business 33001: microeconomicsfaculty.chicagobooth.edu/owen.zidar/teaching/fall 2016/lectures... ·...

Business 33001:Microeconomics

Owen Zidar

University of Chicago Booth School of Business

Week 1

Owen Zidar (Chicago Booth) Microeconomics Week 1 1 / 92

Outline for Week 1

Overview, logistics, schedule, etc

Material

1 Demand

2 Supply

3 Market Equilibrium

4 Elasticities

5 Applications


Introduction

1 My backgroundPh.D. from UC Berkeley, BA from DartmouthStaff Economist at Council of Economic AdvisersConsultant for US Treasury on Business TaxationAnalyst at Bain Capital

2 Research fiscal policy topicsIncidence and efficiency costs of corporate taxationEconomic impacts of taxing high-income earnersEffect of state tax system on U.S. economyImpact of patents on firms and workersBusiness Income and US Income InequalityProductivity, technological change, and the future of fiscal policy


About Microeconomics

1 MarketsAllocation of scarce resourcesVoluntary transactions between Buyers and SellersPrices and quantities

2 Individuals: Choice under constraints

Who are the relevant parties? (Buyers, Sellers)What are their incentives?What actions can they take?

3 Equilibrium: How choices interact and determine outcomes

Aggregate individual decisions


Motivation: why should you care?

1 Improve Business DecisionsConsumer behaviorCost minimization, hiring, investment decisions, and profitabilityEntry & market potential, pricing, competitor behavior

2 Understanding MarketsCapital MarketsLabor Markets and rising income inequalityTechnological change and economic growth

3 This is ChicagoExpectations of employers that you know economics, esp. microeconFoundation for many other classes at Booth

4 Model-based ThinkingModel is a simplified representation of reality that gets to the essenceof what is going onComplex global economy, big data


Applied Micro is increasingly valuable in big data world

Yet data is merely the raw material of knowledge. We’re rapidlyentering a world where everything can be monitored andmeasured [...] but the big problem is going to be the ability ofhumans to use, analyze and make sense of the data.

– Erik Brynjolfsson, director of the Massachusetts Institute ofTechnology’s Center for Digital Business


The Microeconomic Approach

To understand how and why something happens, look at individualbehavior

Who are the people making the choices?

What does each person want?

What decisions are optimal, given constraints?

How do the decisions of different players interact?

What adjusts if the choices aren’t mutually consistent?


Applying the Microeconomic Approach

Main application: Markets

Big picture: Supply and Demand

Consumers choose what to buy

Maximize utility subject to prices & budgets

Firms choose how much to produce, what prices to charge

Maximize profit subject to demand curves, costs

Equilibrium: prices and quantities adjust to clear market

What predictions do we make about the impact of supply/demandshifts, taxes, etc?

What can we say about consumer and producer welfare?


Example: Uberworld

The world’s most valuable startup is leading the race to transform thefuture of transport

The Economist. September 2016Owen Zidar (Chicago Booth) Microeconomics Week 1 9 / 92

Outline for Course

1 Supply and Demand

2 Consumer and Producer Surplus, Efficiency, Government & Taxes

3 Consumer Behavior, Market Demand, & Consumer Welfare

4 Economic Cost, Factor Demand, and Technological Change

5 Industry Supply, Market Equilibrium, and Profits

6 Capital Markets, Investment, and Economic Growth

7 Monopoly, Innovation, Advertising

8 Oligopoly, Strategic Interaction, and Product Differentiation

9 Uncertainty, Insurance, and Adverse Selection

10 Price Discrimination, Behavioral Economics, and Other topics


Logistics


Expectations

This course is analytically challenging

You will likely not understand everything fully the first (or second)time you encounter some of the concepts

The way you are going to learn is to really struggle with the material

The goal is not memorization, but, by the end of the class, that youreally understand the concepts and learn how to use economics

to make better decisionsto change the way you thinkto understand how the world works


Logistics (1/2)

1 TAsChris Cruickshank: [email protected] Gelb: [email protected]

2 GradingProblem Sets (20%)

Due at the beginning of each class (excluding Weeks 1 and 5)8 required problem setsThey will challenge youGoal is to understand key economic concepts by end of the courseDrop 2 lowest scoresMay work in groups, but turn in your own PS

In-class midterm Week 5 (30%)Cumulative Final Exam (50%)

