# Henry George theorem

The Henry George theorem states that under certain conditions, aggregate spending by government on public goods will increase aggregate rent based on land value (land rent) more than that amount, with the benefit of the last marginal investment equaling its cost. The theory is named for 19th century U.S. political economist and activist Henry George.

## Theory

This general relationship, first noted by the French physiocrats in the 18th century, is one basis for advocating the collection of a tax based on land rents to help defray the cost of public investment that helps create land values. Henry George popularized this method of raising public revenue in his works (especially in Progress and Poverty), which launched the 'single tax' movement.
In 1977, Joseph Stiglitz showed that under certain conditions, beneficial investments in public goods will increase aggregate land rents by at least as much as the investments' cost. This proposition was dubbed the "Henry George theorem", as it characterizes a situation where Henry George's 'single tax' on land values, is not only efficient, it is also the only tax necessary to finance public expenditures. Henry George had famously advocated for the replacement of all other taxes with a land value tax, arguing that as the location value of land was improved by public works, its economic rent was the most logical source of public revenue.
Subsequent studies generalized the principle and found that the theorem holds even after relaxing assumptions. Studies indicate that even existing land prices, which are depressed due to the existing burden of taxation on income and investment, are great enough to replace taxes at all levels of government.
Economists later discussed whether the theorem provides a practical guide for determining optimal city and enterprise size. Mathematical treatments suggest that an entity obtains optimal population when the opposing marginal costs and marginal benefits of additional residents are balanced.
The status quo alternative is that the bulk of the value of public improvements is captured by the landowners, because the state has only (unfocused) income and capital taxes by which to do so.

## Derivation
### Stiglitz (1977)

The following derivation follows an economic model presented in Joseph Stiglitz’ 1977 theory of local public goods.
Suppose a community where production, a which function of the population size of the workforce N, renders private and public goods. The community seeks to maximize the utility function:
$U=U(c,G)\;,$
subject to the corresponding resource constraint:
$Y=f(N)=cN+G\;.$
Where Y is output, c is the per capita consumption of private goods, and G is the aggregate consumption of local public goods reflected by its government expenditure on its provision.
Land rents in this model are calculated using the ‘Ricardian rent identity,’ (See Luigi Pasinetti’s “A Mathematical Formulation of the Ricardian System,”):
$R=f(N)-f^{\prime }(N)N\;.$
Where $f^{\prime }(N)=\partial Y/\partial N=$ marginal product of laborers.
From the resource constraint f(N)=cN+G{\textstyle f(N)=cN+G}, it follows that:
$c={\frac {f(N)-G}{N}}\;.$
The community’s utility maximization problem becomes:
$\max _{N}U=U\left({\frac {f(N)-G}{N}},G\right)\;.$
Differentiation of the utility function with respect to N yields:
${\frac {dU}{dN}}={\frac {\partial U}{\partial c}}\cdot {\frac {Nf^{\prime }(N)-f(N)+G}{N^{2}}}=0\;.$
Since $\partial U/\partial c\neq 0$:
${\frac {Nf^{\prime }(N)-f(N)+G}{N^{2}}}=0\;,$
From which we derive first-order conditions:
c=f$^′$(N), {\textstyle c=f^{\prime }(N)\;, }
G=f(N)−f$^′$(N)N, {\textstyle G=f(N)-f^{\prime }(N)N\;, }
N$^∗$=f(N)−G/f$^′$(N).{\textstyle N^{*}={\frac {f(N)\;-\;G}{f^{\prime }(N)}}\;.}
Comparison of the FOC for G and the Ricardian rent identity reveals an equality, but only when size of the local workforce is optimal. Thus:
${\frac {dU}{dN}}=0\Rightarrow R=G\;.$
In summation, land rents would be just sufficient to finance a provision of local public goods, given that certain conditions are satisfied, since the population size that is such that utility is maximized, is also such that land rents equals government expenditures on local public goods.