## wpiea2022263-print-pdf

## Source details

**Canonical URL:** [wpiea2022263-print-pdf](https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022263-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2022/english/wpiea2022263-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2022/english/wpiea2022263-print-pdf.pdf.json)

---

### Overview and motivation
- Urban land scarcity is driven indirectly by zoning, which preserves land for agriculture and conservation; land rents therefore reflect natural scarcity at least indirectly.
- The paper models a static economy with heterogeneous households who supply labor and consume a homogeneous good; the government sets labor income taxes and land rent taxes (equivalent to land value taxes).
- Assumptions: constant returns to scale, perfectly substitutable labor types, fixed prices (partial equilibrium results generalize to general equilibrium following Diamond and Mirrlees (1971)).

### Land value versus land rent taxation (Section 2.1)
- Land value v that generates land rent a and is subject to land value taxation τv:
  - v = sum_{t=1}^{∞} (a − τv v) / (1 + r)^t = 1/r (a − τv v).
  - Equivalently v = a / (r + τv).
- Key implications:
  - The value of land decreases with the level of land value taxation.
  - Increasing land value taxes reduces the tax base; tax revenue does not increase linearly with the tax rate.
  - Land value taxation can in principle exceed 100% of land value because the tax reduces land value below a single year’s land rent.
- One-to-one correspondence between land value taxes τv and land rent taxes τa:
  - If τv v = τa a and using v = a / (r + τv), τa = τv / (r + τv).
  - Land value taxation and land rent taxation are equivalent; land rent taxation is used in the model for mathematical tractability.
  - Example mapping: τa = 1 corresponds to τv = infinity.

### Model structure and equilibrium (Sections 2.2–2.3)
- Households:
  - Three income sources: labor, capital, and land.
  - Own fixed amounts of land and do not trade land; land rent tax is capitalized instantaneously so no incentive to trade land.
  - Capital is untaxed (capital taxes set to zero by assumption).
  - Taxes are linear on both labor and land rents (except Section 5 where lump-sum transfers are considered).
- Household optimization:
  - Utility u_n = u(c_n, l_n) with u_c, −u_l > 0 and u_cc, u_ll < 0, ∀n.
  - Consumption: c_n = (1 − τl) f_n l_n + k_n + (1 − τa) a_n.
  - First-order condition: −u_l / u_c = (1 − τl) f_n, ∀n.
  - Private marginal utility of income λ_n = u_c = −u_l / ((1 − τl) f_n).
  - Roy’s identity derivatives:
    - ∂v_n / ∂τl = −λ_n f_n l_n
    - ∂v_n / ∂τa = −λ_n a_n

### Government objective and FOC structure (Section 2.4)
- Government maximizes ∫_N Ψ(u_n) dn with Ψ′(u_n) > 0, Ψ′′(u_n) ≤ 0.
- Budget constraint: G = ∫_N (τl l_n + τa a_n) dn.
- Rent taxation bounded: τa ≤ 1.
- Lagrangian and first-order conditions:
  - L = ∫_N Ψ(v_n(τl, τa)) dn + η(∫_N (τl f_n l_n + τa a_n) dn − G) + μ(1 − τa).
  - ∂L/∂τl = ∫_N (−Ψ′ λ_n f_n l_n + η(f_n l_n + τl f_n ∂l_n/∂τl)) dn = 0.
  - ∂L/∂τa = ∫_N (−Ψ′ λ_n a_n + η(τl f_n ∂l_n/∂τa + a_n)) dn − μ = 0.
  - Complementary slackness: μ(1 − τa) = 0.
  - Budget constraint binding: ∫_N (τl f_n l_n + τa a_n) dn − G = 0.
- Slutsky decomposition for labor supply responses:
  - ∂l_n/∂τl = ∂l*_n/∂τl − f_n l_n ∂l_n/∂T.
  - ∂l_n/∂τa = ∂l*_n/∂τa − a_n ∂l_n/∂T.
  - T is a hypothetical transfer representing wealth effect; l*_n is compensated labor demand.

