Pubp720 2011 spring-lecture-03_full-page

PUBP720: Managerial Economics Auerswald

Prof. Philip Auerswald

Lecture No. 3:

Utility maximization;

income and substitution

effects

2/7/2011 6:38:49 PM

Lecture #3 2


Agenda for today

• Utility maximization subject to a budget constraint (approach 1:graphical) and properties of the demand function

• Optimization (two variables)• Utility maximization subject to a budget constraint (approach 2:

Constrained optimization: Lagrangian Method)• Aplia

Lecture #3 3


Utility maximization subject to a budget constraint (approach 1: graphical) and properties of the

demand function

Lecture #3 4

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 #3 5


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

(Cobb-Douglas)

Maximization Subject to Budget Constraint

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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 #3 7


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 #3 8


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

(Cobb-Douglas)

Properties of the Demand Function

Quan

tity

dem

ande

d of

goo

d x

Lecture #3 9


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 #3 10


• 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 microeconomics.Finding answer means defining the properties of the demand function.


Lecture #3 11


Case 1: Change in income.

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


Next week—Case 2: Change in Price

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Utility maximization subject to a budget constraint (approach 2:

constrained optimization)

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Steps: Constrained Optimization

• Step 1. State the Optimization problem (Max ..…, s.t. …..). Use the expression, g(…) = 0 for the constraint.

• Step 2. Set the Lagrangian expression:

• Step 3. Obtain the first order condition of the Lagrangian expression by taking the first (partial) derivatives.

• Step 4. Solve the first order conditions simultaneously.

• Step 5. Replace xi and λ in the objective function with x*i and λ* to obtain the optimal level of the objective.

Lagrangian multiplier

Optimal amount of inputs: xi∗ where i 1, . . . ,n and ∗

Constraint

y fx 1, x2 , x3, . . . , x n gx1 , x2, x 3, . . . , xn

Objective fn.

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Constrained Maximization

• Set up:

fL, KTC p LL pK K

u fx1,x2I px 1x1 px2x2

EX.1 EX.2

Max: y fx 1, x2 , x3, . . . , x ns.t.: gx 1, x2 , x3, . . . , x n 0

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Max: y fx1,x2,x3, . . . ,xnsubject to (s.t.): gx1,x2,x3, . . . ,xn 0

Constrained Optimization• Lagrangian Setup:

∂ℒ∂x 1

∂f∂x 1

∂g1∂x 1

0∂ℒ∂x 2

∂f∂x 2

∂g2∂x 2

0

∂ℒ∂x n

∂f∂x n

∂x 2∂x n

0∂ℒ∂

gx 1, x2 , x3, . . . , x n 0

F.O.C.

Lagrangian multiplier Constraint

Solve the simultaneous equations

<= objective fn.

<= constraint

ℒ f (x1 , x2 , x3 , . . . , xn g x1 , x2 , x3 , . . . , xn

Lecture #3 16


ℒ −x 12 2x1 − x2

2 4x 2 5 1 − x1 − x2

y −x 12 2x1 − x2

2 4x 2 5x 1 x2 1

Max:

s.t.:

You can take only one vitamin a day (either vitamin A or vitamin B)

Constrained Optimization: Health

constraint

1 − x1 − x2 0

F.O.C.

Solve the simultaneous equations

−x12 2x1 − x2

2 4x2 5 − x1 − x2

−2x 1 2 − 0

−2x2 4 − 0

1 − x1 − x2 0

∂ℒ

∂x 1

∂ℒ

∂x 2

∂ℒ

∂

Lecture #3 17



−2x 1 2

∂ℒ∂x 1

−2x1 2 − 0∂ℒ∂x 2

−2x2 4 − 0∂ℒ∂ 1 − x1 − x2 0

1

2

1

2

−2x 1 2 −2x2 4Thus:

−2x1 2

3

−2x 2 4 4

Replace in using gives us:x1 3 5

1 − x2 − 1 − x2 0 1 − x2 1 − x2 0 −2x2 −2 x2

∗ 1 6

Replacing in using gives us:5 6x2

x1∗ x2

∗ − 1 1 − 1 0 x1

∗ 0

Replacing in using gives us:4 6x2

∗ −21 4 2 ∗ 2 7

Divide both sides by -2:

5

x1 − 1 x2 − 2x1 x2 − 1

Lecture #3 18


Leave it for now.

y −x12 2x1 − x2

2 4x2 5

y∗ −1 4 5 8

x 2∗ 1, x 1

∗ 0, ∗ 2


Optimal health achieved under the constraint

y∗ −1 2 − 4 8 5 10

Compare to the case without constraint (last week):

y fx 1, x2 −x12 2x 1 − x2

2 4x2 5∂f∂x 1

−2x 1 2 0 x1∗ 1

∂f∂x 2

−2x 2 4 0 x2∗ 2

Lecture #3 19


Steps: Constrained Optimization

• Step 1. State the Optimization problem (Max ..…, s.t. …..). Use the expression, g(…) = 0 for the constraint.

