CHAPTER FOUR ELASTICITY
We have seen in chapter three how a change in the price of the good results in change in quantity demanded of that good in the opposite direction (movement along the same demand curve); and how a change in income results in a change in quantity demanded at every price. The same thing is said about the changes in the price of related goods and other nonprice determinants. The question is now how to measure the magnitude of each change in quantity demanded or supplied as a response to a change in one of the independent variables. The same argument can be applied to the quantity supplied. In order to have a better picture of the degree of responsiveness of quantity to a change in one of the independent variable we have to understand the concept of elasticity.
The Economic Concept of Elasticity Elasticity is a measurement of the degree of responsiveness of the dependent variable to changes in any of the independent variables. In general elasticity is the percentage change in one variable in response to a percentage change in another variable. Elasticity =
% ∆ dependentVariable % ∆Y = = Elasticity Coefficient % ∆ IndependentVariable % ∆X
Elasticity coefficient includes a sign and a size. We need to interpret the sign and the size of the coefficient. Sign shows the direction of the relationship between the two variables. A positive sign shows a direct relationship while a negative sign shows an inverse relationship between the two variables.
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Size illustrates the magnitude of this relationship. In other words, it shows how large the response of the dependent variable to the change in the independent variable. Large elasticity coefficient means that a small change in the independent variable will result in a large change in the dependent variable (the opposite is true). Elasticity coefficient is a unitfree measure because in calculating the elasticity we use the percentage change rather than the change to avoid the difficulty of comparing different measurement units, and the percentages cancel out. Changing the units of measurement of price or quantity leave the elasticity value the same Elasticity is an important concept in economic theory. It is used to measure the response of different variables to changes in prices, incomes, costs, etc. In addition to price and income elasticities of demand, you may estimate the elasticity with respect to any of the other variables like advertisement and weather conditions. You may even measure the elasticity of production to various inputs or the elasticity of your grades in managerial economics to hours of study. This chapter covers some of the important types of elasticities.
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THE PRICE ELASTICITY OF DEMAND (Ed): In the previous chapter we have discussed the movement of the quantity demanded along a given demand curve as a result of change in the price of the good. The direction of the movements reflects the law of demand that shows an inverse (negative) relationship between P and Qd; the lower the price the greater the quantity demanded. When supply increases while demand stays
P
constant, the equilibrium price falls and the
P0
equilibrium quantity increases. But does
P2
the price fall by a large amount or a little?
P1
S0 S1
D2
And does the quantity increase by large amount or a little? The answer depends on
Q0
Q2
D1 Q1
Q
the responsiveness of quantity demanded to a change in price. We are now going to discuss the question of how sensitive the change in quantity demanded is to a change in price. The response of a change in quantity demanded to a change in price is measured by the price elasticity of demand. Price elasticity of demand (Ed) is an economic measure that is used to measures the degree of responsiveness of the quantity demanded of a good to a change in its price, when all other influences on buyers’ plans remain the same. The price elasticity of demand is calculated by dividing the percentage change in quantity demanded by the percentage change in price. Ed =
%∆Q d ∆Q d / Q d = %∆P ∆P / P
Example: Suppose P1 = 7, P2 = 8, Q1 = 11, Q2 = 10, then If P from 8 to7, Ed = 0.8 If P from 7 to 8, Ed = 0.64 You can see that the value of Ed is different depending on direction of change in P even with the same magnitude. Page 3 of 34
To solve this problem we use Arc elasticity The arc elasticity of demand is measured over a discrete interval of a demand (or a supply) curve. To calculate the price elasticity of demand (Ed): We express the change in price as a percentage of the average price—the average of the initial and new price, and we express the change in the quantity demanded as a percentage of the average quantity demanded—the average of the initial and new quantity. By using the average price and average quantity, we get the same elasticity value regardless of whether the price rises or falls. ∆Q d Q 2 − Q1 Q 2 − Q1 %∆Q d Q avg ( Q 2 + Q1 ) / 2 Q 2 + Q 1 = = Ed = = P2 − P1 P2 − P1 ∆P %∆P Pavg P2 + P1 (P2 + P1 ) / 2 =
Q 2 − Q1 P2 + P1 Q 2 − Q1 P2 + P1 × = × =− Q 2 + Q1 P2 − P1 P2 − P1 Q 2 + Q1
Where, Q1 = the original (the old) quantity demanded, Q2 = the new quantity demanded P1 = the original (the old) price, P2 = the new price Qavg = the average quantity, Pavg = the average price The formula yields a negative value, because price and quantity move in opposite directions (law of demand). But it is the magnitude, or absolute value, of the measure that reveals how responsive the quantity change has been to a price change. Thus, we ignore the minus (negative) sign and use the absolute value because it simply represents the negative relationship between P and Qd Example: Suppose P1 = 7, P2 = 8, Q1 = 11, Q2 = 10, then Ed =
10 − 11 8 − 7 ÷ = −0.71 10 + 11 8 + 7 2 2
Now how to interpret the elasticity coefficient? What Ed=  0.71 means?
