Henry George theorem

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.{{cite book |author1=Stiglitz, Joseph |author-link1=Joseph Stiglitz |editor1-last=Feldstein |editor1-first=M.S. |editor2-last=Inman |editor2-first=R.P. |title=The Economics of Public Services |date=1977 |publisher=Palgrave Macmillan, London |isbn=978-1-349-02919-8 |pages=274–333 |chapter=The Theory of Local Public Goods|doi=10.1007/978-1-349-02917-4_12 }} 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.{{cite journal|last=Arnott|first=Richard J.|author2=Joseph E. Stiglitz|title=Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size|journal=Quarterly Journal of Economics|date=Nov 1979|volume=93|issue=4|pages=471–500|jstor=1884466|doi=10.2307/1884466|s2cid=53374401}} 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.{{cite book |title=Progress and Poverty |last=George |first=Henry |year=1879 | url= http://www.henrygeorge.org/pchp19.htm}}

Subsequent studies generalized the principle and found that the theorem holds even after relaxing assumptions.{{cite journal|author-last1=Behrens|author-first1=Kristian|author-last2= Kanemoto |author-first2=Yoshitsugu |author-last3= Murata |author-first3=Yasusada |title=The Henry George Theorem in a Second-Best World |journal=Journal of Urban Economics |date=Jan 2015|volume=85|pages=34–51|doi=10.1016/j.jue.2014.10.002|s2cid=52904689 |url=https://grips.repo.nii.ac.jp/record/1147/files/DP14-11.pdf }} 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.{{Cite web |url=http://www.masongaffney.org/publications/G1Adequacy_of_land.CV.pdf |title=Adequacy of Land as a Tax Base |access-date=2018-08-29 |archive-url=https://web.archive.org/web/20150415114523/http://www.masongaffney.org/publications/g1adequacy_of_land.cv.pdf |archive-date=2015-04-15 |url-status=dead}}{{cite web|last=Gaffney |first=Mason |author-link=Mason Gaffney |url=http://www.masongaffney.org/publications/G2009-Hidden_Taxable_Capacity_of_Land_2009.pdf |title=The Hidden Taxable Capacity of Land: Enough and to Spare |date= 2009}}{{cite web |last=Foldvary |first=Fred |author-link=Fred Foldvary |url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1103586 |title=The Ultimate Tax Reform: Public Revenue from Land Rent |date=January 2006|ssrn=1103586 }}

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.{{Cite web|last=Doucet|first=Lars|date=2021-12-09|title=Does Georgism Work?, Part 1: Is Land Really A Big Deal? |url=https://astralcodexten.substack.com/p/does-georgism-work-is-land-really|access-date=2021-12-26|website=Astral Codex Ten}}{{Cite web|last1=Kumhof|first1=Michael|last2=Tideman|first2=T. Nicolaus|last3=Hudson|first3=Michael|last4=Goodhart|first4=Charles|date=2021-10-20|title=Post-Corona Balanced-Budget Super-Stimulus: The Case for Shifting Taxes onto Land|url=https://papers.ssrn.com/abstract=3954888|language=en|location=Rochester, NY|ssrn=3954888 }}

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

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

$Y\; =\; f(N)\; =\; cN\; +\; G$

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

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

$R\; =\; f(N)\; -\; f^\backslash prime(N)N$

Where $f^\backslash prime(N)=\; dY/dN\; =$ marginal product of laborers.

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

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

Differentiating using the quotient rule yields:

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

From which we derive first-order conditions:

$c\; =\; f^\backslash prime(N)\; \backslash ;\; ,$

$G\; =\; f(N)\; -\; f^\backslash prime(N)N\; \backslash ;\; ,$

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

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

$R\; =\; G\; \backslash ;\; .$

The following derivation follows 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. {{cite journal|last=Arnott|first=Richard J.|author2=Joseph E. Stiglitz|title=Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size|journal=Quarterly Journal of Economics|date=Nov 1979|volume=93|issue=4|pages=471–500|jstor=1884466|doi=10.2307/1884466|s2cid=53374401}}

Essential Assumptions

Let "A" stand for assumption.

1. The city has an optimal number of residents.
2. Production exhibits constant returns-to-scale.
3. Transportation costs are linear with respect to the distance from the urban center.
4. Land is homogeneous so land rents only reflect differences in transportation costs.
5. The shape of the city is a two-dimensional circle, and the radius is optimally chosen by an urban planner.
6. The urban population is evenly distributed across the area of the city, so the population size is also the same as the area of the city.

Additional Assumptions

• Individuals have identical tastes.
• No congestion externalities.
• No impure public goods.
• Differential land rents are well defined, meaning opportunity land rents are uniform along the perimeter of the city.
• 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 choses the city’s geometry and population size.

The Model

By (A.3), transport costs $f$ at distance $t$ are linear with respect to distance:

$f(t)\; =\; f^\backslash prime(t)t$

Since land is homogenous à la (A.4), the rent gradient is found by:

$R^\backslash prime(t)\; =\; -f^\backslash prime(t)\; =\; -\backslash frac\{f(t)\}\{t\}$

Because we are integrating over a circular region (A.5), we can use shell integration to calculate aggregate land rents:

$ALR\; =\; \backslash int^\{t^*\}\_0\; R(t)2\backslash pi\; t\; \backslash ;\; dt$

Where $t^*$ is the distance of the urban boundary from the urban center.

