Das House-Kapital
A Long Term Housing & Macro Model†
  • 1 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund
  • | 2 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

There are, by now, several long term, time series data sets on important housing & macro variables, such as land prices, house prices, and the housing wealth-to-income ratio. However, an appropriate theory that can be employed to think about such data and associated research questions has been lacking. We present a new housing & macro model that is designed specifically to analyze the long term. As an illustrative application, we demonstrate that the calibrated model replicates, with remarkable accuracy, the historical evolution of housing wealth (relative to income) after World War II and suggests a further considerable increase in the future. The model also accounts for the close connection of house prices to land prices in the data. We also compare our framework to the canonical housing & macro model, typically employed to analyze business cycles, and highlight the main differences.

Abstract

There are, by now, several long term, time series data sets on important housing & macro variables, such as land prices, house prices, and the housing wealth-to-income ratio. However, an appropriate theory that can be employed to think about such data and associated research questions has been lacking. We present a new housing & macro model that is designed specifically to analyze the long term. As an illustrative application, we demonstrate that the calibrated model replicates, with remarkable accuracy, the historical evolution of housing wealth (relative to income) after World War II and suggests a further considerable increase in the future. The model also accounts for the close connection of house prices to land prices in the data. We also compare our framework to the canonical housing & macro model, typically employed to analyze business cycles, and highlight the main differences.

1 Introduction

Research on housing and macroeconomics has prospered over the last decade.1 One outcome of this research effort is the increasing availability of long term, time series data, such as data on house prices and land prices (Knoll, Schularick and Steger, 2016), data on rents for dwellings (Knoll, 2016), and data on housing wealth (Piketty and Zucman, 2014a). The importance of this new data is widely discussed in the literature (Bonnet et al., 2014; Davis, Fisher and Whited, 2014; Gennaioli, Shleifer and Vishny, 2014; Jórda, Schularick and Taylor, 2016; Stiglitz, 2015).

Thinking about such data over the long term has appeared difficult, if not impossible, given that an appropriate and consistent long term housing & macro model has been lacking. The present paper aims to fill this void. We propose a Long Term Housing & Macro Model that enables us to analyze the mechanisms that produce the stylized facts on long term trends in house prices, land prices, the housing wealth-to-income ratio, and the split of private wealth into housing wealth and non-residential wealth.2 We place considerable emphasis on modelling the housing sector and land as an input factor. Our new framework rests on three premises:

Premise 1 (Fixed Land Endowment). The overall land endowment is given by nature. The total amount of land that can be used economically, therefore, is fixed in the long run.

Premise 2 (Land Rivalry). Land that is used as an input in the production of new houses is permanently withdrawn from alternative economic uses, unless existing houses are demolished.

Premise 3 (Land in Housing Production). (i) A house is a bundle of the underlying land plot and the residential structure. Setting up new housing projects requires land as an essential input. (ii) Investments in structures do not, however, require land as an input.

Premise 1 represents a general law of nature. This has been acknowledged by previous researchers who model land and long run economic growth, starting with Nichols (1970).3 A sceptic may argue that land augmenting technical change, land reclamation, and land development due to infrastructure investment can all enlarge the available amount of land. This is indeed plausible in the short to medium term. In the long run, however, the total amount of economically usable land is certainly fixed by nature. Increasing land scarcity is presently (and has been for decades) compatible with rising prices of farmland and urban residential land, as documented by Knoll et al. (2016) for 14 advanced countries, since World War II. This view is also compatible with the findings of Saiz (2010) who shows, by employing geographical, satellite generated data for U.S. metropolitan cities, that residential development is effectively curtailed by the availability of suitable land.4 Premise 2 simply describes the fact that land represents a rivalrous input: a plot of land that is underneath a house cannot, at the same time, host a manufacturing plant. As a result, land that is employed for setting up new housing projects resembles an exhaustible resource. Premise 3 (i) appears largely undisputed and is taken into account by existing theories (e.g., Davis and Heathcote, 2005; Favilukis, Sydney and Van Nieuwerburgh, 2016) that are discussed in Section 5 under the label Canonical Housing & Macro Model. Premise 3 (ii) has not yet been taken into account by existing models that treat land as an input for residential investment. Violating Premise 3 (ii) implicitly assumes that even replacement investments in depreciated structures (e.g., broken windows) require land. It also implies, however, a long run inconsistency in the sense that the cumulated amount of land devoted to the housing sector converges to infinity for time approaching infinity. In turn, this violates Premise 1 as well. We overcome this shortcoming by distinguishing between the extensive and the intensive margin of residential investment. Only an enlargement of the housing stock along the extensive margin, i.e., setting up new housing projects, requires land.

Our framework can be outlined as follows. We consider a two-sector Ramsey growth model. There is a numeraire goods sector and a housing sector. The paper’s innovative contribution is the elaborate modeling of the housing sector. The housing stock can be expanded along the extensive margin (new housing projects), and along the intensive margin (larger residential buildings). Real estate development firms purchase non-residential land and transform it into residential land. There is land rivalry in the sense that land devoted to real estate development is permanently withdrawn from its alternative economic use, i.e. as an input in the numeraire goods sector. Housing services firms set up housing projects (extensive margin) by purchasing a developed real estate and combine it with residential buildings to produce housing services that are sold to households. The production of housing services, at the level of a single housing project, is characterized by decreasing returns to scale, as explained below. This creates profits for housing services producers, which provide incentives for real estate development in the first place, despite perfect competition. Construction firms produce residential structures (intensive margin) that represent an accumulable factor. The land price and the house price are fully endogenous and respond to economic growth that is driven by rising population density and technical change.

As an illustrative application, we show that the calibrated model replicates, with remarkable accuracy, the historical evolution of housing wealth (relative to income) after Word War II. We also examine its implications for the future trajectory of housing wealth. Moreover, the model provides insights into the dynamics of non-residential wealth (relative to income), the second major private wealth component, consisting of physical capital and non-residential land wealth. The evolution of the wealth-to-income ratio and its decomposition is important for at least two reasons. First, Piketty (2014) stresses that a rising wealth-to-income ratio, assuming that the interest rate remains largely constant, changes the functional income distribution to the advantage of capital income recipients. Moreover, given that wealth is not uniformly distributed across the population, a rising wealth-to-income ratio is associated with a more unequal distribution of personal income.5 Second, the rising wealth-to-income ratio appears to be a key trigger for the surge in the size of the finance industry, as documented by Philippon (2015). Jordá et al. (2016) argue that the growth of finance has been closely linked to an explosion of mortgage lending to households in the last quarter of the twentieth century. By analyzing the long run evolution of housing wealth, the paper therefore also provides a deeper understanding of the real and fundamental driving forces behind the process of financialization.6

Instead of providing an exhaustive overview of the theoretical literature on housing and macroeconomics, we highlight some of the more recent contributions. Hornstein (2008, 2009) employs a general equilibrium model to explain the surge in house prices in the US between 1975 and 2005. Davis and Heathcote (2005) build a multi-sector stochastic growth model with a housing sector to examine the business cycle dynamics of residential investment. Favilukis et al. (2016) construct a stochastic two-sector general equilibrium model of housing and non-housing production to explain the surge and the subsequent decline of the price-to-rent ratio in the US housing market between 2000 and 2010. Li and Zeng (2010) employ a two-sector neoclassical growth model with housing in order to explain a rising real house price driven by comparably slow technical progress in the construction sector. Borri and Reichlin (2016) use a two-sector, life-cycle economy with bequests to explain the increasing wealth-to-income ratio and wealth inequality driven by rising construction costs that result from comparably slow technological progress in the construction sector. We depart from the previous literature in two important respects that are related to each other. First, none of the previous contributions rests on the assumption that Premises 1-3 hold simultaneously. Second, we distinguish between the extensive and the intensive margin of housing production. As a result, we are able to provide a consistent and appropriate model that can be employed to analyze the long term.

The Canonical Housing & Macro Model, described by the previously mentioned papers, is well suited for analyzing phenomena at the business cycle frequency. However, it appears less suited to think about long term macroeconomic developments. The main reasons are that it does not capture rivalry between residential and non-residential land use (violating Premise 2) and leads to the long run inconsistency (by violating Premise 1 and Premise 3 (ii)), as mentioned above. Consequently, there are many important and intrinsically long term research questions that could not be addressed so far, such as: (1) How does a secular increase in housing demand affect the distribution of income and wealth? (2) How do zoning regulations (targeting the extensive margin of the housing stock) and building restrictions (targeting the intensive margin of the housing stock) affect housing affordability in a growing economy? (3) How does the taxation of housing wealth affect the dynamics of distribution of income and wealth? The Long Term Housing & Macro Model enables us to think about such important questions and we expect that our model will be employed in the future to address these and related research questions.

The paper is organized as follows. Section 2 presents our model. Section 3 derives important implications and defines the general equilibrium. Section 4 demonstrates how the model can be applied in order to understand the long term evolution of housing wealth, land prices, and house prices. It also discusses properties of the long run equilibrium.7 Section 5 compares the Long Term Housing & Macro Model to the Canonical Housing & Macro Model. The final section concludes.

2 The Model

Consider a perfectly competitive, closed economy in general equilibrium. Time is continuous and indexed by t ≥ 0. The innovation of our macroeconomic framework is the careful modeling of the housing sector. This sector encompasses different types of firms that interact in the production of housing services. Real estate development firms purchase a piece of land and conduct infrastructure investment to develop land for residential purposes. Real estate development provides the technical and legal prerequisite to produce housing services. It diminishes the amount of land that can be employed elsewhere in the economy. The overall supply of land, Z, is exogenously fixed. Housing services firms set up new housing projects by purchasing a developed real estate.8 They produce housing services by combining a developed real estate and residential buildings (“structures”). Construction firms manufacture structures by employing materials and labor.

The numeraire sector produces a final good by combining physical capital, labor, and land. Like in standard (one-sector) models, the numeraire good can be used for (non-residential) consumption or for physical capital investments. It can also be transformed into materials that serve as input for building structures.

2.1 Firms

2.1.1 The Numeraire Good Sector

The non-residential (Y) sector produces a final good, chosen as numeraire, according to a standard Cobb-Douglas production function:

Yt=BtY(KtY)α(LtY)β(ZtY)1αβ,(1)

where KtY,LtY and ZtY denote the amounts of physical capital, labor and land employed as input in the Y sector at time t, respectively. Total factor productivity (TFP) in the Y sector, BtY>0, may change over time and α, β > 0 denote constant technology parameters that satisfy α + β < 1. Physical capital is broadly defined to include non-residential structures, in addition to machines, and is only employed in the Y sector. The capital resource constraint is given by KtYKt, where Kt denotes the total supply of physical capital in terms of the numeraire good. In equilibrium KtY=Kt will turn out to hold. Given that capital depreciates at rate δK ≥ 0, gross physical capital investment read as ItKK˙t+δKKt.9 K0 is given.

2.1.2 Housing Sector and the Characteristics of Land

There is free entry into the housing sector. Producing housing services requires to combine activities along the extensive margin (i.e., real estate development that constitutes an ex ante investment in the stock of houses) and along the intensive margin (i.e., producing residential buildings that may depreciate over time). Enlarging the stock of houses along the extensive margin absorbs additional parcels of land that are incorporated into the newly built houses. Real estate development, therefore, inevitably implies that land is withdrawn from the alternative use in the Y sector.

Real Estate Development (Extensive Margin) The “number” of housing projects (houses) at time t is denoted by Nt, a real number. This variable captures the extensive margin of the housing stock. To set up one additional housing project (an activity that is conducted by housing services firms) it requires one developed real estate. We use the term “real estate development” (conducted by real estate development firms) to describe the activity of purchasing (non-residential) land and incurring (private) infrastructure investment that transform non-residential land into residential land. The number of land units that must be put underneath each house is given by ψ > 0. Total land usage in the housing sector is given by ZtN=ψNt. The land resource constraint reads as ZtN+ZtYZ, implying that the alternative use of land in the Y sector is limited by ZtYZψNt. Consistent with Premise 2 - land rivalry, only land that has not been used in the process of real estate development can be devoted to the alternative land use (land area ZY), such as office space and land devoted to goods production, including agriculture, manufacturing (except construction), and services (except housing).

