Microeconomics and Policy Analysis PUAF 640

1
Microeconomics and Policy Analysis PUAF 640

• Professor Randi Hjalmarsson
• Fall 2009
• Lecture 3

2
Class Outline Chapter 2

• Announcements
• Read Waldfogel paper The Deadweight Loss of
  Christmas for next class. Be prepared to discuss
  Section A.
• Problem Set 1 due today.
• Problem Set 2 due October 1 (available tomorrow).
• Lecture is on Tuesday next week at 700.
• Todays lecture
• Budget Constraints
• Preferences and Utility
• Utility Maximization

3
Where do household S/D curves come from?

• How do we know that at a price of P1, households
  or individuals demand Q1?
• Go behind individual demand curve.
• How does a consumer choose how much of different
  goods to buy?
• Given his income level, prices of different
  goods, and his preferences for goods

4
Consumer Decision Model

• Taking his constraints into account, individual
  attempts to reach highest feasible level of
  satisfaction.

Budget Constraint what he can afford to do
Preferences what the individual wants to do
Decision
5
Budget Constraints - Notation

• I Income of the Individual (sometimes see Y
  used)
• K types of goods that an individual can buy
• x1, x2, , xk are the quantities of each of the k
  goods purchased
• P1, P2,,Pk are the prices of those goods

6
Budget Constraints Assumptions

• Make it easier to present the model. Relaxing the
  assumptions would not alter basic results.
• Assume I is exogenous.
• Income is a given in the model. The individual
  does not decide how much to work.
• Assume no saving or borrowing and that all goods
  are consumed when purchased. No stockpiling.
• Abstracting from intertemporal decisions. Just 1
  time period.

7
Budget Constraints Assumptions

• Assume only two goods. Common to label them x1
  and x2 or x and y.
• Primarily for convenience. Much easier to draw
  two dimensional graphs!
• Assume the consumer is a price taker.
• Each consumer faces a budget constraint.
• They cant spend more than their income.

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Budget Constraint

• Amount spent Amount earned (income).
• Budget constraint for two commodities
• Pxx Pyy I
• More generally, for k commodities
• P1x1 P2x2 Pkxk I

9
Graph Budget Constraint (BC) for 2 goods

• Pxx Pyy I
• What will BC look like?
• May help to rewrite BC such that y is a function
  of x.
• Pyy I - Pxx
• y (I/Py) - (Px/Py)x
• What are x and y intercepts?
• What is the slope?
• Always label intercepts and axes!

y
I/Py
Slope -Px/Py
x
I/Px
10
Budget Constraint Graph contd

• What does the graph tell us?
• Shows the feasible set of bundles that can be
  consumed.
• Individual can afford to buy any bundle that lies
  on or below the budget constraint (shaded area).
• Budget constraint itself represents the different
  bundles of goods that would exhaust the
  individuals income, I.

y
I/Py
Budget Constraint
Feasible Set
x
I/Px
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Important Notes about the BC

• The corner points (x and y intercepts) represent
  bundles in which the consumer spends all of her
  income on one good.
• What does the slope of the budget constraint tell
  us? (Recall m -Px/Py)
• Negative of the slope indicates the rate at which
  the market permits you to substitute one good for
  another.
• Let Px2 and Py 3, then m 2/3.
• For every unit of x you give up, you get 2/3 unit
  of y.
• Or give up 2 units of y to get 3 units of x.
• Px/Py is the price of x in terms of y.
• The price of one unit of x is 2/3 units of y.

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Steve has income of 100 and faces prices of
Px4 and Py 10.

