produced or used, on the other.
The
various decisions a business enterprise makes
about its productive activities can be classified into three layers of
increasing complexity. The first layer includes decisions about methods of
producing a given quantity of the output in a plant of given size and
equipment. It involves the problem of what is called short-run cost
minimization. The second layer, including the determination of the most
profitable quantities of products to produce in any given plant, deals with
what is called short-run profit maximization. The third layer,
concerning the determination of the most profitable size and equipment of plant,
relates to what is called long-run profit maximization.
Minimization of
short-run costs
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The production function
However
much of a commodity a business firm produces, it endeavours to produce it as
cheaply as possible. Taking the quality of the product and the prices of the
productive factors as given, which is the usual situation, the firm’s task is
to determine the cheapest combination of factors of productionthat can produce the
desired output. This task is best understood in terms of what is called theproduction function, i.e., an equation that
expresses the relationship between the quantities of factors employed and the
amount of product obtained. It states the amount of product that can be
obtained from each and every combination of factors. This relationship can be
written mathematically as y = f (x1, x2, . . . , xn; k1, k2, . . . , km).
Here, y denotes
the quantity of output. The firm is presumed to use n variable factors of production; that is, factors like hourly paid production workers and raw materials, the quantities
of which can be increased or decreased. In the formula the quantity of the
first variable factor is denoted by x1 and so
on. The firm is also presumed to use mfixed
factors, or factors like fixed machinery, salaried staff, etc., the quantities
of which cannot be varied readily or habitually. The available quantity of the
first fixed factor is indicated in the formal by k1 and so
on. The entire formula expresses the amount of output that results when
specified quantities of factors are employed. It must be noted that though the
quantities of the factors determine the quantity of output, the reverse is not
true, and as a general rule there will be many combinations of productive
factors that could be used to produce the same output. Finding the cheapest of
these is the problem of cost minimization.
The
cost of production is simply the sum of the costs of all of the various
factors. It can be written:
in
which p1 denotes
the price of a unit of the first variable factor, r1 denotes
the annual cost of owning and maintaining the first fixed factor, and so on.
Here again one group of terms, the first, covers variable cost (roughly“direct
costs” in accounting terminology), which can be changed readily; another group,
the second, covers fixed cost (accountants’
“overhead costs”), which includes items not easily varied. The discussion will
deal first with variable cost.
The principles involved in selecting the cheapest combination of
variable factors can be seen in terms of a simple example. If a firm
manufactures gold necklace chains in such a way that there are only two
variable factors, labour (specifically, goldsmith-hours) and gold wire, the
production function for such a firm will be y = f (x1, x2; k), in which the symbol k is
included simply as a reminder that the number of chains producible by x1 feet of
gold wire and x2 goldsmith-hours
depends on the amount of machinery and other fixed capital available. Since
there are only two variable factors, this production function can be portrayed
graphically in a figure known as an isoquant diagram.
Substitution of factors
The isoquants also illustrate an
important economic phenomenon: that of factor substitution. This means that one variable factor can be substituted
for others; as a general rule a more lavish use of one variable factor will
permit an unchanged amount of output to be produced with fewer units of some or
all of the others. In the example above, labour was literally as good as gold
and could be substituted for it. If it were not for factor substitution there
would be no room for further decision after y, the number of chains to be
produced, had been established.
The shape of the isoquants shown,
for which there is a good deal of empirical support, is very important. In
moving along any one isoquant, the more of one factor that is employed, the
less of the other will be needed to maintain the stated output; this is the
graphic representation of factor substitutability. But there is a corollary:
the more of one factor that is employed, the less it will be possible to reduce
the use of the other by using more of the first. This is the property known as
“diminishing marginal rates of substitution.” The marginal rate of
substitution of factor 1 for factor 2 is the number of units by which x1 can be reduced per unit increase in x,
output remaining unchanged. In the diagram, if feet of gold wire are indicated
by x1 and goldsmith-hours by x2, then the
marginal rate of substitution is shown by the steepness (the negative of the
slope) of the isoquant; and it will be seen that it diminishes steadily as x2 increases because it becomes harder and harder
to economize on the use of gold simply by taking more care. The remainder of
the analysis rests heavily on the assumption that diminishing marginal rates of
substitution are characteristic of the production process generally.