### Distributional definitions and characteristics (Section 3.1)
- Social marginal value of transferring a marginal unit of income to individual n:
  - λ*_n = Ψ′ λ_n + η τl f_n ∂l_n/∂T.
  - Interpretation: Ψ′ λ_n = direct social welfare gain; η τl f_n ∂l_n/∂T = lost tax revenue due to reduced labor supply after transfer.
- Marginal cost of public funds (MCF):
  - MCF = η / λ̄*, where λ̄* = ∫_N λ*_n dn.
- Feldstein-style distributional characteristics:
  - ξ_l ≡ −cov[λ*_n, z_n] / (λ̄* z̄).
  - ξ_a ≡ −cov[λ*_n, a_n] / (λ̄* ā).
  - Interpretations:
    - ξ_l measures marginal gain in social welfare, as fraction of taxed labor income, from marginally increasing labor tax revenue.
    - ξ_a measures marginal gain in social welfare, as fraction of land rents, from marginally increasing land rent tax revenue.
    - ξ_l > 0 in models where households differ only by labor productivity; with asset income present, sign may vary.
- Compensated elasticities:
  - ε_{lτl} ≡ (∂l*_n / ∂t) / ((1 − τl) l_n) < 0.
  - ε_{lτa} ≡ (∂l*_n / ∂τa) / ((1 − τa) a_n) = 0.

### Optimal tax rules (summary of Proposition 1, Section 3.2)
- Policy rules:
  - 1 − 1/MCF + ξ_l/MCF = τ_l/(1−τ_l) (−ε_lτ_l).
  - 1 − 1/MCF + ξ_a/MCF − μ 1/η̄_a = 0.
  - μ(1−τ_a) = 0.
- Definitions and interpretation:
  - ε_lτ_l = ∫_N ε_lτ_l f_n l_n dn / ∫_N f_n l_n dn is the income-weighted average labor supply elasticity.
  - 1 − 1/MCF is the revenue-raising Ramsey term.
  - ξ_l/MCF and ξ_a/MCF capture distributional concerns: taxes that redistribute from high to low income earners justify higher rates.
  - τ_l/(1−τ_l) (−ε_lτ_l) captures the deadweight loss of labor taxation; relevant only for labor due to fixed land supply.
- MCF expressions:
  - MCF = 1 − ξ_l / [1 − τ_l/(1−τ_l) (−ε_lτ_l)].
  - MCF = 1 − ξ_a / [1 − μ/η̄_a].
- Key implication:
  - MCF increases with tax deadweight loss and decreases with distributional benefits. ξ_a is minus the normalized covariance of land rents a_n and λ*_n; if high labor income households also have high land rent income then ξ_a > 0.

### Equity-efficiency trade-off and possibility of negative labor tax
- Optimal land rent tax is generally neither necessarily 0 nor 1; households with high welfare weight (low asset endowment) should not have assets fully taxed away.
- If land rents exceed government expenses, G < ∫_N a_n dn, and land value is more concentrated than labor income, government could set a high land rent tax and return excess revenues as labor subsidies (negative labor income taxes).

### Level of land rent taxes (Section 4)
- Rent tax bound: τ_a ≤ 1.
- Under standard utility functions and empirically relevant cases, expect MCF > 1 and ξ_a > 0.
- Proposition 2:
  - If optimum has MCF > 1 and ξ_a ≥ 0, then full land rent taxation is optimal, τ_a = 1.
  - Sketch: μ = η̄_a (1 − (1−ξ_a)/MCF). With η > 0, η̄_a > 0, MCF > 1 and ξ_a > 0, μ > 0 ⇒ τ_a = 1 by μ(1−τ_a)=0.
- Corollaries:
  - Corollary 1: If all households identical (ξ_a = ξ_l = 0) and ε_lτ_l < 0 so MCF > 1, then τ_a = 1.
  - Corollary 2: If MCF > 1 and either all households have same welfare weight or all households receive same land rents (cov[λ*_n, a_n] = 0 so ξ_a = 0), then τ_a = 1.
  - Corollary 3: If capital and labor income are non-decreasing functions of land rent income, then in an optimum with MCF > 1 land rents are fully taxed, τ_a = 1 (cov[λ*_n, a_n] ≤ 0 implies ξ_a ≥ 0).
- Breadth of optimal τ_a:
  - Proposition 3: Any level 0 ≤ τ_a ≤ 1 can be an optimal tax rate.
  - Constructive example (selected details):
    - f_n = 1, u(c_n, l_n) = c_n − 1/(1+1/ε) l_n^{1+1/ε} with 0 < ε < 1, yielding l = (1−τ_l)^ε.
    - For ε = 0.5, derived:
      - τ_l/(1−τ_l) = 2 − (Ψ'_k + 1)(3/2).
      - τ_l = [2 − (Ψ'_k + 1)(3/2)] / [3 − (Ψ'_k + 1)(3/2)].
      - As Ψ'_k → 0, τ_l → 1/3.
    - By varying G, Ψ'_k and ε, all τ_a ∈ [0,1] reachable.
  - Implication: if land is owned mainly by low income households (e.g., subsistence landowners), linear full land rent taxation may not be welfare optimal.

