*See
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