Let PtZ denote the price per unit of land. The costs C(I˜tN,PtZ) of increasing the number of developed real estates (equal to the number of housing projects) in any period t by

N˙t=I˜tN(2)

amounts to

C(I˜tN,PtZ)=ψPtZI˜tN+ξ2(I˜tN)2,(3)

ξ > 0. The cost function (3) has two components. The first component, ψPtZI˜tN, shows the costs associated with the purchase of ψN˙t units of land. The second component, ξ2(I˜tN)2ItN, gives the transformation costs (or adjustment costs) that equals the private infrastructure investment in terms of the numeraire good to transform non-residential land into residential land. These adjustment costs are convex in the number of newly developed real estates per period of time, N˙t=I˜tN, which makes Nt a state variable. If we assumed that ξ = 0, then Nt would turn into a jump variable, which appears economically less plausible. N0 is given.

As will become apparent, the land requirement per house (measured by parameter ψ) does neither affect the long run land price nor important long run ratios, such as the price-to-rent ratio and the different wealth-to-income ratios.10 The distinction between the enlargement of the housing stock along the extensive margin (which requires a fixed amount of land) and the intensive margin (which does not require land) is consistent with Premise 3 - land in housing production. It allows us to treat the total available amount of land as fixed (Premise 1 - land endowment) despite continuous depreciation of the housing stock (along the intensive margin). This feature provides a key advantage vis-à-vis the Canonical Housing & Macro Model that is important when it comes to long term analysis, as explained in detail in Section 5.

Construction and Housing Services (Intensive Margin) Producing housing services requires to purchase a developed real estate (i.e., setting up a housing project) and combine it with residential buildings (“structures”). At the microeconomic level, the developed real estate represents the fixed input, whereas structures represent the variable input in the production of housing services. The amount of housing services produced increases with the amount of residential structures employed. However, reflecting the developed real estate as fixed input, it increases less than proportionate with the amount of residential structure.11 That is, the production of housing services, at the level of single housing project, is characterized by decreasing returns to scale. Let xt denote the amount of structures per housing project at time t. An amount xt of structures produces housing services ht per house according to

ht=Bth(xt)γ(4)

0 < γ < 1, where Bth>0 is a (possibly time-variant) productivity parameter. Total supply of housing services is Ntht.12 There are two admissible institutional settings describing the relationship between housing services producers and real estate developers. First, both activities are organized within the same firm. Second, housing services producers buy real estates from real estate developers at price qtN that, in equilibrium, equals the present discounted value (PDV) of future profits accruing from a housing project.

There is a representative construction firm producing structures that are rented out to the housing services producers. It combines materials Mt and labor LtX according to a constant-returns-to-scale technology. The production of one unit of construction materials (e.g., cement) requires one unit of the numeraire good at time t.13 That is, the extraction of construction material is implicitly assumed to require capital, labor, and land with the same technology as in the numeraire good sector. Let δX > 0 denote the depreciation rate of structures (residential buildings) and

ItX=BtX(Mt)η(LtX)1η(5)

gross investment in structures, 0 < η < 1, where BtX>0 is a (possibly time-variant) productivity parameter. The total stock of residential structures, denoted by Xt, therefore evolves according to

X˙t=BtX(Mt)η(LtX)1ηItXδXXt(6)

with X0 given. The amount of residential buildings that is employed by all housing services firms cannot exceed the overall stock of residential buildings, i.e., NtxtXt.

Let qtX denote the shadow price per unit of structures associated with constraint (6). The house price is conceptualized as the sum of the value of a housing project (qtN) and the value of the employed structure associated with a real estate (valued at qtXxt), i.e.,14

PtHqtN+qtXxt.(7)

2.2 Households

There is an infinitely living, representative household with intertemporal utility

U=0(logCt+θlogSt)eρtdt,(8)

where Ct and St denote total consumption of the numeraire good and housing services at time t, respectively, ρ > 0 is the subjective discount rate, and θ > 0 indicates the relative preference for housing services.15 In equilibrium, St = Ntht.

Households supply inelastically Lt units of labor at time t to a perfect labor market. The labor resource constraint is LtX+LtYLt. We allow Lt to increase temporarily and assume that it remains constant in the long run.

Households own the entire stock of financial assets, consisting of bonds that provide firms in the numeraire sector with physical capital (Kt), shares issued by housing services firms (qtNNt), and ownership claims on construction firms (qtXXt). The total financial asset holding, At, of the representative individual is thus given by

At=Kt+qtNNt+qtXXt.(9)

The household supplies inelastically the non-residential land, i.e., the amount of land not (yet) purchased by the housing sector, to the numeraire good sector. Hence, in equilibrium, it holds that ZtY=ZψNt>0. Total private wealth, Wt, is the sum of financial asset holdings, At, and the value of non-residential land PtZZtY. Thus, according to (9) and (7), total private wealth may be expressed as

WtAt+PtZZtY=PtHNthousing wealth+Kt+PtZZtYnon-residential wealth,(10)

where PtHNt (the house price times the number of houses) represents housing capital and Kt+PtZZtY captures non-residential wealth as the sum of physical capital and non-residential land wealth. Although initial values of stocks K0, N0, X0 and Z0Y are given, total initial wealth, W0=q0NN0+q0XX0+K0+P0ZZ0Y, is endogenous because initial asset prices q0N,q0X, and the initial land price, P0Z, are endogenous.

To enable a careful model calibration, it is important to account for capital income taxation. The reason is that a tax on capital income affects the rate at which the profit stream of firms and land returns are discounted. We assume that both capital income and returns from land ownership are taxed at the constant rate τr.16 For simplicity, we do not model government consumption or public investment and assume that the tax revenue is redistributed lump sum to households.

Let wt,rt,RtZ,pt and Tt denote the wage rate, the interest rate, the rental rate of land, the (relative) price per unit of housing services, and the lump sum transfer at time t, respectively. The intertemporal household budget constraint may then be expressed as

W˙t=(1τr)rtWt+wtLtCtptSt+Tt.(11)

2.3 Sources of Economic Growth

Sectoral productivities, as described by the vector Bt(BtY,Bth,BtX), and population size, Lt, may (exogenously) grow for an arbitrary long period of time but approach a constant as t → ∞. These processes are anticipated by all agents. We think it is appropriate to assume that population size cannot grow without bounds in an economy with fixed land endowment. The numerical analysis below (Section 4) is based on the assumption that BY, Bx and L increase for more than a century according to a logistic function that is chosen as part of our calibration strategy (keeping Bh time invariant).17 One could, alternatively, allow for permanent exponential growth of BY and Bx. In this case, a steady state still exists, given that a specific parameter condition holds, as is typical for multi-sector growth models.

3 Equilibrium Analysis

This section highlights important equilibrium implications that result from the decisions of firms in the housing sector and defines the general equilibrium.

3.1 Decisions in the Housing Sector and Asset Prices

Let the rental rate of residential structures at time t be denoted by RtX. The instantaneous profit resulting from a housing project that accrues to housing services producers, noting production function (4), depends on the amount of employed structures and is given by πtptBth(xt)γRtXxt. The necessary first order condition for profit maximization yields the inverse demand schedule for structures which is articulated by the representative housing services firm

RtX=ptBthγ(xt)γ1,(12)

implying positive equilibrium profits that amount to

πt=(1γ)ptBth(xt)γ.(13)

The representative real estate development firm maximizes, at each time t, profits, defined as the value of newly developed real estates, qtNN˙t, minus total real estate development costs, C.18 Thus, using (2), the representative firm solves

maxI¯tNqtNI˜tNC(I˜tN,PtZ) s.t. (3),(14)

taking the price of developed real estates, qtN, and the land price, PtZ, as given. The associated first order condition implies the following law of motion (with N0 given) for the number of developed real estates:

[I˜tN=]N˙t=qtNψPtZξ.(15)

According to (15), if the value of a housing project is sufficiently large relative to the land price (i.e., if qtN/PtZ>ψ ), the number of houses is being enlarged, i.e., N˙t>0. Because the market for developed real estates is competitive, the price qtN will be bid up until it is equal to the PDV of the profit stream that the representative housing services firm can realize.19 At every date t, it must therefore be true that qtN=tπsetsrvdvds. Hence, the following no-arbitrage condition must hold at each t:

q˙tNqtN+πtqtN=rt.(16)

It says that the sum of the share price growth rate (capital gains) and the dividend (per unit of numeraire good invested) paid to the owners of housing services firms must equal the rate of return to bonds.20 Therefore, households are indifferent between investing in housing projects or purchasing bonds.

Construction firms rent out the entire stock of residential structures to housing services producers by charging RtX per unit of structures. The representative construction firm maximizes the PDV of the cash flow, defined as the difference between rental income RXX and the costs of gross investment, M + wLX. That is, the representative construction firm solves

max{Ms,LsX}s=tt(RsXXsMswsLsX)etsrvdvds s.t. (6),X00 given.(17)

From (17), it is easy to see (shown in Appendix 7.3) that the equilibrium rate of return per unit of structures satisfies

q˙tXqtX+RtXqtXδX=rt,(18)

where qtX denotes the value of one unit of X. Ruling out bubbles by imposing an appropriate endpoint condition, the value per unit of structures equals the PDV of future rental returns, accounting for the depreciation rate δX of structures, i.e.,

qtX=tRsXets(rv+δX)dvds.(19)

Implied by the constant returns to scale technology (5) and perfectly competitive markets, the value of total gross output in the construction sector must equal the total factor costs in construction, i.e.,

qtXItX=Mt+wtLtX.(20)

Finally, denoting by RtZ the rental rate of land, the price of land equals the PDV of income from renting out one unit of land to the producers in the Y sector, i.e., PtZ=tRsZetsrvdvds. Hence, the following no-arbitrage condition must hold:

P˙tZPtZ+RtZPtZ=rt.(21)

3.2 Definition of Equilibrium and GDP

Definition 1. A general equilibrium is a sequence of quantities, a sequence of prices, and a sequence of operating profits of housing services producers

{Yt,KtY,Xt,Nt,xt,ht,Mt,LtY,LtX,ZtY,Ct,St,At,Wt}t=0,{pt,PtZ,qtN,qtX,wt,rt,RtZ,RtX}t=0,{πt}t=0

for initial conditions K0 > 0, N0 > 0, X0 > 0 and given {Lt,BtY,BtX,Bth}t=0 such that

  1. the representative individual maximizes lifetime utility, i.e., solves
    max{Ct,St}t=00(logCt+θlogSt)eρtdt s.t. (11),Wtexp(0t(1τr)rsds)0;(22)
  2. the representative firm in the numeraire goods (Y) sector maximizes profits taking factor prices as given (i.e., factor prices equal marginal products);

  3. the representative real estate developer solves profit maximization problem (14), taking the land price, PtZ, and the price of housing projects, qtN, as given;

  4. housing services producers maximize profits at each time t, taking the price of housing services, p t , and the rental rate of structures, RtX, as given;

  5. the representative firm in the construction (X) sector solves profit maximization problem (17), taking the sequences of rental rate of structures and wage rates {RtX}t=0as given;{wt}t=0 as given;

  6. there are no arbitrage possibilities from purchasing land, i.e., (21) holds;

  7. the bond market, the land market, the market for structures, and the land market clear at any t, i.e.,
    KtY=Kt,(23)
    ZtY=ZψNt,(24)
    Ntxt=Xt,(25)
    LtX+LtY=Lt;(26)
  8. the financial asset market clears at any t, i.e., (9) holds, and total wealth is given by (10);

  9. the market for housing services clears at any t, i.e.,
    St=Ntht=NtBh(xt)γ;(27)
  10. the market for the numeraire good clears at any t, i.e.,21
    Yt=Ct+ItK+ItN+Mt.(28)

The reduced form dynamic system that results from this equilibrium definition comprises seven differential equations plus a set of algebraic equations.22

The gross domestic product (GDP) is given by the value of total consumption, CtCt + ptSt, plus the value of total investment, ItItK+ItN+qtXItX. Using (20), (27) and (28) in GDPt = Ct + It gives us the GDP as the sum of value added of all sectors:

GDPt=Yt+ptNtht+wtLtX.(29)

4 Thinking Long Term

How can we employ the model to think about the, by now available, time series data on housing and macro over the long term? In the main text, we focus on a few selected variables that appear particularly interesting in light of recent discussions, namely the housing wealth-to-income ratio, the house price, and the land price.

