1 ec 500 chapter 3 quantitative demand analysis. 2 headline winners of wireless auction to pay $7...

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EC 500

Chapter 3Quantitative Demand Analysis

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Headline

• Winners of Wireless Auction to Pay $7 Billion The CEO of a regional telephone company picked up the

March 14 New York Times and began reading on page D1:The Federal Government completed the biggest auction in history today, selling off part of the nation’s airways for $7 billion to a handful of giant companies that plan to blanket the nation with new wireless communications networks for telephones and computers…

The CEO read the article with interest because his firm is scrambling to secure loans to purchase of the licenses the FCC plans to auction off in his region next year.

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The region serviced by the firm has a population that is 7 percent greater than the average where licenses have been sold before, yet the FCC plans to auction the same number of licenses. This troubled the CEO, since in the most recent auction 99 bidders caught up to a total of $7 billion-an average of $70.7 million for a single license.

Fortunately for the CEO, the New York Times article

contained a table summarizing the price paid per license in 10 different regions, as well as the number of licenses sold and the population of each region. The CEO quickly entered this data into his spreadsheet, clicked the regression tool button, and found the following relation between the price of a license, the quantity of licenses available, and regional population size .

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InP= 2.23 - 1.2 InQ + 1.25 InPop

(price and population figures are expressed in millions of dollars and people, respectively):

Based on the CEO’s analysis, how much money does he expect his company will need to buy a license? How much confidence do you place in this estimate?

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Overview

I. The Elasticity Concept– Own Price Elasticity– Elasticity and Total Revenue– Cross-Price Elasticity– Income Elasticity

II. Demand Functions– Linear – Log-Linear

III. Regression Analysis

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1. The Elasticity Concept

• How responsive is variable “G” to a change in variable “S”

If EG,S > 0, then S and G are directly related.If EG,S < 0, then S and G are inversely related.

S

GE SG

%

%,

If EG,S = 0, then S and G are unrelated.

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The Elasticity Concept Using Calculus

• An alternative way to measure the elasticity of a function G = f(S) is

,G S

dGGE

dSS

If EG,S > 0, then S and G are directly related.

If EG,S < 0, then S and G are inversely related.

If EG,S = 0, then S and G are unrelated.

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Own Price Elasticity of Demand

• Negative according to the “law of demand.”

Elastic:

Inelastic:

Unitary:

,

%

%X X

d d dX X X

Q PX X X

Q Q QE

P P P

1, XX PQE

1, XX PQE

1, XX PQE

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Perfectly Elastic & Inelastic Demand

)( ElasticPerfectly , XX PQE

D

Price

Quantity

D

Price

Quantity

)0, XX PQE( Inelastic Perfectly

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Own-Price Elasticity and Total Revenue

• Elastic – Increase (a decrease) in price leads to a

decrease (an increase) in total revenue.

• Inelastic– Increase (a decrease) in price leads to an

increase (a decrease) in total revenue.

• Unitary– Total revenue is maximized at the point where

demand is unitary elastic.

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

0 010 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

0 10 20 30 40 50

80

800

0 10 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

80

800

60 1200

0 10 20 30 40 500 10 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

80

800

60 1200

40

0 10 20 30 40 500 10 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

80

800

60 1200

40

20

0 10 20 30 40 500 10 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

80

800

60 1200

40

20

Elastic

Elastic

0 10 20 30 40 500 10 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

PTR

100

80

800

60 1200

40

20

Inelastic

Elastic

Elastic Inelastic

0 10 20 30 40 500 10 20 30 40 50

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Elasticity, Total Revenue and Linear Demand with P = -2Q + 100

QQ

P TR100

80

800

60 1200

40

20

Inelastic

Elastic

Elastic Inelastic

0 10 20 30 40 500 10 20 30 40 50

Unit elastic

Unit elastic

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Another example Q = 80 – 2P (or P = 40 - 0.5Q)

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• At point B,

• At Point E,

,

10 700.14

5 5X X

d dX X

Q PX X

Q QE

P P

,

10 401.0

5 20X X

d dX X

Q PX X

Q QE

P P

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When E = -1,

• From Q = 80 – 2P (or P = 40 - 0.5Q)• Revenue = P*Q = (40 - 0.5Q)Q = 40Q – 0.5Q2

• MR = dR/dQ = 40 – Q MR = 0 implies Q* = 40.

