Demand Theory

n

V = (TRt - TCt)/(1 + i )t

t=1

n

V = (P X Qt - TCt)/(1 + i)t

t=1

Objectives: Understand revenue portion of valuation model

Demand functions and demand curves Important demand determinants Elasticity

• the percentage change in quantity demanded divided by the percentage change in one of its determinants (% in Q / % in X)

Why are elasticities used? Optimal Prices and Advertising Levels

• marginal concepts• elasticity concepts

Demand Functions and Curves

Demand Function- Q = f(price, income, …) relationship between quantity demanded and all principal determinants of demanda. priceb. incomec. price of complements and substitutesd. advertisinge. demographics and socioeconomic characteristicsf. stock of existing goodsg. price expectationsh. tastes...

We can set price and advertising!

Two forms of demand functions

logarithmic or multiplicative linear

Cigarette Demand Function Example (Logarithmic or Multiplicative Form)

ln Ct = 2.116 + 1.289 ln yt - 0.724 ln pt + 0.032 ln At

or in logarithmic or multiplicative form:

Ct = 8.298 yt1.289 pt

- 0.724 At

0.032

where

Ct = per capita consumption of cigarettes in year t

yt = real per capita income in year t

pt = real retail price of cigarettes in year t

At = Advertising goodwill stock in year t (Advertising expenditures depreciated at an annual rate of 0.33)

For logarithmic or multiplicative functions, coefficients or exponents are elasticities

Coefficients of log relationship are elasticities

Income elasticity is 1.289• a 10% increase in income increase demand by 10 X

1.289 or 12.89 %• a 1% increase in income increase demand by 1 X 1.289

or 1.289 % Price elasticity is -0.724 (inelastic because it is

less than one in absolute value)• raising prices raises revenues

Advertising elasticity is 0.032

Linear Airline Demand ModelQuarterly Revenue Passenger-Miles

Q = 38,266,435 + 24,136 Income + 147,987 Gas - 2,359,938 Fare - 82,914 Dscnt + ...

where Q = quarterly revenue passenger-miles (000?) Income = real disposable personal income Gas = real price of gasoline (cost competitiveness

of the private automobile) Fare = Standard airline fare per mile Dscnt = discount fare (Yield/Std. Fare)

Easy to interpret coefficient as a marginal effect but difficult to generalize about its sensitivity• A one unit increase in income increases demand by 24,136

units holding all other variables in the model constant

Quantity

Demand Curve

Demand Curve- relationship between price and quantity holding all other factors (except price) constant.

Price

Demand CurveDemand curve: quantity demanded as a

function of price holding all other variables constant

Example: a. If the demand function is

Q = 10 - 2 P + A.5 and A = 4, then the demand curve is

Q = 12 - 2 P or P = 6 - .5 Q b. If the demand function is

Q = 10 - 2 P + A.5 and A = 36, then the demand curve is Q = 16 - 2 P or P = 8 - .5 Q

Demand Curve- relationship between price and quantity given other determinants

Q= 10 - 2 P + A.5

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20Q

P

Demand Curve for A = 4Demand Curve for A= 36

A change in a non-price variable like Advertisingshifts the demand curve

Demand ElasticitiesDefinition of elasticity: (% in Q / % in X)

the percentage change in the quantity demanded (Q) divided by the percentage change in X where X is price, income, other prices, advertising.

Point on the demand function: Point Elasticities: = Q x

x QOver a range between (Q1,X1) and (Q2,X2):Arc Elasticity: E = (Q2-Q1) (x2+x1)

(x2-x1) (Q2+Q1)

Q

Q

X

X

Point Price ElasticityPrice

Quantity

If P = 4 – Q, then Q = 4 – P

3

1

Point Price Elasticity () = (dQ/dP) (P/Q) = (-1) (3/1)

= -3

4

4

At a price of 3, a ten percent increase in price will decrease quantity demanded by thirty percent.(%Q)/(%P) = -3 (%Q) = (-3)(%P)

(%Q) = (-3)(10) = -30

Arc Elasticity Example

Price

Quantity

(Quantity =19.5,Price = 50.5)

(Quantity = 20.5,Price = 49.5)

50.5

49.5

19.5 20.5

Demand Curve

Arc Price Elasticity ExampleP2 = $49.5 , P1= $50.5 , Q2 = 20.5 , Q1= 19.5

(Q2-Q1)

(Q2+Q1)/2 = (Q2-Q1)/(Q2+Q1)

(P2-P1) (P2-P1)/(P2+P1)

(P2+P1)/2

= (Q2-Q1) X(P2+P1)

(P2-P1) X (Q2+Q1)

= (-1)/(40) = (-1)(100) = -2.5

(1)/(100) (1)(40)

Marginal Revenue when charging a single price to all customers

Total Revenue is price X quantity (TR = P X Q)

Marginal Revenue is the additional revenue from selling an additional unit of output• MR = dTR/dQ = P + Q (dP/dQ)

• MR is the Price from the additional unit minus the revenues that we give up by lowering price to units that could be sold at a higher price.

