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8/11/2019 Course 4 First Part BE 2013 28th of October
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The Firm: Basics
MICROECONOMICSPrinciples and Analysis
Frank Cowell
October 2011
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Overview...
The setting
Input require-ment sets
Returns to scale
Marginal
products
The Firm: Basics
The environmentfor the basic
model of the firm.Isoquants
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The basics of production...
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...
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Notation
Quantities
zi amount of input iz= (z
1
, z2
, ..., zm
) input vector
amount of outputq
Prices
price of input i
w= (w1, w2, ..., wm) Input-price vector
price of outputp
For next
presentation
wi
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Feasible production
The basic relationship between
output and inputs:
q f (z1, z2, ...., zm)
This can be written more compactly
as:
q f (z)
single-output, multiple-input
production relation
fgives the maximumamount of
output that can be produced from a
given list of inputs
Note that we use and not
= in the relation. Why?
Consider the meaning of f
Vector of inputs
The production
function
distinguish two
important cases...
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Technical efficiency
The case where
production is technically
efficient
The case where
production is
(technically) inefficient
Case 1:
q =f (z)
Case 2:
q < f (z)
Intuition: if the combination (z,q)is inefficient you can
throw away some inputs and still produce the same output
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z2
q>f (z)
Bou ndary po ints are feasib le
and eff icient
The function fq
0
The product ion funct ion
Interio r poin ts are feasible
but inef f ic ient
Infeasib le poin ts
q
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Overview...
The setting
Input require-ment sets
Returns to scale
Marginal
products
The Firm: Basics
The structure ofthe production
function.Isoquants
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The input requirement set
remember, we must
haveq f (z)
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 set of input
vectors that meet the
technical feasibility
condition for output q...The shape ofZdepends on the
assumptions made about production...
We will look at four cases. First, thestandard case.
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z1
z2
The input requirement set
Feasib le but ineff icient
Feasib le and technical ly
eff icient
Infeasib le points .
Z(q)
q f(z)
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z1
z2
Case 1:Zsmooth, strictly convex
Pick two boundary po in ts
Draw the l ine b etween them
Intermediate points l ie in the
interio r of Z.
A combination of twotechniques may produce
more output.
What if we changed
some of the assumptions?
z
z
Z(q)
q
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Case 2:ZConvex (but not strictly)
z1
z2
A combination of
feasible techniques is
also feasible
Z(q)
z
z
Pick two boundary po in ts
Draw the l ine b etween them
Intermediate points l ie in Z
(perhaps on the boun dary).
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Case 3:Zsmooth but notconvex
z1
z2
in this region there is an
indivisibility
Jo in two po in ts across the
dent
Z(q)
Take an intermediate point
Highl ight zone where this can
occur.
This point is
infeasible
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z1
z2
Case 4:Zconvex but not smooth
Slope of the boundary isundefined at this point.
q =f(z)
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z1
z2
z1
z2
z1
z2
z1
z2
Summary: 4 possibilities forZ
Only one
efficient pointand not
smooth. But
not perverse.
Problems:
the "dent"represents an
indivisibility
Standard case,
but strong
assumptions
about
divisibility andsmoothness
Almost
conventional:
mixtures may
be just as
good as singletechniques
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Overview...
The setting
Input require-ment sets
Returns to scale
Marginal
products
The Firm: Basics
Contours of the
production
function.Isoquants
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IsoquantsPick a particular output level q
Find the input requirement setZ(q)
The isoquantis the boundary ofZ:
{ z : f(z) = q }
Think of the isoquant as
an integral part of the set
Z(q)...
f(z)fi(z) :=
zi .
fj (z)fi (z)
If the function fis differentiable at z
then the marginal rate of technical
substitutionis the slope at z:
Where appropriate, use
subscript to denote partial
derivatives. So
Gives the rate at which you can trade
off one input against another along the
isoquant, to maintain constant output q
Lets look at
its shape
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z1
z2The set Z(q).
Isoquant, input ratio, MRTS
A conto ur of the funct ion .
{z: f (z)=q}
An eff ic ient point.
Input ratio describes one
production techniquez2
z1
z
The input rat io
z2 /z1= constant
Marginal Rate of Techn ical
Subst i tut ion
The isoquant is the
boundary of Z
Increase the MRTS
z
MRTS21=f1(z)/f2(z)
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 log(z1/z2)= log(f1/f2)
prop change input ratio
prop change in MRTS
Dinput-ratio DMRTS=
input-ratio MRTS
z1
z2
s=
z1
z2
s= 2
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Elasticity of substitution
z1
z2
structure of the
contour map...
A cons tant elast ic i ty of
subst i tu t ion isoquant
Increase the elast icity o f
substitut ion.. .
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Homothetic contours
The isoquants
O z1
z2Draw any ray through the
origin
Get same MRTS as it c uts
each isoqu ant .
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Contours of a homogeneous function
The isoquants
O
z1
z2 Coordinates of input z
Coordinates of scaled up
input tz
O tz1
tz2
z2
z1
tz
z
q
trq
f (tz)= trf (z)
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Overview...
The setting
Input require-ment sets
Returns to scale
Marginal
products
The Firm: Basics
Changing all
inputs together.
Isoquants
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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 scaleof the production function.
Examine effect of varying all inputs together: Focus on the expansion path.
qplotted 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
z2
q
0
An increasing returns to
scale funct ion
t>1 impliesf(tz)> tf(z)
Pick an arbit rary point on the
surface
The expansion path
Double inputs
and you more
than double
output
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Case 2: DRTSq
A decreasing returns to
scale funct ion
Pick an arbit rary point on the
surface
The expansion path
z2
0
t>1 impliesf(tz)< tf(z)
Double inputs
and output
increases by less
than double
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Case 3: CRTS
z2
q
0
A constant returns to s cale
funct ion
Pick a point o n the surface
The expansion path is a ray
f(tz)= tf(z)
Double inputs
and output
exactly doubles
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Relationship to isoquants
z2
q
0
Take any on e of the three
cases (here it is CRTS)
Take a horizontal slice
Project down to g et the
isoquant
Repeat to get isoq uant map
The isoquant
map is the
projection of the
set of feasible
points
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Overview...
The setting
Input require-ment sets
Returns to scale
Marginal
products
The Firm: Basics
Changing one
input at time.
Isoquants
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Marginal products
Measure the marginal change in
output w.r.t. this input
f(z)MPi= fi(z) =
zi .
Pick a technically efficient
input vector
Keep all but one input constant
Remember, this means
a zsuch that q= f(z)
The marginal product
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CRTS production function again
z2
q
0
Now take a vertical slice
The result ing path for z2=
constant
Lets look at
its shape
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MP for the CRTS function
z1
q
f(z)
The feasible set
A section of the
production function
Technical ly ef f ic ient po ints
Input 1 is essential:
Ifz1= 0 thenq = 0
Slope of tangent is the
marginal produc t of input 1
f1(z)
Increase z1
f1(z) falls withz1(or
stays constant) iff is
concave
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Relationship between qandz1
z1
q
z1
q
z1
q
z1
q
Weve just taken the
conventional case
But in general thiscurve depends on the
shape of f.
Some other
possibilities for the
relation betweenoutput and one
input
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Key concepts
Technical efficiency
Returns to scale
Convexity
MRTS
Marginal product
Review
Review
Review
Review
Review
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What next?
Introduce the market
Optimisation problem of the firm
Method of solution
Solution concepts.
http://localhost/var/www/apps/conversion/tmp/scratch_9/FirmOptimisation.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_9/FirmOptimisation.pptRecommended