Pubp720 2011 spring-lecture-02_full-page

PUBP720: Managerial Economics Auerswald

Professor Philip Auerswald

Lecture No. 2:

Utility Functions and Utility Maximization

1/31/2011 6:40:05 PM

Lecture #2 2

PUBP720: Managerial Economics AuerswaldOverview of first lecture

• Current economic science far more concerned efficiency (a “positive” question) than equity (a “normative” question)

• Fundamental theorem of economics states that competition will yield an “efficient”outcome.“Efficient” in this context means that no one can be made better off without someone else being made worse off. This criterion is known as “Pareto optimality.”It has nothing to do with distributional equity.

• Core model involves– initial allocation (constraint)– maximization subject to constraint– equilibrium (e.g. of price and quantity exchanged)– something fundamental changes (e.g. preferences, technology)– shift to new equilibrium (e.g. new price and quantity exchanged)

Lecture #2 3

Plan for today

• Axioms of consumer preference• Indifference curves and the utility function• Marginal utility and marginal rate of substitution• Optimization

– Find the maximum or minimum of an “objective function”– Maximum (or minimum) are found where the slope of the function

equals zero.• Utility maximization subject to a budget constraint [Course CD]• Expenditure minimization subject to constraint that utility is held

constant• Working through page 2 of the Clavell handout [Course CD]

Lecture #2 4

Axioms of consumer preference

• At its foundations, formal economic theory is built upon four fundamental axioms of consumer preference:

– preferences are known (people know what they like)– preferences are reflexive and transitive (basic rationality)– preferences are continuous (smoothness assumption)

• Two additional axioms simplify the use of tools of calculus to derive “refutable implications”

– non-satiation (more is preferred to less)– diminishing marginal rate of substitution (balanced bundles of

goods are, in general, preferred to unbalanced ones)

Lecture #2 5

Indifference curves and utility functions

• Can represent a “level set” of utility function as an indifference curve.

indifference curvestim

cool, refreshing, non-alcoholic beverages ( )

moving this way means more of both

Lecture #2 6

PUBP720: Managerial Economics AuerswaldIndifference curves and the utility function

U(x,y)= x0.5y0.5

(Cobb-Douglas)

-slope = marginal rate of substitution!

Lecture #2 7

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Indifference curves and the utility function

Lecture #2 8

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Lecture #2 9

Take the derivative of the following functions

(a) v (t) = 8 · t

(b) f (x) = x2

(c) g (x) = 43 · x3

(d) h (x) = 12 · x0.5

Drill IDerivatives

Lecture #2 10

(a) v (t) = 8 · t→ dv

(b) f (x) = x2→ df

dx= 2x

(c) g (x) = 43 · x

3→ dg

dx= 4x2

(d) h (x) = 12 · x

0.5→ dh

dx= 14 · x

−0.5

Drill IDerivatives (Answers)

Lecture #2 11

Optimization

• Optimization (or maximization) is the most commonly solved mathematical problem in economics.

– Examples: • Utility maximization: You’re a consumer and interested in

finding the commodity/service quantity that maximizes your utility level (given the income level).

• Profit maximization: You’re a manufacturer or a service provider and interested in finding the most profitable supply level given the amount of inputs you can purchase.

Optimization problem is intuitive and clear graphically.

Lecture #2 12

Optimization: A coffee shop

• Profit function: fq 1000q − 5q 2

q ∗ 100

Hours of labor at a coffee shop

Profit

profit labor

Any way to know without drawing a graph?

Derivative

qis ’s function

Lecture #2 13

Optimization: general concept

• Derivatives dfdq or f q

dfdq 0

(=slope):We use this!

“derivative of function f with respect to q”

IMPORTANCE: The sign of derivatives (positive, 0 or negative)

MAX Profit

Lecture #2 14

Optimization: A coffee shop

fq 1000q − 5q2

10q 1000dfdq 0

q∗ 100�

1000 − 10q 0

Lecture #2 15

y f x 1 , x 2 − x 1 − 1 2 − x 2 − 2 1 0Med A Med BHealth

− x 12 2 x 1 − x 2

2 4 x 2 5

Optimization: Health

• Health function:

Medication A (x1) and Medication B (x2)

“Partial”Derivative

Health(e.g. cholestorollevel)

“While keeping the intake of medication A constant, what is the optimal intake of medication B (vice versa)?”

