Shahar Rotberg. University of Toronto August 23, Abstract

Transcription

1 Should Productive Wealth and Housing Wealth Be Taxed Differently? Latest Version Can Be Found Here: Shahar Rotberg University of Toronto August 23, 2017 Abstract I utilize a model that mimics a realistic wealth distribution in order to study how different types of wealth should be taxed when government revenue neutrality is required. My framework incorporates labor income risk, incomplete financial markets, heterogeneous investment abilities, imperfect inter-generational investment ability transmission, a realistic housing market, and two types of wealth: productive wealth, and unproductive (housing) wealth. I calibrate the model to U.S. data and run simulations in which the current U.S. capital income tax is replaced with a wealth tax. I document the following main findings: first, consistent with previous research in which housing is not considered, taxing wealth remains superior to taxing capital income even when housing is included. Second, it is optimal to only tax productive wealth. Third, while welfare gains in previous research emerge mainly from higher wages, welfare gains in my model are a result of higher wages, larger homes, and a more profitable housing construction sector. Fourth, larger homes combined with a more profitable construction sector account for more than 60% of welfare gains. Intuitively, wealth taxation is superior to capital income taxation because it increases the tax burden on wealthy but unproductive investor-households. This reduces the tax burden on productive investor-households, enabling them to accumulate productive wealth faster. As a result, productive wealth is used more efficiently, thus increasing wages and welfare. Furthermore, housing wealth should be left un-taxed for the following reason: lower tax rates on productive wealth enable more productive investor-households to accumulate productive wealth faster, and thus, increase wages and welfare. However, to lower productive wealth tax rates, higher tax rates must be imposed on housing wealth. The latter causes housing prices to decline, the construction sector to slow down, home sizes to shrink, and, ultimately, welfare to decline. Ceteris paribus, welfare is maximized where the marginal benefit of higher wages is balanced against the marginal cost of smaller homes and a less profitable construction sector. Quantitatively, the optimum is achieved at the corner. JEL Classification: E21, E22, E62, H21, R21, R31 I would like to thank my supervisors, Professor Burhanettin Kuruscu and Professor Gueorgui Kambourov, for their excellent guidance, advice, and support throughout this research project. This paper also benefited from insightful feedback and advice from my committee member, Professor Joseph Steinberg. I thank the Brown Bag seminar s participants at University of Toronto for their helpful comments. As well, I thank Sabina Georgescu for her editing help. This paper is dedicated to my parents for their unwavering support. All errors are my own. 1

2 Keywords: Wealth Tax, Wealth Distribution, Housing, Optimal Taxation, Rate-of-Return Heterogeneity, Capital Income Tax. I Introduction The distortionary nature of capital taxation combined with its pivotal role in generating government revenues, has, over the years, generated an extensive body of research aimed at finding the most efficient way to tax capital 1. Considering the vast interest in this topic, it is perhaps surprising to discover that only one research paper has rigorously dealt with the question of whether capital income (flow) or wealth (stock) should be taxed 2. Guvenen, Kambourov, Kuruscu, Ocampo, & Chen (2017) (henceforth GKKOC) show that wealth taxation is less distortionary than capital income taxation when there is heterogeneity in investment ability. Their results are an outcome of incomplete markets and unproductive investor children inheriting significant wealth from their productive investor parents. And so, on occasion, significant wealth is not put to its most productive use. GKKOC show that wealth taxation shifts the tax burden from productive investors to unproductive ones, allowing productive investors to accumulate wealth faster. Thus, wealth is more efficiently used, generating positive aggregate outcomes for wages, output, and welfare. However, a shortcoming of their work is their treatment of all types of wealth as identical and productive in the real sector. In practice, residential housing wealth is not the same as productive wealth. Although residential housing structures generate utility flow, once built, their wealth is not used to produce output. According to data from the Survey of Consumer Finances (SCF), between , on average, 23% of total net wealth was tied up in households primary residences (which is unproductive wealth) 3. As such, the framework in GKKOC is unable to answer questions such as: does productive wealth taxation remain less distortionary than capital income taxation when households have the option to shift their resources to housing wealth? Should housing wealth be taxed given its potential to distort the housing market, and if so at what rate should it be taxed? To answer these questions I combine several frameworks from previous research into one. I begin by using the framework in GKKOC to model heterogeneity in investment ability. Then, to model a realistic housing market, I utilize a similar housing set up to that in Arslan, Guler, & Takin (2015) (hereafter AGT). In contrast to AGT s model, in which the price of rental units is normalized to zero, I explicitly model a rental market with non-zero rent. In doing so, I take the rental market structure used in Kaplan, Mitman, & Violante (2016). My model integrates six inter-related components: overlapping generations of households, competitive mortgage lenders, a competitive rental firm, a final goods producer, a housing con- 1 In 2006, in 28 European Union countries, government revenues raised through taxation of different types of capital as a percentage of total government revenues, ranged from a low of 15.5% in Sweden to a high of 36% in Norway. In 2011, the U.S. government raised 27% of its revenues from capital taxation. 2 The reason the literature has mostly been silent on this dilemma is that it did not, for the most part, incorporate investment ability heterogeneity into models, and without heterogeneity in investment ability capital income taxation and wealth taxation are equivalent. 3 Net housing wealth tied up in primary residences as a fraction of total net wealth has been quite stable in the past 25 years (with a slight downward trend). Its lowest was in 2012 (20% of total net wealth) and its highest was in 1988 (26% of total net wealth). 2