The cost
data and the technological data can now be brought together. The variable cost
of using x1,x2 units of the factors of production is written p1x1 + p2x2, and this
information can be added to the isoquant diagram (
Figure 2).
The straight line labelled v2, called
the v2-isocost
line, shows all the combinations of input that can be purchased for a specified
variable cost, v2. The
other two isocost lines shown are interpreted similarly. The general formula
for an isocost line is p1x1 + p2x2 = v, in which v is some particular variable cost. The
slope of an isocost line is found by dividing p2 by p1 and depends only on the ratio of the prices of
the two factors.
Figure 2).
The straight line labelled v2, called
the v2-isocost
line, shows all the combinations of input that can be purchased for a specified
variable cost, v2. The
other two isocost lines shown are interpreted similarly. The general formula
for an isocost line is p1x1 + p2x2 = v, in which v is some particular variable cost. The
slope of an isocost line is found by dividing p2 by p1 and depends only on the ratio of the prices of
the two factors.
Three isocost lines are shown,
corresponding to variable costs amounting to v1, v2, and v3. If 200
units are to be produced, expenditure of v1 on variable factors will not suffice since the v1-isocost
line never reaches the isoquant for 200 units. An expenditure of v3 is more than sufficient; and v2 is the lowest variable cost for which 200
units can be produced. Thus v2 is found to be the minimum variable cost of
producing 200 units (as v3 is of 300 units) and the coordinates of the
point where thev2 isocost line touches the 200-unit isoquant are
the quantities of the two factors that will be used when 200 units are to be
produced and the prices of the two factors are in the ratio p2/p1. It may
be noted that the cheapest combination for the production of any quantity will
be found at the point at which the relevant isoquant is tangent to an isocost
line. Thus, since the slope of an isoquant is given by the marginal rate of substitution,
any firm trying to produce as cheaply as possible will always purchase or hire
factors in quantities such that the marginal rate of substitution will equal
the ratio of their prices.
The isoquant–isocost diagram (or
the corresponding solution by the alternative means of the calculus) solves the
short-run cost minimization problem by determining the least-cost combination
of variable factors that can produce a given output in a given plant. The
variable cost incurred when the least-cost combination of inputs is used in
conjunction with a given outfit of fixed equipment is called the variable cost
of that quantity of output and denoted VC(y).
The total cost incurred, variable plus fixed, is the short-run cost of
that output, denoted SRC(y).
Clearly SRC(y) =
VC(y) + R(K), in which the second term
symbolizes the sum of the annual costs of the fixed factors available.
Marginal cost
Two other concepts now become
important. The average variable cost, written AVC(y), is the variable cost per unit of output.
Algebraically, AVC(y)
= VC(y)/y. The marginal variable
cost, or simply marginal cost [MC(y)]
is, roughly, the increase in variable cost incurred when output is increased by
one unit; i.e., MC(y)
= VC(y + 1) - VC(y). Though for theoretical purposes a more
precise definition can be obtained by regarding VC(y) as a continuous
function of output, this is not necessary in the present case.
The usual
behaviour of average and marginal variable costs in response to changes in the
level of output from a given fixed plant is shown in Figure 3.