3 Weekly Review SessionsOptional – go over past hwWednesday, 6pm-7pm, Gleacher Center, Room 306


Logistics (2/2)

1 TA and I are available by emailAsk any questions about the courseArrange meetingTA is primary contact for problem set questions

2 Optional Textbook: Goolsbee, Levitt, Syverson

3 If necessary, can go to class in one of other sections

4 This course will require some mathSolve system of two linear equationsTake derivatives of simple functions

Read math review online and do practice problems

For math support, please contact Student Services

5 Course websiteIf you need Chalk access: email me!


Today’s Class

Overview, logistics, schedule, etc

Material

1 Demand

2 Supply

3 Market Equilibrium

4 Elasticities

5 Applications


Demand


Demand

Definition

The relationship of the quantity of good x purchased and its price Px

holding money income and the prices of all other goods constant is calledthe demand curve for good x .


Individual Demand

The demand curve:

tells us how much an individual will consume at each price

is simply a schedule that summarizes an individual’s choices (andhence underlying preferences)

is derived holding income and prices of other goods constant

To find out how much an individual will purchase at a given price, wesimply extend a horizontal line at the price we are interested in until itreaches the demand curve


Graphing Demand: Pizza Example

Quantity

Price


Graphing Demand: Willingness to Pay

Quantity

Price


Graphing Demand

0 20 40 60 80 100Q0

20

40

60

80

100

P

Total Demand


Characterizing Demand Mathematically

0 20 40 60 80 100Q0

20

40

60

80

100

P

Demand Equation: QD = 100− P


Add Horizontally to Obtain Aggregate Demand

Price # 1 #2 Total$5 0 0 04 1 0 13 1 1 22 2 1 31 3 2 50 3 8 11

0 2 4 6 8Q0

1

2

3

4

5P

Consumer 1

Consumer 2

0 2 4 6 8 10 12Q0

1

2

3

4

5P

Aggregate


Demand Curves

“Law of Demand”: quantity demanded decreases in price

Demand curve is downward sloping

Q

P

D

Any counterexamples?


Supply


Individual Firm Supply

Just as demand represents the willingness of people to buy a good atvarious possible prices, supply represents the willingness of people tosell goods at various possible prices.

The height of the supply curve of a price-taking individual seller ismarginal cost.

Supply curves slope up because of increasing marginal cost– the morethe seller is producing, the greater the cost of producing more.


Supply

How much quantity would firms produce, at a given market price?

Individual Firm Supply Curve

“Intensive” margin of adjustment:When price goes up, increase productionWhen price goes down, decrease production

“Extensive” margin: Entering/exiting a marketWhen price goes up, start a new firmWhen price goes down, go out of business

Industry Supply Curve

Sum of individual firm supplies


Graphing Supply

Quantity

Price


Graphing Supply

Q

P

S1

S2

Q

P

S1

S2

S1�S2

Firm Supply Curves Industry Supply


Graphing Supply

Q

P

S

The supply curve is increasing

Any counterexamples?How should we think about economies of scale (manufacturing)?


Market Equilibrium


Market Equilibrium

Definition

A market equilibrium is a situation in which there is no pressure for priceor quantity to change. The market equilibrium price equates quantitysupplied and quantity demanded, so there is zero excess demand and zeroexcess supply.


Price is too high ⇒ Excess Supply

Quantity

Price


Price is too low ⇒ Excess Demand

Quantity

Price


Market Equilibrium

Putting together Supply and Demand:

Supply Quantity = Demand QuantitySupply Price = Demand Price

qQ

p

P

S

D

Solving:

Graph: find intersectionMath: solve system of equations


Market Equilibrium

Putting together Supply and Demand:

Supply Quantity = Demand QuantitySupply Price = Demand Price

qQ

p

P

S

D

How do markets find the “right” price and quantity?

What if the price started too high?

What if the price started too low?