3 struction sector, and a government. Moreover, there are two types of wealth in the model: productive wealth and housing wealth. Productive wealth is used to produce output, while housing wealth only generates utility flow for households. Throughout their lives, households differ in their ability to invest productive wealth. To increase their investments and capital income, households can borrow productive wealth from each other. However financial markets are incomplete, which means that households can only borrow up to a certain fraction of their net wealth. Given households own and borrowed productive wealth, they produce intermediate goods, which are sold to a final goods producer for a price (to be discussed below). During their labor force participation period, households face labor income risk. At a certain age, households are forced to retire from the labor force, at which point they receive social security benefits. The presence of labor income risk and reduced income in retirement motivate households to save. Upon their death, households net wealth is inherited by their offspring (inheritance includes the house and mortgage debt). As well, when households die they imperfectly transmit their investment ability to their offspring. However, the intergenerational transmission of investment ability is mean-reverting across a household s lineage. Therefore, sometimes, significant wealth is held by unproductive investors. Since markets are incomplete, some of this wealth is not being put into its most productive use. To account for the presence of unproductive wealth, my model features a housing market. Since in reality housing cannot be bought in very small amounts, I model housing as a lumpy asset. To simplify the analysis, housing is homogeneous in size. Households can be renters or homeowners. To become homeowners households must pay at least the minimum down-payment and, if necessary, take on mortgage debt, on which interest is paid, to cover the remainder of the purchase. Homeowners who have mortgage debt can deduct their mortgage interest payments from their taxable income. The difference between renting and home-owning lies in home-owning generating higher utility flow than renting. Mortgage lenders are exogenous in the model. They lend perfectly elastically at the market going mortgage interest rate. Households who choose to be renters, pay a rental price for their rental unit to a competitive rental company. The rental company collects rents and incurs management and upkeep costs on their rental units. The final goods producer hires labor inputs from households (for a wage) and buys intermediate inputs from households (for a price). Using these inputs, the final goods producer produces a final good that is sold to households. The housing supply is determined in the construction sector, in which a construction company chooses a level of housing investment each period. Housing investment is sold to households and the rental company at the market price of housing. Given that in my model the population of households is fixed and that each household lives in one home, the housing stock varies only when the size of homes changes. Home size is determined based on the equilibrium price of housing and the price elasticity of the housing stock. The government raises revenues by taxing labor income, non-durable consumption, and capital. Government revenues are then used to pay social security benefits to retired households and 3

4 for the provision of public goods. As is standard in the Ramsey literature, the government in the model must meet a minimum revenue requirement in order to provide an exogenous amount of public goods. Public goods, however, do not affect private consumption nor enhance future production. I calibrate the model to U.S. data from the Survey of Consumer Finances collected between The main target of the calibration is the fraction of wealth owned by the top 1% of the wealth distribution, which I use to pin down the variability in investment ability. This target is important because the more variability in investment ability there is, the more misallocation of wealth there is for the government to correct, and the more welfare can be enhanced. To understand how housing affects the calibrated value of investment ability variability, I calibrate my model first with housing and then without housing. I find that housing depresses wealth inequality because it is more equally distributed than productive wealth. Therefore, when housing is included in the model more dispersion in productive wealth is required to generate the wealth concentration at the top 1% of the wealth distribution observed in the data. This means that, larger variability in investment ability is required when housing is accounted for. To answer the questions of interest in this paper, I start by running an experiment in which I replace the currently prevailing capital income tax with a pure productive wealth tax (keeping housing wealth un-taxed). I keep labor income tax and non-durable consumption tax constant. Government revenue neutrality is preserved. The experiment is repeated across four different versions of my model. In my first simulation, the model has no housing market (version one). In the second simulation, I keep the housing stock fixed and normalize the rental price to zero (version two). The third simulation features a varying housing stock and, again, zero rental prices (version three). The fourth simulation has a varying housing supply but non-zero rental prices (version four). Comparing the four versions of my model highlights the relative importance of the housing market, the housing construction sector, and the rental market, for achieving welfare gains through wealth taxation. I briefly detail the results of my simulations in the next several paragraphs (a convenient summary of my results is also presented in table 5 on page 21). Consistent with GKKOC, the simulation of version one indicates that when housing is not included pure productive wealth taxation increases welfare by 2.8% and raises both output and the wage rate by 7.4%. These results are a direct consequence of wealth taxation increasing the tax burden on wealthy, but unproductive investor-households, and lessening the burden on productive investor-households. The resulting re-allocation of resources increases the productivity of capital, which increases output, the wage rate, and welfare. The simulation of version two, which features a fixed housing stock and zero rents, shows that pure productive wealth taxation increases welfare, output, and the wage rate by about 15% less relative to version one, while causing housing prices to rise by 49%. Housing prices rise when households shift their resources to housing wealth in response to the introduction of the productive wealth tax. This shifting of resources causes productive wealth to further decline, resulting in relatively lower gains. My simulation of version three, which has a varying housing stock and zero rental prices, shows that introducing a pure productive wealth tax increases welfare by 5.8%, output by 9%, 4

