Frank Cowell: Microeconomics
October 2011
The Firm: Basics
MICROECONOMICS
Principles and Analysis
Frank Cowell
Overview...
The Firm: Basics
Frank Cowell: Microeconomics
The setting
The environment
for the basic
model of the firm.
Input requirement sets
Isoquants
Returns to scale
Marginal
products
The basics of production...
Frank Cowell: Microeconomics

Some of the elements needed for an analysis of the
firm
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Technical efficiency
Returns to scale
Convexity
Substitutability
Marginal products
This is in the context of a single-output firm...
...and assuming a competitive environment.
First we need the building blocks of a model...
Notation
Frank Cowell: Microeconomics

Quantities
zi
z = (z1, z2 , ..., zm )
•amount of input i
q
•amount of output
•input vector
For next
presentation

Prices
wi
w = (w1, w2 , ..., wm )
•price of input i
p
•price of output
•Input-price vector
Feasible production
Frank Cowell: Microeconomics
The basic relationship between
The production
output and function
inputs:

•single-output, multiple-input
production relation
q  f(z1, z2, ...., zm )
This can be written more compactly
•Note that we use “” and not
Vector of inputs
as:
“=” in the relation. Why?

q  f(z)

•Consider the meaning of f
f gives the maximum amount of
output that can be produced from a
given list of inputs
distinguish two
important cases...
Technical efficiency
Frank Cowell: Microeconomics
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Case 1:
q = f(z)
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Case 2:
q <f(z)
•The case where
production is technically
efficient
•The case where
production is
(technically) inefficient
Intuition: if the combination (z,q) is inefficient you can
throw away some inputs and still produce the same output
The function f
Frank Cowell: Microeconomics
q
q >f (z)
q =f (z)
0
q <f
 The production function
 Interior points are feasible
but inefficient
 Boundary points are feasible
(z) and efficient
 Infeasible points
z2
We need to
examine its
structure in
detail.
Overview...
The Firm: Basics
Frank Cowell: Microeconomics
The setting
The structure of
the production
function.
Input requirement sets
Isoquants
Returns to scale
Marginal
products
The input requirement set
Frank Cowell: Microeconomics

Pick a particular output level q

Find a feasible input vector z
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Repeat to find all such vectors
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Yields the input-requirement set
Z(q) := {z: f(z) q}
The shape of Z depends on the
assumptions made about production...
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
We will look at four cases.
remember, we must
have q  f(z)

The set of input
vectors that meet the
technical feasibility
condition for output q...

First, the
“standard” case.
The input requirement set
Frank Cowell: Microeconomics
 Feasible but inefficient
z2
 Feasible and technically
efficient
 Infeasible points.
Z(q)
q < f (z)
q = f (z)
q > f (z)
z1
Case 1: Z smooth, strictly convex
Frank Cowell: Microeconomics
 Pick two boundary points
 Draw the line between them
z2
 Intermediate points lie in the
interior of Z.
Z(q)

q = f (z')
z
Note important role of
convexity.
q< f (z)

A combination of two
techniques may produce
more output.
z
q = f (z")
z1
What if we changed
some of the assumptions?
Case 2: Z Convex (but not strictly)
Frank Cowell: Microeconomics
 Pick two boundary points
 Draw the line between them
z2
 Intermediate points lie in Z
(perhaps on the boundary).
Z(q)

z

z
z1
 A combination of
feasible techniques is
also feasible
Case 3: Z smooth but not convex
Frank Cowell: Microeconomics
 Join two points across the
“dent”
 Take an intermediate point
z2
 Highlight zone where this can
occur.
Z(q)
This point is
infeasible
 in this region there is an
indivisibility

z1
Case 4: Z convex but not smooth
Frank Cowell: Microeconomics
z2
q = f (z)
 Slope of the boundary is
undefined at this point.

z1
Summary: 4 possibilities for Z
Frank Cowell: Microeconomics
Standard case,
but strong
assumptions
about
divisibility and
smoothness
z2
z1
z1
Problems:
the "dent"
represents an
indivisibility
z2
z1
Almost
conventional:
mixtures may
be just as
good as single
techniques
z2
Only one
efficient point
and not
smooth. But
not perverse.
z2
z1
Overview...
The Firm: Basics
Frank Cowell: Microeconomics
The setting
Contours of the
production
function.
Input requirement sets
Isoquants
Returns to scale
Marginal
products
Isoquants
Frank Cowell: Microeconomics

Pick a particular output level q

Find the input requirement set Z(q)

The isoquant is the boundary of Z:
Think of the isoquant as
an integral part of the set
Z(q)...

{ z : f (z) = q }
If the function f is differentiable at z  Where appropriate, use
subscript to denote partial
then the marginal rate of technical
derivatives. So
substitution is the slope at z: fj (z)
f(z)
——
f
(z)
:=
——
i
fi (z)
zi .

