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also: List of Economic Topics

PRODUCTION
FUNCTION
– microeconomics

In
microeconomics, a production function expresses the relationship between an organization’s
inputs and its outputs. It indicates, in mathematical or graphical form, what
outputs can be obtained from various amounts and combinations of factor inputs.
In particular it shows the maximum possible amount of output that can be produced
per unit of time with all combinations of factor inputs, given current factor
endowments and the state of available technology. Unique production functions
can be constructed for every production technology.

Alternatively,
a production function can be defined as the specification of the minimum input
requirements needed to produce designated quantities of output, given available
technology. This is just a reformulation of the definition above.

The
relationship is non-monetary, that is, a production function relates physical
inputs to physical outputs. Prices and costs are not considered. (For a primer
on the fundamental elements of physical production theory, see production
theory basics).

The
production function as an equation

In
its most general mathematical form, a production function is expressed as:

Q=
f(X1,X2,X3…)
where:
Q= quantity of output
X1, X2, X3, etc.= factor
inputs (such as capital, labour, raw materials, land, technology, or management)

There are several ways of specifying this function. One is as an additive
production function:

Q=
a + b X1 + c X2 + d X3
where a, b, c, and d are parameters that are determined
empirically.
Another is as a Cobb-Douglas production function (multiplicative):

Q=
aX1b X2c
Other forms include the constant elasticity of substitution production
function (CES) which is a generalized form of the Cobb-Douglas function, and the
quadratic production function which is a specific type of additive function. The
best form of the equation to use and the values of the parameters (a, b, c, and
d) vary from company to company and industry to industry. In a short run production
function at least one of the Xs (inputs) is fixed. In the long run all factor
inputs are variable at the discresion of management.

The
production function as a graph

Any
of these equations can be plotted on a graph. A typical (quadratic) production
function is shown in the following diagram. All points above the production function
are unobtainable with current technology, all points below are technically feasible,
and all points on the function show the maximum quantity of output obtainable
at the specified levels of inputs. From the origin, through points A, B, and C,
the production function is rising, indicating that as additional units of inputs
are used, the quantity of outputs also increases. Beyond point C, the employment
of additional units of inputs produces no additional outputs, in fact, total output
starts to decline. The variable inputs are being used too intensively (or to put
it another way, the fixed inputs are under utilized). With too much variable input
use relative to the available fixed inputs, the company is experiencing negative
returns to variable inputs, and diminishing total returns. In the diagram this
is illustrated by the negative marginal physical product curve (MPP) beyond point
Z, and the declining production function beyond point C.

Quadratic
Production FunctionFrom the origin to point A, the firm is experiencing increasing
returns to variable inputs. As additional inputs are employed, output increases
at an increasing rate. Both marginal physical product (MPP) and average physical
product (APP) is rising. The inflection point A, defines the point of diminishing
marginal returns, as can be seen from the declining MPP curve beyond point X.
From point A to point C, the firm is experiencing positive but decreasing returns
to variable inputs. As
additional inputs are employed, output increases but at a decreasing rate. Point
B is the point of diminishing average returns, as shown by the declining slope
of the average physical product curve (APP) beyond point Y. Point B is just tangent
to the steepest ray from the origin hence the average physical product is at a
maximum. Beyond point B, mathematical necessity requires that the marginal curve
must be below the average curve (See production theory
basics for an explanation.).

The
stages of production

To
simplify the interpretation of a production function, it is common to divide its
range into 3 stages. In Stage 1 (from the origin to point B) the variable input
is being used with increasing efficiency, reaching a maximum at point B (since
the average physical product is at its maximum at that point). The average physical
product of fixed inputs will also be rising in this stage (not shown in the diagram).
Because the efficiency of both fixed and variable inputs is improving throughout
stage 1, a firm will always try to operate beyond this stage. In stage 1, fixed
inputs are underutilized.

In
Stage 2, output increases at a decreasing rate, and the average and marginal physical
product is declining. However the average product of fixed inputs (not shown)
is still rising. In this stage, the employment of additional variable inputs increase
the efficiency of fixed inputs but decrease the efficiency of variable inputs.
The optimum input/output combination will be in stage 2. Maximum production efficiency
must fall somewhere in this stage. Note that this does not define the profit maximizing
point. It takes no account of prices or demand. If demand for a product is low,
the profit maximizing output could be in stage 1 even though the point of optimum
efficiency is in stage 2.

