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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 N

Lbs P2O5 80 120 160 200

40 82.3 86.7 88.5 88.6

80 95.9 102.1 105.4 106.8

120 102.4 110.1 114.5 116.9

160 105.4 114.2 119.6 122.9

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 of

workers each and every year.

Technique Hand

Pound

Small

Mill

Investment 0 9,359

Laborers 45.83 13.95

Technique Large

Mill

Small

Bulk

Large

Bulk

Investment 29,675 44,335 77,835

Laborers 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

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 10 20 30 40 50

Laborers

Invest

men

t C

ost

s

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

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 10 20 30 40 50

Laborers

Invest

men

t C

ost

s

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

80

100

120

0 20 40 60

Air

Oth

er S

tu

ff Low

IsocostMed.

IsocostHigh

Isocost

Price of “Other Stuff” = 2

Equations for 3 lines. Cost of

Chosen bundle?

Isocost Lines:

0

20

40

60

80

100

120

0 20 40 60

Air

Oth

er S

tu

ff Low

IsocostMed.

IsocostHigh

Isocost

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

20

40

60

80

100

120

0 20 40 60

Air

Oth

er S

tu

ff Low

IsocostMed.

IsocostHigh

Isocost

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 60

CAS

OS

Low

IsocostMed.

IsocostHigh

Isocost

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

– Use a specific technology

• catalytic converters on cars.

• scrubbers on coal fired power plants.

• One chooses a technology standard to reduce emissions

Effluent Standard

• Effluent (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

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 standard for that amount.

– Used in both Clean Air Act and Clean Water Act

TBES

0

50

100

150

0 20 40 60

Air

Oth

er S

tu

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

20

40

60

80

100

0 20 40 60

Air

Oth

er S

tu

ff

(20,50)

Technique (20,50) costs 180 and is least

cost way to make Q* using 20 units of air

Technique (20,80), the basis for the regulation, costs

240 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.

• 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

50

100

150

200

250

0 20 40 60

Air

Oth

er S

tu

ff

P1 = 4;

A=24

P1=16.7;

A= 16

Price of “Other Stuff” = 2

Conditional Factor Demand

0

5

10

15

20

0 10 20 30

Quantity of Clean Air Used

Pric

e of C

le

an

A

ir

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

Using Prices

0

20

40

60

80

100

0 20 40 60

Air

Oth

er S

tu

ff

(20,50)

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

A price for air of 13.3

achieves the same level of

clean air as the TBES of 20

units of air.

Using Prices

0

20

40

60

80

100

0 20 40 60

Air

Oth

er S

tu

ff

(20,50)

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

Before pollution charge, it

already cost $4/unit to use

the 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