Chapter_2C 消费者行为理论

a
b
∂U a −1 b MU1 = = ax1 x2 ∂ x1 ∂U a b −1 MU 2 = = bx1 x2 ∂ x2
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Ordinary Demands: Cobb-Douglas Example
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So the MRS is
b dx2 ∂ U /∂ x1 ax1a −1 x2 ax2 MRS = =− = − a b −1 = − . dx1 ∂ U /∂ x2 bx1 x2 bx1
n
7
How to Get Individual Demand Function by Algebra Approach(代数方法)
n
With 2 goods, the individual’s objective is to maximize utility from these 2 goods: U=U(x1,x2)
* 2
* x1 =
aI ( a + b) p1
x1
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Ordinary Demands: Cobb-Douglas Example
n
Note that for Cobb-Douglas utility function
n n n
Demands are linear in income Expenditure shares are constant Expenditure shares sum to one
* 1
n
Substitute x1* into (2) and solve for x2* to get
bI x = (a + b) p2
* 2
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Ordinary Demands: Cobb-Douglas Example
x2
b U ( x1 , x2 ) = x1a x2
bI x = ( a + b) p2
aI px = (a + b) bI * p2 x2 = . (a + b)
* 1 1
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How to Get Demand Curve by Graphic Approaches
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By the way of comparative static analysis, if we let the price of a good change holding other factors constant, we get ordinary demand curve Example: the price of a good increase
p1
x2
p1’’’
demand curve
p1’’ p1 ’ x1*(p1’’’) x1*(p1’)
x1*
x1*(p1’’) x1*(p1’’’) x1*(p1’’) x1*(p1’)
x1
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the price of a good Changes and Price -consumption curve(价格--消费线) p1
2
Overview of Last Class
n n n
Budget Constraint(预算约束) Application of Budget Constraint Utility Maximization (Consumer’s Optimal Choice) Types of Optimal Solution
x2
p1x1 + p2x2 = I p1 = p1’
p1= p1’’ x1
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the price of a good Increases
x2
p1x1 + p2x2 = I p1 = p1’
p1= p1’’’
p1= p1’’ x1
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the price of a good Changes
x2
p1 = p1’
x1*
x1*(p1’’’)
x1*(p1’) x1*(p1’’)
x1 1
24
the price of a good Changes
p1
xx2 2
p1’’’ p1’’ p1 ’ x1*(p1’’’) x1*(p1’)
x1*
x1*(p1’’) x1*(p1’’’) x1*(p1’’) x1*(p1’)
x1 x1
p1 x1 + p2 x2 − m = 0
9
How to Get Individual Demand Function by Algebra Approach
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From 3 first-order conditions, we can get three equations, and then solve them to get individual demand functions for two goods. x1*=x1(p1,p2,I) x2*=x2(p1,p2,I) xi*= xi(p, I) ( i=1,2) is called ordinary demand function(普通需求函数), is also called Marshallian Demand Function（马歇尔需求函 数）
L( x1 , x2 , λ ) = u( x1 , x2 ) − λ [ p1 x1 + p2 x2 − m]
Take first order necessary conditions for maximum
∂u ( x1 , x2 ) − λp1 = 0 ∂x1 ∂u ( x1 , x2 ) − λp 2 = 0 ∂x2
x2
p1’’’ p1’’
demand curve
P.C.C
p1 ’ x1*(p1’’’) x1*(p1’)
x1*
x1*(p1’’) x1*(p1’’’) x1*(p1’’) x1*(p1’)
n
At (x1*,x2*), MRS = -p1/p2 so
* ax 2 p1 bp1 * * − * =− ⇒ x2 = x1 bx1 p2 ap 2
(1)
n
Also, at (x1*,x2*), the budget is exhausted, so
p x + p x = I ( 2)
* 1 1 * 2 2
6
Representation of Individual Demand Function
dx = d x ( PX , P , I ; preference) s Y
n
The three elements that determine the quantity demanded are the prices of X and Y, the person’s income (I), and the person’s preferences for X and Y. Preferences appear to the right of the semicolon because we assume that preferences do not change during the analysis.
4
n n
n
n
A Survey of This Section(2.3)
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This section studies how people change their choices when conditions such as change in income or changes in the prices of goods affect the amount that people choose to consume. This section then compares the new choices with those that were made before conditions changed The main result of this approach is to construct an individual’s demand curve
Chapter 2
Consumer Behavior and Demand Theory
1
Chapter 2 includes:
n n n n
2.1 Preference and Utility 2.2 Utility Maximization and Choice 2.3 Income and Substitution Effects 2.4 Market Demand and Elasticity
n
Subject to the budget constraint: I=x1P1+x2P2
n
For maximizing a utility function subject to a
constraint, we set up the Lagrangian expression:
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How to Get Individual Demand Function by Algebra Approach
12
Ordinary Demands: Cobb-Douglas Example
n
Substitute for x2* from (1) into (2) to get
bp1 * p x + p2 x1 = I ap2
* 1 1
n
Solve for x1* to get
aI x = (a + b) p1
ห้องสมุดไป่ตู้
x1
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the price of a good Changes
p1
x2
p1’’ p1 ’ x1*(p1’) x1*(p1’’) x1*(p1’’) x1*(p1’)