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Market Demand 市场需求 Think of an economy containing n consumers, denoted by i = 1, … ,n. Consumer i’s ordinary demand function for commodity j is x*ji (p1 , p2 , mi ) 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. The market demand curve is the “horizontal sum” of the individual consumers’ demand curves. Denoted by demand function D=D(P) or inverse demand function P=P(D) p1 p1 p1’ p1” p1’ p1” p1 20 x*A 1 15 p1’ p1” 35 x*1A xB 1 x*1B 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 Economists use elasticities to measure the sensitivity of quantity demanded of commodity i with respect to the price of commodity i (ownprice elasticity of demand,需求的自价 格弹性) demand for commodity i with respect to the price of commodity j (cross-price elasticity of demand,需求的交叉价格 弹性). demand for commodity i with respect to income (income elasticity of demand 需求 的收入弹性) quantity supplied of commodity i with respect to the price of commodity i (ownprice elasticity of supply 供给的自价格弹 性) 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? p1 10 slope =-2 5 p1 10 X1* slope = - 0.2 50 X * 1 In which case is the quantity demanded X1* more sensitive to changes to p1? p1 10 slope =-2 5 p1 10 X1* slope = - 0.2 50 X * 1 In which case is the quantity demanded X1* more sensitive to changes to p1? p1 10 10-packs slope =-2 5 p1 10 X1* Single Units slope = - 0.2 50 X * 1 In which case is the quantity demanded X1* more sensitive to changes to p1? p1 10 10-packs slope =-2 5 p1 10 X1* Single Units slope = - 0.2 50 X * 1 In which case is the quantity demanded X1* more sensitive to changes to p1? It is the same in both cases. 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. * % 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. Measuring increases in percentage terms keeps the elasticity unit-free x x y y 或 dx y dy x Price elasticity of demand X * i * i , pi dX pi * dpi X i E.g. Suppose pi = a - bXi. Then Xi = (a-pi)/b and dxi pi ( p) dpi xi ( pi ) dx i 1 dpi b pi 1 pi pi 1 xi b (a pi ) / b b a pi pi pi = a - bXi* a a/b Xi* pi pi = a - bXi* pi X* ,p i i a pi a a/b Xi* pi a pi = a - bXi* pi X* ,p i i a pi p 0 0 a/b Xi* pi a pi X* ,p i i a pi pi = a - bXi* p 0 0 0 a/b Xi* pi a pi X* ,p i i a pi pi = a - bXi* a a/2 p 1 2 aa/2 0 a/b Xi* 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* pi a a/2 pi X* ,p i i a pi pi = a - bXi* a pa aa 1 0 a/2b a/b Xi* pi = a - bXi* pi a a/2 pi X* ,p i i a pi a pa aa 1 0 a/2b a/b Xi* pi pi X* ,p i i a pi pi = a - bXi* a own-price elastic (有弹性) a/2 1 own-price inelastic (缺乏弹性 0 a/2b a/b Xi* pi pi X* ,p i i a pi pi = a - bXi* a own-price elastic (own-price unit elastic) 1 a/2 单位弹性 own-price inelastic 0 a/2b a/b Xi* dX*i X* ,p * i i dpi Xi pi E.g. X*i kpia . Then so dX*i apia 1 dpi a pi pi a 1 X* ,p kapi a a. a a i i kpi pi pi 2 everywhere along the demand curve. Xi* If raising a commodity’s price causes little decrease in quantity demanded, then sellers’ revenues rise. Hence own-price inelastic (缺乏弹性) demand causes sellers’ revenues to rise as price rises. If raising a commodity’s price causes a large decrease in quantity demanded, then sellers’ revenues fall. Hence own-price elastic (富有弹性) demand causes sellers’ revenues to fall as price rises. * R ( p ) p X (p). Sellers’ revenue is * R ( p ) p X (p). Sellers’ revenue is * dR dX So X* (p) p dp dp * R ( p ) p X (p). Sellers’ revenue is * dR dX So X* (p) p dp dp * p dX * X (p )1 * X (p ) dp * R ( p ) p X (p). Sellers’ revenue is * dR dX So X* (p) p dp dp * p dX * X (p )1 * X (p ) dp X* (p)1 . dR X* (p)1 dp dR X* (p)1 dp so if 1 then dR 0 dp and a change to price does not alter sellers’ revenue. dR X* (p)1 dp dR 0 but if 1 0 then dp and a price increase raises sellers’ revenue. dR X* (p)1 dp And if 1 dR 0 then dp and a price increase reduces sellers’ revenue. 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. 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 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 q dp( q) MR( q) p( q) 1 . p( q) dq and so dq p dp q 1 MR( q) p( q) 1 . 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 of the 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. 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. An example with linear inverse demand. p( q) a bq. Then R( q) p( q)q ( a bq)q and MR( q) a 2bq. p a p( q) a bq a/2b a/b q MR( q) a 2bq p a MR( q) a 2bq p( q) a bq $ a/2b a/b q a/b q R(q) a/2b Recall that price elasticity of demand is x 1 , p1 %x1 %p1 Hence income elasticity of demand is m ,p 1 %m %p1 X * i ,m * dX m i * Xi dm Normal good: >0 Inferior good: <0 Luxury good: >1 Necessary good: 0<<1 From Individual to Market Demand Functions Elasticities Revenue and own-price elasticity of demand Marginal revenue and price elasticity 消费者对商品 x 和在其它商品上的开支 y(价 格为1)的效用函数为 1 2 u ( x, y ) x x y 2 1)市场上有完全同样的消费者100人,写出x 的市场需求函数。 2)x该如何定价使销售收入最大?此时价格弹 性是多少?