Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Preliminary Lecture notes for a Micro Course Based on
Microeconomics 8ed by Pindyck & Rubinfeld
Prepared by Houston H. Stokes.
Goal of the Notes: Allow the student to have an outline of the key ideas and solutions
to a number of problems that will be discussed in class. Since the notes are distributed
in WORD® format, students can edit the notes.
Introduction
Quote from Robert Mundell Man and Economics 1968
"Economics is the science of choice. It began with Aristotle but got mixed up with ethics in the
Middle Ages. Adam Smith separated it from ethics, and Walrus mathematized it. Alfred Marshall
tried to narrow it, and Keynes made is fashionable. Robbins widened it, and Samuelson dynamized it,
but modern science made it statistical and tried to confine it again.
But the science won't stay put. It keeps cropping up all over the place. There is an economics of
money and trade, of production and consumption, of distribution and development. There is also an
economics of welfare, manners, language, industry, music, and art. There is an economics of war and
an economics of power. There is even an economics of love.
Economics seems to apply to every nook and cranny of human experience. It is an aspect of all
conscious action. Whenever decisions are made, the law of economy is called into play. Whenever
alternatives exist, life takes on an economic aspect. It has always been so. But how can it be?
It can be because economics is more than just the most developed of the sciences of control. It is a
way of looking at things, an ordering principle, a complete part of everything. It is a system of
thought, a life game, an element of pure knowledge.
Chapter 1 Preliminaries

Economics is concerned with scarcity. If something is not scarce, there is no economic
problem.

Microeconomics => Study of behavior of individual economic units. How they react and
how they interact to form larger units.

Economics uses theory to explain actions and predict future actions. For example if: the
Bulls are in the playoffs, then there will be a larger number of people wanting tickets
than if they have a poor season. If there is high unemployment, then at the university
more students enroll in business courses because their opportunity costs (pay loss due
to being in class) are less.

Theory does not work well all the time. The market test of theory is how well it does in
comparison to another theory. A theory must have the possibility to being proved wrong.
The statement "all unmarried men are bachelors" is a tautology and cannot be proved
1
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
wrong. The statement "If American Airlines lowers the ticket price to Bermuda, an
increased number of people will fly this weekend." Can be proved wrong or right.

Marginal utility theory can be used to derive the demand curve. However it will be
shown later that to get a downward sloping demand curve the only assumption needed is
that consumers buy randomly along the budget line. => a minimalist approach to
deriving an important economic concept - downward sloping demand.

Normative Economics => What should be done. "Microsoft should be allowed to
bundle the IE with Windows XP because it benefits consumers."

Positive Economics => What will happen, not what should be done. "An increase in the
property tax in the area of UIC will tend to lower the price of apartments, everything
else equal."

Many decisions involve multiple assumptions. Book example of 1985 decision of Ford
to produce the Taurus involved:
1. Consumer tastes vs demand "would they like the car?"
2. How sensitive would demand be to price changes? (elasticity),
3. What would be the production costs? (Depends on assumptions of #'s of cars
produced, union demands, inflation, how fast workers learn).
4. How would competitors react (market structure).
5. How much new capital would be needed? (interest rates, engineering).
6. How would the decision be changed if oil prices moved favorably,
unfavorably?
7. How should Ford organize the production?
8. How might anticipated Government regulation changes influence the
decision (Gas mileage requirements).

What is a Market. A market is the collection of buyers and sellers that, through their
actual or potential interactions, determine the price of a product or set of products. A
market includes more than an industry (a collection of firms that sell the same or closely
related products). Key business decision: Determining the market for the product!

If prices differ in two markets, arbitrage may be possible. If the price of a T-bill
maturing in period t in NY is > than the price of the same T-bill in Chicago => people
will buy in Chicago and sell on the NY market. If the price of a Big-Mac is less in
Norfolk VA than Chicago, arbitrage is NOT feasible, even with fast planes.
2
Microeconomic Study Notes (17/7/2012
Houston H. Stokes

Markets can be competitive (wheat) or noncompetitive (electric power) due to entry
costs.

In a competitive market with many producers no one producer can change the industry
price. Product differentiation => power to alter price within limits. Successful
advertising =>  in perceived product differentiation.

Extent of the market => the boundaries, both geographical and physical of a market.

Real vs Nominal Price. Economic decisions should be made on the basis of real prices.
Money illusion => consumer looks at the nominal price not the real price. Assume two
groups: debtors and creditors. Creditors and debtors set the interest rate depending on
their expectations of price movements. Unexpected price increases (decreases) favor
debtors (creditors).

Measuring Price The CPI measures price changes by buying a market basket of goods
at different times. Three known problems are:
1. Tastes may change over time but same basket is bought.
2. Relative prices may change but the same basket is bought
3. Price changes involve subtle income changes that give rise to income effects
that will alter the mix of goods bought.

The CPI for period t, Pt is calculated with the Laspeyres formula:
n
n
i 1
i 1
Pt   pti q1i /  p1i q1i

(1)
If in place of the above equation the quantities could have been adjusted every year as in
the Paasche formula
n
n
i 1
i 1
Pt   pti qti /  p1i qti
(2)
which is not possible to calculate but would avoid the tastes bias and the relative price bias.
Table 1.2 lists the CPI and education and egg prices in both nominal and real terms. How
you get data in 2010 prices? See file ch1_1.xls for the answer? Why would one want to
make this calculation?
3
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Study Questions:
1. Over the past year the price inflation has been 10%. The price of a used Ford SUV has
fallen from $6,000 to $5,000. How much as the real price fallen:
((1.1 * 6000) - 5000)/(1.1*6000) = 24.24
Since last year the price of gold has risen from $120 to $420. What annual rate of
inflation would hold the price of gold fixed in rea1 terms?
(420-120)/120
= 250%
If this was over 5 years and we assume yearly compounding, what would the rate of
inflation have to be?
420
= 120(1. + r)5
(1. + r)5 = (420/120)=3.5,
= > ln(1.+r) * 5 = ln( 3.5), => ln(1. + r) = ln(3.5)/5
or (1. + r) = exp(ln(3.5)/5),
(1.+ r) = 1.2847 or 28.47% annual inflation.
Check: (1.2847)5 = 3.5, 120*3.5 = 420
2. Suppose that the Japanese yen raises against the U. S. dollar; that is, it now takes more
dollars to buy any given amount of Japanese yen. Explain why this simultaneously
increases the real price of Japanese cars for U. S. consumers and lowers the real price of
U. S. automobiles for Japanese consumers.
4
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 2 Basics of Supply and Demand
The basic model postulates that Supply responds positively to price and Demand
responds negatively to price. Qs = Qs(P) and Qd = Qd(P). At equilibrium P (P0) = Q (Q0)
and the market clears. Qs(P) > Qd(P) implies excess supply while Qs(P) < Qd(P) implies
excess demand. This is shown below
S
P
P1
P0
P2
D
Q
Points where price is above P0 represent excess supply, while points below P0 represent
excess demand. The laws of motion of Walrus state "If there is excess demand, price will
rise." The laws of motion Marshall state "If there is excess demand, quantity will rise." P0
is the "market clearing price." Here supply = demand. Assume that somehow the price goes
to P1. Now demand < supply. According to Marshall adjustment will proceed by a reduction
in supply, while according to Walrus price will fall. Similar arguments can be made for P2.
In a market for Old Master Paintings clearly Walrus adjustment is the only legal adjustment
mechanism. In 1969 the first man walked on the moon. The New York times printed the
edition showing this historic moment for three days until everyone got a copy. This was
Marshall adjustment.
In the above diagram it does not matter which adjustment model you use, price will go to P0.
Assume that supply now is negatively sloped or that economics of scale exist. The producer
can increase his production and lower his price. Two Models are possible.
5
Microeconomic Study Notes (17/7/2012
P
Houston H. Stokes
D
P
S
D
S
Q
Q
The left graph is stable according to Marshall and unstable according to Walrus.
The right graph is stable according to Walrus and unstable according to Marshall. Why?
There is a shift in supply if one of the variables held constant (not on the axis) changes and
that in turn changes the amount supplied. Assume the supply of wheat is a function of P and
R where R = rainfall.
Qs = a
+
1 P
+
2 R
(3)
where 1 > 0 and 2 > 0. An increase in R would, everything else equal move the supply
curve right and lower the price for wheat assuming the demand curve is not flat. Assume the
demand for wheat is a function of price and income.
Qd = a'
+
3P
+
4Y
(4)
Where 3 < 0 and 4 > 0. An increase in Y moves the demand curve right and results in an
increase in price assuming that the supply curve is not flat and that Y => quantity
demanded .
Demand and supply curves hold tastes, prices of all other good, incomes and technology etc
constant. Over time supply and demand usually shift right. An exception would be the
demand for buggy whips! Why?
Key Concept: Given a demand curve, it can be said that there is a change in demand or
shift in the demand curve if the price of another good changes, income changes or tastes
change. If the price of the good itself changes, then there is a change in the quantity
demanded or a movement along the demand curve. In the above example Y will cause
a change in demand while P will cause a change in the quantity demanded. For supply,
P causes a change in the quantity supplied while R causes a change in supply. It is
always important to understand the difference with what moves a curve and what
causes a movement along the curve.
6
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
.
Solve simple system:
Qs= 1800 + 240P
Qd= 3550 - 266P
In equilibrium Qd = Qs. or 1800 + 240P = 3550 - 266P
506P = 1750 => P = $3.46 and Q = 1800 + (1750/506)*240 = 2630
Excel file ch2_1.xls solves this system. Using this technology. Solve the system assuming:
-
An advertising campaign raises 3550 to 4023.
A lack of rainfall lowers 1800 to 1500
A change in taste changes 266 to 280
This is the basic template for micro analysis of supply and demand. A major research
objective is to determine what will change a, a', 1,…4. It will be made more complex as
we move forward.
Ed
=
=
=
=
elasticity of demand
(%Q)/(%P)
(Q/Q)/(P/P)
(Q/P)*(P/Q)
(5)
Usually d is used in place of Ed. The price elasticity is measured at a point. On a linear
demand curve the slope (Q/P) is constant.
Q = 8-2P
=>
(Q/P) = -2
As Q  0
=>
Ed  -
As P  0
=>
Ed  0
If |Ed| > 1
=>
P => Total Revenue (TR) up.
TR = P * Q
A firm should never operate where |Ed|
(6)
< 1.
Advertising is designed to make |Ed| decrease for every price. After a successful
advertising campaign the firm will have more product differentiation and can either raise
price or increase sales at the current price. Careful analysis will tell us what to do.
7
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
S
b
a
P2
P1
c
d
O
Q1
e f
Firm started with demand curve a e which implied price P1 and Q1. A major advertising
campaign moved the demand curve from a e to b f. For purposes of this example a e is | | to
b f. Price rises to P2.
Proof that |d | at d is more inelastic than at c.
At c
d
= (Q/Q)/(P/P)
= (Q1e/OQ1)/(OP1/OP1)
= (Q1e/OQ1)
= ce / ac
At d
d
(| | lines produce proportionate segments)
= (Q/Q)/(P/P)
= df / bd
Since df = ce and bd > ac = > at d |d| is less than at c
Note that at a d = - while at e d = 0.
A parallel shift right of a demand curve makes is more inelastic at each price.
Income elasticity measures the percent change in the quantity of a good for a percent
change in income.
8
Microeconomic Study Notes (17/7/2012

