1/12/15
first 3 or 4 classes we will cover the apendex to the book (but
read chapters 1-3)
p.573-558 of 9th edition
p. 542-565 of 7th edition
intro micro
*demand curve
a change in Px (from 7 to 5) causes a movement along the demand
curve (a change in quantity demanded, from 20 to 25, but not a
change in demand)
suppose the income of consumers increase:
D shifts outward, assuming butter is a normal good
*D2D^
increase in demand from D to D'
an increase in quantity deamnded for each value of Px (from 30
to 38 at Px=7)
suppose the price of margerine increases:
what happens to D for butter?
D shifts inward (decrease in demand for butter), since butter
and margerine are substitutes
*S
height of S = MC of each firm in the industry
suppose the wager rate of farm workers increases:
*S2S^
farm labor is an input used in producing butter
there is a decrease in the supply of butter
What effect does this have on the market for butter?
*S3S^ > (P^ & Xv)
the equillibrium quantity of butter decreases from Xe to Xe'
the equillibrium price of butter incerases from Pe to Pe'
an elastic demand curve
*Erelatively flat
an inelastic demand curve
*E2relatively steep
ECO 5000 material
consumer preferences
X and Y are two goods. We consider a single consumer
X = # of hamburgers consumed per month
Y = # of bags of french fries consumed per month
*CPthe more preferred bundle is the one with a higher level of
utility attached to it
any point uch as (2,2), can be called a consumption bundle
(commodity bundle)
as long as X and Y are both goods, then more is better than
lessgood= an item you can purchase for which more is better than
less
more is better than less means the same thing as nonsatiation or
monotonicity
more is better than less = if there are two consumption bundles,
and one has more of both goods, or more of one good and the same
amount of the other good, then it is preferred (has a higher level
of utility than the other consumption bundle)
*CBA = (3, 3)B = (4, 4)C = (5, 3)D = (6, 2)
which of the following is necessarily true?
B is preferred to A? Yes
C is preferred to A? Yes
D is preferred to A? No, D has 3 more hamburgers but 1 less
french fries. So it is not necessarily true
*Uall points on the same indifference curve have the same
utility
we have an indifference curve, which is a curve that connects
consumption bundles between which the consumer is indifferent
there are many different indifference curves, corresponding to
different levels of utility
*U2infinitely many and they do not cross
a higher indifference curve corresponds to a higher level of
utility
B is preferred to A since more is better than less (from
*U2)
C is preferred to AC is indifferent to B (same indifference
curve)B is preferred to A (more is better than less)consequently C
is preferred to A (we're using something called transitivity)
any point on a higher indifference curve is preferred to any
point on a lower indifference curve. (The point on the higher
indifference curve corresponds to a higher level of utility).
It is important to note that two different indifference curves
do not intercept
*U3does not happen
indifference curves normally have the following sort of
curvature (convex to the origin, bowed out towards the origin)
*Iindifference curve convex to origin
marginal rate of substitution (MRS)
MRS = (absolute value of) the slope of an indifference curve =
how many units of Y a consumer is willing to give up in order to
get one more unit of x
MRS = marginal rate of substitution
normal assumption about indifference curves is that there's a
diminishing MRS
this means that the MRS gets smaller as you move down and to the
right along an indifference curve
*I2bc change in X-axis is on bottom of denominator the MRS gets
smaller as you move down an indifference curve
the MRS represents how many units of Y the consumer is willing
to give up to get one more nit of x
indifference curves do NOT (except under unusual cases) have the
following sort of shape:
*I3concave to the origin
*I4a line segmeant btwn to ponts on an indifference curve lies
entirely above the curve
if U(A) = U(B) and C is half way between A and B, then U(C) >
U(B). (C is on a higher indifference curve than A or B)
a consumer tends to prefer averages to extremes
the budget line
what consumption bundles can the consumer afford?
X = one good (hamburgers)
Y = another good (french fries)
Px = price of X = price of one unit of X
Py = price of Y = price of one unit of Y
I = money income of the consumer (in dollars)
expenditures of the consumer:
=Px(X) + Py(Y)
=expenditures on X + expenditures on Y
the budget constraint is:Px(X) + Py(Y) > basic efficiency
condition for a pure exchange economy
in an economy with both exchange and production, things are more
complicated, there are several efficiency conditions (don't worry
about details)
an economy is a pareto efficient (pareto optimal) state if the
only way in which one consumer can be made better off is to make
another consumer worse off
if an economy is not in a pareto efficient state, then things
can be improved on by making one (or more) consumers better off
without making anyone worse off
it is desirable for an economy to be in a pareto efficient state
(sometimes we just say efficient state)
some pareto efficient states might not be very fair, because a
few people might have most of the goods
it might be desirable to have the government move the economy
from a pareto efficient state that is very unequal to a pareto
efficient state that is somewhat more equitable
first fundamental theorem of welfare economics
an economy in which all consumers and firms take prices as
given, and in which there are no taxes, subsidys, public goods, or
externalities, is pareto efficient
under what conditions does the first fundamental theorem not
apply?
1 market power is present
a a monopolist does not take prices as given
b oligopolists do not take prices as given
c a monopsonist does not take input prices (wages) as given
2 public goods
goods which are consumed in such a way that one consumer's
enjoyment of the good does not reduce another consumers enjoyment
of the good
- chapter 4
it is very hard for private markets to produce an appropriate
level of a public good (or in some cases to produce the public good
at all)
3 externalities
a situation where one persons actions affect other people in a
way that is not taken into account in the market system
air polution
water polution
traffic congestion
noise
chapter 5
appendix
consumer and producer surplus
*E
if P = P0, then CS = consumer surplus = area 1 (on diagram)
CS is a measure of how much more the good is worth to consumers
than what they actually pay for it
height of D (demand cure) is the MV = marginal value
MV = how much one more unit of the good is worth to a
consumer's
if P = P1, then CS = 1 + 2 (from *E)
as the result of a decrease in P from P0 to P1, the CS increases
by area 2
consider S, the supply cure.
Height of S = MC = marginal cost
*F
if P = P2, then PS = producer surplus = 3 (from *F)
if P = P3, then PS = 3 + 4
If P increases from P2 to P3, then PS increases by area 4
PS is a measure of the gain to firms from being able to sell
their good at some price as opposed to not being allowed to sell
anything = profit of firms + fixed cost of firms
if the PS goes up by some amount, then profit goes up by the
same amount
*G
Pe = equillibrium price
xe = equillibrium quantity
1 = CS
2 = PS
overall benefits from exchange in the market for x = CS + PS = 1
+ 2
this supply and demand equillibrium is efficient
if u had:1 a monopolization of the x-industry, or2 a tax is
imposed, than either of those might change the overall benefits
from exchange
chapter 4
public goods
a private good is a good which, if consumed by one person, that
particular unit of the private good cannot be consumed someone
else
if I eat a hamburger u cannot eat the same hamburger
a pure public good is a good for which one person's use of the
public good does not diminish another person's use of the public
good (consumption of the good is nonrival, or noncongestible)
ex:-natural defense-lighthouse-t broadcast
sometimes a pure public good is defined as a good which is
nonrival and nonexcludable (it is not feasable to prevent the
person from using the public good if the person does not pay)
a good might be a private good even if the good is provided by
the government (public housing, public health care)
an example of supply and demand for a private good
x = number of hamburgers
50 consumers of type 1, with the individual demand curve
MV = 10 - 0.1x
(MV = marginal value)
P = 10 0.1x
0.1x = 10 P
x = 10 P/0.1 = 10(10 P) = 100 10P
x = 100 10P
*Hdemand curve
50 consumers of type 2, with the individual demand curve
P = 10 0.2x
0.2x = 10 P
x = (10 P)/0.2 = 5(10 P) = 50 5P
*Isteeper demand cure
how do we find the aggregate (or market) deamand function for
x?
How do we add the demand curves of individual consumers?