### Lump-sum transfers (Section 5)
- Model extension introduces lump-sum transfers T:
  - Household budget: c_n = (1−τ_l) f_n l_n + k_n + (1−τ_a) a_n + T.
  - Government budget: G + T = ∫_N (τ_l f_n l_n + τ_a a_n) dn.
- Policy rules with lump-sum transfers (Proposition 4):
  - MCF = 1.
  - ξ_l = τ_l/(1−τ_l) (−ε_lτ_l).
  - ξ_a = μ 1/η̄_a.
  - μ(1−τ_a) = 0.
- Interpretation:
  - With lump-sum instrument available, MCF = 1.
  - Labor tax balances distributional benefits and labor supply losses.
  - Land rent taxation has no distortionary effect and is used to reduce inequality from unequal land distribution.
- Level of land rent taxes with lump-sum transfers:
  - Proposition 5: If ξ_a > 0 then τ_a = 1.
  - Proposition 6: Even with lump-sum transfers, any level 0 ≤ τ_a ≤ 1 can be optimal (constructive example similar to Proposition 3).

### Empirical evidence (Sections 6 and 6.1)
- Central empirical variable: ξ_a, reflecting covariance between land rents and social welfare weight; sign depends on which income groups own most land rents relative to welfare weights.
- Data limitations: exact measurement of ξ_a requires government social welfare function parameters; proxy evidence uses share of housing in total wealth across income/wealth deciles.
- Price trends (Knoll et al. (2017) findings summarized):
  - Housing prices increased in France and the United States between 1998 and 2008 at a historically unprecedented speed.
  - Since 2008, housing prices remained at the historical high value in France and returned roughly to the 1998 value in the US.
  - The rise was driven by an even more pronounced increase in land prices; the land price index more than tripled in that period in France.
  - The share of land in housing prices doubled between 1930 and 2010 in both France and the US.
  - In France, the share of land in housing prices rose rapidly from 40 to 60% between 2000 and 2010.

### France—Asset composition and implications for land value taxation (Section 6.2)
- Asset composition by wealth deciles:
  - Wealthier households have a higher absolute amount of housing wealth.
  - Analysis displays gross housing wealth plus net financial asset position (total financial assets less debt).
  - From Knoll et al. (2017): 59% of housing value is given by land in France.
  - If the land share in housing value does not vary much between wealth deciles, absolute land wealth increases with total wealth.
- Financial leverage by decile:
  - Financial leverage = ratio between debt and net personal wealth.
  - First three wealth deciles: average leverage below 10%.
  - Fourth decile: average leverage of 74%.
  - Leverage in following deciles drops off to below 10% at the top decile.
- Relative asset composition and clustering:
  - First three deciles: each owns less than 12% of their wealth in housing on average; wealth concentrated in “currency, bonds and deposits”.
  - Fourth to seventh deciles: own between 99 and 127% of net wealth in housing and achieve values above 100% with debt financing.
  - Top three deciles: own between 40 and 81% of wealth in housing; hold between 13 and 41% in equity, fund shares, offshore wealth, pension funds and life insurance.
- Implications for land value taxation:
  - In absolute terms, land value taxes would increase with wealth on average.
  - Progressivity depends on relative tax payments; middle deciles hold very high shares of wealth in housing (and thus land) and are often highly leveraged (notably the fourth decile with 74% leverage).
  - Some deciles own more than 50% of net wealth in land; a high linear land value tax would have an uneven impact and could raise concerns about debt servicing capacity.
  - Distributional effect must consider both direct land tax and effects of reductions in other taxes (e.g., labor income tax cuts financed by land tax revenue):
    - Low income households would gain strongly as they pay hardly any land value tax and benefit from lower labor income taxes.
    - Middle income households would benefit modestly if they receive an average labor income tax cut (they pay a high share of their income in land value tax but less in absolute terms).
    - High income households’ net tax burden would increase, though aggregate efficiency gains could mean they lose less than lower-income groups gain.
  - For leveraged households financing housing with debt, increasing land value taxes while reducing labor income taxes (in particular for low and middle incomes) could address servicing concerns because net labor income would increase substantially even though the land value tax is added to regular debt servicing.