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It means that if the price of the good increases (decreases) by 1% the quantity demanded of the good decreases (increases) by 0.71% Example: Price($)
Qd(bushels of Wheat)
8 20 7 40 6 60 5 80 What is the Ed if P increases from 6 to 7? Ed =
40 − 60 7 + 6 × = −2.6 60 + 40 7 − 6
A 1% increase in P would result in a 2.6% decrease in Qd Example: If a rightward shift in the supply curve leads to an increase in Qd by 10 % as a result of a decrease in P by 5%. a. Calculate Ed. Ed =
%∆Q d 10 = = −2 %∆P −5
b. Interpret Ed Ed = 2 means that a decrease in P by 1% results in an increase in Qd by 2% c. What would be the increase in Qd if P decreases by 4%? Since E d =
%∆Q d , then %∆Q d = ( E d ) ( %∆P ) = (2) (4%) = + 8 %, %∆P
Thus, a decrease in P by 4% results in an increase in Qd by 8% d. What would be the decrease in P if Qd increases by 6% Since E d =
%∆Q d 6% %∆Q d , then % ∆P = = = −3% , %∆P Ed −2
Thus, if Qd increases by 6%, P decreases by 2% However, if we want to measure Ed at a single point rather than between two
points we should use point elasticity of demand
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The point elasticity of demand measured at a given point of a demand (or a
supply) curve. It is the price elasticity for small changes in the price or for changes around a point on the demand curve. dQ d %∆Q d dQ P εd = = Q = × dP %∆P dP Q P
The point elasticity of a linear demand function can be expressed as: εd =
∆Q P × ∆P Q
Notice that the first term of the last formula is nothing but the slope of the
demand function with respect to the price. Having this fact in mind you will easily remember that:
1. The value of the elasticity; varies along a linear demand curve, as P/Q change even though, the slope is constant. 2. The value of the elasticity varies along a nonlinear demand curve as both terms in the above equation varies from as we move along a nonlinear demand curve. 3. The value of the elasticity is constant along the demand curve only in the case of an exponential function in the form: Qd = aPb, where the price elasticity of demand equals –b, which can be proved as follows: εd =
dQ P P × = −baP −b−1 × = −b dP Q aP −b
This type of nonlinear equations can be expressed in linear form using logarithm
log Q = log a – b (log P) Example:
Calculate the elasticity of demand using the following equation: Qd = 50P3, Page 6 of 34
εd =
dQ P P − 150 P × P 3 − 150 P 4 × = −150(P − 4 ) × = × = × 4 = −3 dP Q 50 50 50P −3 P4 P
Example:
Given Qd = 2000  20P, Find εd when P=70 At P=70, Qd = 2000 – 20(70) = 2000 – 1400 = 600 εd =
70 dQ P × = ( −20) × = −2.33 600 dP Q
Example:
If Qd =200  300P + 120I + 65T – 250Pc + 400Ps, and if I=10, T=60, Pc =15, Ps j=10, Find: a. Ed for the price range $10 and $11 b. Εd at P =$10 Qd = 200  300P + 120(10) + 65(60) – 250(15) + 400(10) = 200  300P + 1200 + 3900 – 3750 + 4000 Qd = 5550 – 300P a. Ed for the price range $10 and $11 (Arc Elasticity) At P=10, Qd = 5550 – 300(10) = 2550 At P=11, Qd = 5550 – 300(11) = 2250 Ed =
2250 − 2550 11 + 10 × = −1.31 11 − 10 2250 + 2550
b. Εd at P =$10 (Point Elasticity) εd =
10 dQ P × = ( −300 ) × = −1.2 2550 dP Q
Example:
Assume a company sells 10,000 units of its output at price of $100. Suppose competitors decrease their price and as a result the company’s sale decrease to 8,000 units. Ed in this pricequantity range is 2. What must be the price if the company wants to sell the same number of units before its competitors decrease their price? Using E d =
Q 2 − Q1 P2 + P1 × = −2 P2 − P1 Q 2 + Q1 Page 7 of 34
Q1=8000, Q2= 10000, P1= 100, P2=? −2= =
P2 + 100 (10,000 − 8,000 )(P2 + 100 ) 10,000 − 8,000 × = P2 − 100 10,000 + 8,000 (P2 − 100 )(10,000 + 8,000 )
2,000(P2 + 100 ) 2,000P2 + 200,000 = (P2 − 100 )(18,000 ) 18,000P2 − 1,800,000
For simplification, divide numerator and denominator of lefthand side by 1000 −2=
2P2 + 200 18P2 − 1,800
− 2(18P2 − 1800 ) = 2P2 + 200
36P2 + 3600 = 2P2 + 200 38P2 = 3400 P2 = 3400/38 = 89.5 TR1 = 100 X 8,000 = 800,000 TR2 = 89.5 X 10,000 = 895,000 Since TR2 > TR1 Ö TR Ö it is good to cut price but π is not known since we do not the TC Example:
