# Review of Demand Elasticities

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```					          Lecture 5:
Elasticity of Demand and Supply

Text: Chapter 5 ( pages 101-105)
Responsiveness of the Quantity Demanded to
Price Changes

   Earlier, we indicated that, ceteris paribus, the quantity of a
product demanded will vary inversely to the price of that
product. That is, the direction of change in quantity
demanded following a price change is clear.

   What is not known is the extent by which quantity
demanded will respond to a price change.
 To measure the responsiveness of the quantity

demanded to change in price, we use a measure called
PRICE ELASTICITY OF DEMAND.
Own Price Elasticity of Demand

   The price elasticity of demand is a measure of the
responsiveness of quantity demanded to a price change
   Own Price Elasticity of Demand: The percentage change in
the quantity demanded relative to a percentage change in its
own price.
Q P Q P
ED               
Q    P P Q
   For a smooth (differentiable) demand curve, the price
elasticity of demand is given by
Q P
ED    
P Q
Using Price Elasticity of Demand
   Elasticity is a pure ratio independent of units.

   Since price and quantity demanded generally
move in opposite direction, the sign of the
elasticity coefficient is generally negative.

   Interpretation: If ED = - 2.72: A one percent
increase in price results in a 2.72% decrease in
quantity demanded
Own Price Elasticity of Demand (ED)

    A numerical example

Quantity   Q2-Q1   Price   P2-P1                    ED

1               125

2         1     100      -25    (1/-25)X(125/1) = - 0.04x125 = - 5

4         2      50      -50     (2/-50)x(100/2) = - 0.04x50 = - 2

5         1      10      -40    (1/-40)x(50/4) = - 0.025x12.5 = - 0.3
Price Elasticity of Demand …

   Classifications of Own Price Elasticity of Demand:
 Inelastic demand ( |ED| < 1 ): a change in price brings about a
relatively smaller change in quantity demanded (ex. gasoline).
 Total Revenue = P×Q rises as a result of a price increase

   Unitary elastic demand ( |ED| = 1 ): a change in price brings
about an equivalent change in quantity demanded.
 TR= P×Q remains the same as a result of a price increase

   Elastic demand ( |ED| > 1 ): a change in price brings about a
relatively larger change in quantity demanded (ex. expensive
wine).
 TR = P×Q falls as a result of a price increase
Total Revenue and ED

Price Inelastic Demand   Price Elastic Demand
| ED | < 1                | ED| > 1

P                       P

Q                     Q

TR                    TR
Price Elasticity along
Linear Demand Curves
   Linear Demand Curve:                        P
Q = a – bP                            a/b
   Price elasticity of this demand
Ed = (∂Q/ ∂P)(P/Q) = − b(P/Q)            a/2b              M
   Any downward sloping demand
curve has a corresponding
inverse demand curve.                       0                             Q
a/2       a
   Inverse linear Demand Curve:
P = a/b – (1/b)Q
   At P= a/b, Ed = − ∞; at P = 0, Ed = 0; at P= a/2b, Ed = −1
   In the region of the demand curve to the left of the mid-point M, demand is
elastic, that is − ∞ ≤ Ed < – 1
   In the region to the right of the mid-point M, demand is inelastic, – 1 < Ed ≤ 0
Constant Elasticity Demand Curve
   Another commonly used demand curve is the constant
elasticity demand curve, given by
Q = aP-b
   For this demand curve, the price elasticity of demand is
Ed = (∂Q/ ∂P)(P/Q) = − b
   Thus, the price elasticity of demand is always the same
(-b) on every point of this demand curve.
   However, such a demand curve could be elastic, or
inelastic, or unit-elastic (depending on the value of b) .
Own Price Elasticity of Demand (ED)

   Calculating Own Price Elasticity of Demand from a Demand
Function:                            Q P
 Using calculus:            Ed       
P Q
   Given a demand function:
Qb = 100 – 30 Pb – 20 Pc + .005I, where, Qb = Quantity
demanded of beer in billion 6-packs, Pb = Price of beer per 6-
pack (\$5), Pc = Price of a pack of chips (\$1), and I = Annual
household income (\$25,000).
   Qb = 100 – 30*(5) – 20*(1) + .005*(25000) = 55
   Taking partial derivative of the demand function with respect to
price and substituting values for P and Q we get:
Q P                    5
Ed              ( 30 )       2.7272
P Q                  55
Price Elasticity of Demand for some
Commodities in the US

Product           Own Price Elasticity (Demand)
Turkey                        - 1.56
Margarine                     - 0.84
Beef                          - 0.64
Cheese                        - 0.46
Potatoes                      - 0.30
Cross Price Elasticity of Demand

   Shows the percentage change in the quantity demanded of
good Y in response to a change in the price of good X.
   EDYX = % Change in QDY / % change in PX
   Algebraically:          Qy Px        Qy Px
EdYX                                
Qy       Px        Px          Qy

Read as the cross-price elasticity of demand for commodity Y
with respect to commodity X.
Units of Y demanded     Price of X         EDYX___________
60                 \$10

