Public Debt

Abstract

During the 1980s and 1990s, we observed many public debt crises in Latin America and Asia. Prior to, during, and after the financial crisis of 2007–2008, we also observed many countries in the European Monetary Union (EMU) with severe public debt problems. Government debt has increased to unprecedented levels in the post-World War II era in the so-called “GIIPS” countries (Greece, Ireland, Italy, Portugal, and Spain). The Greek, Spanish, and Italian governments have had to pay premiums of 16, 3, and 3 percentage points on their public debt relative to Germany. Consequently, the president of the European Central Bank (ECB), Mario Draghi, initiated a program whereby the ECB extended 3-year loans in the amount of 1 trillion (!) euros to the Eurozone banking sector. Interest rates subsequently converged, but debt levels remain high and still amounted to 179%, 100%, and 132% of GDP in Greece, Spain, and Italy in 2015, almost 10 years after the onset of the crisis. As a second major second case of severe public debt, Japan had accumulated a gross public debt equal to approximately 240% of GDP by 2015.

Appendices

Appendix 7.1: Government Budget with Money Finance

In Sect. 7.3, we assumed that the government only finances its expenditures by means of taxes and public debt. In this appendix, we also consider money financing of the government budget. Therefore, we include high-powered money or central bank money $\tilde H_t$ in the government budget constraint (7.2):

$$\displaystyle \begin{aligned} \tilde H_{t+1}- \tilde H_t + P^B_t \tilde B_{t+1} - \tilde B_t = P_t G_t - P_t T_t.\end{aligned} $$

(7.58)

The revenue from money creation, $\tilde H_{t+1}- \tilde H_t$, is called seignorage and can be written in real terms as⁶⁵

$$\displaystyle \begin{aligned} S_t = \frac{\tilde H_{t+1}-\tilde H_t}{P_t} = (1+\pi_{t+1} )H_{t+1} - H_t= \pi_{t+1} H_{t+1} + \varDelta H_{t+1} ,\end{aligned}$$

where ΔH _t+1 = H _t+1 − H _t denotes the change in real central bank money with $H_t\equiv \tilde H_t/P_t$. Notice that only central bank money (inside money) is included in the definition of H _t, while outside money that is created in the banking sector with the help of credit does not provide revenues for the government.

With this approximation, the real government budget (7.3) is therefore given by

$$\displaystyle \begin{aligned} \frac{1}{1+r^B_t}B_{t+1} + \pi_{t+1} H_{t+1} + \varDelta H_{t+1} = B_t + G_t-T_t.\end{aligned} $$

(7.59)

Similarly, one can divide (7.59) by real GDP Y _t to derive

$$\displaystyle \begin{aligned} \frac{1+\gamma_{t+1}}{1+r^B_t} \frac{B_{t+1}}{Y_{t+1}} + \left( \gamma_{t+1} +\pi_{t+1} \right) \frac{H_{t+1}}{Y_{t+1}} + \varDelta \frac{H_{t+1}}{Y_{t+1}}= \frac{B_t}{Y_t} + \frac{G_t-T_t}{Y_t},\end{aligned} $$

(7.60)

where we have used the approximation πγ ≈ 0. In steady state with constant B∕Y  and H∕Y , revenues from seignorage (relative to GDP) become

$$\displaystyle \begin{aligned} \frac{S}{Y} = (\gamma+\pi) \frac{H}{Y}.\end{aligned}$$

Let us consider a rough approximation of real seignorage revenues for the US in 2016. For simplification, let us assume that the US is in steady state in 2016. Nominal GDP in 2016 amounted to $18,621 billion, while high-powered money was equal to $3,744 billion as of July 6, 2016, according to data from the Federal Reserve Bank of St. Louis.⁶⁶ The inflation rate (for consumer prices) and real GDP growth amounted to 2.18% and 0.81%, respectively, in the US economy during the period 2001–2016. Therefore, as a very crude approximation,

$$\displaystyle \begin{aligned} \frac{S}{Y}= (\gamma+\pi) \frac{H}{Y}\approx (0.0201+0.0081) \times 0.201 =0.60\%.\end{aligned}$$