Integration by parts and substitution of $R^\backslash prime(t)\; =\; -f(t)/t$ yields:

$$\backslash begin\{align\}$$

ALR&{}= R(t^*)\pi t^{*2} - \int^{t^*}_0 R(t)\pi t^2 \, dt \\

&{}= R(t^*)\pi t^{*2} + \int^{t^*}_0 f(t)\pi t \, dt

\end{align}

$R(t^*)\backslash pi\; t^\{*2\}$ are land rents at the urban boundary times the area of the city. Therefore, the rest are differential land rents:

$DLR\; =\; \backslash int^\{t^*\}\_0\; -R^\backslash prime(t)\backslash pi\; t^2dt\; =\; \backslash int^\{t^*\}\_0\; f(t)\backslash pi\; t\; \backslash ;\; dt$

Aggregate transportation costs are calculated as:

$ATC\; =\; \backslash int^\{t^*\}\_0\; f(t)2\backslash pi\; t\; \backslash ,\; dt$

Therefore, in the limiting case where transportation costs are linear, land is homogenous, and the shape of the city is circular, we obtain:

$DLR\; =\; \backslash frac\{ATC\}\{2\}$

Since transport costs are linear, we may write $e\; =\; f^\backslash prime(t)\; =\; f(t)/t$ as a constant. Bringing the constants outside ATCs integral and computing yields:

$ATC\; =\; 2e\; \backslash pi\; \backslash int^\{t^*\}\_0\; t^2\; \backslash ,\; dt\; =\; \backslash frac\{2e\}\{3\}\; \backslash pi\; t^\{*3\}$

Because units characterizing the geometry of the city are given by (A.5) and (A.6), we can calculate the urban radius as:

$t^*\; =\; \backslash left(\backslash frac\{N\}\{\backslash pi\}\backslash right)^\{1/2\}\; =\; N^\{1/2\}\backslash pi^\{-1/2\}$

Substitution of $t^*$ into $ATC$ yields:

$ATC\; =\; \backslash frac\{2e\}\{3\}\backslash pi^\{-1/2\}N^\{3/2\}\; =\; kN^\{3/2\}$

Where $k\; =\; \backslash frac\{2e\}\{3\}\backslash pi^\{-1/2\}$ is a composite constant since it's only made of constants.

The resource constraint for large urban economies can therefore be written as:

$yN\; =\; cN\; +\; G\; +\; kN^\{3/2\}$

Where $y$ is treated as constant under (A.2).

(A.1) requires that the urban planner solves the following maximization problem:

$\backslash max\_N\; c\; =\; y\; -\; \backslash frac\{G\}\{N\}\; -\; kN^\{1/2\}$

Take first-order conditions using the quotient and power rules:

$\backslash frac\{dc\}\{dN\}\; =\; \backslash frac\{G\}\{N^2\}\; -\; \backslash frac\{k\}\{2\}N^\{-1/2\}\; =\; 0$

Or:

$G\; =\; \backslash frac\{k\}\{2\}N^\{3/2\}\; =\; \backslash frac\{ATC\}\{2\}$

Therefore, comparing $DLR$ and the first-order condition for $G$ yields the Henry George theorem for large urban economies:

$DLR\; =\; \backslash frac\{ATC\}\{2\}\; =\; G$

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 $G$, deriving the optimal level of $G$ that satisfies the Samuelson condition isn’t necessary. {{Cite journal | last1 = Arnott | first1 = Richard.| title = Does the Henry George Theorem Provide a Practical Guide to Optimal City Size? | journal = The American Journal of Economics and Sociology | volume = 63 | issue = 3 | pages = 1057–1090 | date = November 2004 | jstor = 3488064 | url = https://www.jstor.org/stable/3488064 }}

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

{{Reflist}}

• {{cite web|url=http://inord.laurentian.ca/6_02/Henry_George.htm|title=A Rule Called George: Fixing the Property Tax System|author=David Robinson|publisher=The Institute for Northern Ontario Research and Development|date=2002-06-07|access-date=2007-11-03|archive-url=https://web.archive.org/web/20030524073948/http://inord.laurentian.ca/6_02/Henry_George.htm|archive-date=2003-05-24|url-status=dead}}
• {{cite book|url={{google books |plainurl=y |id=7kwq4U-hjVUC|page=140}}

|title=Economics of Agglomeration: Cities, Industrial Location, and Regional Growth, p.140|author=Masahisa Fujita and Jacques-François Thisse|publisher=Cambridge University Press|year=2002|access-date=2007-11-04|isbn=9780521805247}} {{isbn|978-0-521-80524-7}}

• {{cite web|url=http://findarticles.com/p/articles/mi_m0254/is_5_63/ai_n8642234|title=Does the Henry George Theorem provide a practical guide to optimal city size?|author=Richard Arnott|publisher=The American Journal of Economics and Sociology|date=November 2004|access-date=2007-11-03}}
• {{cite web|year=2013|url=http://www.econstor.eu/bitstream/10419/77659/1/cesifo_wp4280.pdf |title=Financing Public Capital through Land Rent Taxation: A Macroeconomic Henry George Theorem CESifo Working Paper, No. 4280|last1=Mattauch |first1=Linus |last2=Siegmeier |first2=Jan |last3=Edenhofer |first3=Ottmar |last4=Creutzig |first4=Felix |author3-link=Ottmar Edenhofer |author4-link=Felix Creutzig}}
• {{Cite web|date=2016-11-11|title=Provision of Infrastructure: Self-financing as Sustainable Funding – DOC Research Institute|url=https://doc-research.org/de/report2/provision-infrastructure-self-financing-sustainable-funding-2/|access-date=2021-12-25|last=Löhr|first= Dirk |publisher=DOC Research Institute, Expert Comment|archive-url=https://web.archive.org/web/20161111061253/https://doc-research.org/de/report2/provision-infrastructure-self-financing-sustainable-funding-2/ |archive-date=2016-11-11 }}

Category:Economics theorems

Category:Georgism

Category:Land value taxation