A 50% decrease in the price of salt caused the quantity demanded to increase by10%. Calculate the price elasticity of demand for salt, explain the meaning of your result and tell if the demand for salt is elastic or inelastic? Ed = 10/50 = 0.20 which means a10% change in price results in a 2% Change in the quantity demanded in the opposite direction. Example:
Qd = 50 – P3, is the demand curve equation for apple, calculate the price elasticity of demand when P =3 and Q = 9. εd =
dQ P 3 1 × = −3(3 2 ) × = −27 × = −9 dP Q 9 3
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Categories of Demand Elasticity Using absolute value of Ed, we differentiate between five categories of elasticity
that range between zero and infinity. 1. Relatively Elastic Demand (Ed > 1) If E d =
%∆Q d > 1 ⇒ % ∆Qd > % ∆P ⇒ demand is elastic. %∆P
Consumers are very responsive to changes in P. Demand curve is flatter ⇒ 1% change in P results in a more than 1% change in Qd (in the opposite direction). (if Ed = 2 that means if P by 1% Qd by 2%.) Examples of elastic goods: cars, furniture, vacations, etc. 2. Relatively Inelastic Demand (Ed < 1) If E d =
%∆Q d < 1 ⇒ % ∆Qd < % ∆P ⇒ demand %∆P
is inelastic. Consumers are not very responsive to changes in P. Demand Curve is steeper ⇒ 1% (or ) in P results in a less than 1% (or) in Qd (if Ed = 0.70 that means if P by 1% Qd by 0.7%.) or (if P by 10% Qd by 7%.) Examples of inelastic goods: medicine, food, etc. If the price elasticity is between 0 and 1, demand is inelastic. P
More Elastic
More Inelastic Qd
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3. Unitary Elastic Demand (Ed = 1) If E d =
P
%∆Q d = 1 ⇒ % ∆Qd = % ∆P ⇒ demand is %∆P
unitelastic 1% in P results in a 1% in Qd
D
0
4. Perfectly Elastic Demand (Ed = ∞) %∆Q d = ∞ ⇒ demand is perfectly elastic If E d = %∆P
P
Qd
S1 S2 D
⇒ horizontal demand curve ⇒ the same price is
P
charged regardless of Qd (perfect competition). Qd
Any price increase would cause demand
to fall to zero. Shifts in supply curve results in no change in price. Examples: identical products sold side by side, agricultural products.
5. Perfectly Inelastic Demand (Ed = 0) If E d =
P
D
%∆Q d = 0 ⇒ demand is perfectly inelastic %∆P
S1 S2
⇒ a vertical demand curve ⇒ demand is
completely inelastic. Qd remains the same regardless of any change in price. Shifts in supply
Qd Qd
curve results in no change in Qd. Examples: medicine of heart diseases or diabetes such as insulin A good with a vertical demand curve has a demand with zero elasticity. We conclude from the five categories above that the more flatter is the demand
curve the more elastic is the demand and the more steeper is the demand curve the more inelastic is the demand Page 10 of 34
Elasticity along straight line demand curve Elasticity of demand (Ed) is not the slope of the demand curve.
Slope =
∆P , ∆Q d
Elasticity: E d =
%∆Q d %∆P
For a straightline (linear) demand curve the slope is constant (i.e., the slope is
the same at every point along the curve). It is equal to the change in price over the change in quantity demanded. Although the slope is constant, price elasticity varies along a linear demand
curve.
P
Ed = ∞
Ed > 1 Ed=1 Ed < 1 Ed = 0
Q
The following equation shows the relationship between the elasticity and the
slope of a straight line demand curve Ed =
∆Q d P %∆Q d ∆Q d / Q d ∆Q d P 1 P × = × = = × = %∆P Qd ∆P Q d slope Q d ∆P / P ∆P
Since the slope of straightline demand curve is constant, constant Ö elasticity varies as a result of variation of
1 is also slope
P ; i.e. straightline Qd
demand curve elasticity depends on the values of Qd and P
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1. When P = 0, Ed = 0 (perfectly inelastic) 2. When Q = 0, Ed = ∞ (perfectly elastic) 3. Ed increases as we move upward along a straightline demand curve (from the inelastic range to the elastic one) (as P ↑ and Q↓) 4. Ed decreases as we move downward along the straightline demand curve (as P↓ and Q↑). Thus, along downward sloping demand curve, demand is elastic when price is
high, inelastic when price is low and unitelastic at the midpoint of the demand curve.