EI
I
=
Houston H. Stokes
(Q/Q)/(I/I) = (Q/I)(I/Q)
(7)
A good is normal if I > 0 and inferior if I < 0 . Baked beans are the classic inferior good
and the BMW car a classic normal good.
Assume an individual faces goods a to n. Let Pa be the price of good a, Pb be the price of
good b. Let A be the change in the amount of A bought and B be the change in the
amount of B bought. Assume the individual gets an increase in income I.
Pa A +
PbB + ….. +
PnN =
I
(8)
If we assume I = 0, the above equation defines the budget line which will be used in
indifference curve analysis.
Divide (8) through by I and then multiplying all terms on the left by I A / I A gives a key
expression. In steps
[ Pa A / I ]  [ Pb B / I ],...,[ Pn N / I ]  [ I / I ]
[ Pa A AI / I AI ]  [ Pb B BI / I BI ],...,[ Pn N NI / I NI ]  [ I / I ]
Finally
Ka Ia +
Kb Ib + … + Kn In
=
1
(9)
Or the sum of the weighted income elasticities = 1
The cross price elasticity of demand  APB measures the effect of the price of good B on
the quantity of good A.
 AP  ( QA / QA ) / ( PB / PB )  QA PB / QA PB
B
(10)
If A is a substitute for B, then  APB is > 0 or an increase in the price of B => the quantity
of A . If A is a complement of B then  APB is < 0. Coke is a substitute for Pepsi while
gas is a complement for an auto.
In the long run demand is usually more elastic. When the price of gas went up in the 70's
people did not just dump their cars that got poor gas mileage BUT when they replaced them
they made sure they got a car with a higher MPG rating.
In the long run the supply of a product is usually more elastic than in the short run since in
the long run fixed factors can be changed.
9
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Given QD = a - bP
(11)
QS = c + dP
(12)
And P* and Q* are equilibrium P and Q, then at equilibrium the elasticity of demand D and
elasticity of supply S are
D
=
-b(P*/Q*)
(13)
S
=
d(P*/Q*)
(14)
=> if we know D , S, P* and Q* we can get the supply curve and the demand curve. In the
market we observe P* and Q*.
Book shows P* and Q* are $.75 and 7.5 million D , S are -.8 and 1.6 respectively. From
equation (13) and (14) we get -b = (-.8) * (7.5/.75) = -8. and d = (1.6)*(75/.75) = 16.
Excel file ch2_2.xls solves the general case of determining the supply and demand functions
given the elasticity and P* and Q*.
In the long run supply and demand are usually more elastic. When OPEC raised price =>
discoveries went up. Users shifted to more efficient cars over time.
An effective price control will cause a shortage. If the supply curve is NOT vertical, the total
consumed will go down.
Demand for E
Problem 1.
P
60
80
100
120
Qd
22
20
18
16
Qs
14
16
18
20
d = (Q/P)*(P/Q)
Q/P  -2/20 =>
d = (-.1)*(80/20)
=
-.40
at 100
d = (-.1)*(100/18)
=
-.56
Problem 2. T or F "Since tuition has doubled in real terms in the last 15 years this implies
the demand curve for education is vertical." FALSE may be seeing shifts of both supply
and demand.
10
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Problem 3. File Ch2_3.xls contains simulated Data from 25 Families obtained from Stigler
Theory of Price Edition 4 page 37. Variables are expend = expenditure, pincome =
permanent income, Tincome = transitory income, oincome = observed income.
The data was generated from the model expend = f(1+.02*pincome). Tincome was
randomly added such that pincome + tincome = oincome
Run expend=f(constant pincome)
Expend=g(constant oincome)
11
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 3 Consumer Behavior
Key Assumptions of Theory of Consumer Behavior:
Preferences are complete => Consumers can rank market baskets.
Preferences are transitive => If consumer prefers basket A to B and B to C then the
consumer prefers A to C.
More of any good is always better.
Indifference Curve - Locus of points showing different combinations of goods to which the
consumer is indifferent.
X
II
I
III
IV
a
b
Y
Consumer starts with a of X and b of Y. Points in I => more satisfaction. Points is III => less
satisfaction.
Indifference curves must be in II and IV.
X
Complements
X
Y
Substitutes
Y
12
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Y
Usual Case Utility along b > than along a
2
b
1
a
X
Indifference curves cannot cross unless there is a change in tastes.
Ordinal ranking
=> A is better than B but cannot tell by how much.
Cardinal ranking
=> Can tell how much better A is than B.
Slope of indifference curve measure marginal rate of substitution between goods. In
terms of above example = ( Y / X ) .
At 1 ( Y / X )  0
At 2 ( Y / X )  
=> along an indifference curve there is diminishing marginal rate of substitution.
Given you have relatively more X than Y => will give less Y for every additional X.
Budget line = locus of points showing different combinations of goods that can be bought
given income. Where budget lines is tangent to highest indifference curve => desired point
(a).
Y
A
a
3
2
1
B
X
13
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
A = I / Py, B = I / Px. At a on highest indifference curve. MRS = Y/X =Px / Py
1
2
if income increases budget line moves out.
If price of X falls budget line rotates
Y
a
I
Px
=> Budget line moves from 1 to 2 above
=> Budget line moves from ab to ac on left.
B
b
c
X
Total spending fixed along budget line. Object is to reach highest Indifference curve (not
drawn).
Unless there is a corner solution, MRS = Px / Py. Of in words slope of the indifference curve
= slope of budget line.
Corner solution => consume only one good
Utility = satisfaction one gets for consuming a good.
Marginal Utility declines as consumption of a good increases unless lumpyness problem (4
tires on a car).
MRSxy = MUx / MUy
In equilibrium
MUx / MUY
=
PX / PY
MUx / PX
=
MUY / PY
(MUx / PX) > (MUY / PY)
=> Have too much Y relative to X
Marginal utility per dollar last dollar spent must be the same for all goods.
14
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Gas rationing => loss of welfare. Rationing done to maintain price. In NYC rent controls
led to vast sections of the city being abandoned. Rationing once in place is hard to remove.
See Figure 3.22
Other
Goods
OA
C
Gas
Consumer wants OC of gas. The government forces person to take OA and thus be on a
lower indifference curve.
The Laspeyrse price index is defined as
n
n
i 1
i 1
Pt   pti q1i /  p1i q1i
which overstates the amount of a price increase because the person is given enough money
to buy their old bundle even though this is NOT what they will buy since relative prices
have changed.
a
Y
1
2
3
II
III
I
X
Consumer was on indifference curve 2 and budget line I. The price of X increases and the
consumer ends up on indifference curve 3 and budget line II. A subsidy is given to allow
consumer to buy old bundle of goods BUT consumer now buys relative more of good Y and
less of good X since Px/Py has risen.
15
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
# 2 page 99 Draw indifference curves for:
Al likes beer, hates hamburgers
U3
U2
U1
beer
U3 > U2 > U1
Hamburger
Betty indifferent between 3 beers or 2 hamburgers. Nothing changes no matter what
the level of consumption.
beer
9
6
3
2
4
6 Hamburger
Chris eats one hamburger and one beer in fixed proportions.
beer
Hamburger
16
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
1. Antonio buys 8 new college textbooks at a cost of $50.00 each. Used books were $30.00
each. Next year prices for new books go up 20% and used books go up 10%. Antonio's Dad
sent him $80.00 more. Is he better off?
Cost last year for new books 8 * $50.00 = $400.00
Cost last year for used books 8 * $30.00 = $240.00
Relative price of new and used books last year 50/30 = 1.67
Relative price of new and used books this year (50*1.20)/(30*1.1) = 60 / 33 = 1.82
 New books are relative more costly.
Cost this year for the two choices would be 8*60
8*33
= $480
= $264
Dad gave him money to buy 8 new books this year. Will he still buy this bundle now that
prices of new books are relatively higher?
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
2. Jane has a utility function U(F,c) = FC. Set up a table for U=12 and U = 24
Suppose Jane has $12.00 to spend and Pf = $1.00 and Pc = $3.00 Show budget line
What should she buy? (F=6 C =2 ).
Ch3_1.xls
Jane U(Fmc)=FC
U(F,C) = FC
F
1
1.5
2
3
4
6
8
12
P food
U=12
C
12
8
6
4
3
2
1.5
1
1 P Cloth
3
U=24
Cost
37
25.5
20
15
13
12
12.5
15
F
1
2
3
4
6
8
12
24
Budget line is Y = Pf * F + Pc * C
MRS = -1/3
17
C
24
12
8
6
4
3
2
1
Cost
73
38
27
22
18
17
18
27
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 4 Individual Market Demand
Demand Curve holds constant:
Prices of other goods
Income
Tastes
Assume Income available is I:
A = I /P1y ,
B = I / P1x,
C = I / P2x,
D = I / P3x
As Px falls the budget line moves from AB to AC to AD. The possible set of goods
increases. The consumer increases x purchases and Y purchases. The above diagram is
same as figure 4.1.
Points to remember: As price of X falls there is a substutution effect that is due to the
relative price change and an income effect which is due to the increase in real income
18
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
due to the price decrease. If the price of X falls, the consumption of Y can increase, or
decrease.
Income Consumption Curve Traces out points of consumption of X and Y that occurs
as income increases.
If X and Y are both superior goods, as I increases the consumption of X and Y increases.
=> Demand curves for both goods shift right.
Y
ICC
X
Here X is inferior at high income = > as I increases, consumption of X eventually falls.
Y
3
2
1
X
As Income rises X consumption first increases, then falls. As income rises the
consumption of Y increases.
19
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Y is a normal good. X is a normal good for low incomes, an inferior good for high
prices.
In an N good world all goods can be normal goods. As a upper limit only N-1 can be
inferior. => In a 2 good world cannot have two inferior goods.
Income and substitution effects can be drawn 4 ways for a price fall and 4 ways for a
price rise assuming normal goods. For the inferior not giffen and inferior giffen there are
8 other ways each. Total number of cases = 24.
Inferior not Giffen = > good is inferior but not so inferior that the income effect out
weighs substitution effect.
Four ways to draw cases:
Hicks method (American assumptions). => Assume base case. Prices change
causing the budget line to rotate (See figure 4.6). Helping budget line drawn tangent to
old indifference curve but parallel to new budget line.
Slutsky method (American assumptions) => Assume base case. Prices change
causing the budget line to rotate . Helping budget line drawn through old basket but
parallel to new budget line.
Hicks method (European Assumptions). => Assume base case. Prices change
causing the budget line to rotate . Helping budget line drawn tangent to new indifference
curve but parallel to old budget line.
Slutsky method (European Assumptions) => Assume base case. Prices change
causing the budget line to rotate . Helping budget line drawn through new basket but
parallel to old budget line.
For a price fall Hicks (Slutsky) method with American assumptions looks like a price
increase for Hicks (Slutsky) with European assumptions.
We assume American assumptions for this course!!!
Define
AB
AB*
BC
B*C
AC
=
=
=
=
=
substitution effect using Hicks
substitution using Stutsky
income effect using Hicks
income efffect using Stutsky
Total Effect
Assume P X falls.
20
Microeconomic Study Notes (17/7/2012
Normal good
Inferior not Giffen
Inferior Giffen
=>
=>
->
Houston H. Stokes
ABB*C
ACB*B
CB*BA
Y
a
2
1
II
III
I
A
B B*
Cc
d
X
We assume income is fixed. Original budget line is ac. Consumer consumes A of X.
Price falls => budget line shifts out to ad. Consumer now obtains C of X.
Hicks method draws helping budget line 1 tangent to old indifference curve I but
parallel to new budget line. AC broken into AB (Hicks substitution effect) and BC Hicks
income effect.
Slutsky method draws helping budget line 2 through old point. Consumer is
overcompensated and substitution effect is now AB* and income effect B*C.
Slutsky American method is how Lasperse price index (  Pjt Qj1 /  Pj1Qj1 ) is
constructed.
Slutsky European method is how Paasche price index (  Pjt Q jt /  Pj1Q jt ) is
constructed.
Figure 4.6 shows Hicks method for a price fall of a normal good. In terms of
ABC, F1 = A, E = B and F2 = C. Figure 4.6 can be used to show the Hicks method using
European assumptions for a price increase.
Figure 4.7 shows Hicks method for an inferior good not Giffen good.
21
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Figure 4.8 shows Hicks method for an inferior good that is Giffen.
Figure 4.9 shows the effect of a Gas tax with total rebate. Gas tax => relative price of gas
rises => relative consumption of gas falls. BUT the gas tax involves an negative income
effect. State elects to rebate the full tax to each person on an average basis. => heavy
users of gas will be at less utility, low users of gas will benefit since they will be over
compensated. Graph shows effect on heavy user. For all users relative price of gas has
increased.
Market demand = sum of all private demands
Price
1
2
3
4
5
A
6
4
2
0
0
B
10
8
6
4
2
C
16
13
10
7
4
Market
32
25
18
11
6
At price > 3 consumer A is out of the market.
Assuming demand curve is elastic (inelastic) => price fall inplies that revenue increases
(decreases).
Point elasticity
=
(P/Q)*(1/slope)
For straight line demand curves elasticity = point elasticity. Not all demand curves are
straight lines.
Arc elasticity
=
( Q / P)( P / Q )
Arc Income elasticity
=
(Q / I )( I / Q )
Demand curves have kinks as consumers jump into the market. See figure 4.11
P
Domestic Demand
World D
Market Demand
Q
22
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Consumer surplus = total benefit of consuming = difference between what a consumer
is willing to pay for a product and what consumer actually pays.
A
B
E
C
D
Consumer surplus
=
area ABE
=
AB * BE / 2
=
(P * Q) / (d * 2)
d
=
BC / AB,
P*Q
=
BC * CD
=>
(P * Q) / (d * 2)
=
=
=
(BC * CD)/(2*BC/AB)
(BC*CD*AB)/(2*BC)
(AB*BE/2)
Value of clean air: Would consumers maintain their cars better if the exhaust was on
front of car? Why?
Bandwagon effect => You like the good better if others get it. Bandwagon effect
makes market more elastic. Retail marketing based on creating bandwagon effect. MJ
paid to wear a special shoe.
Snob Effect
=> You like the good less the more others have it. Snob effect
makes market demand less elastic.
Table 4.5 data is in Excel file Table4_5. Estimate
Q  a  B1 P  B2 I
23
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
and
ln(Q)  a   B1 ln( P)  B2 ln( I )
Do you get what the book gets? (Hint I do not!!!)
Math Treatment
Assuming goods X and Y and income I, Customer wants to
(1)
Max U(X,Y)
Given budget constraint
(2)
I = PxX + PyY
Form Lagrangian or the function to be maximized
(3)
  U ( X , Y )   ( PX X  PY Y  I )
If the budget constraint is satisfied => second term = 0. Differentiate Lagrangian with
respect to X, Y and  and set to zero. Obtain
(4)
MU X ( X , Y )  PX  0
(5)
MU Y ( X , Y )  Py  0
(6)
PX X  PY Y  I  0
where MU X ( X , Y )  U ( X , Y ) / X etc
Equations (4) and (5) indicate that the consumption of X and Y will proceed until the
marginal utility is a multiple  of P or
(7)
  [ MU X ( X , Y ) / PX ]  [ MU Y ( X , Y ) / PY ]
or
(8)
MU X ( X , Y ) / MU Y ( X , Y )  PX / PY
Equation (8) expresses an equilibrium condition. This equation is satisfied where the
indifference curve is tangent to the budget line.
Along an indifference curve the utility is fixed or
24
Microeconomic Study Notes (17/7/2012
(9)
Houston H. Stokes
U ( X ,Y)  U *
As we move along the indifference curve
(10)
MU X ( X , Y )dX  MUY ( X , Y )dY  dU *  0
can express the MRSXY as
(11)
dY / dX  MU X ( X , Y ) / MU Y ( X , Y )  MRS XY
or the ratio of the marginal utilities.
Practical case using Cobb-Douglas utility function. This section based on Appendix
to Chapter 4.
(12)
U ( X , Y )  X aY 1a
which can be estimated as
(13)
log(U ( X , Y ))  a log( X )  (1  a ) log(Y )
forming the Lagrangian and solving we find
(14)
  a log( X )  (1  a ) log(Y )   ( PX X  PY Y  I )
(15)
 / X  a / X  PX  0
(16)
 / Y  (1  a )Y  PY  0
(17)
 /   PX X  PY Y  I  0
Solving (15) and (16) for PXX and PYY and substituting in the budget equation gives
(18)
a /   (1  a ) /   I  0
which implies that   1/ I . Form the first two equation we get
(19)
X  (a / PX ) I
and
(20)
Y  [(1  a ) / PY ]I .
25
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
The Cobb-Douglas utility function implies that the consumption of each good depends
only on the price of that good and income, not the price of the other good or that cross
elasticities of demand are zero.
The Lagrange multiplier  represents the extra utility generated by one more dollar.
Taking the total derivative
(21)
dU / dI  MU X ( X , Y )(dX / dI )  MU Y ( X , Y )(dY / dI )
since any increase in income is spent
(22)
dI  PX dX  PY dY
since from (4) and (5) PX  MU x ( X , Y ) and PY  MU Y ( X , Y ) then
(23)
dU / dI  PX (dX / dI )  PY (dY / dI )   ( PX dX  PY dY ) / dI
substituting for dI gives
(24)
dU / dI  ( PX dX  PY dY ) / ( PX dX  PY dY )  
An example. Assume a  1 / 2 , PX=$1.00 and PY=$2.00 and I = $100. From (19) & (20)
X  (a / PX ) I and Y  [(1  a ) / PY ]I . Thus X = 50 and Y = 25. Since   1/ I , then
 =1/100. If income were to increase to $101, X=50.5 and Y=25.25. The original utility
.
was .5 *log(50) .5 *log(25)  3565
while the new utility was
.5 *log(50.5) .5 *log(25.25)  3575
.
.  was .01 which is the exact difference in utility.
(12) implies that if the consumption of X and Y were to increase by  , then total utility
goes up by  . In a more general case
(25)
U ( X ,Y)  X Y 
where in general     1 . Here if X and Y increase by  , then utility goes up by
 (   ) . Direct differentiation of (25) with respect to X and Y gives
(26)
MU X ( X , Y )  dU ( X , Y ) / dX  aX ( a 1)Y B  aU ( X , Y ) / X
(27)
MUY ( X , Y )  dU ( X , Y ) / dY  X aY (  1)  U ( X , Y ) / Y
Equations (27) and (28) show that if Y is fixed and X is increased the marginal utility of
each additional X will decline. An additive utility function will not have this property.
26
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Duality in Consumption Theory states that maximizing utility for a given budget is
equivalent to minimizing the cost of obtaining a given level of utility. This can be best
seen with indifference curves and budget lines.
Discussion Problem # 1
Sally Henin has an arc price elasticity of demand for gasoline of -.8. Her income
elasticity for gas is .5. Sally has a current income of $40,000 per year and spends $800
per year for gas. The current price of gas is $1.00.
a. The government is concerned about energy usage and is contemplating an
excise tax which will cause gas prices to increase to $1.40. What will happen
to Sally's consumption?
b. The Government is considering a $200.00 tax rebate. How will this impact
Sally?
c. Assume both the tax and the rebate are implemented. Is Sally worse off or
not?
a. Solve arc elasticity formula for new Q
b. Using new Q and arc income elasticity formula sub in new income and recalculate Q
c. See how much she spends before and after
(Q2  800) /(1.40  1.00) (Q2  800) (1.00  1.40)

