choice. economic rationality u the principal behavioral postulate is that a decisionmaker chooses...
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Choice
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Economic Rationality
The principal behavioral postulate is that a decisionmaker chooses its most preferred alternative from those available to it.
The available choices constitute the choice set.
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Economic Rationality
How is the most preferred bundle in the choice set located?
The answer is simple: the consumer chooses the bundle within the choice set that is on the indifference curve that provides the highest level of utility.
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Rational Constrained Choice
x1
x2
x1*
x2*
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Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is the mostpreferred affordablebundle.
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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 For well-behaved (monotonic and
convex) preferences, tangency is a necessary and sufficient condition for optimality.
At the optimum the consumer substitutes one good for the other at a rate identical to the market’s.
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Computing Ordinary Demands - a Cobb-Douglas Example
Suppose that the consumer has Cobb-Douglas preferences.
U x x x xa b( , )1 2 1 2
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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
12 1 1 2 2
11 2 1 2 1
/
/
a b
a b
dx U x ax x axMRS
dx U x bx x bx
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Computing Ordinary Demands - a Cobb-Douglas Example.
So the MRS is
At (x1*,x2*), MRS = -p1/p2 so
12 1 1 2 2
11 2 1 2 1
/
/
a b
a b
dx U x ax x axMRS
dx U x bx x bx
** *2 1 12 1*
1 2 2
ax p bpx x
bx p ap (A)
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Computing Ordinary Demands - a Cobb-Douglas Example.
(x1*,x2*) also exhausts the budget so
* *1 1 2 2p x p x m (B)
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Computing Ordinary Demands - a Cobb-Douglas Example.
So now we know that
* *12 1
2
bpx x
ap (A)
* *1 1 2 2p x p x m (B)
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Computing Ordinary Demands - a Cobb-Douglas Example.
So now we know that
xbpap
x21
21
* * (A)
* *1 1 2 2p x p x m (B)
Substitute
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Computing Ordinary Demands - a Cobb-Douglas Example.
So now we know that
xbpap
x21
21
* * (A)
* *1 1 2 2p x p x m (B)
* *11 1 2 1
2
bpp x p x m
ap
Substitute
and get
This simplifies to ….
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Computing Ordinary Demands - a Cobb-Douglas Example.
*1
1( )
amx
a b p
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Computing Ordinary Demands - a Cobb-Douglas Example.
*2
2( )
bmx
a b p
Substituting for x1* in p x p x m1 1 2 2
* *
then gives
*1
1( )
amx
a b p
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Computing Ordinary Demands - a Cobb-Douglas Example.
So we have discovered that the mostpreferred affordable bundle for a consumerwith Cobb-Douglas preferences
U x x x xa b( , )1 2 1 2
is * *1 2
1 2
( , ) ,( ) ( )
am bmx x
a b p a b p
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Computing Ordinary Demands - a Cobb-Douglas Example.
x1
x2
*1
1( )
amx
a b p
*2
2( )
bmx
a b p
U x x x xa b( , )1 2 1 2
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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* = m
(b) the slopes of the budget constraint, -p1/p2, and of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).
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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.
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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
MRS = -1
Slope = -p1/p2 with p1 > p2
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
*2
2
mx
p
x1 0*
MRS = -1
Slope = -p1/p2 with p1 > p2
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Examples of Corner Solutions -- the Perfect Substitutes Case
x1
x2
*1
1
mx
p
x2 0*
MRS = -1
Slope = -p1/p2 with p1< p2
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Examples of Corner Solutions -- the Perfect Substitutes Case
So when U(x1,x2) = x1 + x2, the mostpreferred affordable bundle is (x1*,x2*)where
* *1 2
1
( , ) ,0m
x xp
and
* *1 2
2
( , ) 0,m
x xp
if p1 < p2
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
1
m
p
2
m
p
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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
2
m
p
1
m
p
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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
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
Which is the mostpreferred affordable bundle?
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
x1
x2U(x1,x2) = min{ax1,x2}
x2 = ax1
The most preferredaffordable bundle
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Examples of ‘Kinky’ Solutions -- the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*
Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = mwhich gives
* *1 2
1 2 1 2
,m am
x xp ap p ap
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Application: choosing a tax
Which is less harmful, a quantity tax or an income tax, both allowing the same revenue?
1. We can show that an income tax is always better in the sense that given any quantity tax, there is an income tax that leaves the consumer at a higher indifference curve.
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Application: choosing a tax
2. Outline of the argument:
a) original budget constraint: p1x1 + p2x2 = m
b) budget constraint with quantity tax:
(p1 + t)x1+p2x2 = m
c) optimal choice with tax: (p1+t)x1* + p2x2
* = m
d) revenue raised is tx1*
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Application: choosing a tax
e) income tax (lump-sum) that raises the same amount of revenue leads to budget constraint: p1x1 + p2x2 = m - tx1*i) this line has the same slope as the original budget line, but lies inwards
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Application: choosing a tax
ii) also passes through (x1*;x2*)
iii) proof: p1x1*+ p2x2* = m - tx1*
iv) this means that (x1*; x2* ) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x1 *; x2*)