Public debt, positional concerns, and wealth inequality

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Public debt, positional concerns, and wealth inequalityR Kirill Borissov a, Andrei Kalk b,∗ a b

European University at St. Petersburg, 6/1A Gagarinskaya Str., St. Petersburg 191187, Russia Paris School of Economics, 48 Boulevard Jourdan, Paris 75014, France

a r t i c l e

i n f o

Article history: Received 1 February 2019 Revised 11 November 2019 Accepted 24 November 2019 Available online xxx JEL classification: E62 E21 D31 O41 Keywords: Public debt Wealth distribution Positional concerns Growth

a b s t r a c t We consider an AK growth model with positional concerns and public debt financed by distortionary income taxes. We show that if positional concerns are not too strong, there is a threshold level of the debt-to-GDP ratio. For the debt-to-GDP ratio below this level, the economy converges to a unique egalitarian balanced-growth equilibrium, whereas if this ratio is above the threshold level, the economy eventually settles on a two-class balancedgrowth equilibrium. The rate of growth in the egalitarian equilibrium is higher than that in any possible two-class equilibrium. Thus, a reduction in public debt may cause the economy to switch from the two-class regime to the egalitarian regime and accelerate growth. Our results also suggest that policies aimed at reducing initial inequality using public debt may, in fact, increase wealth inequality in the long run. © 2019 Elsevier B.V. All rights reserved.

1. Introduction While there is a large literature on the effects of public debt on economic growth,1 only a few studies have examined its effects on inequality. The purpose of our paper is to investigate the role of public debt in the dynamics of wealth inequality. For this, we propose a simple endogenous-growth model with altruistic dynasties that differ only in their initial endowments of capital and government bonds. A key mechanism through which the long-run wealth inequality arises in our model relies on the presence of positional externality. We adopt this mechanism from Borissov (2016). In particular, we assume that agents’ decisions depend not only on their absolute level of consumption but also on the perception of their social status, i.e., how their consumption relates

R We are grateful to Hippolyte d’Albis, Jean-Pierre Drugeon, Axelle Ferriere, Kai Konrad, Mikhail Pakhnin, Thomas Piketty, Olga Podkorytova, Clemens Puppe, Xavier Raurich, Gilles Saint-Paul, Abhijit Tagade, Bertrand Wigniolle, and an anonymous referee for their comments. We also thank the participants of seminars at Karlsruhe Institute of Technology, Max Planck Institute for Tax Law and Public Finance, the International Workshop “Economic Growth, Macroeconomic Dynamics and Agents Heterogeneity” (St. Petersburg, May 2017), the PET 2018 conference (Hue, June 2018), and the 3rd European Workshop on Political Macroeconomics (Tbilisi, June 2018) for their feedback on earlier versions of this paper. ∗ Corresponding author. E-mail addresses: [email protected] (K. Borissov), [email protected] (A. Kalk). 1 See Panizza and Presbitero (2013) for a survey.

https://doi.org/10.1016/j.jebo.2019.11.029 0167-2681/© 2019 Elsevier B.V. All rights reserved.

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to that of others.2 If positional concerns in the economy are strong enough, relatively poor dynasties consume so much that they become even poorer. Eventually, they save nothing, while the richest dynasties accumulate the entire wealth. Apart from consumption, agents care about the disposable income of their offspring.3 In our model, there are two possible regimes depending on the strength of positional externality: egalitarian and twoclass. In the egalitarian regime, the wealth distribution tends asymptotically toward full equality regardless of the initial state. In the two-class regime, the population is divided in the long run into two groups, the rich and the poor. The entire stock of capital and public debt is eventually owned by the dynasties which were the richest in the beginning and all other dynasties become poor. It follows in particular that the two-class regime allows any proportion between the rich and the poor. We show that if positional concerns are not too strong, there is a threshold level of the debt-to-GDP ratio. For the ratio below this value, the economy converges to the egalitarian balanced-growth equilibrium, and the closer the ratio to the threshold, the slower the convergence. For the ratio above this value, the economy eventually settles on a two-class balanced-growth equilibrium with the speed of wealth divergence increasing in the debt-to-GDP ratio. A nice aspect of these results is that the balanced growth rate associated with the egalitarian equilibrium is always higher than those associated with two-class equilibria. That said, the rate of growth in the two-class regime is increasing in inequality. Thus, the government in our model does not face a trade-off between equality and efficiency.4 If the economy is in the two-class regime, then a long-run reduction in public debt will not only lead to a higher rate of growth, but it may also cause the economy to switch to the egalitarian regime. Furthermore, our results suggest that for a given uneven initial distribution of wealth, policies aimed at reducing inequality by increasing public debt may, in fact, increase wealth inequality in the long run. Distributional effects of public debt have been studied by Mankiw (20 0 0) and Michel and Pestieau (1998, 1999). Introducing two types of agents — altruists and non-altruists — they show that although public borrowing is neutral in aggregate terms, it redistributes income from the poorer non-altruists to the richer altruists.5 These authors share the assumption that the taxes used to finance public debt are lump-sum. While this case is important for studying income and consumption inequality, the effect of public debt on wealth inequality is straightforward under non-distortionary taxation. With two types of agents, it simply says that the rich altruists, who own the entire capital stock and public debt, increase their steady-state wealth by exactly the amount of increase in public debt. The poor non-altruists still have zero wealth, and so the wealth inequality increases unambiguously. When distortionary taxes are present, such as an income tax, the distributional effect of public debt does not work in the same way. First, the rich altruists have to pay taxes for their additional bond holdings and, second, public debt crowds out private investment. To the best of our knowledge, Maebayashi and Konishi (2019) is the only paper that studies, among other questions, how public debt affects wealth inequality under distortionary taxation. They present several analytical results about the impact of inequality on debt sustainability and use numerical analysis to show that a rise in the public deficit ratio enlarges inequality and decreases the growth rate. In their model, the division of the population into two income classes cannot be influenced by the public debt policy. It occurs due to two assumptions: the heterogeneity in agents’ discount factors and a “joy-of-giving” bequest motive. Contrary to this, we assume that agents are identical in every respect except for their initial wealth. Furthermore, we allow zero optimal bequests using a less stringent form of altruism. The paper is organized as follows. In Section 2, we present the model. Section 3 introduces the definitions of equilibria and analyses their long-run dynamics. Section 4 examines how the debt-to-GDP ratio affects the rate of growth and the dynamics of wealth distribution. In the final section, we briefly discuss the results and conclude.

2. The model 2.1. Demographics We consider a closed economy inhabited by successive to infinity. Population is constant over time and consists of dynasty, therefore an agent j is equivalent to a member of offspring. Dynasties are identical in every respect except for

generations of agents. Time is discrete and runs from t = −1 L agents indexed by j ∈ {1, 2, . . . , L}. Each agent is born into a the dynasty j. She lives for one period and gives birth to one their initial wealth.

