Firm: Basics
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Transcript Firm: Basics
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
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
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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
The function f
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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
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...
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
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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
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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)
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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
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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
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z2
q = f (z)
Slope of the boundary is
undefined at this point.
z1
Summary: 4 possibilities for Z
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Standard case,
but strong
assumptions
about
divisibility and
smoothness
z2
z1
z2
z1
Problems:
the "dent"
represents an
indivisibility
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
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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
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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
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Elasticity
of
substitution
z
2
A constant elasticity of
substitution isoquant
Increase the elasticity of
substitution...
structure of the
contour map...
z1
Homothetic contours
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The isoquants
Draw any ray through the
origin…
z2
Get same MRTS as it cuts
each isoquant.
O
z1
Contours of a homogeneous function
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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
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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:
Take three standard cases:
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
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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
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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
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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
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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
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
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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
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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
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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
What next?
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Introduce the market
Optimisation problem of the firm
Method of solution
Solution concepts.