This amount of seignorage is a very high value in the context of the post-World War II monetary history of the United States. Prior to the financial crisis of 2007–2008, high-powered money relative to GDP was much smaller. In the wake of this crisis and the start of the “Quantitative Easing” program of the Federal Reserve, however, high-powered money increased almost fivefold, from $848 billion on January 8, 2008, to $4,150 billion on September 17, 2014. Therefore, seignorage (relative to GDP) used to be much smaller in the US. In particular, King and Plosser (1985) report that seignorage only equalled 0.3% of GDP during the period 1952–1982.

Appendix 7.2: Computation of the Large-Scale OLG Model in Sect. 7.5

In this Appendix, we first define the stationary equilibrium for the model in Sect. 7.5 before we describe the computation of the transition.

2.1 Stationary Equilibrium

To describe the model in stationary variables, we define the following individual stationary variables:

$$\displaystyle \begin{aligned} \begin{aligned} &\tilde c_t^s \equiv \frac{c_t^s}{A_t},\;\; \tilde \omega^s_t \equiv \frac{\omega^s_t}{A_t}, \;\; \tilde k^s_t \equiv \frac{k^s_t}{A_t}, \;\; \tilde b^s_t \equiv \frac{b^s_t}{A_t}, \;\; \tilde tr_t \equiv \frac{tr_t}{A_t}, \;\; \tilde pen_t \equiv \frac{pen_t}{A_t},\\ & \tilde \lambda^s_t \equiv \frac{\lambda^s_t}{A_t^{-\sigma}}, \end{aligned} \end{aligned}$$

and aggregate stationary variables:

$$\displaystyle \begin{aligned} \begin{aligned} & \tilde k_t \equiv \frac{K_t}{A_t N_t}, \;\;\tilde y_t \equiv \frac{Y_t}{A_t N_t}, \;\;\tilde b_t \equiv \frac{B_t}{A_t N_t},\;\; \tilde beq_t \ \equiv \frac{Beq_t}{A_t N_t}, \;\;\tilde tax_t \equiv \frac{T_t}{A_t N_t},\\ & \tilde L_t \equiv \frac{L_t}{N_t}, \end{aligned} \end{aligned}$$

implying the factor prices

$$\displaystyle \begin{aligned} r_t &=\alpha \tilde k_t^{\alpha-1} \tilde L_t^{1-\alpha}-\delta, \end{aligned} $$

(7.61a)

$$\displaystyle \begin{aligned} w_t &= (1-\alpha) \tilde k_t^{\alpha} \tilde L_t^{-\alpha}. \end{aligned} $$

(7.61b)

Stationary non-capital income $\tilde x^s_t=x^s_t/A_t$ of the s-year-old household in period t is represented by:

$$\displaystyle \begin{aligned} \tilde x^s_t = \left\{ \begin{array}{ll} (1-\tau^w_t-\tau^p_t) w_{t} \bar y^s l^{s}_{t} & \quad s=1,\ldots,R-1,\\ \tilde pen_t & \quad s= R,\ldots,J. \end{array}\right. \end{aligned} $$

(7.62)

The stationary budget constraint of the household at age s = 1, …, R − 1 is given by

$$\displaystyle \begin{aligned} (1+\tau^c_t) \tilde c^{s}_{t} = \tilde x^s_t+ \left[1+(1-\tau^K_t) r_t \right] \tilde \omega^{s}_{t} + \tilde tr_{t} - (1+\gamma) \tilde \omega_{t+1}^{s+1},\end{aligned} $$

(7.63)

where individual wealth $\tilde \omega ^s_t$ is equal to the sum of the two assets: capital $\tilde k^s_t$ and government bonds $\tilde b^s_t$, $\tilde \omega _t^s = \tilde k^s_t + \tilde b^s_t$.