We add them horizontally (for a private good). In other words,
we express x as a function of P for each consumer, and add all of
the values of x to find the total quantity deamanded at each value
of P
total quantity demanded = 50(100 10P) + 50(50 5P) = 5000 500P +
2500 250P = 7500 750P = 750(10 P)
x = 7500 750P
consider two cases
case 1 MC = 2
market supply cure is horizontal at a height of 2
case 2
market supply cure is x = 2250P 4500
2/9/15
an example of suply and demand for a private good
50 consumers of type 1
MV = 10 0.1X
P = 10 0.1X
X = 100 10P
50 consumers of type 2
MV = 10 0.2X
P = 10 0.2X
X = 50 5P
aggregate (market) demand curve
X = 50(100 10P) + 50(50 5P)
X = 7500 750P
case 1
MC = 2
the supply curve has the equation P = 2 (a horizontal line of
height 2)
*Jfirm demand curve,
if P = 0 then X = 7500
if X = 0 then 10 = 7500 750P
P = 7500/750 + 10
what are the equillibrium price and quantity (the coordinates of
point E)?
two equations
X = 7500 750P
P = 2
thus we have P = 2substitute this into the demand equation
X = 7500 750P
=7500 750(2) = 6000
what about a single consumer of type 1?
P = 10 0.1X
X = 100 10P
use the second equation
X = 100 10(2) = 100 20 = 80
what about a single consumer of type 2?
X = 50 5P = 50 5(2) = 4050(80) + 50(40) = 4000 + 2000 = 6000
sp things check out okay
different consumers can consume different amounts
the equillibrium quantity in th emarket is the sum of the
quantities consumed by different consumers
it is important to note that:
MV1 = marginal value of a consumer of type 1
= 10 0.1X
=10 (0.1)(80)
=10 8
=2
MV2 = marginal value of consumer of type 2
= 10 0.2X = 10 (0.2)(40)
=10 8 = 2
we have the condition that
MV1 = MV2 = P = MC
true in the competitive equillibrium for a private good
note the conditions
MV1 = MV2 = MC
is to the books condition...
case 2the market supply curve is x = 2250P 4500
if x = 0 then
0 = 2250P 4500
P = 4500/2250 = 2
*K
X = 2250P 4500
2250P = 4500 + X
P = 4500/2250 + X/2250
P = 2 + X/2250
MC = 2 + X/2250
what are the equillibrium price and quantity?
We have two equations
D: X = 7500 750P
S: X = 2250P 4500
set quantity demanded equal to quanity supplied
X = X
7500 750P = 2250P 4500
12000 750P = 2250P
12000 = 3000P
P = 1200/300 = 4
Xe can be found by substituting P = 4 into the supply curve or
demand curve
Xe = 7500 750P
= 7500 750(4)
=7500 3000 = 4500
for a single consumer of type 1, x1 = 100 10P = 100 10(4) =
60
for a single consumer of type 2
Xe = 50 5P = 50 5(4) = 30
50(60) + 50(30) = 3000 + 1500 = 4500
so everything is consistent
once again, it can be shown that it is true that
MV1 = MV2 = P = MC
each of these is 4 in this example
when you're dealng with a private good, you do nto add the
demand curves of consumers vertically (you do not find the total
price at each quantity)
when you derive the aggregate demand curve
HW:200 consumers with the MV function:
MV = 20 0.02X
marjet supply function is
X = 80,000 + 5,000P
what are the market equillibrium values of X and Pif X is a
private good? How much of X is consumed by each consumer?
Suppose we have a public good
firework display in a stadium
x = number of fireworks
fireworks are a public good (asuming no congestion)
50 consumers of type 1
MV1 = 10 0.1X
*Lthe maximum amount a consumer of type 1 could be willing to
pay for the 70th firework is $3
the striped area is the maximum willingness to pay (for a
consumer of type 1) for a whole fireworks display of 70
fireworks
50 consumers fo type 2
MV2 = 20 0.2X
*Mthe maximum amount that a consumer of type 2 would be willing
to pay for the 70th firework is $6
suppose that the marginal cost of an extra firework is MC =
300
*N
how much is the 70th firework worth to all of the consumers?
Mvt = total marginal value to all 100 consumers
= 50(3) + 50(6)=150 + 300 = 450
450 > 300 so its worthwhile to have the 70th firework
(since MVt > MC)
we want to find the total willingness to pay at each value of
X
Mvt = total marginal value of all 100 consumers
=50MV1 + 50MV2
=50(10 0.1X) + 50(20 0.2X)
=500 5X + 1000 10X
Mvt = 1500 15X
this would be equivelent to finding the total price at each
value of X
Pt = 50P1 50P2
=50(10 0.1X) + 50(20 0.2X)
=500 5X + 1000 10X
= 1500 15X
either way, what we are basically doing is to add vertically the
demand curves of all the individual consumers (total marginal
value, or price, at each value of X)
basic efficiency conditionsMV t = MC
50MV1 + 50MV2 = MC
1st exam: March 9 (pushed back from the 2nd due to snow day)
through the chapter on taxation
fireworks example (public good)
MVt = MC
where
Mvt = sum of the marginal values of all the different
consumers
=50MV1 + 50MV2
note this is essentially the same as the books condition that:
sum of the MRS of all consumers = MRT
what does the basic efficiency condition lead to in this
example?
Mvt = 1500 15X
MC = 300
Mvt = MC
1500 15X = 300
1500 = 300 +15X
1200 = 15X
X = 1200/15 = 80
80 = efficient number of fireworks
with a public good different consumers consume the same
amount
int his example each consumer consumes (views) 80 fireworks
Mvt = 1500 15X
=1500 15(80)
=1500 1200 = 300
for each consumer of type 1,
MV1 = 10 0.1X
=10 (0.1)(80) = 2
for each consumer of type 2,
MV2 = 20 0.2X
=20 (0.2)(80) = 4
note that
50(2) + 50(4)1) =100 + 200 = 300
as expected
the Mvt of 300 is made up of 50 consumers who value the 80th
firework at $2and 50 consumers who valuethe 80th firework at $4
In contrast to the case of a private good, in the case of a
public good (such as the fireworks example just given)
i different consumers have different marginal values for the
last unit of X
ii diferent consumers have the same value of X (X = 80)
to pay for the fireworks display, the government could charge
each consumer of type 1 $2 for each firework
ii each consumer of type $4for each firework and each
consumerwould have a significantamount of consumer surplus from
this firework display
there are some problems with charging cusgtomers this way,
however
how does the government know the mv function (demand function)
of each consumer?
Each consumer has the incentive to pretend he (she) doesnt like
fireworks very much so he doesnt have to pay very much
the government has a difficult time charging each consumer
different prices even if it knows their MV functions
the governemnt may have a hard time figuring out the optimal
value of X (since it doesnt know the consumers MV functions)
these problems did not exist in the case of a private good
can a public work (sych as a fireworks display) be funded by the
contributions of private individuals)
maybe. But see the books discussion of the free rider
problem
public versus private education
the choice of a family to send their kid to a private school or
public school
assumptions:
family income = I
the family pays income taxes, T, to support the public school
systemwhether or not they go to public school
if the kid goes to public school, educational quality = Eq
education can be purchased in the private school system at a
price of P per unit of E
the utility of a family is a function of E and Y, where Y is all
other goods which have a price of 1.0there are indifference curves
with e and Y on the two axes
suppose the kid goes to public school then
E = Ep
y = I T
suppose the kid goes to private school the family chooses E to
maximize its utility subject to the budget constraint
y + pe = I T
choice of e for private school (the family chooses a school
which provides the level of e shown in the following diagram).
*O
problem:
200customes with the MV function
MV = 20 0.02X
where X is the number of fireworks (a public good)
the MC function for fireworks is
MC = 1000
what is the efficient number of fireworks
suppose E0 < Ep
*Pthe family sends the kid to the public school system (point
B)
the family can choose point B (public school) or any point along
the budget line (private school) but the overall budget
constraintconsists of pointB and the budget line. In thiscase, the
family chooses point B (public school)
suppose e0 > Ep:
case 1: (case shown in book)
*Q
this family send their kid to public schools
at point A you have more of e but less of y than at point B
but B is on a hgiher indifference curve so the family chooses B
over A
case 2:
*Rthe family send their kid to private school (point A)
different families might have different values of income
a family with a higher value of I might be more likely to send
its kid to a private school than a family with a lowe value of I
(assuming the families have the same set of indifference
curves)
chapter 14
partia equillibrium analysis of a per unit tax (unit tax) in a
perfectly competitive market
suppose there is a tax u on each unit x that is sold
let Q0 and P0 be the mrket equillibrium quantity and price in
the absence of the tax
*S
the statuatory incidence of the tax can be on consumers or
producers
stauatory incidence = who pays the tax directly to the
government
economic incidence = who was hurt by the tax and by how much
the economic incidence is oftensignificantly different than the
statuatory incidence
for example, if the statuatory incidence is on producers, they
might shift the burden of the tax onto consumers (tax shifting)
statuatory incidence on producers:
the unit tax (u) causes the supply curve to shift uniformly up
by an amount u
why? The height of the supply curve is equal to the MC (marginal
cost) and if a f9irm has to pay the government u for eachunit, this
increases the effective MC by u
*T
new intersection of supply and demand is at
quantity Q1
price Pg
the price paid by consumers to producers is Pg. This is the
efective price for consumers
Pg = gross price = demand price = consumer price
what is the effective price received by producers?