• What is Steves budget constraint?
• Pxx Pyy I 4x 10y 100
• Graph BC. First, solve for y
• 10y 100 - 4x
• y 10 (4/10)x
• What is the slope?
• -4/10 -2/5
• What does the slope tell us?
• One has to give up 2 units of y to get 5 units of
  x.
• Rate at which market trades good x for good y.

y
Slope -2/5
10
x
25
13
Budget Constraint and Income Changes

• Pxx Pyy I
• What happens to BC if income changes?
• Parallel shift of the budget constraint.
• Increase in income shifts BC out can afford
  more
• What happens to the slope?
• Doesnt change!
• Ratio of prices hasnt changed!

y
I2/Py
I1/Py
BC1
BC2
I1/Px
x
I2/Px
14
Back to Steve

• Initially I 100, Px4 and Py 10
• What if I increases to 200?
• New budget constraint
• I2 Pxx Pyy
• 200 4x 10y
• y 20 (4/10)x
• Slope is the same -2/5
• What happens to the intercepts?
• Y-intercept 20, x-intercept 50
• They doubled. Income doubled, so the max you can
  buy of one good (if spend all money on that good)
  also doubles.

15
Budget Constraint and Price Changes

• What happens to BC if price of good x increases?
• Think about the corners.
• If spend all money on x, can you buy more or less
  x?
• Less x.
• If you spend all money on y, can you buy more or
  less y?
• Same y.
• So, BC rotates in when price increases! (See BC2)
• When price decreases?
• Rotates out. (See BC3)

y
I/P1y
BC3
BC1
BC2
I/P3x
I/P1x
x
I/P2x
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Rotate around x axis if price of y changes
I/P3y
Price y decreases, rotate out
y
I/P1y
BC3
BC1
I/P2y
BC2
Price y increases, rotate in
I/P1x
x
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Preferences and Utility

• Now know which bundles are available to
  individual, but still need to figure out how he
  decides on a bundle.
• The bundle that is chosen is determined by an
  individuals preferences.
• How do we represent an individuals preferences?
• Build preferences according to three assumptions
  or axioms (rules generally accepted as true).

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Axiom 1 - Completeness

• Completeness When confronted with two bundles,
  the consumer can always tell which one is
  preferred, or if she is indifferent.
• While any set of preferences can be complete,
  they might not be rational.
• Thus, axioms 2 and 3 rule out some
  inconsistencies.

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Axioms 2 and 3

• Axiom 2 Transitivity
• If x is preferred to y, and y is preferred to z,
  then x is preferred to z.
• Likewise, if indifferent between x and y, and
  indifferent between y and z, then must be
  indifferent between x and z.
• Axiom 3 Non-satiation
• More is always better (gives more satisfaction)!
• This assumption is often for convenience, as
  there is nothing irrational about getting
  satiated.

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Utility Function

• Axioms allow definition of a utility function on
  all goods defines an individuals preferences.
• Utility function Measures how well off an
  individual is given that the individual consumes
  various amounts of the consumption goods.
• Utility U(x1, x2) or U(x, y)
• Utility U(x1, x2, x3,, xk)
• Utility function is a way of ranking bundles.
• Note that magnitude of difference between U1 and
  U2 dont tell us anything.

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Extra Assumptions about Utility Functions

• As long as x and y are goods, utility increases
  as x and/or y increases.
• x is hamburgers and y is all other goods
• Assumption is that holding y fixed, an increase
  in hamburgers results in an increase in utility
  level.
• In other words, marginal utility of x is positive
• Marginal utility is the change in utility
  associated with increasing consumption of x,
  holding y constant.
• MUx gt 0
• For a utility function U(x,y)

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Extra Assumptions about Utility Functions

• Diminishing Marginal Utility
• As consumption of a good increases, utility
  increases but at a slower and slower rate i.e.
  at a decreasing rate.
• First hamburger may be great, second hamburger
  satisfying, third hamburger okay,
• Utility functions are continuous and smooth.

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What does utility curve look like?

• Why does the curve slope up?
• Assumption 1 positive marginal utility
• Why is the curve bowed?
• Assumption 2 diminishing marginal utility
• MU at A gt MU at B, i.e. as x increases
• At A, MU is high small change in x yields big
  change in U.
• At B, MU is low.

utility
B
A
x
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Indifference curves

• Often more useful than utility function
• Indifference curves plot out all possible
  combinations of the various goods that yield an
  identical utility level.
• Indifference curve for utility level, U, is set
  of all x and y such that
• UU(x,y)

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What do indifference curves look like?
y
Individual is indifferent between all bundles on
U0.
Increasing U2 gt U1 gt U0
U2
But, individual prefers any bundle on U1 to one
on U0 since he has higher utility.
U1
U0
x
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Can we have the following indifference curve?