*Content based on wpiea2022263-print-pdf - 3.2 Results (source PDF).*

### Section 6 we use empirical data from France and the US to obtain an indication of how land

### Section 6 we use empirical data from France and the US to obtain an indication of how land rents are distributed among households. In Section 7, we discuss possible practical obstacles to the introduction of land value taxes with the aim of showing that these obstacles can be addressed with reasonable efforts. We conclude in Section 8.

### Overview and motivation
- Urban land scarcity is driven indirectly by zoning, which preserves land for agriculture and conservation; land rents therefore reflect natural scarcity at least indirectly.
- The paper models a static economy with heterogeneous households who supply labor and consume a homogeneous good; the government sets labor income taxes and land rent taxes (equivalent to land value taxes).
- Assumptions: constant returns to scale, perfectly substitutable labor types, fixed prices (partial equilibrium results generalize to general equilibrium following Diamond and Mirrlees (1971)).

### Land value versus land rent taxation (Section 2.1)
- Land value v that generates land rent a and is subject to land value taxation τv is:
  - v = sum_{t=1}^{∞} (a − τv v) / (1 + r)^t = 1/r (a − τv v).(1)
  - Equivalently v = a / (r + τv).(2)
- Key implications:
  - The value of land decreases with the level of land value taxation.
  - Increasing land value taxes reduces the tax base; tax revenue does not increase linearly with the tax rate.
  - Land value taxation can in principle exceed 100% of land value because the tax reduces land value below a single year’s land rent.
- One-to-one correspondence between land value taxes τv and land rent taxes τa:
  - If τv v = τa a and using (2), τa = τv / (r + τv).(3)
  - Land value taxation and land rent taxation are equivalent; land rent taxation is used in the model for mathematical tractability.
  - Example mapping: τa = 1 corresponds to τv = infinity.

### Model assumptions (Section 2.2)
- Households have three income sources: labor, capital, and land.
- Households own fixed amounts of land and do not trade land; land rent tax is capitalized instantaneously so no incentive to trade land.
- Capital is untaxed (capital taxes set to zero by assumption) to focus analysis on land value taxes in a static framework.
- Taxes are linear on both labor and land rents (except in Section 5 where lump-sum transfers are considered).
  - Rationale: linear taxes yield simple, intuitive tax rules; non-linear land rent taxes would create incentives to trade land and distortions.
- Distinction: unit tax (by area) would distort land supply; value (rent) tax does not distort land supply as long as τa < 1.
- Possible extensions noted but not modeled: overlapping generations, dynamic capital taxation, market frictions (e.g., apartment rental markets).

### Household optimization and equilibrium (Section 2.3)
- Population mass = 1; individuals indexed by n ∈ N = [0,1]; individual earning ability f_n; wage per efficiency unit normalized to 1.
- Individual variables indexed by subscript n; prices normalized to 1.
- Individuals supply labor l_n and receive capital income k_n and land rents a_n.
- Utility function u_n = u(c_n, l_n) with u_c, −u_l > 0 and u_cc, u_ll < 0, ∀n.(4)
- Capital taxes set to zero; consumption equals net income:
  - c_n = (1 − τl) f_n l_n + k_n + (1 − τa) a_n.(5)
- Labor income z_n = f_n l_n.
- Household first-order condition:
  - −u_l / u_c = (1 − τl) f_n, ∀n.(6)
- Private marginal utility of income λ_n = u_c = −u_l / ((1 − τl) f_n).(7)
- Indirect utility v_n(τl, τa) = u(ĉ_n, l̂_n); Roy’s identity gives:
  - ∂v_n / ∂τl = −λ_n f_n l_n
  - ∂v_n / ∂τa = −λ_n a_n