40                 \$12             (-20/2)x(10/60) = - 1.66
Cross Price Elasticity of Demand (Edyx)
   Calculating Cross Price Elasticity of Demand from a Demand
Function:                         Qy Px
 Using calculus:       Edyx         
Px Qy
   Given a demand function:
Qb = 100 – 30 Pb – 20 Pc + .005I, where, Qb = Quantity
demanded of beer in billion 6-packs, Pb = Price of beer per 6-
pack (\$5), Pc = Price of a pack of chips (\$1), and I = Annual
household income (\$25,000).
   Qb = 100 – 30*(5) – 20*(1) + .005*(25000) = 55
   Taking partial derivative of the demand function for beer with
respect to price of chips and substituting values for Pc and Q
we get:
Qb Pc              1
Edbc           ( 20 )      0.3636
Pc Qb             55
Classification of
Cross-price elasticity of Demand
   Interpretation:
 If Edyx = - 0.36: A one percent increase in price of chips results in
a 0.36% decrease in quantity demanded of beer

   Classification:
 If (Edyx > 0): implies that as the price of good X increases, the
quantity demanded of Good Y also increases. Thus, Y and X are
substitutes in consumption (ex. chicken and pork).
   If (Edyx < 0): implies that as the price of good X increases, the
quantity demanded of Good Y decreases. Thus Y & X are
Complements in consumption (ex. bear and chips).
   If (Edyx = 0): implies that the price of good X has no effect on
quantity demanded of Good Y. Thus, Y & X are Independent in
Income Elasticity of Demand (EI)
   Shows the percentage change in the quantity demanded of
good Y in response to a percentage change in Income.
   EI = % Change in QY / % change in I

   Algebraically:         Qy I    Qy   I
EI              
Qy   I   I    Qy

Units of Y demanded      Income             EI
100             \$1200
150              \$1600 (50/400)x(1200/100) = 1.5
Income Elasticity of Demand (EI)
   Calculating Income Elasticity of Demand from a Demand
Function:                      Qy    I
 Using calculus:      EI         
I    Qy
   Given a demand function:
Qb = 100 – 30 Pb – 20 Pc + .005I, where, Qb = Quantity
demanded of beer in billion 6-packs, Pb = Price of beer per
6-pack (\$5), Pc = Price of a pack of chips (\$1), and I =
Annual household income (\$25,000).
   Qb = 100 – 30*(5) – 20*(1) + .005*(25000) = 55
   Taking partial derivative of the demand function with
respect to income and substituting values for Q and I we
get:
Qb     I                 25000
EI               (0.005 )              2.2727
I     Qb                   55
Income Elasticity of Demand (EI)
   Interpretation:
 If EI = 2.27: A one percent increase income results in a

2.27% increase in quantity demanded of beer
   Classification:
 If EI > 0, then the good is considered a normal good

(ex. beef).
 If EI < 0, then the good is considered an inferior good

(ex. ramen noodles)
 High income elasticity of demand for luxury goods

 Low income elasticity of demand for necessary goods
Income Elasticity of Demand for Some
Commodities in the US

Product                  Income Elasticity of
Demand
Total food                       0.31
Food away from home               0.53
Fresh fruits                      0.25
Fish                              0.20
Beef                              0.10
Pork                              0.04
Managerial decisions and elasticities

   You are a marketing manager and your costs have
increased (energy, salaries) reducing your net revenues.
the effect on sales, total and net revenues.

   You are a supermarket manager and you want to offer a
10% price cut in margarine this week. You want to know
how much more margarine you need to have to satisfy
Example of price elasticity use

   Your supermarket is selling 1000 containers of
margarine a week at \$ 1.50 each. You know that the
own price elasticity for margarine is –0.8. If you
decide to reduce the price by 10%, how many more
margarine containers would you be selling that week?
Example..
Since E= Q/Q / P/P = -0.8, and
P/P=-0.10
Q/Q= -0.8 *- 0.10
Q=-0.8*-0.10 * 1000 = 80 more margarine containers to be
sold

What would be the total revenue gain?
Revenue without price reduction= 1000*1.50 = 1500
New revenue= 1080 *1.35 = 1458
Was it a good decision?
Forecasting price using price elasticity of
demand

   Example using Price Elasticity of Demand
   Elasticity of Demand for Beef: Ed = − 0.67
   Beef disappeared in quarter 3 of 2009 = 6.64 bill. Lbs
   Beef disappeared in quarter 4 of 2009 = 6.45 bill. Lbs
   % change in Eq. Quant. Dem. = ∆Q/Q = (6.45-6.64)/6.64 = − 0.029
   Ed = (∆Q/Q ) /(∆P/P )
   − 0.67 = − 0.029 /(∆P/P )
   ∆P/P = − 0.029/ − 0.67 = 0.043
    If price of beef in quarter 3 of 2010 is 277 cents/lb, then
forecasted beef price for quarter 4 of 2010 is
   P* = 277 + 0.043×277 = 277+11.98 = 288.98 cents/lb.
Equilibrium Displacement Model