Market Equilibrium

Putting together Supply and Demand:Supply Price = Demand PriceSupply Quantity = Demand Quantity

ExampleDemand: QD = −2PD + 50Supply: QS = 3PS − 45

Q

P

S

D

Equilibrium: Q = 12, P = 19


Two Quick Examples

1 Why are lobsters so cheap?

2 What should we do about poachers?


Example # 1 – Maine Lobsters

The Maine lobster industry is being crippled by a glut of supply

that many attribute to climate change, sending the price per pound

plummeting and turning the crustacean into something it has rarely

been before: affordable. The volume of the state’s lobster harvests has

skyrocketed as oceans have warmed, helping the shellfish and their

larvae to grow faster. But demand for what has long been considered a

luxury dish has not kept pace, especially since the global recession hit

in 2008, even as fuel and equipment costs rise. – FT (August, 2013)


Shifts in Supply and Demand

How do shifts in the Supply Curve affect Price and Quantity?

Q

P

S

D


Shifts in Supply and Demand

How do shifts in the Supply Curve affect Price and Quantity?

Q

P

D

S

S'

Supply increases (shift out): Price down, Quantity up

Supply decreases (shift in) : Price up, Quantity down


Example #2 – Elephant Ivory

Elephant Ivory:

To restrict poaching, US banned commercial trade in ivory in 1989

Illegal ivory is seized. What to do with it?

November, 2013: six tons of accumulated ivory destroyed

“[We] decided to destroy this material as a demonstration of our

commitment to combating wildlife trafficking,” the U.S. Fish and

Wildlife Service said. ”We want to send a clear message that the

United States will not tolerate ivory trafficking and the toll it is taking

on elephant populations, particularly in Africa.” – CNN

Economists: Maybe we shouldn’t destroy it.Maybe we should sell it instead!

What would be the market impact of selling the seized ivory?


Changes in Demand and Supply


There are two ways the quantity demanded can change1. Moves along demand curve vs. 2. Shifts of demand curve

“Demand goes up” can mean one of two things.

Move along a demand curve:Price falls, so quantity goes up

Q

P

D

A

B

Shift of a demand curve:Any P gives a higher Q

Q

P

D

D'


Factors that can shift demand curves

QD1 (P1,P2, ..., Income)

1 Prices of other goodsButter and Margerine are substitutes:Price of M increases ⇒ Increases B demand

Balls and Raquets are complements:Price of R increases ⇒ Decreases B demand

2 IncomeNormal good:An increase in income leads to an increase in demand for the good

Inferior good:An increase in income leads to a reduction in demand for the good

3 Expectations about the future4 Size of the market5 Tastes changed (as a last resort)Owen Zidar (Chicago Booth) Microeconomics Week 1 45 / 92

Factors that can shift supply curves

1 Prices of inputs

2 Technology

3 Expectations about the future

4 Number of Sellers


Supply and Demand Shifts

What determines relative changes in P and Q when conditions change?

If supply shifts out-wards, how much doP and Q move?

If demand shifts out-wards, how much doP and Q move?

Q

P

S'

S

D

Q

P

S'

S

D

Q

P

SD

D'

Q

P

S

D'

D

Relative changes driven by slope of supply and demand.


Elasticities


Elasticity (of Supply or Demand)

Price Elasticity: measure of how fast Q changes with PElastic: small change in P yields large change in QInelastic: large change in P yields small change in QWhat do elastic / inelastic Supply and Demand look like?

Q

P

S

D

Q

P

S

D

Q

P

S

D

Q

P

S

D

Informally: Inelastic = Steep, Elastic = FlatOwen Zidar (Chicago Booth) Microeconomics Week 1 49 / 92

Elasticity (Informally: Inelastic = Steep, Elastic = Flat)

Q

P

S'

S

D

Q

P

S'

S

D

Q

P

SD

D'

Q

P

S

D'

D

If demand is elastic [inelastic], a supply shift leads to...Large [small] change in quantitySmall [large] change in price

If supply is elastic [inelastic], a demand shift leads to...Large [small] change in quantitySmall [large] change in price


Price Elasticity

Price Elasticity: Percent change in Q for a given percent change in P

ε =%∆Q

%∆P

We will define two convenient notions of elasticity.

Arc Elasticity:Elasticity between two points

When we move from A to B

Point Elasticity:Elasticity at a given point

At A, or at B

Q

P

D

A

B


Basics of Price Elasticityε = %∆Q

%∆P

Elasticity is Positive for Supply curve, Negative for Demand curve

Why percent changes?