### Government objective and constraints (Section 2.4)
- Government maximizes Bergson-Samuelson social welfare: ∫_N Ψ(u_n) dn with Ψ′(u_n) > 0, Ψ′′(u_n) ≤ 0.(8)
- Government budget constraint (per person and total, since population = 1):
  - G = ∫_N (τl l_n + τa a_n) dn.(9)
- Rent taxation bounded by 100%:
  - τa ≤ 1.(10)
- Lagrangian for welfare maximization:
  - max_{τl,τa} L = ∫_N Ψ(v_n(τl, τa)) dn + η(∫_N (τl f_n l_n + τa a_n) dn − G) + μ(1 − τa).(11)
  - η is marginal social value of public resources; μ is multiplier for τa ≤ 1.
- First-order conditions:
  - ∂L/∂τl = ∫_N (−Ψ′ λ_n f_n l_n + η(f_n l_n + τl f_n ∂l_n/∂τl)) dn = 0.(12)
  - ∂L/∂τa = ∫_N (−Ψ′ λ_n a_n + η(τl f_n ∂l_n/∂τa + a_n)) dn − μ = 0.(13)
  - Complementary slackness μ(1 − τa) = 0.(14)
  - Budget constraint binding: ∫_N (τl f_n l_n + τa a_n) dn − G = 0.(15)
- Slutsky decomposition for labor supply responses:
  - ∂l_n/∂τl = ∂l*_n/∂τl − f_n l_n ∂l_n/∂T
  - ∂l_n/∂τa = ∂l*_n/∂τa − a_n ∂l_n/∂T
  - T is a hypothetical transfer representing wealth effect; l*_n is compensated labor demand.

### Definitions and distributional characteristics (Section 3.1)
- Social marginal value of transferring a marginal unit of income to individual n:
  - λ*_n = Ψ′ λ_n + η τl f_n ∂l_n/∂T.(16)
  - Interpretation: Ψ′ λ_n = direct increase in social welfare from marginal income to n; η τl f_n ∂l_n/∂T = lost tax revenue due to reduced labor supply after transfer.
- Marginal cost of public funds (MCF):
  - MCF = η / λ̄*, where λ̄* = ∫_N λ*_n dn.(17)
  - MCF is the ratio between social marginal value of one unit of public income and the average social marginal value of one unit of private income (with adjusted definition).
- Feldstein-style distributional characteristics for labor and land rent taxes:
  - ξ_l ≡ − [∫_N λ*_n z_n dn − ∫_N λ*_n dn ∫_N z_n dn] / [∫_N λ*_n dn ∫_N z_n dn] = −cov[λ*_n, z_n] / (λ̄* z̄).(18)
  - ξ_a ≡ − [∫_N λ*_n a_n dn − ∫_N λ*_n dn ∫_N a_n dn] / [∫_N λ*_n dn ∫_N a_n dn] = −cov[λ*_n, a_n] / (λ̄* ā).(19)
  - Interpretations:
    - ξ_l measures the marginal gain in social welfare, expressed as a fraction of taxed labor income, from marginally increasing labor tax revenue.
    - ξ_a measures the marginal gain in social welfare, expressed as a fraction of land rents, from marginally increasing land rent tax revenue.
    - In models where households differ only by labor productivity, ξ_l > 0; with asset income present, sign may vary.
- Compensated elasticities of labor supply:
  - ε_{lτl} ≡ (∂l*_n / ∂t) / ((1 − τl) l_n) < 0
  - ε_{lτa} ≡ (∂l*_n / ∂τa) / ((1 − τa) a_n) = 0

*Source: wpiea2022263-print-pdf - Section 6 we use empirical data from France and the US to obtain an indication of how land rents are distributed among households.*

### 3.2    Results

### wpiea2022263 - 3.2 Results

### Optimal tax rules (summary of Proposition 1)
- Policy rules for the optimal labor and land rent tax:
  - 1 − 1/MCF + ξ_l/MCF = τ_l/(1−τ_l) (−ε_lτ_l)  (equation (20))
  - 1 − 1/MCF + ξ_a/MCF − μ 1/η̄_a = 0  (equation (21))
  - μ(1−τ_a) = 0  (equation (22))
- Definitions and interpretation:
  - ε_lτ_l = ∫_N ε_lτ_l f_n l_n dn / ∫_N f_n l_n dn is the income weighted average of the labor supply elasticity.
  - The term 1 − 1/MCF is a revenue-raising Ramsey term.
  - ξ_l/MCF and ξ_a/MCF capture distributional concerns: a tax that redistributes from high to low income earners is higher than without distributional preferences.
  - τ_l/(1−τ_l) (−ε_lτ_l) captures the deadweight loss of labor taxation; relevant only for labor due to fixed land supply.
- Marginal cost of public funds (MCF) solved from policy rules:
  - MCF = 1 − ξ_l / [1 − τ_l/(1−τ_l) (−ε_lτ_l)]  (equation (23))
  - MCF = 1 − ξ_a / [1 − μ/η̄_a]  (equation (24))
- Key implications:
  - MCF increases with the deadweight loss of the tax and decreases with distributional benefits (standard for labor tax from (23)).
  - ξ_a is minus the normalized covariance of land rents a_n and the net social welfare weight λ*_n. If high labor income households also have high land rent income on average, then ξ_a > 0, an empirically relevant case.