5 the wage rate by 8.5%, and housing prices by 20.6%. Version three of the model focuses the attention on the importance of the housing construction sector for welfare gains. As in version two, housing prices rise because households re-allocate resources to housing wealth to avoid the tax on productive wealth. However, in version three, the higher housing prices also stimulate the housing construction sector and increase its profitability, which results in larger homes. The latter two, which are absent in version two, cause welfare gains to more than double relative to version two. Furthermore, housing prices rise by less than they do in version two because the more profitable construction sector encourages households to retain more productive wealth (since the amount of productive wealth a household owns determines the amount of profits they receive from the housing construction company). Lastly, the simulation of version four, in which there is a varying housing supply and non-zero rental prices, shows that implementing a pure productive wealth tax increases welfare by 7%, output by 10.1%, the wage rate by 9.5%, housing prices by 20.6%, and rental prices by 5.4%. As well, the productive wealth tax rate is about 8% lower relative to version three. The same mechanisms that are at play in version three are also at play in this version. However, the rise in rental prices provides an additional incentive for households to save productive wealth, which increases the tax base and reduces the tax rate on productive wealth relative to version three. Consequently, increases in the wage rate and welfare are more substantial relative to version three. In my second experiment I search for the optimal combination of productive wealth and housing wealth tax rates (the optimum is achieved when the expected welfare of a newly born household is maximized). I do so across three different versions of my model in order to highlight how the housing construction sector and the rental market affect the optimal tax combination and welfare gains. As in the first experiment, I keep taxes on labor income and non-durable consumption constant. As well, government revenue neutrality is preserved. I start with version two, which has a fixed housing stock and zero rental prices. Then I examine the optimal tax combination in version three, which features a varying housing stock and zero rental prices. Lastly, I use version four, which includes both a varying housing stock and rental prices that are greater than zero. I briefly discuss my results in the paragraphs below (summaries of the results of the optimal wealth tax combinations of the three versions are also presented in table 7 on page 24 (version two), in table 9 on page 26 (version three), and in table 10 on page 26 (version four)). In version two, the optimum is achieved when housing wealth is taxed at a rate of 100% and productive wealth is taxed at a rate of 0.36%. This tax system generates a 7.1% increase in welfare and a 13% increase in both output and the wage rate. However, it causes housing prices to drop by a staggering 97%. While increasing homeownership rates at the bottom of the wealth distribution plays a role in welfare maximization, higher wages are what matter most for welfare maximization in this version of the model. Thus, fixing homeownership rates, the objective is to maximize revenues from housing wealth taxation and reduce the tax on productive wealth to its lowest bound in order to achieve the highest possible increase in wages 4. Contrary to the result obtained in version two, the optimum in version three is achieved when housing wealth is left un-taxed and productive wealth is taxed at a rate of 1.07%. This tax 4 Not accounting for other factors such as interest rates. 5

6 system increases welfare by 5.8%, output by 9%, and the wage rate by 8.5%. Furthermore, it causes housing prices to increase by 20.6%, the construction sector to be 75% more profitable, and home size to increase by 1.5%. Housing wealth should not be taxed due to the following: lower tax rates on productive wealth enable more productive investor-households to accumulate productive wealth faster, and thus, increase wages and welfare. However, in order to lower productive wealth tax rates, housing wealth must be taxed more heavily. The increased tax on housing wealth makes housing less attractive. This, in turn, causes housing prices to decline, the construction sector to slow down, home sizes to shrink, and welfare to decline. As such, in this version of the model, the main objective is to balance the cost of smaller homes and a less profitable construction sector with the benefit of higher wages. Quantitatively, the optimum is achieved at the corner. In version four, the optimum is reached when housing wealth is not taxed and productive wealth is taxed at a rate of 0.99%. This tax system increases welfare by 7%, output by 10.1%, and the wage rate by 9.5%. Moreover, it causes the housing price to rise by 20.6%, which raises the rental price by 5.4% and increases the profitability of the construction sector by 75%. The latter results in homes that are 1.5% larger. The difference between version three and version four is that in version four the increase in rental prices creates an extra incentive for households to save, which increases aggregate productive wealth and thus, creates a larger tax base. The larger tax base enables the government to impose a productive wealth tax rate that is 8% lower relative to version three, leading to higher wages and an additional 20% increase in welfare gains relative to version three. II Related Literature Early work on capital taxation has shown that capital income should not be taxed in the longrun (Judd (1985), Chamley (1986)). This result is an outcome of strong assumptions such as, infinite lives, no labor income risk, and complete financial markets. Due to these assumptions, the Chamley-Judd class of models cannot mimic empirical wealth distributions. Given that the distribution of wealth is pivotal in the analysis of capital taxation, it renders this class of models ill-suited for studying optimal capital taxation. In their seminal papers, Huggett (1993) and Aiyagary (1994) incorporate more realistic assumptions into the Chamley-Judd class of models, such as finite lives, labor income risk, and incomplete financial markets. Using these more realistic environments, Conesa, Kitao, & Krueger (2009), Aiyagary (1995), Imrohoroglu (1998), and others, show that the optimal capital income tax is large even in the log-run. This result is due to capital income taxation being less distrotionary in this class of models than in the Chamley-Judd class of models. However, this class of models does not account for heterogeneity in households rate-of-return on capital. As such, it can neither be used to study whether it is more efficient to tax capital income or wealth nor can it generate the realistic wealth distributions needed for proper quantitative work on optimal capital taxation 5. 5 This class of models fails at mimicking empirical wealth distributions because precautionary savings motives taper off quickly as wealth increases. As a result, this class of models generates wealth distributions with thin tails. 6

