Isoquant to Cost Curves

© 1998 Peter Berck

Review TechniqueTechnique to make Q*: bundle of

inputs that makes Q*Efficient technique to make Q*:

x is an efficient technique if there is no technique, y,that also makes Q* such that y has less of one input and not more of any input

Isoquant and Production FunctionThe Q* isoquant: { x | x is an efficient

technique and x produces Q*}Production function: Q = F(x). Output

as function of (efficient) input bundles {x| F(x) = Q*, x efficient} is also isoquant Isoquant is level curve of production

function see the physical model

Bressler 1952 Exampley = 20+ 6.67 x2 + 10 x3 - .5 x3

x2

Cost functionMinimum amount of money necessary to

buy the inputs that will produce output Q. Answer is amount of money as function of Q

Isocost line, I: {x | I = p1x1 + p2x2} Straight line

Intercept I/p2

Slope - p1/p2

Problem AgainGiven that we want to produce on

isoquant Q*, which technique on Q* should we choose?

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

The graph: Q* isoquant and many isocost lines.

GraphTwo goods

Other stuff Clean Air Services

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

Cost Min Technique

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IsocostMed.IsocostHighIsocost

Price of “Other Stuff” = 2

Equations for 3 lines. Cost ofChosen bundle?

Isocost Lines:

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IsocostMed.IsocostHighIsocost

Price of “Other Stuff” = 2

Blue Isocost:slope -2=- p1/p2; p1 = 4; I = 80*2=160; 160 =4 Air + 2 OS Green Isocost: 200 = 4 Air + 2 OS .Red Isocost: 120 = 4 Air + 2 OS

C(Q*) = 160

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Price of “Other Stuff” = 2

cost 200

cost 160

Chosen(24,32)

C(Q1)=120, C(Q*)=160, C(Q2)=200

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OS

LowIsocostMed.IsocostHighIsocost

C(Q)Plot Q1, Q2,Q3 against 120,160,200.That is your cost curve.You can choose any set of increasing

Q’s give the information you have been given.

Pollution Control

Technology StandardTechnology 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 to reduce emissions

Effluent StandardEffluent (or emissions) Standard

Can emit no more than X tons per (choose one)megawatt hour (output)per year (absolute!)per ton of coal burned (per input)

Obviously get very different results depending on what you choose

TBESTechnology Based Effluent Standard

First find a technology that reduces emissions at a reasonable cost

Find out how much emissions would go down

Then set an emissions standard for that amount.

Used in both Clean Air Act and Clean Water Act

TBES

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Regulator knowsof technique touse only 20 units of Air and makeQ*. InefficientTechnique

Price of “Other Stuff” = 2

The Regulation:When you make Q*, you may use no

more than 20 units of clean air services. You may use the technique the regulatory engineers have discovered (20,100) or any other technique that uses no more than 20 units of air and has output Q*

Why this way?Regulator knows that it can be doneRegulator has upper bound on costRegulator is assured of cleaning up

the air.

Response to TBES

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Technique (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*.

Back Door EconomicsBest Practicable Technology

used for water pre 1977 means known technology at reasonable cost

Best Available Technology used for water post 1983 means any technology; but in practice is

limited by cost Intent: Cleaner water under BAT.

What to readChapter 8 in BH. Example is

agricultural pollution.If you are interested, get

Environmental Law and Policy and read it.

An exerciseLet Q = k x, where x is an input and

k is a positive number. Let w be the price of the input x.

What is the least cost way of making Q?

What is C(Q)?

Conditional Factor DemandHow much of an input will be used as

a function of output required and prices of inputs?

X(Q,p)How could changing the price of

clean air result in the same usage of clean air / unit output as the TBES regulations?

Our AssumptionFirm’s need to dispose of waste gas,

which they vent to the air. It is never free to 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 another input. For instance, by buying capital in the form of an afterburner.

Another applicationIn India the ratio of the price of labor

to capital is much less than in the USThe USSR consistently priced capital

below its true value to the “evil empire.” Does this help explain the emphasis on heavy industry and big dams?

Air as a function of price

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P1 = 4; A=24

P1=16.7;A= 16

Price of “Other Stuff” = 2

Conditional Factor Demand

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Quantity of Clean Air Used

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

Using Prices

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Slope on High Price line is -100/15 = -p1 /2 so p = 13.3.

A price for air of 13.3achieves the same level ofclean air as the TBES of 20units of air.

Using Prices

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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

Summary So FarBoth a TBES and a pollution charge

can produce 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 the TBES and charge result in the same use of air

AC is lower with a Quota!

Does MC increase?The case of the quota adjusted for

outputAssume quota increases from 20 to 28

units with additional output. Next slide is the Q and Q+1 isoquant.MC under price or quantity regulation

is the cost of the inputs to go from Q to Q+1

What of MC?

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(20,50)

(28,60)

QQ+1

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

MC is cost of added inputsMC 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 $0 MC is $200

Tax and Quota Same?MC is steeper under an output

increasing quotaAC is lower under (any) quotaLong run: Long Run Supply is where

P = AC With entry of new firms industry output

goes up Requires pollution quota for new firms

Some RealityQuotas are often per unit of output

allows expansion of output more cheaply but pollution expands too

Another plan is a fixed quota across industries that is tradeable. expansion in one industry means

contraction in anotheror more efficient use of pollution quotapollution remains fixed

Review: C(Q) and X(Q*,p)

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Price of OS is 3