Producers receive Pg from consumers and pay u to the government
for each unit of the good
let Pn be the net price = supply price = producer price
Pn = amount kept by producers after paying the tax = Pg u
Pn = Pg u
note that in the diagram Pn is below Pg by exactly u since the
vertical distance between S and S' is u
2/16/15
200 customers with the MV function
MV = 20 0.02x
market supply function isx = 80,000 + 5,000P
what are the market equillibrium values of X and P if x is a
private good? How much of x is consumed by each consumer?
MV = 20 0.02x
demand function of one consumer is
P = 20 0.02x
P + 0.02x = 20
0.02x = 20 P
x = 1/0.02(20 P)
=50(20 P) = 1000 50P
the market demand function is
x = 200(1000 50P)
=200,000 10,000P
set x = xsing the market demand and supply functions:
x = x
200,000 10,000P = 80,000 5,000P
120,000 10,000P = 5,000P
120,000 = 15,000P
P = 120,000/15,000 = 8
use market demand or supply equation
x = 80,000 + 5,000P
=80,000 + 5,000(8)
=80,000 + 40,000
=120,000
for an individual consumer,
x = 1,000 50P
=1,000 50(8)
=1,000 400 = 600
the tax reduces the equillibrium quantity from Q0 to Q1
what are the efficiency effects of the tax?
Method 1:
before the tax is imposed
CS = consumer surplus = 1 + 2 + 3 + 7 (area on graph from last
week)
PS = producer surplus = 4 + 5 + 6 + 8 (area on graph from last
week)
after the tax is imposed
CS = 1
PS = 6
tax revenue = u (Q1) = area of a rectangle on the diagram = 2 +
3 + 4 + 5 (area on graph from last week)
as the result of the tax:
CS = change in CS = 1 (1 + 2 + 3 + 7) = -(2 + 3 + 7) (area on
graph from last week)
PS = change in producer surplus = 6 (4 + 5 + 6 + 8) = -(4 + 5 +
8) (area on graph from last week)
you get tax revenues you didn't have before
efficiency gain of the tax = CS + PS + tax revenue
= -(7 + 8) (area on graph from last week)
the tax results in an efficiency loss (deadweight loss, excess
burden) = 7 + 8
the amount by which consumers and producers are hurt is greater
than the amount of the tax revenue
economic incidence of the tax
some of the tax burden is on consumers, since Pg > P0 (CS has
decreased)
some of the tax has been shifted onto consumers
some of the burden is on producers, since Pn < P0 (PS has
decreased)
method 2 (of examining efficiency issues):
suppose output is reduced from Q0 to Q1
the change in the total value of the output produced to
consumers = area underD
= TV = -(7 + 8 + 9)
the change in the total cost of producing output
=the change in the area under the marginal cost curve (the
supply curve) = TC = - 9
efficiency gain = TV TC
= -(7 + 8 + 9) - (-9)
= -(7 + 8)
once again, efficiency loss = deadweight loss = excess burden =
7 + 8
in an efficiency sense, the tax makes the economy worse off
problem
200 customers with the MV function
MV = 20 0.02x
where x is the number of fireworks (a public good) the MC
function (rest of problem is in 2/9 notes)
the total marginal value is the number of consumers times the MV
of each consumer's
Mvt = 200MV
= 200(20 0.02x)
= 4000 4x
set MVt = MC
4,000 4x = 1,000
4,000 = 1000 4x
3,000 = 4x
x = 3000/4 = 750
first exam is march 9
the MV of each consumer for the 750th firework is
MV = 20 0.02x
= 20 (0.02)(750)
=20 15 = 5
note that 200MV = 200(5) = 1,000
wich is consistent with the fact that MC = 1000 and the fact
that MVt = 1000 for the 750th firework
partial equillibrium of a unit tax (perfectly competitive
market)
statuatory incidence on consumers
the unit tax u causes the demand curve to shift uniformly down
by an amount u (see book)
*Udemand curve shifts down by the amount of the tax
P0 and Q0 = equillibrium price and quantity with no tax
u = unit tax
once the tax is imposed,
Q1 = equillibrium quantity
Pn = net price
Pg = gross price
the price paid by consumers to producers is Pn this is the
effective price for producers (suppliers)
what is the effective price paid by consumers?
Consumers pay Pn to producers
and pay u to the government (for each unit of the good)
Pg = total amount paid by consumers = Pn + u
Pg = Pn + u
note that in the diagram Pg is above Pn by exactly u, since the
vertical distance between D and D' is exactly u
the price paid from consumers to producers is different
depending on weather the statuatory incidence is on producers or
consumers
however, Q1, Pg, and Pn are exactly the same whether the
incidence is on consumers or producers
the economic incidence of a unit tax is independent of the
statuatory incidence
consumers are hurt the same amount (bear the same burden) either
way
producers are hurt the same amount (bear the same burden) either
way
example with straight-line supply and demand curves
S: P = 20 + 0.1Q
D: P = 80 0.2Q
P = P
20 + 0.1Q = 80 0.2Q
0.1Q = 60 0.2Q
0.3Q = 60
Q = 60/0.3 = 600/3 = 200
Q0 = 200
substitute that value of Q into the demand or supply curve
P = 20 + 0.1Q
=20 + (0.1)(200)
=20 + 20 = 40
P0 = 40
now impose a tax u = 6 with a statutory incidence on
producers
S' (new supply curve) is: P = 20 + u + 0.1Q
= 20 + 6 + 0.1Q
= 26 + 0.1Q
D: P = 80 0.2Q
P = P
26 + 0.1Q = 80 0.2Q
0.1Q = 54 0.2Q
0.3Q = 54
Q = 54/0.3 = 540/3 = 180
Q1 = 180
note that Q1 < Q0, as expected
use equation of D to calculate Pg
Pg = 80 0.2Q1
= 80 (0.2)(180)
= 80 36 = 44
use equation of S (the old supply curve) to calculate Pn
Pn = 20 + 0.1Q1
= 20 + (0.1)(180)
= 20 + 18 = 38
two general formulae (the two equations will be given on test,
but the variables will not be defined)
let
ms = slope of the supply curve
md = absolute value of the slope of the demand curve
(Pg P0)/u = function of the burden of the tax that falls on
consumers = md/(md + ms)
(P0 Pn)/u = fraction of the burden of the tax that falls on
producers = ms/(md + ms)
note that the sum of these to fractions is equal to 1
do these formulae work for the example?
(Pg P0)/u = (44 40)/6 = 2/3
md/(md + ms) = 0.2/(0.2 + 0.1) = 2/3
(P0 Pn)/u = (40 38)/6 = 1/3
ms/(md + ms) = 0.1/(0.2 + 0.1) = 1/3
so the formulae do work for the example
two more general formulae (the two equations will be given on
test, but the variables will not be defined)
let
Es = elasticity of supply
Ed = elasticity of demand
(Pg P0)/u = Es/(Es + Ed)
P0 Pn)/u = Ed/(Es + Ed)
sheet on exam:
(Pg P0)/u = md/(md + ms) = Es/(Es + Ed)
(P0 Pn)/u = ms/(ms + md) = Ed/(Es + Ed)
not handed out on exam:
Pg = gross price = consumer price (effective price to
consumers)
Pn = net price = producer price (effective price to
producers)
Problem
D: P = 600 0.2Q
S: P = 300 + 0.3Q
what are Q0 and P0?
Suppose u = 40
what are Pg, Pn, and Q1 after the tax is imposed
note
Es = (Q/Q)/(P/P) = (P/Q)(Q/P) = (P/Q)/(P/Q) = (P/Q)/ms
and similarly for Ed
If ms = 0 then Es =
horizontal supply curve
if my = then Es = 0
vertical supply curve
elastic suply
ms relatively small
Es relatively large
relatively flat upward sloping supply curve
1 < Es <
inelastic supply
ms relatively large
Es relatively small
relatively steep upward sloping supply curve
0 < Es < 1
Ed = (P/Q)/md
if md = 0 then Ed =
horixontal demand curve
if md = then Ed = 0
vertical demand curve
elastic demand
md relatively small
Ed relatively large
relatively flat downward-sloping demand curve
1 < |Ed| <
inelastic demand
md relatively large
Ed relatively small
relatively steep downward sloping demand curve
0 < |Ed| < 1
(pg Pn)/u = Es/(Es + Ed) = ([P/Q]/ms)/([P/Q]/ms +
[P/Q]/md)...