Doesn’t depend on units (dollars vs. pesos, barrels vs. gallons)Can compare across markets“A 10% increase in price changes quantity by 10 · ε%”

Classifying Supply/Demand:“Elastic”: Magnitude > 1

Perfectly elastic (flat): Magnitude of ∞

“Inelastic”: Magnitude < 1

Perfectly inelastic (vertical): Magnitude of 0


Arc Price Elasticityε = %∆Q

%∆P

Suppose US Demand curve for oil is

QD = 15000− 50P

for prices in $/bbl,quantities in 1000 bbl/day.

8000 9000 10000 11000 12000Q

80

100

120

140

160P

A

B

Say we start at an equilibrium of P = 100 and Q = 10000,then after a supply shift we move to P = 120, Q = 9000.

What are %∆Q and %∆P?What is the implied arc elasticity of demand?

Seems like elasticity can be − 12 , or − 2

3 , or something in the middle.For percent changes, should use use new or old values of P and Q?


Arc Price Elasticityε = %∆Q

%∆P

We evaluate percent change relative to average of old and new values

Consider a move (on Supply or Demand) from (P1,Q1) to (P2,Q2)

Let %∆P =P2 − P1

12P1 + 1

2P2

Let %∆Q =Q2 − Q1

12Q1 + 1

2Q2

We define Arc Elasticity as

ε =%∆Q

%∆P.

Q

P

D

�P1 ,Q1 ��P2 ,Q2 �

Why the average? Convenience: same answer in both directions.


Calculating Arc Elasticityε = %∆Q

%∆P

Calculating arc elasticity of demand:

Old price is 100, quantity 10,000

New price is 120, quantity 9,000

Percent change in price:

%∆P =120− 100

Avg Price=

20

110

Percent change in quantity

%∆Q =9000− 10000

Avg Quantity=−1000

9500 8000 9000 10000 11000 12000Q

80

100

120

140

160P

A

B

Arc Elasticity:%∆Q

%∆P=−1000/9500

20/110' −.58


Point Elasticityε = %∆Q

%∆P

Point Elasticity is about small changes from a given level

If we increase price by 1%, how much does quantity change?

Approximates Arc elasticity for very small moves

Consider an infinitesimal move from (P,Q) to (P + dP,Q + dQ)

Let %∆P =dP

PLet %∆Q =

dQ

Q

Define Point Elasticity as ε =%∆Q

%∆P=

dQ/Q

dP/P=

P

Q

dQ

dP

The notation dQdP = Q ′(P) is the derivative (slope) of the

supply/demand curve.


Calculating Point Elasticityε = %∆Q

%∆P= P

QdQdP

Calculating point elasticity of demand:

Demand is QD = 15000− 50P.

Start from P = 100, Q = 10000.

Calculate derivative: dQdP = −50

8000 9000 10000 11000 12000Q

80

100

120

140

160P

A

Point Elasticity:P

Q

dQ

dP=

100

10000· −50 = −1

2

“A 1% increase in price leads to a .5% reduction in demand”

“A 10% increase in price leads to a 5% reduction in demand”Owen Zidar (Chicago Booth) Microeconomics Week 1 57 / 92

Factors Affecting Elasticity of Demand

1 Preferences: how much do you need the good?

Electricity vs. Plastic SurgeryInelastic demand for small components of large goods – batteries

2 Availability of substitutes: More substitutes ⇒ More elasticNumber and similarity of competing products in your marketDefinition of market – broad vs. narrow

Alcoholic Beverages vs. Beer vs. Bud LightGasoline vs. Gasoline from a particular gas station

3 Short-term vs. Long-term: More elastic over long-termWhy might demand for gasoline be more elastic in the long-term?What about Canon laser printer toner cartridges? Cigarettes?


Examples of Estimated Demand Elasticities

1 Broad Food Groups

Eggs -0.06Beef -0.35Fish -0.39Juice -1.05

2 Specific Breakfast Cereals

Cap’N Crunch -2.28Fruit Loops - 2.34Cheerios -3.66

3 Specific Automobiles

Jeep Grand Cherokee -3.06Toyota Corolla -3.92

Source: Table 2.3 of Goolsbee Levitt Syverson textbook


Factors Affecting Elasticity of Supply

1 Production costs (technology, inputs)

2 Short-term less elastic than Long-term

3 Market Supply more elastic than Firm Supply


Difficulty in measuring elasticities

Suppose we observe the price of gas from month to month:Sample Data:

Prices are up 2% since last monthQuantity sold at gas stations is down 1% since last month

What is the implied price elasticity of demand?What is the implied price elasticity of supply?What can we learn from this data?