2 Several recent papers have studied the implications of this idea for inequality (García-Peñalosa and Turnovsky, 2008; Alvarez-Cuadrado and Long, 2012; Mino and Nakamoto, 2016), household debt (Alvarez-Cuadrado and Japaridze, 2017), taxation (Aronsson and Johansson-Stenman, 2010; 2014) and other public policies (Wendner and Goulder, 2008; Dioikitopoulos et al., 2019). 3 This type of altruism is known as “family altruism” (see, e.g., Michel et al. (2006)) originally described by Becker and Tomes (1979), though in a more general form. In the context of public debt, it is studied in Lambrecht et al. (2006). 4 See Okun (1975) for the classical view of this trade-off. More recently, García-Peñalosa and Turnovsky (2007) argue that, in a model with elastic labor supply, any growth-enhancing policies tend to make the (gross) distribution of income less equal. 5 Similar results were obtained in a more general model by Michel and Pestieau (2005), who consider a finite variety of agents differing in their degree of altruism and their productivity level. An alternative generalization was developed in Pestieau and Thibault (2012), where the heterogeneity comes from the degree of altruism and the preference for wealth.

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2.2. AK-technology In every period, the economy produces a unique final good that is either consumed or invested. Technology is given by

Y = F (K, AL ), where Y is output, K is the stock of capital, L is the labor input, and A is the economy-wide stock of knowledge. Each individual is endowed with one unit of labor, which they supply inelastically. The production function F : R2+ → R+ is assumed to be continuous, concave, continuously differentiable on int R2+ and homogeneous of degree one. Technological progress is Harrod-neutral with AL being effective labor units. For simplicity, capital fully depreciates each period. The knowledge stock of the economy is proportional to the capital stock:

A=

K , k¯ L

(1)

where k¯ > 0 is exogenously given. Borrowed from Frankel (1962) and Romer (1986), this assumption specifies the class of AK models. Using the fact that the production function is homogeneous of degree one, we can rewrite F(K, AL) as F (1, 1/k¯ )K. That is, output is linear in the capital stock:

Y = aK, with a ≡ F (1, 1/k¯ ). In other words, as physical capital is accumulated, output grows in proportion. In what follows, we assume that all markets are competitive. This implies that the amount of capital and labor used by firms must equate the marginal products of capital and labor to their respective prices, which firms take as given. Therefore, in intensive terms, the capital intensity of effective labor, denoted by k ≡ K/AL, the interest rate, r, and the wage rate per unit of effective labor, w, are related as follows:

r = f  (k ) − 1, w = f (k ) − f  (k )k, where f(k) ≡ F(k, 1) is the intensive-form production function. By assumption (1), capital per unit of effective labor is constant (i.e., k = k¯ ). Hence, the factor prices are also constant and determined by

r¯ ≡ f  (k¯ ) − 1, w¯ ≡ f (k¯ ) − f  (k¯ )k¯ .

(2)

Due to the Euler’s theorem combined with competitive factor markets, national product equals the sum of the factor incomes:

Y = (1 + r¯ )K + w¯ AL = (1 + r¯ )K +

w¯ K. k¯

Let

α ≡ (1 + r¯ )K/Y be the share of capital in the national output. Then the wage of each individual is

w ≡ w¯ A =

Y w¯ K = (1 − α ) . L k¯ L

(3)

2.3. Public sector The public sector of the economy is represented by the domestic public debt, denoted by D. We assume that the government controls the ratio of public debt to GDP,

χ ≡ (1 + r¯ )D/Y,

(4)

by issuing government bonds. This ratio is a long-run policy parameter in the model. Our goal is to examine the effects of χ on economic growth and wealth inequality. Given the above assumptions about production, equation (4) implies

D=

χ K, α

(5)

that is, the amount of public debt is proportional to the capital stock at each period. We assume that the government budget is balanced and there is no government spending. Taxes are raised with a proportional income tax with rate τ ∈ [0, 1), which is imposed on all sources of income in the economy, including bonds. Thus, the government budget constraint can be written as6

(1 + r¯ )Dt = τt (Yt + (1 + r¯ )Dt ) + Dt+1 , where the current public debt and interest payments on this debt are on the left-hand side, while tax revenues and new borrowing from the population are on the right-hand side. Then, for any given policy χ ≥ 0, the tax rate must be adjusted in accordance with the following rule:

τt = 6

 Kt+1 χ  1− . 1+χ (1 + r¯ )Kt

(6)

The subscript t is the time index.

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2.4. Consumer problem Consider the agent of dynasty j ∈ {1, . . . , L} who lives in period t. Her disposable income is defined as an after-tax sum of the wage earnings, wt , identical across agents and the current value of bequest (in the form of physical capital and j j j government bonds) left by her parent, st−1 . Out of this, she consumes ct ≥ 0 and leaves the rest st ≥ 0 to her offspring. Hence, her budget constraint is given by





j ctj + stj = (1 − τt ) wt + (1 + r¯ )st−1 .

The agent cares about her consumption and the disposable income of her offspring. We assume that consumption is j subject to positional concerns, meaning that the agent derives utility from it by comparing ct with the consumption of some reference group. For such reference consumption, we use the average consumption of generation t,

ct ≡

L 

ctj /L.

j=1

The offspring’s disposable income is assumed, for simplicity, to be non-positional.7 More formally, the preferences of agent j living at time t are represented by the following utility function:





ln(ctj − γ ct ) + δ ln (1 − τt+1 ) wt+1 + (1 + r¯ )stj



,

where γ ∈ [0, 1) is a strength of positional concerns and δ > 0 is a degree of altruism. Thus, the utility maximization problem of agent j living in period t is as follows:

max

ctj ≥0,stj ≥0

s.t.





ln(ctj − γ ct ) + δ ln (1 − τt+1 ) wt+1 + (1 + r¯ )stj



j ctj + stj = (1 − τt )(wt + (1 + r¯ )st−1 ).

(7)

Three points are worth emphasizing. First, the positional externality provides incentives to consume more and save less than in the case with no externality. It therefore acts as a force against leaving the bequest, while altruism acts as a force toward it. Given the budget constraint, the trade-off between these two opposing effects underlies the consumer problem. Second, the agent derives utility from the offspring’s disposable income rather than directly from the bequest left to her offspring. This assumption allows a zero optimal bequest. As a consequence, some agents in the economy may behave like pure spenders, consuming their entire after-tax income. Third, both the measure of positional externality and the degree of altruism are identical for all agents. In each generation, agents differ only in bequests left by their ancestors in period t − 1. The tax rates, as well as all other aggregate variables, are taken by the agent as given. Due to the our log specification of the utility function, we can rewrite the objective function in (7) as

ln(ctj − γ ct ) + δ ln(wt+1 + (1 + r¯ )stj ). This implies that the optimal solution of the consumer problem depends on the current tax rate τ t , but does not depend on the future tax rates. 3. Temporary, intertemporal and balanced-growth equilibria In this section, we give the definitions of equilibria in our model and analyze their long-run dynamics. Our definitions are fairly standard. First we define a temporary equilibrium where all consumers maximize their utility, firms maximize their profits, the government balances the budget each period, and all markets clear. An intertemporal equilibrium is defined as a sequence of temporary equilibria. j j Formally, let the bequests st−1 ≥ 0 be given for all j at some time t. Let further Kt + Dt = Lj=1 st−1 and Dt = χα Kt . A tuple

