Firm: basics
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Transcript Firm: basics
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
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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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