Course 4 First Part BE 2013 28th of October

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


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


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


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


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.ppt

Documents

Course 4 First Part BE 2013 28th of October