{(ctj , stj )Lj=1 , Kt+1 , Dt+1 } constitutes a time t temporary equilibrium if j

j

(i) for every j = 1, · · · , L, (ct , st ) is a solution to the consumer problem (7) at wt and wt+1 given by (3) with K = Kt and j K = Kt+1 respectively, τ t and τt+1 given by (6), (r¯, w¯ ) coming from profit maximization of firms (2) and ct = Lj=1 ct /L considered as exogenous by agent j; j (ii) Kt+1 + Dt+1 = Lj=1 st > 0; χ (iii) Dt+1 = α Kt+1 . Note that physical capital and public debt are perfect substitutes for agents as wealth owners. When γ is too high (e.g., γ is close to 1), no temporary equilibrium exists. To guarantee their existence, we need to j j j be sure that ct − γ ct > 0, j = 1, . . . , L, for tuples {(ct , st )Lj=1 , Kt+1 , Dt+1 } that are suspected to be equilibria. We show in 7 Similar to Alvarez-Cuadrado and Long (2012), introducing offspring-related positional concerns into the model gives essentially the same results when consumption-related positional concerns are stronger.

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Appendix A that the sufficient condition for this to occur is8

γ < γ (χ ) ≡

[δ ( α + χ ) + 1 + χ ] ( 1 − α ) . δ (α + χ )(1 − α ) + (1 + χ )2

(8)

Moreover, if it holds, then a time t temporary equilibrium is unique. j j j Now suppose that we are given {(s−1 )Lj=1 , K0 , D0 } such that s−1 ≥ 0 for all j and K0 + D0 = Lj=1 s−1 > 0. A recursively ∞ of temporary equilibria is called an intertemporal equilibrium starting from constructed sequence {(ct , st )Lj=1 , Kt+1 , Dt+1 }t=0 j

j

j {(s−1 )Lj=1 , K0 , D0 }. Its existence and uniqueness follows from the existence and uniqueness of temporary equilibria.

We next turn to balanced-growth equilibria, where all variables grow at the same constant rate. Consider a tuple

{(s j )Lj=1 , K, D} and a number g > −1. We call {(s j )Lj=1 , K, D} a balanced-growth equilibrium with the rate of balanced growth g j j ∞ given for all t = 0, 1, . . . by if the sequence {(ct , st )Lj=1 , Kt+1 , Dt+1 }t=0 j j ct+1 = (1 + g)ctj , stj = (1 + g)st−1 , j = 1, . . . , L,

Kt+1 = (1 + g)Kt , Dt+1 = (1 + g)Dt , is an intertemporal equilibrium starting from {(s j )Lj=1 , K, D}. Taking into account (6), we know that the tax rate in this equilibrium must be equal to

τ=

χ (r¯ − g) . (1 + χ )(1 + r¯ )

To describe the structure of balanced-growth equilibria, we define the following threshold:

γ ∗ (χ ) ≡

δ (1 − α ) . δ (1 − α ) + 1 + χ

(9)

It is easy to see that the right-hand side of (9) increases with the share of labor income (1 − α ) and the degree of altruism (δ ) but decreases with the public debt-to-GDP ratio (χ ). Also, one can easily verify that

γ ∗ ( χ ) < γ ( χ ), χ ≥ 0. It turns out that the long-run properties of the economy crucially depends on whether γ < γ ∗ (χ ) or γ > γ ∗ (χ ). If γ > γ ∗ (χ ), then there are multiple balanced-growth equilibria. In these equilibria, the set of dynasties is divided into two groups. One group leaves positive bequests and owns the entire capital stock and public debt. All other dynasties leave no bequests, and the members of these dynasties spend their disposable income (represented by the after-tax wage) only on personal consumption. For this reason, we can interpret the first group of dynasties as the rich, the second group as the poor, and corresponding balanced-growth equilibria as two-class equilibria. What is important is that two-class equilibria with any proportion between the rich and the poor are possible, except for the degenerate equilibrium where all dynasties are poor.9 For given γ and χ , the rate of growth associated with a balanced-growth equilibrium is fully determined by the share of the rich in the population, which we denote by m. It is given by

g( γ , χ , m ) ≡

δ (1 + r¯ )[α + χ + m(1 − α − (1 + χ )γ )] − 1. (1 + δ + χ )(α + χ ) + m[(1 + χ )(1 − α − δγ α ) − (1 − α )δχ ]

Conversely, if γ < γ ∗ (χ ), then there is a unique balanced-growth equilibrium. In this equilibrium, all agents in the economy leave positive bequests to their offspring and, moreover, these bequests are the same. Therefore we can treat this equilibrium as egalitarian. Since there are no poor agents, the population share of the rich is equal to 1 and the economy grows at the rate g(γ , χ , 1). The following proposition formalizes the above description of the structure of balanced-growth equilibria. Proposition 1. If γ < γ ∗ (χ ), then there exists a unique (up to a constant) balanced-growth equilibrium {(s j )Lj=1 , K, D}. In this equilibrium,

sj 1 = , j = 1, . . . , L, K+D L and the associated rate of growth is

g = g( γ , χ , 1 ) . 8 One can verify that, if the population is sufficiently large and γ > γ (χ ), the consumer problem has no solution for at least one j, and therefore there exists no temporary equilibrium. 9 The two-class equilibrium where all dynasties are rich is possible, however. It occurs in a very special case when the initial wealth of all dynasties is identical at t = −1.

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If γ > γ ∗ (χ ), then for any non-empty subset J ⊆ {1, . . . , L} with cardinality |J|, there exists a unique (up to a constant) balancedgrowth equilibrium {(s j )Lj=1 , K, D}. In this equilibrium,

sj 1 = , j ∈ J; s j = 0, j ∈ / J, K+D |J | and the associated rate of growth is

g = g(γ , χ , |J|/L ). Proof. It follows from Proposition 2.



In the rest of the section, we characterize the dynamic and asymptotic properties of the equilibrium distribution of wealth across dynasties. These properties depend on the relationship between γ and γ ∗ (χ ) and determine the structure of balanced-growth equilibria. In particular, we show that if γ < γ ∗ (χ ), then any intertemporal equilibrium converges to the egalitarian balanced-growth equilibrium. In other words, the wealth distribution tends asymptotically toward equality regardless of its initial state. Conversely, if γ > γ ∗ (χ ), then, from some time onward, an intertemporal equilibrium path coincides with a two-class balanced-growth equilibrium. All dynasties stop leaving bequests (i.e., save nothing) and become poor except for the dynasties which were the richest in the beginning (at t = 0). j j j ∞ Consider an intertemporal equilibrium {(ct , st )Lj=1 , Kt+1 , Dt+1 }t=0 starting from {(s−1 )Lj=1 , K0 , D0 }. Without any loss of generality, let us put s−1 in order such that the number of the richest dynasties at t = 0 is equal to L ≤ L. Then we have j





s1−1 = s2−1 = · · · = sL−1 > sL−1+1 ≥ · · · ≥ sL−1 . j

j

i Since, at each subsequent period, agents differ only in bequests left by their ancestors, st−1 ≥ st−1 implies st ≥ sti . That is the

order of dynasties according to their accumulated wealth remains the same. Moreover, if Hence, we obtain 

j st−1

i > st−1 and

j st

j

> 0, then st > sti .