Piketty and Zucman (2014b) document the long term evolution of the wealth-to-income ratio, while the long term evolutions of the aggregate real land price and the aggregate real house price have been investigated by Davis and Heathcote (2007) and Knoll et al. (2016). The long term surge in real asset prices and wealth (relative to income) after World War II is of macroeconomic importance as it has the clear potential to unfold first order distributional consequences. For instance, rising land prices point to a modern version of Ricardo’s (1817) famous principle of scarcity.23 While Ricardo was mainly concerned with agricultural land and the production of corn to feed a growing population, societies in modern times are to a larger extent confronted with the need for residential investments to meet the increasing demand for housing services under the constraint of scarce land.

Thinking long term requires two ingredients: First, a model that allows for the determination of a consistent steady state (Section 4.1). Second, a model that can also be employed to trace the evolution of the variables under study over time (Section 4.2).

4.1 Long Run Equilibrium

According to Proposition 1 in Appendix 7.2, a long run equilibrium (steady state) typically exists, is unique and characterized by zero growth rates of all stock variables.24 Here we highlight only a few selected steady state implications.

Define by D = L/Z the population density and by DYLY/ZY the labor-to-land ratio in the Y sector. The long run land price, PZ*, and the long run house price, PH*, read as25

PZ*=(1αβ)r*(αr*+δK)α1α(BY)11α(DY*)β1α,(30)
PH*=ψ(1αβ)(αr*+δK)α1α(BY)(DY*)β1α(11γ+δXr*)r*+δX,(31)

where the long run real interest rate is given by r*=ρ1τr and the long run labor-to-land ratio in the Y sector, DY*, is proportional to D, but independent of TFP parameters, such as BY.26 Thus, both PZ* and PH* depend positively on population density, mirroring Ricardo’s (1817) principle of scarcity.

We follow Piketty (2014) and Piketty and Zucman (2014a, 2014b, 2015) and employ the net domestic product (NDP) as aggregate income in the denominator to report wealth-to-income ratios (Appendix 7.1). NDP equals GDP net of depreciation, NDP = GDPδKKδXqXX. Housing wealth is given by PHN. The housing wealth-to-income ratio, HNDPPHNNDP, is in the long run given by:

HNDP*=(1+(1γ)δXr*)[r*+δX+(r*+δXθ+ηγδX)r*+δKr*+(1α)δK+(1η)γδX][r*+(r*+δXθ+ηγδX)r*+δKr*+(1α)δK+(1+(1η)γ)δX](1+1θ)(r*+δX).(32)

Importantly, HNDP* neither depends on population density, D, nor on TFP parameters. Appendix 7.2 (Steady State Properties) and Appendix 7.3 (Proofs) demonstrate that this property applies, in the long run, to all interesting ratios of endogenous variables, such as the non-residential wealth-to-income ratio, NNDPK+PZZYNDP, the NDP-to-GDP ratio, NDPGDP, the labor share in GDP, wLGDP, the investment rate, IGDP, and the ratio of total housing consumption to total housing wealth, pSPHN (i.e., the inverse of the house price-to-rent ratio). The same holds for the long run allocation of labor and land, characterized by the fractions LX/L, LY/L, ZN/Z, and ZY/Z. These steady state properties imply that we can assess, say, the long run housing wealth-to-income ratio without knowing D (the ratio of the population size to the amount of economically usable land) or TFP levels in the long run.27

4.2 Transitional Dynamics - Post World War II

The calibrated Long Term Housing & Macro Model is solved numerically. Appendix 7.4 describes the calibration strategy and documents the data sources. We trace the historical evolution of the housing wealth-to-income ratio, the house price, and the land price after World War II and provide projections until the year 2100. The numerical analysis employs the relaxation algorithm to solve the model for transitional dynamics (Trimborn, Koch and Steger, 2008). This technique allows us to calculate exact numerical solutions for the transitional dynamics implied by the non-linear differential equation system.28

4.2.1 The Experiment

The macroeconomic, model-based experiment — conducted separately for France, Germany, UK and the USA — can be described as follows: First, we feed country-specific population growth, as reported by Piketty and Zucman (2014b), into the model.29 We also feed in country-specific time paths for TFP parameters BtX and BtY according to a logistic function, such that, given the observed (exogenous) increases in population size and endogenous capital accumulation, implied aggregate income growth rates between 1955 and 2010 coincide with the actual growth rates, reported by Piketty and Zucman (2014b).30 That is, economic growth is driven by exogenous population growth, exogenous technical progress, and endogenous capital accumulation. Growth is transitory, but may extend over several centuries. Second, initial state variables are set to match initial wealth-to-income ratios. Specifically, N0 and X0 are set such that the model-based initial housing wealth-to-income ratio, H0NDP, matches the respective empirical value in 1955, assuming identical proportional deviations from the initial steady state (given L0,B0Y and B0X) for both Nt and Xt. Similarly, K0 is set such that model-based non-residential wealth-to-income ratio, N0NDP, matches the respective empirical value in 1955.31 We let the simulation run from 1955 until 2100 to yield country-specific time paths for HtNDP for that time period.

4.2.2 Housing Wealth-to-Income Ratio

Figure 1 displays the evolution of the housing wealth-to-NDP ratio, HtNDP, for France, Germany, UK, and the US, over time. The solid (blue) lines display the model-based time paths, whereas the dotted (red / purple) lines represent the empirical data.32 Overall, the model matches the empirical series quite accurately, given that only information about initial state variables, population growth and aggregate income growth enters the experiment and the calibration of the model (described in Appendix 7.4) does not rely on housing wealth. There are, of course, deviations between the empirical data and the results of the model. The strongest deviation can be observed for Germany that starts in 1955 with H1955NDP0.65 and in recent times shows H2005NDP2.19, whereas the model implies H2005NDP3.33. Given H1955NDP0.65, Germany starts with the lowest value of the housing stock, which appears reasonable with respect to war destructions during World War II.33 The model economy then builds up the housing stock quite rapidly, despite convex (quadratic) land transformation costs. The model-based values nearly match the empirical observations in 2005 for FRA and UK. For the US, the empirical value in 2005 is about H2005NDP2.18, while the model implies H2005NDP2.50, a deviation of about 15 percent. The long run value is the same for every economy, given that we assume a unique capital income tax rate of τr = 0.2, and amounts to about 410 percent. The model therefore gives us, for the first time, a notion about the long run housing wealth-to-income ratio and the specific trajectory that converges to this asymptotic value.34

In a similar fashion, one can investigate the evolution of non-residential wealth (relative to income) over time, NtNDP.35 The Long Term Housing & Macro Model therefore enables us to analyze the major private wealth-to-income ratios, as discussed by Piketty (2014) and Piketty and Zucman (2014a, 2015), within a unifying theoretical model of different wealth components.

Figure 1.
Figure 1.

Housing wealth (relative to NDP) from 1955 until 2100.

Citation: IMF Working Papers 2017, 080; 10.5089/9781475590302.001.A001

Notes. (1) Dotted line: empirical series (red: linear interpolation between decadal averages starting in 1955 until 2005; purple: linear interpolation between the 2005 value and the actual 2010 value). Solid (blue) line: implied series resulting from the calibrated Long Term Housing & Macro Model. (2) Country specific parameters: N0 and X0 are set such that model-based and empirical values for P0HN0NDP0 coincide, assuming the same proportional deviations from the initial steady state. Ko is set such that model-based and the empirical values for K0+P0ZZ0YNDP0 coincide. Population grows according to a logistic function in line with empirical population growth. TFP parameters BtY and BtX increase over time according to logistic functions such that GDP grows in accordance with empirical data between 1955 and 2010, given (exogenous) population growth and (endogenous) capital accumulation. Data are taken from Piketty and Zucman (2014b). Land size is normalized to Z = 1 except for Germany where Z = 1.4 after the reunification in 1990. (3) Common parameters: α = 0.28, β = 0.69, γ = 0.9, η = 0.38, θ = 0.22, δX =0.015, δK =0.07, ρ = 0.025, τr =0.2, ψ = 1, ξ =100.

4.2.3 Land Prices and House Prices

The model-based explanation for the evolution of housing wealth (relative to income) provided above has an important correlate, namely the evolution of land prices and house prices. The factor land plays a prominent role. Given that the overall land endowment is fixed (Premise 1) and that land represents a rivalrous factor (Premise 2), land is becoming scarcer and more expensive as the economy grows. Our analysis thus suggests that the price channel of increased land valuations plays a critical role in the process of pushing wealth-to-NDP ratios up over time. Rising land prices push the house price up and this price channel triggers an increase in the ratio of housing wealth to NDP, HtNDP.36

Figure 2 displays the evolution of the land price, PtZ, and the house price, PtH, as implied by the calibrated Long Term Housing & Macro Model, assuming that the labor force grows, in total, by a factor of 1.72 and TFP parameters BtY and BtX rise by a factor of 2.7 and 2.16 between 1955 and 2010, respectively. These are average values across France, UK, and the US.37 The model yields an overall increase in the land price by a factor of 7.5 and an overall increase in the house price by a factor of 3.4 between 1955 and 2010. This pattern is largely consistent with the empirical data. Considering averages across France, UK and the USA yields a growth factor of 5.2 for land prices and a growth factor of 3.2 for house prices between 1955 and 2010 (Knoll et al., 2016). Hence, the calibrated model is consistent with an additional important stylized fact, namely that the land price increases by more than the house price and therefore explains the major share of the surge in house prices.38

Figure 2.
Figure 2.

Land prices and house prices (“average economy”)

Citation: IMF Working Papers 2017, 080; 10.5089/9781475590302.001.A001

Notes: Evolution of the land price, PtZ, and the house price, PtH, as resulting from the model (1955 values = 1). Lt increases, in total, by a factor of 1.72, while BtY and BtX increase by a factor of 2.7 and 2.16, respectively, according to logistic functions. State variables (Kt, Nt and Xt) start below their (conditional) steady state, the proportional deviations are averaged across FRA, UK, and the US, according to Figure 1. Other parameters as for Figure 1.

5 Canonical Housing & Macro Model

We finally sketch and discuss the canonical macroeconomic model with a housing sector, as it has been recently employed, among others, in Davis and Heathcote (2005), Hornstein (2009), Iacoviello and Neri (2010), Li and Zeng (2010), Favilukis et al. (2016), and Borri and Reichlin (2016). The goal is to highlight the major differences between the Canonical Housing & Macro Model and our Long Term Housing & Macro Model.

