Discussion Paper No. 1032

SECULAR STAGNATIONIN AN ECONOMY WITH LAND

Matthias Schlegl

June 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Secular Stagnation in an Economy with Land

Matthias Schlegl
JSPS International Research Fellow, Osaka University
Ph.D. Candidate, University of Munichemail: matthias.schlegl@econ.lmu.de

This Version: June 2018∗

Abstract
I present a model of secular stagnation with land and infinitely-lived agents with wealthpreferences. Land is the prime example of a non-producible productive asset and rules out
a negative real interest rate in steady state. With standard wealth preferences, higher land
prices stimulate consumption and full employment is always feasible in steady state unlessthe central bank follows a deflationary policy. In contrast, secular stagnation emerges as theunique equilibrium of the monetary economy if wealth preferences are insatiable. In contrast
to conventional wisdom, it is very existence of land itself that prevents full employment frombeing feasible in this case. Increases in the inflation target are no longer effective in restoringfull employment as stagnation is a real phenomenon. These conclusions hold if households
require a risk premium on land. Then, a higher risk premium can restore full employment
when wealth preferences are insatiable.

Keywords: Secular Stagnation, Wealth Preferences, Rational Bubbles, LandJEL Classification: E31, E44∗

I am grateful to Gerhard Illing, Jean-Baptiste Michau and Yoshiyasu Ono for helpful comments and suggestions. Financial support from the “JSPS Postdoctoral Fellowship for North American and European Researchers(Short-Term)” of the Japan Society for the Promotion of Science (JSPS) is gratefully acknowledged as is the
Grants-in-Aid for Scientific Research (S) Humanities and Social Sciences (Social Sciences) ’Behavioral-Economic
Analysis of Long-Run Stagnation (15H05728)’ (2016-).

1

1

Introduction
The term “secular stagnation” describes the persistent failure of an economy to produce atfull capacity as a result of an oversupply of savings. Originally postulated by Hansen (1939)
in the wake of the Great Depression, the hypothesis seemed to have lost its relevance in the
post-war economic system as downturns were limited to short-lived recessions that could be
sufficiently well addressed by monetary policy. However, the unexpected fall of Japan, thethen second-largest economy, into decades of inadequate growth and particularly the sluggish
recovery in most economies during the Great Recession have led to a resurrection of this idea,
most famously by Summers (2013).1
The secular stagnation hypothesis relies on two claims. First, the natural rate of interest,
which is the short-term interest rate consistent with full employment, has permanently declined
into negative territory as a result of high desired savings and low investment demand. Economies
hence overaccumulate capital. Second, the real return on savings cannot fall sufficiently toaccommodate the decline of the natural rate as a consequence of the zero lower bound on thenominal interest rate and the commitment of central banks to maintain price stability. Thismismatch results in an oversupply of savings, which depresses aggregate demand and prevents
markets from clearing at full employment. Therefore, stagnation is a monetary phenomenon.
There is, however, a major objection to this argument. The existence of a non-producible
productive asset prevents overaccumulation and restricts the real interest rate to be positive asshown by Homburg (1991). The argument is simple: An asset with positive rent payments andfinite price has a strictly positive yield, which constitutes a real return. Excess savings can beinvested into this asset, the natural rate of interest remains positive in steady state and secularstagnation does not occur. Land is the prime example of such an asset with high empiricalrelevance. According to Homburg (2014), land values exceed GDP and public debt in many
major economies and land is almost as important as capital from a macroeconomic perspective.
However most models of secular stagnation exclude the possibility of investment in land.2
I propose a theory of secular stagnation in an economy with land based on strong wealthpreferences of households. Specifically, I extend the standard neoclassical growth model with
infinitely-lived agents along the following dimensions.
1
In fact, Summers (2014) argues that weak aggregate demand has been present in the United States for several
decades but was masked by the stock market boom of the 1990s and the credit boom of the early 2000s, both ofwhich initiated unsustainable consumption spending of households. Illing et al. (2017) formally show that credit
booms can temporarily cure aggregate demand deficiency but come at the cost of more severe stagnation in thelong-run due to the resulting debt overhang.
2
The existence of equity also prevents overaccumulation of capital if the dividend on stocks is strictly positive.
This is the case if labor is not an input factor in the production process as in Zhou (2013) or in case of imperfectfinancial markets that prevent negative dividend payments and stock dilution.

2

First, I introduce wealth preferences of households. The idea of wealth accumulation forthe sake of itself as an aspiration of human behavior has a long tradition in economic thought
and is embraced among others in the works of Adam Smith, Karl Marx, Max Weber and John
Maynard Keynes as described in great detail by Zou (1994), who refers to the concept as “spiritof capitalism”. In addition, it can be related to behavioral factors like status concerns. Wealthpreferences have first been formally introduced into the neoclassical growth model by Kurz (1968)
and Bose (1971) and their implications are analyzed in several contributions.3 Wealth preferences
create a strong desire for savings which lowers the natural interest rate below the time preference
rate and potentially into negative territory. Preferences for wealth therefore present an alternative
to the overlapping generations model that can also generate a negative real rate, as is well-known
since Samuelson (1958). Recent contributions by Michaillat and Saez (2014) and Michau (2017)
study similar models of secular stagnation based on wealth preferences. Ono (1994, 2015) offersa theory of stagnation based on the additional assumption that wealth preferences are insatiable.
However, these papers do not account for the availability of a non-producible productive asset.
I extend these models by introducing land as a means of saving for households in addition tobonds, physical capital and money. Land is a non-producible asset in fixed supply and pays apositive rent.4 The land value affects the natural rate of interest via the wealth preference andits rental yield represent a strictly positive real return on savings, which rules out a negative realinterest rate in steady state.
I also introduce money into the model via a money-in-the-utility framework, in which thecentral bank directly controls the nominal money supply. In addition, I impose a limit on
nominal wage declines in periods of aggregate demand shortage to prevent a deflationary spiral.
This friction is borrowed from Schmitt-Grohé and Uribe (2016, 2017) and results in a non-linear
inflation process.5
What determines the value of land? Under what conditions does secular stagnation occur
in steady state? And what is the role of monetary policy? I analyze these questions in both aneoclassical model with land and its extended monetary version. The analysis focuses primarilyon the steady state properties of these models. The conclusions differ markedly depending on
the nature of the preference for wealth.
3

Bakshi and Chen (1996) and Smith (2001) study the implications of wealth preferences for asset pricing.
Kumhof et al. (2015) explain the rise in inequality in recent decades in a model with wealth preferences of richhouseholds. Michau et al. (2017) analyze their effects on the natural interest rate and the occurrence of rational
bubbles in a model with infinitely-lived assets. Similarly, Kamihigashi (2008) and Zhou (2013, 2015) study rational
equity bubbles in a model with wealth preferences.
4
The constant supply property of land is one of the main reasons why it is typically omitted in standard growthmodels with technological progress. An exception is Nichols (1970), who incorporates land into a balanced growthmodel with land-augmenting technological progress.
5
Similar nominal rigidities are used in other models of secular stagnation like Michau (2017), Eggertsson,
Mehrotra, and Robbins (2017), Caballero and Farhi (2018) or Ono and Ishida (2014).

3

With standard wealth preferences, the case for secular stagnation is substantially weakened by
the introduction of land as investments in land absorb the oversupply of savings. Full employment
prevails in the neoclassical model as increases in the land value stimulate household consumption
and increase the natural interest rate into positive territory. In the monetary version of thismodel, secular stagnation could in principle occur in steady state. However, this requires quite
restrictive assumptions. Specifically, the economy has to be deflationary at full capacity to ruleout full employment. In addition, it requires that the real money stock does not affect the desire
for savings via the wealth preference. Then, the real interest rate is determined solely in the real
economy. In contrast, stagnation cannot occur if the real money stock is considered net wealthby households as increases in the money supply stimulate consumption until full employment isrestored.6 In addition, the land price always equals its fundamental value and rational land pricebubbles cannot occur.
These conclusions change profoundly if the desire for wealth accumulation is insatiable.
Specifically, I follow the literature initiated by Ono (1994) and assume a strictly positive lower
bound on the marginal utility from wealth. Then, full employment cannot be maintained insteady state if the desire for savings is sufficiently strong and secular stagnation occurs. This
steady state is not well-defined in the neoclassical economy, but uniquely determined in the monetary economy by the binding nominal wage rigidity. Stagnation persists even though households
can invest in land at a positive real rate. In fact, it is the very existence of land that creates
stagnation as it prevents the actual return on savings from falling sufficiently to clear markets
at full employment. As households become sufficiently wealthy, their consumption ceases to beaffected by further increases in wealth, which is why the natural interest rate can permanently
remain below the return on savings. In addition, land price bubbles can occur.
Consequently, secular stagnation is a real phenomenon and increases in the inflation target
are no longer effective in restoring full employment. The steady state with aggregate demand
shortage is unique and multiple equilibria are ruled out. This is in contrast to monetary models
of secular stagnation like Michau (2017) or Caballero and Farhi (2018), in which both stagnationand full employment steady states coexist for a sufficiently high money growth rate.
These findings continue to hold when households require a risk premium on land investments.
Increases in the inflation target are only effective if the return on money, rather than the returnon land, constrains the return on savings. With standard wealth preferences, an increase in therisk premium can prevent the economy from operating at full employment. In contrast, a higher
risk premium can help to restore full employment when wealth preferences are insatiable.
6This mechanism would also restore full employment in Michau (2017). However, money is not part of net
household wealth in the steady state of his model and hence does not affect the natural rate.

4

Related Literature: This paper contributes to the growing literature on secular stagnationand offers a theory of stagnation based on the existence of a real asset, namely land. In contrast,
stagnation is considered a monetary phenomenon throughout the literature starting with Keynes(1936), who argues in Chapter 17 of the General Theory that permanent demand shortage canexist as a steady state phenomenon in a monetary economy.7
In the liquidity trap literature, stagnation occurs temporarily following severe demand shocks
that result in a negative natural rate of interest (see among others Eggertsson and Krugman,2012; Eggertsson and Woodford, 2003; Krugman, 1998). As the zero lower bound and wellanchored inflation expectations prevent the real return on money from falling sufficiently, anoversupply of savings depresses demand. However, full employment is eventually restored evenin the absence of policy measures.
In secular stagnation models, no natural recovery occurs because the natural interest rate ispermanently negative as a result of structural factors.8 In standard infinitely-lived agent models
the natural interest rate in steady state is determined by the positive time preference rate of theleast impatient agent as shown by Becker (1980). Models of secular stagnation therefore rely onan overlapping generations framework or wealth preferences, both of which allow for a negative
natural rate in steady state.
In Eggertsson, Mehrotra, and Robbins (2017), the natural rate is negative as a consequenceof deleveraging shocks, demographic developments, inequality or a fall in the price of capital
goods. Kocherlakota (2013) highlights the role of asset price declines that can worsen stagnation
if labor markets are incomplete, while Caballero and Farhi (2018) show that a shortage of safe
assets depresses the natural interest rate in a perpetual youth model.9 Alternatively, wealthpreferences of households can result in a negative natural rate. Then, secular stagnation canoccur in the steady state of a monetary economy as shown by Michaillat and Saez (2014) andMichau (2017). A related strand of stagnation models is based on the stronger assumption of
insatiable wealth or liquidity preferences. Initiated by Ono (1994) and substantially extended
by Ono and Ishida (2014), these contributions assume that there exists a strictly positive lower
bound on the marginal utility from wealth. A limit on the feasible inflation rate, for example
via a restrictive inflation target of the central bank, rules out full employment.
7

A modern formalization of Chapter 17 is provided by Ono (2001) in a continuous dynamic optimization modelin which stagnation occurs as a consequence of strong liquidity preferences of households.
8
In Ono (2001) and Illing et al. (2017), it is not the zero lower bound but strong liquidity preferences thatcreate a lower bound on the return on money holdings. Then stagnation can occur even at a positive natural rateof interest, but it remains a monetary phenomenon.
9
The open economy versions of these secular stagnation models are analyzed in Eggertsson, Mehrotra, andSummers (2016) and Caballero, Farhi, and Gourinchas (2016). International considerations include the notionof a “global savings glut”, postulated by Bernanke (2005). For an overview of these and other factors, see the
VoxEU.org eBook edited by Baldwin and Teulings (2014).

5

Yet, what most of these contributions omit is the importance of non-producible and productive real assets, like land. There are two notable exceptions though.
In Kocherlakota (2013), the existence of land also implies a strictly positive real rate in steadystate, which can result in stagnation if the wage inflation rate does not adjust sufficiently in caseof imperfect labor markets. Then, exogenous land price declines further reduce the labor input.
However, once capital is introduced into the model, stagnation requires the stronger assumptionthat the real rate cannot fall below the time preferences rate of the household. In contrast, I do
not require any exogenous assumptions on the real rate. All restrictions are derived from the
No-Arbitrage condition in asset markets.
The perpetual youth model of Caballero and Farhi (2018) features a non-producible Lucastree. However, the return on this asset includes a risk-premium in the stagnation equilibrium. It
is then again the existence of money that accounts for stagnation as the inflation rate restricts
the real return on the safe asset. In contrast, I show that secular stagnation can result from theexistence of both money and land when there is an exogenous risk premium on land.10
As stagnation is typically a monetary phenomenon, a sufficiently large and credible increasein the inflation target or the money growth rate is a common remedy (see for example Caballero
and Farhi, 2018; Eggertsson, Mehrotra, and Robbins, 2017; Michaillat and Saez, 2014; Michau,2017). This lowers the return on money and can restore full employment. Then, both full
employment and stagnation steady states can occur. In an economy with land, increases in the
inflation target cease to affect the economy under stagnation and cannot restore full employment
as stagnation is a real phenomenon.
The negative real rate in secular stagnation models implies that economies accumulate toomuch capital, a state known as dynamic inefficiency.11 Under this condition, a rise in government
debt can be a Pareto improvement, as shown by Diamond (1965). In fact, Michau (2017) argues
that public debt cannot be unsustainable under secular stagnation. Moreover, rational asset pricebubbles can occur as analyzed by Tirole (1985). In this paper, dynamic efficiency is restored
once land is available. However, land price bubbles can still occur and are not ruled out by
the transversality condition when the marginal utility of wealth is bounded from zero either inabsolute terms or relative to consumption as also shown by Ono (1994), Kamihigashi (2008) andZhou (2013, 2015)10

In this paper, agents have perfect foresight, which excludes stagnation equilibria based on self-fulfilling expectations as in Benhabib et al. (2001). Secular stagnation results from fundamental factors.
11
Empirical tests of dynamic inefficiency are imperfect as temporary shortfalls of the real rate below the growthrate are possible in a dynamically efficient economy as argued by Homburg (2014). In a seminal paper, Abel et al.
(1989) compare capital income with investment expenses as a criterion for efficiency and conclude that all major
economies are dynamically efficient. This result is challenged by Geerolf (2017) after adjusting the efficiency
criterion for rent income from land.

