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UNIVERSIT

OF TORONTO SCARBOROUGH
DEPARTMENT OF MANAGEMENT

MGEB02: Price Theory: A Mathematical Approach

Instructor: A. Mazaheri
Sample Test-1.1 (Solutions)

Instructions: This is a closed book test.

You have

Hours.

Good Luck!

Last
Name:

First
Name:

FOR MARKERS ONLY:

Q1 Q2 Q3 Q4 Q5 Total
Marks
Earned

Maximum
Marks

Possible
40

15 1

100

Page 2 of 1

Answer all following 5 questions:

Question-1 [40 Points] Answer the following Short Questions:

a) [4 points] You are analyzing the market for Crude oil in the last decade or so, you know that
the price has risen from $20 or so to $100+ during this period. Show what must have happened to
the demand and supply to lead to such an equilibrium.

Demand shifts right, P & Q increase

b) [6 Points] In the following, the initial equilibrium is given. Suppose price of X declines.
Assuming X is Giffen, draw the new equilibrium. On the same graph show the income and the
substitution effects.

Page 3 of 15

c) [6 Points] Draw representative indifference curves for the followings:

i) Sandra has a strange habit and she insists on it; she likes to eat an apple and two bananas
together.

Perfect complement:

ii) Adam does not care about orange juice or apple juice as long as he has juice.

Perfect Substitute:

Banana

Apple

Orange Juice

Apple Juice

Page 4 of 15

d) [6 Points] Assume you have a fixed budget of $10. Further assume that you spend your entire
budget. Both good X and good Y cost $1 each. You are spending all your money on X. At this
bundle, your marginal utility of X is 10 while your marginal utility of Y is 5.

Are you optimizing your utility? Why or why not? Using a graph explain your answer.

Solution:

(1) spend your entire income because you are on the budget line
(2) The MRS = 10/5=2 > 1, or (MUx/ Px) > (MUy/ Py).

That is, the marginal utility of X per dollar spent is higher than that of Y. However, you
already gave away all Y and cannot get more X. Therefore your optimal consumption
bundle is a corner solution where you consume no Y.

X
Y

Page 5 of 15

e) [8 Points] Assume a utility function that is given by U(X,Y) = X0.5Y0.5. Further assume a
budget of $50. When the prices where PX=1 and PY=1, you consumed X= 25 and Y = 25, while
when the prices changed to PX=2 and PY=1, you consumed X= 12.5 and Y = 25. With the help of
the following graph decompose the total effect of the price change into the substitution and
income effects.

Solution:

We know TE = 25-12.5. We need to find the SE. Having SE we can solve for IE as
TE=SE+IE. SE is the change in the quantity of

demanded, if (1) the individual remains at
the same indifference curve and (2) if MRS is equal to the new price ration.

18.568.175.1

32.72568.17

68.17)2(25)2(),1(

2
1
2

252525:1

5.05.

5.05.05.0

5.0

−=−

−=

>−=

==>==>

==>

==×=

IE
SE

XXX

Y
X

MRS

YXU

Graphically:

12.5

50 17.68

IE
SE

Page 6 of 15

f) [5 Points] You have 5 spent on two products, a composite product (Y) with a price of py = 1
and coffee (X) with a price of px = 1. If you purchase three cups of coffee, you will be offered one
coffee for free and a 50% discount on each additional cup of coffee purchased. Show graphicall

how this affects your budget line.

8
5

3 4

y
x

Page 7 of 15

g) [5 Points] Ahmed only consumes hamburger (Y) and coffee (X). He wants two cups of coffee.
If he gets less than two cup of coffee he will not care about anything else and if he is given more
than two cups of coffee he will discard it. Graph his representative indifference map.

Coffee

Hamburger

Page 8 of 15

Question-2 [15 Points] As a manufacturer you are interested in obtaining quick estimates of the
supply and demand curves for your product. You have done some research and you know that for
your product the elasticity of supply is 2, the elasticity of demand is -1.5. You also know that the
current price and quantity are $50 and 1,000, respectively. Assume that both demand and supply
are linear.

a) [6 Points] What is the supply and demand curves at the current price and quantity.
b) [4 Points] What impact would a 10% decline in demand have on the equilibrium price and
quantity?
d) [5 Points] Ignore part (b). Suppose the government subsidizes your product by 5 dollars per
unit. What would be the new equilibrium price? Use a graph to show your answer.

Solution:
a) Demand curve

Q = a0 + b0P
Ed = b0 × P/Q = -1.5 = b0 × 50/1000

b0 = – 30 => 1000 = a0 – 30(50) => a0 = 2500

Qd = 2500 – 30P

Next, we estimate the supply curve
Q = a1 + b1P

E1 = b1 × P/Q = 2 = b1 × 50/1000
b1 = 40 => 1000 = a1 + 40(50) => a1 = -1000

Qs = -1000 + 40P

b) Multiply demand equation by 0.9
Qdʹ = 0.9 (2500 – 30P)
Qdʹ = Qs and solve

2250 – 27P = -1000 + 40P

P = 48.51, => Qdʹ = 2250 – 27(48.5) Qdʹ = 940.3

c) PD = PS -5, QD =QS = Q
Qd = 2500 – 30P => PD = 2500/30-(1/30)Q
Qs = -1000 + 40P => PS = 25 +(1/40)Q
PD = PS -5 => 2500/30-(1/30)Q = 25 +(1/40)Q –5 => Q =1085.7=> PD = 47.14

Page 9 of 15

Question-3 [15 Points] Suppose your preferences for Gasoline (X) and a composite good
(Y) is set in accordance to U(X,Y) = 2X0.25 + Y0.25. You have an annual income of
$30,000 and that the price of the composite good is $1.
a) [5 Points] If the government introduces a rationing system such that you can only
consume 15,000 liters a year at $1 a liter. What would be your optimal consumption
bundle?
b) [6 Points] If the government removes the rationing system and the free market price of
gasoline jumps to $2. What would be your new optimal consumption bundle? Are you
better off with or without the rationing?
c) [4 Points] Illustrate your solution in a clearly labeled graph.