Pricing Strategy: The Relationship between P, Ed, and TR Managers of profit maximizing firms are usually concern with the best pricing
strategy. There is a relationship between the price elasticity of demand and revenue
received. Total revenue (TR) equals the total amount of money a firm receives from the
sales of its product TR = P X Q. TR is affected by changes in both P and Qd. But as we know by now the law of
demand implies that an increase in P will result in a decrease in Qd. Thus, an increase in P may or may not lead to greater TR. This depends on
which effect is the largest, price effect or the effect of quantity demanded. The size of the price elasticity of demand coefficient, tells us which of these two
effects is largest. o If demand is elastic (Ed >1) ⇒ % ∆Qd > % ∆P 10 %↑ in P results in more than 10 %↓ in sales ⇒ TR ↓ 10 %↓ in P results in more than 10 %↑ in sales ⇒ TR ↑ o If demand is inelastic (Ed 1
E MC and Mπ is positive. The optimal production reached when MR = MC and Mπ = 0 Units produced over and above the optimal level will have negative Mπ because
for these units MR < MC When TR is maximized, MR = 0 and MC (positive value) is definitely greater
than MR. The conclusion here is that if the manager maximizes TR the firm will make less
than max profit. Ed
Demand
MR
P
TR
Ed >1
Elastic
MR >0
Ed < 1
Inelastic
MR < 0
Ed = 1
Unit elastic
MR = 0

Max.
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P ∞
Ed> 1
Ed = 1
P*
Ed< 1 0
0
Q
MR TR E = 1; E > 1;
MR=0
MR > 0
E < 1; MR< 0
TR 0
Q*
Q
The above graph shows that: o Ed >1 ⇒ Demand elastic ⇒ MR>0 ⇒ P and TR move in the opposite
direction (negative relationship) o Ed < 1 ⇒ Demand inelastic ⇒ MR < 0 ⇒ P and TR move in the same
direction (positive relationship) o Ed = 1 ⇒ Demand unit elastic ⇒ MR = 0 ⇒ TR is maximum Example:
If a company wants to ↑ its TR when Ed = 0.75, it should ↑ P Example:
If a company wants to ↑ its TR when Ed = 1.5, it should ↓ P Page 15 of 34
Example:
If Ed = 1, an ↑ in P by 15%, ⇒ Qd ↓ by 15%, ⇒ TR will not change Example:
Given Qd = 20 – 2P, Find the price range for which a. D is elastic b. D is inelastic c. D is unit elastic d. If the firm increases P to $7, is TR increasing or decreasing? Answer: Qd = 20 – 2P ⇒ P = 10 – 0.5Q TR = 10Q – 0.5Q2 MR = 10 –Q When MR = 0, 10 – Q =0 ⇒ Q = 10 and P = 10 – 0.5(10) = 5 At this P and Q, Ed = 1 a. D is elastic for price range above 5 (or Q less than 10) b. D is inelastic for price range below 5 (or Q above 10) c. D is unit elastic at P = 5 and Q = 10 d. If the firm chooses to increase the price to $7 and 7 is in the elastic part, TR will be decreasing Example:
Given Qd = 150 – 10P, find Q and P at which εd = 1 Since Qd = 150 – 10P ⇒ P = (150/10) – (1/10) Q = 15 – 0.1Q ⇒ TR = 15Q – 0.1Q2 MR = dTR/dQ = 15 – 0.2Q (Note that the slope of MR equation is twice the slope of the inverse demand equation). TR reaches maximum when MR = 0 (Q that max TR is the same as Q that makes MR= 0 Set MR =0 ⇒ 15 – 0.2Q = 0 ⇒ Q =15/0.2= 75 and P = 15 – 0.1(75) = 7.5 Page 16 of 34
So, at Q=75 and P=7.5 MR = 0 and ε d =
dQ P ⎛ 7.5 ⎞ − 75 × = ( −10 )⎜ = −1 Ö TR is maximized ⎟= dP Q 75 ⎝ 75 ⎠
Find P and Q at which Ed >1 and Ed 1 at P > 7.5, and Q < 75 Ed 75 Example: P
Q
TR MR
Ed
10
1
10


9
2
18
8
6.33
8
3
24
6
3.40
7
4
28
4
2.14
6
5
30
2
1.44
5
6
30
0
1.00
Ed = 1 (unitary elastic), TR is
4
7
28
2
0.69
max and MR is zero
3
8
24
4
0.47
2
9
18
6
0.29
1
10 10
8
0.16
Ed > 1 (elastic demand
Ed < 1 (inelastic demand
Exercise:
P
From the graph to the right a. calculate Ed b. When P increases what would happen to TR? Ed = 1 and TR remains the same. The area (05a20) = the area (04b25)
Page 17 of 34
5
a b
4 0
D 2
2
Q
MR and Elasticity The relationship between Marginal Revenue (MR), price (P), and price the
elasticity of demand (Ed), can be stated using the formula: ⎡ ⎛ 1 ⎞⎤ MR = P ⎢1 + ⎜⎜ ⎟⎟⎥ ⎣ ⎝ ε d ⎠⎦
Clearly the equation shows that if Ed < 1, MR must be positive: if Ed > 1, MR
must be negative; and if Ed = 1, MR must be zero. To proof the relationship between MR and Ed, (for your information only)
We know that TR = P X Q dP dP dQ dTR d = +Q =P+Q (PXQ) = PX dQ dQ dQ dQ dQ Multiply the second term by P/P P dP 1 dP Q dP = P(1 + XQ MR = p + XQ ) = P(1 + X ) P dQ P dQ P dQ But dQ P εd = X dP Q 1 dP Q = X So, ε d dQ P MR =
Thus, MR = P (1 +
1 ) εd
If P = 20 and εd = 4 find MR ⎡ ⎛ 1 ⎞⎤ MR = 20 ⎢1 + ⎜ ⎟⎥ ⎣ ⎝ − 4 ⎠⎦
MR = 20 (1  0.25) = 20 (0.75) = 15
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Factors Affecting Demand Elasticity Demand for some goods and services is elastic whereas for other goods and
services is inelastic. Elasticity does not only differ from one good to another but also it may differ for
a particular product at different prices. The elasticity of demand is computed between points on a given demand curve.