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2

A Neoclassical Economy with Land
Throughout this paper, I assume perfect foresight and abstract from uncertainty and tech-

nological progress. Time is continuous and denoted by t. The model consists of homogeneous
households and competitive firms, which produce the single output good using labor and capital. There is no government. I make two extensions to the standard neoclassical model. First,
households have preferences for wealth. Second, there is the possibility of investment in land.

2.1

Asset Markets and the Real Return on Savings

There are three assets in the neoclassical model: Bonds bt , physical capital Kt and land Ht .
All of them serve as a store of value for households to transfer income over time.
Bonds are financial claims between households that yield a real return of rt . Financial markets
are perfect and there are no asymmetric information or contract enforcement problems. Bond
market equilibrium requires bt = 0 as bonds are in zero net supply.
Physical capital is an input in the production process. It is owned by households, who rent
it at price Rt to firms. Capital depreciates at rate δ > 0. One unit of output can be transformed
into one unit of consumption or one unit of investment and there are no adjustment costs.
Therefore, the relative price of capital is one.12 The return on capital rtK is then given by
rtK = Rt − δ .

(1)Land is a non-producible productive asset. Unlike physical capital, land cannot be produced
and does not depreciate. It is in fixed supply, normalized to Ht = 1, and trades at real price qt .
Each unit of land pays a real rent z > 0 to its owner, which is exogenous. Households can freely
dispose of land, which implies that its price cannot become negative, i.e. qt ≥ 0. The return onland rtL consists of a rental yield and potential capital gains or losses with
rtL =

q̇t
z
+
.
qt qt

(2)

Consider the decomposition of the land price qt into a fundamental component qtF and a
bubble component qtB , i.e.
qt = qtF + qtB ,12(3)

If investment is irreversible, we have K̇t + δKt ≥ 0. Then, the price of capital equals one whenever investment
is positive, but it can drop below one once this constraint is binding. Changes in the price of capital then affectthe return on capital, which is analyzed in Michau (2017). I abstract from this occasionally binding constraint
as I focus on steady state analysis, in which investment is strictly positive due to depreciation. The same holds
during the adjustment dynamics if the initial capital stock is below the steady state level.

7

where the fundamental price is associated with rent payments and the bubble is a speculative
price component, which is not backed by claims on income. The fundamental return is given by
rtL in (2) whereas the return on the bubble equals the capital gain or loss given by
rtB =

q̇tB
.
qtB

(4)

Equilibrium in the bond market, the market for physical capital and the market for land
requires the equalization of returns across these assets.13 If bubbles exist, the bubble component
has to grow with this rate as well, which is a standard requirement in the literature (see Tirole,1985). This is formalized in the No-Arbitrage condition
rt = rtK = rtL (= rtB ) ,(5)

where rtK , rtL and rtB are defined in (1), (2) and (4) respectively. Condition (5) implies thathouseholds are indifferent between bonds, physical capital and investment in land as a means ofsavings in equilibrium. All assets yield the same real return, which I will denote by rtS . This
return determines the actual incentives of households to save.

2.2

Competitive Firms

Firms rent physical capital Kt from households at rental price Rt and employ labor Lt atreal wage wt . They produce output Yt with the neoclassical production function
Yt = F (Kt , Lt ) ,(6)which exhibits constant returns to scale and fulfills the standard assumptions of strictly positiveand declining marginal products of both factors. Let kt = Kt /Lt and yt = Yt /Lt denote thecapital stock per worker and output per worker respectively. Using (6), I rewrite output per
worker as yt = F (kt , 1) = f (kt ) with f 0 (kt ) > 0 and f 00 (kt ) < 0.
Firms take wt and Rt as given. In competitive factor markets, capital and labor are paid
their marginal products, which implies that the rental price and the real wage are given by
Rt = FK (Kt , Lt ) = f 0 (kt )(7)wt = FL (Kt , Lt ) = f (kt ) − kt f 0 (kt )

(8)

13

The models of Caballero, Farhi, and Gourinchas (2016) and Caballero and Farhi (2018) also assume a nonproducible productive asset. However, the return on this asset can include a premium in the presence of aggregate
risk depending on the marginal investor. I discuss the case of a risk premium in section 5.

8

Since the rental price always equals the marginal product of capital, the return on physicalcapital rtK is given by its net marginal product. Using (7), rtK in (1) is rewritten as
rtK = f 0 (kt ) − δ .

(9)

By the No-Arbitrage condition (5), this return equals the real rate on bonds and the returnon land rtL . Hence, (9) constitutes the actual real return on savings rtS for households.14

2.3

Households and the Natural Rate of Interest

There is a mass one of identical and infinitely-lived households and no population growth.
Households own physical capital, which they rent to firms. Each household is endowed with one
unit of labor Lt , which is supplied inelastically at wage wt . Therefore, Lt = 1 represents full
employment. Households can save via bonds bt , investments in physical capital Kt or investment
in land at price qt . Hence, total real wealth at is decomposed as
at = bt + Kt + qt Ht .

(10)

Households receive wage income and interest or rental income on their investments. This
income is used to finance consumption spending ct . Real wealth evolves asȧt = rt at + rtK − rt Kt + rtL − rt qt Ht − ct + wt Lt .

(11)

By the No-Arbitrage condition (5), all assets have to yield the same return in equilibrium.
Then, (11) simplifies toȧt = rt at − ct + wt Lt .

(12)

The representative household has preferences over consumption and wealth. His lifetime
utility is given by
Z ∞
U0 =

[u(ct ) + v(at )] e−ρt dt ,

(13)

0

where ρ > 0 is the time preference rate. Utility u(c) satisfies the standard assumptions u0 (c) > 0,
u00 (c) < 0, lim u0 (c) = ∞ and lim u0 (c) = 0. In contrast, the desire for wealth accumulation is
c→0

c→∞

insatiable. Specifically, I follow the literature initiated by Ono (1994) and assumev 0 (a) > 0 , v 00 (a) ≤ 0 , lim v 0 (a) = v0 ,a→0

lim v 0 (a) = β > 0 .

a→∞

14Note that the same result is obtained when firms own the capital stock directly. In this case, households areshareholders and receive dividend income equal to firm profits. As firms have to replace the depreciated capitalstock, the net marginal product still determines the return on household savings rtS .

9There exists a lower bound β on the marginal utility from wealth, which stays positive evenas wealth becomes infinitely high. This assumption is borrowed from behavioral economics andrelated to status preferences as discussed in Murota and Ono (2011).15
Maximizing (13) subject to (12) yields the Euler Equation and the transversality condition
as
ηc

ċt
v 0 (at )u00 (ct )ct= rt − ρ + 0, where ηc ≡ − 0,ct
u (ct )u (ct )

(14)

lim at u0 (ct )e−ρt = 0 .

(15)

t→∞

Compared to the standard Euler Equation, wealth preferences create an incentive for savingsin addition to the impatience motive. I define the natural rate of interest rtN as the real interest
rate that clears the goods market at full employment. It is given by (14) with Lt = 1. As saving
becomes a virtue by itself, goods market clearing requires a real rate below the time preference
rate. The natural rate can even become negative if the preference for wealth is sufficiently strong.
Finally, the transversality condition (15) implies that household wealth cannot grow at a rate
greater or equal to ρ in a stationary steady state.

2.4

Market Clearing Conditions

The real wage perfectly adjusts to clear the labor market with Lt = 1 at full employment.
The No-Arbitrage condition (5) implies bond market clearing with bt = 0, land market clearing
with Ht = 1 and equilibrium in the market for physical capital. Then real wealth is given byat = Kt + qt = kt Lt + qt .

(16)

Aggregate supply consists of output Yt and the rent z from land. Aggregate demand equals
consumption demand ct and gross investment in capital K̇t +δKt . Goods market clearing requires
Yt + z = ct + K̇t + δKt ,(17)

which can be rearranged and expressed in per worker terms as
K̇t = k̇t Lt + kt L̇t = [f (kt ) − δkt ] Lt + z − ct .

(18)

Under full employment, it holds that Lt = 1 and L̇t = 0. Then kt = Kt and K̇t = k̇t . In thenext section, I analyze the conditions, under which the full employment steady state exists.
15

Ono, Ogawa, et al. (2004) offer empirical support for the insatiability of preferences for monetary wealthbased on quarterly data in Japan using parametric and non-parametric methods.

103

Analysis of the Neoclassical Economy
The neoclassical model economy is characterized by the Euler Equation (14), the wealthcomposition (16), the return on physical capital (9), the return on land investment (2), thegoods market clearing condition (18) and the No-Arbitrage condition (5) together with thetransversality condition (15) for any given initial value of capital K0 > 0. In case of a bubbly
equilibrium, equations (3) and (4) also apply for a given initial bubble value q0B > 0. Under full
employment, it holds that Lt = 1 and consequently Kt = kt . Each stationary steady state ischaracterized by ċt = 0 and k̇t = 0.
The feasibility of full employment in steady state is determined by the actual real return on
savings rS and the natural rate of interest rN . The actual return on savings follows from the
No-Arbitrage condition (5), which requires the equalization of the real returns of all available
assets. Using the definitions of the returns on land in (2) and capital in (9), the return on savings
is given by
rS = f 0 (k) − δ =

z
q˙t
+
,
qt qt

(19)

where q̇t ≥ 0 in steady state depends on the presence of land price bubbles, which in turn depends
on the specification of the utility from wealth. The natural rate of interest rN is derived from
(14) with ċt = 0 as
rN = ρ −v 0 (a),u0 (f (k) − δk + z)

(20)

where I use the goods market clearing condition (18) with k̇t = 0 and L = 1 to substitute for
consumption under full employment as a function of the capital stock, i.e.
c = f (k) − δk + z .

(21)

Note that consumption is maximized when the capital stock is at the Golden Rule level k̃, which
is defined by f 0 (k̃) = δ.
The real rate rS in (19) determines the actual return on assets and hence the saving incentives
for households, while the natural rate rN in (20) describes the real return on savings that isnecessary to achieve full employment. Hence, full employment requires rS = rN . An persistentoversupply of savings occurs for rS > rN in steady state as the incentives to save are too high.
This depresses aggregate demand and prevents full employment from materializing.
In the following, I analyze the feasibility of full employment and the role of land for different
assumptions on the preference for wealth. I start with the neoclassical model without land as abenchmark. Most of the derivations and proofs are relegated to the appendix.

11

3.1

The Benchmark Model without Land

Consider first the neoclassical model augmented for wealth preferences but without land.
In equilibrium, household wealth is given by the capital stock as bonds are in zero net supply
and a = k in (20).16 The return on savings rS in (19) equals the net marginal product of
capital without any further restrictions. Both rN and rS are continuous functions of the capitalstock and illustrated in Figure 1 for the case of standard wealth preference (left panel) and aconstant marginal utility of wealth v 0 (a) = β (right panel). Full employment requires rS = rNor equivalently
f 0 (k) − δ = ρ −v 0 (k).
u0 (f (k) − δk + z)

(22)

Let k̄ and r̄ denote the steady state values of the capital stock and the real interest rate. The
following proposition shows that the full employment steady state is always realized in thiseconomy irrespective of the nature of the preference for wealth.
Proposition 1 The steady state under full employment, characterized by (22), always exists
irrespective of the lower bound on the marginal utility of wealth. The steady state is unique under
mild restrictions on preferences or model parameters. The dynamic system exhibits saddle-point
stability around this steady state.
The proof is in Appendix A. I assume that the steady state is unique, which holds under mild
restrictions as discussed in Kurz (1968) and Bose (1971). Graphically, uniqueness requires thatthe slope of rN exceeds the slope of rS at each intersection.
Full employment is sustainable even with insatiable wealth preferences because physical capital is both a store of value and a producible asset whose replacement affects aggregate demand.
Suppose there was an excess supply of savings, i.e. rN < rS . As no other assets are available toabsorb the excess savings, these have to be invested in bonds and physical capital. As a consequence, the capital stock increases and the net marginal product of capital declines as does thereturn on savings rS . The depreciation rate constitutes a lower bound this return, i.e. rS > −δ.
In addition, the higher capital stock affects the natural rate via two channels. As households
become wealthier, they prefer lower savings and higher consumption. The natural rate increases
via this wealth channel as v 0 (a) declines. Higher values of the capital stock also affect the supply
of goods. Increase in k increase the net supply if k is below the Golden Rule level k̃, which lowers
the natural rate. The opposite holds if k is above the Golden Rule, since more and more output
is used for the replacement of the depreciating capital stock.
16
Note that I assume that households receive the income z despite the exclusion of land. In this case, z should
be interpreted as a lump-sum transfer. Hence, the expression for consumption in (21) continues to hold and there
are no further changes to the natural rate in (20).

12Figure 1: The Steady State of the Benchmark Modelrrr S (k) = f ′ (k) − δ′r N (k) = ρ − vu′(k)(c)

r S (k) = f ′ (k) − δρr N (k) = ρ − u′β(c)ρr S (k̃) = 0

r S (k̃) = 0
k̄

k̄
k

k

k̂

k̂

r̄

r̄

-δ

-δ

Note: This figure shows the natural interest rate rN and the real return on savings rS as a function of thecapital stock k. The left graph shows the case of a declining marginal utility of wealth, i.e. v 00 (a) < 0. The
right graph shows the case of a constant marginal utility, i.e. v 0 (a) = β.

For k < k̃, the net effect depends on the relative strength of these channels and the naturalrate might increase (left panel) or decrease (right panel) for higher k. For k > k̃, the naturalrate increases because of both wealth and income effects. Eventually, output is fully used up forinvestment and the natural rate approaches the time preference rate ρ (at k̂ in Figure 1). With aconstant marginal utility of wealth, the wealth channel ceases to operate and increases in wealthno longer stimulate consumption. In contrast, the income channel continues to operate and thenatural rate increases with the capital stock once it exceeds the Golden Rule level.
Taken together, any excess supply of savings is invested into physical capital and bonds. The
associated increase in the capital stock reduces the return on savings and eventually increases
the natural interest rate. Market clearing (22) is always feasible at full employment, even for aconstant marginal utility of wealth. The steady state real rate satisfies r̄ ∈ (−δ, ρ), which can be
negative for sufficiently strong wealth preferences. Then overaccumulation of capital above theGolden Rule level occurs.
Secular stagnation results from the existence of a non-producible store of value that affects
the incentives for savings without, however, affecting full employment consumption.17 Money isan example of such an asset in a monetary economy, while land is an example of such an asset
in a real economy.
17

This is also the reason why the possibility of storage or the existence of non-depreciable capital does notallow for deviations from full employment. In this case, the lack of demand would be substituted by investment
demand and K̇t > 0 in steady state, which is sustainable if wealth preferences are insatiable.