To learn about demand elasticity, find cases where only supply shifts

Individual firm: can experiment with prices to find elasticity of owndemand, not market

To learn about supply elasticity, find cases where only demand shifts


Applying what we learned today


A Unified Framework and Three Applications

We are going to incorporate supply, demand, equilibrium, and elasticitiesinto a unified framework and then use it to answer 3 questions:

1 Why is rent in San Francisco so high?

2 Why is wage inequality increasing?

3 Why aren’t prime-age men working?


Recall the two ways the quantity demanded can change1. Moves along demand curve vs. 2. Shifts of demand curve

“Demand goes up” can mean one of two things.

Move along a demand curve:Price falls, so quantity goes up

Q

P

D

A

B

Shift of a demand curve:Any P gives a higher Q

Q

P

D

D'


Two ways the quantity demanded can change (Math)

The quantity demanded can change in two ways:

%∆QD = %∆D︸︷︷︸Shift

+ εD%∆P︸︷︷︸Movement Along

%∆QD is the percentage change in the quantity demanded

%∆D is the shift in demand in percentage terms

%∆P is the percentage change in price

εD is the elasticity of demand

Note that the shift and movement along are in terms of percent changesin quantities


Two ways the quantity supplied can change (Math)

Similarly, the quantity supplied can change in two ways:

%∆QS = %∆S︸︷︷︸Shift

+ εS%∆P︸︷︷︸Movement Along

%∆QS is the percentage change in the quantity supplied

%∆S is the shift in supply in percentage terms

%∆P is the percentage change in price

εS is the elasticity of supply

Note that the shift and movement along are in terms of percent changesin quantities


Unified Framework

What do we know?

1 %∆QD = %∆D + εD%∆P

2 %∆QS = %∆S + εS%∆P

In equilibrium, the change in quantity demanded and supplied have to bethe same:

%∆QD = %∆QS

%∆D + εD%∆P = %∆S + εS%∆P


Implications for Prices and Quantities

The magnitude of price changes reflect four forces:

%∆P =%∆D −%∆S

εS − εD

We can use this price change to determine the quantity change:

%∆Q = %∆S + εS(

%∆D −%∆S

εS − εD

)%∆Q =

−εD%∆S + εS%∆D

εS − εD

Bottom line: the quantity change is a an elasticity-weighted average ofshifts in supply and demand


Application#1: Rents for San Francisco Apartments


Using Framework: San Francisco Rent Example

Suppose we have: %∆D = .4,%∆S = 0, εD = −1, εS = 0 What will

ultimately happen to prices (at the new equilibrium)?

%∆P =?

%∆P =%∆D

εS − εD

%∆P =.4

0− (−1)= .4



What will ultimately happen to quantities (at the new equilibrium)?

We have: %∆D = .4,%∆S = 0, εD = −1, εS = 0

%∆Q =?

%∆Q =εS%∆D

εS − εD

%∆Q = 0


Using Framework: San Francisco Rents in Real World

What do we know?

1 Tech booming, so %∆D large

2 Little new construction, so %∆S ≈ 0.

%∆P =%∆D

εS − εD

%∆Q =εS%∆D

εS − εD

Takeaways:

1 If supply is not very elastic, then price responses will be large

2 Reducing housing supply restrictions could effectively increase εS

3 Higher εS would lower rents and enable more people to live in SF


Evidence: Inelastic Supply Cities

Source: Trulia


Evidence: Elastic Supply Cities

Source: Trulia


Application #2: Rise in Wage Inequality (from D. Autor)

From David Autor. Science 23 May 2014: Vol. 344 no. 6186 pp. 843-851


College vs. High-school Gap in Median Earnings (D. Autor)


What do we actually observe (Katz-Murphy Example)

Quantity

Price


Supply has increased, but outpaced by demand

There’s a “race between education and technology” (Goldin and Katz)


Using Framework: %∆D > %∆S

To fix ideas, suppose the extreme case of ∆S = 0. Then we have:

%∆P =%∆D

εS − εD

%∆Q =εS%∆D

εS − εD

Takeaways:

1 When demand increases, price and quantity increase (if supply isupward sloping)

2 If supply is not very elastic, then price responses will be large

3 Rise in wage inequality partially reflects higher demand for skill1

1Note that %∆D > %∆Q > %∆S in this example. In general, you can use the

quantity expression, i.e., %∆Q = −εD%∆S+εS%∆DεS−εD

, to compare demand and supply shiftswith the magnitude of quantity changes.


