消费者最优选择

x2*
x1*
x1
Rational Constrained Choice
The
most preferred affordable bundle is called the consumer‟s ORDINARY DEMAND at the given prices and budget. Ordinary demands will be denoted by x1*(p1,p2,m) and x2*(p1,p2,m).
Suppose
that the consumer has Cobb-Douglas preferences.
a U( x1 , x 2 ) x1 xb 2
Computing Ordinary Demands a Cobb-Douglas Example.
Suppose
that the consumer has Cobb-Douglas preferences.
* * p1x1 p2x 2 m.
(B)
Computing Ordinary Demands a Cobb-Douglas Example.
So we have discovered that the most preferred affordable bundle for a consumer with Cobb-Douglas preferences
Chapter Five
Choice 消费者最优选择
Structure
Rational
constrained choice（理性约 束性选择） Computing ordinary demands – Interior solution （内部解） – Corner solution （角点解） – “Kinky” solution（特殊解） Example: Choosing taxes（选择税收 类型）
Where Are We Doing in This Chapter?
After
modeling a consumer‟s choice set and his preference (represented by utility functions), we now put them together and model how he/she makes optimal choice. In mathematical terms, this is a constrained maximization problem; In economics, this is a rational choice problem.
* x2
bm ( a b )p 2
x* 1
am ( a b)p1
x1
Rational Constrained Choice
When
x1* > 0 and x2* > 0 and (x1*,x2*) exhausts the budget, and indifference curves have no „kinks‟, the ordinary demands are obtained by solving: (a) p1x1* + p2x2* = y (b) the slopes of the budget constraint, -p1/p2, and of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).
At
(x1*,x2*), MRS = -p1/p2 so
Computing Ordinary Demands a Cobb-Douglas Example.
So
the MRS is
a1 b dx 2 U/ x1 ax1 x 2 ax 2 MRS . a dx1 U/ x 2 bx1 bx1 xb 1 2
x2*
x1*
x1
Rational Constrained Choice
(x1*,x2*)
satisfies two conditions: (a) the budget is exhausted; p1x1* + p2x2* = m (b) the slope of the budget constraint, -p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).
a U( x1 , x 2 ) x1 xb 2
Then
U a MU1 ax1 1xb 2 x1
U a MU2 bx1 xb 1 2 x2
Computing Ordinary Demands a Cobb-Douglas Example.
So
the MRS is
y p1
x1
Examples of Corner Solutions -the Perfect Substitutes Case
Rational Constrained Choice
But
what if x1* = 0? Or if x2* = 0? If either x1* = 0 or x2* = 0 then the ordinary demand (x1*,x2*) is at a corner solution to the problem of maximizing utility subject to a budget constraint.
Examples of Corner Solutions -the Perfect Substitutes Case
x2 MRS = -1
Slope = -p1/p2 with p1 > p2. x1
Examples of Corner Solutions -the Perfect Substitutes Case
Rational Constrained Choice
x2 More preferred bundles
Affordable bundles
x1
Rational Constrained Choice
x2
x2*
x1*
x1
Rational Constrained Choice
x2 (x1*,x2*) is the most preferred affordable bundle.
消费者的边际支付 意愿要大于市场交 换比率，消费者通 过增加消费X1,使消 费者的境况改善。 即MRS>P1/P2（这 里是指斜率）
x2
x1
x1
Computing Ordinary Demands （计算正常需求）
Solve
for 2 simultaneous equations.（解联 立方程） – Tangency – Budget constraint The conditions may be obtained by using the Lagrangian （拉格朗日的）multiplier method, i.e., constrained optimization in calculus（微积分、结石）.
x2 (x1*,x2*) is interior. (a) (x1*,x2*) exhausts the budget; p1x1* + p2x2* = m.
x2*
x1*
x1
Rational Constrained Choice
x2 (x1*,x2*) is interior . (b) The slope of the indiff. curve at (x1*,x2*) equals the slope of the budget constraint.
Meaning of the Tangency Condition
切点的含义 Consumer‟s marginal willingness to pay equals the market exchange rate.（消费者的 边际支付意愿等于市场交换率） Suppose at a consumption bundle (x1, x2), 市场交换率 MRS=-2, -P1/P2=-1 – The consumer is willing to give up 2 unit of x2 to exchange for an additional unit of x1 – The market allows her to give up only 1 unit of x2 to obtain an additional x1 (x1, x2) is not optimal choice She can be better off increasing her consumption of x1.
At
(x1*,x2*), MRS = -p1/p2 so
ax* 2
* bp1 * x2 x1 . ap2
p1 * p2 bx1
(A)
Computing Ordinary Demands a Cobb-Douglas Example.
(x1*,x2*)
also exhausts the budget so
a1 b dx 2 U/ x1 ax1 x 2 ax 2 MRS . a b 1 dx1 U/ x 2 bx1 bx1 x 2
Computing Ordinary Demands a Cobb-Douglas Example.
So
the MRS is
a1 b dx 2 U/ x1 ax1 x 2 ax 2 MRS . a b 1 dx1 U/ x 2 bx1 bx1 x 2