st1 = st2 = · · · = stL > stL +1 ≥ · · · ≥ stL

(10)

for all t = 0, 1, . . .. Denote the number of agents who leave positive bequests in period t = −1, 0, 1, . . . by Mt , so that

stj > 0, j = 1, . . . , Mt ; stj = 0, j = Mt + 1, . . . , L, and their population share by mt :

mt ≡ Mt /L. Following Mankiw (20 0 0), we will call the agents leaving positive bequests savers. Now we can describe the dynamics of wealth inequality. The next proposition should be read as follows. In the case where γ < γ ∗ (χ ), irrespective of the initial distribution of wealth, all agents are savers starting from time t = 0, and eventually the economy converges to the egalitarian balanced-growth equilibrium. On the contrary, in the case where γ > γ ∗ (χ ), the number of savers is non-increasing over time, and eventually only the agents belonging to the dynasties that were the richest at t = 0 leave positive bequests and own the entire capital stock and public debt. Proposition 2. If γ < γ ∗ (χ ), then

Mt = L (or mt = 1 ), t = 0, 1, . . . ,

lim

t→∞

stj = 1, j = 1, . . . , L. (Kt+1 + Dt+1 )/L

(11)

(12)

∞ is non-increasing over time and there exists T such that for t = T , T + 1, . . . , If γ > γ ∗ (χ ), then the sequence {Mt }t=0





Mt = L or mt = L /L , j st−1 =

Kt + Dt j , j = 1, . . . , L ; st−1 = 0, j = L + 1, . . . , L. L

Proof. See Appendix B.

(13)

(14)



It is noteworthy that the number of savers can increase only in the case where γ < γ ∗ (χ ) and only at time t = 0. This happens if at time t = −1 some agents have no savings. In all other cases, the number of savers cannot increase. As we show in the proof of Proposition 2, if the number of savers does not decrease at some time t, i.e., Mt ≥ Mt−1 (mt ≥ mt−1 ), then

Kt+1 = (1 + g(γ , χ , mt ))Kt , Dt+1 = (1 + g(γ , χ , mt ))Dt , Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029

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and for all j = 1, . . . , Mt , j st−1 stj = A(γ , χ , mt ) + B(γ , χ , mt ) , (Kt+1 + Dt+1 )/L (Kt + Dt )/L

where the functions A :  → R and B :  → R++ are defined on

 ≡ {(γ , χ ) : 0 < γ < γ (χ ), 0 < χ < +∞} × [0, 1] and have the following properties: (i) for any m ∈ (0, 1],

γ ≶ γ ∗ ( χ ) ⇔ A ( γ , χ , m ) ≷ 0 ⇔ B ( γ , χ , m ) ≶ 1;

(15)

(ii) for any χ ≥ 0 and 0 ≤ γ < γ (χ ), the solution to the equation

x = A ( γ , χ , m ) + B ( γ , χ , m )x is given by

x = 1/m. In particular, if γ < γ ∗ (χ ), then the growth rate of the economy does not depend on the initial wealth distribution; from time t = 0 onwards, it is equal to g(γ , χ , 1):

Kt+1 = (1 + g(γ , χ , 1 ))Kt , Dt+1 = (1 + g(γ , χ , 1 ))Dt , t = 0, 1, . . . . As for the dynamics of wealth inequality in this case, for all t = 0, 1, . . . , we have j st−1 stj = A (γ , χ , 1 ) + B (γ , χ , 1 ) , j = 1, . . . , L, (Kt+1 + Dt+1 )/L (Kt + Dt )/L

and hence the speed of convergence to the egalitarian balanced-growth equilibrium is determined by B(γ , χ , 1). A higher value of B(γ , χ , 1) implies a lower speed of convergence. The case where γ > γ ∗ (χ ) is radically different. Until some time period, the group of savers remains the same as at time t = T0 ≡ −1. More precisely, there is a time T1 such that Mt = M−1 for all t = −1, 0, . . . , T1 − 1. Some agents drop out from the group of savers at time T1 , and there is a time T2 such that Mt = MT1 for all t = T1 , T1 + 1, . . . , T2 − 1, and so on. Eventually, there is a time TS such that, from TS onwards, only those who were the richest in the beginning (i.e., at t = 0) continue to leave positive bequests: Mt = L for all t = TS , TS + 1, . . .. For each time span from Ts to Ts+1 − 1 with s < S, the number of savers remains constant and the wealth share of dynasty j, whose bequest is positive at time Ts , evolves according to j st−1 stj = A(γ , χ , mTs ) + B(γ , χ , mTs ) . (Kt+1 + Dt+1 )/L (Kt + Dt )/L

If dynasty j is currently richer than the average agent in the group of savers, that is,



j st−1 > (Kt + Dt )/MTs



j st−1

1 or > (Kt + Dt )/L mTs

,

dynasty j increases its share in the aggregate wealth.10 In contrast, if dynasty j is currently poorer, that is,



0<

j st−1

< (Kt + Dt )/MTs

j st−1

1 or 0 < < (Kt + Dt )/L mTs



,

its share in the aggregate wealth falls. Thus, the economy is characterized by wealth divergence. The speed of divergence over the time span from Ts to Ts+1 − 1 is determined by B(γ , χ , mTs ). A higher value of B(γ , χ , mTs ) implies a higher speed of divergence. Note that starting from time TS , the economy grows at the constant rate g(γ , χ , L /L):





Kt+1 = 1 + g

    γ , χ , L /L Kt , Dt+1 = 1 + g γ , χ , L /L Dt , t = TS , TS + 1, · · · .

10 Interestingly, as long as this dynasty is poorer than the richest in the beginning, its share in the aggregate wealth will increase for some time and yet it will eventually decrease to zero. The dynamics of the wealth share can therefore be non-monotonic.

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Fig. 1. Effect of γ on growth in two types of balanced-growth equilibrium.