To derive the first-order conditions, the household maximizes the Lagrange function

$$\displaystyle \begin{aligned} \begin{aligned} \mathscr{L} = \sum_{s=1}^J & \beta^{s-1} \left(\prod_{j=1}^s \phi_{t+j-2,j-1}\right)\; \bigg[ \frac{1}{1 -\sigma } \left( ({c}_{t+s-1}^s)^{1 -\sigma } \left[1 -\nu_0 (1 -\sigma ) \left(l_{t+s-1}^s\right)^{1 +1/\nu_1 }\right]^{\sigma} -1\right) \\ & +\lambda_{t+s-1}^s \bigg(x^s_{t+s-1}+ \left[1+(1-\tau^K_{t+s-1}) r_{t+s-1} \right] \omega^{s}_{t+s-1} + tr_{t+s-1} - \omega_{t+s}^{s+1} \\ & - (1+\tau^c_t) c^{s}_{t+s-1} \bigg) \bigg] \end{aligned} \end{aligned}$$

with respect to $c^s_{t+s-1}$, $l^s_{t+s-1}$, and $\omega ^{s+1}_{t+s}$. The first-order conditions of the s-year-old household are represented by (7.47).⁶⁷ In terms of stationary variables, (7.47) can be expressed as follows:

$$\displaystyle \begin{aligned}\tilde \lambda^{s}_t (1+\tau^c) &= (\tilde c^{s}_t)^{-\sigma} \left[ 1-\nu_0 (1-\sigma) (l_t^{s})^{1+1/\nu_1}\right]^{\sigma},\;\; s=1,\ldots,J \end{aligned} $$

(7.64a)

$$\displaystyle \begin{aligned}\tilde \lambda^{s}_t (1-\tau^w_t-\tau^p_t) \bar y^s w_t &= \nu_0 \sigma \left(1+\frac{1}{\nu_1}\right) (\tilde c^{s}_t)^{1-\sigma} \left[ 1-\nu_0 (1-\sigma) (l^{s}_t)^{1+1/\nu_1}\right]^{\sigma-1} \end{aligned} $$

$$\displaystyle \begin{aligned} & \cdot (l^{s}_t)^{1/\nu_1},\;\; s=1,\ldots,R-1 \end{aligned} $$

(7.64b)

$$\displaystyle \begin{aligned} (1+\gamma)^{\sigma} \tilde \lambda^{s}_t &=\beta \phi_{t,s} \tilde \lambda^{s+1}_{t+1} \left[ 1+(1-\tau^K_{t+1}) r_{t+1}\right],\;\; s=1,\ldots,J-1. \end{aligned} $$

(7.64c)

The stationary budget constraint of the government in per capita terms is represented by:

$$\displaystyle \begin{aligned} \tilde g_t + \tilde tr_t + (1+r_t) \tilde b_t = (1+\gamma) (1+n) \tilde b_{t+1} + \tilde tax_t+ \tilde beq_t.\end{aligned} $$

(7.65)

The resource constraint of the economy in stationary equilibrium is given by:

$$\displaystyle \begin{aligned} \tilde y_t = \tilde c_t +\tilde g_t + (1+\gamma) (1+n) \tilde k_{t+1}-(1-\delta) \tilde k_t.\end{aligned} $$

(7.66)

2.1.1 Steady State Computation

To compute the steady state, we solve a non-linear equations problem in 28 variables consisting of the 14 individual asset levels, $\tilde \omega ^s=\tilde k^s+\tilde b^s$, s = 1, …, 15, (with $\tilde \omega ^1\equiv 0$), the 9 individual labor supplies, l ^s, s = 1, …, 9, and the aggregate variables, $\tilde k$, $\tilde L$, $\tilde \omega$, τ ^p, and $\tilde tr$.