13

3.2

The Neoclassical Model with Land

Consider first the immediate effect of introducing land, which pays a fixed rent z to its owner,
into the neoclassical model. The existence of land prevents the real rate of interest from becoming
negative in steady state as summarized in the following proposition.
Proposition 2 The steady state real interest rate in the presence of land cannot become negative,
i.e. r > 0, as a consequence of the No-Arbitrage condition (5) and the possibility of free disposal.
The intuition is as follows (see Appendix B for the proof). The land price cannot become negative
due to free disposal, which also excludes the possibility of a declining price in steady state. As
land yields a positive rent to its owner, the rental yield on land remains strictly positive andconstitutes a real return which households can earn on their savings. The No-Arbitrage condition
then implies that the returns on other assets cannot become negative as well. This result does
not depend on the exogenous income component z, but holds as long as there is positive income
or utility associated with ownership of the asset in steady state.18
It follows that the actual return on savings rS in (19) and hence the net marginal product of
capital have to be strictly positive in an economy with land. Thus, the steady state capital stock
will always be strictly below the Golden Rule level k̃ and I obtain the same result as Homburg(1991): The existence of a non-producible productive asset rules out dynamic inefficiency as wellas the possibility of Golden Rule growth. Any excess savings can be invested in land at a positive
rate of return. This result does not depend on the specification of the wealth preference.
In a purely fundamental steady state with q̇t = 0, households earn the rental yield on land
and the land value q adjusts to equalize this yield to the net marginal product of capital. In a
bubbly steady state with q̇t > 0, the rental yield asymptotically approaches zero and the returnon land is purely driven by the capital gain.
In addition, private wealth no longer coincides with the capital stock as households can invest
in land. The land value then affects the natural interest rate rtN via the wealth preference asv(a) = v(k + q). As households are more wealthy, their desire for savings is reduced comparedto the benchmark model and the natural rate increases for a given value of the capital stock.
Both the feasibility of full employment and the possibility of rational bubbles in the land
price depend on the nature of the wealth preference.
18

Alternatively, land can be modeled as a factor input in the production function (6) similar to capital, as in
Homburg (1991, 2014). Then, z is the rental price of land and equals the marginal product of land, which isstrictly positive. Similarly, the real interest rate cannot become negative in the presence of housing that yields a
positive user cost in terms of utility as in Illing et al. (2017) or in the presence of positive dividends from stock
ownership as in Zhou (2013).

14

3.3

The Steady State with Standard Wealth Preferences

Consider first the case of standard wealth preferences. The land price is constant in steadystate and bubbles do not occur as summarized in the following proposition.
Proposition 3 With β = 0, there cannot be an ever-growing or ever-decreasing land price.
Therefore, we must have q̇t = 0 in steady state. Moreover, the land price is purely fundamental
and bubbles cannot exist, i.e. qt = qtF and qtB = 0.
An ever-declining land price is not possible as the land price would eventually become negative
as shown in the proof of Proposition 2 in the appendix. In contrast, an ever-growing land priceimplies that the land value becomes infinitely high and so does household wealth. With β = 0,
the marginal utility of wealth decreases to v 0 (a) = 0 and the natural rate converges to the time
preference rate ρ as can be seen in (20). The return on land eventually consists purely of thecapital gain in (2), which also approaches ρ. As a consequence, the growth rate of both theland price and household wealth converges to ρ as the capital stock is constant. However, thisviolates the transversality condition (15). Therefore, we must have q̇t = 0 in steady state. It
directly follows that there cannot be a bubble in equilibrium. Any bubble has to grow with thereal interest rate as is clear from (4). This rate is strictly positive in steady state by Proposition(2). This implies q̇tB > 0, which is not possible.19
With q̇t = 0, the return on land is given by the rental yield in (2). The land price adjusts
such that this return equals the net marginal product of capital. Full employment requiresf 0 (k) − δ = ρ −zv 0 k + f 0 (k)−δ
u0 (f (k) − δk + z)

>0.

(23)

This is the equivalent to condition (22) in the benchmark model. The return on savings is
restricted to be positive due to the availability of land as a store of value.
∗∗
Both, rN and rS are continuous functions of k and illustrated in Figure 2. Let kN
C and rN C

denote the steady state values of the capital stock and the real rate. The following propositionshows that full employment is always feasible in this economy.
Proposition 4 The steady state under full employment, characterized by (23), always exists if
β = 0. This steady state is unique under mild restrictions on preferences or model parameters.
It is characterized by a lower capital stock and a higher real rate than in the benchmark model
without land. In addition, the model exhibits saddle-point stability around this steady state.
19This result depends on the assumption of no depreciation in the value of land. I do not consider the possibilityof depreciation in the rent z as the focus is on the nature of the wealth preferences. For the general case withdepreciation, see Michau et al. (2017).

15

Figure 2: The Steady State of the Neoclassical Model with Land for β = 0rrr S (k) = f ′ (k) − δ > 0

r S (k) = f ′ (k) − δ > 0ρρ′r N (k) = ρ − v u(k+q)
′ (c)

′

r N (k) = ρ − v u(k+q)
′ (c)

∗
rN
C
′

r N (k) = ρ − vu′(k)
(c)

∗
rN
C

k

k

∗
kN
C

∗
kN
C

k < k̃

k < k̃

-δ′r N (k) = ρ − vu′(k)(c)-δ

Note: This figure shows the natural rate rN for the model with land (solid line) and the benchmark model(dotted line) as well as the return on savings rS as a function of the capital stock k. The left graph shows
the case of a positive real interest rate r̄ > 0 in the benchmark model, whereas the right graph shows thecase of a negative real interest rate r̄ < 0 in the benchmark model. The return on savings is strictly positive
as explained in Proposition 2, which is illustrated by the vertical line at k̃.

The proof is in Appendix C. As before, I assume uniqueness of the steady state.
For a given value of k, the return on savings rS is unaffected by the introduction of land, but
restricted to be positive, which is represented by the vertical line at k̃. In contrast, household
wealth is higher in the presence of land, which reduces the desire for savings and increases thenatural rate rN . Hence, the rN curve shifts upwards in Figure 2, which results in a lower capital∗∗stock and a higher real rate. Hence, kN
C < k̄ and rN C > r̄. This holds for both a positive (leftpanel) and a negative (right panel) real rate r̄ in the benchmark model.
The existence of land establishes a zero lower bound on the real return on savings as households can always invest in land at a positive return. Excess savings that could only be invested
in physical capital in the benchmark model are channeled into land investment. Therefore,
overaccumulation of capital cannot occur despite the preference for wealth.
In addition, the value of land also affects the natural rate. As the capital stock increases,
the fundamental value of land increases and so does household wealth. As households become
wealthier, their desire for savings declines and the natural rate increases. In fact, the land pricebecomes infinitely high when k approaches k̃ and the natural rate rN approaches ρ, while the∗return on savings converges to zero. It follows that there always exists a capital stock kN
C < k̃

for which the return on savings matches the natural rate at full employment. This real rate is∗strictly positive at rN
C ∈ (0, ρ).

16

3.4

The Steady State with Insatiable Wealth Preferences

With β > 0, a negative real interest rate is still ruled out by Proposition 2. However,
the insatiability of wealth preferences gives rise to a continuum of land price paths that arecompatible with equilibrium, including paths with exploding land price bubbles.
Proposition 5 With insatiable wealth preferences, there is a continuum of equilibrium paths forthe land price. These include a constant steady state land price q̇t = 0 as well as steady stateswith exploding land prices q̇t > 0.
Similar to before, we cannot have a steady state with a declining land price as qt ≥ 0. In caseof q̇t = 0, the land price is simply given by its fundamental value, namely the capitalized valueof rents z. This is the only solution if β = 0. An ever-increasing land price in steady state
implies that household wealth expands indefinitely. However, the marginal utility of wealth now
converges to the lower bound β > 0. As a consequence, the real interest rate remains strictly
below the time preference rate as is clear from (20) with v 0 (a) = β. Then the return on landis purely given by the capital gains component in (19) as the rental yield becomes infinitely
small. Therefore, the growth rates of the land price and household wealth are below the time
preference rate ρ. The transversality condition (15) is not violated despite the expansion of theland value. Intuitively, this is because the strong preference for wealth prevents households fromexpanding their desired consumption spending in response to an increase in wealth. Hence, theresource constraint of the economy is not violated as wealth is accumulated without expanding
consumption. Households prefer to keep their land investments rather than selling it to realize
parts of the increase in value.20
This implies that rational bubbles are sustainable, which is in stark contrast to the case of
β = 0. For any initial bubble value q0B > 0, the bubble component of the land price will growwith the real return on savings as can be seen from (4). As this rate is strictly positive by
Proposition 2, the bubble component will grow in steady state, thereby expanding household
wealth. Eventually, the lower bound on the wealth preference is obtained and we have v 0 (a) = β.
In the steady state of the model with land and insatiable wealth preferences, full employment
cannot always be realized. In that case, there exist a range of values L < 1 that are compatible
with market clearing but the resulting secular stagnation steady state is not well-defined. This
result is summarized in the following proposition.
20

Similar conclusions are obtained by Ono (1994), Kamihigashi (2008) and Zhou (2013, 2015) about the dynamics of equity prices based on non-standard behavior of wealth preferences. The first two papers also assume
a lower bound on the marginal utility of wealth, while the last one restricts the behavior of the marginal rate of
substitution between wealth and consumption. In all cases, the real rate does not converge to the time preference
rate as wealth expands and the transversality condition is not violated.

17

Proposition 6 In the model with land and β > 0, the economy operates at full employment insteady state if
β < ρu0 c(k̃) ,(24)

where k̃ is the Golden Rule capital stock. The steady state then has the same stability properties
as in the case of standard wealth preferences. In contrast, the economy fails to operate at full
employment and there exist a range of values L < 1 compatible with market clearing if condition(24) is violated and if the rent from land ownership z is sufficiently low, specifically if
ρu0 (z) > β ≥ ρu0 c(k̃) .

(25)This result holds for both a constant land price and in the presence of land price bubbles. However,the secular stagnation steady state with L < 1 is not well-defined.
I give an intuitive outline here (see Appendix D for the proof). Consider first the case of land
price bubbles. In a bubbly steady state, we have q̇tB > 0 in (4) as rS > 0 by Proposition 2. Boththe land price and real wealth expand indefinitely and v 0 (a) = β in the asymptotic steady state.
Hence, the natural rate in (20) no longer depends on household wealth and attains a minimumat the Golden Rule capital stock k̃. In contrast, the return on savings is unaffected compared tothe previous case. Both returns are illustrated in Figure 3.21 Full employment requiresf 0 (k) − δ = ρ −βu0 (f (k) − δk + z)

>0.

(26)

This is the equivalent to conditions (22) and (23) before. Full employment is not feasible if thepreference for wealth is sufficiently strong. With β = 0, increases in the land price reduce thedesire of households for savings and increase the natural rate into positive territory. In contrast,with insatiable wealth preferences, increases in the land price at some point cease to stimulate
consumption. The desire for saving remains sufficiently strong. Investment demand does not
expand as excess savings are invested in land instead of capital. As a consequence, an oversupplyof saving occurs and the goods market cannot clear at full employment.
The same argument holds for the fundamental steady state with a constant land price. In
this case, households wealth is finite and we have v 00 (a) < 0. Full employment requiresf 0 (k) − δ = ρ −zv 0 k + f 0 (k)−δ
u0 (f (k) − δk + z)

21

>0,

(27)

Note that the bubbly steady state with insatiable wealth preferences is identical to the benchmark modelwith a constant marginal utility of wealth that is depicted in the right panel of Figure 1, except for the additional
restriction of rS > 0. In the model without land, we have instead r̄ ∈ (−δ, ρ).

18

Figure 3: The Steady State of the Neoclassical Model with Land for β > 0rrr S (k) = f ′ (k) − δ > 0

r S (k) = f ′ (k) − δ > 0ρρr N (k) = ρ − u′β(c)r N (k) = ρ − u′β(c)

∗
rN
C

∗
(kN
C)

k
∗
kN
C

k < k̃

k

∗
(rN
C)

-δ

-δ

k < k̃

Note: This figure shows the natural interest rate rN for the model with land and v 0 (a) = β in the bubbly
steady state as well as the return on savings rS as a function of the capital stock k. In the left graph,condition (24) holds and rS > 0 is not violated under full employment since β is sufficiently low. In theright graph, condition (24) is violated. Market clearing at full employment would require a negative interest
rate, which is precluded by the possibility of land investment.

which is identical to (23) and the dynamics and stability properties of the model economy are thesame as in the case of β = 0. Yet, the realization of full employment depends on the lower boundβ. Specifically, the marginal utility of wealth approaches its lower bound once the steady state
capital stock approaches k̃ since the value of land becomes infinitely high. Full employment at
a positive real rate can only occur if the natural rate at k̃ exceeds the actual return on savings,
which is zero at the Golden Rule level. Existence of the full employment steady state thendepends on the same condition as in the bubbly steady state since the value of the natural rate
at k̃ is identical in both cases.
Hence, condition (24) determines the feasibility of full employment in the steady state with
β > 0. In Figure 3, this condition holds in the left panel, while it is violated in the right panel.
Consider a concrete example to illustrate the factors that determine the feasibility of full
employment. Assume the standard production function f (k) = k α and a constant intertemporalelasticity of substitution σ > 0 with u0 (c) = c−σ . The Golden Rule level of the capital stock is
given by f 0 (k̃) = δ and the Golden Rule level of consumption satisfies c̃ = f (k̃) − δ k̃ + z. Full
employment is no longer feasible if condition (24) is violated or alternatively if condition (25)holds. In this example, full employment is not feasible if
−σ

α α
1−α
+z
.
β ≥ ρ (1 − α)δ19Intuitively, factors that lower the desire for savings and increase the natural rate make full
employment easier to sustain, while factors that reduce the natural rate make it more difficult.
Household preferences affect the feasibility of full employment. Specifically, a lower time
preference rate ρ, a more concave utility function u(c) and a stronger preference for wealth β
increase the desire of households for savings and reduce the natural rate.
In addition, a higher depreciation rate δ implies that more output is used for investment
to replace the depreciated capital stock, which lowers consumption under full employment fora given k. This increases the natural interest rate. In contrast, a higher capital intensity of
production α increases the supply of goods available for consumption, which lowers the naturalrate and makes full employment harder to sustain.22
Finally, higher rental income z from land ownership increases consumption under full employment as well as desired savings. The natural rate decreases.
To summarize the analysis so far: Full employment is always feasible in a purely neoclassicaleconomy with land and standard wealth preferences. In contrast, full employment cannot always
be sustained despite the existence of land if wealth preferences are insatiable. In fact, it is thevery existence of land that rules out the full employment steady state by preventing the real
return on savings from becoming negative.
In a monetary economy, the inflation rate also creates a lower bound on the real return on
savings due to the zero lower bound on the nominal rate. Yet, secular stagnation can occur inthe monetary economy with standard wealth preferences as shown by Michaillat and Saez (2014)
and Michau (2017), while it is not possible in the economy with land unless wealth preferences
are insatiable. The reason is that money does not affect the utility from wealth in these models
as it is not considered net wealth. The real money stock then does not affect the natural rate.
In the neoclassical economy with land, it is not the preference for wealth itself that preventsfull employment in steady state but its insatiable nature.23 It results in a desire for savings
that is too strong to clear markets at full employment. The oversupply of savings then causesa shortfall of the labor supply. In fact, there are multiple levels of the labor supply that areconsistent with market clearing. Specifically, any L < 1 that fulfills condition (25) in Proposition
6 defines a secular stagnation steady state. Secular stagnation is then not a monetary, but a real
phenomenon. The steady state is however indeterminate in the absence of real wage rigidities.
Therefore, I introduce money and downward nominal wage rigidity to determine the steady state.
22

Note that higher depreciation or a lower capital intensity also lower the net marginal product of capital andshift the rS curve. However, it is the return on savings at the Golden Rule level that determines the feasibility offull employment. This return is always zero.
23
Note that the insatiability of wealth preferences in the benchmark model is consistent with full employment
as there is no non-producible productive store of value.

