Production and Technology
Philip A. Viton
May 1, 2012
Philip A. Viton
CRP 781 —() Production
May 1, 2012
1/1
The Basic Model
Inputs
Technology
Output
A …rm uses inputs (or factors of production) to produce one or more
outputs.
The inputs are transformed into outputs via a production technology.
(Of course, this varies by product/industry).
We will generally restrict attention to a single output.
Notation: inputs are z1 , z2 , . . . , zN ; output is q.
Philip A. Viton
CRP 781 —() Production
May 1, 2012
2/1
Technology
The production technology is governed by the state of
knowledge/practice in science and engineering (for a particular
industry).
Technology changes over time: this is the question of technical
change. Though it is very important, we do not consider it here: we
consider only the state of technology at a given time (today).
Philip A. Viton
CRP 781 —() Production
May 1, 2012
3/1
Production Function
We focus on production of a given output (cars, wheat, bus transit,
housing, etc).
We summarize the state of technology via a function giving, for a
given input bundle, the maximum output that can be produced using
that bundle.
This function is called a production function.
Write it as:
q = f (z1 , z2 , . . . , zN )
= f (z )
where f is the production function and z = (z1 , z2 , . . . , zN ), an input
bundle.
Philip A. Viton
CRP 781 —() Production
May 1, 2012
4/1
Production Possibilities
Di¤erent input bundles can
usually be used to produce the
same quantity of output.
z2
For a 2-input production
function q = f (z1 , z2 ) …rst …x
output at q1 units.
Now consider all input bundles
that can produce this output.
q1
Connect up the bundles.
z1
Philip A. Viton
The result is an isoquant for
output level q1 .
CRP 781 —() Production
May 1, 2012
5/1
Isoquants
Any bundle on an isoquant is a di¤erent way to produce the given
quantity of output (the label of the isoquant)
“Higher” isoquants represent greater levels of output
Isoquants are dense in input space, slope downwards, and cannot cross
Isoquants are much like indi¤erence curves with the important
di¤erence that their labels represent actual physical quantities of
output.
Isoquants (production functions), unlike indi¤erence curves (utility
functions), can be estimated from data.
Philip A. Viton
CRP 781 —() Production
May 1, 2012
6/1
Factor Substitution I
Factor substitution refers to the possibility of substituting one input
for another while maintaining the same level of output (ie remaining
on a given isoquant).
The possibilities for factor substitution are revealed by the curvature
(shape) of an isoquant, or by its slope at a given input bundle.
The slope of an isoquant is called the Marginal Rate of Technical
Substitution (MRTS).
It is analogous to the MRS for isoquants.
Philip A. Viton
CRP 781 —() Production
May 1, 2012
7/1
Factor Substitution II
z2
Starting from bundle A on the
q1 -isoquant we attempt to
reduce utilization of z1 by ∆z1
∆z2
We can do that at a “cost” of
using ∆z2 more of z2
A
q1
∆z1
Philip A. Viton
z1
Since ∆z2 is large, we conclude
that substituting z2 for z1 is
di¢ cult relative of ∆z1 .
CRP 781 —() Production
May 1, 2012
8/1
Factor Substitution III
The …gure shows a di¤erent
isoquant for output q1 .
z2
We attempt the same
substitution as before (reduce
utilization of z1 by ∆z1 ).
A
∆z2
q1
∆z1
Philip A. Viton
z1
We see that substituting z2 for
z1 is easy in the sense that
relatively little more z2 is needed
if we want to utilize less z1 .
CRP 781 —() Production
May 1, 2012
9/1
Factor Substitution IV
z2
∆z2
Conversely, in the previous case,
trying to cut back on z2 by ∆z2
requires that we utilize a lot
more z1 .
A
q1
∆z1
Philip A. Viton
It is di¢ cult to substitute z1 for
z2 .
z1
CRP 781 —() Production
May 1, 2012
10 / 1
Marginal Product of a Factor
Given q1 = f (z1 , z2 ) we increase our utilization of z1 by a small
amount ∆z1 holding all other inputs …xed at their original levels.
Normally this will result in being able to produce more output:
q2 = f (z1 + ∆z1 , z2 ).
The change in output is ∆q = q2
q1 .
We de…ne the marginal product of the factor (here, z1 ) as
MP (z1 ) = .
∆q
∆z1
The marginal product tells us the change in output we can get for
each unit increase in the input.
Marginal products are usually taken to be positive (if you use more
labor, you can usually produce more product, for example) but not
always: a marginal product can be negative (for a speci…c factor).
Philip A. Viton
CRP 781 —() Production
May 1, 2012
11 / 1
Returns to Scale I
Given q1 = f (z1 , z2 ) or more generally q1 = f (z ) we consider scaling
up all inputs by the same factor λ.
Let this result in output q2 , so q2 = f (λz1 , λz2 ), or q2 = f (λz ).
Let the new output be related to the previous output by q2 = θq1 .
That is, if we scale up all inputs by the factor λ, and we …nd that
output increases by a factor of θ. (We ought strictly to write
q2 = θ (λ)q1 , to remind us that θ will depend on our pre-selected λ).
We now categorize the relation between λ (the scale factor for inputs)
and θ (the resulting observed scale factor for output, given that all
inputs have been scaled by λ).
Philip A. Viton
CRP 781 —() Production
May 1, 2012
12 / 1
Returns to Scale II
If θ > λ we say that production is subject to increasing returns to
scale (IRTS).
Example: we scale up inputs by λ = 2 and output increases by
θ = 2.5. This is IRTS.
If θ < λ we say that production is subject to decreasing returns to
scale (DRTS).
Example: we scale up inputs by λ = 2 but we …nd that output
increases by only θ = 1.5. This is DRTS.
If θ = λ we say that production is subject to constant returns to scale
(CRTS).
Example: we scale up all inputs by λ = 2 and we …nd that output also
doubles. This is CRTS.
Philip A. Viton
CRP 781 —() Production
May 1, 2012
13 / 1
Returns to Scale: Alternative Notation
The usual textbook notation for returns to scale is a bit di¤erent.
Instead of θ (the scale factor for output) we write λα : this has the
advantage of relating the change in output directly to the change in
inputs (λ). It also links the economics notion of returns to scale to
the mathematical concept of a homogeneous function, which is useful
in advanced treatments.
So, in this notation, scaling all inputs by λ results in output scaling by
a factor of λα .
Under this notation:
α = 1 : CRTS
α > 1 : IRTS
α < 1 : DRTS
Philip A. Viton
CRP 781 —() Production
May 1, 2012
14 / 1
Returns to Scale and Technology
Note that returns to scale (and marginal product) is a property of a
production technology, hence is the determined by the (current) state
of science and engineering.
We do not choose whether we are subject to IRTS, DRTS or CRTS :
the technology tells us (dictates) what the possibilities are.
It is entirely possible that (the degree of) returns to scale changes as
the output level changes: for example, we could go from IRTS at low
levels of output to DRTS at high levels of output.
Philip A. Viton
CRP 781 —() Production
May 1, 2012
15 / 1