pubp720 2011 spring-lecture-02_full-page
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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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
Indifference curves and utility functions
• Can represent a “level set” of utility function as an indifference curve.
indifference curvestim
e on
the
beac
h (
)
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)
dy
dx
-slope = marginal rate of substitution!
Lecture #2 7
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
Indifference curves and the utility function
Lecture #2 8
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
Indifference curves and the utility function
Lecture #2 9
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
(a) v (t) = 8 · t→ dv
dt= 8
(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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
Optimization: general concept
• Derivatives dfdq or f q
dfdq 0
dfdq 0
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
PUBP720: Managerial Economics Auerswald
Optimization: A coffee shop
fq 1000q − 5q2
dfdq
10q 1000dfdq 0
q∗ 100�
1000 − 10q 0
Lecture #2 15
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
Partial Derivatives: Health
y fx1,x2 −x1 − 12 − x2 − 2 10
∂f∂x 1
Vitamin A Vitamin B
−2x1 2 0
2x1 2
x1∗ 1
∂f∂x 2
−2x2 4 0
2x2 4
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
PUBP720: Managerial Economics Auerswald
Optimization
fq −4q1/2 − 8q 10 fq q3 50
dfdq 0 df
dq 0
Lecture #2 22
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
Indifference curves and utility functions
• Can represent a “level set” of utility function as an indifference curve.
indifference curvestim
e on
the
beac
h (
)
cool, refreshing, non-alcoholic beverages ( )
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)
dy
dx
-slope = marginal rate of substitution!
Lecture #2 26
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
Indifference curves and the utility function
Lecture #2 27
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
Indifference curves and the utility function
Lecture #2 28
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
+∂U
∂y· dy| {z }
∆U due to ∆y
Hold U constant (by assumption); this means, set dU = 0
Lecture #2 30
PUBP720: Managerial Economics Auerswald
Definition: Marginal Rate of Substitution
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
+∂U
∂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
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
dy
dx
-slope = marginal rate of substitution!
Definition: Marginal Rate of Substitution
Lecture #2 32
PUBP720: Managerial Economics Auerswald
indiference curvestime
on th
e be
ach
()
cool, refreshing, non-alcoholic beverages ( )
moving this waymeans more of bothAND higher utility
dy
dx
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):
dU
dx≡ Ux ≡ fx ≡ f1
dU
dy≡ Uy ≡ fy ≡ f2
Lecture #2 33
PUBP720: Managerial Economics Auerswald
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
X ?
X
X
QUESTION:
XX− X−
Lecture #2 34
PUBP720: Managerial Economics Auerswald
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
YX 20
5 4
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
PUBP720: Managerial Economics Auerswald
Leontief (fixed coefficient) utility
functionperfect complements
U1
U2
Quantity of x
Quantity of y
Perfect Complements
U(X,Y) = min(aX, bY)
y1
y2
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
PUBP720: Managerial Economics Auerswald
Linear utility functionperfect substitutes
U1
U2
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.
MRS
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
YX
MRS 14
YX 1
41010 1
4
U X0.2Y0.8
Lecture #2 38
PUBP720: Managerial Economics Auerswald
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
M/py
M/px
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
(0,2)
(4,0)
(2,1)
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
M/py
M/px
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
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
Maximization Subject to Budget Constraint
Lecture #2 42
PUBP720: Managerial Economics Auerswald
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
M/py
M/px
feasible
income expansion path
Example:M=$4
M’ =$6M’’ =$8
px=$1M’/px=6M’’/px=8
Lecture #2 44
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
U(x,y)= x0.5y0.5
(Cobb-Douglas)
Properties of the Demand Function
Quan
tity
dem
ande
d of
goo
d x
Lecture #2 46
PUBP720: Managerial Economics Auerswald
Comment on graphical conventions in economics
px
quantity of good xdemanded
conventional notation in mathematics and engineering: • input on x-axis
• output on y-axis
px
quantity of good xdemanded
conventional notation in economics (… because, later, supply and demand will jointly determine prices and
quantities demanded)
Lecture #2 47
PUBP720: Managerial Economics Auerswald
• 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.
Properties of the Demand Function
Lecture #2 48
PUBP720: Managerial Economics Auerswald
• 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
M/py
M/px
feasible
income expansion path
Properties of the Demand Function
Next week—Case 2: Change in Price
Lecture #2 49
PUBP720: Managerial Economics Auerswald
• 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?
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PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
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
PUBP720: Managerial Economics Auerswald
2Kelli
6Teri
Inkyoung
6Colleen
6Sagar
5Chris C
5
5SeungWon
5Chris P
4Jennifer
4Nazia
4Ian
4Valentia
3Geoff
3James
3Martha
3Kristi
2Matt
2Kashae
2Maziar
1Steven
1Britney
1Peter
1Lisa
GROUPNAME_FIRSTNAME_LAST
Group assignments