20

4

A Monetary Economy with Land
I introduce money via a money-in-the-utility framework following Sidrauski (1967), where

the central bank controls the nominal money supply. In addition, nominal rigidities limit the
downward adjustment of wages and prevent deflationary wage-price spirals.

4.1

Monetary Policy, Wage Setting Frictions and Inflation

Money serves as a means of savings for households, in addition to bonds, capital and land, and
it is a unit of account for the price level. Money does not pay interest, but provides householdswith a convenience yield due to its liquidity and transaction value.
The central bank directly controls the nominal money supply MtS , while the nominal interest
rate it is endogenously determined in the money market. Let Pt denote the price level of theoutput good and mst ≡ MtS /Pt the real money supply. Similarly let mdt denote the real money
demand of households. In equilibrium, the money market clears such thatMtS
≡ mst = mdt = mt
Pt

(28)

Let µ denote the growth rate of the nominal money supply. The real money supply thenevolves asṁt = (µ − πt )mt ,(29)

where πt denotes the rate of inflation. In the monetary economy, bonds pay a nominal interest
rate it , which is related to the real rate of return via the Fisher Equation as
rt = it − πt .

(30)

As land and capital are real assets, their returns are unaffected by the introduction of money
and the No-Arbitrage condition (5) continues to hold, with the return on bonds given by (30).
In addition, the firm problem is unaffected and the rental price of capital and the real wage
are still determined by the marginal products of capital and labor in (7) and (8). The latter
implies that the real wage is constant in steady state. Therefore, the inflation rate always equals
the change in the nominal wage Wt in steady state.
I assume downward nominal wage stickiness. Specifically, I follow Schmitt-Grohé and Uribe(2016, 2017) and postulate that nominal wages cannot fall by more than an exogenous rate γ,i.e.
Ẇt≥ −γ .
Wt
21(31)

A value of γ = 0 implies that nominal wages cannot fall at all, whereas γ = 1 allows for perfectly
flexible wages.24 Equation (31) is only binding in case of stagnation, hence the complementaryslackness constraint(1 − Lt )

Ẇt
+γ
Wt

!
=0.

(32)

Under full employment we have Lt = 1 and (31) is not binding. The inflation rate is determined by the money supply growth rate µ as can be seen from (29) with ṁt = 0. The central
bank can perfectly control the inflation rate and money is fully neutral. The price level adjusts
to clear the money market in (28). It is then the dynamics of the price level that determine
nominal wage dynamics.
In contrast, we have Lt < 1 under stagnation and (31) is binding. Nominal wages decline ata constant rate γ. The dynamics of the nominal wage determine the inflation rate π as the real
wage is given by (8). Taken together, the rate of inflation is determined as

πt =




 µ if Lt = 1 ,

Ṗt
=

Pt

 −γ − ẇt if Lt < 1 .
wt

(33)

I assume µ ≥ 0 > −γ. Hence, the economy is deflationary under stagnation, but not underfull employment. Under stagnation, inflation is related to changes in the real wage, which are
determined by changes in the capital-labor ratio as ẇt = −f 00 (kt )kt k̇t from (8). In steady state,
the inflation rate is constant as k̇t = 0 and hence ẇt = 0.25

4.2

Households and the Natural Rate of Interest (Revisited)

Money is a store of value for households. Total real wealth at is now given by the household’s
real bond holdings bt ≡ Bt /Pt , with bt = 0 in equilibrium, the value of real capital, the value ofland and the real money holdings as
at = bt + kt Lt + qt Ht + mt ,
24

(34)

The same assumption is adopted by Hanson and Phan (2017) and Biswas et al. (2017). Similar relationshipare used among others in Ono and Ishida (2014), Eggertsson, Mehrotra, and Robbins (2017), Caballero and Farhi(2018) or Michau (2017). They are derived from nominal frictions in the wage setting process like fairness concernsof workers or the prevalence of a wage norm or a reference wage. For a review of the empirical evidence on wage
stickiness, see section 8 in Schmitt-Grohé and Uribe (2016).
25
The asymmetry in the inflation process based on nominal wage stickiness is a common feature of models of
secular stagnation and can be found among others in Ono and Ishida (2014), Michau (2017) or Illing et al. (2017).
In this setting, the inflation rate in the secular stagnation steady state is determined by structural factors, which
affect the degree of wage stickiness γ, and hence unaffected by the labor supply gap. In contrast, the inflation
rate depends on the steady state output gap in Ono and Ishida (2014), Illing et al. (2017) and Schmitt-Grohé andUribe (2017).

22

and, using the No-Arbitrage condition (5) for the returns on capital and land, real wealth evolves
asȧt = rt at − it mt − ct + wt Lt + z̃t ,(35)

where I assume that all seignorage income z̃t ≡ µmt is redistributed to households in a lump-sum
fashion. Holding money creates opportunity losses equal to the nominal interest rate it .
I follow the money-in-the-utility literature and I assume that real money balances generate
utility ω(mt ) to households as they are a liquid means of payment. This function fulfills thestandard assumptions of ω 0 (m) > 0, ω 00 (m) < 0, lim ω 0 (m) = ∞ and lim ω 0 (m) = 0. The
m→0

m→∞

lifetime utility of households is given by
Z ∞
U0 =

[u(ct ) + ω(mt ) + v(ãt )] e−ρt dt ,

(36)

0

which is the standard Sidrauski (1967) model except for the wealth preference v(ãt ). Households maximize (36) subject to (35), which gives the optimality condition for the intertemporal
allocation of consumption and optimal money demand asηcċtu00 (ct )ct
v 0 (ãt ), where ηc ≡ − 0,= rt − ρ + 0
ct
u (ct )u (ct )(37)

ω 0 (mt )= it .
u0 (ct )(38)

In addition, the transversality condition (15) continues to apply to real household wealth.
The Euler Equation in (37) is unaffected by the introduction of money, except for the difference in the wealth composition in (34). It is up for debate if the real money stock constitutes
net wealth of households and generates utility via the wealth preference. Both views can befound in the literature. Ono (1994, 2015) makes the case for including money in the utility fromwealth. Then v(ãt ) = v(at ) as before and the natural rate increases for a given k compared to
the neoclassical model. As households feel wealthier, their desire to save declines. In contrast,
Michaillat and Saez (2014) and Michau (2017) argue that only wealth generated by real assets
affects the utility from wealth.26 Then v(ãt ) = v(at − mSt ) and the real money stock does notaffect the wealth preference. The natural rate is the same as in the purely neoclassical economy.
From the perspective of the individual household, the assumption v(ãt ) = v(at ) seems morenatural. However, I will analyze the implications of both views.
26
Michaillat and Saez (2014) assume that the central bank issues money to buy household bonds. The wealthcreated by money is therefore equal to the bond liability and net wealth equals the stock of physical capital.
Similarly, Michau (2017) argues that both government debt and money do not constitute net wealth of households
as both need to be redeemed eventually.

23

The nominal rate it determines optimal money demand in (38). Households equate theliquidity premium, which is the marginal rate of substitution between money and consumption,
to the nominal interest rate on bonds. In fact, (38) represents a No-Arbitrage relation between
money and bonds and establishes a zero lower bound on the nominal interest rate. The wealthpreference itself does not affect money demand directly.
The real return on money rtM affects the return on savings rtS as money is a store of value
for households. The nominal return on money consists of the liquidity premium in (38). Using
the Fisher Equation (30) and inflation in (32), the real return on money satisfies

rtM =

 0
ω (mt )

− µ if Lt = 1 , u0 (ct )ω 0 (mt )− πt =
u0 (ct )ω 0 (mt )ẇtif Lt < 1 .
+γ+
u0 (ct )wt(39)

Asset market clearing requires the equalization of the real returns on money, bonds, land andcapital. The No-Arbitrage condition (5) is extended to include money as
rt = rtM = rtK = rtL (= rtB ) ,(40)

where rtK , rtL , rtB and rtM are given by (9), (2), (4) and (39). This constitutes the real return on
savings rtS , which households receive irrespective of the asset in which they invest.

5

Analysis of the Monetary Economy
The equilibrium of the economy is characterized by the Euler Equation (38), the return onland (2), the return on capital (9), the return on money (39), the goods market clearing condition(18), the money supply process in (29) and the No-Arbitrage condition (40), together with thetransversality condition (15), where household wealth is given by (34), for a given initial valueof capital K0 > 0. Under full employment Lt = 1 and Kt = kt . Under stagnation Lt < 1 andK0 > 0 is given while kt > Kt . In steady state, it holds that ċt = 0 and k̇t = 0, which implies
K̇t = 0 and L̇t = 0. The properties of q̇t and ṁt depend on v(ã).
The actual return on savings rS in steady state is determined by the No-Arbitrage condition(40) that requires the equalization of real returns across assets as

rS =



ω 0 (M S /P )





 u0 (f (k) − δk + z) − µ if Lt = 1 











γ if Lt < 1

24= f 0 (k) − δ =

z
q̇t
+ >0,
qt qt

(41)

where I used money market equilibrium (28) and goods market equilibrium (18) to substitutefor optimal money demand and consumption. Note that Proposition 2 continues to hold, which
is why rS is required to be strictly positive.
Under full employment, the real money stock increases in the capital stock of the economy.
An increase in k lowers the net marginal product of capital and increases consumption as k < k̃.
This makes holding money more attractive and increases money demand of households. For agiven money supply M S , the price level P declines to clear the money market. As a consequence,
the real money stock increases and the nominal interest rate is lowered until rM = rK . As thenominal interest rate is strictly positive, the money growth rate µ - or equivalently the inflation
target of the central bank - represents a lower bound on the return on savings when the economy
operates at full employment. This lower bound is what prevents the return on savings fromadjusting to the natural rate in other models of secular stagnation. In the model with land, thislower bound is only binding in an economy that is deflationary under full employment, i.e. ifµ < 0. For positive inflation rates, it is always the non-negativity restriction imposed by theexistence of land that represents the minimum return on household savings.
In a secular stagnation steady state, the economy is deflationary and nominal wages andprices decline with rate γ. As a consequence, the real money supply expands indefinitely sinceµ + γ > 0. In this asymptotic steady state, money demand of households becomes satiated andthe nominal interest rate converges to zero. The return on savings is then determined by therate of deflation and the capital stock adjusts accordingly.
The existence of a steady state requires the equalization of the return on savings rS definedin (41) with the natural rate of interest rN in (37) with ċt = 0, taking into account the wealthcomposition. The full employment steady state exists if
ω 0 (M S /P )v 0 (ã)
0
−
µ
=
f
(k)
−
δ
=
ρ
−
>0,
u0 (f (k) − δk + z)
u0 (f (k) − δk + z)

(42)

where I use the goods market clearing condition under full employment (18) to substitute for
consumption in (37) and where wealth is defined by the wealth composition (34) with bt = 0,
Lt = 1 and Ht = 1. For v(ã) = v(a), the natural rate increases for a given k compared to theprevious section as money is part of net wealth in v(a). In contrast, for v(ã) = v(a − mS ), thereal rate is determined in the real economy and unaffected by the introduction of money. Oncethe real rate is determined, and strictly positive for full employment to be feasible, households
adjust their desired money holdings such that rM = rS from (40) and the price level adjusts toclear the money market in (28). Money is fully neutral as an exogenous increase in the money
supply will simply result in a proportional increase in the price level.

25In addition, for a secular stagnation steady state to exist, it has to hold thatγ = f 0 (k) − δ = ρ −v 0 (ã)>0,u0 ([f (k) − δk]L + z)

(43)

where I use goods market clearing condition (18) with L < 1 to substitute for consumption in(37). Under stagnation, the inflation rate is determined by the binding wage rigidity and thereturn on money equals the rate of deflation as the nominal rate becomes asymptotically zero.
The capital-labor ratio is determined by the No-Arbitrage condition between the net marginal
product of capital and the return on money. The expansion of the money supply reduces themarginal utility of wealth to its lower bound in the asymptotic steady state for v(ã) = v(a).
The feasibility of full employment and stagnation depend on the nature of the utility fromwealth. As before, I distinguish again between standard and insatiable wealth preferences.

5.1

The Case of Standard Wealth Preferences

The steady state with standard wealth preferences is characterized by a constant land priceas Proposition 3 continues to hold. Hence, the land price is given by the capitalized value ofrents and bubbles do not exist. The same argument applies to the real money stock. Under full
employment, we have πt = µ from (33) and the real money supply is constant from (29). Ina secular stagnation steady state, the real money supply expands indefinitely since µ + γ > 0.
The expansion of the money supply increases the natural rate to ρ, which eventually violates thetransversality condition for v(ã) = v(a).27
Full employment is always feasible in a neoclassical economy with land and β = 0 as shownin Proposition 4. The respective condition for the monetary economy is as follows.
Proposition 7 In the monetary model with β = 0, the full employment steady state in (42) isthe unique equilibrium if the real money stock affects the utility from wealth, i.e. if v(ã) = v(a).
The secular stagnation steady state in (43) cannot exist for v(ã) = v(a).
For the case of v(ã) = v(a − mS ), the full employment steady state is not feasible if∗µ < −rN
C <0

(44)

∗
where rN
C > 0 denotes the steady state real rate in the neoclassical model with land.

The full employment steady state is unique under mild restrictions and the dynamic system
exhibits saddle-point stability around this steady state.
27

The expansion of the real money supply under stagnation does not affect the real interest rate for v(ã) =v(a − mS ). Then, money can expand without violating the transversality condition if µ + γ < ρ.

26

Figure 4: The Steady State of the Monetary Model with Land for β = 0rrSr S (k) = f ′ (k) − δ > max{0, −µ}′r (k) = f (k) − δ > max{0, −µ}

ρ

r

ρ

r N (a − mS )∗r N (a − mS )

r∗
∗
rN
C

∗
rN
C

k

r S > −µkk < kµr S > −µk < k̃-δ-δ

k < k̃

r N (a)r N (a)Note: This figure shows the natural rate rN for the monetary economy (solid line) and the neoclassicalmodel with land (dotted line) as well as the return on savings rS as a function of the capital stock k. The
dotted line also represents the natural rate for v(ã) = v(a − mS ). The left graph assumes µ ≥ 0, whereas
the right graph shows 0 > µ > −γ. k̃ is the Golden Rule capital stock, kµ is defined by f 0 (kµ ) = δ − µ.