Marginal revenue for a linear demand curve

If the demand curve is linear• P = 8 - Q, • TR = P X Q = 8 Q - Q2

• MR = dTR/dQ = 8 - 2Q• For a linear demand curve, the marginal

revenue curve has 1. the same intercept as demand curve

2. twice the slope

0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10

Quantity

P a

nd M

R

Demand and Marginal Revenue

MR

Demand Curve (P)

2. Relationship between price, price elasticity and marginal revenues (for all demand curves)

Total revenues = P Q where P is a function of Q

(MR) = dTR/dQ = P+Q(dP/dQ) MR = P(1 + 1 )

eP

A. If eP = -1 (unitary elasticity), MR = 0

Revenue unaffected by a price change

B. If eP < -1 (-2, -3, …) (elastic), MR > 0

Revenue increases with a price decrease

C. If eP > -1 (-.5, -.3, …) (inelastic), MR < 0

Revenue decreases with a price decrease

Two Optimal Pricing Rules

To Maximize Profit, 1. Set marginal revenue = marginal cost

Solve for quantity and then price or

2. Set Price = marginal cost X markupwhere the markup = [ep/(ep+1)]:Price = MC [ep/(ep+1)]

if consumers are price sensitive, low markupif consumers are less sensitive, higher markup

C. Examples of optimal pricing rules:

1. Linear Demand Curve: set MR = MC

P = 10 - QTR = Total Revenue = P Q = 10 Q - Q2

MR = dTR/dQ = 10 - 2 Q

TC = 5 QMC = dTC/dQ = $5

MR = MC MR = 10 - 2 Q = 5 = MC, Q = 5/2 = 2.5, P = 15/2 = $7.50

Price

Q

10

105

MR

MC5

$7.50

2.5

MaximumProfit Cont.

Q = 5P-1.2A.8 and the price elasticity is -1.2

TC = 5 Q and MC = $5

Profit maximizing P = [ep/(ep+1)] MC

= [-1.2/(-1.2 + 1)] 5 = 6(5) = $30

The profit maximizing price is $30

Pricing with Multiplicative Demand Curve or (constant) Arc Elasticity

If the price elasticity is only -2, the optimal price drops to $10

P = [(-2)/(-2+1)](5) = $10The higher the price elasticity,

the lower the optimal mark up and the lower the optimal price.

This is a variant of the inverse elasticity rule.

QQ

PP

MCMC

DemandDemand

55

30

10 MRMR

Point and Arc Advertising Elasticities

Point Advertising Elasticity:eA = (Q/A) (A/Q) = 2 (4/20) = .4

Arc Advertising Elasticity: given two points(Q1,A1) and (Q2,A2):EA = (Q2 - Q1) (A2 + A1)

(Q2+ Q1) (A2 - A1)= (22-20)(5+4) = .43 (22+20)(5-4)

A

Q

AA1,Q14,20

A2,Q25,22

Q

2

4

20

Q = 12 + 2 A

1. Optimal Advertising Policy:

Profits = revenues - production costs - distribution costs

= P Q - C - A

where P = priceQ = Q(P,A) = quantity demandedC = C(Q) = C(Q(P,A)) = production cost

A = advertising expenditures

Advertising expenditures

adds to total cost but also

stimulates demand which

increases revenues and

further increases production costs.

Two Advertising Rules

Two Rules:

(1) Increase Advertising if the profit contribution of added output is greater than the advertising expenditures necessary to generate the added output.

(2) Increase Advertising if marginal revenue of advertising is greater than (negative of the) price elasticity of demand.

Example of First Rule:

Flow Motors estimates the marginal profit contribution of selling Ford automobiles at $1,000.

If it needs to increase advertising by $500 to increase its sales by one unit, should the outlay for promotion be made?

Yes.

The Wharton Corporation estimates that the price elasticity of demand for the tennis rackets it produces is -1.9.

Wharton’s managers estimate that an additional $125,000 in advertising outlays will lead to $260,000 in additional sales. Hence, the marginal revenue from an additional dollar of advertising outlays is 2.08.

Wharton can increase its profits by increasing its advertising outlays, because the marginal revenue from an additional dollar of advertising is greater than the price elasticity.

Example of the Second Rule

Other Elasticities: Income Elasticity

Point eI = (Q/I) (I/Q) or Arc EI = (Q2-Q1) (I2+I1)

(Q2+Q1)(I2-I1)

If income elasticity is negative, the good is an inferior good. (Mass transit)

If income elasticity is positive, the good is a normal good.

If the income elasticity is greater than zero but close to zero, the good is relatively recession-proof.

If the income elasticity is greater than one, the good is a growth good.

Cross Elasticity: Relationship between the price on one product and quantity of another

Assume Good w and Good XecX = Qw/Px (Px/Qw)

where Qw is the quantity of one good and Px is the price of another goodIt is the percentage change in the quantity of w

with a percentage change in the price of X.If the cross elasticity is a. positive (ecx > 0), x and w are substitutes

b. zero ( = 0), independent c. negative( < 0), complements

Cross Elasticity Example:

Qx1 = 200 when Pw1 = 5

Qx2 = 120 Pw2 = 4

Ecx = (200-120) (5+4) = 2.25

(200+120) (5-4)X and W are Substitutes