11.8523
89.21476

000,303969.0
3969.05.0

2
1
1

25.0
5.0

75.0

75.0
75.0
75.0
75.0

==>
==>

=+=>
==>==>

===

−

−
Y
X

XX
XYXY

X
Y

Y
X
MRS

But cannot consume more than 15000 L, therefore corner solution:

000,15
000,15

000,30

==>
=

Y
X

b) Without the rationing and with the new price:

000,10
000,10

000,302

2
2

75.0
75.0
==>
==>

=+=>
==>

==
Y
X

XX
XY
X
Y

MRS

Page 10 of 15

You are better off with the rationing because:

without

Ration

without

Ration

UU
U
U

=+×=

30000,10000,102
20.33000,155000,12

25.025.0

10 15

Page 11 of 15

Question-4 [18 Points] Mary has the following utility function: U(x, y) = 4y + 2×0.5. Let px and

py be the corresponding prices and

her income.

a) [6 Points] Setup the Lagrangian function and find the first order conditions (FOCs). Use these
FOCs to find the expression for the marginal rate of substitution (MRS). Use the MRS to graph
the indifference map. What is “special” about these indifference curves?

0)(

)(24

5.0
5.0
5.0
−
−
=

=−+−=
∂
∂

=−=
∂
∂

−+−+=

MRS

IYpXp
L

x
x

IYpXpxyL

y
x
yx

λ
λ
λ

The indifference curve can cross the horizontal line since MRS does not depend on y.

Page 12 of 15

b) [6 Points] Find the demand functions for x and y. Graph the Engle curve for x.

0)(P if

16
4

if
4

*
x

*
2

2
*

2/1

==>>=

−=










−

=
−

==>≤







=








==>=

−

yIx
P
I

p
p

p
I

p
p

p
pI

p
xpI

yIxP
p

p
x

p
p

x
p
px

x
x
y

y
x

x
X

x
y
x
y
y
x

(Py/4Px)2

Slope = Px

Page 13 of 15

c) [6 Points] Suppose Mary lives in a city where px = 1, py = 4 and her job offers her I = 0.5.
Find the optimal consumption levels for x and y? Graph your solution.

1
4
4

=





==>







= x

p
p
x
x
y

cannot afford this so hr consumption will be

0.5

X and no y.

0.5

Page 14 of 15

Question-5 [12 Points] There are 50 consumers in the economy. Half of them live in city A and
demand Orange according to the individual inverse demand curve P = 2− Q. The other half live in
city B and demand Orange according to the individual inverse demand curve P = 6−3Q. Suppose
that the market-clearing price for Orange is $1.

a) [6 Points] Write down the market demand for Orange in this economy and then graph it?
b) [6 Points] Suppose the price increases from $1 to $2, how does the consumer surplus change?

Solution:

[ ]

PP
Q

PPQ

33.33100
3

25
50

3
22525

255022525

−=

−

=



 −=

−=−=

But demand seize to exist in city A when the P >= 2 while in city B the demand is zero when
P>=6 therefore the demand will be kinked at P=2 or:

)2(33.33100

)2(
3

25
50

≤−=

>−=
PPQ

P
P

Q
M

33.33

100

Page 15 of 15

[ ]

505.375.12
5.37

67.66)

3
4

45.0(25

3
4

3
2:2

17.104)

3
5

55.0(25

3
5

3
2:1

25
0

0,2
5.12)5.011(25

12:1

−=−−=∆
−=∆

=×××=

=



 −==

=×××=
=


 −==

−=∆
=

==
=×××=

=−==

P
QP

CS
P
QP

CS
CS

QP
CS

PQP

A
B
B
B
B
A
A
A
A
A

Market Demand

Market demand curve Curve relating the quantity of a good that all consumers in a market will buy to its price.

If you have the individual demand then the market demand is the horizontal sum of the individual demands.

Market Demand
Summing to Obtain a Market Demand Curve
Three consumers (A, B, C) with given demands
=> Market demand curve:
Sum the three demand curves: DA, DB, and DC.
=> At each price, the quantity demanded by the market is the sum of the quantities demanded by each consumer.
Price of $4,
=> Market demand = 11 units (sum of the quantity demanded by A (no units), B (4 units), and C (7 units).

2
Market Demand
Example, we might obtain information about the demand for home computers by adding independently obtained information about the demands of the following groups:
Households with children
Households without children
Single individuals
Two Notes:
The market demand curve will shift to the right as more
consumers enter the market.
Factors that influence the demands of many consumers will also
affect market demand.

3
Market Demand
Elasticity of Demand
Denoting the quantity of a good by Q and its price by P, the price elasticity of demand is
Inelastic Demand
|Ep |< 1 (absolute value is less than one): The quantity demanded not responsive to changes in price. => Total expenditure on the product increases when the price increases.
Elastic Demand
|Ep |> 1 (absolute value is more than one): The quantity demanded responsive to changes in price.
=> Total expenditure on the product declines when the price increases.

4
Market Demand
Elasticity of Demand
Isoelastic Demand
Isoelastic demand curve Demand curve with a constant price elasticity.
Unit-Elastic Demand Curve
When the price elasticity of demand is −1.0 at every price, the total expenditure is constant along the demand curve D.

5
Domestic demand for wheat:
QDD = 1430 – 55P
QDD: number of bushels (in millions) demanded domestically, and P is the price in dollars per bushel.
Export demand is:
QDE = 1470 − 70P
QDE: number of bushels (in millions) demanded from abroad.

Market (aggregate) Demand:

QDD + QDE = (1430 − 55P) + (1470 − 70P) = 2900 − 125P
Market Demand

6
Market Demand
The Aggregate Demand for Wheat
Horizontal sum of the domestic demand AB and the export demand CD.
Note: Each individual demand curve is linear, but the market demand curve is kinked.
– No export demand when the price of wheat is greater than $21.