Lecture #2 16

Partial Derivatives

• f(x) ⇒ derivative is the slope• f(x,y) ⇒ partial derivative is the slope along one input dimension

holding the other constant

Lecture #2 17

Take the derivative with respect to x of the following functions

(a) U (x, y) = x2 · y2

(b) V (x, y) = 12 · x

0.5 · y0.5

Drill II:partial derivatives

Lecture #2 18

Take the derivative with respect to x of the following functions

(a) U (x, y) = x2 · y2→ ∂U

dx= y2 · 2 · x

(b) V (x, y) = 12 · x

0.5 · y0.5→ ∂V

dx= 14 · x

−0.5 · y0.5

Drill II:partial derivatives

Lecture #2 19

Partial Derivatives• Often, a function often involves more than one variable.

• For instance, demand of car can be expressed as:

• How does the sales change if the price goes up by $1 while keeping other factors constant?

Partial derivative is used to answer this kind of question.

What is the change in expected sales if the price of the car goes up by $1, ceteris paribus.

“Ceteris Paribus”

Q f price , age, color , type ,...)

Lecture #2 20

Partial Derivatives: Health

y fx1,x2 −x1 − 12 − x2 − 2 10

∂f∂x 1

Vitamin A Vitamin B

−2x1 2 0

x1∗ 1

∂f∂x 2

−2x2 4 0

x2∗ 2

Max health condition is achieved when consuming 1 unit of medication A and 2 units of medication B.

F.O.C. (First Order Condition)

Health

−x 12 2x1 − x2

2 4x 2 5

y∗ −1 2 − 4 8 5 10 Max health condition

Lecture #2 21

Optimization

fq −4q1/2 − 8q 10 fq q3 50

dfdq 0 df

Lecture #2 22

Optimization

• How do you make sure that your (profit) function looks like this?

Second-order derivative

= Change in slope

= Slope (derivative) of a slope function

Concave

“How do you make sure the

concavity of your function?”

Lecture #2 23

Axioms of consumer preference

• At its foundations, formal economic theory is built upon four fundamental axioms of consumer preference:

– preferences are known (people know what they like)– preferences are reflexive and transitive (basic rationality)– preferences are continuous (smoothness assumption)

• Two additional axioms simplify the use of tools of calculus to derive “refutable implications”

– non-satiation (more is preferred to less)– diminishing marginal rate of substitution (balanced bundles of goods

are, in general, preferred to unbalanced ones)

Lecture #2 24

Indifference curves and utility functions

• Can represent a “level set” of utility function as an indifference curve.

indifference curvestim

moving this way means more of both

Lecture #2 25

PUBP720: Managerial Economics AuerswaldIndifference curves and the utility function

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Lecture #2 26

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Lecture #2 27

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Lecture #2 28

Definition: Marginal Utility

U = f ( xgood number 1

; ygood number 2

Marginal utility (the “uphill” slope of increasing utility as x and/or y increase):

dUdx≡ Ux ≡ f x ≡ f 1

dUdy≡ Uy ≡ f y ≡ f 2

For “normal” goods, marginal utility is positive f 1; f 2 > 0.

Lecture #2 29

U = f ( xgood number 1

; ygood number 2

Definition: Marginal Rate of Substitution

• QUESTION: Suppose we increase the amount of good x a little bit, how does the

consumer have to change her consumption of y to keep utility constant? In other

words, what is the consumer’s marginal rate of substitution?

(HINT : Since both goods are normal, we’re going to consume

less of y.) Recall that a small change in U caused by a small

changes in x and y is given by:

dU =∂U

∂x· dx| {z }

∆U due to ∆x

∂y· dy| {z }

∆U due to ∆y

Hold U constant (by assumption); this means, set dU = 0

Lecture #2 30

Set dU = 0 (along an indifference curve)

dU = 0 =∂U

∂x· dx+ ∂U

∂y· dy

Utility = U(x, y)

Total differential:

dU =∂U

∂x· dx| {z }

∆U due to ∆x

∂y· dy| {z }

∆U due to ∆y

−∂U∂y

· dy = ∂U

∂x· dx [given that dU = 0]

−dydx=∂U/∂x

∂U/∂y[given that dU = 0]

Lecture #2 31

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Lecture #2 32

indiference curvestime

moving this waymeans more of bothAND higher utility

2. Marginal rate of substitution (the slope of the trade-off

between y and x): dydx

We’re dealing with two “slopes” at the same time:

1. Marginal utility (the “uphill” slope of increasing utility as

x and/or y increase):

dx≡ Ux ≡ fx ≡ f1

dy≡ Uy ≡ fy ≡ f2

Lecture #2 33

Review: Basic Math Rules

Basic Math Rule I:X YX Y 1

Basic Math Rule II:1

X X−

Basic Math Rule III:X−X X−

What is X

QUESTION:

