BUS 525: Managerial Economics

Lecture 3

Quantitative Demand Analysis

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

The Elasticity Concept

• Elasticity is a measure of the responsiveness of a variable to a change in another variable: the

percentage change

in one variable that arises due to a given

percentage change

in another variable – How responsive is variable “G” to a change in variable “S”?

E G

,

S

 % 

G

% 

S

If

E G,S

> 0, then

S

and

G

are directly related.

If

E G,S

< 0, then

S

and

G

are inversely related.

If

E G,S

= 0, then

S

and

G

are unrelated.

The Elasticity Concept Using Calculus

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

E G

,

S



dG dS S G

If

E G,S

> 0, then

S

and

G

are directly related.

If

E G,S

< 0, then

S

and

G

are inversely related.

If

E G,S

= 0, then

S

and

G

are unrelated.

Own Price Elasticity of Demand

A measure of the responsiveness of the demand for a good to changes in the price of that good: the percentage change in the quantity demanded of the good divided by the percentage change in the price of the good

E Q X

,

P X

 % 

Q X

% 

P X d

• Negative according to the “law of demand.” Elastic:

E Q X

,

P X

 1 Inelastic:

E Q X

,

P X

 1 Unitary:

E Q X

,

P X

 1

Perfectly Elastic & Inelastic Demand

Price Price D D Quantity Perfectly Elastic (

E Q X

,

P X

  ) Quantity Perfectly Inelastic (

E Q X

,

P X

 0 )

P

100

Elasticity, Total Revenue and Linear Demand

TR

0 10 20 30 40 50

Q

0

Q

P

100 80

Elasticity, Total Revenue and Linear Demand

TR

0 10 20 30 40 50

Q

800 0 10 20 30 40 50

Q

Elasticity, Total Revenue and Linear Demand

P

100 80 60

TR

1200 0 10 20 30 40 50

Q

800 0 10 20 30 40 50

Q

Elasticity, Total Revenue and Linear Demand

P

100 80 60 40

TR

1200 0 10 20 30 40 50

Q

800 0 10 20 30 40 50

Q

Elasticity, Total Revenue and Linear Demand

P

100 80 60 40 20 0 10 20 30 40 50

Q TR

1200 800 0 10 20 30 40 50

Q

Elasticity, Total Revenue and Linear Demand

P

100 80 60 40 20 Elastic 0 10 20 30 40 50

Q TR

1200 800 0 10 20 Elastic 30 40 50

Q

Elasticity, Total Revenue and Linear Demand

P

100 80 60 40 20 Elastic Inelastic 0 10 20 30 40 50

Q TR

1200 800 0 10 Elastic 20 30 40 50

Q

Inelastic

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.

Elasticity, Total Revenue and Linear Demand

P

100 80 60 40 20 0 10 20 30 40 50

Q TR

Elastic Unit elastic Inelastic 1200 800 Unit elastic 0 10 Elastic 20 30 40 50

Q

Inelastic

Class Exercise I

• Research department of an airline estimates that the own price elasticity of demand for a particular route is -1.7. If the airline cuts price by 5 percent, will the ticket sales increase enough to increase overall revenues?

• If so, by how much?

Factors Affecting Own Price Elasticity

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

• Broader categories of goods have more inelastic demand than more specifically defined categories.

– 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.

Some Elasticity Estimates

Table 3-2 Selected Own Price Elasticities Market Transportation Motor vehicles Motorcycles and bicycles Food Cereal Clothing Women’s clothing Own Price Elasticity -0.6

-1.4

-2.3

-0.7

-1.5

-0.9

-1.2

Table 3-3 Selected Short and Long-Term Own Price Elasticities Market Short-Term Own Price Elasticity Long-Term Own Price Elasticity Transportation Food Alcohol and tobacco Recreation Clothing -0.6

-0.7

-0.3

-1.1

-0.9

-1.9

-2.3

-0.9

-3.5

-2.9

1-18

The Arc Price Elasticity of Demand

How can the percentage changes in Q and P be calculated in order to derive the own price elasticity of demand ?

E QX,PX  Q  Q -------------- (Q 1 + Q 2 )/2 ------------- (Q 1 + Q 2 ) =------------------ = -------------------  P ------------- (P 1 + P 2 )/2  P ------------ (P 1 + P 2 )

• • Class Exercise II

Consider a Demand Curve Q = 40,000,000 - 2,500P Calculate arc elasticity of demand from the given data

16,000 Price P 2 =12,500 P 1 =12,000 0 B A 8,750,000 10,000,000 40,000,000 Q

How sensitive are consumers to a change in the avg. price of automobiles?

We calculate the arc price elasticity of demand between A and B as:

E p

= 10,000,000-8,750,000 ----------------------------- (10,000,000+8,750,000)/2 -------------------------------- = - 3.267 12,000 - 12,500 ---------------------- (12,000 + 12,500)/2

Interpretation

  Between points A and B (or between the price range from $12,000 to $12,500), a one-percent increase in the average price of cars will bring about, on average, a reduction of sales by 3.267%, ceteris paribus.