=(md)(ms)/ms]/[([md][ms]/ms) + ([md][ms]/md)] = md/(md + ms)
conclusion:
(Pg Pn)/u = Es/(Es + Ed) = md/(md + ms)
suppose we know P0, ms, md, and u. what are Pg and Pn
(Pg P0)/u = md/(md + ms)
Pg P0 = [md/(md + ms)](u)
Pg = (u)(md/[md + ms]) + P0
in the last example,
Pg = 40 + [0.2/(0.2 + 0.1)](6)
= 40 + (2/3)(6)
= 40 + 4 = 44
similarly,
(P0 Pn)/u = ms/(ms + md)
P0 Pn = (u)(ms/[ms + md])
Pn P0 = -(u)(ms/[ms + md])
Pn = P0 - (u)(ms/[ms + md])
in the last example,
Pn = 40 (0.1/[0.1 + 0.2])(6)
= 40 (1/3)(6)
40 2 = 38
3/2/15
problem
suppose Es = 7 and Ed =3
suppose a tax u = 30 is imposed
which changes Pg or Pn?
What is (Pg Pn)/(Po - Pn)
Answer
method 1
(Pg Po)/u = E/(Es + Ed) = 7/(7 + 3) = 7/10
=> Pg P0 = (7/10)u = (7/10) 30 = 21
(P0 Pn)/u = Ed/(Es + Ed) = 3/(3 + 7) = 3/10
=> P0 Pn = 3/10 * 30 = 9
Thus Pg increase by more than Pn decreases and,
(Pg Pn)/(Pg Pn)...
method 2
(Pg P0)/(P0 Pn) = [(Pg Pn)/u]/[(Pn P0)/u]
=[Es/(Es + Ed)]/[Ed/(Ed + Es)]
=Es/Ed = 7/3 > 1
=> (Pg P0)/(Pg Pn) > 1
Pg P0 > Pg Pn
thus Pg increases by more than Pn decreases
exam one week from today
^^^MIDTERM^^^
VVVFINALVVV
chapter 5
in the book and in the lecture, we use the word externality to
refer to a technological externality
there is also something known as a pecuniary externality but we
will ignore it
a situation with an externaility is normally associated with a
market failure
(private markets usually do not lead to an efficient outcome in
the presence of an externality)
government intervention is normally required for efficiency
externality = a situation where one economic agent's actions
(person or firm) affects the welfare of others (usually adversley)
in a way that is not taken into account by market prices
examples of externalities:1-air polution2-water
polution3-traffic congestion4- noise pollution5-smoking in a
crowded room
first example (from book)
*Vthe water pollution from bart's factory adversley affects
lisa's fishery
the ususal situation, in the absence of any government
intervention, is that bart's factory will pollute the river beyond
the efficient level of pollution since bart has no incentive to
care about the fact that his pollution hurts lisa's fishery
second example
suppose that air pollution results from the manufacturing of
steel
simplifying assumption: for each unit of steel produced, there
is a given fixed amount of air pollution that is genereated
consequence of the simplifying assumption is that the only way
to reduce air pollution is to reduce the production of steel
note: in the real world various thkngs can be done to reduce the
pollution that results from the production of each unit of
steel
we assume the following three types of economic agents in a
metropolitan area1- steel producing firms2- demanders of steel
(probably firms that use steel as an input, such as constructions
firms or auto plants3- residents of the metropolitan area who are
hurt by pollution
*W
d = demand curve for steel
the height of D is the MV (marginal value) or MB (marginal
benefit) of an extra unit of steel to demanders
MPC = marginal private cost
the hieght of MPC is the MC (marginal cost) of the extra
capital, labor and raw materials
the MPC curve does not take the costs of pollution into
account
MD = marginal damages
the height of MD is the dollar amount which corresponds to how
much residents are hurt by the extra ammount of pullution that is
generated by the production of one more unit of steel
note: in the diagram and in the books diagram, it is assumed
that MD = 0 for the first unit of steel produced (this need not be
the case)
also it is assumed that one extra unit of pollution causes more
incrimental damage if the elvel of pollution is already higher
(since MD slopes up)
MSC = marginal social cost
= MPC + MD
=marginal private costs + marginal damages
= overall costs to society of the prodduction of one extra unit
of steel
*X
Q1 and P1 are the market equillibrium quantity and price in the
absence of any government action (MPC = MB)
Q* = the socially optimal output of steel (MSC = MB)
MB = height of the demand curve = marginal benefit = marginal
value
at output Q1 MSC > MB
so it makes sense to reduce output until you are at Q*
is there an efficiency gain from reducing Q from Q1 to Q*?
ignore residents for a minute and consider only steel-producing
firms and demanders of steel
if Q decreases from Q1 to Q1* reduction in pollution costs = (3)
+ (4)
= area under MPC
the value to demanders of steel of the Q1 Q* units that are no
longer produced = area under D (MB)
=(2) + (3) + (4)
tentative efficiency loss = (2)
but we have not takena ccount of the pollution yet
the reduction in pollution damages to residents = area under
MD
= (4)
= area between the marginal private costs and the marginal
social costs
= (1) + (2)
overall, the reduction in Q from Q1 to Q* is associated with an
efficiency gain of (1) + (2) tentative efficiency loss
= (1) + (2) (2)
= (1)
if it is possible to vary the amount of pollution produced per
unit of steel produced, then it would also be the case that in the
socially efficient situation, there would be less pollution per
unit of steel produced than in the free market situation
what can be done to reduce the level of output from Q1 to
Q*?
1.) one possibility is a merger between the polluter and the
victim of the pollution. This might very well work in example 1
(bart and lisa)
as discussed in the book, this is an example of internalizing an
externality
however this could not really work in examples 2 (steel
market)
what can be done to reduce the level of output and pollution
from Q1 to Z1 (free market levels) to Q* and Z* (the socially
optimal levels)
1.) merger between the polluter and the victim of pollution2.)
pigouvian taxation
the government imposes a tax on each firm that is proportional
to the ammount of steel produced (or, better yet) proportional to
the ammount of pollution produced
draw the previous diagram
the government now imposes an optimal piguvian tax, t, which is
a unit tax which is equal to the height of MD at Q*, and also equal
to the height difference between MSC and MPC at Q*
the effective marginal (private) cost to steel producers goes up
by t, becoming MPC + t
the new market equillibrium value of Q is determined by the
interaction...
As the result of the tax, the market ends up producing at Q =
Q*
the new price of steel is P*, which is higher than the price in
the absence of the tax (see earlier diagram)
note: P* is the new gross price
*YP* new gross pricePn new net price
tax revenues = (5) + (6)
=(t)(Q*)
steel-producing firms are worse off, since Psv by area (6) plus
bottom part of area (2)
demanders of steel are worse off, since Csv by area (5) plus top
part of area (2)
resdents are better off (since pollutionv)
there is a net efficiency gain equal to area (1) as a result of
the taxation
thicontradicts with chapter 14 where a tax was associated with a
net efficiency loss
the tax gives firms an incentive to produce less steel and (if
the tax is on pollution itself) it also gives firms an incentive to
emit less pollution per unit of steel
another way to see efficiency gain
it is equal to
tax rev + CS + PS + benefits of less pollution to residents
= (5) + (6) [(5) + (6) + (2)] + [(1) + (2)]
= (1)
1.) a subsidy
the government gives each firm a subsidy for each unit of steel
that it reduces its production by (or each unit of pollution that
it reduces its pollution by)
there are a number of reasons that a subsidy program is less
desireable than a piguvian taxation program. Some of these examples
are given in the book
midterm:
9th edition:ch 4, 7, 14, appendix
7th edition:ch 4, 12, appendix
do not worry about chapters 1 and 2
in chapter 3, focus on the alst part, beginning with market
failure
there's gonna be some choice on the exam
there might be a couple questions based on the book that will
not be from the lecture notes though these will be avoidable
because of the choices on the exam
diagrams and problems but could some short essays
continuing...