The system of non-linear equations consists of the 23 first-order conditions of the household (the 14 Euler conditions and the 9 first-order conditions of the household with respect to the labor supply) as presented in (7.64b) and (7.64c) (after the substitution of $\tilde \lambda ^s_t$ from (7.64a)) and the following 5 aggregate equilibrium conditions⁶⁸:

$$\displaystyle \begin{aligned} (1+n) \tilde \omega &=\sum_{s=1}^{J} \mu_s \tilde \omega^{s+1}, \end{aligned} $$

(7.67a)

$$\displaystyle \begin{aligned} \tilde L & = \sum_{s=1}^{R-1} \mu^s \bar y^s l^s, \end{aligned} $$

(7.67b)

$$\displaystyle \begin{aligned} \tilde k & =\tilde \omega- \tilde b, \end{aligned} $$

(7.67c)

$$\displaystyle \begin{aligned} \tilde tr & = \tilde tax +\tilde beq+(n+\gamma + n\gamma -r) \tilde b -\tilde g, \end{aligned} $$

(7.67d)

$$\displaystyle \begin{aligned} \tau^p &= \frac{\sum_{s=R}^{J} \mu_s \tilde pen }{w \tilde L}, \end{aligned} $$

(7.67e)

where μ ^s represents the stationary share of the population N _t(s)∕N _t and $\tilde tax= \tau ^w w \tilde L+\tau ^K r \tilde \omega + \tau ^c \tilde c$ with

$$\displaystyle \begin{aligned} \tilde c =\sum_{s=1}^{J} \mu_s \tilde c^s, \; \; (1+n) \tilde b =\sum_{s=1}^{J} \mu_s \tilde b^{s+1}.\end{aligned} $$

(7.68)

Accidental bequests in steady state (with ϕ _t,s = ϕ _s) amount to⁶⁹

$$\displaystyle \begin{aligned} (1+n) \cdot \tilde beq = \sum_{s=1}^J \mu_s (1-\phi_s) \left[ 1+ (1-\tau^K) r\right]\tilde \omega^{s+1}.\end{aligned}$$

To compute $\tilde b^s$, we used the condition that all agents hold the two assets $\tilde k^s$ and $\tilde b^s$ in the same proportion.

All other variables, e.g., individual consumption, factor prices, and aggregate bequests and taxes, can be computed with the help of the 28 endogenous variables. For example, for the computation of individual consumption levels $\tilde c^s$, we can use the individual budget constraint. For the computation of the factor prices w and r, we use the first-order conditions of the firms.

We solve this non-linear equations problem with a modified Newton-Rhapson algorithm as described in Section 11.5.2 and applied to a large-scale OLG model in Section 9.1.2 of Heer and Maußner (2009). The main challenge for the solution is to determine good initial values for the individual and aggregate state variables.

Therefore, we start from a simple nine-period OLG model with exogenous labor in which all cohorts are workers. The exogenous labor supply is set equal to 0.3, and the initial value for the aggregate capital stock is set equal to the corresponding value in the Ramsey model. Next, we add one additional cohort of retirees in each step and use the solution of the model in the previous step as an input for the initial value of the next step. Finally, we introduce endogenous labor into the model. During these initial computations, we compute the solution for the individual optimization problem in an inner loop and update the aggregate capital variables in an outer loop with a dampening iterative scheme as described in Section 3.9 of Judd (1998) that helps to ensure convergence. For the final calibration and the computation of the steady states for different tax rates, we apply the modified Newton-Rhapson algorithm to the complete set of the 28 individual and aggregate equilibrium conditions. The algorithm is implemented in the Gauss programs Ch7_US_debt.g.

The transition dynamics are computed as described in Appendix 6.2. Different from this case, however, the final steady state for policies 3 and 4 is not known until we have computed the transition because we do not know the accumulated government debt in 2215 in these cases. We begin with an initial guess that debt remains constant. We update the final steady state using the solution from the transition dynamics in each iteration.

The transition is computed much faster than in the case of the large-scale OLG model in Sect. 6.4.1 and only amounts to approximately 2 min (rather than several hours). The computation is executed by the Gauss program Ch7_US_transition.g. In the absence of individual income uncertainty, we can compute the solution of the individual optimization problem with the help of the Newton-Rhapson algorithm and do not have to apply a time-consuming algorithm that is based upon value function iteration. In addition, we use a period length of 5 years rather than 1 year.