Hence, the price elasticity of demand is influenced by all determinants of demand. We can summarize the main factors that affect Ed as: 1. Availability and closeness of Substitutes When a large number of substitutes are available, consumers respond to a
higher price of a good by buying more of the substitute goods and less of the relatively more expensive one. So, we would expect a relatively high price elasticity of demand for goods or services with many close substitutes, but would expect a relatively inelastic demand for goods with few close substitutes. Example:
Dell computer, for example, has many substitutes. So its price elasticity of demand is highly elastic because the consumers can easily shift to the other substitutes if the price of Dell computer increases Example:
Pepsi and Coke are very close substitutes. So, the availability of Pepsi makes the price elasticity of demand of Coke very high. Any increase in the price of Coke will result in a huge shift of consumers to Pepsi’s purchase. Furthermore, the broader the definition of the good, the lower the elasticity since
there is less opportunity for substitutes. The narrower the definition of the good the higher the elasticity, since there are more substitutes.
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Example:
A buyer who likes Japanese cars and has relative preference for Toyota products may have higher price elasticity of demand for Camry than the price elasticity of demand for Toyota cars. His price elasticity of demand for Toyota cars is higher than the price elasticity of demand for Japanese cars. And his price elasticity of demand for Japanese cars is higher than the price elasticity of demand for cars in general. Why? Example:
Consider the relative price elasticity of demand for a good such as apples compared to a good such as fruits. What is the difference between apples and fruits? Apples are, of course, a fruit but so are lots of other goods as well. Hence, more substitutes exist for apples than exist for the broader category of fruits. We have already determined that as the number of substitutes increase then so does that goods relative price elasticity of demand. 2. Proportion of total expenditures to Income The higher the proportion of income spent on the good, the higher the elasticity
of demand. Expensive good take a greater proportion of an individual’s income and expenditures than the inexpensive goods; so expensive good are more elastic. Example:
Consider the price elasticity of demand for a good such as a pen compared to that for a good such as a car. One of the big differences between these two types of goods is that the price of a pen is small as a proportion of the income while the price of a car is typically a large percentage of income. Doubling the price of pens will not, therefore, have a big impact on one’s income. However, doubling the price of cars will have a large impact on one’s income. Thus, the demand for highpriced goods such as cars tends to be more price elastic than the demand for lowpriced good such as bread or salt.
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3. The Time Elapsed Since Price Change (Length of Time) Over time, demand tends to be more elastic because time is available to search
for substitutes for a good when a longer time period is considered. Example:
Consider what happens as the price of a good such as gasoline doubles. People respond to the higher price by decreasing their use of gas. However, in just a short time period it is more difficult to do this than in a longer period. Essentially, the longer the time period people have to adjust, the more alternatives they can find to reduce their consumption of gas. For example, they might be able to move closer to work, buy a more fuelefficient car, use public transportation, arrange with friends to go in on car, etc. Thus, in short run, the response is very limited ⇒ demand is less elastic; over
time, demand tends to be more elastic because time is available to search for substitutes and adjust to the new situation 4. Necessary vs. Luxury goods Demand for necessary goods, goods that are critical to our everyday life and
have no close substitute, is relatively inelastic (food, medicine). Demand for luxury goods, goods with many substitutes and we would like to
have but are not likely to buy unless our income jumps or the price declines sharply, is relatively elastic (cars, traveling to foreign countries for vacation). Nevertheless, what is one person's luxury is another person's necessity 5. Durability of the product: The demand for durable goods (such as cars) tends to be more price elastic
than the demand for nondurable goods, such as foods. This is because durable goods have the possibility of postponing purchase,
have the possibility of repairing the existing ones, and the possibility of buying used ones.
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As a result a small percentage change in the prices of durable goods cause
larger percentage change in the quantity demanded. The Elasticity of Derived Demand: The demand for intermediate goods (goods used in producing the final good) is
called a derived demand, since the demand for these goods is directly associated with the demand for the final good. The derived demand for a specific intermediate good will be more inelastic: 1. The more essential is that good to the production of the final good. 2. The more inelastic the demand for the final good. 3. The smaller the share of that good in the cost of producing the final good. 4. The shorter the time passes after the price changes.