XX− X−

Lecture #2 34

USE THE FACT THAT MRS MUXMUY

∂U/∂X∂U/∂Y

Utility 10 XY X0.5Y0.5

MUX 0.5X−0.5Y0.5

MUY 0.5X0.5Y−0.5

MUXMUY

0.5X−0.5Y0.5

0.5X0.5Y−0.5 0.50.5

X−0.5

X0.5Y0.5

Y−0.5

Since 1X0.5 X−0.5 and 1

Y−0.5 Y0.5

MUXMUY

1 X−0.5X−0.5 Y0.5Y0.5 X−0.5−0.5 Y0.50.5 X−1 Y YX

Suppose X,Y 5,20

Then MRS MUXMUY

Numerical Example: MU and MRS

Say X is the no. of dinners at a fancy restaurant, and Y is vacation days.

If (X,Y)=(5,20), you need 4 vacation days (y) to compensate a dinner loss (x).

Utility Function:

Lecture #2 35

Leontief (fixed coefficient) utility

functionperfect complements

Quantity of x

Quantity of y

Perfect Complements

U(X,Y) = min(aX, bY)

Additional y does not increase your utility, if

holding x constant

Examples:

1. Coffee and donuts (for donut-dipping coffee drinkers)

2. Buns and hamburgers (for people who only eat hamburgers in buns)

From the graph, y2 > y1

Then it should be U(y2) > U(y1)

But the graph shows U(y2) = U(y1) = U1

Lecture #2 36

Linear utility functionperfect substitutes

Quantity of X

Quantity of Y

Perfect Substitutes

U(X,Y) = aX + bY

Exactly the same (constant) at any points along the same difference curve.

NO DIMINISHING MRS

(The balanced consumption of the two goods doesn’t do any good).

Example:

1. Sony and Toshiba flat screen TVs

2. BP gasoline and Exxon gasoline

Lecture #2 37

PUBP720: Managerial Economics AuerswaldPractice

• Find the MRS of the utility function:

USE THE FACT THAT MRS MUXMUY

∂U/∂X∂U/∂Y

• Find the MRS at specific point (X,Y) = (10,10):

MUX 0. 2X0.2−1Y0.8 0. 2X−0.8Y0.8

MUY 0.8X0.2Y0.8−1 0. 8X0.2Y−0.2

MRS MUXMUY

0.2X−0.8Y0.8

0.8X0.2Y−0.2 0.20.8

X−0.8

X0.2Y0.8

Y−0.2 14 X−0.8X−0.2Y0.8Y0.2

14 X−0.8−0.2Y0.80.2 1

4 X−1Y1 14

MRS 14

41010 1

U X0.2Y0.8

Lecture #2 38

The Budget Constraint

• So far we’ve talked about preferences. What about constraints?– budget (initial allocation): M (for “money”)

prices: px (price of good x) and py (price of good y)– constraint: can’t spend more than income (no savings or

borrowing here)px • x + py • y≤ M

If assuming non-satiation, then px • x + py • y = M

Lecture #2 39

PUBP720: Managerial Economics AuerswaldThe Budget Constraint

• If you spend everything on good x, how much could you buy?px • xmax = M

xmax = M/px

Likewiseymax = M/py

quantity of good y

quantity of good x

Feasible

Budget constraintIf no quantity

discounts (e.g. “buy two, get the third

free”), then budget constraint is linear

Example:M=$4

px=$1/unitpy=$2/unit

M/px=4 units

M/py=2 units

Lecture #2 40

PUBP720: Managerial Economics AuerswaldMaximization Subject to Budget Constraint

• Economics assumes agents maximize something. Consumer max. utility subject to the budget constraint.Intuition: Get the best possible bundle for the given budget constraint.

• Graph: Pick the best possible indifference curve.

quantity of good y

quantity of good x

Feasible can do better than this

not feasiblemax. subject to constraint

Note. At point of tangency slope

of budget line

-(px/py)_= -MRS= -(U1/U2)

Lecture #2 41

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Maximization Subject to Budget Constraint

Lecture #2 42

U(x,y)= x0.5y0.5

(Cobb-Douglas)

budget line

Note:We only require of

indifference curves: • no crossing

• convex to the origin

Maximization Subject to Budget Constraint

Lecture #2 43

PUBP720: Managerial Economics AuerswaldMaximization Subject to Budget Constraint

• What if M changes?