• No! Why?
• Non-satiation assumption
• At B, consume higher levels of x and y, therefore
  should have higher utility than at A (not the
  same)
• Exceptions?
• Suppose x is pollution and y is all other goods
• If person doesnt like pollution, then could have
  such a curve.
• But, means we are relaxing the assumption that
  all goods are goods.

y
U0
B
A
x
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Can we have the following indifference curves?

• No. Why?
• Transitivity property
• According to graph, you are
• Indifferent between A and C
• Indifferent between B and C
• So, should also be indifferent between A and B
• But, B is preferred to A (since get more x and
  more y)
• Transitivity assumption is not satisfied!

y
y
B
A
C
C
U1
U0
x
x
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Properties of Indifference Curves

• Indifference curves always slope down.
• If you consume more of x, then you need less of y
  to stay at the same utility level
  (non-satiation).
• Indifference curves cannot cross.
• By transitivity.
• Indifference curves are bowed in towards the
  origin.
• Because of diminishing marginal utility.
• Understand by looking at indifference curve slope.

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Marginal Rate of Substitution (MRS)

• Negative of the slope of the indifference curve
• So, curves are bowed in implies that
• MRS decreases as x increases and y decreases.
• What does MRS measure?
• Willingness to trade x for y while remaining just
  as well off (indifferent)
• MRSyx 5
• MRS of y for x
• Individual needs 5 units of y in exchange for 1
  more unit of x.
• MRSyx is of units of y willing to give up for 1
  more x.

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MRS contd

• At A
• MRS is (slope of dashed line)
• y is high relative to x.
• MRS is high since individual would be willing to
  give up a lot of y just to get 1 more unit of x.
• Because of diminishing marginal utility.
• When have just a little x, the next unit of x is
  worth a lot. But, when have a lot of y, next unit
  of y is worth just a little.

y
A
B
U1
x
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MRS contd

• At B
• x is high relative to y.
• MRS is low
• Individual is only willing to give up a little y
  to get 1 more unit of x.
• Because of diminishing marginal utility.
• When have a lot of x, the next unit of x is not
  worth much. But, when have little y, next unit of
  y is worth a lot.
• Suppose slope at A -2.
• Individual is willing to give up 2 more units of
  y to get 1 more x and stay at same U.

y
A
B
U1
x
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MRS Relationship to MU

• Where does this come from?
• Recall that MUy is the change in utility
  associated with consumption of additional unit of
  y, holding x constant.

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MRS Relationship to MU (contd)

• Take away enough y so that you move from f to g
• How does utility change?
• Change is ?yMUy
• Give you enough x to bring you from g to h.
• How does utility change?
• Change is ?xMUx
• What happens to total utility?
• ?U0 (f and h on same curve)
• ?U 0 ?yMUy ?xMUx

y
f
?y
h
U1
g
?x
x
34
MRS Relationship to MU (contd)

• ?U 0 ?yMUy ?xMUx
• Rearrange terms
• The negative of the slope of the indifference
  curve equals ratio of marginal utilities.

35
Special types of indifference curves

• Perfect Substitutes
• How would you characterize MRS?
• MRSyx is constant.
• Amount of y individual needs to be given in
  exchange for one unit of x is the same at points
  A and B.
• Note that it is not necessarily one for one.

y
U2
U1
A
U0
B
x
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Special types of indifference curves

• Perfect Complements
• x and y must be consumed in fixed proportion.
• Proportion is defined by slope of line through
  origin.
• Individual doesnt get extra utility from more x
  until he gets more y.
• B makes individual no better off than A.

y
U2
U1
A
B
U0
x
37
Describe each consumers preferences for x and y

• Give up a lot of y for one more x. y is less
  important than x

• Give up just a little y for more x. x is less
  important than y.

y
y
x
x
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Utility Maximization

• Can now answer question of how individual chooses
  bundle to buy from all affordable bundles.
• What bundle will he choose?
• He will choose from the affordable bundles
  (feasible set) the one that gives the highest
  level of utility.
• Maximizes utility subject to his budget
  constraint.