Point: Revenue is maximized when E = -1 (implying MR = 0).

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Decision of Singapore Airlines

• Should it increase fares to boost cash flow, or adopt a “cut price and make it up in volume”?– Price elasticity is -1.7. What is your

suggestion? Why?

• If it cuts fares by 5%, how much sales will increase? -1.7 = %change in Q / 5% ; thus, %change in Q = 8.5%

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Factors Affecting Own Price Elasticity

– Available Substitutes• The more substitutes available for the good, the more elastic

the demand.

– Time• Demand tends to be more inelastic in the short term than in

the long term.• Time allows consumers to seek out available substitutes.

– Expenditure Share• Goods that comprise a small share of consumer’s budgets

tend to be more inelastic than goods for which consumers spend a large portion of their incomes.

• [Are foods more elastic than transportation?]

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Price and MR

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• Point: When E = -1, MR = 0 ; Revenue is maximized.

• Formula

If E = -1, MR = 0

If E < -1, MR > 0

If E > -1, MR < 0

1 EMR P

E

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• How was the formula derived?R = P*Q

MR = dR/dQ = P + Q*dP/dQ

= P[1 + (Q/P) (dP/dQ)]

= P[1 + 1/E] = P[(1+E)/E]

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Cross Price Elasticity of Demand

If EQX,PY > 0, then X and Y are substitutes.

If EQX,PY < 0, then X and Y are complements.

Y

dX

PQ P

QE

YX

%

%,

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Example

• You are the manager of Publix. Suppose that the price of recreation increases by 15%. Then, how it will affect the sales of foods?– Cross elasticity of food and recreation = 0.15

0.15 = %change in Qfood / 15%

Thus, %change in Qfood = 2.25%

– Is it a substitute?

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Income Elasticity

If EQX,M > 0, then X is a normal good.

If EQX,M < 0, then X is a inferior good.

M

QE

dX

MQX

%

%,

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Example

• Suppose that the income elasticity of nonfed ground beef is –1.94. If income increases by 10%, how it will affect the demand for nonfed ground beef? 1.94 = %change in Qg_beef / 10%

Thus, %change in Qg_beef = -19.4%

– Is it a normal good?

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Uses of Elasticities

• Pricing.

• Managing cash flows.

• Impact of changes in competitors’ prices.

• Impact of economic booms and recessions.

• Impact of advertising campaigns.

• And lots more!

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Example 1: Pricing and Cash Flows

• According to an FTC Report by Michael Ward, AT&T’s own price elasticity of demand for long distance services is -8.64.

• AT&T needs to boost revenues in order to meet it’s marketing goals.

• To accomplish this goal, should AT&T raise or lower it’s price?

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Answer: Lower price!

• Since demand is elastic, a reduction in price will increase quantity demanded by a greater percentage than the price decline, resulting in more revenues for AT&T.

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Example 2: Quantifying the Change

• If AT&T lowered price by 3 percent, what would happen to the volume of long distance telephone calls routed through AT&T?

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Answer

• Calls would increase by 25.92 percent!

%92.25%

%64.8%3

%3

%64.8

%

%64.8,

dX

dX

dX

X

dX

PQ

Q

Q

Q

P

QE

XX

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Example 3: Impact of a change in a competitor’s price

• According to an FTC Report by Michael Ward, AT&T’s cross price elasticity of demand for long distance services is 9.06.

• If competitors reduced their prices by 4 percent, what would happen to the demand for AT&T services?

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Answer

• AT&T’s demand would fall by 36.24 percent!

%24.36%

%06.9%4

%4

%06.9

%

%06.9,

dX

dX

dX

Y

dX

PQ

Q

Q

Q

P

QE

YX

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3. Interpreting Demand Functions

• Mathematical representations of demand curves.