(1.00  1.40) /(Q2  800) (Q2  800) (1.40  1.00)
.8(Q2  800)  (Q2  800)6
6.80Q2  4160
Q2  61176
.  612
.8 
.5  ((Q3  612) / (40,200  40,000))((40000  40200) / (Q3  612))
.5(Q3  612)  (Q3  612)401
400.5Q3  306  245412
Q3  613.53
On final indifference curve she spends 613.53 * 1.40 = $858.94 . With her income and the
rebate she would have 40200-858.94 = $39,341 to spend on goods.
If she bought 614 at the old price she would have had 40000-614 = $39386
Sally is NOT happy with the government!!
27
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Problem # 2
The San Francisco Chronicle reported that the toll on the Golden gate bridge was raised
from $2.00 to $3.00. Following the toll increase traffic fell 5%.
What is the point price elasticity of demand?
Steve Leonoudakis chairmen of the bridge's finance auditing committee
warned that the toll increase could cause toll revenues to decrease by $2.8
million per year. Is this statement consistent with Economic Theory?
Elasticity = (Q / Q)( P / P) = -.05/.50 = -.1 or inelastic demand
Steve is wrong. Inelastic demand => revenue increases as price increases.
Chapter 5 Uncertainty and consumer behavior not covered in detail.
Fines: Individual balances “gain from breaking the law” against expected fine
Double park
Fine
Fine
=> utility gain $5.00
=> $50.00 where probability of getting caught = .1
=> $500.00 where probability of getting caught = .01
These cases imply enforcement costs = 0.0. Most drivers who do not like risk will not
double park.
In case of music piracy difficult to catch. => can have very high fines.
28
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 6 Production
Firms turn inputs or factors of production into outputs via a production function
Q=F(K,L).
A production function implies a given technology.
Firm uses each input as efficiently as possible. Isoquant (production indifference curves)
show different quantities of inputs (say L and K) such that output is fixed.
L
Q=30
Q=20
Q=10
K
In short run not all inputs can be changed. In the long run all inputs can be changed.
Given K, Average product = Q/L, Marginal product = Q/L. Q = Total product.
When Marginal Product > Average Product => average product is increasing.
Figure 6.1 shows relationship between AP, TP and MP. Why would a firm never operate
where MP < 0? MP reaches a peak at point of inflection. of TP curve.
Figure 6.2 shows how output per time period goes up as technology improves.
Figure 6.4 shows that cereal yield and the food price index appear to be negatively
related. Why might this be so? Except for a brief period in the 70's, food prices have
declined.
Law of diminishing returns => with other inputs fixed, MP of an input declines as the
amount of that input used increases.
Malthus believed that the earth would not support population growth. To date he has been
wrong.
29
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Table 6.3 shows Annual Growth of Labor Productivity in various countries. It is shown
below:
Year
1960-1973
1974-1982
1983-1991
1992-2000
2001-2009
US
2.29
.22
1.54
1.94
1.90
Japan
7.86
2.29
2.64
1.08
1.50
GDP per hour worked in 2009 dollars
$56.90
$38.20
France
4.70
1.73
1.50
1.40
.90
Germany
3.98
2.28
2.07
1.64
.80
UK
2.84
1.53
1.57
2.22
1.30
$54.70
$53.10
$45.80
A technological change raises the production function. Here it is occurring over time.
Q  Aet L K 
In any particular year the aggregate value of goods and services produced by en economy
is equal to the payments made to all factors of production.
We can estimate a production function in log form with Excel. Take Data from 6.4 and fit
model. See ch6_1.xls
ln( q)  ln( a )   ln( L)   ln( k )
Production Function data from Table 6.4 of Pindyck-Rubinfeld (2005)
Q
L
20
40
55
65
75
40
60
75
85
90
55
75
90
100
105
65
85
100
110
115
75
K
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
lnq
2.995732
3.688879
4.007333
4.174387
4.317488
3.688879
4.094345
4.317488
4.442651
4.49981
4.007333
4.317488
4.49981
4.60517
4.65396
4.174387
4.442651
4.60517
4.70048
4.744932
4.317488
30
lnL
0
0
0
0
0
0.693147
0.693147
0.693147
0.693147
0.693147
1.098612
1.098612
1.098612
1.098612
1.098612
1.386294
1.386294
1.386294
1.386294
1.386294
1.609438
LnK
0
0.693147
1.098612
1.386294
1.609438
0
0.693147
1.098612
1.386294
1.609438
0
0.693147
1.098612
1.386294
1.609438
0
0.693147
1.098612
1.386294
1.609438
0
Microeconomic Study Notes (17/7/2012
90
105
115
120
5
5
5
5
Houston H. Stokes
2
3
4
5
4.49981
4.65396
4.744932
4.787492
1.609438
1.609438
1.609438
1.609438
0.693147
1.098612
1.386294
1.609438
2
22
24
SS
3.769633
0.306771
4.076404
MS
1.884817
0.013944
F
135.1693
Significance
F
4.38E-13
Coefficients
3.394232
0.483056
0.483056
Standard
Error
0.061017
0.041549
0.041549
t Stat
55.62726
11.62623
11.62623
P-value
3.66E-25
7.28E-11
7.28E-11
Lower 95%
3.267689
0.396889
0.396889
Yhat
29.79175
41.63997
50.64917
58.20024
64.82428
41.63997
58.20024
70.79241
81.34655
90.60499
50.64917
70.79241
86.10902
98.94665
Actual
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.961637
R Square
0.924745
Adjusted R
Square
0.917903
Standard
Error
0.118085
Observations
25
ANOVA
df
Regression
Residual
Total
Intercept
lnL
LnK
Upper
95%
3.520774
0.569223
0.569223
RESIDUAL OUTPUT
Observation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Predicted
lnq
3.394232
3.72906
3.924923
4.063889
4.17168
3.72906
4.063889
4.259752
4.398718
4.506509
3.924923
4.259752
4.455614
4.594581
Residuals
-0.3985
-0.04018
0.08241
0.110498
0.145808
-0.04018
0.030455
0.057736
0.043933
-0.0067
0.08241
0.057736
0.044195
0.010589
31
20
40
55
65
75
40
60
75
85
90
55
75
90
100
Error
9.79175
1.639968
-4.35083
-6.79976
-10.1757
1.639968
-1.79976
-4.20759
-3.65345
0.604987
-4.35083
-4.20759
-3.89098
-1.05335
Lower
95.0%
3.267689
0.396889
0.396889
Upper
95.0%
3.520774
0.569223
0.569223
Microeconomic Study Notes (17/7/2012
15
16
17
18
19
20
21
22
23
24
25
4.702372
4.063889
4.398718
4.594581
4.733547
4.841338
4.17168
4.506509
4.702372
4.841338
4.949129
Houston H. Stokes
-0.04841
0.110498
0.043933
0.010589
-0.03307
-0.09641
0.145808
-0.0067
-0.04841
-0.09641
-0.16164
110.2082
58.20024
81.34655
98.94665
113.6982
126.6387
64.82428
90.60499
110.2082
126.6387
141.0521
105
65
85
100
110
115
75
90
105
115
120
5.208238
-6.79976
-3.65345
-1.05335
3.698178
11.63871
-10.1757
0.604987
5.208238
11.63871
21.05206
The above Excel output shows how data can be used. Figure 6.4 graphs this data.
Marginal rate of technical substitution = K / L for a given Q (K on vertical axis).
MRTS   K / L  ( MPL / MPK )
since along isoquant
( MPL )( L)  ( MPK )( K )  0
When inputs are substitutes => isoquants are straight lines.
When inputs are complements => isoquants are right angles
Assuming a Cobb-Douglas production function sum of coefficients determines returns
to scale. Assume all inputs increase by  . Get
Q*  aet (L) a (K )   aet  ( a   ) La K    ( a   ) Q
Example: MPL= 50, MRTS = .25. What is MPK? .25  MPL / MPk  50 / MPk .
=> MPK  200
Example: Determine whether decreasing, constant or increasing returns.
A. Q .5KL, Q* .5(K )(L), Q*  Q2 or increasing returns.
B. Q  2K  3L, Q*  2 K  3 L  Q or constant returns.
Example: Assuming Q=100(K.8L.2) starting K=4 and L=49 show MP of labor and K are
declining
32
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
K  4, L  49,  Q  100 * 4.8 * 49.2  660.21
K  5, L  49,  Q  100 * 5.8 * 49.2  789.25,  MPK  129.04
K  6, L  49,  Q  100 * 6.8 * 49.2  91319
. ,  MPK  123.94
K  7, L  49,  Q  100 * 7.8 * 49.2  1033.04,  MPK  119.85
K  4, L  49,  Q  100 * 4.8 * 49.2  660.21
K  4, L  50,  Q  100 * 4.8 * 50.2  662.89,  MPL  2.68
K  4, L  51,  Q  100 * 4.8 * 51.2  66552
. ,  MPL  2.63
K  4, L  52,  Q  100 * 4.8 * 52.2  66811
. ,  MPL  2.59
33
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 7 Cost of Production
Accounting Cost - concerned with historical expenditures.
Economic Cost takes into account opportunity cost that has to be taken into
consideration in the decision process (It is more costly for a doctor to mow the lawn on
Monday at 10:00 than a janitor).
Sunk costs - costs that cannot be recovered.
Valuable Alternative use of land => high opportunity cost of present use.
Zoning in many cases reduces opportunity cost.
Total Cost (TC) = Variable Cost (VC) + Fixed Cost (FC)
Marginal Cost (MC) = VC / Q  TC / Q
Average Cost (AC) = TC / Q
Table 7.1 illustrates FC, AC VC MC, AFC, AVC, ATC
In the short run can change labor to increase output. Given the wage w ,
MC  VC / Q  wL / Q  w / MPL
Assuming labor is the only variable input
AVC  wL / Q  w / APL
where APL is the average product of labor.
Figure 7.1 The MC curve goes through the minimum of the ATC and AVC but not the
AFC. As Q  , AFC  0 .
In the long run all inputs are variable.
Isocost line (= production budget line) is locus of points shows different combinations of
inputs such that cost is fixed.
C  wL  rK
solving for K gives
34
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
K  (C / r )  (( w / r ) L)
Isocost slope
K / L   ( w / r )
of ratio of wage rate to rental cost of capital.
Want to maximize production for a given total cost TC.
K
A=TC/Pk
A
B=TC/PL
Q
B
L
The MRTS   K / L  MPL / MPK . In equilibrium the slope of the isocost PL / PK =
MRTS
MPL / MPk  PL / PK  w / r
or
MPL / w  MPK / r
Figure 7.5 show effect of an effluent fee raising the relative price of the input water.
Production moves to be less water intensive.
35
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Expansion path shows the amounts of labor and capital used as firm grows (figure 7.6).
This is a long run concept.
K
Long Run
Short Run
L
Economies of scale includes increasing returns to scale as a special case. Increasing
returns to scale requires inputs be used in fixed proportions. Economies of scale allows
the ratio of inputs to change.
Figure 7.6 Long-Run Cost with Constant Returns to scale
Figure 7.10 Long-Run Cost with first Economies then Diseconomies of Scale. The LAC
traces tangency points of SAC curves.
At minimum point of LAC the LMC and SMC intersect.
Economics of scale => MC < AC. Diseconomies of scale => MC > AC. Define
EC
= elasticity of total cost
EC  [ TC / TC] / [ Q / Q]  [ TC / Q] / [TC / Q]  MC / AC
Economics of scale => E c  1, of MC < AC.
Economies of scope are present when joint output of a single firm producing two
products is greater that the output of two firms each producing one product. Define
C ( Qi ) as cost of producing Qi of the ith good. Economies of scope => SC > 0
SC  [ C(Qi )  C(Q1 ,..., Qn )] / C(Q1 ,..., Qn )
i
Diseconomies of scope SC < 0
36
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Learning Curve => Over time labor costs go down as more units are produced. Define L
as labor per unit, A> 0, B>0, 0<  <1 and N = # of units produced. Learning curve
implies
L  A  BN  
As N  , L  A
If N=1 then L=A+B or the amount of labor for first unit. If   0 there is no learning.
B
When   0 then second unit costs A   which converges to A as N increases.
2
Costs can decline because of increasing returns to scale or learning. See figure 7.12
which shows cumulative number of machines on the x axis. Figure 7.13 shows
production rate.
$
Economies of scale => A to B
Learning Curve => A to C
A
B
C
AC1
AC2
Q
In the early life of a product the learning curve is steep. After while the "learning" slows
down. In chip business had a 20% learning curve. => 10% increase in cumulative
production => costs fall 2%. In aircraft industry learning curve rate was 40%.
Estimating a cost curve
Linear Model used if MC is constant at  . MC  
VC  Q
37
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Quadratic Model used if have straight line MC. MC    2Q.
VC  Q  Q2
Cubic cost function used if have U shaped MC curve of form
MC    2Q  3Q2
VC  Q  Q2  Q3
Can use Excel and other software to estimate the MC curve.
Problem 12 page 272. Computer firm cost is AC  10.1(CQ) .3Q where CQ is
cumulative quantity, and Q is quantity per year.
a. There is a learning curve effect since -.1
b. There is decreasing returns to scale?
TC  10Q.1(CQ)(Q) .3Q2
MC  TC / Q  10.1(CQ) .6Q
Since .6Q .3Q => decreasing returns to scale
c. CQt=40,000, Qt = 10,000, Qt+1= 12,000
ACt  10.1(40) .3(10)  9., ACt 1  10.1(50) .3(12)  8.6
Problem 13 page 272.
Assume TC  a  bQ  cQ2  dQ3 , MC  (TC) / Q  b  2cQ  3dQ2
AC  [a / Q]  b  cQ  dQ2 . AC not defined for Q=0. For a U shaped cost
curve we want c<0 and d>0. At minimum point of AC, MC=AC. Set a=0 to normalize y
axis and equate MC=AC. => b  2cQ  3dQ2  b  cQ  dQ2 or c  2dQ . If c=-1 and
d=1 then Q=1/2
Key Question: How do we use this theory to solve a practical problem?
Firm has a production function F ( K , L)  AK  L which could be estimated using Excel
when we note that log[ F ( L, K )]  log( A)   log( K )   log( L) . Want to determine a
general case that shows the optimum K and L given the cost of capital r and the wage w.
The following is a simpler treatment than 273-278. In equilibrium we note that
(1)
[ MPK / MPL ]  r / w
38
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Given the production function