Application #3: Why aren’t prime age men working?


Supply?


Demand?


Distinguishing Shifts in Supply & Demand in General

Q

P

S

D

From before:

Demand SupplyIncrease P ↑, Q ↑ P ↓, Q ↑

Decrease P ↓, Q ↓ P ↑, Q ↓

Therefore:

Price Up Price DownQuantity Up Demand ↑, Supply ? Demand ?, Supply ↑

Quantity Down Demand ?, Supply ↓ Demand ↓, Supply ?


Wrapping up

Prices convey information about values and costs to buyers andsellers, and they provide the incentive to act on that information.

Demand is determined by buyers’ willingness to pay (marginal value)for a good or service. The height of the demand curve at anyquantity measures marginal value.

Supply is determined by the marginal cost of providing a good orservice. The height of the supply curve at any quantity measuresmarginal cost.

Prices and quantities adjust to clear the market

These simple ideas can take you a very long way


SUPPLEMENTAL MATERIAL


Application: Short Run vs. Long Run Demand for Gas


Extending the framework to two goods: Gas & Cars

The demand system for gas and cars is:2

%∆GAS = a%∆PGAS + b%∆PCAR

%∆CAR = c%∆PGAS + d%∆PCAR

The long-run elasticity is a, but in short-run people can’t adjust as muchso there is an indirect impact from the second term, b%∆PCAR , when theprice of gas changes.

2Note that a = εG ,G , b = εG ,C , c = εC ,G , and d = εC ,C . Assumed no income growth.Owen Zidar (Chicago Booth) Microeconomics Week 1 87 / 92


This indirect impact can come from the demand side. Let %∆CAR = 0.


0 = c%∆PGAS + d%∆PCAR ⇒ %∆PCAR =−cd

%∆PGAS

The overall short-run impact of changes in gas prices reflects two forces:

%∆GAS = a%∆PGAS︸︷︷︸direct effect

+ b

(−cd

%∆PGAS

)︸︷︷︸

indirect effect

Takeaways:

1 Gas price declines can bid up the price of cars, which can reduceoverall responsiveness of %∆GAS to %∆PGAS in the short-run

2 The magnitude of the difference depends on the strength ofcomplementarity (which comes from b and c)



The supply side will respond to high car prices eventually too

Let %∆CAR = εS%∆PCAR .


εS%∆PCAR = c%∆PGAS + d%∆PCAR ⇒ %∆PCAR =c

εS − d%∆PGAS

The overall short-run impact of changes in gas prices reflects two forces:

%∆GAS = a%∆PGAS︸︷︷︸direct effect

+ b

(c

εS − d%∆PGAS

)︸︷︷︸

indirect effect

Takeaways:1 The indirect effect on gas depends on the supply elasticity of cars.

2 If εS = 0, then we get the result from the last slide.

3 If εS =∞, then %∆PCAR = 0 and εLR = a.



1 Suppose Initially:

QD0 = 4− P (1)

QS0 = 2 (2)

⇒ S0 = 2,D0 = 2,P0 = 2 (3)

2 Then there is a shift in demand:

QD1 = 5− P (4)

3 What is the demand now at initial prices?

QD1 = 5− P = 5− 2 = 3 (5)



What are the components of our framework?

%∆D =3− 2

12 3 + 1

2 2=

1

2.5= .4 (6)

%∆S =2− 2

12 2 + 1

2 2= 0 (7)

εD =dQD

dP

P

Q= −1

2

2= −1 (8)

εS =dQS

dP

P

Q= 0

2

2= 0 (9)



Does this line up with what ultimately happens to P and Q?

QD0 = 4− P (10)

QS0 = 2 (11)

⇒ S0 = 2,D0 = 2,P0 = 2 (12)

QD1 = 5− P (13)

QS1 = 2 (14)

⇒ S1 = 2,D1 = 2,P1 = 3 (15)

%∆P =3− 2

12 3 + 1

2 2= .4 (16)

%∆Q =2− 2

12 2 + 1

2 2= 0 (17)


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