Chapter Five
Choice
Structure
Rational constrained choice
Computing ordinary demands
–Interior solution （ 内在解）
–Corner solution （ 角点解）
–“Kinky” solution
Example,Choosing taxes
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.
How is the most preferred bundle in
the choice set located?
Rational Constrained Choice
x1
x2
Affordable
bundles
More preferred
bundles
Rational Constrained Choice
Affordable
bundles
x1
x2
More preferred
bundles
Rational Constrained Choice
x1
x2
x1*
x2*
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is the most
preferred affordable
bundle.
Rational Constrained Choice
The most preferred affordable bundle
is called the consumer?s ORDINARY
DEMAND （ 一般需求） at the given
prices and budget.
Ordinary demands will be denoted by
x1*(p1,p2,m) and x2*(p1,p2,m).
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,
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(x1*,x2*) exhausts the
budget.
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(a) (x1*,x2*) exhausts the
budget; p1x1* + p2x2* = m.
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.
Rational Constrained Choice
(x1*,x2*) satisfies two conditions:
(a) the budget is exhausted;
p1x1* + p2x2* = m
(b) tangency,the slope of the budget
constraint,-p1/p2,and the slope of the
indifference curve containing (x1*,x2*)
are equal at (x1*,x2*).
Meaning of the Tangency Condition
Consumer?s marginal willingness to pay
equals the market exchange rate.
Suppose at a consumption bundle (x1,x2),
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.
x1x1
x2
Computing Ordinary Demands
Solve for 2 simultaneous equations.
–Tangency
–Budget constraint
The conditions may be obtained by
using the Lagrangian multiplier
method,i.e.,constrained
optimization in calculus.
Computing Ordinary Demands
How can this information be used to
locate (x1*,x2*) for given p1,p2 and
m?
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?
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?
MU Ux ax xa b1
1
1 1 2
MU Ux bx xa b2
2
1 2 1
Computing Ordinary Demands -
a Cobb-Douglas Example.
So the MRS is
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
/
/
.
Computing Ordinary Demands -
a Cobb-Douglas Example.
So the MRS is
At (x1*,x2*),MRS = -p1/p2 so
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
/
/
.
Computing Ordinary Demands -
a Cobb-Douglas Example.
So the MRS is
At (x1*,x2*),MRS = -p1/p2 so
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
/
/
.
ax
bx
p
p
x bp
ap
x2
1
1
2
2
1
2
1
*
*
* *,
(A)
Computing Ordinary Demands -
a Cobb-Douglas Example.
(x1*,x2*) also exhausts the budget so
p x p x m1 1 2 2* *,(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,(B)
Substitute
Computing Ordinary Demands -
a Cobb-Douglas Example.
So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,(B)
p x p bpap x m1 1 2 1
2
1* *,
Substitute
and get
This simplifies to ….
Computing Ordinary Demands -
a Cobb-Douglas Example.
x ama b p1
1
*
( ),
Computing Ordinary Demands -
a Cobb-Douglas Example.
x bma b p2
2
*
( ),
Substituting for x1* in
p x p x m1 1 2 2* *
then gives
x ama b p1
1
*
( ),
Computing Ordinary Demands -
a Cobb-Douglas Example.
So we have discovered that the most
preferred affordable bundle for a consumer
with Cobb-Douglas preferences
U x x x xa b(,)1 2 1 2?
is (,)
( ),( ),
* * ( )x x a m
a b p
b m
a b p1 2 1 2
Computing Ordinary Demands -
a Cobb-Douglas Example.
x1
x2
x a ma b p1
1
*
( )
x
b m
a b p
2
2
*
( )
U x x x xa b(,)1 2 1 2?
Rational Constrained Choice,
Summary
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*).
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
x1
x2
MRS = -1
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
The equality of
–p1/p2 and MRS is
impossible.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp2
2
*?
x1 0*?
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp1
1
*?x2 0
*?
MRS = -1
Slope = -p1/p2 with p1 < p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
So when U(x1,x2) = x1 + x2,the most
preferred affordable bundle is (x1*,x2*)
where?
0,
p
y)x,x(
1
*
2
*
1
and?
2
*
2
*
1 p
y,0)x,x(
if p1 < p2
if p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 = p2.
y
p1
y
p2
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.
y
p2
y
p1
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Which is the most preferred
affordable bundle?