(2)
Q0  AK  L
(3)
[ AK  1 L / AK  L 1 ]  [r / w]
(4)
[L / K ]  r / w
(5)
L  [rK / w]
Substituting (5) into the production function (2) gives
(6)
Q0  [ AK    r  K  /   w ]
(7)
K (   )  (w / r )  Q0 / A
The equilibrium amount of capital is
(8)
K  [(w /  r )  /(   ) ](Q0 / A)1/(   )
If the cost of capital, r, increases or the marginal product of capital,  , falls K will fall.
Direct substitution of (8) into (5) gives
(9)
L  [(  r / w) ( /(   )) ](Q0 / A) (1/(   ))
If the wage rate, w, increases or the marginal product of labor,  , falls then less labor
will be used. The firms cost function is
(10)
C  wL  rK
At equilibrium
(11)
C  w(  /(   )) r ( /(   )) [( /  ) (  /(   ))  ( /  ) (  /(   )) ](Q / A) (1/(   ))
Equation (11) shows how C is related to wages, w, cost of capital, r, and returns to scale
(   ) . MATLAB can be used to calculate the appropriate derivatives
Equation (11) shows how costs are related to the production function of the firm as well
and input costs.
See ch7_1.xls for solution.
39
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 8 Profit Maximization and Competitive Supply
Firms want to maximize revenue.
Competitive supply => many firms, all price takers, no product differentiation, free entry
Profit is the difference between revenue and cost for a given level of output q.
(1)
 ( q )  R( q )  C ( q )
To maximize profit differentiate and set to zero. => MR(q) = MC(q)
(2)
 / q   (TR) / Q   (TC) / Q  0
For a competitive firm P is given =>  d  and P=MR. It can be proved that
(3)
MR  AR[1  (1 /  d )]  P[1  (1 /  d )]
(4)
TR / q  [(qp  pq) / q]  p[(qp  pq) / qp]
 p[1  ( pq / qp)]  p[1  (1 /  d )]
Figure 8.1 shows profit maximization. Figure 8.2 shows firm in competitive market is a
price taker.
Figure 8.3 shows a competitive firm (MR=AR) making extra normal profits and
operating with decreasing returns.
In long run new firms will enter and firm will earn normal profits or those profits such
that no firms enter or no firms leave industry.
Figure 8.5 show that in the short run the firms MC curve is the firm supply curve.
If an input increases in price => AC curve shifts up => MC curve shifts up and firm
produces less output.
Book indicates that short run MC of petroleum has flat sections until other refineries get
on the line. See figure 8.8. ComEd's costs are constant until they start jet engine
generators.
Industry supply curve is the sum of firm MC curves in the short run.
40
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Producer Surplus = area above firm supply curve and below price. See figure 8.11.
In long run firm can alter all inputs. If industry is making extra normal profits in the short
run => new firms will enter such that in the long run all firms are making normal profits.
The long run competitive equilibrium point is P=SMC=LMC. Firm is at the minimum
point of its AC curve. Given the firm's technology, it is most efficient. It earns economic
profit (normal profit). See figure 8.13.
In long run firm tries to equate LRMC to price BUT new firms can enter the industry and
drive down price.
Economic profit = R - wL - rK where rK is the opportunity cost of capital. A firm
making zero economic profits may still stay in business.
Economic rent = return to factors of production in limited supply. MJ earns economic
rent. Songs and books are copyrighted to allow economic rent to be earned for a limited
number of years.
The long run supply of a constant cost industry is a straight line. This is a partial
equilibrium argument. As Q increases input prices will increase at some time. General
Equilibrium has all assumptions variable. Some industries are so small that they can be
assumed to have no effect on input prices. See figure 8.15
Increasing Cost Case => If extra normal profits are earned new firms will enter the
industry and input prices will increase. See figure 8.17.
Taxes. An output tax raises the MC curve. A Lump Sum Tax (such as a liquor license)
does not change the MC since it is paid no matter what. The effect of a n output tax in the
long run is to reduce industry supply.
Perfectly Competitive Markets Assume:
-
Price Taking
Product Homogeneity
Perfect Mobility of Assets
Perfect Information
Even if only one firm in the market it may pay to act as if the firm was in a competitive
market since extra normal profits => others firms may enter to bid away profit.
Problem # 1 The Conigan box company is in a competitive market. It sells boxes in
batches of 100 for $100.00. TC  3,000,000.001Q2 .
What is profit maximizing output? Want MC=MR=AR. =>
100 .002Q,  Q  50,000
41
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
What is profit?
  TR  TC,   100 *(50000)  (3000000.001*(50000) 2  $500,000
The firm should stay in business in the short run since AVC < P:
AVC  TVC / Q, TVC  TC  TFC ,
TVC  5,5,00,000  3,000,000  2,500,000
AVC  (2,500,000) / 50,000  $50
Example: Restaurant stays open in “off hours” if it can cover labor costs.
Problem # 2 Competitive market has a market demand curve of
P  75  15
.Q
and supply curve of
P  25..5Q
where Q = where a typical firm has MC  2.5  10q .
What is market price and rate of sales? We set S = D or
25.5Q  75  15
. Q,  Q  25, P  75  15
. * 25  $37.5
Each firm sells where MR = MC . Since perfect competition MR=AR=P=$37.50
37.5  2.5  10q,  q  35
.
Note: If you use ch2_1.xls template to solve system remember that the template is set up
as: Q s  f ( P), Qd  f * ( P) ,  Q s  50  2 P, Qd  50  (1/1.5) P
42
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 9 Analysis of Competitive Markets
Want to look at welfare and efficiency effects of a competitive market.
Figure 9.1 shows producer (area between supply curve and market price) and consumer
(area between demand curve and market price) surplus.
Price Controls: Figure 9.2 and 9.3 show changes in producer and consumer surplus
when the government put on a price control.
The more inelastic the demand curve the more the deadweight loss.
If supply is fixed there is no dead weight loss.
Looking at figure. Producer surplus loss = A+C. Consumer net gain = A-C-B.
Unless demand is totally inelastic a price control => a shortage.
Price controls can have a serious long term effect on market as was case with NY
housing.
Figures 9.4 & 9.5 contrast the effects to price being held below and above the market
price.
Kidney Market. (See figure 9.6) Supply function of Q s  16000  .4P implies that if the
price is held to $0.0 by the 1984 National Organ Transplantation Act, and you cannot
sell an organ, only 16000 will be supplied. The demand function is Qd  32000  .4 P .
16000  .4 P  32000  .4 P,  P  $16000 / .8  $20,000
Equating Q  16000  (.4* 20000)  16000  (.4* 20000)  24000
On figure 9.6 suppliers lost A+C. If supply is reduced due to not being able to sell
kidneys, the market clearing price is $40,000 and many consumers are rationed on a
"willingness to pay" A+D = total value of kidneys at new market clearing price.
Middlemen and hospitals get gain NOT donators.
43
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Minimum Prices (Government restricts market from lowering prices. Fair trade laws.
Tariffs.) (See figure 9.7).
P1
A
B
P2
C
Q3
Q0
Q1
Government raises price from P2 to P1. Supplier produces Q1 not Q3 > "glut"
From free market consumer surplus loss = B. Producer surplus loss = C (Assuming
producer produces Q3)
Minimum Wage Figure 9.8 shows the analysis for minimum wage which causes
unemployment of L2-L1.
Price Supports. In contrast to minimum price laws, price supports require that
government buy farm products. Figure 9.11 shows that as a result of the purchase
program production when from Q0 to Q2 and price from P0 to Ps. Consumer surplus
reduced by = B+A. Producer gain = A+B+D. Government cost = Ps(Q2-Q1).
Change in welfare (A+B+D)-(A+B)-[Ps(Q2-Q1)].
Production Quotas. Government reduces supply (taxi licenses, bar permits). NYC has
113,150 taxi licenses. Roughly same as 1937. Medallion costs $880,000. Cost in 1947
was $2,500. In 1980 was $55,000. If supply went up current medallion owners would see
a loss of value of their medallions. Drives cab ride prices very high. See Figure 9.13. In
NYC cabs are run 24/7, not shut off at night.
Acreage limitation programs give farmers money incentive to leave land idle. Causes
farmers to farm their land more intensively. Figure 9.11 suggests cost to government
must at least equal B+C+D of the gain from planting since farmer sees new higher price
as given.
CS   A  B, PS  A  C  payments to stop producing (at least B  C  D)
Thus PS  A  C  B  C  D  A  B  D
Total change in welfare
44
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
( CS  PS  (Cost to gov ))  (  A  B)  ( A  B  D)  ( B  C  D)   B  C
Society would be better off if Government just gave producers A+B+D since what
government would lose farmers would gain for a zero net effect.
Book Example page 336 figure 9.12 shows how to use above theory:
In 1981
Qs  1800  240 P, Qd  3550  266P
(240  266) P  (3550  1800),  P  $3.46, Q  2630
Government must increase demand to raise the price. Let Qg be government demand
3550  266 P  Qg  1800  240 P,  Qg  506 P  1750
Given desired P, Qg is determined. If P=$3.70, => government buys 122 million 2688566) bushels at cost of $451.4 million. Inspect figure 9.12. Customers lose A((3.73.46)*2566) + B(.5*(3.70-3.46)(2630-2566) or 616+8=624 million. Note that A is
substantially bigger that B in this problem. Total cost = 624+451.4=1075.4
Exact answers for this problem and related ones are in ch9_1.xls.
As an exercise run the problem where the desired price is $3.80. Adjust demand and
supply and see what happens.
If P = $4.00, => government buys 506*4-1750 = 274 at a cost of $1096 million.
Calculate A and B here.
Quotas and Tariffs. Figures 9.14 and 9.15 shows basic setup. (9.14 shows a tariff that
eliminates all imports). Consider figure 9.15. With free trade the price is Pw domestic
supply is Qs, domestic demand is Qd. Imports are (Qd.-Qs). After a tariff (or quota).
Domestic supply jumps to Q's, domestic demand falls to Q'd and imports fall to
Q'd-Q's. Trapezoid A = gain to producers. Due to higher price consumer loss is
A+B+C+D. If a tariff is imposed, Gov gains revenue D. Net domestic loss is B+C+D.
Excel file ch9_2 shows setup to analyze Sugar Case. A tariff of 83% gives book case.
Can adjust tariff to eliminate imports and thus tariff revenue (area D).
Product Taxes. Figure 9.16 and 9.17 shows effect on producers and consumers of a per
unit tax on producers. Area A + B = what buyers lose. Area D + C = what sellers lose.
Are A + D = what government collects.  s = elasticity of supply.  d = elasticity of
demand. A tax falls mostly on the buyer (seller) if  d /  s is small (large).
Pass-through (to consumers) fraction is
 s / ( s   d ) .
45
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Fraction of tax producers bear is  d / ( s   d )
NOTE: These formulas make use of fact that  d  0 .
Most luxury goods have inelastic demand => luxury good tax hits consumers hard.
Example 9.7 Tax on Gas. See figure 9.20 and Excel 9_3.xls
We are given that  d  .5,  s .4, P*  $1.00, Q*  100 . From Chapter 2 and excel
ch2_2.xls we know that
Given QD = a - bP
(11)
QS = c + dP
(12)
And P* and Q* are equilibrium P and Q, then at equilibrium the elasticity of demand D and
elasticity of supply S are
D
=
-b(P*/Q*)
(13)
S
=
d(P*/Q*)
(14)
This implies Qd  150  25P, Qs  60  20 P . With no governmental tax
150  25P  60  20 P
90  45P  P  $2.00. Q  100
After Government places a $1.00 tax the new price is $2.44 and the new quantity is 89.
The producer price falls to $1.44. The steeper the supply curve the less quantity falls and
the more the producer price falls. The more inelastic the demand curve the more the
consumer price increases. In this case a greater proportion of the tax is passed on. The
following calculations replicate the figure 9.20
ps  supply price, p0  no tax price, pb  price including tax
pb  ps  tax  ps  1.00
Q d  150  25 pb
Q s  60  20 ps
Q d  Q s  150  25( ps  1)  60  20 ps
(150  25  60)  45 ps  ps  65 / 45  1.444
pb  $2.44. Q  60  20*1.44  88.8
46
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Problem # 12 page 354 has the demand and supply of hula beans. See p354_12.xls
Qs  50  P, Qd  100.5 P . The world trade price is $.60. Congress puts on a tariff of
$.40. Use of ch9_2.xls implies that Equilibrium P and Q are $1.00 and 50 if there was no
foreign supply. If the tariff is imposed at .666666 then the domestic producer will
produce 10. The domestic consumption will be 50 (instead of 70 at world trade price of
$.60). Deadweight loss = (70-50)*$/40*.5= area C = $4.00. Before the tariff since
domestic production is 10 and consumption is 70 => import 60. After tariff there is no
imports. Consumer surplus loss =$24.
47
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 10 Market Power: Monopoly and Monopsony
Monopoly => One producer MR  AR.
Monopsony => One buyer
Assuming P=6-Q
P
6
5
4
3
2
1
Q
0
1
2
3
4
5
TR
0
5
8
9
8
5
MR
5
3
1
-1
-3
AR
5
4
3
2
1