Consumer Surplus
consumer surplus Difference between what a consumer is willing to pay for a good and the amount actually paid.
Consumer Surplus and Demand
Consumer Surplus
The total benefit from the consumption of a product
Minus
The total cost of purchasing it.
Consumer surplus for the purchase of six concert tickets at $14 =the yellow-shaded area.

8
Consumer Surplus
Consumer Surplus and Demand
Consumer Surplus Generalized
For the market: Consumer surplus is the area under the demand curve and above the line representing the purchase price of the good.
The yellow-shaded triangle =
1/2 × ($20 − $14) × 6500 = $19,500.

Applying Consumer Surplus
A tool for welfare analysis, discussed in ECMC02: In aggregate, it measures the total benefit that consumers obtain from buying goods in a market.
When we combine consumer surplus with the aggregate profits that producers obtain, we can evaluate both the costs and benefits not only of alternative market structures, but of public policies that alter the behavior of consumers and firms in those markets.

9
Consumer Surplus (Application)
The effect of Clean Air Act in 1977.
Valuing Cleaner Air
The yellow-shaded triangle: The consumer surplus generated when air pollution is reduced by 5 parts per 100 million of nitrogen oxide at a cost of $1000 per part reduced.
The surplus is created because most consumers are willing to pay more than $1000 for each unit reduction of nitrogen oxide.

10
Numerical Example
There are 100 consumers in the economy. Half of them live in city A and demand popcorn according to the individual inverse demand curve P = 10 − 2Q. The other half live in city B and demand popcorn according to the individual inverse demand curve P = 16−4Q. Suppose that the market-clearing price for popcorn is $4.

a) At the market-clearing price, how many popcorns does each resident of city A buy? What is the price elasticity of demand by residents of city A at this point?
b) Repeat (a) for residents of city B.
c) What is the market demand for popcorn in this economy? Is the market demand function linear? If not, where is the kink?

11
Numerical Example
d) Using the market demand function derived in part (c), what is the total quantity demanded in this economy at the market-clearing price? What is the price elasticity of market demand at this point? Is the absolute value of the price elasticity of market demand larger than the absolute value of the price elasticity of individual demand?
e) If the price increases from $4 to $10, how does the consumer surplus change? Graph the demand curve with quantity on the horizontal axis and price on the vertical axis, and show the change in consumer surplus.

Consumer Behavior

● Theory of consumer behavior

– How consumers allocate incomes among different goods and services to maximize their well-being.

Three distinct elements required to understand consumer behavior:

Consumer preferences (What she wants)

Budget constraints (How much income she has)

Consumer choices (How she allocates her income)

1
Consumer Preferences
In Economics we assume:
– Preferences are given
– Consumers are rational,
– They have goals, and they make decisions that will enable them to achieve those goals.
– Does not mean that the goals are somehow rational or appropriate, nor does it mean that people do what others might think is right or best for them.
– Consumers maximize their satisfaction (i.e., utility), and as long as they are making decisions that achieve that goal, they are being rational.

2
Consumer Preferences
● Market basket (or bundle) List with specific quantities of one or more goods.
Alternative Market Baskets

A 20 30
B 10 50
D 40 20
E 30 40
G 10 20
H 10 40

Market Basket
Units of Food
Units of Clothing
To explain the theory of consumer behavior, should ask whether consumers prefer one market basket to another.

3
Consumer Preferences
Basic Assumptions about Preferences
Completeness:
– Consumers can compare and rank all possible baskets.
– For any two market baskets A and B, will prefer A to B, B to A, or will be indifferent (equally satisfied).
Transitivity:
– Consumers prefer basket A to basket B and basket B to basket C, then the consumer also prefers A to C.
– Necessary for consumer consistency.
More is better than less: (Nonsatiation)
– Goods are assumed to be desirable — i.e., to be good.
=> Consumers always prefer more of any good to less.
=> Consumers are never satisfied or satiated; more is always better, even if just a little better.
– There are goods, such as air pollution, that are undesirable – ignore these “bads” in the context of our immediate discussion.

4
Describing Individual Preferences
More of each good is preferred to less,
A is clearly preferred to basket G,
E is clearly preferred to A.
However, A cannot be compared with B, D, or H without additional information.
Consumer Preferences

U1: Shows all baskets that give the consumer the same level of satisfaction as does basket A;
– Baskets B and D, etc.
An Indifference Curve
Consumer Preferences
● Indifference curve Curve representing all combinations of market baskets that provide a consumer with the same level of satisfaction.

This consumer prefers:
E, to A,
A to H or G

6
A set of indifference curves that describes a person’s preferences.
An Indifference Map
Consumer Preferences
● Indifference map Graph containing a set of indifference curves showing the market baskets among which a consumer is indifferent.
Any market basket on U3, (A), is preferred to any basket onU2 (B), which in turn is preferred to any basket on U1, (D).

The magnitude of the slope of an indifference curve: Marginal Rate of Substitution (MRS) between two goods.
Consumer Preferences
MRS falls from 6 (between A and B) to 4 (between B and D) to 2 (between D and E) to 1 (between E and G).
Convexity The decline in the MRS reflects a diminishing marginal rate of substitution.
– When the MRS diminishes along an indifference curve, the curve is convex.

● Marginal rate of substitution (MRS): Maximum amount of a good that a consumer is willing to give up in order to obtain one additional unit of another good.

8
MRS is constant.
– The consumer views orange juice and apple juice as perfect substitutes: is always indifferent between a glass of one and a glass of the other.
Consumer Preferences
Perfect Substitutes and Perfect Complements
MRS is zero or infinite;
– The consumer views left shoes and right shoes as perfect complements: An additional left shoe gives no extra satisfaction unless without the matching right shoe.

9
Consumer Preferences
Utility function:
a) can be represented by a set of indifference curves, with numerical indicators.
b) can be represented by a mathematical formula.
Ordinal utility function Utility function that generates a ranking of market baskets in order of most to least preferred.
Cardinal utility function Utility function describing by how much one market basket is preferred to another.
● Utility Numerical score representing the satisfaction that a consumer gets from a given market basket.
● Utility function Formula that assigns a level of utility to individual market baskets.

