Demand Theory n V =  (TR t - TC t )/(1 + i ) t t=1 n V =  (P X Q t - TC t )/(1 + i) t t=1

Objectives: Understand revenue portion of valuation model n Demand functions and demand curves n Important demand determinants n 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? n 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 demand a. price b. income c. price of complements and substitutes d. advertising e. demographics and socioeconomic characteristics f. stock of existing goods g. price expectations h. tastes... We can set price and advertising!

Two forms of demand functions n logarithmic or multiplicative n linear

Cigarette Demand Function Example (Logarithmic or Multiplicative Form) ln C t = ln y t ln p t ln A t or in logarithmic or multiplicative form: C t = y t p t A t where C t = per capita consumption of cigarettes in year t y t = real per capita income in year t p t = real retail price of cigarettes in year t A t = 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 n Income elasticity is a 10% increase in income increase demand by 10 X or % a 1% increase in income increase demand by 1 X or % n Price elasticity is (inelastic because it is less than one in absolute value) raising prices raises revenues n Advertising elasticity is 0.032

Regression Analysis: Demand for Beer ln Q = ln P b lnP w ln P c ln M ln A b ln POP D ln Q t-1 Q = e.793 P b P w.311 P c.358 M.08 A.022 POP e.036D 75 Q t Q = beer consumption Pb = beer price Pw = whiskey price Pc = cola price M = income A = advertising POP = population D75 = 1 if >1975 Q t-1 = last year demand 0 <1975

Interpretation of Beer Demand Function Q = e.793 P b P w.311 P c.358 M.08 A.022 POP e.036D 75 Q t Demand for beer is price inelastic (overall) This is not the same as a firm’s elasticity 2. Whiskey and Cola are substitutes 3. Slightly positive income elasticity (Beer does not share proportionately with economy’s growth) 4. Ad elasticity (like most products) is low 5. Beer is highly sensitive to Population changes effect increases consumption 3.6% 7. Lagged effect indicates substantial habit persistence

Linear Airline Demand Model Quarterly Revenue Passenger-Miles Q = 38,266, ,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

To find the demand curve from a demand function, set nonprice variables Suppose demand is determined only by price and (the square root of ) advertising expenditures: Q = P + A.5 where Q is the quantity demanded per period of time, P is price, and A is advertising expenditure level

Demand Curve Demand curve: quantity demanded as a function of price holding all other variables constant Example: a. If the demand function is Q = P + A.5 and A = 4, then the demand curve is Q = P or P = Q b. If the demand function is Q = P + A.5 and A = 36, then the demand curve is Q = P or P = Q

Demand Curve- relationship between price and quantity given other determinants A change in a non-price variable like Advertising shifts the demand curve

n The consumer surplus is the difference between the height of the demand curve and the actual price charged. It is the dollar estimate of the consumer’s net benefit of consumption. n The demand curve is an average revenue curve if firms charge a single price to all customers, n The height of the demand curve measures a consumer’s willingness to pay or the gross benefit of consuming a particular unit of the good. Consumer Surplus, Average Revenue and Willingness to Pay Q P Price Ave. Revenue

Demand Curve: P = 6 -.5Q If Price = 4, Consumer Surplus =.5(6-4)(4-0)=4 Price Quantity Price New Price

Demand Elasticities Definition 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 Q Over 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 Elasticity Price Quantity If P = 4 – Q, then Q = 4 – P 3 1 Point Price Elasticity (  ) = (dQ/dP) (P/Q) = (-1) (3/1) = 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) Demand Curve

Arc Price Elasticity Example P 2 = $49.5, P 1 = $50.5, Q 2 = 20.5, Q 1 = 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)

Linear Demand Curve Elasticities 1. Elasticities vary along a linear demand curve, P = a - bQ if P = a, elasticity = -  P = a/2, elasticity = -1 P = 0, elasticity = 0 Remember Quantity is generally thought to be a function of Price, Q = f(P) If ‘a’ is the intercept, the elasticity is -[Price/(a-Price)] P Q a a/2 Elas = -1

Marginal Revenue when charging a single price to all customers n 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 n If the demand curve is linear P = 8 - Q, TR = P X Q = 8 Q - Q 2 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

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 ) e P A. If e P = -1 (unitary elasticity), MR = 0 Revenue unaffected by a price change B. If e P 0 Revenue increases with a price decrease C. If e P > -1 (-.5, -.3, …) (inelastic), MR < 0 Revenue decreases with a price decrease

Price Elasticity and Marginal Revenue Along Linear Demand and Total Revenue Curves Demand Curve or Ave. Rev. MR TR Q Q $ $/Q 0 Ep = -1 Elastic Inelastic Price Total Revenue MR

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 markup where the markup = [e p /(e p +1)]: Price = MC [e p /(e p +1)] if consumers are price sensitive, low markup if consumers are less sensitive, higher markup

C. Examples of optimal pricing rules: 1. Linear Demand Curve: set MR = MC P = 10 - Q TR = Total Revenue = P Q = 10 Q - Q 2 MR = dTR/dQ = Q TC = 5 Q MC = dTC/dQ = $5 MR = MC MR = Q = 5 = MC, Q = 5/2 = 2.5, P = 15/2 = $7.50 Price Q 10 5 MR MC5 $ Maximum Profit Cont.