The proof is relegated to Appendix E. The intuition is illustrated in Figure 4, which shows
the natural rate for the neoclassical economy with land (dotted line) and the monetary economy
with v(ã) = v(a) (solid line). The dotted line also represents the natural rate for the case ofv(ã) = v(a − mS ). The natural rate in the monetary economy exceeds the natural rate in the
neoclassical economy due to the higher wealth level for a given k if v(ã) = v(a). The return on
savings is unaffected, but potentially restricted by the introduction of money. The restrictions
rS > 0 from land (k < k̃) and rS > −µ from money (k < kµ ) are also illustrated.
Consider first the case of an inflationary economy with µ ≥ 0, which is shown in the left
panel of Figure 4. It is the existence of land that provides a lower bound on the real interest
rate as the nominal interest rate exceeds the real rate. Full employment is always feasible asland price increases stimulate the natural rate towards ρ once the capital stock approaches theGolden Rule level. This is the same mechanism as in the neoclassical model with β = 0.
In a deflationary economy with µ < 0 (right panel), the existence of money creates a lower
bound on the return on savings. Yet, increases in the real money stock have a similar effectas land for v(ã) = v(a). A lower real rate increases money demand to lower the nominal rate
for a given inflation rate. This requires an increase in the real money stock to clear the money
market. The corresponding increase in monetary wealth stimulates the natural rate towards ρ(for k = kµ ) and full employment is always feasible.

27However, this channel only works when real money balances affect the natural rate in steadystate, which is illustrated by the curve rN (a). If households do not derive utility from monetary
wealth, the natural rate is independent of the real money stock, which is curve rN (a − mS ).
Then, the steady state real interest rate is unaffected by the introduction of money and given∗ , which is the same rate as in the neoclassical model with land of section 3. This rate
by rN
C
∗
satisfies rNC ∈ (0, ρ). If the economy is sufficiently deflationary, the real return on savings can∗ . Then full employment is not
be persistently above the natural rate. This requires µ < −rNCfeasible despite the existence of land as the incentives for savings by holding cash are too high.28
As long as the economy under full employment is characterized by price stability or positive
trend inflation, the full employment equilibrium is always feasible with standard wealth preferences once land is available as a means of investment. In fact, it is the only feasible equilibriumof the model economy when money affects the utility from wealth. Then, secular stagnation
cannot occur as either the increase in land prices for µ ≥ 0 or the expansion of the real money
supply for ρ > −µ > 0 absorbs any excess savings.
If wealth preferences are instead given by v(ã) = v(a − ms ), the possibility of stagnation is
not ruled out by the presence of land. Then the full employment and a secular stagnation steadystate can coexist and it is unclear which one is realized. In a monetary economy without land,
this secular stagnation steady state occurs as the unique outcome if the central bank follows
a low inflation target as in Michaillat and Saez (2014) and Michau (2017). Secular stagnation
then becomes a policy choice and full employment can be restored by choosing a sufficiently highinflation target. In contrast, full employment is always feasible in an economy with land andstandard wealth preferences unless in the implausible scenario of the central bank following astrong deflation target. The secular stagnation steady state can, however, exist as the uniqueequilibrium of the economy when wealth preferences are insatiable.

5.2

The Case of Insatiable Wealth Preferences

With a lower bound on the marginal utility of wealth, rational bubbles with q̇t > 0 are
possible in a stationary steady state with ċt = k̇t = 0 as shown in Proposition 5. The same
argument implies that an infinite expansion of the money supply, i.e. ṁt > 0, is consistent withthe transversality condition in a secular stagnation steady state if the rate of expansion is below
the time preference rate ρ. For simplicity, I will assume in the following analysis that there areno bubbles, i.e. qtB = 0. The steady state with bubbles is identical to the case of v 0 (ã) = β due
to an expanding real money stock for v(ã) = v(a).
28Note that the existence of a full employment equilibrium always requires µ > −ρ since the natural rate cannot
exceed the time preference rate ρ.

28

The following proposition is the equivalent to Proposition 6 in the neoclassical economy.
Proposition 8 In the monetary economy with β > 0, full employment is not feasible if
β > ρu0 c(k̃)

for µ ≥ 0 ,

β > (ρ + µ)u0 (c(kµ )) for µ < 0 and v(ã) = v(a) ,∗µ < −rNC(45)

for µ < 0 and v(ã) = v(a − mS ) ,where k̃ and kµ are defined by f 0 (k̃) = δ and f 0 (kµ ) = δ − µ. In addition, there exists a secularstagnation steady state with 0 < L < 1 ifβu0 (f (k ∗ ) − δk ∗ + z)v 0 k ∗ + γz
u0 (f (k ∗ ) − δk ∗ + z)

>ρ−γ >βu0 (z)v0>ρ−γ >

for v(ã) = v(a) ,
(46)

 
z
γ

u0 (z)

for µ < 0 and v(ã) = v(a − mS ) ,where k ∗ is given by f 0 (k ∗ ) = δ + γ, if the money growth rate is small and if nominal wages are
sticky, specificallyµ<ρ−γ ,

(47)

γ<ρ.

(48)

The full employment and secular stagnation steady states are unique under mild restrictions andthe dynamic systems satisfy saddle-point stability around these steady states.
Consider first the case of µ ≥ 0, for which (45) is identical to (24) in Proposition 6. The lower
bound on the return on savings rS always results from the availability of land and money does not
impose a constraint. The existence of land implies that the real return on savings cannot become
negative and that the capital stock cannot exceed the Golden Rule level. Once it approaches k̃,the land price becomes infinitely high but fails to stimulate the natural interest rate as the lower
bound β is binding. Hence, if the natural rate at the Golden Rule capital stock is positive, full
employment is feasible. This is illustrated in the left panel of Figure 5.
In contrast, if the natural rate is negative at the Golden Rule level, an oversupply of savings
occurs at full employment. Land prevents the real return on saving from falling below zero,
while increases in the land price cease to stimulate the natural rate. The oversupply of savings
depresses demand, the realized labor supply drops below the potential labor supply and secularstagnation occurs. This is illustrated in the right panel. The drop in the labor supply shifts thenatural rate upwards and restores goods market equilibrium at a positive real rate.
29

Figure 5: The Steady State of the Monetary Model with Land for β > 0rrr S (k) = f ′ (k) − δ > 0

r S (k) = f ′ (k) − δ > 0ρρ

k < k̃
r N (L = 1)
r N (L < 1)r∗r∗ = γr N (L = 1)
k
k∗

k
∗
)
(rL=1

k < k̃

Note: This figure shows the natural rate rN for the monetary economy with β > 0 as well as the returnon savings rS as a function of the capital stock k for µ ≥ 0 and v 0 (ã) = β. The vertical lines represent thelower bound on the real rate imposed by the existence of land. Full employment condition (45) holds in theleft graph, while it is violated in the right graph.

In a deflationary economy with µ < 0, the lower bound of the return on money exceeds thelower bound of the return on land. Therefore, money constrains the actual return on savings
from falling below the rate of deflation. This implies that the capital stock under full employment
cannot exceed kµ < k̃, with f 0 (kµ ) = δ − µ, at which both physical capital and money yield thesame real return. Full employment is only feasible if the natural rate exceeds the return on savings
at kµ . Otherwise, secular stagnation occurs. Note that condition (45) is independent of the lower
bound β for v(ã) = v(a − mS ) and simply requires sufficiently weak wealth preferences.29
As the real money stock is part of household wealth, the transversality condition (15) is
affected. In steady state, the growth rate of the real money supply µ − π cannot exceed the time
preference rate ρ. Under full employment, the real money supply is constant and this always
holds. In contrast, the growth rate under stagnation is given by µ − γ > 0. Hence, condition(47) requires ρ > µ − γ.
Finally, condition (48) simply implies that the stagnation steady state is well-defined. If therate of deflation under stagnation exceeds the time preference rate ρ, the return on savings is
persistently higher than the natural interest rate, whose maximum is ρ. Hence, the good market
does not clear for any level of the labor supply L and a steady state therefore does not exist.
Note that conditions (47) and (48) are identical for µ = 0.
29This condition is identical to condition (44) in Proposition 7. Under these assumptions, the secular stagnation
steady state also exists for β = 0. Yet, full employment is always possible in that case, while it is not feasible
with β > 0 although v 00 (a) < 0 in steady state. Hence, it is the lower bound on the marginal utility of wealththat can result in secular stagnation as the unique feasible equilibrium.

30

It is easy to see from Proposition 8 that secular stagnation is a monetary phenomenon only
to the extent that the economy under full employment is deflationary. Then increases in themoney growth rate µ can restore full employment. However, secular stagnation becomes a real
phenomenon once µ ≥ 0. It is the result of the existence of land in combination with insatiable
wealth preferences. Further increases in the money growth rate cannot restore full employment.
Consider briefly the features of the full employment and the secular stagnation steady states.
The Full Employment Steady State: The economy behaves similar to the standard moneyin-the utility model. We have L = 1 and π = µ from (33), which implies a constant real money
supply. The land price can include a bubble as shown in Proposition (5), in which case v 0 (a) = β.
Without bubbles, the land price is finite. The capital stock is determined by rS = rN in (42).
From the capital stock, the real rate is derived as the net marginal product of capital. The land
price is the rent z, capitalized with this rate, while consumption is determined from the market
clearing condition. Given c and k, households adjust their money demand such that rM = rSfrom (40) and the price level adjusts to clear the money market in (28). Money is neutral as anincrease in the money supply results in a proportional increase in the price level.
How does the land value affect the steady state? An exogenous increase in z has two effects
for a given k. On the one hand, the higher rent implies a higher fundamental land price, which
increases household wealth and lowers the desire for savings. Hence, the natural rate increases.
On the other hand, consumption under full employment increases, which reduces the natural rate.
Hence, there is a wealth effect via higher land prices that causes the natural rate to increase andan income effect that lowers the natural rate.30 The wealth effect is stronger the higher the00

elasticity of the utility from wealth ηa ≡ − vv0(a)a(a) . For high values of ηa , the increase in land
prices causes a crowding out of investment and the capital stock declines. For sufficiently low
values of ηa , the capital stock increases as the income effect dominates and there is crowding-in.31
The Secular Stagnation Steady State: The realized labor supply falls short of the potentiallabor supply, i.e. L < 1, and the downward nominal wage constraint is binding. Hence, inflation
is given by π = −γ < 0 as can be seen in (33). The real money supply expands at rate µ + γ,
given that (47) holds. The liquidity demand of households is satiated and the nominal rate
converges to it = 0 from (38). Then, the real return on money in (39) is simply given by the30
In addition, the increase in consumption increases money demand for a given nominal rate. The resulting
increase in the real money stock increases household wealth unless we have v(ã) = v(a − mS ), which reinforces
the wealth effect as it increases ηa .
31
Note that the crowding out effect always dominates if the asset does not provided income directly. For
instance, the value of land in Kocherlakota (2013) is derived from the utility it yields to households. Similarly,
the real house price in Illing et al. (2017) depends on the convenience yield from housing.

31

rate of deflation as rM = γ > 0 and, together with the No-Arbitrage condition (40), the steadystate capital stock of the economy is uniquely determined as
f 0 (k ∗ ) = δ + γ > δ ,(49)which is above the Golden Rule level and hence consistent with the requirement of Proposition
2. In addition, the fundamental value of the land price equals the capitalized value of rents as
z
>0,
γ

q∗ =

(50)

although by Proposition 5 several bubbly steady states with q̇t > 0 exist, where the bubble grows
at rate γ without violating the transversality condition as ρ > γ by (48).
Market clearing requires the real return on savings to be consistent with the natural rate rN .
Since this is not feasible at full employment, households adjust their labor supply according to(43) such thatβu0 ([f (k ∗ ) − δk ∗ ]L∗ + z)

=ρ−γ ,(51)

where k ∗ is defined in (49). Note that we have v 0 (a) = β as ṁt > 0 in the asymptotic steady statewith v(ã) = v(a) and q̇tB > 0 in the presence of bubbles. This condition does not depend on themoney growth rate, but only on the degree of wage rigidity γ. The existence of the steady state
requires wages to be sufficiently sticky, specifically ρ > γ. Condition (51) uniquely determines
the labor supply gap under secular stagnation. In case of v(ã) = v(a − mS ) and in the absenceof bubbles, the respective condition is given byv 0 (k ∗ L∗ + q ∗ )=ρ−γ ,
u0 ([f (k ∗ ) − δk ∗ ]L∗ + z)

(52)

where v 00 (a) < 0 and q ∗ is defined in (50). How do changes in the model parameters affect thesecular stagnation steady state?
Higher wage flexibility worsens realized output under stagnation. An increase in γ causes
higher deflation and a higher return on savings rS . The capital stock per worker decreases
to provide for a higher return on capital in (49). This causes the natural rate rN to increase
for a given labor input. The net effect is an excess supply of savings, which worsens the labor
supply gap. The effect that increasing wage flexibility is detrimental under stagnation is common
in models of stagnation and can be found among others in Eggertsson and Krugman (2012),
Eggertsson, Mehrotra, and Robbins (2017) and Michau (2017).32
32

Similarly, an increase in the depreciation rate increases the capital stock per worker in (49). Yet, it also lowers
the supply of goods available for consumption. It is unclear which effect dominates.

32

In addition, a higher time preference rate ρ implies a higher realized labor supply. As households are less patient, their desire for saving decreases and the natural rate increases. Since
the capital-labor ratio is unchanged, this causes excess demand. Moreover, stronger preferences
for wealth reduce the natural rate without affecting the steady state capital-labor ratio. Consequently, a higher value of β causes the labor supply to decline.
A higher land rent z causes the fundamental value of land to increase. In case of v 00 (a) < 0, thenet effect of the opposing income and wealth effects depends on the elasticities of u(c) and v(a).
However, the wealth effect ceases to affect the natural rate if v 0 (a) = β. Therefore, the income
effect dominates. As aggregate supply increases, the natural rate is lowered, which requires alower labor input in steady state. Hence, land price increases worsen stagnation, though via theindirect effect on aggregate supply.33
To summarize, it is the existence of land together with the insatiable nature of the utility
from wealth that results in the secular stagnation steady state as the unique equilibrium given
the conditions in Proposition 8. Stagnation becomes a real phenomenon for µ ≥ 0, which incontrast to other stagnation models does not rely on restrictive inflation targets of the monetary
authority. It is the possibility of investment in land that prevents the central bank from restoringfull employment. As a consequence, an increase in the inflation target, which is a popular policy
recommendation of other secular stagnation models, ceases to be effective.