10
Utility Function
Mathematical Representation
Examples of utility functions
1) Smooth, symmetric: U(X,Y) = XY, U(X,Y) = X0.5 Y0.5
2) Perfect substitutes (Linear) : U(X,Y) = 2X+3Y
3) (next session) Perfect compliments: U(X,Y) = minimum {X,Y}
4) Quasi-Linear: U(X,Y) = X0.5 + Y/4
Let’s focus on Type-1 utility function first.

11
Y
X
Utility
Utility Function
Graphical Representation

12
Y
X
Utility
The Function: U(X,Y) = X0.5 Y0.5

13
Y
X
Utility
Graphical Representation: The Indifference Curves

14
Y
X
U=1/12
U=6/12
U=9/12

Graphical Representation: The Indifference Curves

15
Clothing (C)
Budget Constraints
The table shows market baskets associated with the budget line F + 2C = $80
● Budget constraints Constraints that consumers face as a result of limited incomes.
● Budget line All combinations of goods for which the total amount of money spent is equal to income.
Market Baskets and the Budget Line

A 0 40 $80
B 20 30 $80
D 40 20 $80
E 60 10 $80
G 80 0 $80

Market Basket
Food (F)
Total Spending

16
Budget Constraints
Budget line: The combinations of goods that can be purchased given the consumer’s income and the prices of the goods.
Line AG: The budget associated with
(I= $80, PF = $1, PC = $2)
The slope of the budget line ( between B and D) is −PF/PC = −10/20 = −1/2.

What is the effect of change in income?

17
Budget Constraints
Price Changes: A change in the price of one good (I unchanged) causes the budget line to rotate about one intercept.
Example:
Price of food falls from $1.00 to $0.50 => The budget line rotates from L1 to L2.
Price increases from $1.00 to $2.00, => The budget the line rotates from L1 to L3.
The Effects of Changes in Income and Prices

18
Consumer Choice
Utility is Maximized at A.
– At this point, the budget line and indifference curve U2 are tangent.
– At point A: MRS between the two goods equals the price ratio.
– What about point B?
Should satisfy two conditions:
Be on the budget line
Maximize utility

19
Consumer Choice
● Marginal benefit Benefit from the consumption of one additional unit of a good.
● Marginal cost Cost of one additional unit of a good.
Satisfaction is maximized when the marginal benefit — the benefit associated with the consumption of one additional unit of food — is equal to the marginal cost — the cost of the additional unit of food. (The marginal benefit is measured by the MRS.)
Satisfaction is maximized (given the budget constraint) at the point where

20
Consumer Choice
Mathematical Representation
An individual consumes two goods, X and Y with a utility function given by the expression:
U(X,Y) = X0.5 Y0.5
The current price of X and Y is 25 and 5 respectively. The individual currently has an income of 750.
To calculate the optimal quantities of X and Y given the income constraint and the. We simply need to maximize the Utility subject to the constraint. (Recall the Math review)

21
Consumer Choice
Mathematical Representation
Max U(X,Y) = X0.5 Y0.5
Subject to 25X+5Y=750
From the Math review:

22
Marginal Utility
● Marginal utility (MU) Additional satisfaction obtained from consuming one additional unit of a good.
– Diminishing marginal utility More of a good is consumed, the consumption of additional amounts will yield smaller additions to utility.

What does this mean?

23
Corner Solution
Corner solution: Is the solution to the consumer choice when the optimal solution is not the tangency.
– Situation in which the marginal rate of substitution of one good for another in a chosen market basket is not equal to the slope of the budget line.
a) The budget line is straight yet due to a specific type of utility function the consumer only consumes one of the goods – consumes on a corner.
b) The budget line is kinked.

24
Corner Solution: Straight budget line
Example-1 (Graphical)
Linear utility function (perfect substitute)
MRS is constant, the slope of budget line is constant too
Therefore:
MRS > (PX/PY) (Left Graph)
MRS < (PX/PY) (Right Graph) 25 Corner solution: Example-1 (Numerical) Consider a consumer and two goods: x, y. The Consumer utility function is: U(x,y) = 3x + 2y. The price of x is $2 and y is $1. The consumer has an income of $10. 26 Corner Solution: Straight budget line Example-2 Some utilities are not linear yet lead to corner solution. The quasi-linear utility quite often leads to corner solution. Graphical representation: The utility is not linear yet for the given budget line of AB, the highest level of satisfaction is achieved at B on indifference curve U1, where the MRS is greater than the ratio of the price of ice cream to the price of frozen yogurt. Question: What is different about the indifference curve in the graph? See next iClicker question for mathematical representation. 27 A College Trust Fund Corner Solution: Kinked budget line Example-1 (Graphical) A college trust fund that must be spent on education: From A to B, a corner solution. If the trust fund could be spent on other consumption as well as education, the student would be better off at C. 28 Corner Solution: Kinked budget line Example-1 (Numerical) A consumer is considering choosing a calling plan for her cell phone. The plan has a fixed monthly fee of $60, and it gives 800 free minutes per month and charges $0.2 for each additional minute. The consumer has a monthly income of $100, and she spend it on cell phone and another composite good y, where Py =$1. Her utility function is given by U(x,y) = x0.1 y0.9, where x is the minutes of cell phone she uses in a month. Graph the budget line for calling. Find her optimal bundle. 29 Inefficiency of Gasoline Rationing Corner Solution: kinked budget line Rationing (Graphical) Gasoline rationed at $1 per gallon to a max of 2000 gallons. - Without gasoline rationing, up to 20,000 gallons of gasoline are available for consumption (point B). => The consumer chooses point C on indifference curve U2, consuming 5000 gallons of gasoline.
With a limit of 2000 gallons of gasoline under rationing (at point E), the consumer moves to D on the lower indifference curve U1.