7 A newer strand of literature has shown that including rate-of-return heterogeneity enables models to generate more realistic wealth distributions (Quadrini (2000), Cagetti & De Nardi (2006, 2009), Benhabib, Bisin, & Zhu (2011, 2015, 2016)). This occurs due to households having saving motives beyond precautionary ones. However, this line of research has neither accounted for housing nor has it attempted to answer the question of whether capital income or wealth should be taxed. Recently, empirical justification for modeling rate-of-return heterogeneity has been put forth by Fagereng, Guiso, Malacrino, & Pistaferri (2016). Using 20 years of Norwegian tax record data, they show that there are vast differences in the rates of return earned across different individuals. These differences have a strong permanent component to them, are correlated with wealth, and are also present within asset classes. They also show that rate-of-return heterogeneity persists across generations, although it does mean revert. As discussed in the introduction, GKKOC (2017) are the first to study whether capital income or wealth should be taxed. They find that wealth taxation is superior to capital income taxation because it shifts the tax burden from productive investors to unproductive ones, which allows productive investors to accumulate more wealth and use it more productively. This increases the wage rate, output, and welfare. Furthermore, because their framework incorporates rate-of-return heterogeneity and incomplete financial markets, it can also generate wealth distributions similar to empirical ones. While their work goes a long way in bettering our understanding of optimal capital taxation, it does not incorporate a housing market, rendering their framework unfit to study the questions outlined in this paper. Lastly, the housing literature has neither looked at weather capital income or wealth should be taxed nor has it incorporated heterogeneity in investment ability (Sommer, Sullivan, & Verbrugge (2013), Arslan, Guler, & Taskin (2015), Kaplan, Mitman, & Violante (2016)). As such, to the best of my knowledge, this paper is the first to have a realistic housing market and a realistic wealth distribution in one model. III Model I Environment Households: There are overlapping generations of households. A newly born cohort of households has a mass of 1. Each household can live up to J periods. Households are forced to retire from the labor force for the last R periods of their lives. The age of a household is denoted by o. The probability of a household who is o years old, to survive to age o + 1, is denoted by φ o. Households care about non-durable consumption and housing 6. Housing is a homogeneous and binary durable consumption good. Households cannot own more than one house. Households pay a consumption tax τ c, on their non-durable consumption. 6 Home-owning does not directly increase utility. Instead, utility from two identical non-durable consumption bundles is larger when a household is a homeowner. 7

8 While in the labor force, households receive a wage ω for every efficiency unit they supply in the labor market. There is a deterministic life-cycle component to labor market efficiency ζ 7 o. Labor market efficiency also has a random component, e. The random component can be thought of as a shock to employment. A household is employed full-time when the random component is e hi and is partially employed when the random component is e lo. The transition probability matrix between employment states is Π e = π e hi,hi π elo,hi π ehi,lo π elo,lo. π eij is the probability of moving from state e i, i {hi, lo}, this year, to state e j, j {hi, lo}, next year. At birth, the unconditional probability of drawing e hi is π 1hi, and the unconditional probability of drawing e lo is π 1lo. There is a proportional labor income tax rate τ l. Thus, households who are in the labor force, with employment status e i and deterministic labor market efficiency ζ o, have an after-tax labor income equal to ωζ o e i (1 τ l ). Retired households receive social security benefits denoted by b. b is a function of the average before-tax labor income in the economy, b = ΦĒ. Φ is a constant (a number) and Ē is the average before-tax labor income in the economy. A household also has an ability to invest productive wealth z. Investment ability is unchanging throughout a household s life-time. When a household dies it imperfectly transmits their investment ability to their child. This is represented by the mean reverting process below: Where ρ z (0, 1) and ɛ z N (0, σ 2 ɛ z ). log(z child ) = ρ z log(z parent ) + ɛ z Households have the choice of renting a house or owning a house. Renting gives a lower utility than owning. Specifically, γ 0 < γ 1, where γ 1 is the utility advantage from owning. Denote the size of a housing unit by h (to be determined in equilibrium). Housing unit size is homogeneous. Household utility is given by u(c, h, κ) = (hγκc)1 σc 1 σ c, κ {0, 1}. Households who choose to be renters pay the rental price p r 8. Households who choose to be homeowners can obtain a mortgage m, that is no larger than a certain fraction of the price of a house (for future reference, the terms mortgage, mortgage debt, and HSD, which is an acronym for home secured debt, will be used interchangeably). In particular, m (1 λ p )p, where λ p p is the minimum down-payment on a house and p is the price of a house. Households are not allowed to take on mortgage debt during retirement. The interest rate charged on mortgage debt is r m = r f + r m, where r m is a premium paid on mortgage debt. Households can deduct a portion δ m, of their mortgage interest payments from their taxable labor income 9. To mimic the costs associated with buying and selling a house, such as Realtor fees, home inspection, and other costs, households who purchase a house have to pay pχ b above the house price and households who choose to sell their house receive pχ s below the house price. Furthermore, each period, homeowners must incur maintenance costs M h. These costs are paid to the housing construction 7 In my calibration I set this parameter to equal to 1 for all ages. 8 Households whose after-tax wealth less rental payments is lower than W min, are assisted by the government. That is, if a renter-household s after-tax wealth (which includes all incomes earned) less the rental price, does not reach a certain minimum threshold level of wealth W min, the government provides the household a subsidy to bring the households to minimum wealth level. However, in equilibrium and all counter-factual simulations this never happens. 9 In the model mortgage interest payments cannot be deducted in excess of a household s total taxable labor income. 8

