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.

Henry George (1839-1897)

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.^[1] 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.^[2] 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.^[3]

Subsequent studies generalized the principle and found that the theorem holds even after relaxing assumptions.^[4] 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.^[5][6][7]

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.^[8][9]

Derivation

Stiglitz (1977)

The following derivation follows an economic model presented in Joseph Stiglitz’ 1977 theory of local public goods.^[1]

The resource constraint for a small urban economy can be written as:

${\displaystyle Y=f(N)=cN+G}$

Where ${\displaystyle Y}$ is output, ${\displaystyle f(N)}$ is a concave production function, ${\displaystyle N}$ is the size of the workforce or population, ${\displaystyle c}$ is the per capita consumption of private goods, and ${\displaystyle G}$ is government expenditures on local public goods.

Land rents in this model are calculated using a 'Ricardian rent identity':

${\displaystyle R=f(N)-f^{\prime }(N)N\;,}$

where ${\displaystyle f^{\prime }(N)=dY/dN=}$ marginal product of laborers.

The community planner wishes to choose the size of N that maximizes the per capita consumption of private goods:

${\displaystyle c={\frac {f(N)-G}{N}}\;.}$

Differentiating using the quotient rule yields:

${\displaystyle {\frac {dc}{dN}}={\frac {Nf^{\prime }(N)-f(N)+G}{N^{2}}}=0}$

from which we derive first-order conditions:

${\textstyle c=f^{\prime }(N)\;,}$

${\textstyle G=f(N)-f^{\prime }(N)N\;,}$

${\textstyle N^{*}={\frac {f(N)\;-\;G}{f^{\prime }(N)}}\;.}$

Comparison of the FOC for G and the Ricardian rent identity yields the equality:

${\displaystyle R=G\;.}$

Arnott and Stiglitz (1979)

The following derivation follows a simplified version of an urban economic model presented in Richard Arnott and Joseph Stiglitz's paper titled Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size. ^[10]

Essential Assumptions

Let "A" stand for assumption.

1. Linear transportation costs
2. The geometry of the city is a two-dimensional circle
3. The urban population is evenly distributed across the area of the city, so the number of residents is equivalent to the area they inhabit.
4. Land is homogenous, so land rents only reflect differences in transportation costs
5. No congestion.
6. Production exhibits constant returns-to-scale.
7. The city has an optimal population that maximizes per capita consumption and representative utility.

Additional Assumptions

• Individuals have identical tastes.
• No impure public goods.
• No land rents are defined along the urban boundary
• Shadow prices equal market prices.
• To simplify the local political process, the local public sector is assumed to be run by a ‘benevolent despot’ who maximizes social welfare functions and optimally chooses the city’s geometry and population size.

The Model

Let ${\displaystyle f(x)}$ represent the transport costs incurred at distance ${\displaystyle x}$ to get to the urban center (= 0). By A.1:

${\displaystyle f(x)=tx\;,}$

where ${\displaystyle t}$ is a constant (same for all ${\displaystyle x}$) representing transport costs per unit distance.

Because the shape of the city is circular A.2, and the population is evenly distributed across the area of a circle A.3, then aggregate transportation costs can be calculated using shell integration:

${\displaystyle {\begin{aligned}ATC&{}\equiv \int _{0}^{b}f(x)2\pi x\,dx\\&{}={\frac {2}{3}}t\pi b^{3}\;,\end{aligned}}}$

where ${\displaystyle b}$ is the distance of the urban boundary from the urban center, also called the urban radius.

Since land is homogeneous A.4, we assume that the sum of the transport cost ${\displaystyle f(x)}$ and land rent ${\displaystyle R(x)}$ paid at ${\displaystyle x}$ is the same everywhere, which means that, provided transport costs are linear, rents at ${\displaystyle x}$ satisfy the following equations:

${\displaystyle {\begin{aligned}R(x)&{}=f(b)-f(x)\\&{}=(b-x)t\;.\end{aligned}}}$

Shell integration of ${\displaystyle R(x)}$ yields aggregate land rents:

${\displaystyle {\begin{aligned}ALR&{}\equiv \int _{0}^{b}R(x)2\pi x\,dx\\&{}={\frac {1}{3}}t\pi b^{3}\;.\end{aligned}}}$

Therefore, for a circular region with a unitary density, aggregate land rents are half of aggregate transportation costs, regardless of the value of ${\displaystyle b}$ :

${\displaystyle ALR={\frac {ATC}{2}}\;.}$

The assumption of unitary density A.3 entails an urban radius:

${\displaystyle b=\left({\frac {N}{\pi }}\right)^{1/2}\;,}$

where ${\displaystyle N}$ is the population size.

Evaluating ${\displaystyle ATC}$ yields:

${\displaystyle {\begin{aligned}ATC&{}={\frac {2}{3}}t\pi ^{-1/2}N^{3/2}\\&{}=mN^{3/2}\;,\end{aligned}}}$

where ${\displaystyle m={\tfrac {2}{3}}t\pi ^{-1/2}}$ is a constant with respect to ${\displaystyle N}$ provided the absence of congestion externalities ${\displaystyle \partial t/\partial N=0}$ A.5.

The resource constraint facing the urban economy is:

${\displaystyle yN=cN+mN^{3/2}+G\;,}$

where ${\displaystyle y}$ is a constant (A.6) representing per capita output, ${\displaystyle c}$ is the per capita consumption of private goods (excluding transport services), and ${\displaystyle G}$ is the government expenditures on pure local public goods.

This allows us to write a maximization problem that is satisfied by A.7:

${\displaystyle \max _{N}c=y-mN^{1/2}-{\frac {G}{N^{2}}}\;,}$

with first-order conditions:

${\displaystyle {\begin{aligned}{\frac {dc}{dN}}&{}=-{\frac {m}{2}}N^{-1/2}+{\frac {G}{N^{2}}}=0\\&{}\Rightarrow G={\frac {m}{2}}N^{3/2}={\frac {ATC}{2}}\;.\end{aligned}}}$

Thus, the optimal population size that maximizes per capita consumption is also such that aggregate land rents equal the expenditures on pure public goods:

${\displaystyle ALR={\frac {ATC}{2}}=G\;.}$

A similar result can be obtained by employing a Lagrangian function. However, since the Henry George theorem is satisfied for any level of expenditure on pure local public goods ${\displaystyle G}$, deriving the optimal level of ${\displaystyle G}$ that satisfies the Samuelson condition isn’t necessary. ^[11]

See also

• Geolibertarianism
• Georgism
• Land value tax
• Public goods
• Value capture