what can be done to reduce the elvel of output from Q1 to Q*
(and reduce the level of pollution from z1 to z*)?
z1 = level of pollution with a free market
z* = optimal level of pollution
1.) transferable pollution permits
which is very similar to cap and trade
(a) suppose there is a fixed amount of pollution for each unit
of output. Then the government can sell Q* permits, each allowing a
firm to produce and sell 1 unit of output, auctioning these off to
the firms that are willing to pay the market-clearing price
supply and demand for permits
*ZPP* = market clearing price of a permit to produce one unit of
output
presumably
PP* = t
where t is the optimal piguvian tax that the government would
have imposed if it used taxation instead of permits
the permit system is equivelant to a piguvian tax system, at
least under perfect information
firms would be allowed to buy or sell the permits among
themselves
note: if there is a fixed amount of pollution for each unit of
output, then Z1/Q1 = Z*/Q*
(b) suppose the amount of pollution per unit of output can be
varied, then
Z*/Q* < Z1/Q1
(less pollution per unit of output in the optimal situation than
in the free market situation)
in this case, the government can sell z* permits, each allowing
a firm to emit one unit of pollution, auctioning them off to the
firms at a market-clearing price
*AAsupply and demand for permits
pz* = market-clearing price of a permit to emit a unit of
pollution
with a piguvian tax or transferable permits, pollution is
reduced in the most cost-effective way
the government needs les information than is the case with
subsidies or regulation
the government still needs a significant amount of information
to choose the appropriate level of the piguvian tax, or to choose
the appropriate level of z*
what is Q*?
what is t?
What is z*?
suppose...
*AB
then the government should just impose a piguvian tax of:
t = MD0
and it does not need any other info
in this case a piguvian tax makes more sense than a permit
system
suppose the MD looks follows:
*AC
in this case it might make more sense to use a permit system
(and choose z* as illustrated)thn to use piguvian taxation
1.) establish property rights
might work in the first example (bart and lisa)
either:
(a) bart has the right to pollute and li his pollution to the
efficient level
(or) lisa has the property rights to the river and barts pays
her to pollute up to the efficient level
under certain assumptions, things work out right as long as one
of them (bart or lisa) has property rights to the river
-coase theorem
this certainly wouldnt work in the sec ond example (steel
market) because there are several firms that pollute and many
residents who are hurt, and negotiation among all thes economic
agents is not practical
1.) regulation of pollution
-see the book for discussion of this (sometimes called command
and control regulation)
-not normally favored by economists
-need much more information than other solutions or else have
much more inefficiencies
-approach that is used in U.S.
3/9/15
chapter 6: public choice
majority voting rules
example 1 zoo
three directors of the zoo: a, b, & c
each ahs one vote
the zoo has enough money and space to buy 1 animal
T jaguar
L lion
T tiger
which animal will they buy?
Example 1.1
choiceabc
1stJJT
2ndLTL
3rdTLJ
Vote on L versus T:
a voges for L
B and C vote for T
T wins 2-1
Vote on L versus J:
A and B vote for J
C votes for L
J wins 2-1
suppose there is a vote between any two and the winner faces the
remaining choice
then the J will end up being the winner
this is an easy case since 2 of the 3 voters have J as their
most prefere choice
example 1.2
choiceABC
1stJTL
2ndLLJ
3rdTJJ
L v T:
c and a vote L
b votes T
L wins 2-1
L v J:
B and C for L
A for J
L wins 2-1
t v. J:
B and C vote T
A votes J
T wins 2-1
the ultimate winner will be the lion
example 1.3
choiceABC
1stJTL
2ndLJT
3rdTLJ
Vote on L v. T:
A and C vote for L
B votes for T
L wins 2-1
vote on L v. J:
C votes for L
A and B vote for J
J wins 2-1
vote on J v. T:
a votes for J
B and C vote for T
T wins 2-1
L beats T
T beats J
J beats L
If you have L v. T first, with winner facing the J, then
L beats T
L versus J
J is the overall winner
If you have L versus J and the winner faces the T, then
J beats L
J versus T
T beats J
T is overall winner
if you have J v. T and the winner faces L, then
L is overall winer
paradoxically, the order of the vote significantly effects the
outcome
in some sense, the community's (zoo leadership) preferences are
inconsistent, even though the preferences of each individual
director are consistent
voting paradoxically-hapens in example 1.3
-not in examples 1.1 or 1.2
cycling = can occur in example 1.3 if votes keep being taken,
with the remaining choice always challenging the winner of the last
vote
L versus T
L wins
J cahllenges L
J wins
T challenges J
T wins
L challenges T
L wins
example 2:
three residents: A, B, & C
three choices for a park
J Small park
L medium park
T large park
examples 2.1, 2.2, and 2.3 have the same tables as examples 1.1,
1.2, and 1.3
if all voters have SINGLE-PEAKED preferences then the various
paradoxes do not occur
example 2.2
same table as example 1.2
*AD
in this case all voters have single peak preferences, so
everything is ok (no paradoxes or cycling)
example 2.3:
same table as 1.3
*AE
voter b does not have single peak preference and this is what
allow the possibility of the voting paradox and cycling
median voter theorem:
suppose that all voters have single-peaked preferences for the
quantity of some public good. Then the result of the political
process, if based on majority voting, will be that the preferences
of the median voter will prevail
example:
5 voters: A, B, C, D, and E
A10 acres
B20 acres
C30 acres
D50 acres
E90 acres
*AFsingle-peaked preferences
the median size of the parks that the different voters prefer is
30 acres
the average (mean size) fo the parks that the different voters
prefer is (10 + 20 + 30 + 50 + 90)/5 = 40 acres
whats relevant for majority voting is the median, not the
mean
the community will end up doing a park of 30 acres, according to
the median voter model
this assumes that each voter has single peak preferences over
park size
see the book for an explanation of why it might be the case that
a voters preferences for the size of the public park might not be
single peaked
3/23/15
chapter 5:
terminology for some of the methods for reducing pollution
2.) piguivian tax = is a tax per unit of output
emissions tax = tax per unit of pollution
efficient tax = a tax per unit of water pollution
1.) transferable pollution permits also called tradeable
pollution permits or cap and tradethis genreal method is sometimes
used for fisheries to prevent over-fishing
1.) what the 7th edition calls regulation is called command and
control regulation in the 9th edition (it calls emissions taxes and
cap-and-trade incentive-based regulations)
Problem 6.4 (see handout)
consider and example
total costs (TC) of reducing emissions for each country are
canadamexico
10.00%10040
15.00%250100
20.00%450200
Overall TC if each country reduces emissions by 15% are
250 + 100 = 350
Overall TC of Canada reduces emissions by 10% and mexico reduces
emissions by 20% are
100 + 200 = 300
suppose canada pays mexico 125 so mexico reduces emissions by
20% and canada has to reduce emissions by only 10%
TC to canada are now
100 + 125 = 225
which is less than 250, so canada is better off
TC to mexico are now
200 125 = 75
which is less than 100, so mexico is better off
chapter 12:
income redistribution
general question: should the government redistribute income from
higher income to lower income consumers, and if so, to what extent
should it do this?
Yi = income of consumer I (after any redistribution)
Ui = utility of consumer I = U(Yi)
w = level of social welfare
utilitarian social welfare function:
W = F(U1, U2, , Un)
where n = number of consumers
F is some function
additive social welfare function (a special case of above)
W = U1 + U2 + + Un
consider the function U(y)
total utility
TU(y) = U(y)
marginal utility = how much extra utility you get if y is
increased by $1
= slope of the U(y) function
one possibility
constant marginal utility
*AG
if you have twice as much income, you have twice as much
utility
MU = slope of TU
but TU(y) is a straight upward sloping line
thus MU is a constant
*AH
another possibility
decreasing marginal utility (diminishing marginal utility)
*AI
if you have twice as much income, then you have less than twice
as much utility
as y increases, the slope of TU(y) decreases
*AJ
suppose there are only to consumers (for simplicity)
A = ann
B = bob
assume an additive social welfare function
W = U(Ya) + U(Yb)
assume that U(y) has diminishing marginal utility
yA0 = value of yA before redistribution
yB0 = value of yB before redistribution
W0 = value of W before redistribution = U(yA0) + U(yB0)
*AK
we are assuming that yA0 < yB0
yc = (1/2)(yA0 + yB0)
U(yA0) = area under MUA = (1) = total utility for Ann
U(yB0) = area under MUB (starting from the right end) = (4) +
(3)
suppose the government taxes an ammount T = Yc yA0
= (1/2)(yA0 + yB0) yA0 = (1/2)(yB0 yA0)away from bob and gives
it to Ann
Assume this can be done without changing yA0 or YB0
then we end up with
yA = yB = yC = (1/2)(yA0 + yB0)
this is complete income redistribution
u(yA) = (1) + (2) + (3)
U(yB) = (4)
note: (1) + (2) + (3) = (4)
after the redistribtion,
W = (1) + (2) + (3) + (4)
this redistribution has increased social welfare by area (2)
this sort of argument suggests that income should be
redistributed until all consumers have the same income after
redistribution
second exam probably april 13th
what is the single bigest problem with redistributing the
income?
