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Lectures in Macroeconomics- Charles W. Upton Wealth and Taxes-The Constraint
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Assumptions People live four periods. Before-tax wage income of w 1, w 2, w 3, w 4 Before-tax interest rate of r Tax rates of w, r
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Wealth and Taxes-The Constraint The Basic Consumption Function Our consumption function
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Wealth and Taxes-The Constraint The Basic Consumption Function Our consumption function log(c 1 ) + log(c 2 ) + log(c 3 ) + log(c 4 )
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Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4
![Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4](http://images.slideplayer.com/17/5349696/slides/slide_5.jpg "Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4")
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Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4
![Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4](http://images.slideplayer.com/17/5349696/slides/slide_6.jpg "Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4")
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Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4
![Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4](http://images.slideplayer.com/17/5349696/slides/slide_7.jpg "Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4")
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Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4
![Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4](http://images.slideplayer.com/17/5349696/slides/slide_8.jpg "Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4")
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Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4 =0
![Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4 =0](http://images.slideplayer.com/17/5349696/slides/slide_9.jpg "Wealth and Taxes-The Constraint Assets a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4 =0")
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Wealth and Taxes-The Constraint Manipulation-Hold a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4
![Wealth and Taxes-The Constraint Manipulation-Hold a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4](http://images.slideplayer.com/17/5349696/slides/slide_10.jpg "Wealth and Taxes-The Constraint Manipulation-Hold a 1 = w 1 (1- w ) – c 1 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 a 4 = a 3 [1+r(1- r )] +w 4 (1- w ) – c 4")
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Wealth and Taxes-The Constraint Manipulation a 3 [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 { a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0
![Wealth and Taxes-The Constraint Manipulation a 3 [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 { a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0](http://images.slideplayer.com/17/5349696/slides/slide_11.jpg "Wealth and Taxes-The Constraint Manipulation a 3 [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 3 = a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 { a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0")
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Wealth and Taxes-The Constraint Manipulation { a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 { [a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0
![Wealth and Taxes-The Constraint Manipulation { a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 { [a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0](http://images.slideplayer.com/17/5349696/slides/slide_12.jpg "Wealth and Taxes-The Constraint Manipulation { a 2 [1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 2 = a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 { [a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0")
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Wealth and Taxes-The Constraint Manipulation { [a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 1 = w 1 (1- w ) – c 1 { [(w 1 (1- w ) – c 1 )[1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0
![Wealth and Taxes-The Constraint Manipulation { [a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 1 = w 1 (1- w ) – c 1 { [(w 1 (1- w ) – c 1 )[1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0](http://images.slideplayer.com/17/5349696/slides/slide_13.jpg "Wealth and Taxes-The Constraint Manipulation { [a 1 [1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 a 1 = w 1 (1- w ) – c 1 { [(w 1 (1- w ) – c 1 )[1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0")
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Wealth and Taxes-The Constraint Manipulation { [(w 1 (1- w ) – c 1 )[1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 w 1 (1- w )[1+r(1- r )] 3 + w 2 (1- w )[1+r(1- r )] 2 + w 3 (1- w )[1+r(1- r )]+ w 4 (1- w ) = c 1 [1+r(1- r )] 3 + c 2 [1+r(1- r )] 2 + c 3 [1+r(1- r )]+ c 4
![Wealth and Taxes-The Constraint Manipulation { [(w 1 (1- w ) – c 1 )[1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 w 1 (1- w )[1+r(1- r )] 3 + w 2 (1- w )[1+r(1- r )] 2 + w 3 (1- w )[1+r(1- r )]+ w 4 (1- w ) = c 1 [1+r(1- r )] 3 + c 2 [1+r(1- r )] 2 + c 3 [1+r(1- r )]+ c 4](http://images.slideplayer.com/17/5349696/slides/slide_14.jpg "Wealth and Taxes-The Constraint Manipulation { [(w 1 (1- w ) – c 1 )[1+r(1- r )] +w 2 (1- w ) – c 2 ][1+r(1- r )] +w 3 (1- w ) – c 3 } [1+r(1- r )] +w 4 (1- w ) – c 4 =0 w 1 (1- w )[1+r(1- r )] 3 + w 2 (1- w )[1+r(1- r )] 2 + w 3 (1- w )[1+r(1- r )]+ w 4 (1- w ) = c 1 [1+r(1- r )] 3 + c 2 [1+r(1- r )] 2 + c 3 [1+r(1- r )]+ c 4")
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Wealth and Taxes-The Constraint Manipulation w 1 (1- w )[1+r(1- r )] 3 + w 2 (1- w )[1+r(1- r )] 2 + w 3 (1- w )[1+r(1- r )]+ w 4 (1- w ) = c 1 [1+r(1- r )] 3 + c 2 [1+r(1- r )] 2 + c 3 [1+r(1- r )]+ c 4 Divide through by [1+r(1- r )] 3
![Wealth and Taxes-The Constraint Manipulation w 1 (1- w )[1+r(1- r )] 3 + w 2 (1- w )[1+r(1- r )] 2 + w 3 (1- w )[1+r(1- r )]+ w 4 (1- w ) = c 1 [1+r(1- r )] 3 + c 2 [1+r(1- r )] 2 + c 3 [1+r(1- r )]+ c 4 Divide through by [1+r(1- r )] 3](http://images.slideplayer.com/17/5349696/slides/slide_15.jpg "Wealth and Taxes-The Constraint Manipulation w 1 (1- w )[1+r(1- r )] 3 + w 2 (1- w )[1+r(1- r )] 2 + w 3 (1- w )[1+r(1- r )]+ w 4 (1- w ) = c 1 [1+r(1- r )] 3 + c 2 [1+r(1- r )] 2 + c 3 [1+r(1- r )]+ c 4 Divide through by [1+r(1- r )] 3")
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Wealth and Taxes-The Constraint Present Value
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Wealth and Taxes-The Constraint Present Value z
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Wealth and Taxes-The Constraint The Wealth Formula
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Wealth and Taxes-The Constraint Modifications are possible Set up the equations for a 1, a 2, a 3, a 4 a 4 =0 Add different taxes, transfer payments. Adjust for different interest rates and tax rates for different periods. Derive z.
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Wealth and Taxes-The Constraint Modifications are possible Set up the equations for a 1, a 2, a 3, a 4 a 4 =0 Add different taxes, transfer payments. Adjust for different interest rates and tax rates for different periods. Derive z.
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Wealth and Taxes-The Constraint Modifications are possible Set up the equations for a 1, a 2, a 3, a 4 a 4 =0 Add different taxes, transfer payments. Adjust for different interest rates and tax rates for different periods. Derive z.
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Wealth and Taxes-The Constraint End ©2003 Charles W. Upton. All rights reserved
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