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

 When all consumers are price-takers, the market demand function for commodity j is  If all consumers are identical then 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) p1p1 p1p1 p1p p1’p1’ p1”p1” p1’p1’ p1”p1” p1’p1’ p1”p1”

 Elasticity measures the “sensitivity” of one variable with respect to another.  The elasticity of variable X with respect to variable Y is

 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 ，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 (own- price 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?

X1*X1* slope = - 2 slope = p1p1 p1p1 In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ? X1*X1*

slope = - 2 slope = p1p1 p1p1 X1*X1* X1*X1* In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ?

slope = - 2 slope = p1p1 p1p1 10-packsSingle Units X1*X1* X1*X1* In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ?

slope = - 2 slope = p1p1 p1p1 10-packsSingle Units X1*X1* X1*X1* In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ? 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.

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 或 Price elasticity of demand

E.g. Suppose p i = a - bX i. Then X i = (a-p i )/b and

pipi Xi*Xi* p i = a - bX i * a a/b

pipi Xi*Xi* p i = a - bX i * a a/b

pipi Xi*Xi* p i = a - bX i * a a/b

pipi Xi*Xi* p i = a - bX i * a a/b

pipi Xi*Xi* a p i = a - bX i * a/b

pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b

pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b

pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b

pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b own-price elastic （有弹性） own-price inelastic （缺乏弹性）

pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b own-price elastic own-price inelastic (own-price unit elastic) 单位弹性

E.g.Then so

pipi Xi*Xi* everywhere along the demand curve.

 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.

Sellers’ revenue is

So

Sellers’ revenue is So

Sellers’ revenue is So

so ifthen and a change to price does not alter sellers’ revenue.

but ifthen and a price increase raises sellers’ revenue.

And ifthen and a price increase reduces sellers’ revenue.

In summary: Own-price inelastic demand; price rise causes rise in sellers’ revenue. Own-price unit elastic demand; price rise causes no change in sellers’ revenue. Own-price elastic demand; 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.

p(q) denotes the seller’s inverse demand function; i.e. the price at which the seller can sell q units. Then so

and so

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.

Ifthen Ifthen Ifthen

Selling one more unit raises the seller’s revenue. Selling one more unit reduces the seller’s revenue. Selling one more unit does not change the seller’s revenue. Ifthen Ifthen Ifthen

An example with linear inverse demand. Then and

a a/b p qa/2b

a a/b p qa/2b q $ a/ba/2b R(q)

 Recall that price elasticity of demand is  Hence income elasticity of demand is

 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 ）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？