M’/py

M’/px

first shift of indifference curve

second shift of indifference curve

initial optimal indifference curve

M”/px

M”/py

quantity of good y

quantity of good x

feasible

income expansion path

Example:M=$4

M’ =$6M’’ =$8

px=$1M’/px=6M’’/px=8

Lecture #2 44

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Price of good x is decreasing

optimal consumption bundle at at first px

optimal consumption bundle at at second px

optimal consumption bundle at at third px

Graphical Derivation of the Demand Curve

Lecture #2 45

U(x,y)= x0.5y0.5

(Cobb-Douglas)

Properties of the Demand Function

Lecture #2 46

Comment on graphical conventions in economics

quantity of good xdemanded

conventional notation in mathematics and engineering: • input on x-axis

• output on y-axis

quantity of good xdemanded

conventional notation in economics (… because, later, supply and demand will jointly determine prices and

quantities demanded)

Lecture #2 47

• We’re gone to a lot of trouble to derive these demand functions. What can we say about them?

– Q. On what parameters does demand depend?A. prices (px, py) and income (M).Note: These parameters are exogenous from the standpoint of the consumer.

– Q. How does demand change when the exogenous parameters change?This is our core question for today.Finding answer means defining the properties of the demand function.

Lecture #2 48

• Case 1: Change in income. We’ve already seen this in a different form.

M’/py

M’/px

first shift of indifference curve

second shift of indifference curve

initial optimal indifference curve

M”/px

M”/py

quantity of good y

quantity of good x

feasible

income expansion path

Next week—Case 2: Change in Price

Lecture #2 49

• Another way to think about the same problem (the twin, or “dual” problem): minimize the expenditure of getting to utility level U (“U” bar). Denote expenditure by E.

Intuition: Spend as little as possible for a given level of satisfaction.

• Graph: Pick the best possible budget line, with utility fixed.

The Dual Problem: Expenditure Minimization

quantity of good y

quantity of good x

Note. At point of tangency

slope of budget line ( px/py)_equals MRS

(U1/U2)

hold utility constant at this level U–

Emin/px

Emin/py

Etoo much/py

Etoo much/px

Lecture #2 50

PUBP720: Managerial Economics AuerswaldThe population problem: Theory and evidence

Challenge: Formally frame one issue/concept raised in the Dasgupta article in the terms of theoretical microeconomics.Do not try to solve the problem. Simply try to structure the optimization decision.Questions to address:

– What is the choice being made?– Who is making the decision?– What is being maximized (minimized)?– What tradeoffs do we expect are involved in the decision?– What are the outcomes for the “decision-maker”? For the group

and region?

Lecture #2 51

Summary: What you need to know

• The “slope” of an indifference curve (at a given point) is minus the ratio of the marginal utilities (at that point). This slope is referred to as the marginal rate of substitution (MRS).

• Assuming that the MRS is decreasing w/ increasing x (or y) is the same as assuming curvature of indifference curves.

• Diminishing MRS (U1/U2) is not the same thing as diminishing marginal utility (U11; U22 ).

Lecture #2 52

Summary: What you need to know (continued)

• Budget constraints define the limits of the feasible set. If there are no quantity discounts, then the budget constraint is linear.

• Graphically, many different indifference curves are possible. Werequire only no crossing, and convexity to the origin.

• Increasing (decreasing) income causes the budget line to shift out (in), parallel to original budget line.

• Increasing (decreasing) the price of a good causes the budget line to rotate or pivot in (out).

• The indifference curve that maximizes utility is the one tangent to the budget line. At the optimal point, the price ratio equals the ratio of the marginal utilities (the marginal rate of substitution).

Lecture #2 53

PUBP720: Managerial Economics AuerswaldSummary (continued)

• “Marshallian” (“uncompensated”) demand functions are the solutions to the constrained utility maximization problem.

• The “dual” of the utility maximization problem (subject to fixed budget) is the expenditure minimization problem (subject to fixed utility level). “Hicksian” (compensated) demand functions are the solutions to the constrained expenditure minimization problem.

• Next week: In the case of Marshallian demand, a shift in quantity demanded cause by a change in price can be decomposed into an income effect and a substitution effect.

Lecture #2 54

Announcements

• Remember—there is an Aplia assignment, problem set, or exam nearly every week in this course.

• Problem set #1 is due on SEPTEMBER 23. • Group assignments follow.

Lecture #2 55

2Kelli

Inkyoung

6Colleen

6Sagar

5Chris C

5SeungWon

5Chris P

4Jennifer

4Nazia

4Valentia

3Geoff

3James

3Martha

3Kristi

2Kashae

2Maziar

1Steven

1Britney

1Peter

GROUPNAME_FIRSTNAME_LAST

Group assignments

Pubp720 2011 spring-lecture-02_full-page

managerial economicsux

utility functions

dtb f x

healthy f x

utility level

t b f x

utility functionx0

x2 c g x

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