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Notice that the,tangency solution”
is not the most preferred affordable
bundle,but the least one.
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = -?
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = -?
MRS = 0
MRS is undefined
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Which is the most
preferred affordable bundle?
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
The most preferred
affordable bundle
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m
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*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
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* = m
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* = m
which gives
21
*
1 app
mx
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* = m
which gives,app
amx;
app
mx
21
*
2
21
*
1
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* = m
which gives
.
app
amx;
app
mx
21
*
2
21
*
1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x mp ap1
1 2
*?
x
am
p ap
2
1 2
*?
Choosing Taxes,Various Taxes
Quantity tax,on x,(p+t)x
Value tax,on px,(1+t)px
–Also called ad valorem tax
Lump sum tax,T
Income tax,
–Can be proportional or lump sum
Income Tax vs,Quantity Tax
Original budget,p1x1 + p2x2 = m
After quantity tax,
(p1+t)x1 + p2x2 = m
At optimal choice (x1*,x2*)
–(p1+t)x1* + p2x2* = m (5.2)
–Tax revenue,R*=tx1*
With an income tax,budget is:
p1x1 + p2x2 = m- tx1*
Income vs,Quantity Tax
Proposition,(x1*,x2*) is affordable under
income tax
Equivalent to,prove that (x1*,x2*) satisfies
budget constraint under income tax,
Or,budget constraint holds at point (x1*,
x2*).
p1x1* + p2x2*= m- tx1*
Which is true according to (5.2).
It is not an optimal choice because prices
are different.
Conclusion,The optimal choice must be
more preferred to (x1*,x2*)
Choice
Structure
Rational constrained choice
Computing ordinary demands
–Interior solution （ 内在解）
–Corner solution （ 角点解）
–“Kinky” solution
Example,Choosing taxes
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.
How is the most preferred bundle in
the choice set located?
Rational Constrained Choice
x1
x2
Affordable
bundles
More preferred
bundles
Rational Constrained Choice
Affordable
bundles
x1
x2
More preferred
bundles
Rational Constrained Choice
x1
x2
x1*
x2*
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is the most
preferred affordable
bundle.
Rational Constrained Choice
The most preferred affordable bundle
is called the consumer?s ORDINARY
DEMAND （ 一般需求） at the given
prices and budget.
Ordinary demands will be denoted by
x1*(p1,p2,m) and x2*(p1,p2,m).
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,
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(x1*,x2*) exhausts the
budget.
Rational Constrained Choice
x1
x2
x1*
x2*
(x1*,x2*) is interior.
(a) (x1*,x2*) exhausts the
budget; p1x1* + p2x2* = m.
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.
Rational Constrained Choice
(x1*,x2*) satisfies two conditions:
(a) the budget is exhausted;
p1x1* + p2x2* = m
(b) tangency,the slope of the budget
constraint,-p1/p2,and the slope of the
indifference curve containing (x1*,x2*)
are equal at (x1*,x2*).
Meaning of the Tangency Condition
Consumer?s marginal willingness to pay
equals the market exchange rate.
Suppose at a consumption bundle (x1,x2),
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.
x1x1
x2
Computing Ordinary Demands
Solve for 2 simultaneous equations.
–Tangency
–Budget constraint
The conditions may be obtained by
using the Lagrangian multiplier
method,i.e.,constrained
optimization in calculus.
Computing Ordinary Demands
How can this information be used to
locate (x1*,x2*) for given p1,p2 and
m?
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?
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?
MU Ux ax xa b1
1
1 1 2
MU Ux bx xa b2
2
1 2 1
Computing Ordinary Demands -
a Cobb-Douglas Example.
So the MRS is
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
/
/
.
Computing Ordinary Demands -
a Cobb-Douglas Example.
So the MRS is
At (x1*,x2*),MRS = -p1/p2 so
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
/
/
.
Computing Ordinary Demands -
a Cobb-Douglas Example.
So the MRS is
At (x1*,x2*),MRS = -p1/p2 so
M R S dx
dx
U x
U x
ax x
bx x
ax
bx
a b
a b
2
1
1
2
1
1
2
1 2
1
2
1
/
/
.
ax
bx
p
p
x bp
ap
x2
1
1
2
2
1
2
1
*
*
* *,
(A)
Computing Ordinary Demands -
a Cobb-Douglas Example.