### Equity-efficiency trade-off and possible negative labor tax
- Optimal land rent tax is generally neither necessarily 0 nor 1; households with high welfare weight (low asset endowment) should not have assets fully taxed away.
- If land rents are more than sufficient to pay government expenses, G < ∫_N a_n dn, and land value is more concentrated than labor income, the government could set a high land rent tax and return excess revenues as labor subsidies (negative labor income taxes).

### The level of land rent taxes (Section 4)
- Land rent taxes limited to 100% (from equation (10)).
- Under standard utility functions and empirically relevant cases, expect MCF > 1 and ξ_a > 0.
- Proposition 2:
  - If optimum has MCF > 1 and ξ_a ≥ 0, then full land rent taxation is optimal, τ_a = 1.
  - Proof sketch: solving (21) for μ gives μ = η̄_a (1 − (1−ξ_a)/MCF). With η > 0, η̄_a > 0, MCF > 1 and ξ_a > 0, μ > 0, then (22) implies τ_a = 1.
- Corollaries:
  - Corollary 1: If all households are identical (ξ_a = ξ_l = 0) and ε_lτ_l < 0 so MCF > 1, then τ_a = 1.
  - Corollary 2: If MCF > 1 and either all households have same welfare weight or all households receive same amount of land rents (cov[λ*_n, a_n] = 0 so ξ_a = 0), then τ_a = 1.
  - Corollary 3: If capital and labor income are non-decreasing functions of land rent income, then in an optimum with MCF > 1 land rents are fully taxed, τ_a = 1 (because cov[λ*_n, a_n] ≤ 0 implies ξ_a ≥ 0).
- Counterexamples and breadth of optimal τ_a:
  - Proposition 3: Any level of land taxes 0 ≤ τ_a ≤ 1 can be an optimal tax rate.
  - Constructive example assumptions:
    - f_n = 1, u(c_n, l_n) = c_n − 1/(1+1/ε) l_n^{1+1/ε} with 0 < ε < 1, yielding l = (1−τ_l)^ε.
    - Specific distribution of capital k_n and land a_n split across population halves.
    - For ε = 0.5, derived expressions:
      - τ_l/(1−τ_l) = 2 − (Ψ'_k + 1)(3/2)  (equation (31))
      - τ_l = [2 − (Ψ'_k + 1)(3/2)] / [3 − (Ψ'_k + 1)(3/2)]  (equation (32))
      - As Ψ'_k → 0, τ_l → 1/3.
      - τ_a expressed as function of G, Ψ'_k and ε (equations (33) and (34)); by varying G, Ψ'_k and ε, all τ_a ∈ [0,1] reachable.
  - Implication: if land is owned mainly by low income households (e.g., subsistence landowners), linear full land rent taxation may not be welfare optimal.

### Lump-sum transfers (Section 5)
- Model extension: introduce lump-sum transfers T.
  - Household budget: c_n = (1−τ_l) f_n l_n + k_n + (1−τ_a) a_n + T  (equation (35))
  - Government budget: G + T = ∫_N (τ_l f_n l_n + τ_a a_n) dn  (equation (36))
- First-order conditions and Slutsky conditions adjusted; Lagrangian includes T and μ for land tax upper limit.
- Proposition 4 (policy rules with lump-sum transfers):
  - MCF = 1  (equation (43))
  - ξ_l = τ_l/(1−τ_l) (−ε_lτ_l)  (equation (44))
  - ξ_a = μ 1/η̄_a  (equation (45))
  - μ(1−τ_a) = 0  (equation (46))
- Interpretation:
  - With lump-sum instrument available, MCF = 1: public funds marginal social value normalized.
  - Labor tax balances distributional benefits and labor supply losses (equation (44)).
  - Land rent taxation has no distortionary effect and is used to reduce inequality from unequal land distribution.
- Level of land rent taxes with lump-sum transfers:
  - Proposition 5: If ξ_a > 0 then full land rent taxation is optimal, τ_a = 1.
  - Proposition 6: Even with lump-sum transfers, any level 0 ≤ τ_a ≤ 1 can be optimal (constructive example similar to Proposition 3; by varying k any τ_a attainable).