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INCOME ELASTICITY OF DEMAND The income has an impact upon demand. Recall that the relationship between income and demand may be direct or
inverse, depending on whether the good is a normal good or an inferior good. Income Elasticity of Demand (EY) measures the responsiveness of Qd of a good
to a change in income. It is the percentage change in quantity demanded divided by percentage change in income. It may be calculated across and arc for big changes in income using the
following formula: EY =
%∆Q d O 2 − Q1 y 2 − Y1 O 2 − Q1 Y2 − Y1 O 2 − Q1 Y2 + Y1 ÷ = ÷ = × = Q 2 + Q1 Y2 + Y1 Q 2 + Q1 Y2 + Y1 Q 2 + Q1 Y2 − Y1 %∆Y 2 2
For small changes in income using the point elasticity: ⎛ dQ d ⎞ ⎜ ⎟ %∆Q d ⎜⎝ Q d ⎟⎠ dQ d Y EY = = X = %∆Y dY Q d ⎛ dY ⎞ ⎟ ⎜ ⎝ Y ⎠
EY > 1 ⇒ Demand is income elastic and the good is normal and luxury. % ∆ Qd
> % ∆ Y (A small percentage change in income results in a large percentage change in Qd) 0 < EY < 1 ⇒ Demand is income inelastic and the good is normal and
necessary. % ∆Qd < % ∆Y (A large percentage change in income results in a small percentage change in Qd) EY < 0 (negative) ⇒ the good is an inferior good. Examples:
Given QA = 3 – 2PA +1.5Y + 0.8PB – 3PC If PA= 2, Y=4, PB=2.5, PC=1 Calculate EY dQA/dY = 1.5, QA = 3 – 2(2) +1.5(4) + 0.8(2.5) – 3(1) = 4 EY =
∆Q A 4 Y = 1.5 X = 1.5 > 1 ⇒ Normal (luxury) good X ∆Y 4 QA Page 23 of 34
Example:
The manager of Global Food Inc heard the news that government plans to give a 15% raise to all its employees who represent 70% of the labor force of the country. If the estimated income elasticity of demand for global food products is 0.85, find the expected change in the demand for the firm products. %UY = 15% X 70% =10.5% %∆Q d = %∆Y %∆Q d 0.85 = 10.5
EY =
⇒ %UQd = 10.5 X 0.85 = 8.9% Examples:
1. If people’s average income increased from BD300 to BD350 per month and as a result their purchase of orange juice increased from 5000 liters to 5800 liters per month, Calculate EY EY = 0.96. The increase in income by 10% results in an increase in the Qd of orange juice by 9.6% .Orange juice is a normal, necessary good. People buy more of it when their income increases. 2. If people’s average income increased from BD300 to BD350 per month and as a result their purchase of used mobiles decreased from 400 units to 300 units per month, Calculate EY EY =  1.86. The increase in income by 10% results in a decrease in the Qd of used mobiles by 18.6%. Since the sign is negative this means the mobile is an inferior good. People buy less of it when their income increases. 3. If income ↑ by 5% and Qd ↑ by 10% ⇒ EY = +2 ⇒ normal, luxury good 4. If income ↑ by 5% and Qd ↓ by 10% ⇒ EY = 2 ⇒ inferior good
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CROSS ELASTICITY OF DEMAND The decision to buy a good depends not only on its price but also on the price
and availability of other goods (substitutes or complements). We know that as the price of related good changes, the demand for the good
will also change. What we want to know here is how much will quantity demanded rise or fall as
the price of the related good changes. That is, how elastic is the demand curve in response to changes in prices of related goods. Cross elasticity measures the responsiveness of Qd of a particular good to
changes in the prices of its substitutes and its complements. If X and Y are two goods, the cross elasticity of demand is the percentage
change in Qd of good X to the percentage change in price of good Y The arc elasticity formula: ER =
%∆Q x Q 2 x − Q1x P2 y − P1y Q 2 x − Q1x P2 y + P1y = ÷ × = Q 2 x + Q1x P2 y + P1y Q 2 x + Q1x P2 y − P1y %∆Py 2 2
For small price changes, the cross elasticity may be calculated as a point
elasticity using the following formula: ⎛ dQ x ⎞ ⎜ ⎟ %∆Q x ⎜⎝ Q x ⎟⎠ dQ x Py ER = = = X %∆Py ⎛ dPy ⎞ dPy Q x ⎜ ⎟ ⎜ P ⎟ ⎝ y ⎠
When the cross elasticity of demand has a positive sign, the two goods are
substitute goods. When the cross elasticity of demand has a negative sign, the two goods are
complementary goods When ER=0 ⇒ no relation between PX and DY The size of cross elasticity of demand coefficient is primarily used to indicate the
strength of the relationship between the two goods in question.