• Step 2. Set the Lagrangian expression:

• Step 3. Obtain the first order condition of the Lagrangian expression by taking the first (partial) derivatives.

• Step 4. Solve the first order conditions simultaneously.

• Step 5. Replace xi and λ in the objective function with x*i and λ* to obtain the optimal level of the objective.

Lagrangian multiplier

Optimal amount of inputs: xi∗ where i 1, . . . ,n and ∗

Constraint

y fx 1, x2 , x3, . . . , x n gx1 , x2, x 3, . . . , xn

Objective fn.

Lecture #3 20


Constrained Optimization: Farm

xy

Max. FarmArea (A)?

Area

We need to enclose a field with a fence. We have P feet of fencing material. Determine the dimensions of the field that will enclose the largest area.

A xys. t.

Fence

P 2x 2y

P 2x 2y P − 2x − 2y 0

xy P − 2x − 2yℒ

Lecture #3 21



ℒ xy P − 2x − 2y xy P − 2x − 2y∂ℒ

∂x ∂ℒ

∂y

∂ℒ

∂

y − 2 0 1

x − 2 0 2

P − 2x − 2y 0 3

1

2

y 2 x 2 4

Replacing in using gives us:3 5yThus: x y 5

Replacing in using gives us:4 6x

2∗ P4

∗ P8

Replacing in using gives us:3 5x y∗ P

4

FarmSquare makes the biggest enclosed

area. x∗ P

46

x∗ y∗ P4

Lecture #3 22



∗ P8

A 4004 400

4 10000A ′ 401

4 4014 10050. 06

∗ P8

Shadow Price: Marginal value change in objective function brought by a unit change in constraint function.

Suppose the length of fence material: 400 ft

Then shadow price is the marginal change in the enclosable area brought by 1 feet increase in fence material (from 400 to 401 ft).

“Shadow Price”

A′ −A 50.06

Approximation for the “shadow price”

4008 50

Lecture #3 23


Constrained Optimization: Back to Health

x 1∗ 0, x 2

∗ 1 and ∗ 2Constraint: x1 x2 1

The marginal health improvement achieved by taking an additional pill should be about 2.

y −x12 2x1 − x2

2 4x2 5Max:

s.t.: x1 x2 12 1 − x1 − x2 02

ℒ −x12 2x1 − x2

2 4x2 5 1 − x1 − x22

What does mean? ∗ 2

Lecture #3 24



ℒ −x12 2x1 − x2

2 4x2 5 1 − x1 − x22

−2x1 2

−2x1 2 −2x2 4

−2x2 4

1

2

1

2

Thus:

−2x1 2

x1 − 1 x2 − 2Divide both sides by -2:

x1 x2 − 1

3

4

5

1 − x2 − 1 − x2 0 1 − x2 1 − x2 0 −2x2 −2 x2

∗ 1 6

2-3

3/2

Replace in using gives us:x 1

3 5

2

x2Replacing in using gives us:5 6

x1∗ x2

∗ − 1 1 − 1 0 x1

∗ 03/2 -1 = 0

1/2

∂ ℒ∂x1

−2x1 2 − 0∂ ℒ∂x2

−2x2 4 − 0∂ ℒ∂ 2 − x1 − x2 0

Lecture #3 25


y ∗ −x12∗ 2x1

∗ − x 22∗ 4x2

∗ 5 − 1

2 2 2 1

2 − 32

2 4 32 5

− 14 1 − 9

4 6 5 − 10

4 12 9. 5


8

9.5 - 8 = 1.5 ≈ 2 ∗ 2

Lecture #3 26


Income and Substitution Effects

Lecture #3 27


From Stanley K. Henshaw, Unintended Pregnancy in the United States, Family Planning Perspectives, Volume 30, No. 1, January/February. [http://www.agi-usa.org/pubs/journals/3002498.html]

Lecture #3 28

PUBP720: Managerial Economics AuerswaldIncome and Substitution Effects:Slutsky Equation

E2

E1

X1 X2

Y1

Y2

Total effect

Substitution effect

Income effect

SUBSTITUTION EFFECT

The change in the consumption associated with a change in price while keeping the utility level constant.