5.3

Extension: A Risk Premium on Land

So far, I have assumed that households require the same real returns on all assets in equilibrium. However, land typically is less liquid and subject to more macroeconomic risk as modeled
by Caballero and Farhi (2018) or political risk than bonds and physical capital as argued by
von Weizsäcker (2014). Therefore, I assume in this section that households require a real risk
premium ∆ > 0 for land investments above the risk-free return on bonds or physical capital.
This risk premium is exogenous.
It follows that the required return on land is given by rtL = rtS + ∆. This holds for both thefundamental land price and a land price bubble. Asset market clearing requires the risk-free rateon savings rS in steady state to satisfy the No-Arbitrage conditionrS =ω 0 (M S /P )
 u0 (f (k) − δk + z) − µ if Lt = 1 ,q̇tz+ − ∆ = f 0 (k) − δ =

qt qt




(53)

γ if Lt < 1 .

33This conclusion is opposite to Kocherlakota (2013), who shows how land price declines worsen stagnation inan overlapping generations model. However, land yields utility rather than income in his setting so that changes
in z do not affect consumption directly.

33

Using the same argument as in Proposition 2, the real return on land cannot become negative.
The risk-premium then implies that the short-term real rate cannot fall below the premium.
Specifically, it has to hold thatrS > −∆ and k < k∆ ,(54)

where k∆ is defined by f 0 (k∆ ) = δ − ∆. It follows that the risk-free rate can indeed become
negative once a risk premium on land is included. The return on land can never be a restrictive
factor for the return on savings if the risk premium is sufficiently high. Hence, I will assume
0 < ∆ < δ. For simplicity, I abstract from the possibility of bubbles and assume a stationary
land price in steady state, which is equal to the real rent z capitalized at the required returnfrom land rS + ∆.34
The risk premium affects the feasibility of full employment with both standard wealth preferences and insatiable wealth preferences. Full employment requires the return on savings in (53)to match the natural rate of interest. Specifically, it requiresf 0 (k) − δ = ρ −v 0 (ã)> max{−µ, −∆} ,u0 [f (k) − δk + z]

(55)

where v(ã) = v(a) includes the real money stock or is net of money for v(ã) = v(a − mS ). The
binding constraint on the return on savings results from the risk premium on land or the money
growth rate, whichever is lower in absolute terms. Proposition 7 is modified as follows.
Proposition 9 In the monetary economy with β = 0 and ∆ > 0, the full employment steadystate in (55) is the unique equilibrium if the real money stock affects the wealth preference, i.e.
if v(ã) = v(a). The secular stagnation steady state cannot exist for v(ã) = v(a).
For the case of v(ã) = v(a − mS ), the full employment steady state is not feasible if∗µ < −rN
C (∆) < ∆(56)

∗ (∆) ∈ (−∆, ρ) denotes the steady state real interest rate in the neoclassical model with
where rNC
0
land and a risk premium ∆ > 0. It holds that rN
C (∆) < 0.

This steady state is unique under mild restrictions and the dynamic system exhibits saddlepoint stability similar to the case of ∆ = 0 in Proposition 7.
The proof is equivalent to the case of ∆ = 0 in Appendix E and graphically illustrated in Figure
6, which shows the different returns in the presence of a risk premium on land.
34
The risk premium affects the conditions for rational bubbles. Each bubble has to grow with rS + ∆ to befeasible. This is not possible with standard wealth preferences due to the transversality condition since r → ρ.
With β > 0, rational bubbles are possible if ∆ < β/u0 (c∗ ), where c∗ is consumption in steady state.

34

Figure 6: The Steady State of the Model with Risk Premium for β = 0rrr S (k) = f ′ (k) − δ > max{−∆, −µ}r S (k) = f ′ (k) − δ > max{−∆, −µ}ρρk < k̃

k < k̃

r N (∆ > 0)r∗r N (∆ > 0)

r∗
k

k

∗
r∆

∗
r∆

r S > −µr S > −∆

k < kµ-δ-δr N (∆ = 0)k < k∆r N (∆ = 0)

Note: This figure shows the natural rate rN for the model with (solid line) and without risk premium (dottedline) as well as the return on savings rS as a function of the capital stock k. The left graph assumes an increasein ∆, whereas the right graph shows an increase in µ. The restrictions r > −∆ and r > −µ are represented by
the dotted vertical and horizontal lines. k∆ and kµ are defined by f 0 (k∆ ) − δ = −∆ and f 0 (kµ ) − δ = −µ.

∗The introduction of the risk-premium lowers the real return rN
C in the steady state of theneoclassical model with land. For a given k, a higher risk premium reduces the land price, which
lowers household wealth. The marginal utility from wealth increases and the natural interest
rate declines. As a consequence, the capital stock is reduced and so is the return on savings,
which might become negative as well. This is illustrated in the left graph of Figure 6.
Full employment is feasible despite the negative interest rate if the real money stock is partof the wealth preference, i.e. v(ã) = v(a), for the same reason as before. Whenever the real rate
approaches its lower bound, the associated increases in the land price or the real money stock
stimulate the natural rate into positive territory via the wealth preference. Hence, rN approaches
ρ whenever k approaches k∆ or kµ in the left graph.
However, if the real money stock does not affect household wealth, the money growth rate canbecome a binding constraint on the real return on savings for ∆ > µ. Hence, secular stagnation
can occur as the result of an increase in the risk premium together with a strict inflation target.
This is illustrated in the right graph of Figure 6, where ∆ increases while µ = 0. As the naturalrate declines following the increase in the risk premium, a low money growth rate prevents thereturn on savings from falling sufficiently. Since the real money stock does not stimulate thenatural rate via the wealth preference for v(ã) = v(a − mS ), full employment cannot be realized∗ (∆ > 0), which is condition (56) in Proposition 9.
if µ < −rNC35

In this case, full employment can be restored by both an increase in the money growthrate µ or a reduction in the risk premium on land ∆. The first measure directly lowers thereturn on money and allows the real return on savings to match the lower natural rate. The
second measure increases the land price and therefore stimulates the natural rate rN . Hence, ahigher money growth rate and a reduction in the risk premium can restore full employment withstandard wealth preferences. Once ∆ = 0, full employment is always feasible unless the economy
is deflationary and v(ã) = v(a − mS ).
Consider now the case of insatiable wealth preferences, which is described in Proposition 8for ∆ = 0. With a risk premium on land, the modified version is as follows.
Proposition 10 In the monetary economy with β > 0 and a risk premium ∆ > 0 on land
investment, full employment is not feasible if
β > (ρ + ∆)u0 (c(k∆ )) for µ ≥ ∆ ,β > (ρ + µ)u0 (c(kµ ))for ∆ > µ and v(ã) = v(a) ,∗ (∆)µ < −rNC

for ∆ > µ and v(ã) = v(a − mS ) ,(57)

where k∆ is given by f 0 (k∆ ) = δ − ∆ and kµ by f 0 (kµ ) = δ − µ. In addition, there exists a secularstagnation steady state with 0 < L < 1 ifββ>ρ−γ > 0
u0 (f (k ∗ ) − δk ∗ + z)
u (z)




z
z
v 0 k ∗ + γ+∆
v 0 γ+∆>ρ−γ >
u0 (f (k ∗ ) − δk ∗ + z)
u0 (z)

for v(ã) = v(a) ,(58)
for ∆ > µ and v(ã) = v(a − mS ) ,where k ∗ is given by f 0 (k ∗ ) = δ + γ. Moreover, conditions (47) and (48) of Proposition 8
continue to apply. The full employment and secular stagnation steady states are unique under
mild restrictions and the dynamic systems satisfy saddle-point stability.
The proof is equivalent to the case of ∆ = 0 in Appendix F and graphically illustrated in Figure
7. Propositions 9 and 10 reduce to Propositions 7 and 8 for ∆ = 0.
With a positive lower bound on the marginal utility from wealth, increases in the land
price or the real money stock cease to stimulate the natural rate at some point, which remains
permanently below the time preference rate and potentially below the return on savings. Thenfull employment is not feasible and stagnation occurs. For a sufficiently low risk premium, it is
the return on land that constitutes a lower bound on the return on savings rS , while the money
growth rate provides the lower bound and stagnation is a monetary phenomenon for µ < ∆.
36

Figure 7: The Steady State of the Model with Risk Premium for β > 0rrr S (k) = f ′ (k) − δ > max{−∆, −µ}r S (k) = f ′ (k) − δ > max{−∆, −µ}ρρk < k̃
r N (k)r N (∆ = 0)kr S > −∆

kr S > −∆

-δ-δk < k̃

k < k∆r N (∆ > 0)k < k∆

Note: This figure shows the natural rate rN as well as the return on savings rS as a function of the capitalstock k. In the left graph, v 0 (a) = β, while v(ã) = v(a − mS ) is shown in the right graph. Both graphs
assume an increase in ∆. The vertical and horizontal lines represent the lower bounds on the real rate
imposed by land.

If stagnation is a monetary phenomenon, increases in the money growth rate can restore full
employment by lowering the return on money. This is the same effect as in the case of β = 0 andv(ã) = v(a − mS ). However, they cease to be effective for µ ≥ ∆ as stagnation becomes a real
phenomenon due to the existence of land. To see this, note that condition (57) is independentof the money growth rate for µ ≥ ∆.
With insatiable wealth preferences, the effects of a reduction in the risk premium ∆ are instark contrast to the case of standard preferences, where lowering the risk premium helps torestore full employment, similarly to an increase in the inflation target. With insatiable wealthpreferences, a reduction in the risk premium makes full employment actually harder to sustain
and can result in secular stagnation.
A lower risk premium increases the minimum return on savings rS . However, the associatedincrease in the land price ceases to stimulate the natural rate sufficiently to be compatible withfull employment. With v 0 (a) = β, the natural rate is unaffected by the risk premium, which isillustrated in the left panel of Figure 7. The steady state interest rate under full employment isnegative, which is only sustainable if the inflation rate µ and the risk premium ∆ are sufficiently
high. Then, lowering the risk premium can make full employment unfeasible. The same effectholds with v(ã) = v(a − mS ) and v 00 (a) < 0, which is illustrated in the right graph. Althougha lower risk premium stimulates the natural rate, the effect on the minimum return on savings
always dominates and full employment is not feasible for a low risk premium under (57).
37

The risk premium also affects the steady state properties. Under full employment, a reductionin the risk premium reduces the capital stock as it increases the natural rate of interest. As aconsequence, consumption falls if the capital stock is below the Golden Rule level or expands if
the capital stock is above the Golden Rule level.
In the secular stagnation steady state, the risk premium does not affect the capital stock per
worker, which is determined by (49).35 In addition, consumption and output are unaffected forv 0 (a) = β as changes in wealth no longer affect the desire for savings. Then variations in the risk
premium simply result in changes in asset prices that do not feed back into the real economy. It
is only for the case of v(ã) = v(a − mS ) that a lower risk premium increases the realized labor
supply and hence consumption under stagnation. However, the reduction in the risk premium
can at the same time prevent a return to full employment.
To summarize, the conclusions of the previous sections hold in the general setting with apositive risk premium. With standard wealth preferences, full employment is feasible if the risk
premium on land is below the money growth rate or if the real money stock yields utility tohouseholds in steady state. If v(ã) = v(a − mS ), full employment is not feasible if the risk
premium is high, but can be restored by a higher money growth rate.
In contrast, secular stagnation occurs as the unique steady state of the model if wealthpreferences are insatiable and the risk premium or the inflation rate is sufficiently low. Then an
increase in the risk premium can actually restore full employment, while a higher money growthrate is only effective if money is the restrictive factor. Lowering the risk premium can potentiallystimulate consumption under stagnation, but it also makes full employment less sustainable.

6

Conclusion
Many advanced economies struggle to maintain full employment despite historically low nom-

inal interest rates. Unconventional monetary policies have narrowed interest rate spreads andcontributed to higher prices of stocks, land and other assets, but failed to stimulate household
consumption and investment demand sufficiently.
I have presented a theory of secular stagnation in a monetary economy with land that canrationalize these observations. Persistent stagnation can occur in steady state as a consequenceof a permanent oversupply of savings. Households are not willing to consume at full employment
because holding money or investing in land is more attractive. The return on these assets might
be too high for full employment to be feasible.
35
Note that the model can be extended to include a risk premium on the return on capital as well. Thenequation (49) is affected by variations in the risk premium on capital. A higher spread decreases the capital stock
under stagnation.

38

In a monetary economy, the rate of inflation constrains the real return from savings and thefeasibility of full employment effectively becomes a policy choice. In contrast, the availability ofland implies that the real interest rate cannot fall below zero or the risk premium on land.
With standard wealth preferences, the introduction of land substantially weakens the casefor secular stagnation as any excess savings are invested in land and the associated increasein the fundamental value of land stimulates household consumption via the wealth preference.
Full employment is not feasible for a sufficiently low inflation rate, but only if monetary wealthdoes not affect the utility from wealth. Then, both an increase in the money growth rate or areduction in the risk premium on land can restore full employment.
In contrast, the case for secular stagnation becomes stronger with insatiable wealth preferences. Then, increases in wealth at some point cease to stimulate consumption and secularstagnation occurs as the unique equilibrium. It is the very existence of land that accounts forthe failure of the economy to operate at full employment since the possibility of land investment
prevents the real return on savings from falling sufficiently.
Stagnation in an economy with land is therefore a real phenomenon and cannot always be
cured by monetary policy. In particular, higher inflation targets cease to be effective in restoringfull employment when spreads are low. In addition, policies to reduce spreads in financial markets
might have unintended negative consequence and prevent full employment from being feasible if
the risk premium on land becomes too low. Therefore, fiscal policy seems to be more appropriate,particularly with respect to tax policy as discussed in detail in Michau (2017).

39

A

Proof of Proposition 1
Let k̂ be defined by f (k̂) − δ k̂ + z = 0 such that c(k̂) = 0 in (21). Define k̃ by f 0 (k̃) = δ

as the Golden Rule level of the capital stock. Define k + by f 0 (k + ) = δ + ρ, such that the net
marginal product and the return on savings at k + equal ρ. It holds that k + < k̃ < k̂.
Full employment condition (22) defines the function F (k) as the difference between the returnon savings and the natural interest rate as
F (k) ≡ f 0 (k) − δ − ρ +v 0 (k).
u0 (f (k) − δk + z)

(A.1)

Existence: The full employment steady state exists if there is k̄ > 0 such that F (k̄) = 0.
F (k) is a continuous function of k in (0, k̂). It holds that F (k) > 0 for k < k + . Also, F (k + ) =v 0 (.)/u0 (.) > 0 and F (k̂) = f 0 (k̂) − δ − ρ = f 0 (k̂) − f 0 (k + ) < 0. It follows that there existsk̄ ∈ (k + , k̂) with F (k̄) = 0, irrespective of β > 0 or β = 0.
Uniqueness: Wealth preferences can result in multiple equilibria, see Kurz (1968) and Bose
(1971). Uniqueness requires restrictions on the third moments of v(a) and u(c) as in Kurz (1968)or restrictions on the model parameters, particularly on ρ, as in Bose (1971). F 0 (k̄) < 0 for allk̄ with F (k̄) = 0 is a sufficient condition for uniqueness.36 Specifically,
F 0 (k̄) < 0 ↔

ρ + δ − f 0 (k̄)

 0
v 00 (k̄)
u0 (c̄)+ 00,
f (k̄) − δ < f 00 (k̄) 00
u (c̄) u (c̄)(A.2)

for all k̄ with F (k̄) = 0. This property has to hold locally at each k̄. All terms on the RHS are
weakly positive. Hence, uniqueness is guaranteed for ρ = 0. The same holds for ρ sufficiently
close to zero, where sufficiently is dependent on the functional form of u(c) and f (k), particularly
their third derivatives. This is also the conclusion of Kurz (1968).
Stability: The dynamic system is given by the differential equations for consumption in (14)and the capital stock in (18) for a given initial value k0 > 0:ctv 0 (kt )0ċt =f (kt ) − δ − ρ + 0,
ηcu (ct )

(A.3)

k̇t = f (kt ) − δkt + z − ct ,

(A.4)

36

Since F (k) is the difference between the return on savings rS and the natural rate rN , this condition requires
the slope of the natural rate to exceed the slope of the return on savings at each intersection. Intuitively, the rS
curve has to intercept the rN curve in Figure 1 from above. The same requirement guarantees uniqueness for all
extensions of the benchmark model.