inf
-4
-1.5
-.667
-.25
TR
0
5
8
9
8
5
MR  AR[1  (1 / )]
Key Formulas:
  [ AR / ( MR  AR)]
If || < 1 => as quantity increases TR decreases. See figure 10.1
A
At B  =-1. AB=BD, AE=EO, OC=CD
E
B
O
C
D
Equilibrium at max profit requires MC=MR where MC hits MR from below. Figure 10.2
is the basic diagram. Various points:
-
If normal profits are made => AC curve will be tangent to AR curve at
a point to left of minimum point. Normal profits with MR < AR =>
operating at less than optimum capacity.
-
If MR=AR => perfect competition. Can have three cases: 1. If making
loss in short run => less than optimum capacity. 2. If making normal
profits => at minimum point of AC curve. 3. If making extra normal
profits => operating at greater than optimum capacity.
48
Microeconomic Study Notes (17/7/2012
-
Houston H. Stokes
If MR < AR can
Possible Cases Table: Study below listed tables and be able to draw all possible cases
and prove that the I cases are not possible.
MR=AR
MC=MR
MC<AC
MC=AC
MC>AC
_____________________________________________________________________
AC>AR
Case 1 P
Case 2 I
Case 3 I
AC=AR
Case 4 I
Case 5 P
Case 6 I
AC<AR
Case 7 I
Case 8 I
Case 9 P
_____________________________________________________________________
MR<AR
MC=MR
MC<AC
MC=AC
MC>AC
_____________________________________________________________________
AC>AR
Case 10 P
Case 11 I
Case 12 I
AC=AR
Case 13 P
Case 14 I
Case 15 I
AC<AR
Case 16 P
Case 17 P
Case 18 P
_____________________________________________________________________
Figure 10.2 is case 16. Here AC<AR, MC=MR, MR<AR. The left hand side of the MC
curve has not been drawn.
A firm is at less than optimum capacity (MC<AC), at optimum capacity (MC=AC) or
greater than optimum capacity (MC>AC).
Monopoly power occurs if MR<AR. I firm is in a competitive market if AR=MR. In a
competitive market the firm can sell as much as it can without the price falling.
A firm always maximizes profit if MC=MR and MC hits MR from below.
49
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
A firm makes an extra normal profit if AR>AC, a normal profit if AR=AC and a short
term loss if AR<AC.
Solution Strategy: If the functions are known set MR=MC and solve for P, Q, and
profit.
Cost of production (TC)
C(Q)  50  Q2
Demand (AR)
P(Q)  40  Q
Revenue (TR)
R(Q)  P(Q)Q  40Q  Q2
MC=MR => C'(Q)=R'(Q) or 2Q  40  2Q, Q  10, P  30
Solution is illustrated in figure 10.3. This is case 18 . a firm making extra normal profit
operating at greater than optimum capacity.
Since in equilibrium MC=MR and MR  AR[1  (1 /  d )] the markup model for pricing
is MC  AR  AR(1 /  d ), or AR  P  [ MC / (1  (1 /  d ))]
The "markup model" fails if  d  1 . If | d |  1 the implied price is negative! No firm
will ever operate here!
In a competitive market  d   and in equilibrium (case 5) P=MC.
Figures 10.4a and 10.4b show special cases where demand can shift and leave Q fixed
and demand can shift and leave P fixed. This is usually not the case.
A per unit tax of  increases MC by the same amount. Here profit is where
MR  MC  
A B
Unless there is a vertical demand curve,   Q . If | d | 1, P 
The more inelastic the demand curve the higher the price after the tax. This can be tested
using toolbox file ch9_4.xls. To make the demand curve more inelastic, reduce the
coefficient on price. Toolbox file ch9_4.xls must be used in place of ch9_3.xls since the
former holds the percent tax fixed while the latter holds the dollar amount of the tax
fixed.
In a multi-plant firm the profit maximization condition is
50
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
MC1  MC2 ,...,  MCn  MR
n
Figure 10.6 shows MCT  horizontal sum of MCi   MCi for a n plant system.
i 1
 QT  Qi
Monopoly Power. A firm in an industry with few sellers will face a more elastic demand
curve than the industry as a whole. The greater the product loyalty for that firms product,
the more inelastic the demand curve. The more inelastic the demand curve the more the
firm can raise price. Lerner's measure of monopoly power L  ( P  MC) / P . L is in the
range 0-1. P=MC => L=0 and demand curve has    . Since MR  P[1  (1 / )] and
B A
MR=MC as ||  P .
  1. in food industry, =-10 in store.
. MC or a 10%-11% markup. The
Manager sets P  MC / (1  (1 / ))  MC / (1.1)  111
better the quality of the store the smaller | | and the more price can be raised. If   5
prices could be raised (1.-(1./5)) of 25%.
B
Designer label clothing => || or higher prices. Advertising is designed to lower the
absolute value of the demand elasticity. To make the consumer less sensitive to a price
change. See figures 10.8a and 10.8b
Firm's elasticity of demand due to:
-
Market elasticity of demand
Number of firms in the market
Interaction between firms
The firm will face a more inelastic demand if the number of firms in the market is small
and there is less interaction between firms (i.e. firms do not follow a price lowering with
a price lowering). If firms collude, then prices will go up. OPEC limited supply. Some
countries cheated!!
Figure 10.10 shows Deadweight loss of monopoly power. Monopolist wants to equate
MC=MR to maximize profit. Consumers lost A+B. Producer gains A loses C.
Deadweight loss = -B-C.
Figure 10.11 shows "kink" in MR curve due to a price ceiling due to regulation. If price
is set at Pc , can get perfect competition solution. What is interesting is MC can move
within the kink and prices will not change. Problems:
-
How do we know where that it?
By price control will we be removing incentives?
51
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Figure 10.12 shows a Natural Monopoly ( are in range of declining MC as Q increases.).
Here want one firm. If left alone firm wants to sell Qm at Pm. Government lowers MR as
seen by firm by setting Pr or the lowest price where the firms stays in business. If price
were set to Pc firm will make a loss since here MC=MR but MC<AC.
Rate of return Regulation. Attempts to regulate to insure a given rate of return leads to
firms that use relative more capital than labor.
Monoposony: Firms is such a large buyer of an input that it faces an upward supply
curve (average expenditure curve). On the average expenditure curve draw a marginal
expenditure curve. At equilibrium equate marginal expenditure to marginal value
product. MVP = marginal physical product * Price of output. The marginal physical
product of capital is the increase in output for one more unit of capital.
MPPk  Q(l , k ) / K
Figure 10.14 shows firm with monopsoly power will buy less of an input and pay less
that if it were in a competitive input market.
The more elastic the supply curve, the less monopsony power (see figure 10.15, 10.16).
ME  AE[1  (1 /  s )]
 s    ME  AE
Figure 10.17 shows deadweight loss due to Monopsony. A +C = producer loss.
Consumer (firm with monopsony power) gains A-B. A-B-A-C = -B-C = deadweight
loss.
Bilaterial Monopoly => two monopolists facing each other. Hard to tell what will
happen. Think of auto producers and parts suppliers.
Problem Firm has a
demand curve
supply curve
P  100.01Q
TC  50Q  30000
What is P and Q to maximize profit? MC = 50. MR  (100Q.01Q2 ) / Q  100.02Q
50  100.02Q, Q  2500, P  100  (.01) * 2500  $.75
Problem
Given
P= 27 24 21 18 15 12 9 6 3 0
Q=0 2 4 6 8 10 12 14 16 18
52
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
MC=10
What is P and Q to max profit?
Inspection or regression analysis (See p365_5.xls) indicates
P  27  1.5Q, TR  PQ  27Q  1.5Q2 , MR  (TR) / Q  27  3Q
TR  (18.5)(5.67)  $104.83
Profit =   104.83  (10)(5.67)  $48.17
Problem Firm faces demand P  700  5Q and has two factories.
C1 (Q1 )  10Q12 , MC1  20Q1
C2 (Q2 )  20Q22 , MC2  40Q2
Q  Q1  Q2  [ MC1 / 20]  [ MC2 / 40]  3 MCT / 40
MCT  40Q / 3, MCT  MR  [40Q / 3]  700  10Q, Q  30
Monopoly price = 700  (5)(30)  550
MR  MC1  MC2  700  (10)(30)  400
Q1  400 / 20  20, Q2  400 / 40  10
Firm produces less in higher cost plant!!
53
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 11 Pricing with Market Power
Figure 11.1 suggests that if one price is charged it would be P.* If firm was able to price
discriminate, it would charge a higher price to more inelastic market. It has been reported
that McDonald's charges more for the same item at Water Tower Place than on the South
Side!
A firm that practices first-degree price discrimination charges each customer his/her
reservation price or the maximum that the consumer would pay. Figure 11.2 shows how
in such a world the firm expands production from Q* to Q**. In this word all consumer
surplus is captured by the firm. In practice this is hard to do. Universities try to
practice this approach by giving just enough aid to get the student. If firm does not have
perfect knowledge, then a number of prices can be charged. See figure 11.3
Second-degree price discrimination occurs when price is reduced if additional amounts
are bought. (See figure 11.4) “Big box” vs small store.
Third-degree price discrimination occurs when firm packages same product under two
different labels and sells at a different price. Using formula MRi  Pi [1  (1 /  di )] and
MC  MR1  MR2 , then
[ P1 / P2 ]  [1  (1/ d 2 )]/[1  (1/ d1 )]
Figures 11.5 and 11.6 illustrate third-degree price discrimination.
Table 11.1 suggests that coupon users have higher elasticity of demand than non coupon
users. Assume the non coupon users have elasticity of demand  d 1 . If P1 = P2, and V =
value of coupon, then in an optimum world
[( P1  V ) / P2 ]  [1  (1/ d 2 )]/[1  (1/ d1 )]
Airlines price discriminate. Table 11.2 on page indicates that industry price elasticity for
First-Class, Unrestricted Coach and Discount are -.3, -.4 and -.9. Income elasticities are
1.1, 1,.2 and 1.8. Firm elasticities are > in absolute value.
Intertemporal Price Discrimination => firm charges first consumers more. Hard cover
edition of a book comes out first. Before too long soft cover comes out at a lower price.
Library buys hard cover! Nikon does this with new models! See figure 11.7
Peak-Load Pricing => charge more at a certain time in the day. See figure 11.8
Two-Part Tariff => pay to get into Great America, then, pay for each ride.
54
Microeconomic Study Notes (17/7/2012
Houston H. Stokes