Gives the rate at which you can trade
off one input against another along the
isoquant, to maintain constant output q

Let’s look at
its shape
Isoquant, input ratio, MRTS
Frank Cowell: Microeconomics
 The set Z(q).
 A contour of the function f.
 An efficient point.
z2
 The input ratio
 Marginal Rate of Technical
Substitution
z2 / z1= constant
MRTS21=f1(z)/f2(z)
z2°
 Increase the MRTS
 The isoquant is the
boundary of Z
 z′
 z°
{z: f(z)=q}
z1°
z1
 Input ratio describes one
production technique
MRTS21: implicit “price”
of input 1 in terms of 2
 Higher “price”: smaller
relative use of input 1
MRTS and elasticity of substitution
Frank Cowell: Microeconomics
z2
 Responsiveness
propsubstitution
change input ratio



prop change in MRTS
of inputs to MRTS is elasticity of
D input-ratio D MRTS
=   

input-ratio
MRTS
z2
s=½
∂log(z1/z2)
= 

∂log(f1/f2)
s=2
z1
z1
Frank Cowell: Microeconomics
Elasticity
of
substitution
z
2
 A constant elasticity of
substitution isoquant
 Increase the elasticity of
substitution...
structure of the
contour map...
z1
Homothetic contours
Frank Cowell: Microeconomics
 The isoquants
 Draw any ray through the
origin…
z2
 Get same MRTS as it cuts
each isoquant.
O
z1
Contours of a homogeneous function
Frank Cowell: Microeconomics
 The isoquants
z2
 Coordinates of input z°
 Coordinates of “scaled up”
input tz°
 tz°
tz2°
z2°
f(tz) = trf(z)
 z°
trq
q
O
z1°
tz1°
z1
Overview...
The Firm: Basics
Frank Cowell: Microeconomics
The setting
Changing all
inputs together.
Input requirement sets
Isoquants
Returns to scale
Marginal
products
Let's rebuild from the isoquants
Frank Cowell: Microeconomics
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The isoquants form a contour map.
If we looked at the “parent” diagram, what would we see?
Consider returns to scale of the production function.
Examine effect of varying all inputs together:
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Take three standard cases:
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Focus on the expansion path.
q plotted against proportionate increases in z.
Increasing Returns to Scale
Decreasing Returns to Scale
Constant Returns to Scale
Let's do this for 2 inputs, one output…
Case 1: IRTS
Frank Cowell: Microeconomics
q
 An increasing returns to
scale function
 Pick an arbitrary point on the
surface
 The expansion path…

0
z2
 t>1 implies
f(tz) > tf(z)
Double inputs
and you more
than double
output
Case 2: DRTS
Frank Cowell: Microeconomics
q
 A decreasing returns to
scale function
 Pick an arbitrary point on the
surface
 The expansion path…

0
z2
 t>1 implies
f(tz) < tf(z)
Double inputs
and output
increases by less
than double
Case 3: CRTS
Frank Cowell: Microeconomics
q
 A constant returns to scale
function
 Pick a point on the surface
 The expansion path is a ray

0
z2
 f(tz) = tf(z)
Double inputs
and output
exactly doubles
Relationship to isoquants
Frank Cowell: Microeconomics
q
 Take any one of the three
cases (here it is CRTS)
 Take a horizontal “slice”
 Project down to get the
isoquant
 Repeat to get isoquant map
0
z2
The isoquant
map is the
projection of the
set of feasible
points
Overview...
The Firm: Basics
Frank Cowell: Microeconomics
The setting
Changing one
input at time.
Input requirement sets
Isoquants
Returns to scale
Marginal
products
Marginal products
Frank Cowell: Microeconomics
Remember, this means
a z such that q= f(z)
Pick a technically efficient
input vector


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Keep all but one input constant
Measure the marginal change in
output w.r.t. this input

f(z)
MPi = fi(z) = ——
zi
.

The marginal product
CRTS production function again
Frank Cowell: Microeconomics
q
 Now take a vertical “slice”
 The resulting path for z2 =
constant
0
z2
Let’s look at
its shape
MP for the CRTS function
Frank Cowell: Microeconomics
q
f1(z)
f(z)
 The feasible set
 Technically efficient points
 Slope of tangent is the
marginal product of input 1
 Increase z1…
 A section of the
production function
Input 1 is essential:
If z1= 0 then q = 0
z1
f1(z) falls with z1 (or
stays constant) if f is
concave
Relationship between q and z1
Frank Cowell: Microeconomics
q
q
 We’ve just taken the
conventional case
z1
 But in general this
curve depends on the
shape of f.
 Some other
possibilities for the
relation between
output and one
input…
q
z1
z1
q
z1
Key concepts
Frank Cowell: Microeconomics
Review
Review
Review
Review
Review
Technical efficiency
 Returns to scale
 Convexity
 MRTS
 Marginal product
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What next?
Frank Cowell: Microeconomics
Introduce the market
 Optimisation problem of the firm
 Method of solution
 Solution concepts.
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