In
Stage 3, too much variable input is being used relative to the available fixed
inputs: variable inputs are overutilized. Both the efficiency of variable inputs
and the efficiency of fixed inputs decline through out this stage. At the boundary
between stage 2 and stage 3, fixed input is being utilized most efficiently and
short-run output is maximum.

Shifting
a production function

As
noted above, it is possible for the profit maximizing output level to occur in
any of the three stages. If profit maximization occurs in either stage 1 or stage
3, the firm will be operating at a technically inefficient point on its production
function. In the short run it can try to alter demand by changing the price of
the output or adjusting the level of promotional expenditure. In the long run
the firm has more options available to it, most notably, adjusting its production
processes so they better match the characteristics of demand. This usually involves
changing the scale of operations by adjusting the level of fixed inputs. If fixed
inputs are lumpy, adjustments to the scale of operations may be more significant
than what is required to merely balance production capacity with demand. For example,
you may only need to increase production by a million units per year to keep up
with demand, but the production equipment upgrades that are available may involve
increasing production by 2 million units per year.

If
a firm is operating (inefficiently) at a profit maximizing level in stage one,
it might, in the long run, choose to reduce its scale of operations (by selling
capital equipment). By reducing the amount of fixed capital inputs, the production
function will shift down and to the left. The beginning of stage 2 shifts from
B1 to B2. The (unchanged) profit maximizing output level will now be in stage
2 and the firm will be operating more efficiently.

If
a firm is operating (inefficiently) at a profit maximizing level in stage three,
it might, in the long run, choose to increase its scale of operations (by investing
in new capital equipment). By increasing the amount of fixed capital inputs, the
production function will shift up and to the right.

Homogeneous
and homothetic production functions

There
are two special classes of production functions that are frequently mentioned
in textbooks but are seldom seen in reality. The production function Q=f(X1,X2)
is said to be homogeneous of degree n, if given any positive constant k, f(kX1,kX2)=knf(X1,X2).
When n>1, the function exhibits increasing returns, and decreasing returns
when n<1. When it is homogeneous of degree 1, it exhibits constant returns.

Homothetic
functions are a special class of homogeneous function in which the marginal rate
of technical substitution is constant along the function.

Aggregate
production functions

In
macroeconomics, production functions for whole nations are sometimes constructed.
In theory they are the summation of all the production functions of individual
producers, however this is an impractical way of constructing them. There are
also methodological problems associated with aggregate production functions.

Criticisms
of production functions

During
the 1950s, 60s, and 70s there was a lively debate about the theoretical soundness
of production functions. (See the Capital controversy.) Although most of the criticism
was directed primarily at aggregate production functions, microeconomic production
functions were also put under scrutiny. The debate began in 1953 when Joan Robinson
complained about the way the factor input, capital, was measured and how the notion
of factor proportions had distracted economists.

According
to the argument, it is impossible to conceive of an abstract quantity of capital
which is independent of the rates of interest and wages. The problem is that this
independence is a precondition of constructing an iso-product curve. Further,
the slope of the iso-product curve helps determine relative factor prices, but
the curve cannot be constructed (and its slope measured) unless the prices are
known beforehand.

See
also:

Production
theory basics
Production, costs,
and pricing
Production
possibility frontier
Microeconomics

References

A
further description of production functions (http://cepa.newschool.edu/het/essays/product/prodfunc.htm)

Heathfield, D. F. (1971) Production Functions, Macmillan studies in economics,
Macmillan Press, New York.
Moroney, J. R. (1967) Cobb-Douglass production
functions and returns to scale in US manufacturing industry, Western Economic
Journal, vol 6, no 1, December 1967, pp 39-51.
Pearl, D. and Enos, J. (1975)
Engineering production functions and technological progress, The Journal of Industrial
Economics, vol 24, September 1975, pp 55-72.
Robinson, J. (1953) The production
function and the theory of capital, Review of Economic Studies, vol XXI, 1953,
pp. 81-106
Shephard, R (1970) Theory of cost and production functions, Princeton
University Press, Princeton NJ.
Thompson, A. (1981) Economics of the firm,
Theory and practice, 3rd edition, Prentice Hall, Englewood Cliffs. ISBN 0-13-231423-1