40where I used (9) to substitute for the real rate in (14). Log-linearization around the steady state
via a first-order Taylor approximation gives

  
h
i 
0
00
00 (k̄) + v (k̄)
ρ + δ − f 0 (k̄) − uu00(c̄)
f
c
−
c̄
ċt
(c̄)
u0 (c̄)   t
 =
 ,
0
k̇t
k
−
k̄
−1
f (k̄) − δtwhere I use the definition of ηc to simplify. The eigenvalues λ of the system solve





v 00 (k̄)
u0 (c̄)+ 00
Λ(λ) = λ2 − ρλ + ρ + δ − f 0 (k̄) f 0 (k̄) − δ − f 00 (k̄) 00
u (c̄) u (c̄)= (φ1 − λ)(φ2 − λ) = λ2 − (φ1 + φ2 )λ + φ1 φ2 = 0 ,where I rewrite the equation in terms of its solutions λ1 = φ1 and λ2 = φ2 . Matching terms
impliesv 00 (k̄)
u0 (c̄)+ 00
φ1 φ2 = ρ + δ − f 0 (k̄) f 0 (k̄) − δ − f 00 (k̄) 00<0,u (c̄) u (c̄)(A.5)

which follows from (A.2). Hence, there is one negative and one positive eigenvalue and thesystem is saddle-point stable around the steady state. Any k0 > 0 uniquely determines c0 (k0 )and the system converges to the steady state along the saddle-path. Saddle-point stability holds
globally, which can be seen in the phase diagram of (A.3) and (A.4).

B

Proof of Proposition 2
Start with the differential equation (2) for the land price, q̇t = rqt − z. Let r denote the real

rate in steady state. Define the integrating factor ξs = e−Rrds = e−rs to modify (2) as[qs˙ξs ] = q̇s ξs − qs ξ˙s = [q̇s − rqs ]ξs = −zξs
Z t

Z t




1 −rt 1+
ξs ds = −ze ds = −z − eqt ξt − A = −z
r
r
0
0


 h

1
1
1
z i rt z
1
qt = ξt−1 A − ξt−1 z − e−rt += Aert − zert − e−rt +
= A−
e +
r
r
r
r
r
r
−rs

Let q0 denote the steady state land price at t = 0. Then, q0 = A and the steady state price attime t is given by:hz i rt z
qt = q0 −
e +
r
r

41

(B.1)

Free disposal of land implies qt ≥ 0 for all t. From (B.1), qt ≥ 0 requires r > 0 as qt approaches
z/r < 0 if r < 0. For the special case of r = 0, equation (B.1) becomes
qt = q0 − zt ,(B.2)

which for any q0 > 0 eventually becomes negative. Hence r = 0 is also not possible in steadystate. We therefore must have r > 0.37 Equation (B.1) with r > 0 rules out q̇t < 0 in steadystate. For any q0 > 0, this eventually results in qt < 0, which is not possible.

C

Proof of Proposition 4
Define k + by f 0 (k + ) = δ + ρ, k̃ as the Golden Rule level and k̂ by c(k̂) = 0 as in Appendix

A with k + < k̃ < k̂.
Full employment condition (23) defines the function G(k) as the difference between the returnon savings and the natural interest rate as

G(k) ≡ f 0 (k) − δ − ρ +zv 0 k + f 0 (k)−δ
u0 (f (k) − δk + z)

.

(C.1)

∗
∗
Existence: The steady state exists if there is kN
C > 0 such that G(kN C ) = 0. G(k) is acontinuous function of k ∈ (0, k̃) with G(k) > 0 for k < k + . Also G(k + ) = v 0 (.)/u0 (.) > 0 and+∗∗G(k̃) = −ρ < 0 as v 0 (∞) = 0. Hence, there exists kNC ∈ (k , k̃) with G(kN C ) = 0.

Uniqueness: The arguments of Proposition 1 apply accordingly. A sufficient condition for∗ ) < 0 for all k ∗∗uniqueness is given by G0 (kNC
N C with G(kN C ) = 0 or equivalently∗  0 ∗
v 00 (a∗N C )dqN
u0 (c∗N C )0 ∗00 ∗ρ + δ − f (kN C ) f (kN C ) − δ < f (kN C ) 00 ∗+1+ ∗C ,u (cN C ) u00 (c∗N C )
dkN C
where

(C.2)

∗ )
∗
zf 00 (kN
dqN
C
C
=
−

2 > 0∗dkN
f 0 (k ∗ ) − δCNCNote that condition (C.2) is less restrictive than condition (A.2) in the benchmark model as the
additional term is strictly positive.
Effects on k and r: Let r̄ and k̄ denote the steady state real rate and capital stock in the∗∗benchmark model and rN
C and kN C in the model with land. For r̄ ≤ 0, it follows directly from∗∗
Proposition 2 that rN
C > 0 ≥ r̄ and from (19) that kN C < k̃ ≤ k̄.
37

If land depreciates at rate τ , we must have r > −τ . See Michau et al. (2017) for the general case.

42

∗
∗
∗
Let r̄ > 0 and k̄ < k̃. In the model with land, kN
C defines qN C = q(kN C ) > 0. Consider the∗rise in q from zero to qN
C as exogenous and recover the effects on k from Ω(k, q) = 0 inΩ(k, q) = f 0 (k) − δ − ρ +v 0 (k + q)=0,
u0 (f (k) + z − δk)which is equivalent to F (k) in (A.1) except it includes q in v(a). It follows thatdk∂Ω/∂q−v 00 (k + q)u0 (c)
=−
=
<0,
dq
∂Ω/∂k
F 0 (k)(C.3)∗∗
as F 0 (k̄) < 0 by (A.2). Therefore, kN
C < k̄ < k̃ and rN C > r̄ > 0.

Stability: The dynamic system is given by (2) for the land price, (14) for consumption and(18) for the capital stock with Lt = 1 for a given initial k0 > 0 as

v 0 (kt + qt )ct
0,f (kt ) − δ − ρ +ċt =ηcu0 (ct )(C.4)

k̇t = f (kt ) − δkt + z − ct ,

(C.5)



q̇t = f 0 (kt ) − δ qt − z ,(C.6)

where I used (5) and (9) to substitute for the real rate in (2) and (14). Log-linearization of thissystem around the steady state via a first-order Taylor approximation gives
h
i
  


00 ∗
0 ∗
v 00 (a∗N C )∗ ) − u (cN C ) f 00 (k ∗ ) + v (aN C )∗−ρ + δ − f 0 (kN
ċt
c
−
c
∗
∗
∗
00
0
00
NC
C
u (cN C )u (cN C )u (cN C )   t
NC 
  
  

∗ 
0
∗
k̇t  =  kt − kN ,−1f (kN C ) − δ
0
C
  

00
∗
zf (kN C )∗∗ )−δ
qt − qNq̇t
f 0 (kN0CC
f 0 (k∗ )−δNC

where I use the definition of ηc . The eigenvalues of the transition matrix solve
 0 ∗ u0 (c∗N C ) 0 ∗∗2Λ(λ) = −λ3 + [ρ + f 0 (kN
G (kN C ) = 0 ,C ) − δ]λ − Ω1 λ − f (kN C ) − δ
u00 (c∗N C ) u0 (c∗N C ) 0 ∗∗where Ω1 ≡ ρ f 0 (kN
F (kN C ) ,C ) − δ − 00 ∗u (cN C )∗ ) < 0 from (A.2) and G0 (k ∗ ) < 0 from (C.2). Let ϕ denote the eigenvalues of thewith F 0 (kNiCNCsystem.
I rewrite the equation in terms of its solutions as Λ(λ) = (ϕ1 − λ)(ϕ2 − λ)(ϕ3 − λ) orΛ(λ) = −λ3 + (ϕ1 + ϕ2 + ϕ3 )λ2 − (ϕ1 ϕ2 + ϕ1 ϕ3 + ϕ2 ϕ3 )λ + ϕ1 ϕ2 ϕ3 = 0 .

43

Matching terms of these two equations defines the eigenvalues as∗
ϕ1 + ϕ2 + ϕ3 = ρ + f 0 (kNC) − δ > 0 ,

(C.7)


 u0 (c∗N C ) 0 ∗∗ϕ1 ϕ2 ϕ3 = − f 0 (kN)−δ
G (kN C ) < 0 .
C
u00 (c∗N C )(C.8)

At least one eigenvalue is positive by (C.7), whereas (C.8) implies one or three negative eigenvalues. Hence, there is exactly one negative eigenvalue and the system exhibits saddle-point
stability.

D

Proof of Proposition 6
Define k + , k̃ and k̂ as before with k + < k̃ < k̂. G(k) is given in (C.1) with v 0 (a) ≥ β andrepresents conditions (26) in the bubbly and (27) in the fundamental steady state.
∗
Existence and Uniqueness: The steady state exists if there exists kN
C > 0 such that∗ ) = 0 in (C.1). G(k) is continuous in k ∈ (0, k̂) since v 0 (a) = β for k ≥ k̃. HowG(kN
C
+
+
0
0
∗
ever, kNC ∈ (0, k̃) by Proposition 2. We have G(k) > 0 for k < k and G(k ) = v (.)/u (.) > 0.

Existence requires G(k̃) = ρ − β/u0 (c(k̃)) < 0 or equivalently
G(k̃) < 0 ↔ β < ρu0 c(k̃) .

(D.1)∗ ) < 0 for all k ∗This is condition (24) in Proposition 6.38 Uniqueness requires G0 (kN
N C withC∗ ) = 0. Hence, uniqueness condition (C.2) applies accordingly.
G(kNC∗ ) = 0 at k ∗Suppose instead G(kNC
N C ∈ (k̃, k̂). Existence follows from Proposition 1 withv 0 (a) = β. Let
Ψ(k, L) ≡ f 0 (k) − δ − ρ +βu0 ([f (k) − δk]L + z),

∗ , L) = 0 in steady state. We havewith L ∈ (0, 1) and Ψ(kNC∗ ) − δk ∗ ]βu00 (c∗N C )[f (kN
dk
∂Ψ/∂L
1
C
NC
=−
=
>0,
∗
0
2
dL
∂Ψ/∂k
[u (cN C )]
∂Ψ/∂k

(D.2)

∂Ψ
∗
∗
as kN
C < k̂ and ∂k < 0 from (A.2). Hence, a decline in L lowers kN C . A stagnation steadystate is defined for any L ∈ (0, 1) that satisfiesΨ(k̃, L) < 0 ↔ ρu0 [f (k̃) − δ k̃]L + z > β ,
38

(D.3)

Intuitively, full employment requires that the natural rate rN at k̃ is above the return on savings rS , which∗is zero at k̃. Then an intersection in k+ < kN
C < k̃ is ensured as both functions are continuous in this region.

44

and takes into account the non-negativity restriction L > 0. Taken together, this implies
ρu0 (z) > β .

(D.4)

Stability: In the fundamental steady state, the bubble component is always zero as it would∗
otherwise grow infinitely as rN
C > 0. Hence, q̇ = 0 in steady state and real wealth is constant.

The dynamic system is given by (C.4), (C.5) and (C.6) and the stability patterns are identicalto the case of β = 0 in the previous proof.
In the bubbly steady state, we have ȧt > 0 as q̇tB > 0 for given q0B > 0 and v 0 (at ) = β in(C.4). Denote xt = 1/qtB with x∗N C = 0 in steady state. The bubble equation (4) becomesẋt = −[f 0 (kt ) − δ]xt ,(D.5)

where I use (5) and (9) to substitute for rtB . The dynamic system is given by (C.4) withv 0 (at ) = β, (C.5), (C.6) for the fundamental price qtF and (D.5) for the bubble. A first-order
Taylor approximation around the bubbly steady state gives
u0 (c∗∗∗
NC )f 00 (kN0ρ + δ − f 0 (kNC)
C ) − u00 (c∗
NC )
  
  
∗
 k̇t  −1
f 0 (kN
0C) − δ
 =
00∗
 F 
zf (kN C )∗q̇t  0
f 0 (kN∗C) − δ
f 0 (kN
  
C )−δ
ẋt
0
0
0
ċt





0

ct − c∗N C






∗ 
  kt − kN
C
 ,

 F
F∗ 
 qt − qN
0
C


∗
0 ∗
x
−
x
−f (kN C + δtNC0where I use the definition of ηc and v 00 (a) = 0. The eigenvalues of this system solve0∗0 ∗Λ(λ) = [−f (kNC ) + δ − λ][f (kN C ) − δ − λ] ·u0 (c∗)∗
ρ + δ − f 0 (kNC) − λ∗− u00 (cN∗ C ) f 00 (kN
C)

−1

∗
f 0 (kNC) − δ − λ

NC

=0,

where . denotes the determinant. From the proof of Proposition 1 with v 00 (a) = 0 it follows∗ )−δ > 0that this matrix has one positive and one negative eigenvalue. In addition, λ3 = f 0 (kNC∗ ) − δ] < 0. The bubbly equilibrium hence satisfies saddle-point stability. For
and λ4 = −[f 0 (kNCgiven k0 > 0 and q0B > 0, which determine c(k0 , q0B ) and q F (k0 , q0B ), the model converges to thesteady state along the unique saddle path.

45

E

Proof of Proposition 7
Define k + , k̃ and k̂ as before with k + < k̃ < k̂. The steady state real money stock under full

employment is defined by (38) and (40) as a function of the capital stock asm(k) = ω 0−1

 0
f (k) − δ + µ u0 (f (k) − δk + z) .

(E.1)

with m0 (k) > 0 and m(k) → ∞ for k → kµ , where kµ satisfies f 0 (kµ ) = δ − µ. Hence, kµ > k̃ forµ > 0 and vice versa.
Monetary Wealth Preferences: Consider first the case of money in v(ã) = v(a). Full
employment condition (42) and (E.1) define H(k) aszv 0 k + f 0 (k)−δ+ m(k)

H(k) ≡ f 0 (k) − δ − ρ +u0 (f (k) − δk + z)

.

