Section 12.3 d i E Elasticity of Demand a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 104: Mathematics for Business II Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 1 / 11
Recall From Economics Class! The demand function is a function between the quantity q and the price p . usually we write d i E p = f ( q ) a l l Under normal circumstances, if the price increases, then the quantity u d should decreases. b A Goal: To determine the effect (consumer’s response) to any increase in price. i.e., how the quantity will change if the price changes? . r D Definition The demand of elasticity is a means to measure how a change in the price of a product will affect the quantity demanded. Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 2 / 11
Three cases of elasticity 1 The demand is elastic if the change in price do effect the quantity d demanded. i E For example, if we have a luxury product! a 2 The demand has unit elasticity if the change in price will result in the l l u same change in the quantity demanded. d b For example, meat product! A 3 The demand has inelastic if the change in price will not effect too . r much the quantity demanded. D For example, electricity! Question: How to measure these mathematically? Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 3 / 11
Point of elasticity of demand We find the point of elasticity of demand which is η : = change in demand d i change in price E p a q η : = l l dp u dp d f ( q ) b A q η : = f ′ ( q ) . r D The point of elasticity of demand η is pronounced as “eta“. 1 When | η | > 1, demand is elastic. 2 When | η | = 1, demand has unit elastic. 3 When | η | < 1, demand is inelastic. Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 4 / 11
Example (Old Exam Question) For the demand equation p = 100 + 0.02 q − q 2 determine whether demand is elastic, is inelastic, or has unit elasticity for q = 5. d Solution: First we find the derivative of p which is i E p ′ = 0.02 − 2 q a l l u Next we find η d b p A q η = p ′ . r 100 + 0.02 q − q 2 D q η = 0.02 − 2 q Now set q = 5 to get that η = − 1.505 → | eta | = 1.505 Hence the demand is elastic . Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 5 / 11
Exercise � 4000 − q 2 (Old Final Exam Question) For the demand equation p = determine whether demand is elastic, is inelastic, or has unit elasticity for q = 20. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 6 / 11
Exercise − q 50 determine whether demand is For the demand equation p = 250 e elastic, is inelastic, or has unit elasticity for q = 50. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 7 / 11
Example 150 (Old Exam Question) For the demand equation p = q + 50 determine whether demand is elastic, is inelastic, or has unit elasticity for q = 50. Solution: First we find the derivative of p which is using the quotient rule d p ′ = ( q + 50 )( 0 ) − 150 ( 1 ) − 150 i E = ( q + 50 ) 2 ( q + 50 ) 2 a l l Next we find η u d p b q A η = p ′ . 150 r q + 50 D q η = − 150 ( q + 50 ) 2 Now set q = 50 to get that η = − 2 → | eta | = 2 Hence the demand is elastic . Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 8 / 11
d Example i E (Old Final Exam Question) If η = − 1.5, then the demand is? a l l u Solution: d η = − 1.5 → | η | = 1.5 b A Hence the demand is elastic . . r D Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 9 / 11
Example (Old Final Exam Question) For what value (or values) of q do the demand equation q = 144 − 4 p has unit elasticity? Solution: First we find the derivative of p which can be written as d p = 144 − q → p ′ = − 1 i E 4 4 a l since we have unit elasticity, we find η and we make it equal to 1 so l u d | η | = 1 b A η = 1 or η = − 1 . p p r D q q p ′ = 1 or p ′ = − 1 144 − q 144 − q 4 4 q q = 1 or = − 1 − 1 − 1 4 4 Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 10 / 11
Continue... d i E a 144 − q = 1 or 144 − q l = − 1 l u − q − q d 144 − q = − q or 144 − q = q b A q = 72 . r D Dr. Abdulla Eid (University of Bahrain) Elasticity of Demand 11 / 11
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