微观经济学15章(范里安) 上财

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).
ed（点）＝lim （△Q/△ P ）（P /Q ）
△ P 0
＝ （dQ/d P ）•（P /Q ）
Xi* ,pi
pi Xi*
dXi* dpi
An example of computing a point own-price
elasticity of demand.
Suppose
dXi*
pi = a - bXi. Then
Consumer i’s ordinary demand function for commodity j is x*ji(p1,p2,mi )
From Individual to Market Demand Functions
When all consumers are price-takers,
So dR X*(p) p dX*
dp
dp
X*(p)1
p X*(p)
dX* dp
Sellers’ revenue is R(p) p X*(p).
So dR X*(p) p dX*
dp
dp
X*(p)1
p X*(p)
dX* dp
X*(p)1 .
dR X*(p)1
A: Because then the value of sensitivity depends upon the chosen units of measurement.
x*1 ,p1
%x*1 %p1
is a ratio of percentages and so has no
units of measurement. Hence own-price
kpi 2
k pi2
2 everywhere along
the demand curve.
Xi*
4、Revenue and Own-Price Elasticity of Demand
If raising a commodity’s price causes almost no decrease in quantity demanded then sellers’ revenues will rise.
MR(q) dR(q) . dq
Marginal Revenue and Own-
Let Pp(rqi)cdeenEoltaesthtiecsiteylleorf’sDinevmersaend
demand function; that is, the price at which the seller can sell q units. Then
elasticity of demand is a measure of the
sensitivity of quantity demanded to
changes in own-price which is
independent of the choice of units of
measurement.
From Individual to Market Demand Functions
The market demand curve is the “horizontal sum” of the demand curves of the individual consumers.
For example, suppose there are only two consumers; i = A,B.
a/2b
0
a/b Xi*
pi
pi = a - bXi*
Xi* ,pi
pi a pi
a
own-price elastic
a/2
1 (own-price unit elastic)
own-price inelastic
0
a/2b
a/b Xi*
Xi* ,pi
pi Xi*
dXi* dpi
Hence own-price inelastic demand will cause sellers’ revenues to rise as price rises.
If raising a commodity’s price causes a very large decrease in quantity demanded then sellers’ revenues will fall.
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
pi a pi
p0 0
0
a/b Xi*
pi
pi = a - bXi*
Xi* ,pi
pi a pi
a
p a a / 2 1
2
aa/2
a/2
1
a/2b
0
a/b Xi*
pi
pi = a - bXi*
Xi* ,pi
pi a pi
a
p a a
aa
a/2
1
A second example of computing a point
own-price elasticity of demand.
Suppose Xi* kpia .
Then
dXi* dpi
a pia 1
so
Xi* ,pi
pi kpia
kapia1
a
pia pia
a.
pi
Xi*
kpia
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.
R(q) p(q) q
so MR(q) dR(q) dp(q) q p(q)
dq
dq
p(q)1
q p(q)
dpd(qq) .
Marginal Revenue and Own-
Price Elasticity of Demand
MR(q)
p(q)
1
q p(q)
dp(q) dq
From Individual to Market Demand Functions
p1
p1
p1’
p1’
p1”
p1”
20 x*1A
15
x*1B
From Individual to Market
p1 Demanpd1 Functions
p1’
p1’
p1”
p1”
p1
20 x*1A
1 “horizontal sum”
p1”
of the demand curves
of individuals A and B.
35 x*1A xB1
二、弹性 Elasticity
Elasticity is a measure of the “sensitivity” of one variable with respect to another.
dR X*(p)1
dp
And if 1
then
dR 0 dp
and a price increase reduces sellers’ revenue.
In summary:
Own-price inelastic demand; 1 0 price rise causes rise in sellers’ revenue.
Elasticity computed for a single value of pi is called a point elasticity.
弧弹性：ed（弧）＝（△Q/Q）/（△P/P）
＝ （△Q/△ P ）（P /Q ）
= （△Q/△ P ）（P1+P2)/2
/(Q1+Q2 )/2
点弹性：
chapter 15 市场需求与弹性
一、如何获得市场需求？
From Individual to Market Demand Functions
商品1的市场需求是消费者个人需求的总 和。
From Individual to Market Demand Functions
Think of an economy containing n consumers, denoted by i = 1, … ,n.
5、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.
dp
so if 1 then
dR 0 dp
and a change to price does not alter sellers’ revenue.
dR X*(p)1
dp
but if
1 0
then
dR 0 dp
and a price increase raises sellers’ revenue.
2、Arc and Point Elasticities
If we measure the “average” ownprice elasticity of demand for commodity i over an interval of values for pi then we compute an arcelasticity, usually by using a midpoint formula.
1. Therefore,
Xi
=
(a-pi)/b
and
dpi b
Xi* ,pi
(
a
pi pi
)
/
b
b1
pi . a pi
pi
pi = a - bXi*
Xi* ,pi
pi a pi
a
a/b Xi*
3、Point Own-Price Elasticity
pi a
pi = a - bXi*
Xi* ,pi
1、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?
the market demand function for
commodity j is
Xj(p1,p2,m1,,mn)
n
x*ji(p1,p2,mi ).
i1
If all consumers are identical then
Xj(p1,p2,M) n x*j (p1,p2,m)
where M = nm.
The elasticity of a variable X with
respect to a variable Y is defined as
x,y
%x . %y
Economic Applications of Elasticity
Economists use elasticities to measure the sensitivity of
Hence own-price elastic demand will cause sellers’ revenues to fall as price rises.
Sellers’ revenue is R(p) p X*(p).
So dR X*(p) p dX*
dp
dp
Sellers’ revenue is R(p) p X*(p).