9 company (to be discussed later), and are proportional to the housing price M h = δ p p, where δ h is the depreciation rate of housing. Households can accumulate productive wealth, a 0, and can borrow and lend productive wealth, denoted by d, at the risk-free interest rate r f. Borrowing can be at most a multiple λ a of a household s net wealth (a + p m). Lending can be no larger than a household s productive wealth a. This amounts to the following constraints: a d (a + p m)λ a. Productive wealth used in production depreciates at rate δ a. Household i with productive wealth a i, who borrowed or lent productive wealth d i, and who has investment ability z i, can produce intermediate output x i = k i z i, where k i = a i +d i. For its intermediate output x i, a household receives a price q xi, from the final goods producer (to be discussed). Furthermore, a household s own productive wealth a, also determines the amount of profits it gets from the housing construction company. In particular, a household with productive wealth a i obtains dividends from the construction company that are equal to π h a i ā, where ā denotes the average productive wealth in the economy and π h denotes the per-household economic profit of the construction company. I drop the subscript i to save on notation. Given the above, I can write a household s before-tax, and after-production wealth as follows: { } 1 max aπ h 0 k (a+p m)(1+λ a) ā + (1 δ a)k + q (x) x (1 + r f )(k a) = 1 a(1 + r f + π h ā ) + max 0 k (a+p m)(1+λ a) a + (r f + π h 1 ā )a + π (a, z) { } q (x) x (r f + δ a )k = (r f +π h 1 ā )a+π (a, z) is capital income earned when k is chosen optimally given the constraints placed on it. Households are also faced with a proportional capital income tax rate τ k, and a proportional productive wealth tax rate τ a. Therefore, a household s wealth after production and after taxes are paid is as follows: a(1 τ a ) + ((r f + π h 1 ā )a + π (a, z))(1 τ k ) Let the density and cumulative distributions of households across age, investment ability, employment status, productive wealth, mortgage debt, and housing status be denoted by ψ(o, z, e, a, m, κ) and Ψ(o, z, e, a, m, κ), respectively. Denote the tax rate on capital income τ k. The tax on wealth is divided into two: a tax rate on productive wealth τ a, and a tax rate on housing wealth τ h. Thus, a household who is o years old (before retirement), with deterministic labor market efficiency ζ o, employment status e, productive wealth a, mortgage debt m, and housing status κ, has the 9

10 following after-tax net wealth: 1 W = ωζ o e τ }{{} l (ωζ o e δ m κmr m ) + ((r f + π h }{{} ā )a + π (a, z)) 1 }{{} τ k ((r f + π h ā )a + π (a, z)) + a(1 τ a ) + κ(p m)(1 τ h ) }{{}}{{}}{{} κmr m }{{} 7 Term 1 on the right-hand-side in the above equation is gross labor income. Term 2 is labor income tax paid after subtracting the deductions allowed for mortgage interest payments. Term 3 is capital income. Term 4 is the tax paid on capital income 10. Term 5 is productive wealth net of productive wealth taxes paid. Term 6 is net housing wealth net of housing wealth taxes paid. Term 7 is mortgage interest payments. Together these terms make up the after-tax net wealth of a household. When a household retires, term 1 plus term 2 are equal to social security benefits b. A household has the following dynamic problem: V (o, z, e, a, m, κ; Ψ) = max c,a,m,κ {0,1} { (hγκ c) 1 σc 1 σ c + βφ o EV (o + 1, z, e, a, m, κ ; Ψ e; Ψ) } subject to (1) (1 + τ c )c + a + p r (1 κ ) + p(1 + χ b )κ = W + κ m, if κ = 0 (2) (1 + τ c )c + a + pκ + (p r + pχ s )(1 κ ) + M h = W + κ m, if κ = 1 (3) m = 0, if κ = 0 or if o J R (4) 0 m p(1 λ p ), if κ = 1 (5) a 0 (6) ψ(o, z, e, a, m, κ ) = ψ π ee 1(o, z, e, a, m, κ; Ψ) o, z, e, a, m, κ 1(o, z, e, a, m, κ; Ψ) is an indicator function. It indicates that for a given Ψ, households with state variables o, z, e, a, m, and κ, optimally choose a, m, and κ. Constraint (1) is for a household who is a renter because κ = 0. For a renter who chooses to become a homeowner (κ = 1), the constraint says that after-tax non-durable consumption expenditures plus productive wealth savings plus the cost of purchasing a house must equal to after-tax net wealth plus mortgage debt. For a renter who chooses to stay a renter (κ = 0), the constraint implies that after-tax non-durable consumption expenditures plus productive wealth savings plus the rental price must equal to after-tax net wealth. Constraint (2) is for a household is who a homeowner since κ = 1. For a homeowner who chooses to remain a homeowner (κ = 1), the constraint says that after-tax non-durable consumption expenditures plus productive wealth savings plus the price of the house plus housing upkeep costs are equal to after-tax net wealth 10 For tax purposes, taxes paid on capital income cannot be negative, i.e. there are no subsidies. 10