4. Changes in the debt-to-GDP ratio χ The main objective of this section is to examine how the debt-to-GDP ratio affects the rate of growth and the dynamics of wealth distribution. To do this, we first need to understand the effects of other parameters in the model. The rate of balanced growth, g(γ , χ , m), depends on the strength of positional concerns, γ , the debt-to-GDP ratio, χ , and the population share of the rich, m. The dependence of the growth rate on γ is negative:

∂ g( γ , χ , m ) < 0, ∂γ i.e., weaker positional concerns lead to a higher growth rate g. As for the dependence of the balanced growth rate on inequality, first note that its dependence on the population share of the rich is negative:

∂ g( γ , χ , m ) < 0 if γ > γ ∗ (χ ). ∂m

(16)

In other words, the growth rate is greater in the two-class equilibrium with a lower m than with a higher m. The intuition is simple. A smaller population share of the rich means that the entire capital stock and public debt are distributed among a smaller number of rich agents. A “representative” rich agent then becomes wealthier, and hence she consumes a smaller fraction of her wealth. This, in turn, increases the aggregate saving rate in the economy and the rate of growth. Thus, (16) implies that the higher is the level of inequality, the higher is the growth rate. All this does not, however, mean that higher inequality is always associated with a higher growth rate. Indeed, it is readily verified that

γ ≶ γ ∗ ( χ ) ⇔ g( γ , χ , 1 ) ≷ g( γ , χ , 0 ) . As was noted above, g(γ , χ , 1) is decreasing in γ for all χ ≥ 0, so the rate of growth in the egalitarian balanced-growth equilibrium obtained at γ < γ ∗ (χ ) must be higher than that in any two-class equilibrium. The relationship among the strength of positional concerns, inequality and growth is illustrated in Fig. 1, where the shaded area corresponds to various two-class equilibria. At this point, we can present the main results of the paper and answer the question of what impact public debt has on growth and wealth distribution. Observe that both γ¯ (χ ) and γ ∗ (χ ) are decreasing in χ . Fig. 2 reflects this fact. It shows that an increase in the debt-to-GDP ratio leads to a broader range of γ over which a temporary equilibrium may fail to exist and a narrower range of γ over which the wealth distribution always converges to full equality. Also, for any m ∈ (0, 1], we have

∂ B (γ , χ , m ) δ > 0 if γ < . ∂χ 1+δ

(17)

Recall that if γ < γ ∗ (χ ), then B(γ , χ , 1) determines the speed of wealth convergence (a lower B(γ , χ , 1) means higher speed of convergence), while if γ > γ ∗ (χ ), then B(γ , χ , m) determines the speed of wealth divergence (a higher B(γ , χ , m) means higher speed of divergence). The implications of these findings are as follows: (1) When γ > γ¯ (0 ) (e.g., γ = γ1 in Fig. 2), a temporary equilibrium may not exist. Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029

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Fig. 2. Effects of γ and χ on the type of balanced-growth equilibrium.

(2) When γ ∗ (0 ) < γ < γ¯ (0 ) (e.g., γ = γ2 in Fig. 2), there is a threshold χ¯ (γ ) ≡ γ¯ −1 (γ ). If the debt-to-GDP ratio is below it, then the economy eventually settles on a two-class balanced-growth equilibrium. If, in addition, γ < δ /(1 + δ ), we know that a higher χ leads to a faster wealth divergence. All values of the debt-to-GDP ratio above χ¯ (γ ) may result in non-existence of temporary equilibria. (3) When 0 < γ < γ ∗ (0) (e.g., γ = γ3 in Fig. 2), there is a threshold χ ∗ (γ ) ≡ (γ ∗ )−1 (γ ). If the debt-to-GDP ratio is below it, then the economy converges to the egalitarian balanced-growth equilibrium, and the closer χ to the threshold, the slower the convergence. If the debt-to-GDP ratio is above χ ∗ (γ ), then the economy eventually settles on a two-class balanced-growth equilibrium with the speed of wealth divergence increasing in χ . Again, if χ becomes too high, we obtain possible non-existence of temporary equilibria. To answer the question about the impact of the debt-to-GDP ratio on the rate of growth, suppose we are given γ < γ ∗ (0). It is straightforward to verify that in this case the dependence of the growth rate on χ is negative for any given population share of the rich, m ∈ [0, 1]:

∂ g( γ , χ , m ) < 0 if γ < γ ∗ (0 ). ∂χ

(18)

Hence, public debt has a negative effect on growth, as illustrated in Fig. 3 (again, the shaded area corresponds to various two-class equilibria). More importantly, we see that the rates of balanced growth are always higher for the debt-to-GDP ratio that gives rise to the egalitarian regime than for the debt-to-GDP ratio leading to the two-class regime. 5. Discussion and conclusion In this paper, we propose an AK endogenous-growth model with public debt financed by distortionary income taxes. Our central question is how public debt affects wealth inequality across altruistic dynasties that evolves endogenously over time and that arises through the positional externality. Dynasties are identical in every respect except for their initial wealth. We show that if positional concerns are not too strong, a low level of the debt-to-GDP ratio ensures that the economy converges to a unique egalitarian equilibrium which exhibits perfect equality and a high growth rate. In contrast, when the debt-to-GDP ratio is above some threshold, the economy eventually settles on a two-class equilibrium, with poor dynasties saving nothing in the long run and rich dynasties which own the entire capital stock and public debt and which were the richest in the beginning. Since the initial endowments of capital and government bonds can be arbitrarily distributed, any proportion between the rich and the poor is possible in the two-class regime. The intuition behind the results is as follows. In our model, the bequest motive works in favor of equality, while the consumption externality works in favor of inequality. If the strength of the former is higher than that of the latter, then the economy is in the egalitarian regime; otherwise, it is in the two-class regime. An increase in public debt crowds out investment in capital stock, leaving more output for consumption each period and hence pushing up the average consumption. Then, for a given strength of positional concerns, all dynasties increase their consumption level and decrease their savings rates. Moreover, the decrease in savings rates is more pronounced for relatively poor dynasties. Thus, public debt works in favor of inequality, and if the debt-to-GDP ratio becomes large enough, the economy may end up in a two-class equilibrium. Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029

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Fig. 3. Effect of χ on growth in two types of balanced-growth equilibrium.

The analysis presented here relies on several simplifying assumptions and some modifications of the model are worth discussing. This concerns, in particular, the tax instruments and the specification for an individual’s relative consumption. In the model, the government has access only to a proportional (linear) tax. What if progressive (non-linear) income taxation is available? A natural example can be one where the tax rate on interest income is higher than that on wage earnings. Characterizing the dynamics of wealth distribution with two different tax rates is a complex task. However, it is not difficult to show that the results regarding balanced-growth equilibria remain qualitatively unchanged in this more general case, even if the tax rate imposed on labor or interest income is zero. Introducing positional preferences in our model, we adopt a subtractive specification of relative consumption. Another popular specification in the literature is multiplicative (see, e.g., Abel (2005)). One can show that for the multiplicative specification, a balanced-growth equilibrium does not generically exist in the model. However, it can also be proved that if we adopt a special version of the multiplicative specification which is compatible with balanced growth, the result about the threshold separating the egalitarian and two-class regimes will still hold. Our paper provides two main policy implications which depend on the starting situation we consider. If the economy is in the two-class regime initially, a decrease in the public debt-to-GDP ratio will not only lead to a higher rate of growth, but it may also cause the economy to switch to the egalitarian regime. Thus, there is no trade-off between equality and efficiency in long-run public debt policy. Another implication of the model is that policies aimed at alleviating inequality using public debt may, in fact, increase wealth inequality in the long run. Indeed, suppose that the economy is in the egalitarian regime, but the initial distribution of wealth is uneven. If the government wants to help poor agents and finances this by increasing public debt, eventually the economy can settle on the two-class equilibrium where the level of wealth inequality will be higher than the initial one. Finally, recent empirical evidence provided by Azzimonti et al. (2014) and Arawatari and Ono (2017), who find a significant positive correlation between inequality and public debt, seems to support our theoretical results. Further empirical verification constitutes an interesting and important issue for future research. Appendix A. Temporary equilibrium The aim of this Appendix is to prove the existence and uniqueness of the time t temporary equilibrium. Suppose that wt+1 > 0 and consider the consumer problem (7). It has a solution if and only if

  j γ ct < (1 − τt ) wt + (1 + r¯ )st−1 .