In this figure costs (in dollars per unit) are measured vertically and output
(in units per year) is shown horizontally. The figure is drawn for some
particular fixed plant, and it can be seen that average costs are fairly
high for very low levels of output relative to the size of the plant, largely
because there is not enough work to keep a well-balanced work force fully occupied. People are either idle much of the time
or shifting, expensively, from job to job. As output increases from a low
level, average costs decline to a low plateau. But as the capacity of the plant
is approached, the inefficiencies incident on plant congestion force average
costs up quite rapidly. Overtime may be incurred, outmoded equipment and
inexperienced hands may be called into use, there may not be time to take
machinery off the line for routine maintenance; or minor breakdowns and delays
may disrupt schedules seriously because of inadequate slack and reserves. Thus
the AVC curve has the flat-bottomed U-shape shown. The MC curve, as might be
expected, falls faster and rises more rapidly than the AVC curve.
Figure 3.
In this figure costs (in dollars per unit) are measured vertically and output
(in units per year) is shown horizontally. The figure is drawn for some
particular fixed plant, and it can be seen that average costs are fairly
high for very low levels of output relative to the size of the plant, largely
because there is not enough work to keep a well-balanced work force fully occupied. People are either idle much of the time
or shifting, expensively, from job to job. As output increases from a low
level, average costs decline to a low plateau. But as the capacity of the plant
is approached, the inefficiencies incident on plant congestion force average
costs up quite rapidly. Overtime may be incurred, outmoded equipment and
inexperienced hands may be called into use, there may not be time to take
machinery off the line for routine maintenance; or minor breakdowns and delays
may disrupt schedules seriously because of inadequate slack and reserves. Thus
the AVC curve has the flat-bottomed U-shape shown. The MC curve, as might be
expected, falls faster and rises more rapidly than the AVC curve.Maximization of short-run profits
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The average and marginal cost
curves just deduced are the keys to the solution of the second-level problem,
the determination of the most profitable level of output to produce in a given plant. The only
additional datum needed is the price of the product, say p0.
The most profitable amount of
output may be found by using these data. If the marginal cost of any given
output (y) is less
than the price, sales revenues will increase more than costs if output is
increased by one unit (or even a few more); and profits will rise.
Contrariwise, if the marginal cost is greater than the price, profits will be
increased by cutting back output by at least one unit. It then follows that the
output that maximizes profits is the one for which MC(y) = p0. This is
the second basic finding: in response to any price the profit-maximizing firm
will produce and offer the quantity for which the marginal cost equals that
price.
Such a conclusion is shown in
Figure 3. In response to the price, p0, shown,
the firm will offer the quantity y* given by the value of y for which the ordinate of the MC curve
equals the price. If adenotes the corresponding
average variable cost, net revenue per unit will be equal to p0 - a, and the total excess of
revenues over variable costs will be y*(p0 - a), which is represented
graphically by the shaded rectangle in the figure.
Marginal cost and price
The conclusion that marginal cost
tends to equal price is important in that it shows how the quantity of output
produced by a firm is influenced by the market price. If the market price is
lower than the lowest point on the average variable cost curve, the firm will
“cut its losses” by not producing anything. At any higher market price, the
firm will produce the quantity for which marginal cost equals that price. Thus
the quantity that the firm will produce in response to any price can be found
in Figure 3 by reading the marginal cost curve, and for this reason the
marginal cost curve is said to be the short-run supply curve for the
firm.
The short-run supply curve for a
product—that is, the total amount that all the firms producing it will produce
in response to any market price—follows immediately, and is seen to be the sum
of the short-run supply curves (or marginal cost curves, except when the price
is below the bottoms of the average variable cost curves for some firms) of all
the firms in the industry. This curve is of fundamental importance for economic
analysis, for together with the demand curve for the product it determines the market price of the
commodity and the amount that will be produced and purchased.
One pitfall must, however, be
noted. In the demonstration of the supply curves for the firms, and hence of
the industry, it was assumed that factor prices were fixed. Though this is fair
enough for a single firm, the fact is that if all firms together attempt to
increase their outputs in response to an increase in the price of the product,
they are likely to bid up the prices of some or all of the factors of
production that they use. In that event the product supply curve as calculated
will overstate the increase in output that will be elicited by an increase in
price. A more sophisticated type of supply curve, incorporating induced changes
in factor prices, is therefore necessary. Such curves are discussed in the
standard literature of this subject.
