消费者最优选择(3)

p1x*1 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
x2
The most preferred affordable bundle
x1
Examples of Corner Solutions --
the Non-Convex Preferences Case
x2
Notice that the “tangency solution” is not the most preferred affordable
Computing Ordinary Demands a Cobb-Douglas Example.
Suppose that the consumer has Cobb-Douglas preferences.
U(x1, x2 ) x1axb2
Computing Ordinary Demands a Cobb-Douglas Example.
(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*).
Rational Constrained Choice
So the MRS is
MRS
dx2 dx1
U/ x1 U/ x2
ax1a1xb2 bx1axb2 1
ax2 . bx1
Computing Ordinary Demands a Cobb-Douglas Example.
So the MRS is
MRS
dx2 dx1
U/ U/
x1 x2
x2
x1
Examples of Corner Solutions -the Non-Convex Preferences Case
x2 Which is the most preferred affordable bundle?
x1
Examples of Corner Solutions -the Non-Convex Preferences Case
(a
am 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:
bundle.
The most preferred affordable bundle
x1
Examples of ‘Kinky’ Solutions -the Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1 x1
Examples of ‘Kinky’ Solutions -the Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1 MRS = 0
x1
Examples of ‘Kinky’ Solutions -the Perfect Complements Case
x2
U(x1,x2) = min{ax1,x2}
MRS = -
x2 = ax1 MRS = 0
Chapter Five
Choice 消费者最优选择
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.
Suppose that the consumer has Cobb-Douglas preferences.
U(x1, x2 ) x1axb2
Then
MU1
U x1
ax1a1xb2
MU2
U x2
bx1axb2 1
Computing Ordinary Demands a Cobb-Douglas Example.
U(x1, x2 ) x1axb2
is
( ) (x*1,x*2)
am , bm . (a b)p1 (a b)p2
Computing Ordinary Demands -
a Cobb-Douglas Example.
x2
U(x1, x2 ) x1axb2
x*2 bm
(a b)p2
x*1
curve at (x1*,x2*) equals the slope of the budget
constraint. x2*
x1*
x1
Rational Constrained Choice
(x1*,x2*) satisfies two conditions: (a) the budget is exhausted;
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
most preferred affordable
when p1 = p2.
y
x1
p1
Examples of Corner Solutions -the Non-Convex Preferences Case
x2
x1
Examples of Corner Solutions -the Non-Convex Preferences Case
Choice of the Consumer
Step 1: Draw the budget set;
Step 2: Draw the indifference curves;
Step 3: Locate the point of optimal choice and calculate the solution.
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ax1a1xb2 bx1axb2 1
ax2 bx1
.
At (x1*b a,x2* ), MRppS2 1=xx-1 p* * 21/p2 so
(A)
Computing Ordinary Demands a Cobb-Douglas Example.
(x1*,x2*) also exhausts the budget so
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.