lectures in macroeconomics- charles w. upton wealth and taxes-the constraint
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Lectures in Macroeconomics- Charles W. Upton
Wealth and Taxes-The Constraint
2
32
))1(1(
)1(
))1(1(
)1(
r
w
r
wz
r
w
r
w
Wealth and Taxes-The Constraint
Assumptions
• People live four periods.
• Before-tax wage income of w1, w2, w3, w4
• Before-tax interest rate of r
• Tax rates of w, r
Wealth and Taxes-The Constraint
The Basic Consumption Function
44
33
22
11
2
13
14
1
zc
zc
zc
zc
Our consumption
function
Wealth and Taxes-The Constraint
The Basic Consumption Function
44
33
22
11
2
13
14
1
zc
zc
zc
zc
Our consumption
functionlog(c1) + log(c2) + log(c3) + log(c4)
Wealth and Taxes-The Constraint
Assets
a1 = w1(1-w) – c1
a2 = a1[1+r(1- r)] +w2(1-w) – c2
a3 = a2[1+r(1- r)] +w3(1-w) – c3
a4 = a3[1+r(1- r)] +w4(1-w) – c4
Wealth and Taxes-The Constraint
Assets
a1 = w1(1-w) – c1
a2 = a1[1+r(1- r)] +w2(1-w) – c2
a3 = a2[1+r(1- r)] +w3(1-w) – c3
a4 = a3[1+r(1- r)] +w4(1-w) – c4
Wealth and Taxes-The Constraint
Assets
a1 = w1(1-w) – c1
a2 = a1[1+r(1- r)] +w2(1-w) – c2
a3 = a2[1+r(1- r)] +w3(1-w) – c3
a4 = a3[1+r(1- r)] +w4(1-w) – c4
Wealth and Taxes-The Constraint
Assets
a1 = w1(1-w) – c1
a2 = a1[1+r(1- r)] +w2(1-w) – c2
a3 = a2[1+r(1- r)] +w3(1-w) – c3
a4 = a3[1+r(1- r)] +w4(1-w) – c4
Wealth and Taxes-The Constraint
Assets
a1 = w1(1-w) – c1
a2 = a1[1+r(1- r)] +w2(1-w) – c2
a3 = a2[1+r(1- r)] +w3(1-w) – c3
a4 = a3[1+r(1- r)] +w4(1-w) – c4=0
Wealth and Taxes-The Constraint
Manipulation-Hold
a1 = w1(1-w) – c1
a2 = a1[1+r(1- r)] +w2(1-w) – c2
a3 = a2[1+r(1- r)] +w3(1-w) – c3
a4 = a3[1+r(1- r)] +w4(1-w) – c4
Wealth and Taxes-The Constraint
Manipulation
a3[1+r(1- r)] +w4(1-w) – c4=0
a3 = a2[1+r(1- r)] +w3(1-w) – c3
{ a2[1+r(1- r)] +w3(1-w) – c3}
[1+r(1- r)] +w4(1-w) – c4=0
Wealth and Taxes-The Constraint
Manipulation
{ a2[1+r(1- r)] +w3(1-w) – c3}
[1+r(1- r)] +w4(1-w) – c4=0
a2 = a1[1+r(1- r)] +w2(1-w) – c2
{ [a1[1+r(1- r)] +w2(1-w) – c2][1+r(1- r)] +w3(1-w) – c3}
[1+r(1- r)] +w4(1-w) – c4=0
Wealth and Taxes-The Constraint
Manipulation
{ [a1[1+r(1- r)] +w2(1-w) – c2][1+r(1- r)] +w3(1-w) – c3}
[1+r(1- r)] +w4(1-w) – c4=0
a1 = w1(1-w) – c1
{ [(w1(1-w) – c1)[1+r(1- r)] +w2(1-w) – c2][1+r(1- r)] +w3(1-w) – c3}
[1+r(1- r)] +w4(1-w) – c4=0
Wealth and Taxes-The Constraint
Manipulation
{ [(w1(1-w) – c1)[1+r(1- r)] +w2(1-w) – c2][1+r(1- r)] +w3(1-w) – c3}
[1+r(1- r)] +w4(1-w) – c4=0
w1(1-w)[1+r(1- r)]3+ w2(1-w)[1+r(1- r)]2+ w3(1-w)[1+r(1- r)]+ w4(1-w) =
c1[1+r(1- r)]3+ c2[1+r(1- r)]2+ c3[1+r(1- r)]+ c4
Wealth and Taxes-The Constraint
Manipulation
w1(1-w)[1+r(1- r)]3+ w2(1-w)[1+r(1- r)]2+ w3(1-w)[1+r(1- r)]+ w4(1-w) =
c1[1+r(1- r)]3+ c2[1+r(1- r)]2+ c3[1+r(1- r)]+ c4
Divide through by
[1+r(1- r)]3
Wealth and Taxes-The Constraint
Present Value
3
4
2
321
3
4
2
321
))1(1(
)1(
))1(1(
)1(
))1(1(
)1()1(
))1(1())1(1())1(1(
r
w
r
w
r
ww
rrr
r
w
r
w
r
ww
r
c
r
c
r
cc
Wealth and Taxes-The Constraint
Present Value
3
4
2
321
3
4
2
321
))1(1(
)1(
))1(1(
)1(
))1(1(
)1()1(
))1(1())1(1())1(1(
r
w
r
w
r
ww
rrr
r
w
r
w
r
ww
r
c
r
c
r
cc
z
Wealth and Taxes-The Constraint
The Wealth Formula
2
32
))1(1(
)1(
))1(1(
)1(
r
w
r
wz
r
w
r
w
Wealth and Taxes-The Constraint
Modifications are possible
• Set up the equations for a1, a2, a3, a4
• a4=0
• Add different taxes, transfer payments.
• Adjust for different interest rates and tax rates for different periods.
• Derive z.
Wealth and Taxes-The Constraint
Modifications are possible
• Set up the equations for a1, a2, a3, a4
• a4=0
• Add different taxes, transfer payments.
• Adjust for different interest rates and tax rates for different periods.
• Derive z.
Wealth and Taxes-The Constraint
Modifications are possible
• Set up the equations for a1, a2, a3, a4
• a4=0
• Add different taxes, transfer payments.
• Adjust for different interest rates and tax rates for different periods.
• Derive z.
Wealth and Taxes-The Constraint
End
©2003 Charles W. Upton. All rights reserved