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Chapter 15 Market Demand 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 *i i x j (p1 , p2 , m ) 2 From Individual to Market Demand Functions When all consumers are price-takers, the market demand function for commodity j is n *i X j (p1 , p2 , m ,, m ) x j (p1 , p2 , mi ). i 1 1 n If all consumers are identical then X j (p1 , p2 , M) n x*j (p1 , p2 , m) where M = nm. 3 From Individual to Market Demand Functions The market demand curve is the “horizontal sum” of the individual consumers’ demand curves. E.g. suppose there are only two consumers; i = A,B. 4 From Individual to Market Demand Functions p p 1 1 p1’ p1” p1 p1’ p1” 20 x*A 1 15 *B x1 The “horizontal sum” of the demand curves of individuals A and B. p1’ p1” 35 x*1A xB 1 5 Elasticities Elasticity measures the “sensitivity” of one variable with respect to another. The elasticity of variable X with respect to variable Y is % x x,y . % y 6 Economic Applications of Elasticity Economists use elasticities to measure the sensitivity of quantity demanded of commodity i with respect to the price of commodity i (own-price elasticity of demand) demand for commodity i with respect to the price of commodity j (cross-price elasticity of demand). 7 Economic Applications of Elasticity demand for commodity i with respect to income (income elasticity of demand) quantity supplied of commodity i with respect to the price of commodity i (own-price elasticity of supply) and so on. 8 Own-Price Elasticity of Demand Q: Why not use a demand curve’s slope to measure the sensitivity of quantity demanded to a change in a commodity’s own price? 9 Own-Price Elasticity of Demand p1 10-packs p1 10 10 slope =-2 5 Single Units slope = - 0.2 X1 * In which case is the quantity demanded X1* more sensitive to changes to p1? It is the same in both cases. 50 X1 * 10 Own-Price Elasticity of Demand Q: 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? A: Because the value of sensitivity then depends upon the (arbitrary) units of measurement used for quantity demanded. 11 Own-Price Elasticity of Demand * % x1 x* ,p 1 1 % p1 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. 12 Own-Price Elasticity dX*i X* ,p * i i dpi Xi pi E.g. Suppose pi = a - bXi. Then Xi = (a-pi)/b and * dXi 1 . Therefore, dpi b pi 1 pi X* ,p . i i ( a pi ) / b b a pi 13 Own-Price Elasticity pi pi X* ,p i i a pi pi = a - bXi* a a/b Xi* 14 Own-Price Elasticity pi a pi X* ,p i i a pi pi = a - bXi* p 0 0 0 a/b Xi* 15 Own-Price Elasticity pi a a/2 pi X* ,p i i a pi pi = a - bXi* a a/2 p 1 2 aa/2 1 0 a/2b a/b Xi* 16 Own-Price Elasticity pi = a - bXi* pi a a/2 pi X* ,p i i a pi a pa aa 1 0 a/2b a/b Xi* 17 Own-Price Elasticity pi = a - bXi* pi a a/2 pi X* ,p i i a pi own-price elastic 1 (own-price unit elastic) own-price inelastic 0 a/2b a/b Xi* 18 Own-Price Elasticity pi X* ,p * i i X i X*i kpia . E.g. So, X* ,p i i pi a kpi * dXi dpi * i dX a 1 ka pi Then dpi kapia 1 a pia a pi a. 19 Own-Price Elasticity pi k * a 2 Xi kpi kpi pi2 2 everywhere along the demand curve. Xi* 20 Revenue and Own-Price Elasticity of Demand If raising a commodity’s price causes little decrease in quantity demanded, then sellers’ revenues rise. Hence own-price inelastic demand causes sellers’ revenues to rise as price rises. 21 Revenue and Own-Price Elasticity of Demand If raising a commodity’s price causes a large decrease in quantity demanded, then sellers’ revenues fall. Hence own-price elastic demand causes sellers’ revenues to fall as price rises. 22 Revenue and Own-Price Elasticity of Demand Sellers’ revenue is So * R(p) p X (p). * dR dX X* (p) p dp dp * p dX * X (p )1 * X (p ) dp X (p)1 . * 23 Revenue and Own-Price Elasticity of Demand dR * X (p)1 dp so if 1 then dR 0 dp and a change to price does not alter sellers’ revenue. 24 Revenue and Own-Price Elasticity of Demand dR * X (p)1 dp but if 1 0 dR 0 then dp and a price increase raises sellers’ revenue. 25 Revenue and Own-Price Elasticity of Demand dR X* (p)1 dp And if 1 then dR 0 dp and a price increase reduces sellers’ revenue. 26 Revenue and Own-Price Elasticity of Demand In summary: Own-price inelastic demand: 1 0 price rise causes rise in sellers’ revenue. Own-price unit elastic demand: 1 price rise causes no change in sellers’ revenue. Own-price elastic demand: 1 price rise causes fall in sellers’ revenue. 27 Marginal Revenue and Own-Price Elasticity of Demand A seller’s marginal revenue is the rate at which revenue changes with the number of units sold by the seller. dR( q) MR( q) . dq 28 Marginal Revenue and Own-Price Elasticity of Demand p(q) denotes the seller’s inverse demand function; i.e. the price at which the seller can sell q units. Then R( q) p( q) q so dR( q) dp( q) MR( q) q p( q) dq dq q dp( q) p( q) 1 . p( q) dq 29 Marginal Revenue and Own-Price Elasticity of Demand q dp( q) MR( q) p( q) 1 . p( q) dq and so dq p dp q 1 MR( q) p( q) 1 . 30 Marginal Revenue and Own-Price Elasticity of Demand 1 MR( q) p( q) 1 says that the rate at which a seller’s revenue changes with the number of units it sells depends on the sensitivity of quantity demanded to price; i.e., upon the own-price elasticity of demand. 31 Marginal Revenue and Own-Price Elasticity of Demand 1 MR(q) p(q)1 If 1 then MR( q) 0. If 1 0 then MR( q) 0. If 1 then MR( q) 0. 32 Marginal Revenue and Own-Price Elasticity of Demand If 1 then MR( q) 0. Selling one more unit does not change the seller’s revenue. If 1 0 then MR( q) 0. Selling one more unit reduces the seller’s revenue. If 1 then MR( q) 0. Selling one more unit raises the seller’s revenue. 33 Marginal Revenue and Own-Price Elasticity of Demand An example with linear inverse demand: p( q) a bq. Then R( q) p( q)q ( a bq)q and MR( q) a 2bq. 34 Marginal Revenue and Own-Price Elasticity of Demand p a p( q) a bq a/2b a/b q MR( q) a 2bq 35 Marginal Revenue and Own-Price p Elasticity of Demand a MR( q) a 2bq p( q) a bq $ a/2b a/b q a/b q R(q) a/2b 36 Elasticities Cross Price Elasticity of demand: p j dxi dxi / xi i dp j / p j xi dp j j Income Elasticity of demand: dxi / xi m dxi dm / m xi dm m i 37 Elasticities Ordinary Good: ii 0 i Giffen Good: i 0 Normal Good: im 0 m Inferior Good: i 0 m 1 Luxury Good: i Necessary Good: 0 im 1 38