lecture_4 - From Individual to Market Demand Functions...

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ECOS2001 Lecture 4 1 From Individual to Market Demand Functions Think of an economy containing n consumers, denoted by i = 1, … ,n. Consumer i’s ordinary demand function for commodity j is When all consumers are price-takers, the market demand function for commodity j is * 12 (, , ) ii j xp p m 1* 1 , , , ) , ) . n ni i jj i Xp p m m x p p m = = "
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ECOS2001 Lecture 4 2 If all consumers are identical then where M = nm. The market demand curve is the “horizontal sum” of the individual consumers’ demand curves. * 12 (, , ) ,) jj X ppM nx ppm
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ECOS2001 Lecture 4 3 Elasticities Elasticity measures the “sensitivity” of one variable with respect to another. The elasticity of variable X with respect to variable Y is Used in a wide range of applications: quantity demanded changes with respect to own price, another good’s price, income; quantity supplied etc , % % xy x y ε Δ = Δ
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ECOS2001 Lecture 4 4 Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to a change in a commodity’s own price? Because the value of sensitivity then depends upon the (arbitrary) units of measurement used for quantity demanded. Elasticity is a ratio of percentages and so has no units of measurement. Hence own-price elasticity of demand is a sensitivity measure that is independent of units of measurement
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ECOS2001 Lecture 4 5 Arc Elasticities where and 12 2 m x x x + = 2 m p p p + = * 21 2 1 , () ( ) / ii xp mm x p ε =
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ECOS2001 Lecture 4 6 Point elasticities What is the own-price elasticity of demand in a very small interval of prices centered on pi’? is the elasticity at the point * * , ' ' ii Xp pd X Xd p ε
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ECOS2001 Lecture 4 7 Point Own-Price Elasticity p i X i * p i = a - bX i * a a/b
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ECOS2001 Lecture 4 8 Point Own-Price Elasticity * * * , ii Xp p dX X dp ε E.g. Then so * 1 a i i i ap = * a Xk p = * 1 , . a a i aa pp ka p a a kp p = ×= =
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ECOS2001 Lecture 4 9 Point Own-Price Elasticity p i X i * everywhere along the demand curve. 2 ε=− *2 2 a iii i k Xk pk p p == = D
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ECOS2001 Lecture 4 10 Revenue and Own-Price Elasticity of Demand If raising a commodity’s price causes little decrease in quantity demanded, then sellers’ revenues rise.
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lecture_4 - From Individual to Market Demand Functions...

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