Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W

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1 Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the ercentage change in quantity demanded, q, divided by the ercentage change in rice,. The formula for the demand elasticity (ɛ) is: q d. Note that the law of demand imlies that /d < 0, and so ɛ will be a negative number. In some contexts, it is common to introduce a minus sign in this formula to make this quantity ositive. You should be careful in all circumstances to check which definition is being used. Why do we care about demand elasticities? One way to see why rice elasticity of demand might be useful is to consider the question: How will revenue change as we change rice? Recall that revenue is rice times quantity demanded. If we write everything in terms of rice (by using the demand equation q = q()), we get R() = q(). Now, the derivative of a function tells us how that function will change: If R () > 0 then revenue is increasing at that rice oint, and R () < 0 would say that revenue is decreasing at that rice oint. So, we comute R () using the general revenue equation: R () = q() + d [ = q() 1 + ]. q() d Note that q() is always ositive, so the sign of R () is comletely determine by the quantity in the brackets. We note that whether this quantity is ositive or negative is determined by the size of the second term, q() d, 1

2 which is recisely the demand elasticity ɛ as we have defined it. Thus, the size, in absolute value, of ɛ relative to 1 will determine how the revenue changes as the rice changes. (This is why some economists refer to use the ositive version of demand elasticity.) Now, suose a manager wants to find out how consumers will react when he increases the rice of a good. If increasing the rice increases revenue, the manager would have made a good decision. If increasing the rice decreases revenue, the manger would have made a bad decision. We use elasticities to see how sensitive are consumers to rice changes. Note that the negative sign of the demand elasticity as we have defined it encodes how demand resonds to rice changes: As rice increases, quantity demanded decreases, and as rice decreases, quantity demanded increases. That is, the fact that ɛ is negative tells us rice and quantity demanded q move in oosite directions! Remember, the negative value of ɛ comes from the factor /d, which is the sloe of the demand curve which is negative by the law of demand. (If you had a good that did not follow the law of demand, then the sign would not necessarily be negative.) Economists use the following terminolgy: 1. When ɛ > 1, we say the good is rice elastic. In this case, % Q > % P, and so, for a 1 % change in rice, there is a greater than 1 % change in quantity demanded. In this case, management should decrease rice to have a higher revenue. 2. When ɛ < 1, we say the good is rice inelastic. In this case, % Q < % P, and so, for a 1 % change in rice, there is a less than 1 % change in quantity demanded. In this case, management should increase rice to have a higher revenue. 3. When ɛ = 1, we say the good is rice unit elastic. In this case, % Q = % P, and so, for a 1% change in rice, there is also an 1% change in quantity demanded. This is the otimal rice which means it maximizes revenue. (Why is this so? Draw a grah of revenue versus rice for the case of a linear demand curve to see what is haening.) 2

3 According to economic theory, the rimary determinant of the rice elasticity of demand is the availability of substitutes. If many substitute goods are available, then demand is rice elastic. For examle, if the rice of Pesi increases, consumers can easily switch to Coke. So the rice elasticity of Pesi is very high. Whereas, if there are few substitutes, then demand is rice inelastic. Even if the rice of a basic food items increases, one has to buy this food item to survive. For examle, in much of the world, rice is a stale food item, and is an item which is highly rice inelastic. There are other tyes of elasticities besides rice elasticity of demand, but we will not consider them in this course. Examle 1 Suose the demand curve for opads is given by q = (a) Comute the rice elasticity of this demand function. Noting that /d = 10, we get = = q() d, ( 10), 50. (b) What is the rice elasticity of demand when the rice is $30? Using our result from (a), we get = 1.5. Since 1.5 is greater than 1, this good is rice elastic, and management should consider lowering the rice to raise revenue. (c) What is the ercent change in the demand if the rice is $30 and increases by 4.5%? We note that ɛ is defined to be the ercentage change in demand divided by the ercentage change in rice. This means =

4 Thus, demand will decrease by 6.75%. (d) How does this change in demand inform management about its rice change? Management should decrease rice because the good is elastic. For an increase in rice of 4.5%, quantity demanded droed by MORE than 4.5% (6.75%). Examle 2 Benson just oened a business selling calculators. The demand function for calculators can be given by q = Find the rice for which he should sell the calculators in order to maximize revenue. Solution We first find an exression for demand elasticity. Since /d = 4, ( 4). Revenue is maximized at unit elasticity, which is when ɛ = 1, so we set 1 and then solve for rice : 1 = , 2 = = 4 2, = 6 2 = 400, 200 = = 3 $8.16. Thus, Benson should sell the calculators for $8.16. Examle3 The demand for uer bowl tickets to watch the Vancouver Canucks can be given by the function = ( 100 q )

5 Find the rice elasticity of demand and determine whether the owner of the Canucks should increase or decrease the ticket rice from the current rice of $100. Solution There are a number of aroaches we could take. First, we note that we have our demand equation written as = (q). We could solve exlicitly for q = q(p ) (using the fact that q 0). With this aroach, we get which gives us This give us the demand elasticity and so at the rice of $100, we have q = , d = ( ), ( ) = (1) Since 1 < 1, the rice elasticity of demand is inelastic. A small 18 % change in rice will cause a smaller % change in quantity demanded. Therefore, the owner should increase the rice until the rice elasticity of demand becomes unit elastic in order to maximize revenue. A second aroach to this roblem would be to use the demand equation to find the demand q corresonding to the rice of $100. We can then use imlicit differentiation to find /d in terms of both and q, and so do not need to exlicitly solve the demand equation for q. In this case, we first note that q = 900 if = 100. Differentiating the demand equation as given with resect to gives us ( 1 = q ) d. We leave it as an exercise for the student to substitute q = 900 into this exression to find the corresonding /d. You can then comute ɛ, which will give you the same value as in our first aroach. 5