Because the price elasticity of demand is calculated between two points on a given demand curve, it is called the arc price elasticity of demand.

Caveat

• Elasticity measure depends on the price at which it is measured.

• It is not generally a constant (because the demand curve is not likely to be a straight line).

The Point Price Elasticity of Demand It measures the price elasticity of demand at a given price or a particular point on the demand curve.

e p  Q  P P Q

Class Exercise III

Q x d = -2,500P x 0.05A

X + 1,000M + 0.05P

Y - 1,000,000H+ Other things being equal, if P

1 = $12,000

, Q 1 = 10,000,000 .

• Calculate point elasticity of demand.

• What's the point elasticity of demand at P 2 = $12,500?

• Calculate arc elasticity of demand.

Calculation of the point elasticity using the demand for automobile equation Q x d = -2,500P x + 1,000M + 0.05P

Y Other things being equal, - 1,000,000H+ 0.05A

X if P

1 = $12,000

, Q 1 = 10,000,000 The point price elasticity is: .

e p  Q P  P Q = (-2,500)(12,000/10,000,000) = - 3

Point price elasticity (cont.)

What's the point elasticity of demand at P 2 = $12,500?

At this price, Q = 8,750,000. Hence, e p  Q P = (-----) (---)  P Q = (-2,500)(12,500/8,750,000) = - 3.571

Two versions of the elasticity of demand – Point vs. Arc Price

16,000 12,500 12,000 e p = -3.571

E p

= -3.267

e p = -3.0

Q 8,750,000 10,000,00 0

From Concept to Applications

We began with a definition of the elasticity of demand based on, E Qx, Px %  in Q x d = -------------- %  in P x If we know the price elasticity of demand (E p ), the formula will let us answer a number of "what if" questions.

Examples

(1) How great a price reduction is necessary to increase sales by 10%?

(2) What will be the impact on sales of a 5% price increase?

(3) Given marginal cost and price elasticity information, what is the profit-maximizing price?

Class Exercise IV

Supposing that the elasticity of demand for diesel is -0.5, how much prices must go up to reduce gasoline use by 1%?

The price increase needed to reduce diesel consumption by 1% Supposing that the elasticity of demand for diesel is -0.5, how much prices must go up to reduce gasoline use by 1%?

- 0.01

- 0.5 = ---------- , %  P d %  P d = (-0.01/-0.5) = + 0.02 or 2%

Marginal Revenue and the Own Price Elasticity Of Demand • Demand and marginal revenue – For a linear demand curve marginal revenue curve lies exactly halfway between the demand curve and the vertical axis – Marginal revenue is less than the price of each unit sold – When demand is elastic (-∞

Cross Price Elasticity of Demand

A measure of the responsiveness of the demand for a good to changes in the price of a

related good

: the percentage change in the quantity demanded of the good divided by the percentage change in the price of a related good

E Q X

,

P Y

 % % 

Q X



P Y d

If

E QX,PY

> 0, then

X

and

Y

are substitutes.

If

E QX,PY

< 0, then

X

and

Y

are complements.

Income Elasticity

A measure of the responsiveness of the demand for a good to changes in consumer income: the percentage change in the quantity demanded divided by the percentage change in income

E Q X

,

M

 % 

Q X

% 

M d

If

E QX,M

> 0, then

X

is a normal good.

If

E QX,M <

0, then

X

is a inferior good.

Some Elasticity Estimates

Table 3-4 Selected Cross-Price Elasticities Transportation and recreation Food and recreation Clothing and food Cross-Price Elasticity -0.05

-0.15

-0.18

1-36

Table 3-5 Selected Income Elasticities Transportation Food Ground beef, nonfed Income Elasticity 1.80

0.80

-1.94

Table 3-6 Selected Long-Term Advertising Elasticities Advertising Elasticity Clothing Recreation 0.04

0.25

Other Elasticities

Advertising elasticity A measure of the responsiveness of the demand for a good to changes in advertising expenditure: the percentage change in the quantity demanded divided by the percentage change in advertising expenditure

E Q X

,

A

 % 

Q X

% 

Ax d

Class Exercise V

• Advertising elasticity of recreation : 0.25

• How much should advertising increase to increase the demand for recreation by 15%?

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!

Example 1: Pricing and Cash Flows

• According to a BTRC Report by Zahid Hussain, BTCL’s own price elasticity of demand for long distance services is -8.64. • BTCL needs to boost revenues in order to meet it’s marketing goals.

• To accomplish this goal, should BTCL raise or lower it’s price?

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 BTCL.

Example 2: Quantifying the Change

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

Answer

• Calls would increase by 25.92 percent!

E Q X

,

P X

  8 .