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Where is the optimal bundle?

• A is optimal. Why?
• It is equilibrium individual has no incentive
  to change bundle (cant achieve higher utility)
• More x would increase utility, but cant afford
  it. Likewise for y.
• At B, taking away x and giving more y would yield
  higher utility (and is affordable).
• B is not equilibrium.

y
I/Py
A
U2
U1
U0
Feasible
B
x
I/Px
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Equilibrium condition

• Indifference curve and budget constraint are
  tangent.
• Have equal slopes.
• Individual and market value goods at the same
  rate.
• Chooses x and y such that the MRS equals the rate
  at which the goods can be traded for each other
  in the market.

y
Slope -Px/Py
I/Py
A
U1
x
I/Px
41
Rewrite equilibrium condition as

• Marginal Utility Principle - Bundle that
  maximizes total utility is such that marginal
  utility of the last spent on each commodity is
  the same.
• Why?
• Consider what would happen if take away last
  spent on x and spend it on y.
• Because of diminishing MU, would lose more
  utility from lost x than would gain from
  additional y. (see diminishing MU graph again)
• Not utility maximizing.

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Practice Problem Discussion/Class?

• Jack and Jill have decided to allocate 1000 of
  their individual incomes to purchasing magazines
  on home theater equipment and magazines on
  running/exercising. Jack prefers home theater
  magazines more than exercise magazines, while
  Jill prefers the exercise option more.
• In the same graph, draw sets of indifference
  curves for Jack and Jill which depict these
  different preferences.
• Discuss why the two sets of curves are different
  from each other using the concept of the marginal
  rate of substitution.
• If Jack and Jill face the same prices for these
  magazines, will their marginal rates of
  substitution of home theater magazines for
  exercise magazines be the same or different at
  their optimal goods basket? Assume no corner
  solutions.

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Policy Application Food Stamps

• How does food stamp program affect consumer
  behavior?
• U.S. food stamp program
• Type of in-kind transfer
• 19 million participants in 1985 and cost gt 11
  billion
• Almost 26 million participants in 2005 and cost gt
  28 billion
• Lets consider a very simplified food stamp
  program.
• Family will pay 80 to get 150 worth of food
  stamps.

44
Food Stamps (contd)

• Let I 250.
• What is budget constraint without food stamps?
• PfoodF Paog(AOG) I
• Dont know prices but we know amount spent on
  each good (like letting prices 1)
• 250 F AOG

AOG
250
BC without food stamps
food
250
45
What does BC look like for family eligible for
food stamps?

• Recall pay 80 for 150 of food stamps.
• If spend all 250 on AOG, then get no food stamps
  (y-intercept is the same).
• Need 80 to purchase food stamps.
• Can consume AOG 170
• If purchase no additional food, then can consume
  150 worth of food (point A).
• If spend remaining 170 also on food (no AOG),
  then can spend 320 on food in total).

AOG
250
BC without food stamps
A
170
BC with food stamps
food
250
80
150
320
46
What would BC look like if family gets 70 cash
transfer instead?
AOG
320
250
BC without food stamps
BC with 70 transfer
food
250
320
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Under which policy is consumer better off?

• Family does better with cash than food stamps.
• Achieves U2 versus U1.
• How would you characterize this household?
• Have weak preferences for food.
• Willing to give up a lot of food to get a little
  more AOG.
• Does this have to be the case?

AOG
320
250
U2
U1
U0
food
250
320
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• Individuals with preferences such as these are
  indifferent between the 2 programs.
• Have a stronger preference for food.
• Always remember to consider solution for
  different types of individuals!

AOG
food