When

(A.1)

  wt+1 j γ ct < (1 − τt ) wt + (1 + r¯ )st−1 − , δ (1 + r¯ )

this solution is given by

stj =

j δ (1 − τt )(wt + (1 + r¯ )st−1 ) − δγ ct wt+1 − , 1+δ (1 + δ )(1 + r¯ )

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ctj =

11

  j (1 − τt ) wt + (1 + r¯ )st−1 + δγ ct wt+1 + , 1+δ (1 + δ )(1 + r¯ )

and when

  wt+1 j γ ct ≥ (1 − τt ) wt + (1 + r¯ )st−1 − , δ (1 + r¯ )

it is given by





j stj = 0, ctj = (1 − τt ) wt + (1 + r¯ )st−1 .

Thus, the solution to (7) can be written as

stj =



δ

1+δ

and

j max 0, (1 − τt )(wt + (1 + r¯ )st−1 ) − γ ct −



wt+1 δ (1 + r¯ )

(A.2)



j ctj = (1 − τt ) wt + (1 + r¯ )st−1 − stj .

(A.3)

j j Lemma 1. Suppose wt+1 > 0. A tuple {(ct , st )Lj=1 , Kt+1 , Dt+1 } is a time t temporary equilibrium if and only if (i) Dt+1 = χα Kt+1 ; (ii) Kt+1 ∈ (0, aKt ) is a solution to

(K ) = 0, where

(A.4)

       L K 1−α K χ δ  1 + r¯ χ K  1 − α Kt 1 + r¯ Kt j (K ) ≡ 1 + K− max 0, + + st−1 − γ − − ; α 1+δ 1+χ 1 + χ Kt α L α L L δα L j=1 

j

j

(iii) for all j = 1, . . . , L, (ct , st )Lj=1 is determined by

stj = and

 

δ

1+δ

 ctj =

max 0,

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

(iv) the inequality



γ

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

Kt+1 1 + r¯ Kt − α L L



 <

 1−α K

α

t

L





j + st−1 −γ

Kt+1 1 + r¯ Kt − α L L

 −

1 − α Kt+1 δα L

 (A.5)



t

α

 1−α K

j + st−1 − stj ;

L

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

(A.6)

 1−α K

t

α

L

j + st−1

 (A.7)

holds. Proof. Consider a temporary equilibrium {(ct , st )Lj=1 , Kt+1 , Dt+1 } in period t and characterize its properties. j

j

First, the equality Dt+1 = χα Kt+1 holds by definition. j

j

Second, we know that, given wt+1 > 0, (ct , st )Lj=1 is a solution to (7) for each j if and only if (A.1) holds. This solution

α is given by (A.2) and (A.3). Using wt = (1 − α )Yt /L and the equality α = (1 + r )K/Y , we obtain wt = 1− α (1 + r¯ )Kt /L. Then, together with (6), the inequality (A.1) is equivalent to



γ ct <

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

 1−α K α

t

L

In turn, (A.2) can be written as

stj

=

δ

1+δ

 

max 0,



j + st−1 .

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

 1−α K α

and (A.3) as (A.6). j Next, combining ct = Lj=1 ct /L and (A.6), we obtain

 ct =

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt



(A.8)

1 − α Kt  j + st−1 /L α L L

i=1

t

L

+

 −

j st−1

L 



1 − α Kt+1 − γ ct − δα L

 (A.9)

stj /L.

i=1

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L

j j=1 st−1

By the definition of temporary equilibrium,

 ct =

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

 1−α K

t

α

L



+ 1+

j = (1 + χα )Kt and Lj=1 st = (1 + χα )Kt+1 . Therefore,

χ  Kt   χ  Kt+1 1 + r¯ Kt Kt+1 − 1+ = − . α L α L α L L

Substituting this equation into (A.8) and (A.9) yields (A.5) and (A.7). j Finally, since Lj=1 st = (1 + χα )Kt+1 > 0, we have

       L χ δ  1 + r¯ χ Kt+1  1 − α Kt 1 + r¯ Kt Kt+1 1 − α Kt+1 j 1+ K − max 0, + + st−1 − γ − − = 0, α t+1 1 + δ j=1 1+χ 1 + χ Kt α L α L L δα L



which means that Kt+1 > 0 is the solution to (A.4). This proves the “only if” part of the lemma. The inverse argument proves the “if” part of the lemma.  Let us now prove that if the inequality (8) holds, then for any {(st−1 )Lj=1 , Kt , Dt } such that st−1 ≥ 0, j = 1, . . . , L and j j j Kt + Dt = Lj=1 st−1 > 0, there exists a unique time t temporary equilibrium {(ct , st )Lj=1 , Kt+1 , Dt+1 }. By Lemma 1, it is sufficient to show that (i) equation (A.4) has a unique positive solution; (ii) if Kt+1 is its solution and j st is determined by (A.5) for each j = 1, . . . , L, then (A.7) holds. j j From Kt + Dt = (1 + χα )Kt = Lj=1 st−1 it is clear that there exists j such that st−1 ≥ (1 + χα )Kt /L. Then, for this j we have j



1 − α K

1 + r¯ max 0, 1+χ

t

α

+

L

j st−1



1 + r¯ Kt −γ α L



j



1 − α K

≥

1 + r¯ 1+χ

≥

1 + r¯ 1 − α Kt χ + 1+ 1+χ α L α



α

j + st−1 −γ

t

L



1 + r¯ Kt α L

K  t

L

−γ

1 + r¯ Kt 1 + r¯ K = ( 1 − γ ) t > 0. α L α L

Since for all other j,

 max 0,

1 + r¯ 1+χ

1 − α K

t

α

L



j + st−1 −γ

1 + r¯ Kt α L

 ≥ 0,

we obtain L 

 max 0,

j=1

1 + r¯ 1+χ

1 − α K α

t

L



j + st−1 −γ

1 + r¯ Kt α L

 >0

and hence (0) < 0. It is clear that the function (K) is increasing. Furthermore,

 (aKt ) = −

1 + r¯

α

δ

1+δ

 Kt

L 

χ  1 + r¯ K α α t  



= 1+ max 0,

j=1

1 + r¯ χ 1 + r¯ + 1+χ 1+χ α

 1−α K

t

α

L



j + st−1 −



1 − α 1 + r¯ Kt . δα α L

r¯ Since (1 + χα ) 1+ α Kt > 0 and



L χ  1 + r¯ δ  1+ Kt − α α 1+δ j=1    1 + r¯ 1 + δ χ

=

1+δ

α



1 + r¯ χ 1 + r¯ + 1+χ 1+χ α

 1−α K α

t

L

+

j st−1



1 − α 1 + r¯ Kt − δα α L



 χ  1 − α χ 1−α (1 + r¯ )(1 + χ ) 1+ − 1+ +1+ + Kt = Kt > 0, α 1+χ α α α α2 ( 1 + δ )α 2 δ



we obtain that (aKt ) > 0. Thus, the equation (K ) = 0 has a unique positive solution in the interval (0, aKt ). Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029