Marginal product
It is now possible to derive the
relationship between product prices and factor prices, which is the basis of
the theory of income distribution. To this end, the marginal product of a
factor is defined as the amount that output would be increased if one more unit
of the factor were employed, all other circumstances remaining the same.
Algebraically, it may be expressed as the difference between the product of a
given amount of the factor and the product when that factor is increased by an
additional unit. Thus if MP1(x1) denotes
the marginal product of factor 1 when x1 units are employed, thenMP1(x1) = f(x1 + 1, x2, . . . ,xn; k)
- f(x1, x2 . . . ,xn; k).
The marginal products are closely related to the marginal rates of substitution
previously defined. If an additional unit of factor 1 will increase output by f1 units, for example, then one more unit of
output can be obtained by employing 1/f1more units
of factor 1. Similarly, if the marginal product of factor 2 is f2, then
output will fall by one unit if the use of factor 2 is reduced by 1/f2 units. Thus output will remain unchanged, to a
good approximation, if 1/f1 units of factor 1 are used to replace 1/f2 units of factor 2. The marginal rate of
substitution is therefore f2/f1, or the
ratio of the marginal products of the two factors. It has already been shown
that the marginal rate of substitution also equals the ratio of the prices of
the factors, and it therefore follows that the prices (or wages) of the factors
are proportional to their marginal products.
This is one of the most
significant theoretical findings in economics. To restate it briefly: factors
of production are paid in proportion to their marginal products. This is not a
question of social equity but merely a consequence of the efforts of businessmen
to produce as cheaply as possible.
Further, the marginal products of
the factors are closely related to marginal costs and, therefore, to product prices. For if one more unit
of factor 1 is employed, output will be increased by MP1(x1) units
and variable cost by p1; so the
marginal cost of additional units produced will be p1/MP1(x1).
Similarly, if additional output is obtained by employing an additional unit of
factor 2, the marginal cost will be p2/MP2(x2). But, as
shown above, these two numbers are the same; whichever factor i is used to increase output, the
marginal cost will be pi/MPi(xi) and,
furthermore, the firm will choose its output level so that the marginal cost
will be equal to the price, p0.
Therefore it has been established
that p1 = p0MP1(x1), p2 = p0MP2(x2), . . . ,
or the price of each factor is the price of the product multiplied by its
marginal product, which is the value of its marginal product. This, also, is a
fundamental theorem of income distribution and one of the most significant
theorems in economics. Its logic can be perceived directly. If the equality is
violated for any factor, the businessman can increase his profits either by
hiring units of the factor or by laying them off until the equality is
satisfied, and presumably the businessman will do so.
The theory of production decisions
in the short run, as just outlined, leads to two conclusions (of fundamental
importance throughout the field of economics) about the responses of business
firms to the market prices of the commodities they produce and the factors of
production they buy or hire: (1) the firm will produce the quantity of its
product for which the marginal cost is equal to the market price and (2) it
will purchase or hire factors of production in such quantities that the price
of the commodity produced multiplied by the marginal product of the factor will
be equal to the cost of a unit of the factor. The first explains the supply
curves of the commodities produced in an economy. Though the conclusions were
deduced within the context of a firm that uses two factors of production, they
are clearly applicable in general.
Maximization of long-run profits
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Relationship between the short run and the long run
The theory of long-run
profit-maximizing behaviour rests on the short-run theory that has just been
presented but is considerably more complex because of two features: (1)
long-run cost curves, to be defined below, are more varied in shape than the
corresponding short-run cost curves, and (2) the long-run behaviour of an
industry cannot be deduced simply from the long-run behaviour of the firms in
it because the roster of firms is subject to change. It is of the essence of
long-run adjustments that they take place by the addition or dismantling of
fixed productive capacity by both established firms and new or recently created
firms.