   The concept if elasticity is also useful for forecasting
changes in prices and quantities resulting from supply and
demand curve shifts.
   Supply of pork decreases due to a tougher regulation on manure
treatment and disposal
   Demand for pork increases due to an increase in the price of beef

   If we know the elasticities of demand and supply, we can
calculate the changes in price and quantity demanded (or
supplied) by incorporating elasticities into a model called an
equilibrium displacement model.
Equilibrium Displacement Model…

   Demand Equation
 Qd = (price, income, tastes and preferences, expectations,

and prices of other goods)
 Demand changes (shifts) if any right hand side factor other

than price of the commodity changes
   Supply Equation
 Qs =  (price, input prices, prices of other goods,

expectations, technological change, number of producers)
 Supply changes (shifts) if any right hand side factor other

than price of the commodity changes
Equilibrium Displacement Model…

   Equilibrium: Price and quantity are determined by the
intersection of the demand and supply curves, where
 Qd = Qs (i.e., quantity demanded = quantity supplied)

   Equilibrium changes (gets displaced) if demand and/or
supply changes because of changes in any right hand side
demand and/or supply factor other than the price of the
commodity.
   Suppose that an increase in the production cost will shift
the supply curve to the left. As a result, new equilibrium
price will be higher and equilibrium quantity will be lower.
Equilibrium Displacement Model…

   We can formalize this concept and write the demand equation as
 %∆Qd = Ed×(%∆P) + Sd

   Where, Ed is the elasticity of demand
   Sd represents any exogenous demand shift – the percentage change
in quantity demanded due to a change in the value of any right
hand side variable other than own price

   Similarly, we can write the supply equation as
   %∆Qs = Es×(%∆P) + Ss
   Where, Es is the elasticity of supply
   Where, Ss represents any exogenous demand shift – the percentage
change in quantity supplied due to a change in the value of any
right hand side variable other than own price
Equilibrium Displacement Model…
   In equilibrium the percentage change in quantity demanded
must equal the percentage changed in quantity supplied, i.e.,
%∆Qd = %∆Qs
Ed×(%∆P) + Sd = Es×(%∆P) + Ss
Es×(%∆P) − Ed×(%∆P) = Sd − Ss
[Es − Ed]×(%∆P) = Sd − Ss
%∆P = [Sd − Ss]/[Es − Ed]
 Thus, once we know the values of percentage change in demand
and/or supply because of an exogenous shock, we can easily
calculate the percentage change in price
Equilibrium Displacement Model…

 %∆P = [Sd − Ss]/[Es − Ed]
   Note that the denominator [Es − Ed] is always positive, because Es is
positive and Ed is negative.
 If Sd >0 and Ss =0, then %∆P >0

   If Sd =0 and Ss >0, then %∆P <0
   If Sd >0 and Ss >0 and Sd > Ss , then %∆P >0
   If Sd >0 and Ss >0 and Sd < Ss , then %∆P <0
   Once we calculate the percentage change in price %∆P, we can
substitute that value into the demand or supply equation to calculate
the percentage change in quantity demanded or supplied
   %∆Qd = Ed×(%∆P) + Sd = Ed× [Sd − Ss]/[Es − Ed] + Sd
Equilibrium Displacement Model…

General steps in solving an equilibrium displacement model
 Step 1: Determine the values of percentage change in demand

(Sd) and supply (Ss)
   Step 2: Specify the % changes in quantity demanded and
supplied as
%∆Qd = Ed×(%∆P) + Sd
%∆Qs = Es×(%∆P) + Ss
   Step 3: Set %∆Qd = %∆Qs and solve for %∆P
   Step 4: Plug the calculated value for %∆P into the %∆Qd or
%∆Qs equation to calculate the percentage change in quantity.
Equilibrium Displacement Model…

Example: Impact of manure regulation in the pork market
 Elasticity of pork demand, Ed = −1.96

 Elasticity of pork supply, Es = 2.15

 Because of a newly introduced tighter manure regulation, pork

supply falls by 4.3%, i.e., Ss= − 4.3%
 There is no change in pork demand, i.e., Sd = 0

 Suppose that the current price for pork is \$100/cwt and the

quantity bought and sold is 1,000 cwt per day.
 What would be the new equilibrium price and quantity?
Equilibrium Displacement Model…
% change in pork price and quantity due to the supply shift
   %∆P = [Sd − Ss]/[Es − Ed] = [0 − (− 4.3)]/[2.15 − (− 1.95)]
= 4.3/4.10 = 1.05%
   Thus the pork price increases by 1.05%
   So, the new equilibrium price would be
   P* = (1+ %∆P)*P = (1+0.0105)*50 = 50.525/cwt.
   %∆Qd = Ed× [Sd − Ss]/[Es − Ed] + Sd
= (− 1.95) × [0 − (− 4.3)]/[2.15 − (− 1.95)] + 0
= (− 1.95) × 4.3/4.10 = (− 1.95) × 1.05 = 2.05%
   Thus the pork price decreases by 2.05%
   So, the new equilibrium quantity would be, Q* = (1+ %∆Q)*Q
= (1- 0.0205)*1000 = 979.5 cwt.

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