11 plus the new mortgage debt. For a homeowner who becomes a renter (κ = 0), the constraint implies that after-tax non-durable consumption expenditures plus productive wealth savings plus the rental price plus the cost of selling the house plus upkeep costs are equal to after-tax net wealth 11. Constraint (3) says that a household who chooses to become a renter, κ = 0, or is in retirement cannot get a mortgage. Constraint (4) indicates that mortgage debt cannot be negative and that a household cannot obtain mortgage debt that is larger than (1 λ p ) fraction of the house price. The next constraint imposes that a household cannot hold negative productive wealth. The last constraint indicates that households know the economy s law of motion. Rental Market: I take the rental market structure from Kaplan, Mitman, & Violante (2016). The rental company owns a stock of housing H r, which it rents out to households. For each rental unit, the rental company obtains the rental price p r, and incurs management costs c m, and upkeep costs δ p p. In my counter-factual simulations, I do not impose the housing wealth tax on the rental company. The rental company can frictionlessly buy housing at the housing price p. Each period, the rental company decides its next period housing stock H r. Thus, the rental company has the following problem: V (H r, Ψ) = max H r { (p r c m )H r p(h r (1 δ h )H r ) + 1 } EV (H 1 + r r, Ψ ) f Ψ describes the distribution of households across age, investment ability, employment status, productive wealth, mortgage debt, and housing status. Then using the fact that in the stationary equilibrium Ψ = Ψ and the housing price is constant, optimization implies the following relationship between the rental price and the housing price 12 : p r = c m + p 1 + r f (r f + δ h ) (1) Equation 1 shows that the rental price increases whenever the depreciation rate, the cost of maintenance, or the risk-free rate increase. Final Goods Producer: I use the final goods producer set up in GKKOC (2017). The final goods producer takes as given intermediate goods prices {q xi } i, for {x i } i, as well as the wage rate ω, paid for 1 unit of labor market efficiency. Intermediate outputs are combined with labor to produce the final good Y. The production function is Y = AQ α L 1 α, where A is aggregate productivity, Q = ( i xν i ) 1 ν aggregates intermediate outputs to produce the final good, and L is aggregate labor market efficiency units. 11 I assume for simplicity that even if a household chooses to sell their house, they are still responsible for paying housing wealth taxes and maintenance costs in the period in which they sell the house. 12 In my calibration, δ h is chosen based on the data. Then I pick p r as a fraction of p to match the net housing wealth to total net wealth in the data. p is adjusted to have a 66% homeownership rate. Then c m is selected to maintain the equality in equation 1. In counter-factual simulations, I hold δ h and c m fixed. Then, for different value of τ h, I pick the housing price and rental price jointly to have a 66% homeownership rate and keep the equality of equation 1. 11

12 to L: The final goods producer has the following problem: { } max A( x ν i ) α ν L 1 α q i x i ωl {x j } j=i,l i i For a given wage rate ω, labor demand is determined by the first order condition with respect ω = (1 α)aq α L α For given prices of intermediate outputs {q xi } i, the demand for intermediate output x i, is determined by the first order condition with respect to x i : Housing Supply: The model has a construction company. q xi = αax ν 1 i Q α ν L 1 α (2) Following Topel, & Rosen (1988), construction is subject to increasing marginal costs. This assumption is supported by empirical evidence (Glaeser, Gyourko, & Saks (2004)). Let the total stock of housing supplied be denoted by H s = H r + H o, where H r is the total stock of housing available for rent and H o is the total stock of housing owned by all households who are homeowners (in other words, it is the owner-occupied housing stock). The total stock of housing supplied H s, is made out of two components: the size of a house h, and the total mass of houses. Since each household lives in one house, the total mass of houses is equal to the total mass of households. As such, the total stock of housing supplied varies only due to changes in house size h. Each period, the construction company generates a mass of housing I h, which is added to the depreciated housing stock. For each housing unit, the construction company receives the housing price p. To generate I h, the rental company incurs a cost. To keep the model realistic, I assume that up to a certain cut-off of housing investment there is a constant and minimal marginal cost of construction that must be incurred. Denote the cut-off level by Ih c. Let the marginal cost follow the piecewise function below: c min MC(I h ) = c 1 (I h ) 1 ɛp if I h I c h if I h > I c h The top part of the piecewise marginal cost function (equation 3) says that MC = c min if I h I c h, where c min is a constant. This implies that if the price of housing falls below c min no construction will take place. As such, the price of housing is bounded below by c min. The bottom part of the piecewise marginal cost function says that if I h > Ih c, marginal costs are increasing, MC = c 1 (I h ) 1 ɛp, where c 1 is a constant and, as I will show below, ɛ p is the price elasticity of the housing stock. There are no fixed costs. Let T C h denote the total cost of construction that is congruent with the cost structure discussed above. As discussed earlier, economic profits are distributed to households as dividends depending on their how much productive wealth they own. (3) 12

