techniques, isoquants, and cost curves © 2010 peter berck

Techniques, Isoquants, and Cost Curves

© 2010 Peter Berck

Definitions

• Output Q; specific amount Q*• Inputs x= (x1…xn)• If using inputs x results in output Q*, then x is

a technique to make Q*.– 1 brisket, 1 pan, 3 hours of oven services at 375, 1

large sheet foil, 4 coarsely sliced onions, salt, pepper, paprika are a technique for making pot roast. (Esther Lipow’s (z’’l) recipe.)

Efficient

• If x is less than or equal to y in every dimension and x and y both produce Q*, then y is not efficient.– xi yi for every i

input 1

input 2 y

x

Isoquant

• Let Q* be some specific output like 4 units• All efficient input combinations that produce

Q* are the Q*th isoquant

Find points on same isoquant forCorn Yield in lbs.

Lbs NLbs P2O5 80 120 160 20040 82.3 86.7 88.5 88.680 95.9 102.1 105.4 106.8120 102.4 110.1 114.5 116.9160 105.4 114.2 119.6 122.9

Rice Milling

• Why mill rice at all?• What is wrong with white rice from a

nutritional point of view?

Techniques for Milling Rice

Notes: Source P. Timmer Choice of Technique in Rice Milling in Java. Techniques to produce Rp 10 Million in Value added . Investment in USD. Laborers is the number ofworkers each and every year.

Technique HandPound

SmallMill

Investment 0 9,359Laborers 45.83 13.95

Technique LargeMill

SmallBulk

LargeBulk

Investment 29,675 44,335 77,835Laborers 5.25 2.64 1.17

Milling

• Does a rice mill and hand pounding produce the same white rice product? Which would you rather buy?

• Why is this in value added rather than tons?– VA = Revenue – Cost of materials

Isoquant for Rice Milling

0100002000030000400005000060000700008000090000

0 10 20 30 40 50

Laborers

Inve

stm

ent

Cos

ts

What Technique Minimizes Cost

• Need prices for labor and investment• Price of investment is 1. Plant is assumed to

last 50 years with no maintenance (urrg.)• Price of 50 years worth of labor is calculated

as the size of bank account (with 24% interest) that would pay a laborer $200 per year for 50 years. It is $833 per laborer

Interest Rates

• Is 24% per year a high interest rate for a developing country?

• Are there investment clubs in the US that charge their members 2.5% per month?

Outlay or Isocost Line

• All input combinations of K, investment, and L, laborers, that cost amount E, an unknown, are given by

• E = K + $833 L• More generally: • E = Pk K + PL L

About equi-cost or outlay lines

• E = Pk K + PL L

• K = E/Pk - L PL/PK

– So varying E (which is not known) gives a family of parallel equi-outlay lines

– The cost of every input bundle on the line is the vertical intercept times PK.

– When PK conveniently equals one, the vertical intercept is the cost of every bundle on the line

Which line?

• The least cost way of producing output Q* is found by finding the equi-cost line tangent to the Q*th isoquant.

• C(Q*) is the cost of any input bundle on that line

• The input bundle (or technique) at the tangency is the least cost way to produce Q*

Small Rice Mill Costs Least

0100002000030000400005000060000700008000090000

0 10 20 30 40 50

Laborers

Inve

stm

ent

Cos

ts

Hand

Small Mill

Story:

• Widows used to hand pound rice.• Now SRM’s are used instead.• How is this bad?• How is this good?

Isoquant and Production Function

• The 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

Cost function is the

• Minimum 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

Pollution in an isoquant world

• Two goods– Other stuff– Clean Air Services• negative of pollution• air has 1 ppm of gunk –polluton• air has 99 ppm of non-gunk – cleanth

Cost Min Technique

0

20

40

60

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120

0 20 40 60

Air

Oth

er

Stu

ff LowIsocostMed.IsocostHighIsocost

Price of “Other Stuff” = 2

Equations for 3 lines. Cost ofChosen bundle?

Isocost Lines:

0

20

40

60

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100

120

0 20 40 60

Air

Oth

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Stu

ff LowIsocostMed.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

0

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60

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120

0 20 40 60

Air

Oth

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ff LowIsocostMed.IsocostHighIsocost

Price of “Other Stuff” = 2

cost 200

cost 160

Chosen(24,32)

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

0

20

40

60

80

100

120

0 20 40 60CAS

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 given

the information you have been given.