Figure 11.9 shows single consumer. Usage fee is set to marginal cost. Entry fee is set
to capture entire consumer surplus. Here firm captures all surplus

Figure 11.10 Profit making usage fee is set > MC. Entry fee T* set to surplus of
customer with smaller demand. Profit = 2*T* + (P*-MC)*(Q1 + Q2).

For n customers this becomes (see figure 11.11) n(T)*T* +(P-MC)*Q(n),
Polaroid Camera cost was entry fee. Film was price.
Bundling. When customers have heterogeneous demands that are negatively
correlated => pays to bundle products. Figure 11.12 and 11.13 show cases
A
II
III
I
I
II
III
IV
Consumers buy both
Consumers buy only A
Consumers buy neither A or B
Consumers buy only B
IV
B
By forcing consumer to get both => move to section I
Effectiveness of bundling depends on how negatively products are related.
Mixed bundling. Sell as a package or alone. MS Office 97 is sold this way. Figure 11.18
shows that with zero MC this is even more profitable.
Luxury cars are sold with standard options bundled into the car. Vacation package
another example. Restaurants sell items a la carte and as dinners.
Tying => force customer to buy all items from one source. McDonalds at one time
required owners to get supplies from one source. Gas station has to carry full range of
products.
Advertising. Want marginal revenue of an additional dollar spent on advertising to equal
full marginal cost of that advertising. We define profit as a function of amount sold Q and
advertising A.
 (Q, A)  PQ( P, A)  C(Q)  A
 / A  P(Q / A)  MC (Q / A)  1  0
We rearrange and get the marginal revenue from advertising.
MR a d s  P(Q / A)  1  MC (Q / A)
55
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
[ P  MC][Q / A]  1,  [( P  MC) / P][( AQ) / (QA)]  [ A / PQ]
Given elasticity of demand = [( AQ) / (QA)]   A , in equilibrium
then
[( P  MC ) / P]a  [ A / PQ]
MC  P[1 
1
p
]  [( P  MC ) / P] 
1
p
[ A / PQ]  [ A /  P ]
Which determines how much advertising should be purchased.
The more inelastic the demand for the firm's output, the more advertising. In perfect
competition since  p   , no advertising is done.
This logic takes into account the effect of advertising on increasing sales but also adding
to costs as added units have to be produced.
Transfer Pricing is between units of a vertically integrated firm. Will show that
profits are maximized if transfer price equals marginal cost of respective upstream
division.
 (Q)  R(Q)  Cd (Q)  C1 (Q1 )  C2 (Q2 ) where each term represents revenue from sales,
firm production costs, costs from intermediate production inputs from plant 1 and costs
from intermediate production inputs from plant 2.
To maximize the net marginal revenue firm earns from one more input of Q1 and Q2
differentiate  (Q) with respect to Q1 and Q2.
d / dQ1  (dR / dQ)(Q / Q1 )  (dCd / dQ)(Q / Q1 )  dC1 / dQ1  0
NMRi  ( MR  MCd ) MPPi  MCi
Each upstream firm takes price Pi as given and equates it to their marginal cost MCi.
NMRi  ( MR  MCd ) MPPi  Pi
56
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Engine problem consists of an auto firm assembling cars with the option of getting
engines from an upstream firm. Demand for autos
P  20000  Q
MR  20000  2Q
Cost of assembly
C A (Q)  8000Q
MC A  8000
engine costs
CE (QE )  2QE2
MCE (QE )  4QE
Net marginal revenue of engines given QE  Q
NMRE  MR  MC A  20000  2Q  8000  12000  2Q
NMRE  MCE
12000  2Q  4QE  QE  2000
Optimal transfer price is marginal cost of the 2000 engines or 4QE  $8000
If engines can be bought outside for $6000, then a case can be made that the firm should
buy all engines outside BUT remember the engine dept has an upward slope to MC. This
suggests equate
NMRE  6000
12000  2QE  6000  Q  3000
Here we have QE  Q since we set MCE (QE )  6000
 4QE  6000
QE  1500
Firm buys 1500 outside and uses 1500 of its own engines.
57
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 12 Monopolistic Competition and Oligopoly
Monopolistic Competition => Large number of sellers each with some market power
due to selling a differentiated product. At the model level in the automotive industry we
have monopolistic competition.  d   in monopolistic competition.
-
Many sellers
Each seller selling a slightly differentiated product
Entry by new firms is NOT restricted
Note: Trade theory suggested that trade would be between countries that were not
similar. However most trade is between countries such as Canada and US and Germany
and France which are similar. Krugman used monopolistic Competition Theory to rescue
trade theory.
Oligopoly => Few sellers each with market power. If one firm in an oligopoly market
were to raise price, the others would not follow and that firm would find sales fall off
rapidly. If the same firm were to lower price hoping to increase sales at the expense of the
other firms, then all other firms would lower price and the expected increase in sales
would not be realized.
-
Few sellers
Each seller selling a differentiated product.
Each with substantial market power
Entry is restricted by barriers (example aircraft industry)
Cartel=> Firms work together. (GE in 50's until they got caught.)
-
Like a monopoly BUT members tempted to cheat.
-
Cartel usually does not control complete market
Under Monopolistic Competition free entry =>
-
No extra normal profits in long run.
-
Firms in long run operate at less than optimum (minimum average cost)
capacity (see case 13) and figures 12.1 and 12.2.
Table 12.1 lists Elasticities
Royal Crown
Coke
-2.4
-5.2 to -5.7 (more close substitutes)
58
Microeconomic Study Notes (17/7/2012
Folgers
Maxwell House
Chock Full o’Nuts
Houston H. Stokes
-6.4
-8.2
-3.6
In perfect competition and monopolistic competition When a market is in equilibrium
firms are doing the best they can and have no reason to change their price or
output.
In oligopoly each firm is doing the best it can, given what its competitors are doing.
Cournot Model. Two firms (duopoly) each making its decision at the same time.
Decision rule: Each treats the output level of its competitor as given.
See figure 12.3 where MC is constant
Firm 1 Q
50
25
12.5
Firm 1 thinks Firm two produces
0
50
75
 Firm # 1 profit-maximizing output is a decreasing schedule of how much it thinks
firm 2 will produce. This defines the reaction curve.
Cournot equilibrium => Intersection of firm 1's and firm 2's reaction curve => how
firms divide up quantity. (See figure 12.4).
Assume problem on page 461-462 where there are 2 firms
MCi = 0
P=30-Q
Define R1 = total revenue of firm 1
59
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
R1  PQ1  (30  Q)Q1
 30Q1  (Q1  Q2 )Q1
 30Q1  Q12  Q2Q1
MR1  R1 / Q1  30  2Q1  Q2
R2  PQ2  (30  Q)Q2
 30Q2  (Q1  Q2 )Q2  30Q1  (Q1 ) 2  Q2Q1
MR2  R2 / Q2  30  2Q2  Q1
Firms 1 and 2 have reaction curves
Q1  15.5Q2 , Q2  15.5Q1
which are derived by setting MR1 and MR2 to zero and solving for Q1 and Q2.
at equilibrium Q1  15.5[15.5Q1 ], Q1  Q2  10, Q  20
If firms could work together =>
  PQ  (30  Q)Q  30Q  Q 2
MR  R / Q  30  2Q, Q  15, Q1  Q2  7.5
Note that MC=0 and Q1  Q2  15 defines the contract curve.
For graphical analysis see Figure 12.5.
Stackelberg Model => One firm moves first BUT has to take into account what other
firms will do.
Firm 2 has reaction curve Q2  15.5Q1 . Firm 1 knows this and knows its revenue
is PQ1  30Q1  Q12  Q2 Q1 . To get solution substitute form 2's reaction curve in firm 1
revenue curve
R1  30Q1  Q12  Q1[15.5Q1 ]  15Q1 .5Q12 . MR1
= 15  Q1 ,  Q1  15
Firm 2 will produce 15.5Q1  7.5
Going first gave firm 1 the advantage.
Stackleberg Model good when one firm is dominant and can move quickly to set
quantity.
60
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Cournot Model good when have more firms with no one firm dominant. Cournot model
is stable because each firm produces what maximizes its profits, given what competitors
are producing.
Bertrand Model => Firm sets price NOT quantity in a homogenious product
market.
Assume MC1=MC2 = $3.00 and market demand is P = 30-Q. Here Cournot equilibrium
(where reaction curves are jointly solved) => Q1  Q2  9 . Proof:
TR1  PQ1  30Q1  QQ1  30Q1  Q12  Q1Q2
TR2  PQ2  30Q2  QQ2  30Q2  Q22  Q1Q2
MR1  30  2Q1  Q2  3, reaction curves
Q1  13.5  .5Q2 , Q2  13.5  .5Q1
MR2  30  2Q2  Q1  3
 Q1  13.5  .5[13.5  .5Q1 ], .75Q1  [13.5  (13.5 / 2)], Q1  9
Under Bertrand model set marginal revenue = marginal cost and divide output.
 MR  TR / Q  ((30Q  Q2 / Q)  30  2Q  3.  Q  27 and Q1  Q2  135
..
 P = 30-Q=3
Price undercutting would occur until price = $3.00
Problem with Bertrand model. When there is a homogenious product firms compete on
quantity not price (orange growers). We assume firms divide Q. This may not be the case.
Price Competition with differentiated product and price
Two firms with fixed cost of $20.00 with demand curves of
Q1  12  2 P1  P2 , Q2  12  2 P2  P1
Excel file ch12_1 solves the system for any fixed cost and marginal cost. Verify that this
file calculates correctly. Such a setup provides a 'template' for a more general decision
model. Template also shows Collusive Equilibrium.
Each firm decides to set price given the other firms price. For firm I
61
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
2
 i  PQ
i i  12 Pi  2 Pi  PP
i j  20
 i / Pi  12  4 Pi  Pj  0
reaction curve is
Pi  3  .25 Pj
Pi  3  .25(3  .25 Pj )
Pi  (3  .75) /(1  .252 )  $4.00
collusion profit
  24 P  4 P 2  2 P 2  40
 / P  0  24  4 P
 P  $6.00,   $16.00
Use the template to solve the system assuming demand curves are now:
Q1  24  2 P1  P2 , Q2  32  2 P2  P1
Which might have resulted due to an Ad effort. Since profit is calculated, the
effectiveness of an ad effort can be calculated.
P & G Pricing Problem
Table 12.2 shows P&G decision model assuming fixed costs of $480,000 and variable
costs of $1.00. Excel file ch12_2.xls generalizes this analysis. The base case assumes the
demand curve is
Q  3375 P 3.5 Pu.25 Pk.25
which could have been estimated using Excel if we assume natural logs of the data have
been used.
Use the template to see what would be the situation at P & G if Unilever charged $1.45
and Keo charged $1.47 and P & G investigated prices $1.10, $2.00, $1.87, $1.44
If P & G does an effective marketing effort the 3375 might be raised or the market
elasticity of -3.5 changed. Assume due to a "You got to have it…" type ad effort the
elasticity is lowered to -3.. Assuming the base case what happens to profit? How would
you determine if the ad effort was worth it?
Prisoners Dilemma
Nash equilibrium is a noncooperative equilibrium. If you hope that firm will set price at