13 The construction company s problem can be described as the following: Π h = max I h {pi h T C h } Assuming that I h > Ih c, the first order condition gives the following relationship between housing investment and the price of housing: I h = ( p c 1 ) ɛp In the stationary equilibrium, the total housing stock H s, is constant. Thus, I h = δ h H s. And so, when I h > Ih c, the latter gives the following housing supply in the stationary equilibrium: H s = 1 δ h ( p c 1 ) ɛp (4) Since H p p H = ɛ p, the price elasticity of the housing supply is simply ɛ p. Furthermore, since p is bounded below by c min, H s is bounded below as well. This lower bound is pinned down by equating the two parts of the piecewise marginal cost function, which gives: Government: Hmin s = 1 ( c min ) ɛp δ h c 1 Gross government revenues in the model are denoted by G g and SSC denotes the sum of social security benefits paid to all retired households. As such, net government revenues are denoted by G n = G g SSC. In my simulations I replace τ k with τ a and τ h while requiring the government to raise G n and provide the same SSC as in the benchmark. In all simulations I keep the tax rates on labor income and non-durable consumption, at their benchmark levels 13. II Equilibrium A recursive competitive equilibrium is a housing price p(ψ), a rental price p r (Ψ), housing size h(ψ), a wage rate ω(ψ), intermediate outputs prices {q xi (Ψ)} i, a housing construction total cost function T C h (I h ), per-household construction sector profits π h (Ψ), a risk-free interest rate on borrowing and lending in the productive wealth market r f (Ψ), an interest rate r m (Ψ), on mortgage debt, tax rates on labor income, non-durable consumption, capital income, productive wealth and housing wealth, τ l, τ c, τ k, τ a, τ h, respectively, a density distribution of households across age, investment ability, employment status, productive wealth, mortgage debt, and housing status, ψ(o, z, e, a, m, κ), a value function V (o, z, e, a, m, κ; Ψ), policy functions g j (o, z, e, a, m, κ; Ψ) j {k, c, a, m, κ } and an indicator function 1(o, z, e, a, m, κ; Ψ), for the household, a competitive rental company, labor and intermediate outputs demands L(Ψ) and {x i (Ψ)} i, respectively, 13 In counter-factual simulations subsidies may have to be provided to households whose after-tax net wealth (after subtracting rent expenses) does not reach W min. However, it turns out that these subsidies are zero in all of my simulations. 13

14 for the final goods producer, an exogenous net government revenue constraint G n, housing investment I h (Ψ), for the construction sector, an exogenous rental unit management cost c m, and an exogenous economy-wide productivity A, such that: 1) Given h(ψ), p(ψ), p r (Ψ), ω(ψ), {q i (Ψ)} i, r f (Ψ), r m (Ψ), and π h (Ψ), V (o, z, e, a, m, κ; Ψ) solves the household s problem, the policy functions g j (o, z, e, a, m, κ; Ψ) j {k, c, a, m, κ } are optimal, and 1(o, z, e, a, m, κ; Ψ) indicates that for a given distribution Ψ, households with state variables (o, z, e, a, m, κ) optimally choose (k, a, m, κ ). 2) Given ω(ψ), {q xi (Ψ)} i, and A, L(Ψ) and {x i (Ψ)} i solve the firm s problem. 3) Given the tax system and the optimal decisions in (1) and (2) above, the government satisfies its net revenue constraint, G n. 4) Given p(ψ) and T C h (I h ), I h (Ψ) solves the construction company s optimization problem. 5) Markets clear: i) ω(ψ) clears the labor market: ψ ζ hedψ = L(Ψ) ii) {q xi (Ψ)} i clear the intermediate outputs market: ψ g k(o, z, e, a, m, κ; Ψ)zdψ = i x i(ψ)di iii) p(ψ) generates a 66% homeownership rate (as in the data): ψ g κ(o, z, e, a, m, κ; Ψ)dψ = iv) p r (Ψ) satisfies the rental company s problem (equation 1 on page 11). v) r f (Ψ) clears the productive wealth market: ψ (g k(o, z, e, a, m, κ; Ψ) a)dψ = 0 6) Consistency: i) ψ(o, z, e, a, m, κ ) = ψ π ee 1(o, z, e, a, m, κ; Ψ)dψ o, e, a, m, κ IV Calibration In this section I choose the values of the model s parameters. First, I choose certain parameters externally, without solving the equilibrium of the model. I separate the externally calibrated parameters into non-housing market parameters and housing market parameters. Second, given the externally chosen parameters values, I choose other parameters values to have the model s stationary equilibrium generate certain statistics as close to the data as possible. My model can precisely match 4 out of the 6 targets and slightly misses 2 targets. The latter happens because my system of equations is non-linear, and thus, there is no guarantee that the model can replicate all the targeted moments exactly. However, the fit is quite good. External Calibration of Non-Housing Market Parameters: J is set equal to 81 to have households live from age 20 to age 100, inclusive. I pick R equal to 35 to have households start retirement at age 66. Survival probabilities {φ o } J o=1 are chosen to equal to the ones provided in the article "United States Life Tables, 2010" (Arias (2014)), published in the National Vital Statistics Report, which is produced by the U.S. Department of Health and Human Services. I follow the standard in the literature and pick the risk aversion parameter σ c, to equal 2. Following Huggett (1993), I set full employment e hi = 1, and unemployment e lo = 0.1. The averages of Shimer s (2005) estimates of the annual job separation rate and job finding rate for are used to set π ehi,hi = 0.86 and π elo,lo = I choose the unconditional probability of employment at birth π 1hi = to reflect the fact that, according to the Bureau of Labor 14