Pollution Control

Technology Standard

• Technology is a way to do something (see above)• Technology Standard– Must use a specific technology

• Building codes-one stud every 18” (done)• Safety codes: must wear your goggles in lab.

• One could choose a technology standard to reduce emissions– One could (but doesn’t) require catalytic converters

Effluent Standard

• Effluent (or emissions) Standard– Can emit no more than X tons per (choose one)• megawatt hour (output)• Grams Nox per 100 km driven (output)• per year (absolute!)• per ton of coal burned (per input)

– Obviously get very different results depending on what you choose

Calif Effluent standards cars:Table 1LEV Emission Standards for Light-Duty Vehicles, FTP-75, g/mi

Category

50,000 miles/5 years 100,000 miles/10 years

NMOGa CO NOx PM HCHO NMOGa CO NOx PM HCHO

Passenger cars

Tier 1 0.25 3.4 0.4 0.08 - 0.31 4.2 0.6 - -

TLEV 0.125 3.4 0.4 - 0.015 0.156 4.2 0.6 0.08 0.018

LEV 0.075 3.4 0.2 - 0.015 0.090 4.2 0.3 0.08 0.018

ULEV 0.040 1.7 0.2 - 0.008 0.055 2.1 0.3 0.04 0.011

LDT1, LVW <3,750 lbs

Tier 1 0.25 3.4 0.4 0.08 - 0.31 4.2 0.6 - -

TLEV 0.125 3.4 0.4 - 0.015 0.156 4.2 0.6 0.08 0.018

LEV 0.075 3.4 0.2 - 0.015 0.090 4.2 0.3 0.08 0.018

ULEV 0.040 1.7 0.2 - 0.008 0.055 2.1 0.3 0.04 0.011

LDT2, LVW >3,750 lbs

Tier 1 0.32 4.4 0.7 0.08 - 0.40 5.5 0.97 - -

TLEV 0.160 4.4 0.7 - 0.018 0.200 5.5 0.9 0.10 0.023

LEV 0.100 4.4 0.4 - 0.018 0.130 5.5 0.5 0.10 0.023

ULEV 0.050 2.2 0.4 - 0.009 0.070 2.8 0.5 0.05 0.013

a - NMHC for all Tier 1 standards

Abbreviations: LVW - loaded vehicle weight (curb weight + 300 lbs) LDT - light-duty truck NMOG - non-methane organic gases HCHO - formaldehyde

Calif has an

• Effluent Standard• Manufacturers meet that standard

by choosing a technology:–Catalytic Converters–Also tuning engines, better burn, etc

• THERE IS NO requirement to choose a particular technology.

Technology Based Effluent Standard(TBES)

– First find a technology that reduces emissions at a reasonable cost

– Find out how much emissions would go down– Then set an emissions (or effluent) standard for

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

TBES

0

50

100

150

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Air

Oth

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ff

Regulator knows

of technique to

use only 20 units

of Air and make

Q*. Inefficient

Technique

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 done• Regulator has upper bound on cost• Regulator is assured of cleaning up the air.

Response to TBES

0

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Air

Oth

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ff

(20,50)

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 Economics

• Best 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 read

• Chapter 8 in BH. Example is agricultural pollution.– Isoquant = equal output– Isopleth = equal pollution

An exercise

• Let Q = k x, where x is an input and k is a positive number. Let w be the price of the input x.

• For a given x how many ways are there of making Q?

• What is the least cost way of making Q?• What is C(Q)?

Conditional Factor Demand

• How 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 Assumption

• Firm’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.

Air as a function of price

0 10 20 30 40 50 600

50

100

150

200

250

Air

Oth

er

Stu

ff

P1 = 4; A=24

P1=8;A= 16

Price of “Other Stuff” = 2

I= 200*2=400P1 = 400/25=8

Conditional Factor Demand

15 16 17 18 19 20 21 22 23 24 250

2

4

6

8

10

Quantity of Clean Air Used

Pri

ce o

f C

lean

Air

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

Using Prices

0

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100

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Air

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(20,50)

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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Air

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(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

Summary

• Both 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