THE FIRM: BASICS
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Firm Basics
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Overview
The Firm: Basics
The setting
The environment
for the basic
model of the firm
Input requirement sets
Isoquants
Returns to scale
Marginal
products
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The basics of production 
 Some elements needed for an analysis of the firm
• 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
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Notation
 Quantities
zi
z = (z1, z2 , , zm )
•amount of input i
q
•amount of output
•input vector
For next
presentation
 Prices
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wi
w = (w1, w2 , , wm )
•price of input i
p
•price of output
•Input-price vector
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Feasible production
 The basic relationship between
output and inputs:
•single-output, multiple-input
production relation
q  f(z1, z2, , zm )
The production
function
 This can be written more
compactly as: Vector of inputs
•Note that we use “” and not
“=” in the relation. Why?
•Consider the meaning of f
q  f(z)
 f gives the maximum amount of
output that can be produced from a
given list of inputs
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distinguish two
important cases...
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Technical efficiency
 Case 1:
q = f(z)
 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
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The function f
q
q > f(z)
q = f (z)
0
q < f(z)
 The production function
 Interior points are feasible
but inefficient
 Boundary points are feasible
and efficient
 Infeasible points
z2
We need to examine
its structure in detail
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Overview
The Firm: Basics
The setting
The structure of
the production
function
Input requirement sets
Isoquants
Returns to scale
Marginal
products
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The input requirement set
 Pick a particular output level q
 Find a feasible input vector z
remember, we must
have q  f(z)

 Repeat to find all such vectors
 Yields the input-requirement set
•
Z(q) := {z: f(z) q}
 The shape of Z depends on the
assumptions made about production
The set of input
vectors that meet the
technical feasibility
condition for output q
 We will look at four cases
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First, the
“standard” case
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The input requirement set
 Feasible but inefficient
z2
 Feasible and technically efficient
 Infeasible points
Z(q)
q < f(z)
q = f(z)
q > f(z)
z1
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Case 1: Z smooth, strictly convex
 Pick two boundary points
 Draw the line between them
z2
 Intermediate points lie in the
interior of Z
Z(q)

q = f(z')
z
q< f (z)

Note important role of convexity
A combination of two techniques
may produce more output
z
q = f(z")
What if we changed some of
the assumptions?
z1
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Case 2: Z Convex (but not strictly)
 Pick two boundary points
 Draw the line between them
z2
 Intermediate points lie in Z
(perhaps on the boundary)
Z(q)

z

z
 A combination of feasible
techniques is also feasible
z1
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Case 3: Z smooth but not convex
 Join two points across the “dent”
z2
 Take an intermediate point
 Highlight zone where this can occur
Z(q)
This point is
infeasible
 in this region there is an
indivisibility

z1
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Case 4: Z convex but not smooth
z2
q = f(z)
 Slope of the boundary is
undefined at this point

z1
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Summary: 4 possibilities for Z
Standard case,
but strong
assumptions
about
divisibility and
smoothness
z2
Almost
conventional:
mixtures may
be just as
good as single
techniques
z2
z1
z1
z2
z2
Only one
efficient point
and not
smooth
Problems:
the dent
represents an
indivisibility
z1
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z1
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Overview
The Firm: Basics
The setting
Contours of the
production
function
Input requirement sets
Isoquants
Returns to scale
Marginal
products
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Isoquants
 Pick a particular output level q
Think of the isoquant as an
integral part of the set Z(q)

 Find the input requirement set Z(q)
 The isoquant is the boundary of
Z:{ z : f (z) = q }
 If the function f is differentiable at z  Where appropriate, use
then the marginal rate of technical subscript to denote partial
fj (z) derivatives. So
substitution is the slope at z: ——
f(z)
fi (z)
fi(z) := ——
zi .
 Gives rate at which you trade off
Let’s look at
one input against another along the
its shape
isoquant, maintaining constant q
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Isoquant, input ratio, MRTS
 The set Z(q)
 A contour of the function 
 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°
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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
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MRTS and elasticity of substitution
 Responsiveness of inputs to MRTS is elasticity of substitution
prop change input ratio
-  =
prop change in MRTS
input-ratio
 MRTS

 
input-ratio
MRTS
z2
∂log(z1/z2)
=  
∂log(f1/f2)
z2
s=½
s=2
z1
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z1
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Elasticity of substitution
z2
 A constant elasticity of
substitution isoquant
 Increase the elasticity of
substitution...
structure of the
contour map...
z1
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Homothetic contours
z2
O
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 The isoquants
 Draw any ray through the origin…
 Get same MRTS as it cuts each
isoquant
z1
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Contours of a homogeneous function
 The isoquants
z2
 Coordinates of input z°
 Coordinates of “scaled up” input tz°
 tz°
tz2°
z2°
f(tz) = trf(z)
 z°
trq
q
O
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z1°
tz1°
z1
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Overview...
The Firm: Basics
The setting
Changing all
inputs together
Input requirement sets
Isoquants
Returns to scale
Marginal
products
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Let's rebuild from the isoquants
 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:
• Focus on the expansion path
• q plotted against proportionate increases in z
 Take three standard cases:
• Increasing Returns to Scale
• Decreasing Returns to Scale
• Constant Returns to Scale
 Let's do this for 2 inputs, one output
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Case 1: IRTS
 An increasing returns to scale function
 Pick an arbitrary point on the surface
q
 The expansion path…

0
 t>1 implies f(tz) > tf(z)
z2
Double inputs and
you more than
double output
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Case 2: DRTS
 A decreasing returns to scale function
q
 Pick an arbitrary point on the surface
 The expansion path…

0
 t>1 implies f(tz) < tf(z)
z2
Double inputs and
output increases by less
than double
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Case 3: CRTS
 A constant returns to scale function
 Pick a point on the surface
q
 The expansion path is a ray

0
 f(tz) = tf(z)
z2
Double inputs and
output exactly doubles
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Relationship to isoquants
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
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Overview 
The Firm: Basics
The setting
Changing one
input at time
Input requirement sets
Isoquants
Returns to scale
Marginal
products
March 2012
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Marginal products
•
 Pick a technically efficient
input vector
Remember, this means a
z such that q= f(z)
 Keep all but one input constant
 Measure the marginal change in
output w.r.t. this input
•
The marginal product
f(z)
MPi = fi(z) = ——
zi
.
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CRTS production function again
q
 Now take a vertical “slice”
 The resulting path for z2 = constant
z2
0
Let’s look at
its shape
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MP for the CRTS function
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
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f1(z) falls with z1
(or stays constant) if
f is concave
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Relationship between q and z1
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
q
z1
z1
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Key concepts
Review
 Technical efficiency
Review
 Returns to scale
Review
 Convexity
Review
 MRTS
Review
 Marginal product
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What next?
 Introduce the market
 Optimisation problem of the firm
 Method of solution
 Solution concepts
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