lecture26revised.pdf

8/19/2019 lecture26revised.pdf

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Economics 101A

(Lecture 26, Revised)

Stefano DellaVigna

December 4, 2003


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Outline

1. Barter

2. Walrasian Equilibrium

3. Example

4. An Example of Excellent Economics

5. Unsolicited advice


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1 Barter

• Consumers can trade goods 1 and 2

• Allocation³

(x1∗1 , x1∗2 ), (x2∗1 , x2∗2 )´

can be outcomeof barter if:

• Individual rationality.

ui(xi∗1 , x

i∗2 ) ≥ ui(ω

i1, ω

i2) for all i

• Pareto Efficiency. There is no allocation ³(x̂11, x̂

12), (x̂

21,

such that

ui(x̂i1, x̂

i2) ≥ ui(x

i∗1 , x

i∗2 ) for all i

with strict inequality for at least one agent.


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• Barter outcomes in Edgeworth box

• Endowments (ω1, ω2)

• Area that satisfies individual rationality condition

• Points that satisfy pareto efficiency

• Pareto set. Set of points where indiff erence curves

are tangent


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• Contract curve. Subset of Pareto set inside the

individually rational area.

• Contract curve = Set of barter equilibria

• Multiple equilibria. Depends on bargaining power.

• Bargaining is time- and information-intensive proce-

dure

• What if there are prices instead?


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2 Walrasian Equilibrium

• Prices p1, p2

• Consumer 1 faces a budget set:

p1x11 + p2x

12 ≤ p1ω

11 + p2ω

12

• How about consumer 2?

• Budget set of consumer 2:

p1x21 + p2x

22 ≤ p1ω

21 + p2ω

22

or (assuming x1i + x2i = ωi)

p1(ω1−x1

1)+ p2³

ω1 − x1

2´≤ p1

³ω1 − ω

1

1´

+ p2³

ω2 −or

p1x11 + p2x

12 ≥ p1ω

11 + p2ω

12


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• Walrasian Equilibrium. ³(x1∗1 , x

1∗2 ), (x

2∗1 , x

2∗2 ), p

∗1, p

∗2´

is a Walrasian Equilibrium if:

— Each consumer maximizes utility subject to bud-

get constraint:

(xi∗1 , xi∗2 ) = arg max

xi1,xi2

ui³

(xi1, xi2

´

s.t. p∗1xi1 + p

∗2x

i2 ≤ p

∗1ω

i1 + p

∗2ω

i2

— Markets clear:

x1∗ j + x2∗

j ≤ ω1

j + ω2

j for all j.


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• Compare with partial (Marshallian) equilibrium:

— each consumer maximizes utility

— market for good i clears.

— (no requirement that all markets clear)


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• Graphical depiction in Edbeworth box. Set of opti-

mal points as prices p1 and p2 vary.

• Draw off er curve for consumer 1 (equivalent of de-

mand curve in partial equilibrium):

(x1∗

1 ( p1, p2, (ω1, ω2)) , x1∗

2 ( p1, p2, (ω1, ω2)))

• Off er curve is set of points that maximize utility as

function of the varying prices p1 and p2.

• Draw off er curve for consumer 2.


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• Walrasian Equilibrium is at intersection of the two

off er curves!

• Walrasian Equilibrium is a subset of barter equilib-rium:

— Does satisfy individual rationality?

— Does it satisfy the Pareto Efficiency condition?

— Is any point in Contract Curve a WE for alloca-tion (ω1, ω2)?


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3 Example

• Consumer 1 has Leontieff preferences:

u(x1,x2) = min³

x11, x12

´

• Bundle demanded by consumer 1:

x1∗1 = x1∗2 = x

1∗ = p1ω

11 + p2ω

12

p1 + p2=

= ω11 + ( p2/p1) ω

12

1 + ( p2/p1)


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• Consumer 2 has Cobb-Douglas preferences:

u(x1,x2) =³

x21

´.5 ³x22

´.5

•

Demands of consumer 2:

x2∗1 =.5³ p1ω

11 + p2ω

12

´ p1

= .5

Ãω11 +

p2 p1

ω12

!

and

x2∗2 =

.5 ³ p1ω11 + p2ω12´ p2 = .5

Ã p1 p2ω

11 + ω

12!


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• Impose Walrasian equilibrium in market 1:

x1∗1 + x2∗1 = ω

11 + ω

21

• This implies

ω11 + ( p2/p1) ω12

1 + ( p2/p1) + .5Ã

ω

1

1 +

p2

p1ω

1

2!

= ω

1

1 + ω

2

1


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4 An example of Excellent Economics

• Savings Rate in the US very low: essentially zero inyear 2,000

• Perhaps: Self-control Problem

• People would like to save but...Not today!

• Credit cards and (too) high borrowing rates


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• Is this testable?

• Prediction of hyperbolic discounting theory:

— people do not like to save today

— people like to save tomorrow

• Save Tomorrow?


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• Benartzi and Thaler (2002): Design of Save More

Tomorrow (SMT) Plan

• 401(k) private savings or retirement

• SMT Plan:

— No increase in savings today

— 3% automatic increase in savings at time of pay-

check raise

— can drop out at any time


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• Advantages:

— No current increase

— Commit today for future

— Use inertia/procrastination the good way!

— No decrease in nominal salary (loss aversion)

— Option out


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• The facts:

— 1998: mid-size company, 315 eligible employees

— ‘you guys are saving too little!’

— 79 employees: increase savings now

— 162 employees: no increase now, will try SMT

— 158 employees: remain in SMT plan for two years

— Eff ect: savings rate up from 3.5 to 11.6 percent!

In three years!


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5 Advice

1. Listen to your heart

2. Trust yourself


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3. Take ‘good’ risks:

(a) hard courses

(b) internship opportunities

(c) research — URAP

(d) (graduate classes?)

4. Learn to be curious, critical, and frank


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5. Be nice to others! (nothing in economics tells you

otherwise)

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lecture26revised.pdf