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13

Now suppose that Kt+1 is the solution to (A.4). Let us show that



γ

Kt+1 1 + r¯ Kt − α L L





<

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt



1 − α Kt α L

(A.10)

j

and hence (A.7) holds. Let st be determined by (A.5) for all j = 1, . . . , L. We have

        L  Kt+1 1 − α Kt+1 χ 1 + r¯ χ Kt+1  1 − α Kt 1 + r¯ Kt j (1 + δ ) 1 + Kt+1 = δ max 0, + + st−1 −γ − − α 1+χ 1 + χ Kt α L α L L δα L j=1      L  Kt+1 1 − α Kt+1 1 + r¯ χ Kt+1  1 − α Kt 1 + r¯ Kt j ≥ δ + + st−1 − δγ − − 1+χ 1 + χ Kt α L α L L α L j=1      1−α 1 + r¯ χ Kt+1 1 − α χ  1 + r¯ =δ + Kt + 1 + K − δγ K − Kt+1 − K . 1+χ 1 + χ Kt α α t α t α t+1

By rearranging,

Kt+1 ≥

δ (1 + r¯ )(1 − γ ) K. δα (1 − γ ) + 1 + χ t

Then the left-hand side of (A.10) is less than or equal to

 γ 1−

 1 + r¯ K δα (1 − γ ) γ (1 + χ ) 1 + r¯ Kt t = , δα (1 − γ ) + 1 + χ α L δα (1 − γ ) + 1 + χ α L

while the right-hand side of (A.10) is greater than or equal to



1 + r¯ 1+ 1+χ

 δχ (1 − γ ) 1 − α Kt . δα (1 − γ ) + 1 + χ α L

Using the easily verifiable fact that

γ < γ (χ ) ⇔ we get



γ

  γ (1 + χ ) 1−α δχ (1 − γ ) < 1+ , δα (1 − γ ) + 1 + χ 1 + χ δα (1 − γ ) + 1 + χ

Kt+1 1 + r¯ Kt − α L L



  γ (1 + χ ) 1 + r¯ Kt 1 + r¯ δχ (1 − γ ) 1 − α Kt ≤ < 1+ δα (1 − γ ) + 1 + χ α L 1+χ δα (1 − γ ) + 1 + χ α L   1 + r¯ χ Kt+1 1 − α Kt ≤ + . 1+χ 1 + χ Kt α L

This proves (A.10). Appendix B. Proof of Proposition 2 Let

     1−α δ 1 + r¯ 1−α 1 + r¯ +χ −γ −α − , (1 + δ )(α + χ ) 1 + g( γ , χ , m ) 1+χ 1 + g( γ , χ , m ) δ   δ 1 + r¯ B (γ , χ , m ) ≡ +χ . (1 + δ )(1 + χ ) 1 + g(γ , χ , m ) A (γ , χ , m ) ≡

Lemma 2. If Mt ≥ Mt−1 for some t, then

Kt+1 = (1 + g(γ , χ , mt ))Kt , Dt+1 = (1 + g(γ , χ , mt ))Dt ,

(B.1)

j st−1 stj = A(γ , χ , mt ) + B(γ , χ , mt ) , j = 1, . . . , L. (Kt+1 + Dt+1 )/L (Kt + Dt )/L

(B.2)

∞ starting from (s )L Proof. Consider an intertemporal equilibrium {(ct , st )Lj=1 , Kt+1 , Dt+1 }t=0 such that −1 j=1 suppose Mt ≥ Mt−1 . By (A.5), for all j = 1, . . . , Mt , j

stj =

δ

1+δ



1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

j

 1−α K α

t

L

j



j + st−1 −γ



Kt+1 1 + r¯ Kt − α L L



L

j j=1 s−1

> 0, and



−

1 − α Kt+1 . δα L

(B.3)

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Also, since Mt ≥ Mt−1 , we have

Mt−1 j=1

j

st−1 =

Mt

j s j=1 t−1

= (1 + χα )Kt . Hence,

Mt   χ (1 + δ ) 1 + Kt+1 = (1 + δ ) stj α j=1     Mt   Kt+1 1 − α Kt+1 1 + r¯ χ Kt+1  1 − α Kt 1 + r¯ Kt j = δ + + st−1 − δγ − − 1+χ 1 + χ Kt α L α L L α L j=1     1−α δ 1−α χ 1 + r¯ = [(1 + r¯ )Kt + χ Kt+1 ] mt + 1 + − δγ m K − mt Kt+1 − mt Kt+1 1+χ α α α t t α

or, equivalently,

δ (1 + r¯ )[α + χ + mt (1 − α − (1 + χ )γ )] K = (1 + g(γ , χ , mt ))Kt . (1 + δ + χ )(α + χ ) + mt [(1 + χ )(1 − α − δγ α ) − (1 − α )δχ ] t Furthermore, Dt+1 = (1 + g(γ , χ , mt ))Dt because of (5). Thus, we have proved (B.1). Kt+1 =

Substituting (B.1) for Kt+1 into (B.3), we obtain

stj =



δ

1+δ



−

γ

1 + r¯ χ + (1 + g(γ , χ , mt )) 1+χ 1+χ

1 + r¯ Kt Kt − (1 + g(γ , χ , mt )) α L L

 1−α K

t

α



L

j + st−1

− (1 + g(γ , χ , mt ))



1 − α Kt δα L



for all j = 1, . . . , Mt . By dividing both sides of this equality by (Kt+1 + Dt+1 )/L = (1 + g(γ , χ , mt ))(Kt + Dt )/L and rearranging, we obtain (B.2).  To complete the proof of Proposition 2, consider two cases: γ < γ ∗ (χ ) and γ > γ ∗ (χ ). B.1. Case 1: γ < γ ∗ (χ ) First, we show that

Kt+1 = (1 + g(γ , χ , 1 ))Kt

(B.4)

is the solution to (A.4). Indeed, for Kt+1 given by (B.4), we have



δ

1+δ ≥

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt



δ

1+δ

 1−α K

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt



α 

δ



1+δ

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

L

α

t

L

Kt+1 1 + r¯ Kt − α L L

Kt+1 1 + r¯ Kt − α L L



j + st−1 −γ

for all j = 1, . . . , L, which implies that





−

Kt+1 1 + r¯ Kt − α L L

 −

1 − α Kt+1 δα L

1 − α Kt+1 δα L

 −





1 − α Kt+1 δα L

 >0

   1−α χ  1 + r¯ K + 1+ K −γ K − Kt+1 − K α t α t α t δα t+1   1−α 1 + r¯ δχ = δ (1 − γ ) K + + δγ − Kt+1 . α t α α

