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
y
input 2
x
input 1
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 for
Corn Yield in lbs.
Lbs P2O5
40
80
120
160
Lbs N
80
82.3
95.9
102.4
105.4
120
86.7
102.1
110.1
114.2
160
88.5
105.4
114.5
119.6
200
88.6
106.8
116.9
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
Technique Hand
Pound
Investment 0
Laborers
45.83
Technique Large
Mill
Investment 29,675
Laborers
5.25
Notes:
Small
Mill
9,359
13.95
Small
Bulk
44,335
2.64
Large
Bulk
77,835
1.17
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.
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
Investment Costs
Isoquant for Rice Milling
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
0
10
20
30
Laborers
40
50
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 + P L 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*
Investment Costs
Small Rice Mill Costs Least
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
Small Mill
Hand
0
10
20
30
Laborers
40
50
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
Price of “Other Stuff” = 2
120
Other Stuff
100
Low
Isocost
Med.
Isocost
High
Isocost
80
60
40
20
0
Equations for 3 lines.
Cost of
0
Chosen bundle?
20
40
Air
60
Isocost Lines:
Price of “Other Stuff” = 2
120
Blue Isocost:
slope -2=- p1/p2; p1 = 4; I = Low
Isocost
80*2=160; 160 =4 Air + 2 OS
Med.
Green Isocost: 200 = 4 Air +
2 OS
Isocost
High
.Red Isocost: 120 = 4 Air + 2 OS
Other Stuff
100
80
60
40
Isocost
20
0
0
20
40
Air
60
C(Q*) = 160
Price of “Other Stuff” = 2
120
Other Stuff
100
cost 200
Chosen
(24,32)
80
60
40
cost 160
Low
Isocost
Med.
Isocost
High
Isocost
20
0
0
20
40
Air
60
C(Q1)=120, C(Q*)=160, C(Q2)=200
120
100
Low
Isocost
Med.
Isocost
High
Isocost
OS
80
60
40
20
0
0
20
CAS
40
60
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
50,000 miles/5 years
Categor
y
NMOGa CO
NOx
100,000 miles/10 years
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
Price of “Other Stuff” = 2
Other Stuff
150
100
Regulator knows
of technique to
50
use only 20 units
of Air and make
0
0
20
Q*. Inefficient
Technique
40
60
Air
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
Technique (20,50) costs 180 and is least
cost way to make Q* using 20 units of air
Other Stuff
100
80
60
(20,50)
40
20
Technique (20,80), the basis for the regulation, costs
0
240 and makes Q*.
0
20
40
Air
60
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
Other Stuff
I=
200*2=400
P1 =
400/25=8 250
Price of “Other Stuff” = 2
P1=8;
A= 16
200
P1 = 4; A=24
150
100
50
0
0
20
40
Air
60
Price of Clean Air
Conditional Factor Demand
9
8
7
6
5
4
3
2
1
0
0
10
20
Quantity of Clean Air Used
In this chart the output is held constant at Q*.
30
A price for air of 13.3
Using achieves
Pricesthe same level of
clean air as the TBES of 20
units of air.
Other Stuff
100
80
60
(20,50)
40
20
0
0
20
40
Slope on High Price line is -100/15 =Air-p1 /2 so p = 13.3.
60
Before pollution charge, it
Using already
Prices
cost $4/unit to use
the air to dispose of waste
Other Stuff
100
80
60
(20,50)
40
20
0
0
20
40
Pollution charge of 13.3- 4 = 9.3 adds
$465 to cost
Air
60
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