Consider a very simple model along the lines of Alesina and Rodrik (1994). Population is normalized to one, and aggregate (as well as per capita) output is
y=Akαg1−α (4)
wherekis the aggregate capital stock andg is the amount of public goods provided. The economy consists of one firm operating under perfect competition, takingg as given, so that the wage rate and the rate of return on capital are
w = (1−α)Akαg1−α (5a)
r = αAkα−1g1−α (5b)
The public good is funded by a tax rate of (τ) on the assessed value (a) of the capital stock so that g=τ a. Note that the assessed value will not necessarily be exactly equal to the capital stock.
An individual’s income depends on their wage income, which is identical across all individuals, plus the return to their personal capital, minus the tax on the assessed value of their personal capital,
yi=w+rki−τ ai. (6)
If the assessed value of personal capital is exactly equal toki, then this simply reduces to the model as originally written by Alesina and Rodrik. The net return on capital for an individual is then r−τ, and as this is identical across all individuals, all individuals make the same choice of savings rates, and so there is no change in the distribution of capital across individuals over time.
To introduce redistribution into this analysis, let us assume that the assessed value of capital is related to the actual value of capital in the following manner,
ai=kγ+1i (7)
and the value of γ ∈ (−1,1). The aggregate assessment is simply a = P
iai and the aggregate capital stock isk=P
iki. To see the effect of different values ofγ, consider the assessment rate of capital, orai/ki,
ai
ki =kγi. (8)
As can be seen, ifγ >0 then the assessment rate on personal property is progressive, or increasing in the amount of personal capital. However, ifγ <0, then assessment rates are regressive and an increase in the amount of personal capital results in a lower assessment rate. A value ofγ =−1 implies a simple head tax. In the specific case that γ= 0 then each person has an assessment of ai=ki, as in the original Alesina and Rodrik model.
So what is the optimal tax rate from the perspective of individuali? Let us assume only that the individual is interested in maximizing income. The individual is presumed to be inconsequential enough to ignore the effect of their choice on the level ofkor the assessment ratea/k. Usingg=τ a along with (5a) and (5b) in equation (6) we get
yi = (1−α)Akα(τ a)1−α+αAkα−1(τ a)1−αki−τ kγ+1i . (9) Maximizing overτ yields the following solution for τi∗, the individual’s optimal tax rate,
τi∗=
In the median voter model, the tax rate implemented will be equal to the optimal tax rate of the median individual, who holdskm in assets. Assuming that there is some inequality in the distribution of assets, then it must be the case thatkm< k, or the median individual has fewer assets than the average individual. A natural measure of inequality is then the ratio k/km. Increasing values ofk/kmindicate increasing inequality.
The question now is how the optimal tax rate changes with inequality. Holding kconstant, it can be shown that the relationship ofk/kmto the implemented tax rate,τm, depends crucially on regressiveness of the tax itself. Ifγ=−1 then we have a head tax, and τm declines as inequality (k/km) increases. If taxes are regressive and−1< γ < α−1 then taxes will decline with inequality up to a point. As long as k/km < γα/((1−α)(−γ−1)) then taxes fall as inequality increases.
Once inequality is high enough to cross this threshold, then the median voter has so little wealth that the punitive tax rate does not offset the gains they get to wages from voting for more public goods. Finally, ifγ > α−1, then taxes increase with inequality no matter initial inequality.
The important point is that for some combinations of γ and k/km, the relationship between
inequality and taxes is negative, even though there is full voting and the median voter decides the tax rate.