5.1 Setup

The economy is perfectly competitive and comprises two sectors. The numeraire (Y) sector combines capital, KtY, and labor, LtY to produce a final output good according to

Yt=BtY(KtY)α(LtY)1α,(33)

BtY>0,0<α<1. Notice that land does not enter the Y technology. The numeraire good can be either consumed or invested. The housing sector produces housing services that are sold to households. Housing services per period of time, St, are proportional to the stock of houses Ht, i.e., St = bHt with b > 0. Without loss of generality we set b = 1. The production of houses employs a fixed amount of (additional) land together with a variable amount of structures. The fixed amount of residential land becomes exogenously available each period. The stock of houses accumulates according to H˙t=ItHδHHt, where ItH denotes gross investment and δH > 0 the depreciation rate of the housing stock. Gross additions to the housing stock are described by a constant-returns to scale technology,

ItH=BtHXtβZ¯1β,(34)

BH > 0, 0 < β < 1, where Xt is the amount of structures that are combined with a fixed quantity of (additional) land, Z¯, which is inelastically supplied each period. Residential structures (a flow) are produced according to

Xt=BtX(KtX)γ(LtX)1γ,(35)

Bx > 0, 0 < γ < 1, by combining capital, KX, and labor, LX. Consequently, the evolution of the housing stock H is described by

H˙t=B˜tH(KtX)βγ(LtX)β(1γ)Z¯1β=ItHδHH,(36)

where B˜tHBtH(BtX)β. The exogenous and time-invariant supply of (additional) residential land, Z¯, used as specific factor in housing production, is supposed to capture the idea that “a constant quantity […] of new land/permits suitable for residential development is available each period” (Favilukis et al., 2016). Davis and Heathcote (2005) point to the “declining relative returns to agricultural use” (p. 756) as a potential source of new land.39

Households maximize utility and firms maximize profits. Intertemporal utility of the representative consumer is again given by (8). Let qtH denote the value per unit of H at time t. Total wealth is equal to the value of financial assets, comprising physical capital and housing wealth, i.e.,

Wt=At=Kt+qtHHt,(37)

where K0 > 0 and H0 > 0 are given. Again, let rt denote the rate of return of financial assets, wt the wage rate, RtZ the land rent, and pt the price of housing services, respectively. Household wealth then accumulates according to

W˙t=rtWt+wtLt+RtZZ¯ptStCt.(38)

The resource constraints (holding with equality in equilibrium) are given by

KtX+KtYKt,(39)
LtX+LtYLt,(40)

and market clearing in the numeraire good sector requires

Yt=Ct+ItK=Ct+K˙t+δKKt.(41)

GDPt for this economy is the sum of value added of the numeraire good sector, Yt, the housing services sector, ptHt, and the construction sector (building new houses with value qtHItH).40

GDPt=Yt+ptHt+qtHItH.(42)

5.2 Comparison with the Long Term Housing & Macro Model

The Canonical Housing & Macro Model, due to its lean structure, represents an attractive and important analytical tool to study a large set of research questions in the business cycle context.41 We argue, however, that this approach is less suited for the long term analysis. The Long Term Housing & Macro Model, on the other hand, is a bit more complex, but appropriate when it comes to research questions that focus on longer time horizons. We now highlight the major differences between the two models.

5.2.1 Land Availability and Land Allocation

Canonical Housing & Macro Model The quantity of (additional) land that is employed every period in the housing sector, Z¯, is an exogenous and time invariant flow variable. Land is not used elsewhere in the economy. That is, there is no endogenous land allocation and non-residential land as a wealth component is not accounted for. This automatically implies that the amount of land allocated to the housing sector does not change along the transition. What may change along the transition to the steady state is the quantity of complementary factors: KtX and LtX.

Long Term Housing & Macro Model The economy-wide amount of land is fixed stock variable. Land can be either employed in the Y sector (ZtY), or in the housing sector (ZtN=ψNt). The quantity of land allocated to the housing sector is time-varying and endogenous. This difference in the land allocation (exogenous and time-invariant vs. endogenous and time-varying) has important implications for the evolution of the land price, as explained below.

5.2.2 Cumulated Amount of Land Absorbed by a Stationary Housing Stock

Canonical Housing & Macro Model Let us assume a stationary steady state with a positive and constant housing stock and a positive depreciation rate δH > 0. The cumulated amount of land that is incorporated in the housing sector converges to infinity as time goes to infinity. This is not compatible with Premise 1 (fixed overall land endowment) above and may be labelled long run inconsistency.

Long Term Housing & Macro Model Let us assume a stationary steady state with a positive and constant housing stock and a positive depreciation rate of residential buildings δX > 0. The cumulated quantity of land that is incorporated in the housing stock is ZtN=ψNt, a finite number, even for time approaching infinity. Hence, the Long Term Housing & Macro Model is consistent with Premise 1 and the long run inconsistency is avoided.

5.2.3 Extensive and Intensive Margin of Housing Production

Canonical Housing & Macro Model The housing stock is a one-dimensional object. It appears appropriate to interpret an increase in the housing stock as an increase along the extensive margin because this process requires land. The alternative interpretation would imply that there is a single house that is enlarged continuously upwards.

Long Term Housing & Macro Model The housing stock can be enlarged along the extensive margin (increasing the number of houses) and along the intensive margin (increasing the size of the typical houses). Only the enlargement along the extensive margin requires land as an input (Premise 3).

The distinction between the extensive and intensive margin has three advantages: (i) It allows us to avoid the long run inconsistency of the canonical model that land input goes to infinity when there is depreciation of residential structures. (ii) It enables a distinction between a change of the housing stock either along the extensive margin (e.g., through real estate development or a natural disaster) or along the intensive margin (e.g., through a depreciated stock of structures or investment in structures). (iii) It enables us to distinguish the effects of regulations targeting the extensive margin of the housing stock (zoning regulations) and the intensive margin (building restrictions) within model-based policy evaluations.

5.2.4 Land Price Dynamics

To illustrate a major difference between the two models in the determination of the land price, which is important for our analysis of the previous section, we ask the following question: How does the land price evolve in response to a destruction in the housing stock?

Canonical Housing & Macro Model Given that land input is a (time-invariant) flow, the land price, PtZ, equals the competitive land rent, RtZ, i.e., PtZ=RtZ=ptItHZ¯ (Davis and Heathcote, 2005; Favilukis et al., 2016). Suppose that the initial housing stock is below the steady state level, H0 < H*. As a consequence, the economy allocates a large amount of capital, KtX, and labor, LtX, to the construction sector. This construction boom implies that the housing stock is built up. As the economy converges to the steady state, the construction boom diminishes, implying that capital and labor are being reallocated to the Y sector. Because the time-invariant flow of land, along the transition to the steady state, is combined with less and less KtX and LtX, the marginal productivity of land, ItHZ¯, declines.42 Moreover, the price of housing services, pt, declines too as the supply in the housing market is being enlarged. Taken together, the land price PtZ=ptItHZ¯ unambiguously decreases along the transition to the steady state, as illustrated in Figure 3 (a).

Figure 3.
Figure 3.

Evolution of the land prices in response to housing stock destruction: Canonical Model versus Long Term Housing & Macro Model.

Citation: IMF Working Papers 2017, 080; 10.5089/9781475590302.001.A001

Notes: Panel (a): Canonical Housing & Macro Model, initial housing stock H0 = 0.8H*, set of parameters: ρ = 0.025, γ = 0.7, δH = 0.015, δK = 0.07, α = 0.4, β = 0.5, θ = 0.22, L = Z¯ = BY = BX = 1. Panel (b): Long Term Housing & Macro Model: N0 = 0.8N* and X0 = 0.8X*; Panel (c): Long Term Housing & Macro Model: N0 = 0.8N* and X0 = X*; Panel (d): Long Term Housing & Macro Model: N0 = N* and X0 = 0.8X*; Panels (b) - (d): set of parameters other than initial states as in Figure 2 except BY = BX = L =1.

Long Term Housing & Macro Model Each unit of land can be either permanently incorporated in a house or can be employed for an infinite sequence of periods in the Y sector. The equilibrium land price, excluding bubbles, equals the PDV of an infinite land rent earned in the Y sector: PTZ=tRτZetsrvdvds.

Figure 3 (b) shows that the land price increases along the transition to the steady state in response to a destruction of the housing stock in both dimensions, the intensive and the extensive margin. To understand why, first, suppose the number of houses N is initially below its steady state level (N0 < N*), given the amount of structures. Then there is initially more land available for the alternative use in the Y sector (Premise 2) than in the long run. A high ZtY implies that its marginal product RtZ=YZtY and thus land price PtZ are low early on. This triggers off a construction boom, implying that (i) labor is reallocated to the construction sector and (ii) physical capital is temporarily decumulated (a standard implication of multi-sector models with multiple state variables). That is, non-residential land, ZtY, is initially combined with a small amount of complementary factors, KtY and LtY, again implying that PtZ is low. As more and more land is used, however, to build houses, ZtY is diminished over time by real estate development (Premise 2). This leads to increasing land scarcity. Consequently, both RtZ=YZtY and thus the land price PtZ increase over time, as shown in Figure 3 (c). (As the construction boom diminishes over time, also labor returns to the Y sector and the physical capital stock accumulates to its steady state level.) Both mechanisms cannot arise in the canonical model because there is neither an alternative use of land nor can land become scarcer over time. Second, suppose that the amount of structures X is initially lower than its steady state level (X0 < X*), for a given number of houses. In this case, due to decreasing returns in the production of housing services, the inverse demand for structures, R0X, is high initially. Again, this triggers off a construction boom that drives up the land price over time, as shown in Figure 3 (d), invoking similar mechanisms as for Figure 3 (c).

5.2.5 Housing Wealth and Non-Residential Wealth

Canonical Housing & Macro Model Housing wealth is given by qtHHt. This wealth component comprises the cumulated quantity of land that is incorporated in the housing stock, similarly to the Long Term Housing & Macro Model (both residential land and structures included). The non-residential wealth-to-NDP ratio is given by NtNDP=KtNDP=KtNDPt. That is, land outside the housing sector, i.e., the value of non-residential land, is missing (i.e., PtZZtY0).

Long Term Housing & Macro Model Residential land, ZtN=ψNt, enters housing wealth, PtHNt=qtNNt+qtXXt, since land is incorporated in houses. Non-residential land, ZtY, enters non-residential wealth, that in total is given by Kt+PtZZtY. This matches the assignment of land in the different wealth components according to national accounting. The Long Term Housing & Macro Model also endogenizes the split between non-residential wealth into the two components physical capital and non-residential land wealth.43

Models that do not capture land as an input in the non-residential sector (Y sector) cannot adequately attribute rising non-residential wealth to rising land prices associated with land scarcity. Rising land prices are, however, of central importance for both the evolution of housing wealth and the evolution of non-residential wealth.

6 Summary

We have presented a novel housing & macro model that is specifically designed to think about long term, time series data on housing and macro variables. The model rests on three premises: (1) fixed overall land endowment; (2) land rivalry between non-residential and residential production; (3) land as an essential input in housing production along the extensive margin (setting up new housing projects), but not along the intensive margin (investing in residential buildings).

To illustrate how this set up can be applied, we have employed the calibrated model to analyze the housing wealth-to-income ratio in four industrialized economies. The model replicates, with remarkable accuracy, the historical evolution of housing wealth (relative to income) after World War II and suggests a further considerable increase. The model also accounts for the close connection of house prices to land prices in the data. The analysis points to the fundamental importance of land for understanding the dynamics of wealth. In line with recent empirical studies, it therefore provides a modern formalization of Ricardo’s (1817) principle of land scarcity, which states that economic growth primarily benefits the owners of the fixed factor land.

There are many important research questions on the long term macroeconomics of housing, such as those mentioned in the introduction, that could not be discussed so far. We hope that the analytical framework developed in this paper will be applied to address these and related research questions.

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7 Appendix

7.1 Wealth-to-Income Ratios

We first define wealth-to-income ratios. We use fraktur (or Gothic) scripture with superscript GDP and NDP to denote wealth-to-GDP and wealth-to-NDP ratios, respectively.

The housing wealth-to-GDP ratio is denoted by HtGDP (speak “fraktur H”) and may be expressed as

HtGDPPtHNtGDPt=qtNNt+qtXXtGDPt=AtKtGDPt.(43)

The non-residential wealth-to-GDP ratio, denoted by NtGDP, is the sum of the ratio of physical capital to GDP, KtGDPKtGDPt, and the ratio of the value of non-residential land (farm land and other productive, non-residential land property) to GDP (“non-residential land wealth-to-GDP ratio”), ZtGDPPtZZtYGDPt, i.e.,

NtGDPKt+PtZZtYGDPt=RtGDP+ZtGDP.(44)

The total (private) wealth-to-GDP ratio, MtGDP, reads as

MtGDPWtGDPt=HtGDP+NtGDP.(45)

To calculate wealth-to-NDP ratios from wealth-to-GDP ratios, we divide HtGDP,KtGDP,ZtGDP,NtGDP,MtGDP by the ratio of net income to gross income (“NDP-to-GDP ratio”), ιt=NDPtGDPt. We denote wealth-to-NDP ratios by HtNDPHtGDPιt,KtNDPKtGDPιt,ZtNDPZtGDPιt,NtNDPNtGDPιt,MtNDPMtGDPιt.

7.2 Steady State Properties

This section provides analytical results for the long-run equilibrium. We omit the time index t in the remainder of this Appendix.