collusive level, the other firm will find that it pays NOT to set at this level. Verify that
62
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
ch12_1 shows that if firm # 2 sets $4.00 and firm # 1 sets $6.00 (collusive price) then
firm # 2 makes $20.00 and firm makes $4.00.
Use the template to verify
Firm # 1 $4.00
Firm # 1 $6.00
Firm # 1 $8.00
Firm # 2 $4.00
$12,$12
$4,$20
-$20,$28
Firm # 2 $6.00
$20,$4
$16,$16
-$4,$28
Firm # 2 $8.00
$28,-$20
$28,-$4.
$12,$12
With $6, $6 the total profit is highest ($32.00). But each firm will look at situation as "If I
charge $4.00 I will do "best" (i.e. make $12.00)." Each firm does not dare trust the other
firm to charge the higher price!!
In Oligopoly there is an incentive to keep price fixed. Figure 12.7 illustrates the kinked
demand curve. MC can move inside discontinuous section and there is no incentive to
change price or quantity. => Oligopolistic firms have price rigidity
Price signaling firm such as American Airlines announces it is raising its price. Firm
hopes that such price leadership will result in the others following. (This seems to be the
case in Hyde Park between Shell and Amoco stations which 'post' prices that seem to
agree all the time.)
Banks operate using price signaling as they "respond" to changes at the Fed.
Dominant Firm Model. Figure 12.9 shows dominant firm model. Dominant firm takes
demand curve DD as
DD = Market demand - Fringe Firm Supply.
Dominant firm draws MR on this derived demand curve and sets price. Fringe firms
take this price as given.
An Oil Cartel would not be legal in the US. Figure 12.10 shows situation. Here OPEC
derives a demand curve DOPEC and draws a marginal revenue curve. Price is set at P*
where MROPEC = MCOPEC. If OPEC were not a cartel, OPEC price would be at Pc.
Universities work together to keep athletics "amateur" by paying players nothing.
Somehow they get away with it!!
63
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 13 Game Theory
Game Theory "If I believe that my competitors are rational and act to maximize theor
own profits, how should I take their behavior into account when making my Own profitmaximizing decisions?"
Cooperative game => players can negotiate binding contracts and enforce them
Noncooperative game => players cannot negotiate and cannot enforce contracts
Want to understand your opponent's point of view and assuming your opponent is
rational deduce how he or she is likely to respond to your actions.
Problem. Company A (acquirer) considering bidding for Company T (target). Company
T is exploring for oil and knows what it has found. It is assumed that a share price of
$0.00 - $100.00 for company T under its own management is equally likely. If A buys T,
T will make 50% more money. A thus can make an offer between $0-$150 a share and T
will accept if the offer is greater than or equal to the current share price. What is the
optimum offer?
Answer. Offer $0.00. Assume A offers $50.00. This suggests that the share price is
between 0-50. Its expected value is $25.00. With A's management the value will rise to
$37.50 (1.5*25) which is less than the offer!!
Dominant Strategy one that is optimal for a player no matter what an opponent does
Firm A
Firm A
Advertise
No Ad
Firm B Advertise
10,5
6,8
Firm B No Ad
15,0
10,2
Dominant t-strategy is for A and B is to advertise
Firm A
Firm A
Advertise
No Ad
Firm B Advertise
10,5
6,8
Firm B No Ad
15,0
20,2
Here A does not have a dominant strategy. Depending on what B does, A will either
advertise and get 15 or not advertise and get 20. But B has a dominant strategy which si
to advertise. => A will figure that B will determine this and will advertise.
Nash equilibrium - I am doing the best I can given what you are doing. You are
doing the best you can given what I am doing.
Dominant Strategy - I am doing the best I can no matter what you do, You are
doing the best you can no matter what I do.
64
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Chapter 14 Markets for Factor Inputs
Firms demands for inputs are "derived" in that they are the result of production decisions
due to market demand for the final product.
Marginal revenue product = added revenue from one more unit of input I in the
production of good x. In general case
MRPi  MPPi MRx
In perfect competition since firm faces  x  ,  MRx  Px .
Theory suggests that demand for labor of a firm with monopoly with have a lower MRP
than a perfectly competitive firm. In equilibrium where i = labor
MRPi  w
or
MRx  w / MPPi
The demand for labor becomes
w
DL
MRP2
MRP1
hours
As other inputs increase, MPPL increases which shifts the MRP curve to the right
Each firm takes the supply price of inputs as given. As a group if aggregate demand
increases the price of inputs will increase.
The supply of labor is usually upward sloping. If wages increase, more labor will be
supplied. Above some wage, the number of hours supplied may decrease as potential
workers chose leisure over work.
65
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Table 14.2 suggests that with home with two earners higher wages reduces (-.086) hours
worked my spouse. In one-earned with children this is seen (-.078) but not when there are
no children (.007).
Economic rent => difference between the payments made to a factor of production and
the minimum amount that must be spent to obtain the use of that factor.
An upward supply curve of labor => at the equilibrium wage some workers are obtaining
a wage that is greater than needed to get them to work. (See figure 14.11)
Land rent can be calculated on a vertical supply function (see figure 14.12). This
maximizes rent.
Figure 14.13 suggests that US Army recruitment is hampered by a wage that is below the
competitive wage. Rather that increasing the wage to the competitive wage (for all
officers and enlisted men of that rank), the military pays an enlistment bonus to keep
people in the service. The military pays a "dependents allowance" and other incentives
for those with families. Such incentives have resulted in early marriages.
Monopsony Power occurs when a firm has power over the cost on an input. In a
competitive market the marginal expenditure (ME) for an input is equal to the average
expenditure (AE). With monopsony
AEi  MEi
In equilibrium
MEi  MRPix
where MRPix = marginal revenue product of input i in the production of good x.
If unions restrict the number of workers, then the wages of those in the trade will be
above the wage that would occur if there was free entry. The wages of workers in the
non union labor market will fall due to the increased workers looking for work. In reality
many union workers work off the books on the weekends and nights. At university
tenured faculty are not allowed to lower their wages for the summer.
In period the relative #'s of union workers have decreased. Wages have not risen as
expected due to the increased substitution of capital for labor.
Assuming a production function of the form
Q   et L K 
then MPPL    et L 1 K    Q / L
66
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
There will be increasing returns, constant returns or decreasing returns depending on
whether (    )  1, (    )  1, or (    )  1 .
Chapter 16 General Equilibrium and Economic Efficiency
Analysis has been based on partial equilibrium. In General Equilibrium prices and
quantities are determined in all markets at the same time.
In an efficient allocation of goods no one can be made better
Figure 16.5 shows an Edgeworth box. At points of tangency of the indifference curves,
such as C and D, trade is optimal. Points B and A are a positions off the contract curve
(locus of points where the indifference curves are tangent). A movement from A to C
make both better off.
In partial analysis we noted that the slope of the indifference curve for individual i was
[ MU xi / MU yi ] which was equal to the ratio of prices. If exchange is at the tangency
point of the indifference curves then the ratio of prices is the same for both individuals.
Here
[ MU xi / Px ]  [ MU yi / Py ] for i  1,2
The position on the contract curve represents an individuals initial endowment.
The above analysis shows exchange given that the individuals have the goods. The same
analysis can be done at production level. Assume country # 1 produces goods X and Y
and in the absence of trade will maximize utility at point A. If trades open up, country # 1
can sell ab of X for bc of Y and get to a higher indifference curve at position c.
Consider 4 possibilities between two countries producing two goods with tastes and
production conditions:




Same Tastes
Same Tastes
Different Tastes
Different Tastes
-- Same Production conditions
-- Different Production conditions
-- Same Production conditions
-- Different Production conditions
In all but first case, trade benefits both countries (individuals). (See figure 16.10 & 16.11
& 16.12).
67
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
X
a
b
A
c
Y
Assuming two countries (individuals) and two goods.
Absolute Advantage => A country # 1 has an absolute advantage in good X if it can
product X at less cost than country # 2. A country can have an absolute advantage in all
goods over another country.
Comparative Advantage => A country # 1 has a comparative advantage in the
production of good X than country # 2, if country # 1 is relatively more efficient in the
production of that good.
Holland
Italy
Hours to product 1 lb Cheese
1
6
1 gal wine
2
3
Holland has absolute advantage in both goods. Italy has comparative advantage in wine.
Holland has comparative advantage in cheese,
Example: JM appears to have an absolute advantage in both baseball and basketball over
Stokes. JM has a comparative advantage in basketball over baseball.
If teams were chosen a rational coach would put JM on the basketball team and Stokes on
the baseball team.
Free trade implies: If everyone trades in a competitive marketplace, all mutually
beneficial trades will be completed, and the resulting equilibrium allocation of
resources will be economically efficient.
Assumes given endowments. Is this "fair." Four views on equity:
68
Microeconomic Study Notes (17/7/2012
Houston H. Stokes

Egalitarian - all members of society receive equal amounts of goods. Assumes
that all indifference functions are the same. (Does Stokes want the same amount
of green peppers, which make him violently ill, than Jones?)

Rawlsian - maximize the utility of the least-well-off person.

Utilitarian - Maximize the total utility of all members of society.