30
Comparing Gasoline Rationing to the Free Market
Corner Solution: kinked budget line
Rationing (Graphical)
If Price of gasoline in a competitive market = $2.00
Max consumption of gasoline is 10,000 gallons per year,
the consumer is better off under rationing
utility of F is lower than utility D
If Price of gasoline in a competitive market = $1.5
Consumer worse off under rationing
Utility of G, is higher than utility of D.
See Q-8 on PS2

0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.2
0.4
0.6
0.8
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
0.75
1
0
0.25
0.5
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Individual Demand

The material for the next two lectures very abstract, but it is important to learn how consumers make purchasing decisions in some detail.

– Gain a deeper understanding of what lies behind demand curves.

– The utility maximizing model is crucial in understanding the remainder of Micro.

Individual Demand (Illustration)
Effect of Price Changes
Reduction in the price of food
(income & the price of C fixed)
=> Consumers choose different market baskets.
(a) The baskets that maximize utility for various prices of food (point A, $2; B, $1; D, $0.50) trace out the price-consumption curve.
(b) The demand curve – relates the price of food to the quantity demanded.
(Points E, G, and H correspond to points A, B, and D, respectively).

2
Individual Demand
Notes about this demand curve:
a) It slopes down. We got a downward sloping demand curve because of the way we drew the consumer’s preferences. Elaborate more later.
b) As we move along the demand curve, the level of utility changes.
– Utility rises as P fall.
– Consumer is happier if his purchasing power increases.
c) At each point on the demand curve:

MRS = PF/PC (utility maximization condition).

=> As PF falls while PC is held constant, MRS must be falling.
– As the consumer acquires more food, the amount of clothing he is willing to give up to get more food is decreasing.

3
Individual Demand (Numerical Example)
An individual consumes two goods, X and Y with a utility function given by the expression:
U(X,Y) = X0.5 Y0.5
Denote the price of X and Y by PX and PY respectively and the individual income by I.
Find the Individual demand curves for X and Y.

4
Individual Demand (Numerical Example)
Do as before but instead of actual prices and income, substitute for by notations:

5
Individual Demand (Numerical Example)
Additional Step:

Individual Demand (Illustration)

Income Changes
If income increases (prices fixed):
Budget line shifts outside
=> consumers alter their choice of market baskets.
(a) The baskets that maximize consumer satisfaction for various incomes (point A, $10; B, $20; D, $30)
(b) => Demand curves shift outside (Points E, G, and H correspond to points A, B, and D, respectively.)

Individual Demand
Normal versus Inferior Goods
An Inferior Good
If income increases
=> Less consumption of one of the two goods.
Figure: Hamburger,
A normal good between A and B,
Inferior good between B and C.

Individual Demand
Engel Curves
Engel curves:
Quantity of a good consumed as a function of to income.
(a) Food is a normal good
– The Engel curve is upward sloping.
(b) Hamburger:
Normal good for income less than $20.
Inferior good for income greater than $20.
– The Engel curve is upward then downward sloping .
● Engel curve Curve relating the quantity of a good consumed to income.

9
Individual Demand
Engel Curves for U.S. Consumers
Health care and entertainment are normal goods.
Rental housing is an inferior good for incomes above $35,000.

10
Individual Demand
Substitutes and Complements
From Lecture-2:
Two goods are substitutes: An increase in the price of one leads to an increase in the quantity demanded of the other.
Two goods are complements: An increase in the price of one good leads to a decrease in the quantity demanded of the other.
Two goods are independent: A change in the price of one good has no effect on the quantity demanded of the other.

11
Income & Substitution Effects
The question of Interest:
– When the price of X changes, consumer changes the consumption due to two things, the relative price of X has changed, his purchasing power has changed.

Example: A fall in the price of a good has two effects:

Consumers will tend to buy more of the good that has
become cheaper and less of those goods that are now
relatively more expensive.
Because one of the goods is now cheaper, consumers
enjoy an increase in real purchasing power.

12
Income & Substitution Effects (Illustration)
Income and Substitution Effects: Normal Good
A decrease in the price of food has (income effect + substitution effect)
Initially at A, on budget line RS.
– Price of food falls,
=> consumption increases by F1F2.
The substitution effect F1E (A to D)
– Changes the relative prices of food and clothing but keeps real income (satisfaction) constant.
– The income effect EF2 (D to B) keeps relative prices constant but increases purchasing power.
=> Food is a normal good because the income effect EF2 is positive.

13
Income & Substitution Effects
Substitution effect (SE) Change in consumption of a good associated with a change in its price, with the level of utility held constant.
– For the most part we work with to goods that are substitute so the SE is negative.
Income effect (IE) Change in consumption of a good resulting from an increase in purchasing power, with relative prices held constant.
– Depends whether the good is normal or inferior.
– For normal good the IE is positive.
– For inferior good the IE is negative.
Total Effect (F1F2) = Substitution Effect (F1E) + Income Effect (EF2)
The total effect of a change in price is given theoretically by the sum of the substitution effect and the income effect:

14
Income & Substitution Effects
Income and Substitution Effects: Inferior Good
Initially at A on budget line RS.
Price of food declines:
=> moves to B.
=> Food demand:
– SE: F1E (from A to D),
– IE: EF2 (from D to B).
Food is an inferior good because the income effect is negative.
But SE > IE,
Decrease in the price of food
Increase in the quantity of food demanded.

15
Income & Substitution Effects
Upward-Sloping Demand Curve: The Giffen Good
If food is an inferior,
If IE> SE, the demand curve will be upward-sloping.
Initially at point A. Price of food falls, moves to B and consumes less food.
IE: (EF2) > SE (F1E),
The decrease in the price of food leads to a lower quantity of food demanded.
Giffen good Good whose demand curve slopes upward because the (negative) income effect is larger than the substitution effect.

16
Income & Substitution Effects
Numerical Example
Back to the original example:
U(X,Y) = X0.5 Y0.5
a) Calculate the optimal basket when Px = 4 and Py = 1, and income is $120.
b) Calculate the income and substitution effects of a decrease in the price of food to $10.

17
Income & Substitution Effects (Numerical)
The initial bundle is:

If the Price changes to 10 the demand for X will be:

Therefore the total effect of the rise in the price is the reduction in the demand from 15 to 6.

