cost from iso ii

8/8/2019 Cost From Iso II

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Isoquant to Cost Curves

1998 Peter Berck


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R eview Technique

Technique to make Q*: bundle of inputsthat makes Q*Efficient technique to make Q*:

x is an efficient technique if there is notechnique, y,that also makes Q* such that y

has less of one input and not more of anyinput


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Isoquant and ProductionFunction

The Q* isoquant: { x | x is an efficient technique and x produces Q*}Production function: Q = F(x). Output asfunction of (efficient) input bundles

{x| F(x) = Q*, x efficient} is also isoquant

Isoquant is level curve of production functionsee the physical model


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Bressler 1952 Example

y = 20+ 6.67 x2 + 10 x3 - .5 x3 x2


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Cost function

M inimum amount of money necessary tobuy the inputs that will produce output Q.

Answer is amount of money as function of QIsocost line, I: {x | I = p 1x1 + p 2x2}

Straight lineIntercept I/p 2Slope - p 1 /p 2


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Problem Again

G iven that we want to produce onisoquant Q*, which technique on Q*should we choose?

Answer: The technique on Q* that is onthe lowest possible isocost line

The graph: Q* isoquant and manyisocost lines.


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G raph

Two goodsOther stuff Clean Air Services

negative of pollutionair has 1 ppm of gunk pollutonair has 99 ppm of non-gunk cleanth


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Cost Min Technique

0

0

40

60

0

00

0

0 0 40 60

Air

O t h e r

S t u f f

LowIsocostMed.IsocostHighIsocost

Price of Other Stuff = 2

Equations for 3 lines. Cost of Chosen bundle?


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Isocost Lines:

0

0

40

60

0

00

0

0 0 40 60

Air

O t e r

S t f f



Blue Isocost:slope -2=- p

1/p

2; p

1= 4; I =

80*2=160; 160 =4 Air + 2 OSGreen Isocost: 200 = 4 Air + 2 OS.Red Isocost: 120 = 4 Air + 2 OS


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C(Q*) = 1

4

1

1

4

A r

O t

r S t f f L

owIsocostMed.IsocostHighIsocost


cost 200

cost 160

Chosen(24,32)


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C(Q 1 )=12 , C(Q*)=1 , C(Q 2 )=2

2

4

1

12

2 4CAS

O S



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C(Q)

Plot Q1, Q2,Q3 against 120,160,200.That is your cost curve.

You can choose any set of increasing Q sgive the information you have been given.


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Pollution Control


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Technology tandard

Technology is a way to do something(see above)Technology Standard

Use a specific technologycatalytic converters on cars.

scrubbers on coal fired power plants.One chooses a technology standard toreduce emissions


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Effluent tandard

Effluent (or emissions) StandardCan emit no more than X tons per (chooseone)

megawatt hour (output)per year (absolute!)per ton of coal burned (per input)

Obviously get very different results dependingon what you choose


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TBE

Technology Based Effluent StandardFirst find a technology that reduces emissionsat a reasonable cost Find out how much emissions would go downThen set an emissions standard for that

amount.Used in both Clean Air Act and Clean Water Act


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TBE

A r

t

r

t f f

Regula t r

f t echnique tu s e onl y 20 uni

o f A i r and m ake

Q*. I ne ff icien tT echnique



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The R egulation:

When you make Q*, you may use nomore than 20 units of clean air services.

You may use the technique the regulatoryengineers have discovered (20,100) orany other technique that uses no more

than 20 units of air and has output Q*


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W hy this way?

Regulator knows that it can be doneRegulator has upper bound on cost Regulator is assured of cleaning up theair.


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R esponse to TBE

0

20

40

60

80

100

0 20 40 60

A ir

t h e r

t u f f

( 20,50 )

T echnique (20,50) costs 180 and is leastcost way to make Q* using 20 units of air

Technique (20,80), the basis for the regulation, costs240 and makes Q*.


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Back Door Economics

Best Practicable Technologyused for water pre 1977means known technology at reasonable cost

Best Available Technologyused for water post 1983

means any technology; but in practice islimited by cost

Intent: Cleaner water under BAT.


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An exercise

Let Q = k x, where x is an input and k is apositive number. Let w be the price of the input x.What is the least cost way of making Q?What is C(Q)?


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Conditional Factor Demand

How much of an input will be used as afunction of output required and prices of inputs?X(Q,p)How could changing the price of clean airresult in the same usage of clean air / unit output as the TBES regulations?


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O ur Assumption

Firm s need to dispose of waste gas,which they vent to the air. It is never freeto vent the gas--it requires fans to push it out.Firm s can dispose of less gas and make

the same output by using more of anotherinput. For instance, by buying capital inthe form of an afterburner.


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Another application

In India the ratio of the price of labor tocapital is much less than in the USThe USSR consistently priced capitalbelow its true value to the evil empire.Does this help explain the emphasis on

heavy industry and big dams?


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Air as a function of price

0

0

00

0

00

0

0 0 40 60

Air

O t e r

S t u f f

4;

A 4

6 7;

A 6



28/37

Conditional Factor Demand

3

ant ity o Clean i r Used

r i c e o

C l e a n

i r

In this chart the output is held constant at Q*.


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U sing Prices

0

20

40

60

80

100

0 20 40 60

Air

t h e r

t u f f

( 20,50 )

Slope on High Price line is -100/15 = -p 1 /2 so p = 13.3.

A price for air of 13.3achieves the same level of clean air as the T BES of 20units of air.


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U sing Prices

0

20

40

60

80

100

0 20 40 60

Air

t h e r

t u f f

( 20,50 )

Pollution charge of 13.3- 4 = 9.3 adds $465 to cost

Before pollution charge, italready cost $4/unit to usethe air to dispose of waste


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S ummary S o Far

Both a TBES and a pollution charge canproduce the same level of use of clean air

services and pollution. A TBES does not cost the firm, so C(Q;TBES) < C(Q; pollution charge) when theTBES and charge result in the same use of air

AC is lower with a Quota!


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Does MC increase?

The case of the quota adjusted for output Assume quota increases from 20 to 28units with additional output.Next slide is the Q and Q+1 isoquant.MC under price or quantity regulation isthe cost of the inputs to go from Q toQ+1


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W hat of MC?

ir

t h

r

t

f f

,

,

+

Additional Inputs = (28,60) - (20,50) = (8,10)


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MC is cost of added inputs

MC under price regulation is:additional inputs (8,10)

at prices (13.3,20)MC is $306

MC under a quota:$2 unit * 10 units is $20$ 0 unit * 8 units is $0MC is $200


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Tax and Quota S ame?

MC is steeper under an output increasingquota

AC is lower under (any) quotaLong run: Long Run Supply is where P =

ACWith entry of new firms industry output goesupRequires pollution quota for new firms


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S ome R eality

Quotas are often per unit of output allows expansion of output more cheaply

but pollution expands too Another plan is a fixed quota acrossindustries that is tradeable.

expansion in one industry meanscontraction in anotheror more efficient use of pollution quotapollution remains fixed


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R eview: C(Q) and X(Q*,p)

0

50

100

150

0 20 40 60CAS

O S

Price of OS is 3

cost from iso ii

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