• Example:

• X and Y are substitutes (coefficient of PY is positive).

• X is an inferior good (coefficient of M is negative).

MPPQ YXd

X 23210

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Linear Demand Functions

• General Linear Demand Function:

HMPPQ HMYYXXd

X 0

Own PriceElasticity

Cross PriceElasticity

IncomeElasticity

X

XXPQ Q

PE

XX,

XMMQ Q

ME

X,

X

YYPQ Q

PE

YX,

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Example of Linear Demand

• Qd = 10 - 2P.

• Own-Price Elasticity: (-2)P/Q.

• If P=1, Q=8 (since 10 - 2 = 8).

• Own price elasticity at P=1, Q=8:

(-2)(1)/8= - 0.25.

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0ln ln ln ln lndX X X Y Y M HQ P P M H

M

Y

X

:Elasticity Income

:Elasticity Price Cross

:Elasticity PriceOwn

Log-Linear Demand

• General Log-Linear Demand Function:

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Example of Log-Linear Demand

• ln(Qd) = 10 - 2 ln(P).

• Own Price Elasticity: -2.

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P

Q Q

D D

Linear Log Linear

Graphical Representation of Linear and Log-Linear Demand

P

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3. Regression Analysis

• One use is for estimating demand functions.

• Important terminology and concepts:– Least Squares Regression: Y = a + bX + e.– Confidence Intervals.– t-statistic.– R-square or Coefficient of Determination.– F-statistic.

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An Example

• Use a spreadsheet to estimate the following log-linear demand function.

0ln lnx x xQ P e

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Summary Output

Regression StatisticsMultiple R 0.41R Square 0.17Adjusted R Square 0.15Standard Error 0.68Observations 41.00

ANOVAdf SS M S F Significance F

Regression 1.00 3.65 3.65 7.85 0.01Residual 39.00 18.13 0.46Total 40.00 21.78

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 7.58 1.43 5.29 0.000005 4.68 10.48ln(P) -0.84 0.30 -2.80 0.007868 -1.44 -0.23

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Interpreting the Regression Output

• The estimated log-linear demand function is:– ln(Qx) = 7.58 - 0.84 ln(Px).– Own price elasticity: -0.84 (inelastic).

• How good is our estimate?– t-statistics of 5.29 and -2.80 indicate that the

estimated coefficients are statistically different from zero (significant).

– R-square = .17 (not much meaningful, though)

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More on Regression

• Using Excel: Example– AUCTION_DATA.XLS

• Goals of Regression– Prediction, marginal effects, and testing hypothesis

• Dummy independent variables– Differences

• Dummy Dependent Variables Models– Choice Models

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Conclusion

• Elasticities are tools you can use to quantify the impact of changes in prices, income, and advertising on sales and revenues.

• Given market or survey data, regression analysis can be used to estimate:– Demand functions.– Elasticities.– A host of other things, including cost functions.

• Managers can quantify the impact of changes in prices, income, advertising, etc.

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Back to Headline• In P = 2.23 – 1.2 In Q + 1.25 In Pop

• The coefficient of InPop(1.25) tells us the percentage change in price resulting from each 1 percent change in population.

• Since the population in the relevant region is 7 percent higher than the average, this means

1.25 = %change in P / %change in Pop 1.25 = %change in P / 7% - %change in P = 1.25 * 7% = 8.75%

In other words, the price the CEO expects to pay in his region is 8.75 percent higher than the average price paid in the March 14th auction. Since that price was $70.7 million, the expected price needed to win the auction in his region is, other things equal, $76.9 million.

• The CEO’s model predicts that the demand for licenses will be greater in his region due to the greater size of the market ultimately serviced by the holders of the licenses

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Exercises and Homework

• Chapter 3– In-class

• Q. 2, Q.7, Q. 11, Q. 13

– Homework• Q. 3, Q. 4, Q. 10 (excel regression)• Q. 12, Q. 19