64  % 

Q X

% 

P X d

  8 .

64 3 %     % 

Q X

 3 % 8 .

64  

d

% 

Q X d

% 

Q X d

 25 .

92 %

Example 3: Impact of a change in a competitor’s price • According to an BTRC Report by Zahid Hussain, BTCL’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 BTCL’s services?

Answer

• BTCL’s demand would fall by 36.24 percent!

E Q X

,

P Y

 9 .

06  % 

Q

% 

P Y X d

9 .

06  % 

Q X

 4 %

d

 4 %  9 .

06  % 

Q X d

% 

Q X d

  36 .

24 %

Interpreting Demand Functions

• Mathematical representations of demand curves.

• Example:

Q X d

 10  2

P X

 3

P Y

 2

M

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

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

Linear Demand Functions

• General Linear Demand Function:

Q X d

  0  

X P X

 

Y P Y

 

M M

 

H H E Q X

,

P X

 

X

Own Price Elasticity

P X Q X E Q X

,

P Y

 

Y P Y Q X

Cross Price Elasticity

E Q X

,

M

 

M M

Income Elasticity

Q X

Class Exercise 6

• Given the demand curve, Q x d = 100- 3P x +4P y -.01M+2A x • If P x =25, P y = 35, M= 20,000, A x =50 • Calculate (a) own price, (b) cross price, and © income elasticity of demand

Example of Linear Demand

• Q x d = 100- 3P x +4P y -.01M+2A x .

• Own-Price Elasticity: (-3)P x /Q x .

• If P x =25, P y = 35, M= 20,000, A x =50 • Q=65 [since 100 – 3(25) +4(35) -.01(20,000)+2(50)] = 65 • Own price elasticity of demand at P x =25, Q=65:

E Q X

,

P X

=(-3)(25)/65= - 1.15

• Cross price elasticity of demand at P y =35, Q65

E Q X

,

P y

=(4)(35)/65= 2.15

• Income elasticity of demand at M=20,000

E Q X

,

M

=(-0.1)(20,000)/65= -3.08

Elasticities for Nonlinear Demand Functions • Q x d = c P x βx P y βy M βM H βH • General Log-Linear Demand Function: ln

Q X d

  0  

X

ln

P X

 

Y

ln

P Y

 

M

ln

M

 

H

ln

H

Own Price Elasticity Cross Price Elasticity Income Elasticity : : :  X  Y  M

Example of Log-Linear Demand

• ln(Q d ) = 10 - 2 ln(P).

• Own Price Elasticity: -2.

Graphical Representation of Linear and Log-Linear Demand

P P Linear D Q Log Linear D Q

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.

An Example

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

ln

Q x

  0  

x

ln

P x



e

Summary Output

Regression Statistics

Multiple R R Square Adjusted R Square Standard Error Observations 0.41

0.17

0.15

0.68

41.00

ANOVA Regression Residual Total Intercept ln(P)

df

1.00

39.00

40.00

SS

3.65

18.13

21.78

Coefficients Standard Error

7.58

1.43

-0.84

0.30

MS

3.65

0.46

F

7.85

Significance F

0.01

t Stat

5.29

-2.80

P-value

0.000005

0.007868

Lower 95%

4.68

-1.44

Upper 95%

10.48

-0.23

Interpreting the Regression Output

• The estimated log-linear demand function is: – ln(Q x ) = 7.58 - 0.84 ln(P x ).

– 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.

– R-square of .17 indicates we explained only 17 percent of the variation in ln(Q x ).

– F-statistic significant at the 1 percent level.

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.

Lessons:

(1) The first lessons in business: Never lower your price in the inelastic range of the demand curve. Such a price decrease would reduce total revenue and might at the same time increase average production cost.

(2)When the demand is inelastic, raise the price to increase revenue and, possibly, profit.

(3)When demand is elastic, price increases should be avoided.

Lessons

(Cont.) (4)But should we always cut price when the demand is elastic? Even over the range where demand is elastic, a firm will not necessarily find it profitable to cut prices; the

profitability of such an action depends on whether the marginal revenues generated by the price reduction exceed the marginal cost of the added production.

Another Example: Optimal Pricing Step 1 – Using the relationship between MR and E p Given, TR = PQ,  TR  Q  (PQ) MR = ------- =--------  Q  = P(-----) + Q (-----)  Q Q   P Q Q P   P = P (1 + ----- -----) = P ( 1 + ----) Q 1 e p

Optimal Pricing (Cont.)

Optimal Price is when MC = MR i.e., MC = P (1 + 1/e p ) MC P = ------------ (1 + 1/e p ) That is, the profit-maximizing price is determined by MC and e p

Predicting Revenue Changes from Two Products

Suppose that a firm sells to related goods. If the price of X changes, then total revenue will change by: 

R

  [

R X

 1 

E Q X

,

P X

 

R Y E Q Y

,

P X

]   % 

P X