Part II Capital Investment Choice Chapter 5 Measuring Investment Value:
Chapter 5 Choice
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Transcript of Chapter 5 Choice
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Chapter 5Choice
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Economic Rationality
The principal behavioral postulate is that a decisionmaker chooses his most preferred alternative from those available to him.
The available choices constitute the choice set.
How is the most preferred bundle in the choice set located?
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Rational Constrained Choice
x1
x2
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Rational Constrained Choice
x1
x2
Affordablebundles
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Rational Constrained Choice
x1
x2
Affordablebundles
More preferredbundles
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Rational Constrained Choice
Affordablebundles
x1
x2
More preferredbundles
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Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is the mostpreferred affordablebundle.
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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 byx1*(p1,p2,m) and x2*(p1,p2,m).
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Rational Constrained Choice
When x1* > 0 and x2* > 0 the demanded bundle is interior.
If buying (x1*,x2*) costs $m then the budget is exhausted.
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Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.(a) (x1*,x2*) exhausts thebudget; p1x1* + p2x2* = m.
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Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior .(b) The slope of the indiff.curve at (x1*,x2*) equals the slope of the budget constraint.
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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*).
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Rational Constrained Choice Condition (b) can be written as:
That is,
Therefore, at the optimal choice, the ratio of marginal utilities of the two commodities must be equal to their price ratio.
2 1 1
1 2 2
dx MU pMRS
dx MU p
1 1
2 2
MU p
MU p
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Mathematical Treatment
The consumer’s optimization problem can be formulated mathematically as:
One way to solve it is to use Lagrangian method:
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Mathematical Treatment Assuming the utility function is
differentiable, and there exists an interior solution.
The first order conditions are:
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Mathematical Treatment
So, for interior solution, we can solve for the optimal bundle using the two conditions:
1 1
2 2
1 1 2 2
,
.
MU p
MU p
p x p x m
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Computing Ordinary Demands - a Cobb-Douglas Example. Suppose that the consumer has
Cobb-Douglas preferences.
Then
U x x x xa b( , )1 2 1 2
MUUx
ax xa b1
1112
MUUx
bx xa b2
21 2
1
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Computing Ordinary Demands - a Cobb-Douglas Example. So the MRS is
At (x1*,x2*), MRS = -p1/p2 so
MRSdxdx
U xU x
ax x
bx x
axbx
a b
a b
2
1
1
2
112
1 21
2
1
//
.
ax
bx
pp
xbpap
x2
1
1
22
1
21
*
** *. (A)
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Computing Ordinary Demands - a Cobb-Douglas Example. So now we have
Solving these two equations, we have
xbpap
x21
21
* * (A)
p x p x m1 1 2 2* * . (B)
xam
a b p11
*
( ).
x
bma b p2
2
*
( ).
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Computing Ordinary Demands - a Cobb-Douglas Example. So we have discovered that the most
preferred affordable bundle for a consumer with Cobb-Douglas preferences
is
U x x x xa b( , )1 2 1 2
( , )( )
,( )
.* * ( )x xam
a b pbm
a b p1 21 2
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A Cobb-Douglas Example
x1
x2
xam
a b p11
*
( )
x
bma b p
2
2
*
( )
U x x x xa b( , )1 2 1 2
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A Cobb-Douglas Example Note that with the optimal bundle:
The expenditures on each good:
It is a property of Cobb-Douglas Utility function that expenditure share on a particular good is a constant.
( , )( )
,( )
.* * ( )x xam
a b pbm
a b p1 21 2
.)(
,)(
),( *22
*11
mba
bm
ba
axpxp
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Special Cases
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Special Cases
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Some Notes
For interior optimum, tangency condition is only necessary but not sufficient.
There can be more than one optimum.
Strict convexity implies unique solution.
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Corner Solution
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.
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
MRS = -1
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
x1 0*
MRS = -1
Slope = -p1/p2 with p1 > p2.
2
*2 p
mx
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
x2 0*
MRS = -1
Slope = -p1/p2 with p1 < p2.
1
*1 p
mx
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Examples of Corner Solutions -- the Perfect Substitutes Case So when U(x1,x2) = x1 + x2, the most
preferred affordable bundle is (x1*,x2*) where
and
0,),(
1
*2
*1 p
mxx if p1 < p2
2
*2
*1 ,0),(
p
mxx if p1 > p2
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 = p2.
1p
m
2p
m
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
All the bundles in the constraint are equally the most preferred affordable when p1 = p2.
1p
m
2p
m
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Examples of Corner Solutions -- the Non-Convex Preferences Case
x1
x2Better
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Examples of Corner Solutions -- the Non-Convex Preferences Case
x1
x2
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Examples of Corner Solutions -- the Non-Convex Preferences Case
x1
x2Which is the most preferredaffordable bundle?
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Examples of Corner Solutions -- the Non-Convex Preferences Case
x1
x2
The most preferredaffordable bundle
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Examples of Corner Solutions -- the Non-Convex Preferences Case
x1
x2
The most preferredaffordable bundle
Notice that the “tangency solution” is not the most preferred affordable bundle.
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2
MRS = -
MRS = 0
MRS is undefined
U(x1,x2) = min{ax1,x2}
x2 = ax1
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2U(x1,x2) = min{ax1,x2}
x2 = ax1
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Which is the mostpreferred affordable bundle?
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
The most preferredaffordable bundle
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m(b) x2* = ax1*
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Solving the two equations, we have
.app
amx;
appm
x21
*2
21
*1
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2U(x1,x2) = min{ax1,x2}
x2 = ax1
xm
p ap11 2
*
x
amp ap
2
1 2
*