Proposition 1. (Existence) Suppose that

ϱβ1αγ(1η)1γη>0(A1)

holds. Then there exists a unique, non-trivial long run equilibrium in which DY = LY/ZY fulfills

DY*>D[=LZ].(46)

Assumption (A1) holds for any reasonable calibration of the model. The allocation of labor and land is characterized by fractions lX = LX/L, lY = LY/L, N = ZN/Z, and YZY/Z. Equilibrium property (46) is equivalent to Y* < lY* and N* > lX*. The share of land devoted to the housing sector, N, exceeds the share of labor devoted to the housing sector, lX*, whereas the opposite holds in the rest of the economy. The housing sector is endogenously land-intensive, whereas the rest of the economy is labor-intensive.

We next define the “house-price-to-rent ratio” as the ratio of the price of one house with x units of structures put on ψ units of land to the cost of renting h units of housing services produced with the same amount of structures, x. Formally,

PPHph=qN+qXxph.(47)

Note that p is also equal to the ratio of house capital, PHN, to housing expenditure, pS.

Proposition 2. (Prices). Under (A1), in long run equilibrium,

(i) the interest rate is given by

r*=ρ1τr;(48)

(ii) the price for housing services, p*, is decreasing in BX, Bh, increasing in D, BY, and independent of ψ;

(iii) the wage rate, w*, is decreasing in D, independent of BX, Bh, ψ, and increasing in BY;

(iv) the house price-to-rent ratio is

p*=1+(1γ)δXr*r*+δX;(49)

(v) the return to land, RZ*, the land price, PZ*, and the house price per land unit, PH*/ψ, are increasing in D, BY, and are independent of BX, Bh, ψ.

The Keynes-Ramsey rule implies that the long run after-tax interest rate equals the subjective discount rate, (1 – τr)r* = ρ. An increase in population density, D, means that labor becomes more abundant and land becomes scarcer, in turn lowering the wage rate and raising both the return per unit of land and its price. Consequently, since the housing sector is land-intensive, the price for housing services rises with D. An increase in TFP of the Y sector, BY, raises output of the numeraire good for a given factor allocation, thus increasing the relative long run price for housing services, p*. It also transmits into higher (long run) factor returns, w*, RZ*, like in standard one-sector models. Higher productivity parameters in the housing sector (increase in BX or Bh) lower the price of housing services. While the direct effects of productivity improvements on the long run wage rate, w*, are positive, there is a counteracting and balancing effect on the value of the marginal product of labor in the construction sector through a decrease in the price for housing services. Thus, w* as well as the long run return to land, RZ*, remain unchanged. In the long run, the land price and the house price change proportionally to the rental rate of land, RZ*, when population density or TFP parameters change.

Proposition 3 (Factor allocation). Under (A1), in long run equilibrium, the factor allocation as characterized by ℨY* = 1 – ℨN* and lX* = 1 – lY* is independent of D, B, ψ.

Proposition 4 (Wealth-to-GDP ratios and NDP-to-GDP ratio). Under (A1), in long run equilibrium, the housing wealth-to-GDP ratio is

HGDP*=PH*N*GDP*=1+(1γ)δXr*r*+(r*+δXθ+ηγδX)r*+δKr*+(1α)δK+(1+(1η)γ)δX,(50)

the physical-capital-to-GDP ratio is

KGDP*=K*GDP*=αr*+δK+(r*+δX[1+(1η)γ])(r*+(1α)δK)r*+δXθ+ηγδX,(51)

the non-residential land wealth-to-GDP ratio is

ZGDP*=PZ*ZY*GDP*=1αβ(1+r*+(1α)δKr*+δKr*+δX+(1η)γδXr*+δXθ+ηγδX)r*,(52)

and the NDP-to-GDP ratio is

ι*=NDP*GDP*=1+1θ1+(1θ+ηγδXr*+δX)(r*+δK)r*+(1α)δK+(1η)γδXr*+δX,(53)

Thus, HGDP*,KGDP*,ZGDP* and ι* are independent of D, B, ψ.

Corollary 1. Under (A1), in long run equilibrium, the non-residential wealth-to-GDP ratio, NGDP*=KGDP*+ZGDP*, the total wealth-to-GDP ratio, MGDP*=HGDP*+NGDP*, and wealth-to-NDP ratios HNDP*,KNDP*,ZNDP*,NNDP*,MNDP* are independent of D, B, ψ.

Recalling HNDP*=HGDP*ι*, we obtain (32) in the main text from (50) and (53).

Proposition 4 and Corollary 1 reflect the result that the long run factor allocation is independent of population density, D, and technology parameters, B and ψ (Proposition 3). An increase in population density, D, changes the marginal product of labor and the return to land equally in the housing sector and the Y sector, leaving the factor allocation unchanged. Moreover, reflecting homothetic preferences, changes in technological parameters B and ψ do not induce structural change and leave the factor allocation unaffected as well.

Proposition 5 (Long run amount of structures and housing consumption).

Under (A1), in long run equilibrium,

(i) the amount of structures per unit of land, X*/Z, is increasing in D, BY, BX, Bh, and independent of ψ;

(ii) the amount of housing services per capita, S*/L, is increasing in BY, Bx, Bh, and decreasing in D, ψ.

Since the long run land allocation, i.e., N* = ψN*/Z, is independent of population density, D, and productivity parameters, B, according to Proposition 3, the number of houses per unit of land, N*/Z, is independent of D and B as well. Thus, an increase in D or B means that more structures are built per unit of land, e.g., houses become “higher” rather than more numerous in more densely populated and in more advanced regions, reflecting the opportunity costs of land in its alternative use in the Y sector. Technological progress therefore implies that the amount of housing services increases as well. That would also hold if on each unit of land more houses could be built (decrease in ψ). Finally, because of decreasing returns in transforming structures on a piece of land into housing services, the per capita amount of housing services is decreasing in population density.

Define H = pNh + wLX as the contribution of the housing sector (housing services and residential construction) to GDP. We finally consider the labor share in total income, LwLGDP, the land income share in the housing sector, ℑ ≡ RZZN/H, and the investment (and savings) rate of the economy, sIGDP. Note that s can be decomposed into a non-residential and residential (housing) investment rate, denoted by sKIKGDP and sHIN+qXIXGDP, respectively, i.e.,

sIK+IN+qXIXGDP=sK+sH.(54)

Proposition 6. (Factor income shares and investment rates) Under (A1), in long run equilibrium, the labor share in total income is

L*=w*LGDP*=β+r*+(1α)δKr*+δK(1η)γδXr*+δXθ+ηγδX1+r*+(1α)δKr*+δKr*+δX(1+(1η)γ)r*+δXθ+ηγδX,(55)

the land income share in the housing sector is

I*=1γ1+(1η)γδXr*+δX.(56)

and the economy’s total investment rate is

s*=1ι*.(57)

Thus, *, * and s* are independent of D, B, ψ. The non-residential investment rate, sK*, and the residential investment rate, sH*, are independent of D, B, ψ as well.

7.3 Proofs

Proof of Proposition 1 (Existence). We first calculate the partial equilibrium by examining the supply side only, i.e. we take the (relative) price for housing services (p) as well as the interest rate (r) as given. Then we turn to the demand side.

The current-value Hamiltonian of the representative construction firm associated with optimization problem (17) together with the necessary first-order conditions can then be expressed as

HXRXXMwLX+qX[BXMη(LX)1ηδXX],(58)
[HXM=]1+ηqXBX(LXM)1η=0,(59)
[HXLX=]w+(1η)qXBX(MLX)η=0,(60)
[HXX=]RX+δXqX=q˙XrqX.(61)

Note that (18) follows from (61).

The typical final output firm maximizes profits given by

Π=BY(KY)α(LY)β(ZY)1αβ(r+δK)KYwLYRZZY.(62)

Using KY = K, the necessary first-order conditions are

[αYKY=]αBYKα1(LY)β(ZY)1αβ=r+δK,(63)
[βYLY=]βBYKα(LY)β1(ZY)1αβ=w,(64)
[(1αβ)YZY=](1αβ)BYKα(LY)β(ZY)αβ=RZ.(65)

Combining (63) and (64), leads to

K=αβwr+δKLY.(66)

Substituting (66) into (63) and (65), we obtain

w=β(αr+δK)α1α(BY)11α(ZYLY)1αβ1α,(67)
RZ=(1αβ)BY(αβwr+δK)α(LYZY)α+β.(68)

respectively. Setting N˙=0 in (15) implies, for the long run,

qN=ψPZ.(69)

Substituting (13) into (16) and using q˙N=0, we have

qN=(1γ)pBhxγr.(70)

Setting P˙Z=q˙X=0 in (21) and (18), respectively, we obtain, for the long run,

PZ=RZr,(71)
qX=RXr+δX.(72)

Setting X˙=0 in (6) and using X = Nx, we have

x=BXMη(LX)1ηδXN,(73)

Next, combine (59) with (72) to obtain

M=LX(ηRXBXr+δX)11η.(74)

Now substitute (73) into (12) to find

RX=pBhγ(BXMη(LX)1ηδXN)γ1.(75)

Substituting (75) into (74) yields a useful expression for M:

M=(LX)γ(1η)1γη(γηpBh(δXN)1γ(BX)γr+δX)11γη.(76)

Combining (60) with (72) and (74) we obtain

w=(1η)ηη1η(RXBXr+δX)11η.(77)

Substituting (75) into (77) leads to

w=(1η)ηη1η(γpBhr+δX)11η(BX)γ1η(δXN)1γ1ηMη(γ1)1η(LX)γ1.(78)

Substituting (76) into (78) gives us a useful expression for w:

w=(1η)ηγη1γη(γBhpr+δX)11γη(BX)γ1γη(NδXLX)1γ1γη.(79)

Combining (79) with (67) leads to

LXN=(1η)1γη1γηγη1γ(γBhpr+δX)11γ(BX)γ1γδX[β(αr+δK)α1α(BY)11α]1γη1γ(LYZY)1αβ1α1γη1γΦ(LYZY,p,r,B),(80)

Note that Φ is increasing as a function of LY/ZY. Moreover, Φ is strictly concave as a function of LY/ZY if and only if (A1) holds. Substituting (70) and (71) into (69), we obtain

[π=](1γ)pBhxγ=ψRZ.(81)

Inserting (73) and (68) into (81), we get

(1γ)pBhw(BXMη(LX)1ηδXN)γ=ψ(1αβ)BYw1α(αβ1r+δK)α(LYZY)α+β.(82)

Substituting (67) and (78) into the right-hand side and left-hand side of (82), respectively, yields

(1γ)Mη(1γη)1η(LX)1ηγ(1η)(ηpBh)η1η(γr+δX)11η(BX)γη1η(δXN)1γη1η=ψ(1αβ)βLYZY.(83)

Now substitute (76) into (83) to find

LXN=ψv(r)LYZY,with(84)
υ(r)(1αβ)(1η)γδXβ(1γ)(r+δX).(85)

Finally, combine the right-hand sides of (80) and (84) to obtain a unique, nontrivial solution for DY = LY/ZY that is given by

DY=(ηγη1γη(BXγγ+δX)γ1γη(Bhp)11γη(1ηβ)γ(1η)1γη(αr+δK)α1αBY11α(ψ(1αβ)1γ)1γ1γη)1ϱD˜Y(p,r,B).(86)

According to (84) and LY/ZY=D˜Y(p,r,B), we have

LXN=ψυ(r)D˜Y(p,r,B).(87)

Next, using LY = DYZY and LX = Nψυ(r)DY, LX + LY = L and ZY = ZψN we obtain

NZ=1ψ[1υ(r)](1DD˜Y(p,r,B)),(88)
zY=ZYZ=11υ(r)(DD˜Y(p,r,B)υ(r)),(89)
NZY=1ψD¯Y(p,r,B)D11υ(r)D¯Y(p,r,B)D.(90)

From LX = Nψυ(r), D = L/Z and (88), we also have

LXL=υ(r)1υ(r)(D˜Y(p,r,B)D1).(91)

Thus, as LY/L = 1 – LX/L, we obtain

LYL=11υ(r)(1υ(r)D˜Y(p,r,B)D).(92)

This closes the supply side analysis.