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Two products are considered good substitutes or complements when the
coefficient is larger than 0.5 (in absolute terms) Example:
If
P1x = 20,
P2x= 30
Q1y = 200
Q2y = 250
Q1z = 150
Q2z = 140
Determine the relationship between X and Y, and the relationship between X and Z ER(xy) = 0.556 ⇒ X and Y are strong substitutes ER(xz) =  0.172 ⇒ X and Z are mild complements Example:
Given QA = 3 – 2PA +1.5Y + 0.8PB – 3PC If PA= 2, Y=4, PB=2.5, PC=1 Calculate a. ER between A and B b. ER between A and C Solution, dQA/dPB = 0.8, dQA/dPC = 3, QA = 3 – 2(2) +1.5(4) + 0.8(2.5) – 3(1) = 4 a. E R =
P dQ A 2 .5 = 0.5 ⇒ Strong Substitutes X B = 0 .8 X 4 QA dPB
b. E R =
P dQ A 1 X C = −3 X = −0.75 ⇒ Strong Complements 4 QA dPC
Example:
Nissan Maxima and Toyota Camry are competing substitutes in the market for small passenger cars. The Nissan Manager would like to predict the negative effect of Toyota’s 15% discount on Camry during Ramadhan. From previous years, Nissan manager has an estimate of the cross elasticity of 2.0 between these two brands.
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Given this information, calculate the expected effect on Nissan sales of Maxima cars. Solution ER = 2=
%∆QMamima = %∆PCamry
%∆QMaxima − 15%
⇒ %UQMaxima = 2 X (15%) = 30% ⇒ Maxima sales are expected to drop by 30% as a result of Toyota discounts during Ramadhan. Exercise
Find the point price elasticity, the point income elasticity, and the point cross elasticity at P=10, Y=20, and PR=9, if the demand function were estimated to be: Qd= 90  8P + 2Y + 2PR Is the demand for this product elastic or inelastic? Is it a luxury or a necessity? Does this product have a close substitute or complement? Find the point elasticities of demand. Solution First find the quantity at these prices and income: Qd= 90  8P + 2Y + 2PR = 90 8(10) + 2(20) + 2(9) =90 80 +40 +18 = 68 Ed = (∂Q/∂P)(P/Q) = (8)(10/68)= 1.17 which is elastic EY = (∂Q/ ∂Y)(Y/Q) = (2)(20/68) = +.59 which is a normal good, but a necessity ER = (∂Qx/ ∂PR)(PR /Qx) = (2)(9/68) = +.26 which is a mild substitute
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NET OR COMBINED EFFECT OF ELASTICITY To find the total effect of change in more than one variable on the quantity
demanded, we may combine the effect of price elasticity of demand (Ed), income elasticity of demand (EY), and cross elasticity of demand (ER), and or any other elasticity, thus calculating the net effect of theses changes. Most managers find that prices and income change every year. By definition we know that: o Ed = %∆Q/ %∆P ⇒%∆Q = Ed (%∆P) o EY = %∆Q/ %∆Y ⇒ %∆Q = EY (%∆Y) o ER = %∆Q/ %∆ PR ⇒ %∆Q = ER (%∆PR) If you knew the price, income, and cross price elasticities, then you can forecast
the percentage changes in quantity. Combining these effects (assuming independent and additive functions) we
have: %∆Q = Ed (%∆P) + EY (%∆Y) + ER (%∆PR) Where, P is price, Y is income, and PR is the price of a related good. Example:
LTC has a price elasticity of 2, and an income elasticity of 1.5 for its laptops. The cross elasticity with another brand is +.50 a. What will happen to the quantity sold if LTC raises price 3%, income rises 2%, and the other brand companies raises its price 1%? b. Will Total Revenue for this product rise or fall? Solution a. %∆Q = Ed (%∆P) + EY (%∆Y) + ER (%∆PR) = 2 (3%) + 1.5 (2%) +0.50 (1%) = 6% + 3% + 0.5% = 2.5%. We expect sales to decline. b. Total revenue will rise slightly (about + 0.5%), as the price went up 3% and the quantity of laptops sold falls 2.5%.
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Example:
AMANA Company is planning to increase its price by 10% next year. At the same period; it is estimated that disposable income will increase by 6%. The company is currently selling two million units. If the estimates of elasticities of next period is Ed = 1.3 and EY= 2, what would be the quantity demanded next period? Solution %UQ1 = Ed (%UP) + EY (%UY) Q2 = Q1 + %UQ1 XQ1 = Q1 + (1 + %UQ1) = Q1 + [1 + Ed (%UP) + EY (%UY)] = 2,000,000 [1 + (1.3) (10%) + (2) (6%)] = 2,000,000 (1 – 0.13 + 0.12) = 2,000,000 (1 – 0.01) = 2,000,000 (0.99) = 1,980,000 The effect of price increase is more than the effect of the increase in income. Example:
If Ed = 1.8, EY = 2.2, ER= 1.5 a. By how much Qd changes if P increases by 8%, income increases by 5%, and a substitute price increases by 6%? b. What is the new Q, if the initial sale is 20,000? Solution a. %UQ = 1.8 (8%) + 2.2 (5%) +1.50 (6%) = 0.144 + 0.11 + 0.09 = 0.056 = 5.6% b. Q2 = Q1 (1+%UQ) = 20,000 (1+0.056) = 20,000(1.056) = 21,120
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PRICE ELASTICITY OF SUPPLY S1
When demand increases, the equilibrium
P1 S2
price rises and the equilibrium quantity increases. But does the price rise by a large or a little amount? And does the
P2 P0
quantity increase by large or a little amount? The answer depends on the
D0 Q0
D1
Q1 Q2
responsiveness of quantity supplied to a change in price. Elasticity of supply measures the responsiveness of quantity supplied to a