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Figure 1: Property Tax Rates, by Level of Government, 1860-1930
Figure 2: Real and Assessed Property Values, 1860-1920
Figure 3: Assessment Rates as a Function of Estate Size
Table 1: County Level Summary Statistics, 1890
Variable Mean Std. Dev. Min Max
Property tax and wealth data:
Total taxes per capita ($) 5.900 4.782 0.441 73.203 All school taxes per capita ($) 1.942 1.577 0.102 16.068 Local school taxes per capita ($) 1.628 1.465 0.002 10.885 Other taxes per capita ($) 3.957 3.529 0.059 57.135
Total tax rate 0.013 0.006 0.001 0.093
School tax rate 0.004 0.003 0.000 0.040
Local school tax rate 0.003 0.003 0.000 0.037
Other tax rate 0.008 0.004 0.000 0.086
Total wealth per capita ($) 501.199 489.242 61.627 9182.882 Inequality and other control data:
Farm Gini 0.529 0.147 0.033 0.926
Log output per capita 4.455 0.648 2.220 6.986
Proportion black 0.118 0.199 0.000 0.940
Proportion urban 0.134 0.211 0.000 1.000
Proportion children 0.379 0.044 0.209 0.466
Log of total population 9.706 0.927 4.890 14.231
Geographic controls:
Avg. annual temperature (C) 12.150 3.982 3.000 23.800 Total annual rainfall (cm) 98.140 24.842 19.100 175.100 Length growing period (days) 180.426 39.808 81.000 345.000
Latitude (degrees) 38.929 4.200 25.535 48.794
Notes:All variables have 1966 observations. Property tax and wealth data is from the U.S. CensusReport on Wealth, Debt, and Taxation, U.S. Department of the Interior (1895). Total taxes per capita are the total of all ad valorem taxes collected within a county, by any level of government, divided by county population. School taxes are the subset of taxes collected explicitly to fund education. Local school taxes are a subset of school taxes collected only by municipal or school district authorities. The other taxes are simply total taxes minus school taxes. Tax rates are calculated as the total tax revenue in the county divided by total wealth. The farm Gini is calculated from farm size information in the U.S. Census of 1890. Output per capita is the sum of manufacturing output and agricultural output, as reported in the U.S. Census of 1890. Demographic variables are from the U.S. Census. Geographic variables are obtained from the GEOECOLOGY database of Olson et al (2003). See the appendix for more detail on the construction of all variables.
Table 2: OLS Regressions for County Property Taxes on Inequality, 1890
Dependent Variables:
(1) (2) (3) (4) (5) (6)
lnT /L lnT /V lnV /L lnT /L lnT /V lnV /L
(Taxes p.c.) (Tax rate) (Wealth p.c.) (Taxes p.c.) (Tax rate) (Wealth p.c.)
= (2)+(3) = (5)+(6)
Panel A: T = Total Property Tax Revenue
Farm Gini 0.274 -0.337** 0.611 0.284 -0.634*** 0.918***
(0.440) (0.159) (0.374) (0.278) (0.125) (0.256)
R-squared 0.634 0.348 0.529 0.810 0.390 0.718
Panel B: T = All School Property Tax Revenue
Farm Gini -0.138 -0.749*** 0.611 0.072 -0.846*** 0.918***
(0.442) (0.221) (0.374) (0.315) (0.224) (0.256)
R-squared 0.734 0.539 0.529 0.830 0.552 0.718
Panel C: T = Local School Property Tax Revenue
Farm Gini -0.078 -0.689*** 0.611 0.027 -0.891*** 0.918***
(0.461) (0.246) (0.374) (0.321) (0.244) (0.256)
R-squared 0.753 0.661 0.529 0.816 0.665 0.718
Panel D: T = Other Property Tax Revenue
Farm Gini 0.460 -0.151 0.611 0.401 -0.517*** 0.918***
(0.441) (0.156) (0.374) (0.278) (0.122) (0.256)
R-squared 0.550 0.266 0.529 0.747 0.323 0.718
Controls No No No Yes Yes Yes
Notes: All regressions have 1966 observations and include state fixed effects. Standard errors, clustered at the state level, are reported in parentheses. * denotes significance at 10%, ** denotes 5%, and *** denotes 1%. The panels vary on the measure of property tax revenue used, and the categories are discussed in the text. Note that results for wealth per capita (V /L) do not vary by panel as the wealth is not contingent on the type of tax collected. Tax and wealth data are from U.S. Census special reports, and the Farm Gini is calculated from U.S. Census data on farm sizes (see text). The final three columns include additional control variables: the log of output per capita, the percent urban, the percent black, the percent of children, and the log of total population. See the appendix for the sources of these variables.