According to (27) and (70), we can write for the long run:

qNN=(1γ)pSr.(93)

Moreover, combining (12) with (27), we have RXNx = γpS. Combining X = Nx, (72) and (73) we thus find that for the long run:

qXX=RXNxr+δX=γpSr+δX.(94)

Using (66), (93) and (94) in (9) leads to long run total asset value:

A=αβwLYr+δK+(1γ)pSr+γpSr+δX.(95)

Next, substituting (73) into condition (27) and using (76) gives us

pS=N(Bhp)11γη(BX)γ1γη(LXN1δX)γ(1η)1γη(γηr+δX)γη1γη.(96)

The total tax revenue that is redistributed to households reads as

T=τrrW.(97)

As W = A + PZ ZY, we have W˙=A˙+P˙ZZY+PZZ˙Y. In steady state, P˙Z=Z˙Y=0, thus, W˙=A˙.

According to (8) and (11), the current-value Hamiltonian for the household optimization problem (equilibrium condition 1 in Definition 1) is given by

HlogC+θlogS+λ[(1τr)rW+wL+TCpS],(98)

where λ is the multiplier (co-state variable) associated with financial asset holding, A. Necessary optimality conditions are ∂ℋ/∂C = ∂ℋ/∂S = 0 (control variables), λ˙=ρλH/W (co-state variable), and the corresponding transversality condition. Thus,

λ=1S,(99)
θS=λp,(100)
λ˙λ=ρ(1τr)r.(101)

Combining (99) and (100), we have

C=pSθ,(102)

whereas combining (99) and (101) yields the Keynes-Ramsey rule

C˙C=(1τr)rρ.(103)

We seek for a steady state without long run growth. Setting C˙=0 in (103) gives us the long run interest rate

r*=ρ1τr.(104)

Combining (102) and (11) with W˙=0 and using (97) we obtain

(1θ+1)pS=rW+wL.(105)

Substituting W = A + PZZY into (105) and using (95) and RZ = rPZ leads to

(1θ+γδXr+δX)pS=w(αβrLYr+δK+L)+RZZY.(106)

Inserting (67) into (68) leads to

RZ=(1αβ)(αr+δK)α1α(BY)11α(LYZY)β1α.(107)

By substituting (67), (96) and (107) into (106) we obtain

(1θ+γδXr+δX)NZY(Bhp)11γη(BX)11γη(LXN1δX)γ(1η)1γη(γηr+δX)γη1γη(LYZY)β1α=(αr+δK)α1α(BY)11α[αrr+δK+β(LLY1)+1α].(108)

Substituting LY/ZY=D˜Y(p,r,B), (87) and (90) into (108) and recalling the definition of ϱ in (A1) implies, for the long run,

(1θ+γδXr*+δX)(1D¯(p*,r*,B)D)(Bhp*)11γη(BX)γ1γη(υ(r*)δX)γ(1η)1γη(γηBMr*+δX)γη1γηψ1γ1γη(υ(r*)D¯Y(p*,r*,B)D1)D˜Y(p*,r*,B)e=(αr*+δK)α1α(BY)11α[αr*r*+δK+βυ(r*)(1D¯Y(p*,r*,B)D)υ(r*)D¯Y(p*,r*,B)D1+1α].(109)

Substituting (85) and (86) in (109) we find that the long run price for housing services, p*=p˜*(r*,B,D), is implicitly given by

D˜Y(p*,r*,B)=Dμ(r*)DY*withμ(r)1θ+ηγδXr+δX+r+(1α)δKr+δK(1η)γδXβ(r+δX)1θ+ηγδXr+δX+1γ1αβr+(1α)δKr+δX.(110)

We have D˜Y(0,r*,B)=0,limpD˜Y(p,r*,B)and D¯Yp>0, according to (86) and (A1). Thus, p* > 0 exists and is unique. Moreover, is easy to show that μ(r) ∈ (υ(r), 1) if and only if υ(r) < 1, which holds for all r according to (A1).44 Thus, also (46) holds under (A1). This concludes the proof.

Proof of Proposition 2 (Prices). Part (i) follows from (104). Using (86) and applying the implicit function theorem to (110) confirms part (ii). To prove parts (iii)-(v), first use (67) to find

w*=β(αr*+δK)α1α(BY)11α(DY*)1αβ1α.(111)

Moreover, recall from (71) that

PZ*=RZ*r*with RZ*=(1αβ)(αr*+δK)α1α(BY)11α(DY*)β1α,(112)

according to (107). Next, use (4), (12), (104), (70) and (72) in (47) to confirm (49). Combining (70) and (121) and using (4) yields

p*h*=ψRZ*1γ.(113)

Using (49) and (113) in PH* = p*h*p*, we obtain

PH*=ψ(1αβ)(αr*+δK)α1α(BY)11α(DY*)β1α(11γ+δXr*)r*+δX.(114)

Using that, according to (110), DY* is independent of B and proportional to D confirms parts (iii)-(v). This concludes the proof.

Proof of Proposition 3 (Factor allocation). Combining (89) and (91) with (110), we obtain

zY*=ZY*Z=μ(r*)υ(r*)1υ(r*)[=1zN*],(115)
lX*=LX*L=υ(r*)1υ(r*)1μ(r*)μ(r*)[=1lY*],(116)

respectively. Recalling (104) as well as the definitions of υ(r) and μ(r) in (85) and (110), respectively, concludes the proof.

Proof of Proposition 4 (Wealth-to-GDP ratios and NDP-to-GDP ratio). According to (88) and (110), we have

N*Z=1μ(r*)ψ[1υ(r*)].(117)

Using (29), we can write the housing wealth-to-GDP ratio, the physical capital-to-GDP ratio, and the non-residential land-wealth-to-GDP ratio as

HGDP=PHNGDP=p1+YpNh+wLXphN,(118)
KGDP=KGDP=KY1+pNhY+wLYLXL,(119)
ZGDP=PZZYGDP=PZZYY1+pNhY+wLYLXL,(120)

respectively. According to (69) and (71), in the long run,

qN*=ψRZ*r*.(121)

Substituting the expression for RZ* as given by (112) into (113) leads to

p*h*=ψ(1αβ)1γ(αr*+δK)α1α(BY)11α(DY*)β1α.(122)

According to (1), (4) and (66), we have

YpNh=BY(αβ+wr+δK)α(LYZY)α+βZYZphNZ.(123)

Moreover, using the definitions of υ(r) and μ(r) in (85) and (110), respectively, we find

μ(r*)υ(r*)1μ(r*)=1αβ1γ(1θ+ηγδXr*+δX)(r*+δK)r*+(1α)δK.(124)

Substituting (111), (115), (117), (122) and LY*/ZY* = DY* into (123), and using (124), we get

Y*p*N*h*=(1θ+ηγδXr*+δX)(r*+δK)r*+(1α)δK(125)

in long run equilibrium. Moreover,

wLXphN=wphlXL/ZN/Z.(126)

Substituting (111), (116) and (117) into (126) and using (104), D = L/Z and υ(r*) as given by (85), we obtain

w*LX*p*N*h*=(1η)γδXr*+δX.(127)

Using (49), (125) and (127) in (118) confirms (50).

According to (63), (64), (65) and LY = L(1 – lX), we have

KY=αr+δK,(128)
wLY=β1lX,(129)
RZZYY=1αβ.(130)

Using that RZ* = r*PZ*, in long run equilibrium,

PZ*ZY*Y*=1αβr*.(131)

Substituting (125), (128), (129) and (131) in (119) and (120), respectively, and using (85), (116) and (124) confirm (51) and (52).

Finally, in the long run, investment equals depreciation. Thus, in the long run, NDP equals consumption expenditure, NDP* = C*. The ratio of consumption expenditure to GDP can be written as

ι*=NDP*GDP*=C*GDP*=C*+p*S*GDP*=C*p*S*+11+Y*p*N*h*+w*LX*p*N*h*.(132)

Using (102), (125) and (127) in (132) confirms (53). This concludes the proof.

Proof of Corollary 1. Results immediately follow from Proposition 4, (44) and (45).

Proof of Proposition 5 (Long run amount of structures and housing consumption). Inserting (76) into (73) and using (87), (117) and (110) implies that the long run equilibrium amount of structure per unit of land is given by

X*Z=1μ(r)1υ(r*)(υ(r*)δXDμ(r*))1η1γη(γηBhp*(r*+δX)ψ1γ)η1γη(BX)11γη.(133)

According to (86) and (110), we have

Bhp*=(Dμ(r*))ϱ(1γη)(αr*+δK)α(1γη)1α(BY)1γη1α(ψ(1αβ)1γ)1γηγη(BXγr*+δX)γ(1ηβ)γ(1η)(134)

Substituting (134) into (133) and using the definition of ϱ in (A1) yields

X*Z=χ(r*)D1η+βη1αBX(BY)η1α,where(135)
χ(r)ηη(β1η)γη(1η)1γη(αr+δK)αη1α(υ(r)δX)1η1γη[1μ(r)](1αβ1γγr+δX)η(1γ)1γη[1υ(γ)]μ(r)1η+βη1α.(136)

According to (27) and X = Nx, the consumption of housing services per capita is given by

SL=NLBh(XN)γ=(NZ)1γBhD(XZ)γ.(137)

Using (117) and (133) in (137), we obtain

S*L=χ(r*)γ(1μ(r*)ψ(1υ(r*)))1γBh(BX)γ(BY)γη1αD1γ+(1αβ)γη1α.(138)

Parts (i) and (ii) follow from (135) and (138), respectively, using that υ(r*), μ(r*) and χ(r*) are independent of D, B and ψ. This concludes the proof.

Proof of Proposition 6 (Factor income shares and investment rates). Using (29), we have

L=wLGDP=wLY1+pNhY+wLYLXL.(139)

Using (116) and (129) together with (85), (104) and (110) in (139) confirms (55). Moreover, using H = pNh + wLX, we have

I=RZZNH=RZZNpNh+wLX.(140)

For the long run, according to (140), we can write

I*=RZ*zN*p*h*N*Z+w*lX*D,(141)

where we used zN=ZN/Z,lX=LX/Z and D = L/Z. Using (111), (112), (113), (115), (116) and (117) in (141) and employing (85) confirms (56). Finally, since total investment is equal to capital depreciation in the long run, I* = δKK* + δXqX*X*,

s*=I*GDP*=1NDP*GDP*,(142)

confirming (57). Note that K˙=0 implies IK* = δKK*; thus, sK*=IK*GDP*=δKKGDP*, confirming its claimed properties by recalling Proposition 4. Finally, recall sH* = s*sK*. This concludes the proof.

7.4 Calibration

The set of country-specific, time-invariant parameters comprises the capital income tax rate, τr, and the initial conditions, K0, N0 and X0. The set of country-specific, time-varying parameters comprises the population density, D, and TFP parameters, BY and BX. (We normalize Bh = 1.) The remaining parameters are viewed as being general and set to match the relevant empirical characteristics of the US economy. The calibration strategy does not assume that the US currently is in long run equilibrium.

According to (102), the marginal rate of substitution between the two consumption goods equals the relative price, p = θ(C/S. Data on housing expenditures for the US indicate that the ratio of households’ housing expenditures to total consumption expenditures,

pSC=pSC+pS=θ1+θ,(143)

is quite stable over time for the period 1960-2012 and equals, on average, about 18 percent (Knoll et al., 2016). The value is very close to the average values for the UK, France and Germany. Setting the expression in (143) to 0.18 suggests θ = 0.22.

