rational choice. choice 1. scarcity (income constraint) 2. tastes (indifference map/utility...
TRANSCRIPT
![Page 1: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/1.jpg)
Rational Choice
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CHOICE
1. Scarcity (income constraint)
2. Tastes (indifference map/utility function)
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ECONOMIC RATIONALITY
The principal behavioral postulate is that a decision-maker chooses its most preferred alternative from those available to it.
The available choices constitute the choice set.
How is the most preferred bundle in the choice set located/found?
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RATIONAL CONSTRAINED CHOICE
Affordablebundles
x1
x2
More preferredbundles
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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.
E
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RATIONAL CONSTRAINED CHOICE
MRS=x2/x1 = p1/p2
Slope of the indifference curve
Slope of the budget constraint
Individual’s willingness to trade
Society’s willingness to trade
At Equilibrium E
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RATIONAL CONSTRAINED CHOICE
The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and income.
Ordinary demands will be denoted byx1*(p1,p2,m) and x2*(p1,p2,m).
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RATIONAL CONSTRAINED CHOICE
x1
x2
x1*
x2*
The slope of the indifference curve at (x1*,x2*) equals the slope of the budgetconstraint.
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RATIONAL CONSTRAINED CHOICE
(x1*,x2*) satisfies two conditions: (i) the budget is exhausted, i.e.
p1x1* + p2x2* = m; and (ii) 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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COMPUTING DEMAND
How can this information be used to locate (x1*,x2*) for given p1, p2 and m?
Two ways to do this
1. Use Lagrange multiplier method
2. Find MRS and substitute into the Budget Constraint
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COMPUTING DEMAND Lagrange Multiplier Method
Suppose that the consumer has Cobb-Douglas preferences
and a budget constraint given by
mxpxp 2211
aaxxxxU 12121 ),(
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COMPUTING DEMAND Lagrange Multiplier Method
Aim
Set up the Lagrangian
mxpxpxx aa 2211
121 tosubject max
22111
21,,
21
xpxpmxxL aa
xx
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COMPUTING DEMAND Lagrange Multiplier Method
Differentiate
(3) 0
(2) 0)1(
(1) 0
2211
2212
11
21
11
xpxpmL
pxxax
L
pxaxx
L
aa
aa
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COMPUTING DEMAND Lagrange Multiplier Method
From (1) and (2)
Then re-arranging
2
21
1
12
11 1
p
xxa
p
xaxλ
aaaa-
1
2
2
1
1 xa
ax
p
p
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COMPUTING DEMAND Lagrange Multiplier Method
Rearrange
Remember
axap
xp
1
2211
mxpxp 2211
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COMPUTING DEMAND Lagrange Multiplier Method
Substitute
axap
xp
1
2211
mxpxp 2211
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COMPUTING DEMAND Lagrange Multiplier Method
Solve x1* and x2*
and 1
*1 p
amx
2
*2
1
p
max
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COMPUTING DEMAND Method 2
Suppose that the consumer has Cobb-Douglas preferences.
U x x x xa b( , )1 2 1 2
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COMPUTING DEMAND Method 2
Suppose that the consumer has Cobb-Douglas preferences.
U x x x xa b( , )1 2 1 2
MUUx
ax xa b1
11
12
MUUx
bx xa b2
21 2
1
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COMPUTING DEMAND Method 2
So the MRS is
MRSdxdx
U xU x
ax x
bx x
axbx
a b
a b
2
1
1
2
11
2
1 21
2
1
//
.
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COMPUTING DEMAND Method 2
So the MRS is
At (x1*,x2*), MRS = -p1/p2 , i.e. the slope of the budget constraint.
MRSdxdx
U xU x
ax x
bx x
axbx
a b
a b
2
1
1
2
11
2
1 21
2
1
//
.
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COMPUTING DEMAND Method 2
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
11
2
1 21
2
1
//
.
ax
bx
pp
xbpap
x2
1
1
22
1
21
*
** *. (A)
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COMPUTING DEMAND Method 2
(x1*,x2*) also exhausts the budget so
p x p x m1 1 2 2* * . (B)
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COMPUTING DEMAND Method 2
So now we know that
xbpap
x21
21
* * (A)
p x p x m1 1 2 2* * . (B)
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COMPUTING DEMAND Method 2
So now we know that
xbpap
x21
21
* * (A)
p x p x m1 1 2 2* * . (B)
Substitute
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COMPUTING DEMAND Method 2
So now we know that
xbpap
x21
21
* * (A)
p x p x m1 1 2 2* * . (B)
p x pbpap
x m1 1 21
21
* * .