χ (1 + δ ) 1 + K =δ α t+1 





χ  Kt . α L

 1−α K





j + st−1 −γ

1 − α Kt −γ α L

= A(γ , χ , 1 )(1 + g(γ , χ , 1 )) 1 + Therefore, due to (15) with m = 1,

t

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt

 1−α



This proves that Kt+1 given by (B.4) is the solution to (A.4). Thus, using Lemma 2 and (5), we obtain that Kt+1 = (1 + g(γ , χ , 1 ))Kt and Dt+1 = (1 + g(γ , χ , 1 ))Dt for all t = 0, 1, . . .. Consequently, (A.5) becomes



stj = (1 + g(γ , χ , 1 )) 1 +

  χ  Kt j max 0, A(γ , χ , 1 ) + B(γ , χ , 1 )st−1 . α L

It follows from (15) that A(γ , χ , 1) and hence st are strictly positive for all j = 1, . . . , L, thereby proving (11). Finally, by Lemma 2, (11) and (15) with m = 1, it can be checked that j

lim

t→∞

j st−1

(Kt + Dt )/L

=

A (γ , χ , 1 ) = 1, j = 1, . . . , L, 1 − B (γ , χ , 1 )

which proves (12). Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029

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15

B.2. Case 2: γ > γ ∗ (χ ) ∞ is non-increasing, i.e., s First, we show that the sequence {Mt }t=0 = 0 implies st = 0 for all j = 1, . . . , L and t = 0, 1, . . .. t−1 j

j

j

j

Assume the converse. Then there are j and t such that st−1 = 0 implies st > 0. The equation (A.5) is rewritten as

stj =

δ

1+δ

 

max 0,

1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt



1 − α Kt −γ α L

for such j and t. Furthermore, the following inequality holds:



1 + r¯ χ Kt+1 + 1+χ 1 + χ Kt



1 − α Kt −γ α L



Kt+1 1 + r¯ Kt − α L L



 −

Kt+1 1 + r¯ Kt − α L L



−

1 − α Kt+1 δα L



1 − α Kt+1 > 0. δα L

j

Accordingly, st > 0 for all j = 1, . . . , L and hence Mt = L > Mt−1 . Applying Lemma 2 with mt = 1, we have





1 + r¯ χ 1 − α Kt + (1 + g(γ , χ , 1 )) −γ 1+χ 1+χ α L − ( 1 + g( γ , χ , 1 ) )



1 + r¯ Kt Kt − (1 + g(γ , χ , 1 )) α L L



1 − α Kt = A(γ , χ , 1 )(1 + g(γ , χ , 1 )) 1 + δα L



 χ  1 Kt 1+ > 0. α δ L

This means that A(γ , χ , 1) must be strictly positive. By (15), this is equivalent to γ < γ ∗ (χ ), which is a contradiction of γ > γ ∗ (χ ). Thus, we obtain that Mt ≤ Mt−1 for all t = 0, 1, . . . and, hence, there exist M > 0 and T such that Mt = M for all t ≥ T. By Lemma 2, j st−1 stj = A (γ , χ , m ) + B (γ , χ , m ) , j = 1, . . . , M, t = T + 1, T + 2, . . . , (Kt+1 + Dt+1 )/L (Kt + Dt )/L j

where m = M/L. Taking into account (15), limt→∞

st−1 Kt /L+Dt /L

= ±∞ for all j = 1, . . . , M, if and only if

j st−1 stj = , t = T + 1, T + 2, . . . . (Kt+1 + Dt+1 )/L (Kt + Dt )/L

Therefore, one can check that j st−1

=

(Kt + Dt )/L

A (γ , χ , m ) 1 = , j = 1, . . . , M, t = T + 1, T + 2, . . . . 1 − B (γ , χ , m ) m

j j Finally, let us show that M = L . From (10) it is clear that if j ≤ L and i > L , then st > sti for each t. Since st > 0 for

j = 1, . . . , M and st = 0 for j = M + 1, . . . , L, we have M = L or m = L /L. Thus, (13) and (14) hold. As a corollary, combining (13) and (B.1) in Lemma 2, we obtain Kt+1 = (1 + g(γ , χ , L /L ))Kt and Dt+1 = (1 + g(γ , χ , L /L ))Dt (due to (5)) for all t = T + 1, T + 2, . . .. j

Appendix C. Proofs of (17) and (18) Some simple but tedious calculations are omitted for the sake of brevity. We begin by verifying (18). We have

∂ g( γ , χ , m ) δ (1 + r¯ ) = ∂χ {(1 + δ + χ )(α + χ ) + m[(1 + χ )(1 − α − δγ α ) − (1 − α )δχ ]}2 × {(1 − mγ )(1 + δ + χ )(α + χ ) + m(1 − mγ )[(1 + χ )(1 − α − δγ α ) − (1 − α )δχ ] − [α + χ + 1 + δ + χ + m(1 − α − δγ α − (1 − α )δ )][α + χ + m(1 − α − (1 + χ )γ )]} δ (1 + r¯ ){γ [mδ (1 − α )2 (1 − m ) + m(1 + χ )2 ] − mδ (1 − α )2 (1 − m ) − [m(1 − α ) + α + χ ]2 } = , {(1 + δ + χ )(α + χ ) + m[(1 + χ )(1 − α − δγ α ) − (1 − α )δχ ]}2 implying

∂ g( γ , χ , m ) mδ (1 − α )2 (1 − m ) + [m(1 − α ) + α + χ ]2 ≶0 ⇔ γ ≶ . ∂χ mδ (1 − α )2 (1 − m ) + m(1 + χ )2 Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029

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To prove (18), it is sufficient to show that

γ ∗ (0 ) =

δ (1 − α ) mδ (1 − α )2 (1 − m ) + [m(1 − α ) + α + χ ]2 < δ (1 − α ) + 1 mδ (1 − α )2 (1 − m ) + m(1 + χ )2

or, equivalently,

δ (1 − α )[mδ (1 − α )2 (1 − m ) + m(1 + χ )2 ] − [δ (1 − α ) + 1]{mδ (1 − α )2 (1 − m ) + [m(1 − α ) + α + χ ]2 } < 0. Rearranging and simplifying yields

−δ (1 − α )(1 − m )[(α + χ )2 + mα (1 − α )] − [m(1 − α ) + α + χ ]2 < 0, which proves (18). Next, we verify (17). We have

     1 + r¯ 1 ∂ B (γ , χ , m ) δ 1 ∂ g( γ , χ , m ) 1 + r¯ = 1− − + χ ∂χ 1+δ 1+χ ∂χ (1 + g(γ , χ , m ))2 ( 1 + χ ) 2 1 + g( γ , χ , m ) =

mδ (1 + r¯ )2 (1 + χ )2 (1 − α ){δ (1 − γ ) − γ − m[δγ (1 − γ ) − γ ]} . {(1 + δ + χ )(α + χ ) + m[(1 + χ )(1 − α − δγ α ) − (1 − α )δχ ]}2

Then

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Please cite this article as: K. Borissov and A. Kalk, Public debt, positional concerns, and wealth inequality, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2019.11.029