### Empirical evidence (Section 6 and 6.1 Price trends)
- Central empirical variable: ξ_a, the distributional characteristic reflecting covariance between land rents and social welfare weight. Its sign depends on which income groups own most land rents relative to social welfare weights.
- Data limitations: exact measurement of ξ_a requires government social welfare function parameters; proxy evidence uses share of housing in total wealth across income/wealth deciles.
- Price trends for housing and land (Figure summaries from Knoll et al. (2017)):
  - Housing prices increased in France and the United States between 1998 and 2008 at a historically unprecedented speed.
  - Since 2008, housing prices remained at the historical high value in France and returned roughly to the 1998 value in the US.
  - The rise was driven by an even more pronounced increase in land prices; the land price index more than tripled in that period in France.
  - The share of land in housing prices doubled between 1930 and 2010 in both France and the US.
  - In France, the share of land in housing prices rose rapidly from 40 to 60% between 2000 and 2010.

*Italic: Content based on wpiea2022263-print-pdf - 3.2 Results (source PDF).*

### 6.2    France

### 6.2    France

### Asset composition of wealth deciles
- Wealthier households have a higher absolute amount of housing wealth.
- For analysis, the paper displays gross housing wealth plus the net financial asset position (total financial assets less debt).
- From Knoll et al. (2017): 59% of housing value is given by land in France.
- If the land share in housing value does not vary much between wealth deciles, the absolute amount of wealth in the form of land increases with total wealth.

### Financial leverage by decile
- Financial leverage is defined as the ratio between debt and net personal wealth.
- The first three wealth deciles have an average leverage of below 10%.
- The fourth decile has an average leverage of 74%.
- Leverage in the following deciles drops off to below 10% at the top decile.

### Relative asset composition and clustering of deciles
- The wealth deciles cluster into three groups by share of housing wealth:
  - First three deciles: each owns less than 12% of their wealth in the form of housing on average; their wealth falls almost exclusively in the category “currency, bonds and deposits”.
  - Fourth to seventh deciles: own between 99 and 127% of their net wealth in the form of housing and achieve values above 100% with debt financing.
  - Top three deciles: own between 40 and 81% of their wealth in the form of housing; this group holds between 13 and 41% in the form of equity, fund shares, offshore wealth, pension funds and life insurance.

### Implications for land value taxation
- In absolute terms, land value taxes would increase with wealth on average.
- Progressivity depends on relative tax payments; middle deciles hold very high shares of wealth in housing (and thus land) and are often highly leveraged (notably the fourth decile with 74% leverage).
- Because some deciles own more than 50% of their net wealth in the form of land, a high linear land value tax would have an uneven impact and could raise concerns that the most exposed households may be unable to service their debt.
- The distributional effect must consider both the direct tax on land value and the effects of reductions in other taxes (for example, labor income tax cuts financed by land tax revenue):
  - Low income households would gain strongly as they pay hardly any land value tax and benefit from lower labor income taxes.
  - Middle income households would benefit modestly if they receive an average labor income tax cut (they pay a high share of their income in land value tax but less in absolute terms).
  - High income households’ net tax burden would increase, though aggregate efficiency gains could mean they lose less than lower-income groups gain.
- For leveraged households financing housing with debt, increasing land value taxes while reducing labor income taxes (in particular for low and middle incomes) could address servicing concerns because net labor income would increase substantially even though the land value tax is added to regular debt servicing.