Table 3: Instrumental Variable Regressions for Log Property Tax Rates on Inequality, 1890 Dependent Variable is the log of the tax rate: lnT /V
(1) (2) (3) (4)
All taxes All School Local School Other Taxes
Taxes Taxes
Panel A: No additional controls
Farm Gini -0.524 -1.013** -0.948* -0.240
(0.437) (0.507) (0.542) (0.438)
Hansen J-stat 0.408 1.822 2.315 0.157
Hansen p-value 0.939 0.610 0.510 0.984
First stage F-stat 27.146 27.146 27.146 27.146 First stage p-value <0.001 <0.001 <0.001 <0.001 Panel B: All additional controls
Farm Gini -1.178** -1.540** -1.446** -0.871
(0.509) (0.628) (0.655) (0.538)
Hansen J-stat 0.526 2.222 2.687 0.671
Hansen p-value 0.913 0.528 0.442 0.880
First stage F-stat 12.863 12.863 12.863 12.863 First stage p-value <0.001 <0.001 <0.001 <0.001
Notes: Standard errors, clustered at the state level, are reported in parentheses. * denotes significance at 10%,
** denotes 5%, and *** denotes 1%. All regressions have 1966 observations and include state fixed effects. The excluded instruments in each regression are the log annual rainfall, the log of annual temperature, the log of the growing period, and latitude, all obtained from the GEOECOLOGY database of Olson et al (2003). Panel B includes log output per capita, the percent urban, the percent black, the percent children, and the log of total population as additional controls. See the appendix for the source of these control variables. The dependent variable in each column is the effective tax rate,T /V, whereT is the total amount of tax collected, andV is total wealth. T varies over the columns based on which taxes are used in the calculation. Tax and wealth data are from special U.S. Census reports, see the text for details. For both panels, the Hansen J statistic is distributedχ2(3) and the first stage F statistic is distributedF(4,35).
Table 4: Instrumental Variable Regressions for Property Tax Rates on Inequality, Expanded Sample, 1890
Dependent Variable is the level of the tax rate: T /V
(1) (2) (3) (4) (5)
All School All School All School Local School Other Taxes
Taxes Taxes Taxes Taxes
Panel A: No additional controls
Farm Gini -0.0057** 0.0018 -0.0079*** -0.0063*** -0.0008
(0.0025) (0.0013) (0.0025) (0.0023) (0.0052)
Hansen J-stat 2.754 0.692 2.072 2.390 1.232
Hansen p-value 0.431 0.874 0.557 0.495 0.745
First stage F-stat 21.724 40.800 27.145 21.724 21.724
First stage p-value <0.001 <0.001 <0.001 <0.001 <0.001 Panel B: All additional controls
Farm Gini -0.0115*** -0.0029** -0.0103*** -0.0127*** -0.0088 (0.0040) (0.0012) (0.0031) (0.0040) (0.0075)
Hansen J-stat 1.923 3.642 2.313 1.672 3.730
Hansen p-value 0.588 0.302 0.509 0.643 0.292
First stage F-stat 5.253 12.576 12.862 5.253 5.253
First stage p-value 0.002 0.001 <0.001 0.002 0.002
Sample Information
Included counties All Local school Local school All All
taxes = 0 taxes>0
N 2303 337 1966 2303 2303
Notes: Standard errors, clustered at the state level, are reported in parentheses. * denotes significance at 10%,
** denotes 5%, and *** denotes 1%. All regressions include state fixed effects. The excluded instruments in each regression are the log annual rainfall, the log of annual temperature, the log of the growing period, and latitude, all obtained from the GEOECOLOGY database of Olson et al (2003). Panel B includes log output per capita, the percent urban, the percent black, the percent children, and the log of total population as additional controls. See the appendix for the source of these control variables. The dependent variable in each column is the effective tax rate, T /V, whereT is the total amount of tax collected, andV is total wealth.T varies over the columns based on which taxes are used in the calculation. Tax and wealth data are from special U.S. Census reports, see the text for details.
For all regressions, the Hansen J statistic is distributedχ2(3). The first stage F statistic is distributedF(4,35) in columns (1),(3)-(5), andF(4,9) in column (2).
Table 5: Regressions for Assessment Rate, 1890
Dep. Variable is log of the assessed prop. val.:
(1) (2) (3) (4)
Log true prop. value (β) 0.894*** 0.891*** 0.596*** 0.599***
(0.021) (0.020) (0.049) (0.050)
Farm Gini 0.066 -0.075
(0.154) (0.198)
Additional controls? No No Yes Yes
F-test stat forβ= 1 25.18 30.34 66.78 64.91 F-test p-value <0.001 <0.001 <0.001 <0.001
Notes: Standard errors, clustered at the state level, are reported in parentheses. * denotes significance at 10%, **
denotes 5%, and *** denotes 1%. All regressions are estimated using OLS, have 1966 observations and include state fixed effects. Additional controls are log output per capita, the percent urban, the percent black, the percent children, and the log of total population as additional controls. See the appendix for the source of these control variables. The dependent variable is the log of total assessed property value, while the “true” value is the reported market value of property. Assessment and wealth data are from special U.S. Census reports, see the text for details. The F-test tests the hypothesis that the coefficient on the log of true value,β, is equal to one. The statistic is distributedF(1,35).