5. Optimal Pricing Rules in terms of elasticities A. For profit maximization: A.MR = P(1 + 1/e P ) = P[(e P + 1)/e P ) B.MR = P[(e P + 1)/e P ) = MC B. Therefore: P = [e P /(e P +1)] MC

Q = 5P -1.2 A.8 and the price elasticity is -1.2 TC = 5 Q and MC = $5 Profit maximizing P = [e p /(e p +1)] MC = [-1.2/( )] 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) = $10 The 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. Q Q P P MC Demand MR

Point and Arc Advertising Elasticities Point Advertising Elasticity: e A = (  Q/  A) (A/Q) = 2 (4/20) =.4 Arc Advertising Elasticity: given two points (Q1,A1) and (Q2,A2): E A = (Q2 - Q1) (A2 + A1) (Q2+ Q1) (A2 - A1) = (22-20)(5+4) =.43 (22+20)(5-4) A Q A A1,Q1 4,20 A2,Q2 5,22 Q Q = A

1. Optimal Advertising Policy: Profits = revenues - production costs - distribution costs = P Q - C - A where P = price Q = Q(P,A) = quantity demanded C = 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.

1. Expand advertising as long as the marginal profit contribution of an additional unit of output is greater than the change in advertising expenditures needed to generate the additional unit of output. P - dC/dQ =  /  Q For profit maximization given price

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.

Same Rule in More General Elasticity Terms Skip to the simpler rule below Rearranging terms and multiplying every term by (A/Q) P [(  Q/  A)(A/Q)] = (  C/  Q)[(  Q/  A)(A/Q)] + 1 (A/Q) P e A = MC e A + 1 (A/Q) [(P - MC)/P] e A = A/PQ for any price 1. Advertising expenditures should be increased until advertising expenditures as a percentage of total revenue (A/PQ) are equal to the percentage markup over price [(P - MC)/P] times the advertising elasticity e A.

Interpretation Skip [(P - MC)/P] e A = A/PQ for any price 1. The greater the advertising elasticity, the more we should advertise 2. The greater the price markup over marginal cost, the more we should advertise

If the firm is also charging the profit maximizing price such that (P-MC)/P = -e P -1, Advertising revenues as a percentage of total revenues (A/PQ) should be equal to the advertising elasticity (e A ) divided by (minus) the price elasticity (- e P ): A/PQ = -e A /e P Example: Q = 5 P Q -1.2 A.8, Advertising should be 2/3 (.8/1.2) of total revenue Skip to below

Alternative Rule: P(dQ/dA) = -e P Increase advertising until the marginal revenue of an additional dollar of advertising equals the (negative of the) price elasticity. Profit maximizing level of advertising

When neither the price nor the marginal cost of production will change as a result of small changes in advertising, advertising outlays should be carried to the level where the marginal revenue obtained from an additional dollar of advertising is equal to the price elasticity of demand. We are assuming the firm is charging the profit maximizing price. We assume that the price elasticity is greater than one in absolute value.

Example of Optimal Advertising: Q=48P -2 A.5, P=2, and TC = Q+A. We will advertise $25, $36, or $49. PA QTRTC  MR of advertising=  Total Revenue/  Total Ad. Cost = ( )/(36-25) = 24/11 > 2 = -Price Elasticity. Therefore, increase advertising to (Profit max.) MR of advertising= ( )/(49-36) =24/13 < 2 = -Price Elasticity. Therefore, do not increase advertising to

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

Setup of Problem 11-17, p. 420

Other Elasticities: Income Elasticity Point e I = (  Q/  I) (I/Q) or Arc E I = (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 X e cX =  Q w /  P x (P x /Q w ) where Qw is the quantity of one good and Px is the price of another good It 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 (e cx > 0), x and w are substitutes b. zero ( = 0), independent c. negative( < 0), complements

Cross Elasticity Example: Q x1 = 200 when P w1 = 5 Q x2 = 120 P w2 = 4 E cx = ( ) (5+4) = 2.25 ( ) (5-4) X and W are Substitutes