Substitute
and get
This simplifies to ….
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COMPUTING DEMAND Method 2
xam
a b p11
*
( ).
![Page 29: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/29.jpg)
COMPUTING DEMAND Method 2
2
*2 )( pba
bmx
Substituting for x1* in p x p x m1 1 2 2
* *
then gives
1
*1 )( pba
amx
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COMPUTING DEMAND Method 2
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 )(21
*2
*1 )(
,)(
),(pba
mb
pba
maxx
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COMPUTING DEMAND Method 2: Cobb-Douglas
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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Rational Constrained Choice
But what if x1* = 0 or 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: Perfect Substitutes
x1
x2
MRS = -1
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Examples of Corner Solutions: Perfect Substitutes
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
![Page 35: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/35.jpg)
Examples of Corner Solutions: Perfect Substitutes
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
![Page 36: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/36.jpg)
Examples of Corner Solutions: Perfect Substitutes
x1
x2
2
*2 p
mx
x1 0*
MRS = -1 (This is the indifference curve)
Slope = -p1/p2 with p1 > p2.
![Page 37: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/37.jpg)
Examples of Corner Solutions: Perfect Substitutes
x1
x2
1
*1 p
mx
x2 0*
MRS = -1
Slope = -p1/p2 with p1 < p2.
ANOTHER EXAMPLE
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Examples of Corner Solutions: Perfect Substitutes
So when U(x1,x2) = x1 + x2, the mostpreferred affordable bundle is (x1*,x2*)where
0,),(
1
*2
*1 p
mxx
or
2
*2
*1 ,0),(
p
mxx
if p1 < p2
if p1 > p2.
![Page 39: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/39.jpg)
Examples of Corner Solutions: Perfect Substitutes
x1
x2
MRS = -1
Slope = -p1/p2 with p1 = p2.
1p
m
2p
m
The budget constraint and the utility curve lie on each other
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Examples of Corner Solutions: Perfect Substitutes
x1
x2
All the bundles in the constraint are equally the most preferred affordable when p1 = p2.
yp2
yp1
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Examples of ‘Kinky’ Solutions: Perfect Complements
X1 (tonic)
X2 (gin) U(x1,x2) = min(ax1,x2)
x2 = ax1 (a = .5)
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Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2
MRS = 0
U(x1,x2) = min(ax1,x2)
x2 = ax1
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Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2
MRS = -
MRS = 0
U(x1,x2) = min(ax1,x2)
x2 = ax1
![Page 44: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/44.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2
MRS = -
MRS = 0
MRS is undefined
U(x1,x2) = min(ax1,x2)
x2 = ax1
![Page 45: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/45.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
![Page 46: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/46.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
Which is the mostpreferred affordable bundle?
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Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
The most preferredaffordable bundle
![Page 48: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/48.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
x1*
x2*
![Page 49: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/49.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
x1*
x2*
and p1x1* + p2x2* = m
![Page 50: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/50.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m(b) x2* = ax1*
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Examples of ‘Kinky’ Solutions: Perfect Complements
(a) p1x1* + p2x2* = m; (b) x2* = ax1*
![Page 52: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/52.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m
![Page 53: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/53.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = mwhich gives
21
*1 app
mx
![Page 54: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/54.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = mwhich gives
21
*2
21
*1 ;
app
amx
app
mx
![Page 55: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/55.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = mwhich gives
21
*2
21
*1 ;
app
amx
app
mx
![Page 56: Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility function)](https://reader036.vdocument.in/reader036/viewer/2022070411/56649f585503460f94c7e2e2/html5/thumbnails/56.jpg)
Examples of ‘Kinky’ Solutions: Perfect Complements
x1
x2U(x1,x2) = min(ax1,x2)
x2 = ax1
xm
p ap11 2
*
x
amp ap
2
1 2
*