15 Statistics (BLS), in 2016, 91.6% of labor force persons aged were employed. To calibrate Φ, I follow the U.S. social security benefits system. According the Social Security Administration website, social security benefits replace 40% of a person s average labor income during their work period. Thus, I set Φ = 0.4. I use McDaniel s (2007) estimates for the U.S. of the average labor income tax rate, average non-durable consumption tax rate, and average capital income tax rate, and set τ l = 0.224, τ c = 0.075, and τ k = ν, which affects the curvature of the intermediate goods production function, is chosen to equal 0.95 (see footnote for more discussion of this parameter) 14. I pick the depreciation rate of productive capital δ a, to equal to 0.05 as in GKKOC (2017). I set the persistency of investment ability ρ z, equal to 0.1. This choice is consistent with the estimate of Fagereng et al. (2016) of the inter-generational persistency of the permanent component of investment ability. I normalize the economy-wide productivity A = 1, and consistent with GKKOC (2017), I set α = 0.6 so that the share of labor income in production is 60%. Lastly, I choose ɛ p equal to 0.08, so that the calibrated model s housing stock elasticity is consistent with the long-run U.S. housing stock price elasticity estimated by Mayer, & Somerville (2000). I show the parameters and their values in table 1 on page 15. Table 1: External Calibration of Non-Housing Market Parameters Variable Value Description J Years of Life R 35 Retirement Starting at Age 66 {φ o } J o=1 Varies Probability of Death (United States Life Tables) σ c 2 Risk Aversion (Standard) e hi 1 Full Employment (Huggett (1993)) e lo 0.1 Unemployment (Huggett (1993)) π ehi,hi 0.86 U.S. Average Annual Separation Rate (Shimer (2005)) π elo,lo U.S. Average Annual Job Finding Rate (Shimer (2005)) π 1hi U.S. Employment Rate of Workforce Persons Aged (BLS) Φ 0.4 U.S. Social Security Benetfits as Fraction of Average Wage (Social Security Administration) τ l Labor Income Tax Rate (McDaniel (2007)) τ c Non-Durable Consumption Tax Rate (McDaniel (2007)) τ k 0.25 Capital Income Tax Rate (McDaniel (2007)) ν 0.95 Curvature of Intermediate Goods Production (GKKOC (2017)) δ a 0.05 Productive Capital Depreciation Rate (GKKOC (2017)) ρ z 0.1 Persistency of Inter-Generational Investment Ability (Fagereng et al. (2016)) A 1 Aggregate Productivity (Normalization) α 0.6 Share of Labor Income in Output (GKKOC (2017)) ɛ p 0.08 Mayer, & Somerville (2000) External Calibration of Housing Market Parameters: I normalize the utility advantage from renting γ 0 = 1. λ p = 0.2 so that the minimum downpayment is 20%. I follow Sommer, Sullivan, & Verbrugge (2013) and set the interest rate premium 14 The reason for this choice is the following: GKKOC (2017) show that to match the Pareto tail of the wealth distribution, ν needs to be larger than 0.9. Furthermore, ν = 0.95 generates a risk-free interest rate closer to the data than ν = 0.9. Hence, ν = 0.95 is a reasonable choice for my model. 15

16 on mortgage debt r m = I choose the deductible portion of mortgage interest payments from taxable income δ m = 1, to be in line with the U.S. tax code. Consistent with the estimates in Gruber, & Martin (2003), I set the cost of selling a house pχ s = 0.07p and the cost of buying a house pχ b = 0.025p. Consistent with estimates in Harding, Rosenthal, & Sirmans (2007) for the depreciation rate of housing, I set δ h = Lastly, I normalize the housing unit size h, to equal 1 in the benchmark. I present these variables in table 2 on page 16. Table 2: External Calibration of Housing Market Parameters Variable Value Description γ 0 1 Utility from Renting (Normalization) λ p 0.2 Minimum Downpayment (Data) r m Mortgage Rate Premium (Sommer et al. (2013)) δ m 1 Mortgage Deductibility (U.S. Tax Code) χ s 0.07 Cost of Selling (Gruber, & Martin (2003)) χ b Cost of Buying (Gruber, & Martin (2003)) δ h Housing Depreciation (Harding et al. (2007)) h 1 Home Size (Normalization) Internal Calibration: My internal calibration is performed for the stationary equilibrium 15. method in the appendix on page 27. I detail the solution The calibrated parameters, their values, and statistics targeted are shown at the top of table 3 on page 17. I pick λ a = to generate a debt-toassets ratio of 0.31 (this debt does not include mortgages). This is consistent with the statistic estimated by Asker, Farre-Mensa, & Ljungqvist (2011) for private firms debt-to-assets ratio 16. I set β = so that the ratio of net wealth to output in the model equals the standard 3. To calibrate the next set of the parameters, I use averages of the statistics of interest from the Surveys of Consumer Finances. I choose σ ɛz = to have the top 1% of the wealth distribution own 33% of all the net wealth 17. Next, I jointly choose the utility advantage from home-owning γ 1, and the rental price to housing price ratio pr p, to get the model to generate owneroccupied net housing wealth to net wealth ratio close to the data s 0.23, and a homeownership rate at the top 10% of the wealth distribution that is close to the data s 95%. The utility advantage and rental price to housing price ratio that generate these statistics are γ 1 = and pr p = The housing price p, is picked to have a 66% homeownership rate 19. Lastly, to pin down cut-off housing investment Ih c (above which the marginal cost of housing investment 15 In my calibration I allow households to save up to a multiple of 60,000 of before-tax annual labor income of an employed household, which gives an upper bound on savings that is equivalent to about 4.5 billion dollars in Asker et al. (2011) estimate that public firms debt-to-assets ratio is 0.2. This implies that my model allows generous borrowing, which means I will be estimating a lower bound for welfare gains from wealth taxation. 17 In the model there are 9 investment abilities. I approximate the log-normal investment ability transmission process using the Tauchen method. 18 The value of γ 1 is lower than other numbers in the literature, for example in AGT (2015) γ 1 = In contrast to their model, in my model mortgage payments are deductible from taxable income and the rental price is greater zero. My added features make housing more attractive, and therefore, the utility advantage from homeowning is quite low. 19 Given the choices of the rental price p r, and the housing price p, c m is chosen to keep equation 1 on page 11 true. 16