Appendix 7.3: Data Sources

In addition to the macroeconomic data presented in Appendices 2.4 and 4.6 and the population data described in Appendix 6.3, we introduce the following variables in our empirical analysis:

• Debt-GDP ratios The gross and net debt-ratios presented in Table 7.1 are retrieved from the IMF database (Accessed on 15 December 2017).

• https://www.imf.org/external/pubs/ft/weo/2017/02/weodata/index.aspx.

• The series for gross debt and net debt as percent of GDP are denoted with the identifier ‘GGXWDG_NGDP’ and ‘GGXWDN_NGDP’, respectively. The series will be read by the Gauss program Ch7_data.g from the Excel files IMF_grossdebt.xls and IMF_netdebt.xls.

• The time series data for the US gross debt-GDP ratio are taken from the series ‘GFDEGDQ188S’ from the Federal Reserve Bank of St. Louis (Accessed on 15 December 2017).

• https://fred.stlouisfed.org/series/GFDEGDQ188S.

• Budget deficits Data on general government deficit are retrieved from the OECD, Government at a Glance, 2017 (Accessed on 15 December 2017). The government deficit is defined as the fiscal position of government after accounting for capital expenditures. The series can be downloaded from the OECD at

• https://data.oecd.org/gga/general-government-deficit.htm.

• Government revenue Data for the 1920s are retrieved from the ‘Historical Statistics of the United States 1789–1945’ provided by the US Bureau of the Census with the cooperation of the Social Science Research Council (Accessed on 15 December 2017). The document can be downloaded at

• https://www.census.gov/library/publications/1949/compendia/hist_stats_1789--1945.html.

• Government bond yields The data displayed in Fig. 7.6 are taken from the FRED data base of the Federal Reserve Bank of St. Louis (Accessed on 15 January 2018). For France, for example, the series name of the 10-year nominal government bond yield is ‘IRLTLT01FRM156N’.

• Implicit debt-GDP ratios European Commission, Eurostat. Calculations: Research Centre for Generational Contracts.

• Real GDP growth The Data for Italy in Fig. 7.9 are taken from the World Bank (Accessed on 20 February 2018) and can be downloaded at https://data.worldbank.org/indicator/NY.GDP.DEFL.KD.ZG,

• and

• https://data.worldbank.org/indicator/NY.GDP.MKTP.KD.ZG.

• The data are included in the download files on my homepage that serve as an input into the Gauss computer program Ch7_data.g.

Problems

7.1

Recompute the dynamics of the OLG model with government debt in Sect. 7.4.2 under the assumption that the government applies the same lump-sum transfers $\tilde tr_t$ to both the young and the old generation such that aggregate transfers are the same in both cases:

$$\displaystyle \begin{aligned} \tilde tr_t = \frac{1+n}{2+n} tr_t.\end{aligned}$$

How does this policy affect equilibrium capital stock and output in the numerical example presented in Fig. 7.10?

7.2

Derive the goods market equilibrium (7.42).

7.3

Barro (1979) argues that even in the presence of Ricardian equivalence, debt financing of government expenditures may be optimal if lump-sum taxes are not available and only distortionary taxes such as income taxes can be used. For this reason, reconsider the numerical example of a temporary increase in government consumption from Sect. 4.3, where the transition dynamics are illustrated by the solid green line in Fig. 4.11. Instead, assume now that only distortionary labor income taxation and bond financing of government expenditures are available. The initial and final levels of public debt in periods 0 and 40 are equal to zero. What is the optimal fiscal policy? Is it characterized by tax smoothing?

7.4

Optimal Debt in the Three-Period OLG Model with Income Uncertainty

Assume that an agent lives for three periods. Each period length is equal to 20 years. In the first two periods, the agent is working; in the third period, he receives a pension. Each generation has mass 1∕3. We will only consider the steady state.