We next turn to depreciation rates (δX and δK). For the housing sector (residential structures), Hornstein (2009, p. 13) suggests, by referring to data from the US Bureau of Economic Analysis (2004), that δX = 0.015. The depreciation rate of physical capital, δK, can be inferred from the definition of gross investment in physical capital, IK=K˙+δKK, i.e.,

δK=IKKK˙K.(144)

We assume that, off-steady state, physical capital investment is 10 percent of the physical capital stock, IK/K = 0.1.45 Assuming that the average annual growth rate of physical capital, K˙/K, was about three percent (the sum of the long term GDP per capita growth rate of two percent and the population growth rate of one percent), we arrive at δK = 0.07, according to (144).46

Turning to concavity parameters of the production function in the Y sector, we start by noting that β equals, in equilibrium, the expenditure share for labor in the Y sector, i.e., β = wLY/Y, according to (64). Recall that H = pNh + wLX is the value-added of the housing sector (housing services and residential construction). According to (29), we can thus write GDP = Y + H. Using LY/L = 1 – lX, Y = GDPH, and recalling that the labor share in GDP is L=wLGDP, we get

β=(1lX)L1=HGDP.(145)

According to Bureau of Economic Analysis (2015b), the average value-added of the housing sector and (residential and non-residential) construction as percentage of GDP in the period 1998-2001 (prior to the housing boom) in the US was, on average, 9.1 percent and 4.4 percent, respectively. In our model, qXIX is the value-added of residential construction. The ratio of residential to total investment in structures during 1998-2001 was 61 percent (calculated from Bureau of Economic Analysis, 2015c), which suggest a value-added of residential construction relative to GDP, qXIXGDP. Thus, we set HGDP to 9.1 + 2.7 ≈ 12 percent. According to Henderson (2015, Tab. 2.1), the US employment share in construction decreased from 4.8 percent in 2004 (before the financial crises) to 4.1 percent in 2014 (after the financial crises). Taking an intermediate value of 4.5 percent and multiplying it by the fraction of residential investment in total investment in structures (61 percent), we arrive at lX = 0.045 × 0.61 = 0.027. According to Karabarbounis and Neiman (2014, “CLS KN merged”), the corporate US labor share in total income was pretty stable in the period 1975-2008, only recently declining in a more pronounced way. The average value for the period 1975-2012 was 62 percent. Using HGDP=0.12, lX = 0.027 and L=0.62, we arrive at β = 0.973 × 0.62/0.88 ≈ 0.69, according to (145).

According to (4), (13) and (27), the ratio of housing services producers’ profits to their revenue is πph=1γ. We approximate with the average for the period 1995-2006 of “output of housing services” minus costs minus the “current surplus of government enterprises” (mostly mortgage finance agencies) and divide it by the “output of housing services” (a measure for Nph).47 We arrive at γ = 0.9.

We next derive the numerical value of the output elasticity of materials in the construction sector, η. Combining the first-order condition with respect to labor in the construction sector (60) with (5), we have wLX = (1 – η)qXIX. Using wLX=lXLGDP, we obtain

η=1wLXqXIX=1lXLqXIX/GDP.(146)

Recalling qXIXGDP=0.027,L=0.62, and lX = 0.027, we obtain η = 0.38.

The long run interest rate, r*=ρ1τr, plays an important role for long run asset prices, including the land price. Albeit not modelled here explicitly, the return to equity contains a risk premium; also, ownership of firms and land is at risk of devaluation because of natural disaster, environmental damage or expropriation by government action. Thus, we shall set the subjective discount rate at a value that is in the middle or at the upper end of the typical range used in calibration exercises, ρ = 0.025. (For numerical long run implications, shown in Online-Appendix A.4, we compare results for ρ = 0.02 vis-à-vis ρ = 0.03.)

The capital income tax rate is reported in Piketty and Zucman (2014b). It slightly fluctuates over time. Considering average tax rates between 1970-2010, for Germany this gives us τr = 0.18, for France τr = 0.19, and for the US τr = 0.22. The capital tax rate for the UK is not available in Piketty and Zucman (2014b). According to the OECD tax database (2015), the net top statutory dividend tax rate “to be paid at the shareholder level, taking account of all types of reliefs and gross-up provisions at the shareholder level” is similar in the US and the UK. Capital tax rates may thus be viewed as similar in the four considered countries and around 0.2. As we cannot predict future changes in tax rates, we employ τr = 0.2 for all displayed transitional dynamics.

According to Propositions 2-6, the amount of land per house, ψ, does not affect the factor allocation, factor prices, wealth-to-income ratios, income shares and investment rates in the long run. We set the arbitrary value ψ = 1. Similarly, the cost parameter ξ (residential land development costs) does not have an impact on the long-run equilibrium. It does, however, affect the speed of N-dynamics along the transition. We set ξ = 100.

Finally, we need to specify a. We relate to the evidence by Rognlie (2015, Fig. 6), who decomposes the net capital share of corporate sector value-added in the US into there turn on equipment, structures, land, and pure profits (time series since 1950). He reports that pure profits fluctuate around zero, whereas the land income share is fluctuating around 0.04. The US corporate sector incorporates the housing sector. Thus, the land income share, LIS, is the average of the land income share in the non-residential sector, RZZY/Y = 1 – αβ (according to (65)), and the land income share in the housing sector, I=RZZN/H. Consequently, α is given by

(1αβ)(1HGDP)+IHGDP=LIS(147)

According to (56), the long run land income share in the housing sector is, for γ = 0.9, η = 0.38, δX = 0.015 and r* =0.031 (implied by ρ = 0.025, and τr = 0.2), given by 8.5 percent. Using I=0.085 together with β = 0.69 and HGDP=0.12 in (147) suggests α = 0.28 for LIS = 0.04.

Online-Appendix A: Long Term Housing & Macro Model

In Online-Appendix A.1, we invoke Walras’ law as consistency check of the analytical derivation of the long run equilibrium. Online-Appendix A.2 summarizes the dynamic system, whereas Online-Appendix A.3 summarizes long run equilibrium values. These two sets of information form the basis for the numerical implementation. Online-Appendix A.4 discusses steady state implications for the various wealth-to-income ratios in the model. It also shows long run implications for the labor share, allocation variables and investment rates. Online-Appendix A.5 shows that the evolution of the housing wealth-to-income ratio is rather insensitive to the amount of available land. Online-Appendix A.6 displays transitional dynamics of the non-residential wealth-to-income ratio for the calibrated model.

A.1 Consistency Check Using Walras’ Law

We show that the market for the numeraire good clears in long run equilibrium, i.e., Y = C + δKK + M holds in steady state (market for numeraire good clears, according to equilibrium condition 10 in Definition 1, and K˙=N˙=X˙=0 ). Recall the following definitions hold.

ϱ=β1αγ(1η)1γη,(148)
μ=1θ+ηγδXr+δX+r+(1α)δKr+δK(1η)γδXβ(r+δX)1θ+ηγδXr+δX+1γ1αβr+(1α)δKr+δK,(149)
υ=(1αβ)(1η)γδXβ(1γ)(r+δX),(150)
χ=ηη(β1η)γη(1η)1γη(αr+δK)αη1α(υδX)1η1γη(1μ)((1αβ)γ(1γ)(r+δX))η(1γ)1γη(1υ)μ1η+βη1α,(151)
D=LZ.(152)

In steady state, the following relationships hold (we suppress superscript (*)):

K=αβwr+δKLY,(153)
KY=αr+δK,(154)
w=β(αr+δK)α1α(BY)11α(μD)1αβ1α,(155)
C=pSθ(156)
LXL=υ(1μ)(1υ)μ=1LYL(157)
LYL=11υυ1υ1μ,(158)
SL=χγ(1μψ(1υ))1γBh(BX)γ(BY)γη1αD1γ+(1αβ)γη1α,(159)
Bhp=(Dμ)ϱ(1γη)(αr+δK)α(1γη)1α(BY)1γη1α(ψ(1αβ)1γ)1γ(ηBM)γη(BXγr+δX)γ(1ηβ)γ(1η),(160)
NZ=1μψ(1ν),(161)
M=(LX)γ(1η)1γη(γηpBh(δXN)1γ(BX)γr+δX)11γη.(162)

We first derive

YCδKKK=YKCKδK=r+δKαpSθ1βαr+δKβ(αr+δK)α1α(BY)11α(μD)1αβ1αLYδK=χγ(1μ1υ)1γ(Dμ)ϱ(1γη)(αr+δK)α(1γη)1α(1αβ1γ)1γ(β1η)γ(1η)θD1γ+(1αβ)γη1α(αr+δK)11α(μD)1αβ1α(11υυ1υ1μ)ηγη(γr+δX)γ+1+(1α)δKα=(ηη(νδX)1η1γη(1αβ1γγr+δX)η(1γ)1γη(β1η)γη(1η)1γη)γ1μ1υμ(1η+βη1α)γθD1γ+(1αβ)γη1ααr+δK(11υυ1υ1μ)ηγη(γr+δX)γ×(Dμ)1βγη1αγ(1η)(1αβ1γ)1γ(β1η)γ(1η)+r+(1α)δKα(163)

Rearranging further, we obtain

YCδKKK=(υδX)(1η)γ1γη(Dμ)1βγη1αγ(1η)(1αβ1γ)1γ1γη(β1η)γ(1η)1γημ(1η+βη1α)γθD1γ+(1αβ)γη1ααr+δK(γr+δX)(1η)γ1γη1μμυμ+r+(1α)δKα=r+(1α)δKαr+δKα1θ(1αβ1γ)1γ1γη×(βυ1η)γ(1η)1γη(r+δXγδX)(1η)γ1γη1μμυ(164)

For Y = C + δK K + M to hold, this must be equal to

MK=(LX)γ(1η)1γη(γη(Dμ)ϱ(1γη)(BY)1γη1α(ψ(1αβ)1γ)1γ(δXN)1γ(BX)γ(r+δX)1γηγη(BXγ)γ(1ηβ)γ(1η))11γηαr+δK(BY)11α(μD)1αβ1αLY=(υ1υ[1μ1])γ(1η)1γηη((BY)1γη1α(1αβ)1γ(δX1μ1υ)1γ(r+δX)1γ(1ηβ)γ(1η))11γη(γ1γ)1γ1γηαr+δK(BY)11α(11υυ1υ1μ)μ1γ1γη=(γ(1αβ)δX(1γ)(r+δX))1γ1γη(βυ1η)γ(1η)1γη1μμυr+δKαη.(165)

Recall that μ is implicitly defined by

1αβ1γ(1θ+ηγδXr+δX)1μμυ=r+(1α)δKr+δK.(166)

The market for the numeraire good thus clears if

r+(1α)δKr+δK=(γ(1αβ)δX(1γ)(r+δX))1γ1γη(βυ1η)γ(1η)1γη1μμυη+1θ(1αβ1γ)1γ1γη(βυ1η)γ(1η)1γη(r+δXγδX)(1η)γ1γη1μμυ(1αβ1γ)γ(1η)1γη=(βυ1η)γ(1η)1γη(r+δXγδX)γ(1η)1γη(1αβ1γ)γ(1η)1γη=((1αβ)γδX(1γ)(r+δX))γ(1η)1γη(r+δXγδX)γ(1η)1γη,(167)

which holds.

A.2 Dynamic System

The Long Term Housing & Macro Model is fully described by seven differential equations plus a set of static equations. The differential equations read as:

X˙=BXMη(LX)1ηδXX,(168)
N˙=qNψPZξ,(169)
W˙=rW+wLCpS,(170)
C˙=C[(1τr)rρ],(171)
q˙N=rqNπ,(172)
q˙X=(r+δX)qXRX,(173)
P˙Z=rPZRZ,(174)

where K0, N0, X0 are given. The set of static equations is given by:

ZY+ψN=Z,LX+LY=L,W=K+qNN+qXX+PZZY,(175)
x=XN,h=Bhxγ,S=Nh,(176)
w=(1η)ηη1η(BXqX)11η,RZ=BY(1αβ)(αβwr+δK)α(LYZY)α+β,(177)
RX=pBhγxγ1,π=(1γ)pBhxγ,(178)
θC=pS,KLYαβwr+δK,LXM=(1qXηBX)11η,(179)
(r+δKα)α(wβ)1α=BY(ZYLY)1αβ.(180)

In total, there are 21 equations and 21 endogenous variables: X, x, N, W, C, π, qN, PZ, qX, K, ZY, LX, LY, M, S, h, r, w, RX, RZ, p.