If bob knows that an amount of income
yB0 yc
is going to be taken away from him, he has an incentive to work
less hard and thus have a lower value of yB0
if ann knows an amount of income
yc yA0
is going to be given to her, she has an incentive to work less
hard and thus have a lower value of yA0 (and thus get a biger
transfer of income)
you could do an analysis of the optimal redistribution of
income, taking into account the incentives to work less as a result
of money being transferred to or from each worker = optimal income
taxation
first done successfully by Mirrlees (1971), published in the
Review of Economic Studies
optimal income taxation conclusion:
you do want to redistribute income -you do not want to
completely equalize incomes
Rawlsian social welfare function
W = minimum(U1, U2, , Un)
the social welfare is equal to the utility of the consumer with
the lowest (after transfer) income
Problem
suppose the demand curve for steel is
MB = 140 0.2Q
the marginal private cost curve for producing steel is
MPC = 80 + 0.2Q
The marginal damages curve from the pollution generated is
MD = 0.1Q
what are the
(I) free market equillibrium values of Q and P (Q1 and P1)
(i) socially optimal value of Q (Q*)
(i) the optimal piguvian tax needed to reach Q* and the
corresponding value of P (P*)
chapter 13
expenditure programs for the poor
before discussing these programs, I present a basic model of a
consumer's choice among labor hours, leisure hours, and consumption
goods
supose that the utility of a consumer depends on
z = hours of leisure per day
y = dollars of consumption goods
the consumer utility function is U(z, y)
*AL
L = number of hours of work per day
L + Z = 24 time constraint
y = dollars of consumption
= dollars of income
= wL
where
w = wage rate
we have two equations:
L + z = 24
y = wL
From the first equation
L = 24 z
we substitute this into the second equation
y = wL
y = w(24 z)
= 24w wz
y = 24w wz
wz + y = 24w budget constraint
24w = full income
= what the consumer would make if they worked all the time
price of y = 1
price of z = w
w is the wage rate, and it is the price of leisure
wz + 1y = 24w
if z = 0, then
1y = 24w
y = 24w
if y = 0, then
wz = 24w
z = 24w/w = 24
*AM
y*, z* and L* are the values of y, z, and L chosen by the
consumer
suppose we have a welfare program with the following
characteristics
if someone does not work they get a welfare benefit equal to B0
(some number of $)
if a consumer works L hours per day at wage w, the welfare
is
B = B0 wL
if B0 wL = 0
B = 0 if B0 wL = 0
in other words, the welfare benefit is reduced by $1 for every
$1 the consumer earns
what does the budget constraint look like with the welfare
program in place?
The book argues carefully what it should look like
problem
zoo directors A, B, and C get the following utilities from their
animals
LTH
A8010010
B403050
C605025
Which animal, if any, would be paid by any sequence of pairwise
vote?
What does the budget constraint look like with the welfare
program in place?
The book argues carefully what it should look like
in the following diagrams imagine that
B0 = 30 ($30/day)
w = 6 ($6/hour)
wL = B0 if L = 5 (hours per day)
*AN
at point A, L = 0
so y = B0 = 30
At point C, L = 2
so,
y = B + wL
= (B0 2w) + 2w
= B0 = 30
at point D, L = 5
so,
y = B + wL
= (B0 5w) + 5w
= B0 = 30
note: as drawn in the diagram, B0 = 5w
B0 = wL
L = B0/w = 30/6 = 5
basic question
what effect does the welfare program have on the number of hours
that the consumer works?
First case
*AO
in the absence of the welfare program, the consumer would choose
point D, with 21 leisure hours and 3 labor hours
with the welfare program, the consumer could work 3 hours,
receive a welfare benefit of B0 3w, and end up at point C
the consumer prefers C to D (U(c) > U(D)), since L is the
same and y is greater at C than D
But A, where the consumer does not work at all and gets benefits
B0, and has consumption y = B0, gives the consumer a higher utility
level than C
the consumer prefers A to C since y is the same but leisure
hours are greater at A than C
thus the consumer chooses point A and does not work at all
L0 b0] is the number of [labor leisure] hours for which B drops
to 0. B0 = 19 from diagram
L0 = 24 z0 = S
in the absence of the welfare program, the consumer would choose
point D, with z1 leisure hours z1 = 21
L1 = 24 z1 = 3
a crucial feature of this first case:
L1 < L0
in this case, there is a significant work disincentive of the
welfare program (the consumer works 0 hours instead of L1
hours)
second case
*AP
in the absence of welfare, the consumer chooses point C, with z
= 14 and L = 10
in the absence of the welfare program, the consumer still
chooses point C, since C is on a higher indifference curve than
A
in this case, the consumer behavior is not effected by the
welfare program (th econsumer does not use welfare)
without welfare, the consumer would choose point C, with
z = z1 = 14
L = L1 = 24 z1 = 10
In this case,
L1 > L0
where L0 = 24 z0
=24 19 = 5
3/30/15
problem
consider a basic welfare program with a benefit given by
B = B0 wL if B > 0
draw carefully with all relevant numbers the budget constraint
for a consumer with wage rate w = 10 assuming that B0 = 60
you should have laisure drawn per day on th ehorizontal axis and
dollars of consumption on the vertical axis
answer
B = B0 wL = 60 10L if this is > 0
B = 0 when 60 10L = 0
60 = 10L
L = 60/10 = 6
when L = 6, z = 24 L = 24 6 = 18
and y = B + wL = 0
10(6) = 60
when L 0, z = 24 L = 24 0 = 24 and y = B + wL
= B0 wL + wL
= 60 (10)(0) + (10)(0) = 60
when L = 24, z = 24 L = 24 24 = 0
and y = B + wL = 0 + 10(24) = 240
*AQ
problem
suppose the demand curve for steal is
MB = 140 0.2Q
the amrginal produce cost for producing steal is
MPC = 80 0.2Q
the marginal damages curve from the pollution generated is
MD = 0.1Q
whatare the 1.) free market equillibrium values of Q and P (Q1
and P1)2.) sociallyoptimal value o Q and P (Q* and P*)3.) the
optimal pigouvian tax needed to reach Q* and the corresponding
value of P*
answer
(1)MB = MPC
140 + 0.2Q = 80 0.2Q
140 = 80-0.4Q
60 = 0.4QQ = 60/0.4 = 180 this is Q1
to find P1, use MB or MPC
P1 = MB hen Q = Q1 = 450
(2)
MB = MSC
= MPC + MD
80 + 0.2Q + 0.1Q
80 + 0.3Q
140 0.2Q = 80 + 0.3Q
140 = 80 + 0.5Q
60 = 0.5Q
Q = 60/0.5 = 120 this is Q*
(3)
the optimal piguvian tax, t*, that is needed to reduce Q to Q*
is
t* = MD when Q = Q*
= 0.1Q*
=(0.1)(120) = 22 this is t*
to find P*, use MB or MSC
P* = MSC of Q = Q*
80 + 0.3Q*
= 80 + (0.3)(120)
= 80 + 36 = 116 this is P*
the effective net price to producers of steal is
Pn = P* - t* = 126 22 = 104
a welfare program with a basic benefit of
B0 = $30/day
we are considering its effect on a consumer with wage rate
w = $6/hour
the benefit is reduced by $1 for each $1 the consumer earns
the budget constraint the consumer faces is
*AR
z = 19 means 19 hours of leisure per day, or 5 hours of labor
per day
third case
*AS
L0 = 24 z0 = 24 19 = 5
= amount o labor hours for which B drops to 0
L1 = 24 z1 = 24 17 = 7
without a welfare system, the consumer chooses point C, with z1
(17) hours of leisure and L1 (7) hours of labor
in this case, as in case 2, L1 > L0
unlike case 2, if welfare is available, the consumer goes on
welfare ending up at point A with utility level U2 (note that U2
> U1)
in this case, the consumer goes on welfare even though wL1 >
B0 (even though earnings at point C are greater than the welfare
benefit at point A)
problem
zoo directors A, B, and C get the following utilities from these
animals
LTH
A8010010
B403050
C605035
Which animal, if any, would be picked by any sequence of
pairwise votes?