change in the price of a good when all other influences on selling plans remain the same. Arc elasticity of supply ∆Q s Q 2 − Q1 Q 2 − Q1 %∆Q s Q avg (Q 2 + Q1 ) / 2 Q 2 + Q1 Q 2 − Q1 P2 + P1 = = = = × =+ Es = ∆P P2 − P1 P2 − P1 %∆P Q 2 + Q1 P2 − P1 Pavg (P2 + P1 ) / 2 P2 + P1
Point elasticity of Supply dQ s %∆Q s dQ s P εs = = Q = × dP %∆P dP Q s P
Elasticity coefficient is positive to show the direct relationship between P and Qs Example:
Suppose you have the following data P1=20
Q1=10
P2=30
Q2=13
Es =
13 − 10 30 − 20 ÷ = 0.65 13 + 10 30 + 20 2 2 Page 30 of 34
Supply Elasticity Categories
1. If Es > 1; % ∆ Qs > % ∆ P (if P ↑ by 1%, Qs ↑ by more than 1%) ⇒ supply is elastic 2. If Es < 1; % ∆ Qs < % ∆ P (if P ↑ by 1%, Qs ↑ by less than 1%) ⇒ supply is inelastic 3. If Es = 1; % ∆ Qs = % ∆ P (if P ↑ by 1%, Qs ↑ by 1%) ⇒ supply is unit elastic 4. If Es = ∞, supply is perfectly elastic with horizontal supply curve. The same price is charged regardless of Qs. Any price decrease would cause supply to fall to zero. Shifts in demand curve results in no change in P. 5. If Es = 0, supply is perfectly inelastic with a vertical supply curve. Qs remains the same regardless of any change in price. Shifts in demand curve results in no change in Qs. P
P
P S S
S D1 D1 D0
D0 Qs
Qs
Perfectly Elastic SC Es = ∞
Unit Elastic SC Es = 1
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Qs
Perfectly inelastic SC Es = 0
Factors that influence elasticity of supply
Price elasticity of supply depends on: 1. Resource substitution possibilities In general, the supply of most goods and services has elasticity between zero
and infinity. The easier it is to substitute among the resources used to produce a good or
service, the greater is its elasticity of supply. If the resources of a good are common and available, the supply is more elastic
and supply curve is almost horizontal (wheat and corn) When goods can be produced in different countries, the supply is more elastic
and supply curve is almost horizontal (sugar, beef, computers) If the resources of a good are unique, the supply of that good is highly inelastic
and the supply curve is vertical. (Paintings) 2.
P
MS
SS
Time Frame for Supply Decision
The more time that passes after a price
LS
change, the greater is the elasticity of supply. We distinguish between three time Qs
frames of supply:
a. Momentary Supply (MS): Immediate response of producers to price change. In general, when price changes, most goods usually have a perfectly inelastic momentary supply with a vertical supply curve. No matter what is the price, production decision is already made earlier and it is difficult to change factors of production and technology immediately. (for example the production of agricultural products such grains and fruits) b. The SR supply curve (SS) is more elastic than momentary supply but is less elastic than long term supply. It shows how the quantity supplied responds to price changes when only some factors and technology affecting production are possible to change. The short response is a sequence of adjustments: firms may increase or decrease the amount of labor force and number of Page 32 of 34
work hours. Firms may plan additional training to the new workers or may buy new tools and equipments Short run supply curve slopes upward because producers can change quantity supplied in response to price changes quickly. c. The long run supply curve (LS) is usually highly elastic. It shows the response of quantity supplied to price change after all necessary adjustments and changes in factors of production and technology (building new plants, expanding the existing plants, training new worker)
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THE ADVERTISING ELASTICITY OF DEMAND: Managers’ decision on how much to spend on advertisement is an investment
decision that has to be justified on some economic base. The decision rule to be followed here is again the marginal one, expand
expenditures on advertisement as long as the marginal revenue exceeds the marginal cost of advertisement and don’t spend more when advertisement MR equates its MC. For a manager to take this decision he should know in advance expected
increase in sales due to the planned additional spending on advertisement. Based on actual data collected from previous years, the firm research center
may calculate the sensitivity or responsiveness of sales to advertisement expenditures. The elasticity of sales (the quantity demanded) will accomplish this task easily,
and may be calculated across an arc or at appoint in the same way used previously. E Ad = E Ad =
Q 2 − Q1 AD 2 − AD1 ÷ Q 2 + Q1 AD 2 + AD1
%∆Q ∆Q AD = X d %∆Ad ∆AD Q
Example:
By how much the manager of Hope Company should increase the firm spending on advertisement in order to increase sales by 20% the coming year, if you know that the elasticity of sales with respect to advertisement is 1.75? %∆Q %∆Ad 20% 20% ⇒ %∆Ad = = 11.4% 1.75 = 1.75 %∆Ad E Ad =
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