(E.2)

(i) Existence: The steady state exists if there exists k ∗ > 0 such that H(k ∗ ) = 0. H(k) is
continuous in k ∈ (0, k̃) if µ ≥ 0 or k ∈ (0, kµ ) if µ < 0. It holds that H(k) > 0 for k < k + . Hence,
we cannot have k ∗ ∈ (0, k + ), which requires kµ > k + or µ > −ρ. Also H(k + ) = v 0 (.)/u0 (.) > 0.
If µ ≥ 0, H(k̃) = −ρ < 0 and there exists k ∗ in (0, k̃) with H(k ∗ ) = 0. If µ < 0, H(kµ ) = −ρ < 0
as m(k) → ∞ and there exists k ∗ in (0, kµ ) with H(k ∗ ) = 0 unless µ < −ρ. Hence, existence
requires µ < −ρ for v(ã) = v(a).
(ii) Uniqueness: A sufficient condition is H 0 (k ∗ ) < 0 for all k ∗ with H(k ∗ ) = 0 or
u0 (c∗ )v 00 (a∗ )dq ∗dm∗[ρ + δ − f (k )][f (k ) − δ] < f (k ) 00 ∗ + 00 ∗ 1 + ∗ +,u (c ) u (c )
dk
dk ∗
0

∗

0

∗

where

and

00

∗

(E.3)

zf 00 (k ∗ )
dq ∗
=
−
>0
dk ∗
[f 0 (k ∗ ) − δ]2



dm∗
1
= 00 ∗ (f 0 (k ∗ ) − δ)(f 0 (k ∗ ) − δ + µ)u00 (c∗ ) + f 00 (k ∗ )u0 (c∗ ) > 0 ,∗dkω (m )

for all k ∗ with H(k ∗ ) = 0. This condition is less restrictive than (A.2) or (C.2) since the additionalterm is positive.
(iii) Impossibility of Secular Stagnation: Under stagnation, we have π = −γ < 0 in steadystate and therefore ṁ > 0 as µ + γ > 0. Then i = 0 and v 0 (a) = 0. Secular stagnation condition(44) then requires γ = ρ. However, the steady state is not well-defined as K, L and c areindeterminate in this case and the transversality condition is violated for µ ≥ 0.

46

Non-monetary wealth preferences: With v(ã) = v(a − mS ), H(k) is identical to G(k) in∗∗+∗(C.1). This implies rNC ∈ (0, ρ) and kN C ∈ (k , k̃). As k is uniquely determined by G(k) if(C.2) holds, P adjusts to clear the money market. From (41), defineJ(P ) =

ω 0 (M S /P )∗− µ − rNC .
u0 (f (k ∗ ) − δk ∗ + z)

(E.4)

Full employment requires J(P ) = 0. J(P ) is continuous and strictly increasing in P for a given∗ . Existence requires J(0) < 0 or
M S . It holds that J(∞) = ∞ > 0 and J(0) = −µ − rNC
∗ . Hence, we have µ > −r ∗S0equivalently µ > −rNC
N C in (44) for v(ã) = v(a − m ). As J (P ) > 0,

the full employment steady state is unique if (C.2) holds.
Stability: The dynamic system is given by (C.4), (C.5), (C.6) and (29), where mt is potentially
included in (C.4). There are no bubbles with β = 0. Using (30) to substitute for πt , (38) for itand (9) for rt , (29) is rewritten asω 0 (mt )0+ f (kt ) − δ mt .
ṁt = µ − 0
u (ct )
(E.5)

A first-order Taylor approximation around the full employment steady state gives


ċt





0

∗

00

∗

00

∗

)
)
− vu00(ãρ + δ − f 0 (k ∗ ) − u (cu00)f(c∗(k
)
(c∗ )

  
  
 k̇t  
−1
 =
  
 q̇t  0  ω 0 (m∗ )u00 (c∗ ) ∗mṁt[u0 (c∗ )]2

00

∗

)
− vu00(ã
(c∗ )

00

∗

)
− vu00(ã
(c∗ )



ct − c∗







  kt − k ∗ 

 ,


  qt − q ∗ 
f 0 (k ∗ ) − δ0ω 00 (m∗ ) ∗∗0− u0 (c∗ ) m
mt − mf 0 (k ∗ ) − δ0zf 00 (k∗ )f 0 (k∗ )−δ

f 00 (k ∗ )m∗0

where I use the definition of ηc . The eigenvalues of the system solveω 00 (m∗ ) ∗m −λ ·
u0 (c∗ )



−

h
i
0 ∗)
00 ∗
00 (k ∗ ) + v (ã )fρ + δ − f 0 (k ∗ ) − λ − uu00(c
∗
00
∗
(c )
u (c )
−1
0

+
where

00

∗

)
− vu00(ã(c∗ )f 0 (k ∗ ) − δ − λ0

zf 00 (k∗ )f 0 (k ∗ ) − δ − λ

f 0 (k∗ )−δ00 ∗ 

2 v 00 (ã∗ )ω 0 (m∗ )u00 (c∗ ) ∗  0 ∗00 ∗∗ v (ã )
m
+
f
(k
)m
f 0 (k ∗ ) − δ − λ = 0 ,
f
(k
)
−
δ
−
λ
0
∗
2
00
∗
00
∗
[u (c )]u (c )u (c ).

denotes the determinant. The eigenvalues ϕi of the transition matrix then solve

Λ(λ) = (ϕ1 − λ)(ϕ2 − λ)(ϕ3 − λ)(ϕ4 − λ) = 0, which is a quartic equation in λ. Matching terms
implies that the eigenvalues are related as
ϕ1 ϕ2 ϕ3 ϕ4 = −[f 0 (k ∗ ) − δ]47

ω 00 (m∗ ) ∗ 0 ∗
m H (k ) < 0 ,u0 (c∗ )(E.6)

where H(k) is defined in (E.2) and H 0 (k ∗ ) < 0 follows from condition (E.3). Hence, the dynamic
system has either one or three negative eigenvalues. In addition, we haveϕ1 + ϕ2 + ϕ3 + ϕ4 = −ω 00 (m∗ ) ∗m + 2[f 0 (k ∗ ) − δ] + [ρ + δ − f 0 (k ∗ )] > 0 ,u0 (c∗ )ω 00 (m∗ )u0 (c∗ )(ϕ1 + ϕ2 )ϕ3 ϕ4 + (ϕ3 + ϕ4 )ϕ1 ϕ2 = −[f 0 (k ∗ ) − δ] 00 ∗ G0 (k ∗ ) − 0 ∗ m∗ Ω1u (c )u (c )

v 00 (ã∗ )u0 (c∗ )+ 0 ∗ m∗ 2[f 0 (k ∗ ) − δ][f 0 (k ∗ ) − δ + µ] + f 00 (k ∗ ) 00 ∗<0,
u (c )u (c )

(E.7)

(E.8)

where the negative sign follows from G0 (k ∗ ) < 0 in (C.2) and from the proof of Proposition 4.
Suppose there are three negative and one positive eigenvalue, ϕ4 > 0. Solving the two equations
for ϕ4 gives−[ϕ1 + ϕ2 + ϕ3 ] < ϕ4 < −ϕ1 ϕ2 ϕ3
(ϕ1 + ϕ2 )ϕ3 + ϕ1 ϕ2[ϕ1 + ϕ2 + ϕ3 ](ϕ1 + ϕ2 )ϕ3 + [ϕ1 + ϕ2 + ϕ3 ]ϕ1 ϕ2 > ϕ1 ϕ2 ϕ3(ϕ1 + ϕ2 )[(ϕ1 + ϕ2 + ϕ3 )ϕ3 + ϕ1 ϕ2 ] > 0 ,which contradicts the assumption of three negative eigenvalues. Hence, the system has exactly
one negative eigenvalue and is saddle-point stable around the steady state.

F

Proof of Proposition 8
Define k + , k̃ and k̂ as before with k + < k̃ < k̂. kµ satisfies f 0 (kµ ) = δ − µ and H(k) is definedin (E.2) as a continuous function of k ∈ (0, k̃) if µ ≥ 0 or k ∈ (0, kµ ) if µ < 0.
Full Employment Steady State: Existence requires H(k) = 0 for some k ∈ (0, k̃) if µ ≥ 0or k ∈ (0, kµ ) if µ < 0. It holds that H(k) > 0 for k < k + . Hence, we cannot have k ∗ ∈ (0, k + ),which requires kµ > k + or equivalently µ > −ρ. Also H(k + ) = ṽ 0 (.)/u0 (.) > 0. If µ ≥ 0,existence requires  
H(k̃) < 0 ↔ β < ρu0 c k̃
.

(F.1)

Then there exists at least one k ∗ ∈ (0, k̃) with H(k ∗ ) = 0. If µ < 0 and v(ã) = v(a), full
employment requires
H(kµ ) < 0 ↔ β < (ρ + µ)u0 (c (kµ )) .

(F.2)

Then, there exists at least one k ∗ ∈ (0, kµ ) with H(k ∗ ) = 0 unless µ < −ρ. For the case of µ < 0and v(ã) = v(a − mS ), the proof is identical to Proposition 7.

48

Uniqueness requires H 0 (k ∗ ) < 0 for all k ∗ that satisfy H(k ∗ ) = 0, which is expressed in
conditions (C.2) and (E.3). In addition, the stability properties under full employment are
identical to the case of β = 0 before.
Secular Stagnation Steady State: A secular stagnation steady state exists if (43) holds.
Then, k = k ∗ , where k ∗ satisfies f 0 (k ∗ ) = δ + γ. Define M (L) by (43) with k = k ∗ as
M (L) = γ − ρ +v 0 (ã).
u0 ([f (k ∗ ) − δk ∗ ]L + z)

(F.3)

M (L) is continuous in L. The secular stagnation steady state exists if there is 0 < L < 1 suchthat M (L) = 0.
If v(ã) = v(a) or if q B > 0, v 0 (ã) = β in (F.3) and we have M 0 (L) > 0. We require
M (L = 0) < 0 < M (L = 1) for an equilibrium to exist. This implies:
γ−ρ+

β
β
<0<γ−ρ+ 0.
u0 (z)u (f (k ∗ ) − δk ∗ + z)

(F.4)

Uniqueness is guaranteed as M 0 (L) > 0. If v(ã) = v(a − mS ) = v(k ∗ L + q S ), then v 00 (ã) < 0.
Existence of the secular stagnation steady state then requiresγ−ρ+v 0 (k ∗ + z/γ)
v 0 (z/γ))
<
0
<
γ
−
ρ
+
.
u0 (z)
u0 (f (k ∗ ) − δk ∗ + z)

(F.5)(F.4) and (F.5) are summarized in (46). Uniqueness requires M 0 (L) > 0 at L = LS or equivalently
M 0 (L) > 0 ↔ (ρ − γ)v 00 (aS )f (k ∗ ) − δk ∗>,k∗u00 (cS )(F.6)

which implies that a reduction in the labor input increases the natural interest rate.
Stability of the Stagnation Steady State: It holds that Lt < 1 and kt > Kt . Inflation andthe price level are determined by the nominal wage rigidity. Hence, mt becomes pre-determinedand evolves with µ − πt . Inflation is given by (33), (39) and (40) as
πt = −γ −f 00 (kt )kt
ω 0 (mt )ẇt= −γ +k̇=− f 0 (kt ) + δtwt
f (kt ) − kt f 0 (kt )u0 (ct )This determines the dynamics of the capital-labor ratio kt and those of Lt together with K̇t in(17). As mt → ∞, let xt = 1/mt with xt → 0. For v(ã) = v(a), the dynamic system is given by
ċt =



ct
v 0 (Kt + qt + 1/xt )f 0 (kt ) − δ − ρ +,ηcu0 (ct )49(F.7)

K̇t = [f (kt ) − δkt ]Kt+ z − ct ,
kt

(F.8)



f (kt ) − kt f 0 (kt ) ω 0 (1/xt )0k̇t =− f (kt ) + δ + γ
f 00 (kt )kt
u0 (ct )q̇t = f 0 (kt ) − δ qt − z ,ω 0 (1/xt )0xt ,ẋt = −µ − f (kt ) + δ +u0 (ct )(F.9)(F.10)(F.11)

A first-order Taylor approximation around the stagnation steady state gives


ċt





  
  
K̇t   −1
  
  
 k̇t  =  0
  
  
 q̇t   0
  
ṁt
0

0

∗

00

∗

0

)
− u (cu00)f(c∗(k)

f (k∗ )k∗ − δ

0f 0 (k∗ )k∗ −f (k∗ ) K ∗k∗k∗k∗ f 0 (k∗ )−f (k∗ )k∗0f 00 (k ∗ )q ∗

γ

0

0

0

ρ−γ

0
0
0

00

∗



) 1
− vu00(ã(c∗ ) x∗2ct − c∗







∗
 Kt − K 0f (k∗ )−k∗ f 0 (k∗ ) ω 00 (m∗ ) 1  
∗


 ,
− f 00 (k∗ )k∗u0 (c∗ ) x∗2   kt − k   qt − q ∗ 0∗−γ − µxt − xwhere I use f 0 (k ∗ ) − δ = γ, v 00 (ã) = 0 and ω 0 (m) = 0. The eigenvalues solve ∗ 0 ∗f (k ∗ )k f (k ) − f (k ∗ )− λ (γ − λ)(ρ − γ − λ)−δ−λ =0 .
(−γ − µ − λ)
k∗
k∗
∗

)
Hence, three eigenvalues, λ1 = γ, λ2 = f (kk∗ − δ and λ3 = ρ − γ, are positive, while two
∗ 0

∗

∗

(k )
, are negative. The system is saddle-path stableeigenvalues, λ4 = −γ − µ and λ5 = k f (k k)−f∗around the asymptotic steady state under persistent stagnation.
0

∗

∗

∗

For v(ã) = v(a − mS ), we have v 00 (ã) < 0. Then λ1 = γ > 0, λ2 = f (k )kk∗−f (k ) < 0 and
λ3 = −γ − µ < 0 are unchanged to above. The remaining two eigenvalues are recovered fromf (k ∗ )f (k ∗ − δk ∗ ) v 00 (ã∗ )λ − ρ−γ+
−
δ
λ
+
(ρ
−
γ)
− 00 ∗ = 0 ,k∗k∗u (c )
2

which can be rewritten as λ2 − [λ4 + λ5 ] λ + λ4 λ5 = 0. Matching terms implies that
λ4 λ5 = (ρ − γ)f (k ∗ − δk ∗ ) v 00 (ã∗ )− 00 ∗ > 0 ,k∗u (c )(F.12)

which follows from M 0 (L) > 0 in (F.6). Hence, λ4 > 0 and λ5 > 0. We have three positive andtwo negative eigenvalues and the stability patterns are identical to v 00 (ã) = 0.

The proofs of Proposition 9 and 10 are equivalent to the proofs of Proposition 7 and 8 inAppendix E and F and not discussed in detail.

50

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