Answer
from these utility numbers we can figure out the most preffered,
second most preffered, and least preffered of each zoo director
ABC
1THL
2LLT
3HTH
In a vote between the lion and the tiger. A votes for the tiger
but B and C vote for the lion so L wins 2-1
in a vote between T and H, A and C vote for T, but B votes for
H, so T wins 2-1
in a vote between L and H, B votes for H, but A and C vote for L
so L wins 2-1
then L would be chosen by any sequence of pairwise votes
basic welfare program in classified more or less the same as the
old AFDC program
more or less the same as a TANF for which there is a 100% tax on
earnings
see book what AFDC and TANF stand for AFDC is the program that
used to be used. TANF is used now, though varies by state
negative income tax program in class
more or less the same as a TANF programfor which there is NOT a
100% tax on earnings (see book)
an alternative programfor low-income consumers
negative income tax
benefit given to a consumer in the form of an income grant =
B
B = B0 a(wL)
as long as this is > 0
(a = 1 in basic welfare program)
let's assume that
B0 = $20/day
a =
thus,
B = 20 (1/4)(earnigns per day)
= 20 (1/4)(wL)
as long as this is >/= 0
suppose for a given consumer w= $8 per hour
B = 20 (1/4)(8)L
= 20 2L
overall income (and expenditure) of the consumer = y
y = B + wL
= 20 (1/4)(wL) + wL
y = 20 + (3/4)wL
= 20 + (3/4)(8)L
= 20 + 6L
B = 20 (1/4)wL = 20 2L
B = 0 if L = 10 (z = 14)
if L >/= 10 (z /= 0
where B0 = 60, a = and the wage rate is w = 20
draw carefully, with all relevant numbers, the budget constraint
for the consumer, with leisure hours per day (z) on the horixaontal
axis and dollars of consumption per day (y) on the vertical
axis
chapter 8
cost benefit analysis
suppose the government is considering an investment project. Is
the project worth while?
Assume until further notice that there is no inflation
r is the annual interest rate
suppose you put $1 in the bank in year 0
in one year u have $(1 + r)
Money in the bank in year 01R1/(1 + r)R/(1 + r)
Money in the bank in year 11 + rR(1 + r)1R
If r = 0.10 = 10% (per year)
Year 01100.909.9.09
Year 11.111110
Present value of $1 received one year from now is $1/(1 + r),
because 1/(1 + r) depositted today grows...
the present value of $R received one year from now is R/(1 +
r)
leaves money in the bank for 2 years (or more)
Year 01
Year 11 + r
Year 2(1 + r)(1 + r) = (1 + r)2
Year 3(1 + r)3
(1 + r)2 = 1 + 2r + r2
where 1 is original dollar, 2r is 2 years of interest and r2 is
interest on interest
the present value of $1 received two years from now is $1/(1 +
r)2, since $1/(1 + r)2 depositted today will grow to become $1 in 2
years
$1 grows to become $(1 + r)n in n years
R grows to become R(1 + r)n in n years
the present value of $1 received n years from nowis 1/(1 +
r)n
the rpesent value of $R received n years from now is R/(1 +
r)n
suppose you have a revenue stream of
R0 in year 0
R1 in year 1
R2 in year 2
(and so on)
RT in year T
what is the PV (present value) of this revenue stream?
PV = R0 + R1/(1 + r) + R2/(1 + r)2 + + RT/(1 + r)T
T could be 1, or 2, or 20, or even infinity. T is the last year
in which you get revenues.
A special case of this formula
R0 = 0
R1 = R
R2 = R
Rn = R for every value of n >/= 1
T = infinity
then PV = R/r
suppose you have a potential investment project
C0 = cost in year 0
C1 = cost in year 1
C2 = cost in year 2
(and so on)
CT = cost in year T
B0 = benefit in year 0
B1 = benefit in year 1
B2 = benefit in year 2
(and so on)
BT = benefit in year T
we can use the previous formula if we set
Pn = Bn Cn
the present value of the project is
PV = B0 C0 + (B1 C1)/(1 + r) + (B2 C2)/(1 + r)2 + + (BT CT)/(1 +
r)T
basic rule adopt the project if and only if the PV > 0
a project is admissible if its PV is positive
if two projects are admissible and they are mutually exclusive,
the project with the higher PV should be adopted
simple two year investment project
C0 = net cost in year 0
B1 = net benefit in year 1
r = interest rate
PV = -C0 + B/(1 + r)
adopt the project if the PV > 0 but not if PV < 0
adopt the project if PV > 0
-C0 + B1/(1 + r) > 0
B1/(1 + r) > C0
B1 > C0(1 + r)
adopt the project if the benefit in year 1 is more than enough
to cover the initial cost plus interest
if you do not adopt the project and put C0 in the bank, then you
have
C0(1 + r) in year 1
if you adopt the project then you have something worth
B1 in year 1
adopt the project if
B1 > C0(1 + r)
example
C0 = 202
B1 = 220
r = 0.10 = 10%
should this project be adopted?
PV = -C0 + B1/(1 + r)
= -202 + 220/(1 + 0.10)
= - 202 + 220/1.1
= -202 + 200 = -2 < 0
PV < 0 so do not adopt the project (the project is not
admissible)
note if the interest rate was 0.05 = 5%, then it would be the
case that PV > 0, so then the project should be adopted
alternative method for this example
C0(1 + r)
=(202)(1 + 0.10)
=(202)(1.1) = 222.2
B1 = 220
222.2 > 220 so the project should not be adopted
problem
consider a govenrment investment project with the cost
stream
C0 = 400 C1 = 480
and let the benefit stream be
B1 = 120 B2 = 1350
should the government adopt this project if r = 50%?
4/6/15
three year investment project
C0 = net cost in year 0
B1 = net benefit in year 1
B2 = net benefit in year 2
PV = C0 + B1/(1 + r) + B2/(1 + r)^2
if PV > 0 you adopt the project
if PV < 0 you do not adopt the project
example
C0 = 100
B1 = B2 = 60
r = 0.10 = 10%
PV = -C0 + B1/(1 + r) + B2/(1 + r)^2
= -(100) + (60)/(1 + [.1]) + (60)/(1 + [.1])^2
= -100 + 60/1.1 + 60/1.1^2
= -100 + 60/1.1 + 60/1.21
= -100 + 59.55 + 49.56
= -100 + 104.14
= 4.14 > 0
PV > so we adopt the project
problem
consider a negative income tax program with a benefit given
by
B = B0 a(wL) if B = 0
where B0 = 60, a = and the wage rate is w = 20
draw carefully with all relevant numbers, the budget constraint
for athe consumer, with leisure hours per day on the horizontal
axis and the dollars of consumption per day on the vertical
axis
answer
B B0 a(wL) = 60 - .5(20L)
= 60 10L (if this is 20)
B = 0 when 60 10L = 0
60 = 10L
L = 60/10 = 6
when L = 6, z = 24 L = 24 6 = 18
and y = B + wL = 0 + 20(6) = 120
when L = 0, z = 24 L = 24 0 = 24
y = B + wL
= B0 0.5(wL) + wL
60 0 + 0 = 60
when L = 24, z = 24 L = 24 24 = 0
and y = B + wL = 0 + 20(24) = 480
*AW
infinite period investment project
C0 = net costs in year 0
B = net benefit every year starting in year 1
PV = -C0 + B/(1 + r) + B/(1 + r)^2 + + B/(1 + r)^n + (n =
infinity)
PV = -C0 + B/r
if PV > 0 then adopt the project
if PV < 0 then do not adopt the project
if PV = 0 you're indifferent
adopt the project if PV > 0
-Co + B/r > 0
B/r > C0
B = C0r
why does this formula make sense intuitively
suppose you start out with Co in period 0
there are two things you can do with the money
1.) adopt the project and get benefits of B each year starting
with year 1
1.) do not adopt the project. Put the money in the bank. Keep C0
in the bank and take out interest C0r every year starting year
1
two possible investment projects
they are mutually exclusive so only one of them can be done
(this is what we are assuming in the following example)
project 1 and project 2
for any given value of r we calculate the present value of each
project
(a) of PV < 0 for both projects , thena nother project is
admissible, so do niether
(b) if PV > 0 for one project only, then only this project is
admissible, so do this project. (the one with the positive PV)
(c) if PV > 0 for both projects, then both are admissible, so
do the one with the higher PV
project 1
C0 = 100 = cost of making fireworks
C1 = 50 = cost of giving a firework display in year 1
B1 = 200 = the value that consumers palce on watching the
display
PV1 = -C0 + (B1 C1)/(1 + r)
= -100 + (200 50)/(1 + r)
= - 100 + 150/(1 + r)
if r = 1.00 = 100%
then
PV1 = -100 + 150/(1 + 1)
= -100 + 75 = -25
if r = 0.25 = 25%
then
PV1 = -100 + 150/(1 + 0.25)
= -100 + 150/1.25
=-100 + 120 = 20 > 0
PV1 < 0 if r = 100%
PV1 > 0 if r = 25%
project 1
rPV1
100.00%-25
80.00%-16.7
60.00%-6.25
50.00%0
40.00%
30.00%
25.00%
20.00%
18.00%
15.00%
PV1 > 0 if r < 50%
PV1 < 0 if r > 50%
if r v pv1 ^
if r v then the benefits that occur in the future are discounted
less (are worth more)
project 2
C0 = 100 = cost of making a public park
C1 = C0 = C1 = C2 = = 10
= C = cost of maintaining the park each year starting year 1
B1 = B2 = B3 = = 30
= B = benefits in each year starting with year 1
PV2 = -C0 + (B + C)/r
= -100 + (30 10)/r
= -100 + 20/r
if r = 0.50 = 50%
then
PV2 = -100 + 20/0.5
= -100 +40 = -60
project 2
rPV2
100.00%-80
80.00%-75
50.00%-60
40.00%-50
30.00%-38.75
25.00%-20
20.00%0
18.00%11.1
15.00%...