At any one time an established
firm with an existing plant will make its short-run decisions by comparing the
ruling price of its commodity with cost curves corresponding to that plant. If
the price is so high that the firm is operating on the rising leg of its
short-run cost curve, its marginal costs will be high—higher than its average
costs—and it will be enjoying operating profits, as shown in Figure 3. The firm
will then consider whether it could increase its profits by enlarging its
plant. The effect of plant enlargement is to reduce the variable cost of
producing high levels of output by reducing the strain on limited production
facilities, at the expense of increasing the level of fixed costs.
In response to any level of output
that it expects to continue for some time, the firm will desire and eventually
acquire the fixed plant for which the short-run costs of that level of output
are as low as possible. This leads to the concept of the long-run cost curve:
the long-run costs of any level of output are the short-run costs of producing
that output in the plant that makes those short-run costs as low as possible.
These result from balancing the fixed costs entailed by any plant against the
short-run costs of producing in that plant. The long-run costs of producing y are denoted by LRC(y). The average long-run cost
of y is
the long-run cost per unit of y [algebraically LAC(y) = LRC(y)/y]. The marginal long-run
cost is the increase in long-run cost resulting from an increase of one unit in
the level of output. It represents a combination of short-run and long-run
adjustments to a slight increase in the rate of output. It can be shown that
the long-run marginal cost equals the marginal cost as previously defined when
the cost-minimizing fixed plant is used.
Long-run cost curves
Cost curves
appropriate for long-run analysis are more varied in shape than short-run cost
curves
and fall into
three broad classes. In constant-cost industries, average cost is about the
same at all
levels of
output except the very lowest. Constant costs prevail in manufacturing
industries in which
capacity is
expanded by replicating facilities without changing the technique of
production, as a
cotton mill
expands by increasing the number of spindles. In decreasing-cost industries,
average cost
declines as
the rate of output grows, at least until the plant is large enough to supply an
appreciable
fraction of
its market. Decreasing costs are characteristic of manufacturing in which
heavy,
automated
machinery is economical for large volumes of output. Automobile and steel
manufacturing
are leading examples. Decreasing costs are inconsistent with competitive
conditions,
since they
permit a few large firms to drive all smaller competitors out of business.
Finally, in
increasing-cost
industries average costs rise with the volume of output generally because the
firm
cannot obtain
additional fixed capacity that is as efficient as the plant it already has. The
most
important
examples are agriculture and extractive
industries.
Criticisms of the theory
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The theory of production has been subject to much criticism. One objection
is that the concept of the production function is not derived from observation
or practice. Even the most sophisticated firms do not know the direct
functional relationship between their basic raw inputs and their ultimate
outputs. This objection can be got around by applying the recently developed
techniques of linear programming, which employ observable data without recourse
to the production function and lead to practically the same conclusions.
On another level the theory has been charged with excessive
simplification. It assumes that there are no changes in the rest of the economy
while individual firms and industries are making the adjustments described in
the theory; it neglects changes in the technique of production; and it pays no
attention to the risks and uncertainties that becloud all business decisions.
These criticisms are especially damaging to the theory of long-run profit
maximization. On still another level, critics of the theory maintain that
businessmen are not always concerned with maximizing profits or minimizing
costs.
Though all of the criticisms have merit, the simplified theory of
production does nevertheless indicate some basic forces and tendencies
operating in the economy. The theorems should be understood as conditions that
the economy tends toward, rather than conditions that are always and
instantaneously achieved. It is rare for them to be attained exactly, but it is
just as rare for substantial violations of the theorems to endure.
Only the simplest aspects of the theory were described above. Without much
difficulty it could be extended to cover firms that produce more than one
product, as almost all firms do. With more difficulty it could be applied to
firms whose decisions affect the prices at which they sell and buy (monopoly, monopolistic competition, monopsony).
The behaviour of other firms that recognize the possibility that their
competitors may retaliate (oligopoly) is still a theory of production subject
to controversy and research.
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