INCOME EFFECT

The change in the consumption brought by the increase in purchasing power while keeping the price level constant.

Total effect = Substitution effect + Income effect

N. Koizumi

Lecture #3 29


E2

E1

X1 X2

Y1

Y2

Total effect

Substitution effect

Income effect

Inferior goods

SMALL - ve income effect

(but the total effect is still positive since +vesubstitution effect is bigger than –ve income effect)

N. Koizumi

Lecture #3 30


E2

E1

X1X2

Y1

Y2 Total effect

Substitution effect

Income effect

Giffen goods

BIG - ve income effect

(the total effect is -vesince –ve income effect dominates the +vesubstitution effect)

N. Koizumi

Lecture #3 31


• 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 #3 32


Start from the identity

x∗H³px, py, U

´≡ x∗M

hpx, py,E

³px, py, U

´iSlightly change price px :

∂x∗H∂px

=∂x∗M∂px

+∂x∗M∂m

∂E

∂px

By Shephard’s Lemma

∂E

∂px= x∗H

³px, py, U

´Then

∂xH∂px

=∂xM∂px

+∂x∗M∂m

x∗H

∂xM∂px

=∂xH∂px| {z }

substitution effect (− always)

− ∂x∗M∂m

x∗H| {z }income effect (+/−)

(SLUTSKY EQUATION

AAll that you need to know from this slide is here

Lecture #3 33

PUBP720: Managerial Economics AuerswaldModeling the fertility choice problem

U = f( xkidschildren

, ystuffeverything else

)

The problem:

MaxU (xkids, ystuff) subject to the constraint praising kidsxkids + pstuffystuff ≤ m

Solve using the Langrangian:

L = U (xkids, ystuff)| {z }Obj. func.

+ λ|{z}“Lagrange mult.”

·

⎛⎜⎝m− praising kidsxkids + pstuffystuff| {z }constraint

⎞⎟⎠First order conditions, necessary conditions for a maximum (but not sufficient)

1.dL

dxkids=

∂U

∂xkids− λ

³praising kids

´= 0

2.dL

dystuff=

∂U

∂ystuff− λ (pstuff) = 0

3.dLdλ

=m− praising kidsxkids + pstuffystuff = 0

Lecture #3 34


Combine 1 and 2

praising kids

pstuff=

∂U/∂xkids∂U/∂ystuff

(COMPARE WITH HANDOUT)

Interpretation of the Lagrange multiplier (λ)

λ =∂U/∂xkidspraising kids

Simultaneously solving the first order conditions (FOCs) yields the following solutions (ref

implicit function theorem)

x∗kids =

demand functionz }| {x∗³praising kids, pstuff,m

´y∗stuff =

demnd functionz }| {y∗³praising kids, pstuff,m

´λ∗ = λ∗

³praising kids, pstuff, m

´

Lecture #3 35


Define the “income elasticity of demand for children” as ηkids (pronounced “eta”):

ηkids ≈% change in x∗kids% change in m

(informal)

More generally, define the own price elasticity of demand for good x as εx (pronounced

“epsilon”):

εx ≈% change in xkids

% change in praising kids(informal)

What about “cross-price” elasticities? define the cross price elasticity of demand for kids

as εxy (pronounced “epsilon”):

εxy ≈% change in x∗kids% change in pstuff

(informal)

This is the responsiveness of x∗to changes in the price of good y (py)

Lecture #3 36


Modeling the fertility choice problem

∂xM∂px

=∂xH,kids

∂prasing kids| {z }substitution effect (− always)

−∂x∗M,kids

∂mx∗H,kids| {z }

income effect (+/−)

(SLUTSKY EQUATION)

Lecture #3 37


End

Lecture #3 38


Summary: What you need to know

• 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. We require 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 #3 39


Summary of the Class: What You Need to Know

• Single vs. multi-variable cases– Single variable: Derivative – Multi-variable: Partial derivative

• Optimization without a constraint– First derivative = 0:– Second derivative for concavity:

• Constrained optimization– Lagrangian Multiplier– The value of lamda*:

“Ceteris paribus”

S.O.C.

“Shadow price”

F.O.C.

Lecture #3 40

PUBP720: Managerial Economics AuerswaldSummary (continued)

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

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

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