Market-oriented - The market outcome is the most equitable.
All but the last view requires that the leaders/government know the utility function of all
individuals.
Rawlsian view suggests that an equal distribution of resources may remove the incentive
that most productive people have to work hard. This view allows inequalities as long as
the lest well off person is better off. How might this be measured?
In individual preferences are convex, then every efficient allocation (every point on
the contract curve) is a competitive equilibrium for some initial allocation of goods.
There is a tradeoff between goals of equity and efficiency.
Does the system take into account negative externalities? Would auto pollution be
reduced if exhaust pipes were mandated to be at the front of cars? Air pollution laws in
California gave a "pass" to old cars. => incentive to repair old "dirty" motors rather than
put in a cleaner new motor!!
Why Markets Fail
Market Power - Producer has market power and sets MR not P equal to MC.
Labor union has power and reduces supply of labor to an
industry.
Any Differentiated product => downward sloping demand curve and MR
< AR. BUT many small firms where AR=MR might be at a higher price.
Incomplete Information - Customers may not know real cost on an item such as
cigarettes and thus buy too many for their own good
Insiders may know more than the public and be able to trade
stocks to their advantage. Very hard to police!!
Exteralities
- Steel plant ruins stream for fishing. => Laws prohibit this.
69
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Government now allows select firms to trade their "rights to pollute." Is this right? How
are rights given out initially?
If the Government wishing to reduce hurricane insurance costs mandates that people not
rebuild their homes near the water, then the value of land is reduced.
"Historical Houses" are lowered in value. Should owners be compensated?
Recycling: Have required bottle deposits. Have imposed fees to allow use of dumps.
Property Rights. Owner of land can require that pipeline builder restore surface.
Coase Theorem 'When parties can bargain without cost and to their mutual
advantage, the resulting outcome will be efficient, regardless of how the property
rights are specified.' Farmer requires that the train company clear a path to curtain
sparks starting a fire and pay farmer for loss of land.
Coase Theorem suggests "buying off" a problem. It assumes bargaining is costless. If
bargaining is costly the theorem does not always work.
Public Goods - Are nonrival (have MC=0 for additional users) and are nonexclusive
(no one can be excluded from using them). Example lighthouse. Such a good is hard to
collect for use. Historically farmers maintained or did not maintain roads across their
places. Tolls were collected. This system was replaced with public roads.
NJ Toll road => get a ticket and pay when you get off.
Garden State Parkway => pay a fixed fee every so many miles.
What determines what system?
Clean air is nonexclusive BUT rival. One person's pollution hurts the car behind!!
Public Goods - Defense. Immunization. Should select religions be allowed to reduce their
risk at the expense of others? How should this be decided?
High School Education is rival. It does not have MC=0.
Why are churches given tax breaks while the same is not done for pubs?
Communities have historically paid for their schools from the tax base. Some
communities have been willing to pay more. Recent policy in a number of states
mandates state level funding. Who benefits? Who loses? What will the effect of such a
policy be on high schools in poor neighborhoods, in high neighborhoods, on private
schools?
70
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Discussion: Historically the US has required service in the military in time of war.

In Colonial times males over 17 were required to maintain arms.

In the Civil war those drafted could buy someone else to serve.

During WWII key occupations (farmers) were not drafted and others with
special skills were placed in certain positions.

During Vietnam the more well to do "hid out" in college. The poor were not
paid to go in their place.

Women have never been required to serve in the US.
Discuss the efficiency of these alternative systems.
71
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Linear Programming
Goal: To equip you to solve a number of problems relating to the organization of
production and the purchase of inputs.
Discussion Example
Assume you are placed in charge of a factory packing fish products. Two products
are produced: fish cakes (for human consumption) and fish meal (for animal
consumption). The production process involves Separating (take light meat off fish for
cakes), shredding, packing and canning. Workers are trained for each task. The sales
price of fish cakes is $.80 per pound and fish meal is $.70 per pound. Labor requirements
per 1,000 lb are:
Separating
Shredding
Packing
Canning
Hours Available
205
106
188
65
Req. Fish Cakes
22.78
10.60
14.46
5.00
Req. Fish Meal
13.67
8.83
17.09
5.00
As manager you want to know how to maximize profit by determining how many fish
cakes and fish mean pounds to produce. In addition you want to know how much profit
would change if you were to retrain workers from separating to shredding etc. The
reading lists shows various way to setup the problem.
Setup
Max   .80C .70 M
s. t .
22.78C  13.67 M  205
10.60C  8.83 M  106
14.46C  17.09 M  188
5.00C  5.00 M  65
The b34s setup for this problem (comments begin with /$) is:
/$
/$
/$
/$
/$
/$
/$
/$
/$
b34s program to solve
Fish Packing Problem
Max .80C + .70M
st
22.78C + 13.67M
10.60C + 8.83M
14.46C + 17.09M
72
LE 205
LE 106
LE 188
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
/$
5.00 + 5.00M LE 65
/$
b34sexec lpmax n=2,m1=4,m2=0;
amatrix(22.78, 13.67,
10.60, 8.83,
14.46, 17.09,
5.00, 5.00
);
bvector(205., 106., 188., 65.);
cvector( .80, .70
);
b34srun;
With answers:
LPMAX Command. Version
1 September 1997
Of
2000000 double precision slots in space,
Number of activities (X)
2
Number of constraints (m1+m2)
4
Number of inquality constraints (m1) 4
Number of equality constraints (m2) 0
93 are being used.
Constraint Matrix (A)
1
2
1
22.7800
13.6700
2
10.6000
8.83000
3
14.4600
17.0900
4
5.00000
5.00000
Constraint vector (B)
205.000
106.000
188.000
65.0000
Cost vector (C)
0.800000
0.700000
Objective function maximized at
8.288942665534615
Primal values
2.83325
8.60335
Dual values
0.00000
0.663896E-01
0.665767E-02
0.00000
The answers indicate that profit = $8,288.94 and the production of fish cakes and fish
meal should be 2.83325 and 8.60335 thousand pounds respectively. Since the shadow
price of separating labor and canning labor was 0.0, this suggests that adding more labor
in this area is not needed. Depending on the prices of labor in shredding and packing,
more labor could be added provided that the cost of the labor is less than the shadow
price.
73
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Given the base case  = 8.288942665534615. If one more hour of shredding is allowed,
 = 8.355332303514723. Note that profit has increased by the shadow price of
.066389. If one more hour of canning is added, then the objective function moves up
.00665767 to 8.295600330639099. The reader is encouraged to run the job fish.b34 to
verify these numbers.
If the primal is
Max   CX
s. t . Ax  B
the dual is
Min V  Bw
s. t . A' w  C
In equilibrium it can be shown that V   . The object it to organize production to that
the profit is maximized. The B vector is the numbers of inputs available. Production
should be organized so as to minimize bottle necks. The shadow price of removing a
constraint, w, is solved from the dual. As shown above w(i) is the increase in  that
would occur if one more unit of the ith input was obtained.
Sample LP programs:
Max   5x  3z
s. t 3x  5z  15
5x  2 z  10
Verify that at the solution
  12.36842105263158,
x = 1.05263
w = 0.263158
2.36842
0.842105
A linear programming setup can have a number of problems.
There will be no unique solution if a constraint is parallel to the objective function such
as:
74
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Max   2.5x  z
s. t .
3x  5z  15
3x  2 z  10
A minimum problem such as
Min V  2 x  3 y
x  y 4
6x  2 y  8
x  5y  4
x
3
y 3
can be written as
Max V   2 x   3 y
x y  4
 6 x  2 y  8
 x  5y   4
x
 3
y  3
Verify that at the solution V = -4.0, the primal is ( 1.14286 , 0.571429) and
the dual is ( 0.0, 0.25, 0.50, 0.0, 0.0).
Difficulties that indicate an error was made in setting up a LP problem include
inconsistent constraints, no feasible solution and unbounded solution. It is beyond
this course to discuss these problems at this time.
75
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Gravel Problem (From Managerial Economics by Douglas):
The Grimes Gravel Company produces three mixtures of sand, pebbles and rocks for
eventual sale in 20-kg bags to the home handy person for use in cement work. The sandy
mixture is composed of 10kg sand, 7k pebbles, and 3 kg rocks. The pebbly mixture is
composed of 6kg sand, 10 kg pebbles and 4 kg rocks. The rocky mixture is composed of
2kg sand, 8 kg pebbles and 10 kg rocks. The market prices prevailing ate $2.75 for the
sandy mixture, $2.50 for the pebbly mixture and $2.25 for the rocky mixture. Games
Gravel feels that the market prices will hold regardless of the volume of each procust it
supplies. Games Gravel wishs to maximize sales revenue from its present plant and
equipment. The constraints on output are the limited size of the storage bins for the sand,
pebbles and rocks. The bins hold 2000kg, 3000kg and 2000kg respectively and
replenishment of supplies can only be made once a week.
Find max profit.
What Constraints are binding:
sm = sandy mixture
pm = pebbly mixture
rm = rocky mixture
s = sand
p = pebbles
r = rocks
10 sm + 6 pm + 2 rm LE 2000
7 sm + 10 pm + 8 rm LE 3000
3 sm + 4 pm + 10 rm LE 2000
Max 2.75 sm + 2.50 pm + 2.25 rm
b34sexec lpmax n=3 m1=3 m2=0$
* Gravel problem Douglas page 280
amatrix(10
6
2
7 10
8
3
4
10);
bvector(2000 3000 2000);
cvector(2.75 2.50 2.25);
b34srun$
LPMAX Command. Version
1 September 1997
Real*8 space available
Real*8 space used
Number of activities (X)
Number of constraints (m1+m2)
Number of inquality constraints (m1)
Number of equality constraints (m2)
Constraint Matrix (A)
1
;
12000000
78
3
3
3
0
2
3
76
Microeconomic Study Notes (17/7/2012
1
2
3
10.0000
7.00000
3.00000
Houston H. Stokes
6.00000
10.0000
4.00000
2.00000
8.00000
10.0000
Constraint vector (B)
2000.00
3000.00
2000.00
2.50000
2.25000
Cost vector (C)
2.75000
Objective function maximized at
868.7500000000000
Primal values
90.0000
145.000
115.000
0.106250
0.106250
Dual values
0.168750
Now expand bin # 1 to 6000. What happens?
LPMAX Command. Version
1 September 1997
Real*8 space available
Real*8 space used
Number of activities (X)
Number of constraints (m1+m2)
Number of inquality constraints (m1)
Number of equality constraints (m2)
Constraint Matrix (A)
1
2
3
1
10.0000
7.00000
3.00000
2
6.00000
10.0000
4.00000
12000000
78
3
3
3
0
3
2.00000
8.00000
10.0000
Constraint vector (B)
6000.00
3000.00
2000.00
2.50000
2.25000
Cost vector (C)
2.75000
Objective function maximized at
1178.571428571428
Primal values
428.571
0.00000
0.00000
0.392857
0.00000
Dual values
0.00000
77
Microeconomic Study Notes (17/7/2012
Houston H. Stokes
Lamp Problem from Douglas
The P. M. D. Light Company produces and sells three standing lamps. The contribution
to profit for these lamps is constant at the following levels regardless of output level:
Model A $8.75, Model B 13.25, Model C $16.80. All three models go through the basic
assembly process; then models A and B go through fabric installation process while
model C goes through the antique-bronzing process. The requirements of each model in
each process and the weekly availability of these processes are as follows:
Should all models be produced?
If overtime labor is available for each process at $12.00 in place of the usual $8.00, what
should be done?
Process
Assembly
Fabric
Bronzing
Hours Total
800
480
160
Hours for A
.2
.1
0
Model
.2 Mod_A
.1 Mod_A
+ .3 Mod_B
+ .12 Mod_B
+ .35 Mod_C
.8
Max
8.75 Mod_A
Mod_c
+ 13.25 Mod_B
+
LE 800
LE 480
LE 160
16.80 Mod_C
b34sexec lpmax n=3 m1=3 m2=0$
* Lamp problem Douglas page 281 ;
amatrix(.2
.3
.35
.1
.12 0.
0.
0.
.8);
bvector(800. 480. 160.);
cvector(8.75 13.25 16.80);
b34srun$
LPMAX Command. Version
1 September 1997
Real*8 space available
Real*8 space used
Number of activities (X)
Number of constraints (m1+m2)
Number of inquality constraints (m1)
Number of equality constraints (m2)
Constraint Matrix (A)
1
12000000
78
3
3
3
0
2
3
78
B
.3
.12
0
C
.35
0
.8
Microeconomic Study Notes (17/7/2012
1
2
3
0.200000
0.100000
0.00000
0.300000
0.120000
0.00000
Houston H. Stokes
0.350000
0.00000
0.800000
Constraint vector (B)
800.000
480.000
160.000
13.2500
16.8000
Cost vector (C)
8.75000
Objective function maximized at
35601.66666666667
Primal values
0.00000
2433.33
200.000
0.00000
1.67708
Dual values
44.1667
Summary: LP analysis can be used to allow micro theory to be used to solve real world
problems. In the oil industry and the airline industry, problems with > than 50,000
variables are common.
79