10.00%
5%
if r > 20%
then PV2 < 0
if r < 20% then PV > 0
as r V PV2 ^ (as was true of project 1 too)
if r > 50% then PV1 < 0 and PV2 < 0
so neither project should be adopted
if 20% < r < 50%
then
PV1 > 0 but PV2 < 0
so project 1 should be adopted
if r < 20% then PV1 > 0 and PV2 > 0
and the project with the higher present value should be
adopted
if r = 18% then adopt project 1 (since 27.12 > 1.1)
if r = 15%then adopt project 2 (beause 33.33 > 20.43)
for what value of r do the projects have the same PV?
PV1 = PV2
-100 + 150/(1 + r) = -100 + 20/r
150/(1 + r) = 20/r
-100 + 150/(1 + r) = -100 + 20/r
150/(1 + r) = 20/r
150 = 20/r(1 + r)
150r = 20(1 + r)
150r = 20 + 20r
130r = 20
r = 20/130 = .1538 = 15.38%
overall conclusions
if r > 50% then adopt neither project
if 15.38% < r < 50% then adopt project 1
if r < 15.38% then adopt project 2
low interst rates are especially benefitial for a project whose
benefits extend far into the future
internal rate of return
the internal rate fo return is the, delegated as p(roe), is the
value for r for which the PV of a project is 0
usually it is the case that
(a) if r > p(roe) then PV < 0, so the project is not
admissible
(b) if r < p(roe) then PV > 0, so the project is
admissible
example project 1
PV = -100 + 50/(1 + r)
to claculate p(roe), set PV = 0 and replace r by p(roe)
0 = -100 + 150/(1 + p)
100 = 150/(1 + p)
100(1 + p) = 150
100 + 100p = 150
100p = 50
p = 50/100 = .5 = 50%
this is the same as the value for r for which PV = 0 in the
table earlier on
answer (assigned last week)
consider a government nvestment project with this cost
stream
c0 = 400
c1 = 480
and the benefit stream
B1 = 120
B2 = 1350
should the government adopt this project if r = 50%?
answer
PV = B0 C0 + (B1 C1)/(1 + r) + (b2 C2)/(1 + r)^2
= 0 400 + (120-480)/(1 + .5) + (1350 0)/(1 + .5)^2
= -400 + -360/1.5 + 1350/1.5^2
= -400 240 + 1350/2.25
= -640 + 600 = -40 < 0
the project is not admissible. Do not adopt the project
^^^midterm 2^^^VVVFINALVVV
second exam next classchapter 5, 6, 7, 8, 11
or
chapter 5, 6, 12, 13, 8
in the cost-benefit chapter, you can be tested on anything
assigned except for dealing with inflation
effects of inflation
suppose that in the absence of inflation the interest rate
(annual interest rate) is r
suppose instead of 0 inflation there's a steadya nnual inflation
rate of pie (constant over time)
then there are two things we could mean by the interest rate
it is a reasonable assumption that the real interest rate might
remain unchanged at r independent of pie
nominal interest rate = the interest rate actually observed
real interest rate = nominal interest rate corrected for
inflation
rR = rN pieE
if these are all small numbers
rR = real interest rate
rN = nominal interest rate
pieE = expected inflation rate
if inflation is constant over a long period of time then
pieE = pie
where pie = actual inflationit is a reasonable assumption that
if inflation changes from a steady rate of 0 to a steady rate of
pie, then the real interest rate, which we call rR, remains
unchanged. However, the nominal interest rate increases from rN to
a vlaue which is
rN = (1 + r)(1 + pie) 1= 1 + r + pie + rpie 1
r + pie + rpie
= r + pie
if rpie is a very small number
example
(higher values of r and pie than are realistic for the u.s.)
r = 0.25 = 25%
pie = 0.20 = 20%
what is the nominal interest rate for this case?
Nominal interst rate = rN
= (1 + r)(1 + pie) 1
= (1.25)(1.20) 1
= 1.5 1
=0.50 = 50%
note that r + pie = 45% in this case
recall project 1 from earlier tonight, with pie = 0
C0 = 100
C1 = 50
B1 = 200
r = 0.25 = 25%
PV = -100 (200 50)/(1 + .25) = 20
what ahppens if pie = 0.20 = 20% instead of being 0
r = 0.25 = 25%
pie = 0.20 = 20%
rN = 50%
what happens to C0, C1, and B1 as a result of inflation?
Suppose C0 is unchanged at 100
it is reasonable to assume that the real values for C1 and B 1,
corrected for inflation, are unchanged
however the nominal dollar values fo C1 and B1 are multiplied by
(1 + pie)
in the following pages, C1 and B1 are the real values of C1 and
B1 (measured in year 0 dollars)
C1N and B1N are the nominal values of C1 and B1 ( measured in
year 1 dollars)
the nominal value for the costs and benefits are now
C0N = 100
C1N = C1(1 + pie)
=50(1 + 0.2)
= 60
B1N = B1(1 + pie)
= 200(1 + 0.2)
= 240
if you know the values of C1N and B1N, they can be corrected for
inflation (converted into year 0 dollars) as follows
C1 + C1N/(1 + pie) = 60/(1 + .2) = 50
B1 = B1N/(1 + pie) = 240/(1 + 0.2) = 200
two legitimate ways to evaluate this investment project
method1
use real values for C0, C1, B1, and the interest rate
PV = -C0 + (B1 C1)/(1 + r)
P = -100 + (200 50)/(1 + 0.25)
= -100 + 150/1.25
= -100 + 120 = 20 > 0
PV > 0 so the project is admissible
method2
use nominal values throughout
PV = -C0N + (B1N C1N)/(1 + rN)
note that
rN = (1 + r)(1 + pie) 1
1 + rN = (1 + r)(1 + pie)
thus
PV = -100 + (240 60)/(1 + 0.50)
= -100 + 100/1.5
= -100 + 120
=20 > 0
and this is the same value of PV we calculated the first way
there are two wrong methods
method 3 (wrong)
use nominal values for costs and benefits but use a real
interest rate
PV = -C0N + (B1N C1N)/(1 + r)
= -100 + 180/1.25
=-100 + 144
= 44
and this overstates the present value of the project
method 4 (wrong)
use real values of the cots and benefits but use CN
PV = -C0 + (B1 C1)/(1 + rN)
= -100 + (200 50)/(1 + 0.5)
= -100 + 150/1.5
= -100 + 100 = 0
and this method underestimates the PV of the project