18
Income & Substitution Effects (Numerical)
The SE is the movement along the same indifference curve but with the new prices.
=> The SE should satisfy the new MRS = Px/Py and should yield the same utility (30):

Therefore the reduction in the demand from 15 to 9.487 is the SE whereas the rest is the IE:
TE = 15-6 = SE ( = 15-9.487) + IE( = 9.487-6)

19
Income and Substitution Effects
Example
Effect of a Gasoline Tax with a Rebate
A gasoline tax is imposed when the consumer is initially buying 1200 gallons of gasoline (Point C).
Budget line shifts from AB to AD
Preferences maximized at E
Consumption of 900 gallons.
However, when the proceeds of the tax are rebated to the consumer, his consumption increases somewhat, to 913.5 gallons at H.
=> Despite the rebate program, the consumer’s gasoline consumption has fallen, as has his level of satisfaction.

20
Tutorial Question
Quasi-linear Utility Function. A consumer has the following utility function: U(x, y) = y + x1/2. Let px and py be the corresponding prices and I her income.
a) Write down the corresponding Lagrangian function and find the first order conditions (FOCs).
b) Use these FOCs to find the expression for the marginal rate of substitution (MRS) and graph it.
c) Find the demand functions for x and y.
d) Let px = py = 1 and I = 1/9. find the optimal consumption levels for x and y?
e) How does your answer to the previous part change if I = 1 and still px = py = 1 ?
f) For px = py = 1 , draw the income consumption curve and the Engel curve for good x.
g) Is x a normal or an inferior good? In terms of share of income spent on x, what happens to it as I increases?
h) Is y a normal or an inferior good? Justify your answer?
i) If px rises to 2, what find the substitution and the Income effects on x.

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Microeconomics: The Introduction

Trade-Offs

Consumers

Workers

Firms

Have limited incomes,

– Must decide to save or to spend

– Must decide on variety of goods and services,

Face constraints and make trade-offs.

Must decide whether and when to enter the workforce.

Face trade-offs in their choice of employment.

Must sometimes decide how many hours per week they wish to work,

=> Trading off between labor & leisure.

Must decide:

What products to produce,

How to use the resources available to produce them.

1
Microeconomics: The Definitions?
● Market Collection of buyers and sellers that, through their actual or potential interactions, determine the price of a product or set of products.
● Arbitrage Practice of buying at a low price at one location and selling at a higher price in another.
● Perfectly competitive market Market with many buyers and sellers, so that no single buyer or seller has a significant impact on price.
● Market price Price prevailing in a competitive market.
● Extent of a market Boundaries of a market, both geographical and in terms of range of products produced and sold within it.

2
Why Study Microeconomics?
Important for the a private firm, because it must decide:
– How the people would react to a new products.
– The cost of the new product.
– Its relationship to the government and the effects of regulatory policies.
Important for the government because the government:
– Must evaluate the monetary impact of new regulatory programs on consumers.
– Must determine how new standards will affect the production cost.
– Must ask whether the market forces are sufficient or regulations are required.

3
3
The Basics of Supply and Demand
How changing world economic conditions affect market price and production
The impact of government price controls, minimum wages, price supports, and production incentives
How taxes, subsidies, tariffs, and import quotas affect consumers and producers
Demand and Supply functions will be developed in the remainder of the course. In this lecture we focus on their implication:
S/D analysis: powerful tool can be applied to the analysis of:

4
4

Supply Curve
● Supply Curve: Relationship between the quantity of a good that producers are willing to sell and the price of the good.
Shows how the quantity of a good offered for sale changes as the price of the good changes.
– It is upward sloping: The higher the price, the more firms produce and sell.

Note: If production costs fall, firms can produce the same quantity at a lower price or a larger quantity at the same price. => S shifts to S’.

5
5
Supply Curve
The supply curve: a relationship between the quantity supplied and the price or =>
Supply Function: QS = QS(P)
Inverse Supply: P = PS(QS)
Other Variables That Affect Supply
– Production costs (wages, interest charges, the costs of raw materials.)
– Example: Production costs decrease => output increases for every given price => Shifts to the right.
Change in supply: shifts in the supply curve,
Change in the quantity supplied: movements along the supply curve.
=> more generally the supply curve can be written as:
QS = QS(P,Raw Material, W, Interest rate, etc )

6
6

Demand curve: The quantity of a good demanded as a function of its price.
– Downward sloping;
=> holding other things equal, will want to purchase more of a good as its price goes down.
Demand Function: QD = QD(P)
Inverse demand: P = PD(QD)
Other variables: Income, the weather, and the prices of other goods.
For most products, a higher income level shifts the demand curve to the right (from D to D’).
QD = QD(P, I, Pj, etc)
Demand Curve

7
7
Supply & Demand
Shifting the Demand Curve
● Substitutes Two goods for which an increase in the price of one leads to an increase in the quantity demanded of the other.
● Complements Two goods for which an increase in the price of one leads to a decrease in the quantity demanded of the other.

8
8
The Market Mechanism

● Equilibrium (or market clearing) price
Price that equates the quantity supplied to the quantity demanded.
(P0 )
– Market mechanism Price changes until the market clears
– Surplus Quantity supplied > quantity demanded.
(P1 )

Shortage Quantity demanded > quantity supplied.
(P2 )

9
9

Changes in Market Equilibrium
Supply and demand curves shift over time as market conditions change.
– Rightward shifts of the supply and demand curves => a bit higher price and a much larger quantity.
=> Changes in price and quantity depend on the shift in the curves and the shape of each curve.

10
10

Changes in Market Equilibrium
Example-1

Supply curve for eggs shifted rightward (production costs fell);
Demand curve shifted to the left (consumer preferences changed)

=> The real price of eggs down sharply while consumption is up.

11
11
Changes in Market Equilibrium
Example-2
Supply curve for a college education shifted to the left (Why?).
Demand curve shifted to the right (Why?)
both price and enrollments rose sharply.

