Isoquant to Cost Curves
© 1998 Peter Berck
Review Technique
Technique 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
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
Bressler 1952 Example
y = 20+ 6.67 x2 + 10 x3 - .5 x3 x2
Cost function
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
Problem Again
Given 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.
Graph
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
give 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
TBES
Technology 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
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.
If you are interested, get Environmental
Law and Policy and read it.
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.
Another application
In India the ratio of the price of labor to
capital is much less than in the US
The 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
Price of “Other Stuff” = 2
P1 =16.7;
A= 16
Other Stuff
250
200
P1 = 4;
A=24
150
100
50
0
0
20
40
Air
60
Price of Clean Air
Conditional Factor Demand
20
15
10
5
0
0
10
20
Quantity of Clean Air Used
In this chart the output is held constant at Q*.
30
Using Prices
Other Stuff
100
A price for air of 13.3
achieves the same level of
clean air as the TBES of 20
units of air.
80
60
(20,50)
40
20
0
0
20
40
60
Slope on High Price line is -100/15 =Air
-p1 /2 so p = 13.3.
Using Prices
Before pollution charge, it
already 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 So Far
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
AC is lower with a Quota!
Does MC increase?
The case of the quota adjusted for output
Assume 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?
Other Stuff
100
(28,60)
80
60
40
Q+1
20
(20,50)
0
0
Q
20
40
Air = (8,10)
Additional Inputs = (28,60) - (20,50)
60
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 $0
MC is $200
Tax and Quota Same?
MC is steeper under an output increasing
quota
AC is lower under (any) quota
Long run: Long Run Supply is where P =
AC
With entry of new firms industry output goes
up
Requires pollution quota for new firms
Some Reality
Quotas 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 another
or more efficient use of pollution quota
pollution remains fixed
Review: C(Q) and X(Q*,p)
Price of OS is 3
150
OS
100
50
0
0
20
CAS
40
60