(x1*,x2*) also exhausts the budget so
p x p x m1 1 2 2* *,(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,(B)
Computing Ordinary Demands -
a Cobb-Douglas Example.
So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,(B)
Substitute
Computing Ordinary Demands -
a Cobb-Douglas Example.
So now we know that
x bpap x2 1
2
1
* *?
(A)
p x p x m1 1 2 2* *,(B)
p x p bpap x m1 1 2 1
2
1* *,
Substitute
and get
This simplifies to ….
Computing Ordinary Demands -
a Cobb-Douglas Example.
x ama b p1
1
*
( ),
Computing Ordinary Demands -
a Cobb-Douglas Example.
x bma b p2
2
*
( ),
Substituting for x1* in
p x p x m1 1 2 2* *
then gives
x ama b p1
1
*
( ),
Computing Ordinary Demands -
a Cobb-Douglas Example.
So we have discovered that the most
preferred affordable bundle for a consumer
with Cobb-Douglas preferences
U x x x xa b(,)1 2 1 2?
is (,)
( ),( ),
* * ( )x x a m
a b p
b m
a b p1 2 1 2
Computing Ordinary Demands -
a Cobb-Douglas Example.
x1
x2
x a ma b p1
1
*
( )
x
b m
a b p
2
2
*
( )
U x x x xa b(,)1 2 1 2?
Rational Constrained Choice,
Summary
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*).
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
x1
x2
MRS = -1
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
The equality of
–p1/p2 and MRS is
impossible.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp2
2
*?
x1 0*?
MRS = -1
Slope = -p1/p2 with p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
x yp1
1
*?x2 0
*?
MRS = -1
Slope = -p1/p2 with p1 < p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
So when U(x1,x2) = x1 + x2,the most
preferred affordable bundle is (x1*,x2*)
where?
0,
p
y)x,x(
1
*
2
*
1
and?
2
*
2
*
1 p
y,0)x,x(
if p1 < p2
if p1 > p2.
Examples of Corner Solutions --
the Perfect Substitutes Case
x1
x2
MRS = -1
Slope = -p1/p2 with p1 = p2.
y
p1
y
p2
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.
y
p2
y
p1
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
Which is the most preferred
affordable bundle?
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Examples of Corner Solutions --
the Non-Convex Preferences Case
x1
x2
The most preferred
affordable bundle
Notice that the,tangency solution”
is not the most preferred affordable
bundle,but the least one.
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = -?
MRS = 0
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2
MRS = -?
MRS = 0
MRS is undefined
U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
Which is the most
preferred affordable bundle?
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
The most preferred
affordable bundle
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x1*
x2*
(a) p1x1* + p2x2* = m
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*
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
(a) p1x1* + p2x2* = m; (b) x2* = ax1*.
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* = m
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* = m
which gives
21
*
1 app
mx
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* = m
which gives,app
amx;
app
mx
21
*
2
21
*
1
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* = m
which gives
.
app
amx;
app
mx
21
*
2
21
*
1
Examples of ‘Kinky’ Solutions --
the Perfect Complements Case
x1
x2 U(x1,x2) = min{ax1,x2}
x2 = ax1
x mp ap1
1 2
*?
x
am
p ap
2
1 2
*?
Choosing Taxes,Various Taxes
Quantity tax,on x,(p+t)x
Value tax,on px,(1+t)px
–Also called ad valorem tax
Lump sum tax,T
Income tax,
–Can be proportional or lump sum
Income Tax vs,Quantity Tax
Original budget,p1x1 + p2x2 = m
After quantity tax,
(p1+t)x1 + p2x2 = m
At optimal choice (x1*,x2*)
–(p1+t)x1* + p2x2* = m (5.2)
–Tax revenue,R*=tx1*
With an income tax,budget is:
p1x1 + p2x2 = m- tx1*
Income vs,Quantity Tax
Proposition,(x1*,x2*) is affordable under
income tax
Equivalent to,prove that (x1*,x2*) satisfies
budget constraint under income tax,
Or,budget constraint holds at point (x1*,
x2*).
p1x1* + p2x2*= m- tx1*
Which is true according to (5.2).
It is not an optimal choice because prices
are different.
Conclusion,The optimal choice must be
more preferred to (x1*,x2*)