12
12
Elasticities of Demand & Supply
Elasticity Percentage change in one variable resulting from a 1-percent increase in another.
Percentage change in quantity demanded of a good resulting from a 1-percent increase in its price.
Price Elasticity of Demand

12
12
12

13
13
Elasticities of Demand & Supply
Linear demand curve Demand curve that is a straight line.
The price elasticity of demand depends on:

the slope of the demand curve
the price and quantity.
=> varies along the curve as price and quantity change.
Near the top: elasticity is large.
Near the bottom: elasticity is small.

14
14
Elasticities of Demand & Supply
Infinitely Elastic Demand (Horizontal Demand)
Horizontal demand curve: ΔQ/ΔP is infinite.
– A tiny change in price leads to an enormous change in demand.
– An individual buys as much of a good as they can get at a single price, but for any higher price the quantity demanded drops to zero, while for any lower price the quantity demanded increases without limit.
Completely Inelastic Demand (Vertical Demand)
Vertical demand curve: ΔQ/ΔP is zero.
– The quantity demanded is the same no matter what the price, the elasticity of demand is zero.

15
15
Elasticities of Demand & Supply
Income elasticity of demand Percentage change in the quantity demanded resulting from a 1-percent increase in income.
Other Demand Elasticities
Cross-price elasticity of demand Percentage change in the quantity demanded of one good resulting from a 1-percent increase in the price of another.
Price elasticity of supply Percentage change in quantity supplied resulting from a 1-percent increase in price.
Elasticities of Supply

16
16
Elasticities of Supply & Demand
Point elasticity of demand Price elasticity at a particular point on the demand curve.
Arc elasticity of demand Price elasticity calculated over a range of prices.

17
17
Elasticities of Supply & Demand
Numerical Example
The following supply & demand curves are provided.

18
18
Elasticities
Short term vs. Long Term
The short term and the long term demand curves are not identical
=> The short term and long term elasticities of demand will not be identical.
For some goods, in the short term demand curve is steep:
– Demand in the short run does not adjust if the price changes.
=> Short term elasticity of demand will be lower than the long run.
For some goods, in the short term demand curve is flat:
– Demand in the short run does adjust quickly if the price changes.
=> Short term elasticity of demand will be higher than the long run.

19
19
Elasticities
Short term vs. Long Term
Gasoline: Short-Run and Long-Run Demand Curves
Short run: increase in price has small effect on the demand.
=> Elasticity small
Long run: car owners shift to other sources of transportation, to more fuel-efficient cars, etc..
=> Elasticity large

20
20
Demand
Elasticities
Short term vs. Long Term
Automobiles: Short-Run and Long-Run Demand Curves
Short run: increase in price large effect demand. (Why?)
=> Elasticity large
Long run: owners have to buy cars.
=> Elasticity smaller

21
21
Income Elasticity of Demand
Income elasticity of Demand is:

– For most good is positive. (Normal Goods)
– For some goods it is negative (Inferior Goods)
For most goods and services: Larger in the long run than in the short run.
– For a durable goods (car, refrigerator, etc), the opposite is true. The short-run income elasticity of demand will be much larger than the long-run elasticity.

22
22
Supply Elasticities
Short term vs. Long Term
The short term and the long term supply curves are not identical either.
=> The short term and long term elasticities of supply will not be identical.
For some goods, in the short term, supply curve is more steep
=> The supply in the short run does not adjust if the price changes.

23
23

Supply Elasticities
Short term vs. Long Term
Copper: Short-Run and Long-Run Supply Curves
The supply of primary copper, shown in part is more elastic in the long run.
When price increases, firms would like to produce more but are limited by capacity constraints in the short run.
In the longer run, they can add to capacity and produce more.

These goods can be recycled.

24
24
Elasticities
Short term vs. Long Term
Supply and Demand for Coffee
A freeze or drought in Brazil causes the supply curve to shift to the left.
Short run:
supply is completely inelastic.
Demand is relatively inelastic.
=> Sharp increase in price, from P0 to P1.

25
25

Elasticities
Short term vs. Long Term
Supply and Demand for Coffee
Intermediate run:
Supply and demand are both more elastic
=> Price falls part of the way back, to P2.

26
26
Elasticities
Short term vs. Long Term
Supply and Demand for Coffee
Long run:
Supply is extremely elastic:
=> Price returns to P0.

27
27
UNDERSTANDING AND PREDICTING THE
EFFECTS OF CHANGING MARKET CONDITIONS
Fitting Linear Supply and Demand Curves to Data
Given:
a) P* and Q*,
b) ED and ES,
Calculate c and d for the supply curve and a and b for the demand curve.
Use the curves to analyze the behavior of the market quantitatively.
Follow the numerical example

28
28
Example
In 1998, Americans smoked 470 billion cigarettes, or 23.5 billion packs of cigarettes. The average retail price was $2 per pack. Statistical studies have shown that the price elasticity of demand is –0.4, and the price elasticity of supply is 0.5.
Find the supply and the demand assuming both are linear.
Suppose a new government policy results in a 20% reduction in demand (the quantity demanded is 80% of what it was for every value of price). What would be the equilibrium price?

29
Math Review
Basic calculus: Slope of a line is the “rise over the run”.
– While for a straight line this slope is a constant number, for curves it is not.
– Calculus provides tools to find the slope for a variety of functions.

30
Math Review
One variable framework:

Rules:

31
Math Review
Common Derivatives:

32
Math Review
Two Variable Function:

Partial Derivatives:

33
Math Review
Two Variable Function (Examples)

35
Math Review
Example:

The equation denoted by MRS which quantifies the slope of y with respect to x is used extensively in the Microeconomics.

36
Math Review
Optimization (Max or Min):
– One variable function:

This is a Max because?

37
Math Review
Optimization (Max or Min):
– Two variable function:

This is a Max because?

38
Math Review
Constrained Maximization:

Define a Lagrange multiplier ( )?