Lifetime utility is given by

$$\displaystyle \begin{aligned} U= \mathbb{E}\left\{ \sum_{s=1}^3 \beta^{s-1} u(c^s,1-l^s)\right\}.\end{aligned} $$

(7.69)

Instantaneous utility is represented by

$$\displaystyle \begin{aligned} u(c,1-l)=u(c,1-l)=\frac{\left( c (1-l)^\iota \right)^{1-\sigma}}{1-\sigma},\end{aligned}$$

with ι = 2.0 and σ = 2.0. Assume that β = 0.50. Time is allocated to either work or leisure.

During the first two periods, households work; in the third period, they retire (l ³ ≡ 0). Agents are born without assets, a ¹ = 0. Furthermore, households cannot borrow, a ^s ≥ 0 for s ∈{2, 3}. In addition, the workers pay contributions to the pension system equal to τ = 10% of their gross labor income. Gross labor income depends on individual labor productivity, which amounts to e ^s at ages s ∈{1, 2}. Therefore, the budget constraint at age s = 1, 2 is given by

$$\displaystyle \begin{aligned} (1-\tau) w e^s l^s +(1+r) a^s = c^s + a^{s+1}. \end{aligned}$$

In the first period of life, e ¹ = 1.0. In the second period of life, the individual faces income uncertainty, and insurance markets are missing. Due to health problems, 10% of the agents in the second cohort experience a decline in their labor productivity to $e^2_l=0.1$, while the remaining 90% of the cohort maintain constant productivity with $e^2_h=1.0$. Therefore, average productivity drops to $\bar e^2=0.91$ for the two-period-old agent.

During retirement, agents receive pensions pen that do not depend on the individual contribution history but are provided lump-sum. The budget of the social security authority is balanced:

$$\displaystyle \begin{aligned} \frac{pen}{3}= \tau w \frac{1}{3}\left( e^1 l^1 + 0.9 e^2_h l^2_h + 0.1 e^2_l l^2_l \right),\end{aligned}$$

where $l^2_j$, j ∈{h, l} denotes the labor supply of the agent with high and low productivity at age 2. The budget constraint of the retired worker is given by:

$$\displaystyle \begin{aligned}pen +(1+r) a^3 = c^3. \end{aligned}$$

Production is described by a Cobb-Douglas function:

$$\displaystyle \begin{aligned} Y=K^\alpha L^{1-\alpha},\end{aligned}$$

with

$$\displaystyle \begin{aligned} L= \frac{e^1 l^1+0.9e^2_h l^2_h+ 0.1 e^2_l l^2_l}{3},\end{aligned}$$

and α = 0.36.

Factors are rewarded by their marginal products:

$$\displaystyle \begin{aligned} w_t & = (1-\alpha) \left(\frac{K_t}{L_t}\right)^{\alpha},\\ r_t& =\alpha \left(\frac{K_t}{L_t}\right)^{\alpha-1}-\delta. \end{aligned} $$

The depreciation rate is set equal to δ = 0.5.

• 1. Consider the equilibrium without government debt in which total assets are equal to the capital stock. Solve the problem with the help of direct computation (solving a system of non-linear equations).

• 2. Consider the economy with a government sector that is subject to the public budget constraint

  $$\displaystyle \begin{aligned} Tr_t + r B_t = B_{t+1}-B_t.\end{aligned}$$

  Aggregate transfers are equal to individual transfers, Tr _t = tr _t. In equilibrium, aggregate assets Ω _t are equal to K _t + B _t. Higher government debt helps to insure the one-period-old agent against negative income shocks because it increases interest rates and, hence, savings. However, it crowds out capital and, hence, lowers per capita consumption. In addition, steady-state transfers decrease with higher debt. Compute the optimal level of steady-state public debt, B ≥ 0, that maximizes the expected lifetime utility of the households.

• 3. Compute the optimal level of debt B that maximizes the ex post lifetime utility of the individuals facing a decline in labor productivity (according to the maximin criterion).

• 4. Is your result for the optimal level of public debt robust with respect to the assumption that labor supply is exogenous, l ¹ = l ² = 0.3, where households cannot insure themselves against income uncertainty by adjusting their labor supply?