*Italic: Content based solely on the provided source excerpt.*

### References

### References and Appendices (wpiea2022263-print-pdf - References)

### Major citation themes
- Land taxation, land value taxation, and rent taxation:
  - Arnott and Stiglitz (1979); Edenhofer et al. (2015); Fernandez Milan et al. (2016); Franks et al. (2018); Hughes et al. (2020); Kalkuhl et al. (2018); Koethenbuerger and Poutvaara (2009); Kumhof et al. (2021); Medda (2012); Oates and Schwab (2009); Plummer et al. (2010); Tideman (1982); Schwerhoff et al. (2020).
- Optimal taxation, redistribution, and public finance:
  - Atkinson and Stiglitz (1976); Diamond (1975); Diamond and Mirrlees (1971); Feldstein (1972); Jacobs and de Mooij (2015); Mirrlees (1971); Mirrlees et al. (2011); Stigler (1970); Stiglitz (2015).
- Wealth, housing, and inequality dynamics:
  - Acemoglu et al. (2015); Bach et al. (2016); Bricker et al. (2017); Garbinti et al. (2021); Knoll et al. (2017); Piketty (2014); Ryan-Collins et al. (2017).
- Mathematical and methodological references:
  - Egozcue et al. (2009); Schmidt (2003); Varian (2006).

### Appendix A — Proofs (A.1 Proof of Proposition 1)
- The appendix presents the derivation transforming the government first order condition (12) with respect to the labor tax into the tax rule (20). Key displayed steps and equations:
  - Start: ∫_N (−Ψ′ λ_n f_n l_n + η (f_n l_n + τ_l f_n ∂l_n/∂τ_l)) dF(n) = 0 (52)
  - Intermediate transformations through equations (53)–(56).
  - Elasticity and marginal cost of public funds (MCF) appear in steps leading to (57) and (58):
    - (57) and (58) express the tax rule in terms of ξ_l, MCF, τ_l, and ε_{l,τ_l}:
      - ξ_l / MCF − 1/MCF + 1 = τ_l/(1−τ_l) (−ε_{l,τ_l})  (58)
  - Parallel derivation for the asset tax first order condition starting from (13) and using ∂l*_n/∂τ_a = 0:
    - Equations (59)–(63) reduce to (64) and (65):
      - ξ_a / MCF − 1/MCF + 1 − μ/(η ā) = 0  (65)
- The proof relies on definitions and substitutions for λ*_n, l*_n derivatives, aggregation over n, and identification of ξ_l, ξ_a, MCF, μ, η, and ā as used in the paper's main text.

### Appendix B — Data appendix
- B.1 Data in Section 6.1
  - Figure 1 data source: Knoll et al. (2017), table KSSNPLHDecomposition.xlsx.
  - Figure 2 data source: Knoll et al. (2017), Table 2.
- B.2 Data in Section 6.2
  - Figures 3, 4, and 5 data source: Garbinti et al. (2021), downloaded from the World Wealth and Income Database (WID.world).
  - Data file: WIDFRInequalityData.csv (full data set for France).
  - Variable names used:
    - Housing assets: "Personal housing assets|equal-split adults|Average|Adults|constant 2015 local currency"
    - Business assets: "Personal business and other non-financial assets|equal-split adults|Average|Adults|constant 2015 local currency"
    - Financial assets (net of debt): "Personal financial assets|equal-split adults|Average|Adults|constant 2015 local currency" less "Personal debt|equal-split adults|Average|Adults|constant 2015 local currency"
    - Net personal wealth: "Net personal wealth|equal-split adults|Average|Adults|constant 2015 local currency"
  - Figure constructions:
    - Figure 4 is the ratio of "personal debt" to "net personal wealth".
    - Figure 5 gives the ratio of the three asset categories used in Figure 3 to net personal wealth.
- B.3 Data in Section 6.3
  - Source: Bricker et al. (2017), Survey of Consumer Finances (SCF).
  - SCF web page: https://www.federalreserve.gov/econres/scfindex.htm
  - Table used: scf2016_tablesinternalnominalhistorical.xlsx
  - Table contents:
    - For each asset category: a sheet "Percentage of families holding asset" and a sheet "Mean value of holdings for families holding asset (thousands of dollars)".
  - Figure constructions:
    - Figure 6 uses the "Percentage of families holding asset" table directly.
    - Figure 7 averages obtained by multiplying percentages and mean holdings for families holding asset.
    - "Other non-financial assets" = "Any nonfinancial asset" less "Primary residence", "Other residential property", and "Equity in nonresidential property".
    - "Financial assets (net of debt)" = "Any financial asset" less "Any debt".
    - The sum of "Primary residence", "Other residential property", "Equity in nonresidential property", "other non-financial assets", and "Financial assets (net of debt)" equals the value given in "any asset".
    - Figure 8 values are the ratio between individual category values and the value from "any asset".

*Content unit: wpiea2022263-print-pdf - References*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022263-print-pdf.pdf_