A.3 Steady State Values

According to Section 7.3, we can summarize long run values as follows.

r*=ρ1τr.(181)
ZY*=Zμ(r*)υ(r*)1υ(r*),N*=ZZY*ψ,(182)
LX*=Lυ(r*)[1μ(r*)][1υ(r*)]μ(r*),LY*=LLX,(183)
w*=β(αr*+δK)α1α(BY)11α(Dμ(r*))1αβ1α,(184)
RZ*=(1αβ)(αr*+δK)α1α(BY)11α(Dμ(r*))β1α,(185)
PZ*=RZ*r*,qN*=ψRZ*r*,(186)
p*=(Dμ(r*))ϱ(1γη)(αr*+δK)α(1γη)1α(BY)1γη1α(ψ(1αβ)1γ)1γBhηγη(BXγr*+δX)γ(1ηβ)γ(1η),(187)
X*=Zχ(r*)D1η+βη1αBX(BY)η1α,(188)
x*=X*N*,h*=Bh(x*)γ,S*=N*h*,
C*=p*S*θ,(189)
π*=ψRZ*,(190)
qX*=γp*S*(r*+δX)X*,RX*=(r*+δX)qX,(191)
M*=(ηBXqX*)11ηLX*,(192)
K*=αβw*LY*r*+δK,(193)
W*=K*+qN*N*+qX*X*+PZ*ZY*,(194)

where we used definitions

ϱ=β1αγ(1η)1γη>(A1)0,(195)
υ(r)=(1αβ)(1η)γδXβ(1γ)(r+δX),(196)
μ(r)=1θ+ηγδXr+δX+r+(1α)δKr+δK(1η)γδXβ(r+δX)1θ+ηγδXr+δX+1γ1αβr+(1α)δKr+δK,(197)
χ(r)=ηη(β1η)γη(1η)1γη(αr+δK)αη1α(νδX)1η1γη[1μ(r)](1αβ1γγr+δX)η(1γ)1γη[1υ(r)]μ(r)1η+βη1α.(198)

A.4 Long-Run Implications of the Calibrated Model

Table A.1 shows (in percent) long run implications for wealth-to-NDP ratios as resulting from the calibrated Long Term Housing & Macro Model, assuming alternative subjective discount rates, ρ, and capital income tax rates, τr.

The annual average of the housing wealth-to-NDP ratio, HNDP, in the 2000s was 217 percent in the US with a peak of 254 percent in 2006 before the financial crisis.48 In the UK, Germany and France, HNDP was 271 percent, 217 percent and 285 percent in the 2000s, respectively. The calibrated model under τr = 0.2 implies the long run value, HNDP*, to be 357 percent for ρ = 0.03 and 478 percent for ρ = 0.02.49 Our analysis thus suggests that, in the longer run, the US housing capital will rise considerably above the pre-crisis level if the capital income tax rate remains similar.

Maintaining τr = 0.2, the implied non-residential wealth-to-NDP ratio is NNDP* = 349 percent for ρ = 0.03 and to NNDP* = 440 percent for ρ = 0.02. About three quarters are attributed to physical capital and one quarter to non-residential land wealth. In the 2000s, NNDP was 249 percent in the US, 241 percent in the UK, 139 percent in Germany, and 190 percent in France. The implied future increase from current levels partly reflects the growing importance of land scarcity in the process of economic development. However, as discussed in more detail in Online-Appendix A.6, the fact that we do not distinguish rural and urban land may contribute to an unrealistically high long run non-residential land wealth-to-NDP ratio, ZNDP*. Without further amendments, if the current tax system remains in place, the implied long run wealth-to-NDP ratio, MNDP*, is in the range from 706 to 918 percent, depending on the subjective discount rate, ρ. The sensitivity of long run wealth-to-GDP ratios to ρ is rooted in the fact that the PDV of asset values, particularly land that does not depreciate, is heavily dependent on the rate at which future returns are discounted.

Table A.1.

Long run implications for wealth-to-NDP ratios.

article image

Notes: All values are expressed in percent. Results are based on the following set of parameters: α = 0.28, β = 0.69, γ = 0.9, η = 0.38, θ = 0.22, δX = 0.015, δK =0.07. Recall HNDP*=PH*N*NDP*,KNDP*=K*NDP,ZNDP*=PZ*ZY*NDP*,NNDP*=KNDP*+ZNDP*,MNDP*=W*NDP*=HNDP*+NNDP*.

Table A.1 also displays the sensitivity of wealth-to-income ratios with respect to capital income taxation. We start with lowering τr to 15 percent. As the long run interest rate, r*, is slightly reduced, the implied wealth-to-NDP ratios increase somewhat. The implied housing wealth-to-NDP ratio, HNDP*, now becomes almost 500 percent for ρ = 0.02 and associated long run wealth-to-NDP ratio, MNDP*, is 954 percent without amending non-residential land wealth. These values may be considered as upper bounds. Raising τr to 25 percent, ρ = 0.03 implies that HNDP* is 340 percent, which may be considered as lower bound.

According to Table A.2, the implied long-run labor share in income, L*, is about 60 percent. It does neither critically depend on ρ nor on τr. Comparing it for the latest available year, the labor income share was 57.2 percent in the year 2012 in the US and 62.1 percent in 2011 in the UK (Karabarbounis and Neiman, 2014, “CLS KN merged”). For τr = 0.2, the implied long run employment fraction in residential construction, lX*, is 4.9 and 3.9 percent for ρ equal to three and two percent, respectively. It is slightly decreasing in τr and somewhat higher than the recent US value of 2.7 percent used for calibrating the model. The result is associated with a higher stock of structures, X, in the long run compared to the current (off steady state) one, in line with our prediction of a rising H. A high construction labor employment share copes with the constant depreciation rate of structures, δx, given a high value of X. The implied long run fraction of land devoted to the housing sector in the economy, N*, is about 36-37 percent.

Table A.2.

Long run implications for labor income share, allocation variables, and investment rates.

article image

Notes: All values are expressed in percent. Results are based on the following set of parameters: α = 0.28, β = 0.69, γ = 0.9, η = 0.38, θ = 0.22, δX = 0.015, δK = 0.07. Recall L*=w*LGDP*,lX*=LX*L,zN*=ZN*Z,s*=I*GDP*,sK*=IK*GDP*,sH*=IN*+qX*IX*GDP*.

For τr = 0.2, the long run investment rate for the baseline calibration, s*, is 18.9 and 21.9 percent for ρ equal to three and two percent, respectively, and may be compared to the US gross private domestic investment rate of 16.5 percent for the year 2014 (Bureau of Economic Analysis, 2015a). For the period 1969-2014, the average ratio of US non-residential investment to GDP, a measure for sK, was 12.6 percent, which approximately also was the average value of the 1970s, 1980s, 1990s and 2000s (Bureau of Economic Analysis, 2015c, Tab. 1.1.10). The value is somewhat lower than the long run value for sK* that is implied by our model. The implied long run residential investment rate, sH*=s*sK*, is in the range of 3.6 to 4.9 percent.

A.5 Evolution of Housing Wealth-to-Income under Expansion of Land

Figure A.1.
Figure A.1.

Housing wealth to income (NDP) over 140 years under alternative assumptions on land endowment.

Citation: IMF Working Papers 2017, 080; 10.5089/9781475590302.001.A001

Notes. The economy starts in a steady state. The solid line shows the evolution of Ht in the case of a fixed land endowment; the dashed line shows the case where the amount of economically usable land increase over time, i.e., Zt increases by a factor of 2. Lt,BtY and BtX increase by a factor of 2 according to logistic functions. Other parameters: α = 0.28, β = 0.69, γ = 0.9, η = 0.38, θ = 0.22, δX = 0.015, δK = 0.07, Bh = 1, ρ = 0.025, τr = 0.2, ψ = 1, ξ = 100.

Starting in steady state (contrary to Figure 1), Figure A.1 shows the evolution of housing wealth (relative to income) under two different scenarios. The solid line shows the case of a fixed land endowment, i.e., Z = const. The dashed line assumes that Zt increases over time by a factor of 2. This captures the case where the amount of economically usable land increases over time. The difference for the time path of the housing wealth-to-income ratio is minor. The reason is that there are opposite quantity and price effects. Under scenario “Z = const.” the number of housing projects N changes only slightly, but the house price PtH increases substantially. Under scenario “Zt increases”, the number of housing projects rises substantially, but the house price PtH increases by less compared to scenario “Z = const.”.

A.6 Non-Residential Wealth

Figure A.2.
Figure A.2.

Non-residential wealth (relative to NDP) from 1955 until 2100.

Citation: IMF Working Papers 2017, 080; 10.5089/9781475590302.001.A001

Notes. See Figure 1.

We finally report the implications of the experiment sketched in Section 4.2.1 for non-residential wealth. Figure A.2 displays the evolution of non-residential wealth relative to NDP, ntNDP, for the four countries under consideration. The dotted (red / purple)

lines show the empirical data, whereas the solid (blue) lines display the model-based time paths. The match is less close compared to the housing wealth-to-income ratio (Figure 1). In particular, for Germany the model implies a strong increase, whereas the data show a moderate U-shaped development. Also for France the model implies a somewhat faster increase in non-residential wealth, relative to income, compared to empirical data. This is due to two reasons: First, Germany starts with the lowest value of physical capital (relative to the steady state), which again appears reasonable with respect to war destructions during World War II (N1955NDP105percent). The model economy builds up the capital stock quite rapidly. This implication would be different, if one assumed (convex) adjustment costs for physical capital – a feature that we left out for simplicity, reflecting our focus on housing capital. The same feature would improve the picture for France that starts with only a slightly higher value than Germany (N1955NDP127percent). In 2005 the empirical value is about 190 percent, while the model displays a value of about 255 percent. The match is much better for the U.K. and the U.S. Second, the Long Term Housing & Macro Model rests on one simplifying assumption that is important when it comes to non-residential wealth. It does not distinguish between urban and rural land. That is, there is one single region that hosts both production and housing. This region should in fact be interpreted as predominantly urban, given that about 70 to 80 percent of the population in advanced countries lives in cities. Therefore, the Long Term Housing & Macro Model values the entire non-residential land, which comprises non-residential urban and non-residential rural land in reality, at the urban land price.50 Given the large urbanization rates in advanced countries, this land valuation bias affects housing wealth to a much lesser extent than non-residential wealth. As a result, the non-residential wealth-to-income ratio, ntNDP, is overestimated due to an overvaluation of (non-residential) land, PtZZtY.51

Additional Reference

Sachs, Jeffrey and Peter Boone (1988). Japanese Structural Adjustment and the Balance of Payments, Journal of the Japanese and International Economics 2, 286327.

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Online-Appendix B: Canonical Housing & Macro Model

In Online-Appendix B.1, we characterize analytically the long run equilibrium of the canonical model. In Online-Appendix B.2, we invoke Walras’ law as a consistency check of the analytical derivation of the long run equilibrium. In Online-Appendix B.3, we summarize the dynamical system of the canonical model.

B.1 Long Run Equilibrium

Define lXLX/L, lYLY/L, kXKX/L, kYKY/L, kK/L, hH/L and z¯Z¯/L. Thus, (39) and (40) can be written as kX + kY = k and lX + lY = 1, respectively.

Proposition B.1. In the canonical model, (i) the long run allocation of labor, (lX*, lY*), is independent of both new land per capita, z¯, and productivity parameters BX, BY, BH; (ii) the long run capital-labor ratio (k*) is independent of z¯, BX, BH, and increasing in BY.

Proof of Proposition B.1. Denote by pX the price of structures. Moreover, define kXKX/L, kYKY/L. Profit maximization in the X and Y sector then implies

αBY(lYkY)1α=γpXBX(lXkX)1γ=r+δK,(199)
(1α)BY(kYlY)α=(1γ)pXBX(kXlX)γ=w.(200)

Combining (199) and (200) implies

kX=γ1γwlXr+δK,(201)
kY=α1αwlYr+δK.(202)

Thus, using resource constraints kX + kY = k and lX + lY = 1, we obtain

k=wr+δK(γ1γlX+α1α(1lX)).(203)

Moreover, according to (199), (200) and resource constraints, we find

pX=(1α)BY(kkX1lX)α(1α)BX(kXlX)γ,(204)
w=(1α)BY(kkX1lX)α.(205)

The current-value Hamiltonian of the representative housing firm associated with its profit maximization problem together with the necessary first-order conditions can be expressed as

HHpHpXXRZZH+qH[BHXβZ¯1βδHH],(206)
[HHX=]pX+βqHBH(Z¯X)1β=0,(207)
[HHZ¯=]RZ+(1β)qHBH(XZ¯)β=0,(208)
[HHH