39
Math Review
Constrained Maximization:

Solve

40
Math Review
Constrained Maximization (Example)

Construct the Lagrangian:

Solve:

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=
=
=
=>
+
=
y
y
x
z
xy
z
y
x
y
x
z
y
z
x
z
y
x
z
x
z
y
z
xy
z
z
z
y
x
z
y
x
y
x
y
x
y
x
7
3
,

5
2
)
ln(
7
5
1
,

)
ln(
,

1
,

2
2
3
3
2
2
(
)
x
y
y
x
c
x
c
x
y
y
z
x
z
z
z
MRS
y
x
c
z
x
c
z
y
x
c
z
y
x
y
x
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
=
=
¶
¶
–
=
¶
¶
¶
¶
=
=
=>
÷
÷
ø
ö
ç
ç
è
æ
=
=
=>
=
–
–
–
–
)
(
)
y
(
/
/
)
(
)
y
(

1
1
1
1
2
0
2
4
4
/
2
=
=>
=
–
=
=>
–
=
x
x
f
x
x
f
Max
x
3
13
,
3
14
0
2
4
0
2
5
4
5
2
2
=
=
=>
=
+
–
=
=>
=
+
–
=
=>
+
–
+
–
=
y
x
x
y
f
y
x
f
xy
y
y
x
x
f
y
x
c
by
ax
f(x,y)
z
Max
£
+
=

Subject to

l
0
)
(
)
3
0
)
2
0
)
1
)
(

=
–
+
–
=
¶
¶
=
–
=
¶
¶
=
–
=
¶
¶
–
+
–
=
c
by
ax
L
b
f
y
L
a
f
x
L
c
by
ax
f(x,y)
L
Max
y
x
l
l
l
l
30
2
4

subject to

£
+
=
y
x
xy
f
Max
0
)
30
2
4
(
)
3
0
2
)
2
0
4
)
1
)
30
2
4
(

=
–
+
–
=
¶
¶
=
–
=
¶
¶
=
–
=
¶
¶
–
+
–
=
y
x
L
x
y
L
y
x
L
y
x
xy
L
Max
l
l
l
l
5
.
7
,
75
.
3
=
=
y
x

Microeconomics: The Introduction

Trade-Offs

Consumers

Workers

Firms

Have limited incomes,

– Must decide to save or to spend

– Must decide on variety of goods and services,

Face constraints and make trade-offs.

Must decide whether and when to enter the workforce.

Face trade-offs in their choice of employment.

Must sometimes decide how many hours per week they wish to work,

=> Trading off between labor & leisure.

Must decide:

What products to produce,

How to use the resources available to produce them.

4
4

Note: If production costs fall, firms can produce the same quantity at a lower price or a larger quantity at the same price. => S shifts to S’.

6
6

8
8
The Market Mechanism

Shortage Quantity demanded > quantity supplied.
(P2 )

9
9

10
10

Changes in Market Equilibrium
Example-1

Supply curve for eggs shifted rightward (production costs fell);
Demand curve shifted to the left (consumer preferences changed)

=> The real price of eggs down sharply while consumption is up.

11
11
Changes in Market Equilibrium
Example-2
Supply curve for a college education shifted to the left (Why?).
Demand curve shifted to the right (Why?)
both price and enrollments rose sharply.

12
12
12

13
13
Elasticities of Demand & Supply
Linear demand curve Demand curve that is a straight line.
The price elasticity of demand depends on:

the slope of the demand curve
the price and quantity.
=> varies along the curve as price and quantity change.
Near the top: elasticity is large.
Near the bottom: elasticity is small.

17
17
Elasticities of Supply & Demand
Numerical Example
The following supply & demand curves are provided.

21
21
Income Elasticity of Demand
Income elasticity of Demand is:

23
23

These goods can be recycled.

25
25

Elasticities
Short term vs. Long Term
Supply and Demand for Coffee
Intermediate run:
Supply and demand are both more elastic
=> Price falls part of the way back, to P2.

26
26
Elasticities
Short term vs. Long Term
Supply and Demand for Coffee
Long run:
Supply is extremely elastic:
=> Price returns to P0.

30
Math Review
One variable framework:

Rules:

31
Math Review
Common Derivatives:

32
Math Review
Two Variable Function:

Partial Derivatives:

33
Math Review
Two Variable Function (Examples)

35
Math Review
Example:

The equation denoted by MRS which quantifies the slope of y with respect to x is used extensively in the Microeconomics.

36
Math Review
Optimization (Max or Min):
– One variable function:

This is a Max because?

37
Math Review
Optimization (Max or Min):
– Two variable function:

This is a Max because?

38
Math Review
Constrained Maximization:

Define a Lagrange multiplier ( )?

39
Math Review
Constrained Maximization:

Solve

40
Math Review
Constrained Maximization (Example)

Construct the Lagrangian:

Solve:

)
/
)(
/
(
:
Elasticity
Point
0
0
Q
P
P
Q
E
p
D
D
=
I
Q
Q
I
I
Q
E
I
D
D
=
D
D
=
%
%
)
(
)
(
/
/
x
f
x
y
f
x
f
x
=
¶
¶
=
=
(
)
rule)
(Power

6)
Rule)
(Quotient

)
5
Rule)
(Product

)
(

)
4

)
(

)
3

)
2
0

of
finction

reads

:
)
,
(
y
x
f
z
=
(y)

respect to
with

derivative

Partial

)
,
(
(x)

respect to
with

derivative

Partial

5
2
)
ln(
7
5
1
,

)
ln(
,

1
,

Subject to

l
0
)
(
)
3
0
)
2
0
)
1
)
(

=
–
+
–
=
¶
¶
=
–
=
¶
¶
=
–
=
¶
¶
–
+
–
=
c
by
ax
L
b
f
y
L
a
f
x
L
c
by
ax
f(x,y)
L
Max
y
x
l
l
l
l
30
2
4

subject to

£
+
=
y
x
xy
f
Max
0
)
30
2
4
(
)
3
0
2
)
2
0
4
)
1
)
30
2
4
(

=
–
+
–
=
¶
¶
=
–
=
¶
¶
=
–
=
¶
¶
–
+
–
=
y
x
L
x
y
L
y
x
L
y
x
xy
L
Max
l
l
l
l
5
.
7
,
75
.
3
=
=
y
x

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