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Principles of Public Finance
In order to investigate the burden of public debt, it is useful to explain Ricard’s neutrality theorem. We employ a simple twoperiod model as in Chap. 3. In this regard, a household optimizes consumption for two periods, namely period 1 (current period) and period 2 (future period).
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This appendix examines the economic effect of government debt in a simple dynamic model of economic growth. We first show that taxfinanced transfer payments and public debt have the same effect on longrun equilibrium. We also show that if lump sum taxes are appropriately adjusted, debt policy is not effective; hence, the government deficit is a meaningless policy indicator. We then examine the burden of debt and show that an increase in a constant amount of government debt per worker crowds out capital accumulation in the long run.
This appendix also investigates the role of government debt in the altruism model. As explained in the main text of this chapter, Barro (
1974) extended Ricardian neutrality to the strongest proposition of debt neutrality. The altruism model means that households can be represented by the families that act as though they are infinitely lived. This appendix explains Barro’s (
1974) idea intuitively.
We investigate the burden of debt in a simple growth model of overlapping generations based on Diamond (
1965). This model is useful to investigate the impact of public debt on generations and capital accumulation in a dynamic context. This is a classical model for investigating the role of public debt in a growing economy (see Ihori,
1996).
Consider a closed economy populated by overlapping generations of twoperiodlived consumers and firms. In this model, one young and old generation exist at any point in time. The young have no nonhuman wealth, and the lifetime resources of the young correspond to the labor earnings they receive. There may be population growth. Output is durable and may be accumulated as capital. For simplicity, it is assumed that there is no capital depreciation. The physical characteristics of the endowment are important in overlappinggeneration economics since durable goods represent an alternative technology for transferring resources through time.
An agent of generation t is born at time t and considers him or herself “young” in period t and “old” in period t + 1. The agent dies at time t + 2. When young, the agent of generation t supplies one unit of labor inelastically and receives wages w
_{t}, out of which the agent consumes c
_{t} ^{1} and saves s
_{t} in period t. An agent who saves s
_{t} receives (1 + r
_{t+1})s
_{t} when old, which the agent then spends entirely on consumption,
\( {\mathrm{c}}_{\mathrm{t}+1}^2 \), in period t + 1. r
_{t} is the rate of interest in period t. There are no bequests, gifts, or other forms of net intergenerational transfers to the young. In each period, two generations are alive, the young and the old.
A member of generation t faces the following budget constraints:
From Eqs. (
4.A1) and (
4.A2), her or his lifetime budget constraint is given as
Further, her or his lifetime utility function is given as
The utility function u( ) increases in the vector (c
^{1}, c
^{2}), twice continuously differentiable and strictly quasiconcave. Thus,
Future consumption is a normal good,
where
\( {u}_{12}={\partial}^2u/\partial {c}^1\partial {c}^2 \) and
\( {u}_{11}={\partial}^2u/\partial {c}^1\partial {c}^1 \). Starvation is avoided in both periods,
A consumer born in period t maximizes her or his lifetime utility (
4.A4) subject to the lifetime budget constraint (
4.A3) for given w
_{t} and r
_{t+1}. For simplicity, we assume that the agent is capable of predicting the future course of the economy and that he or she adopts this prediction as her or his expectation of r
_{t+1}. Such rational or perfect foresight expectations are independent of past observations and must be selffulfilling.
Solving this problem for s
_{t} yields the optimal saving function of the agent,
where
\( \partial s/\partial w={s}_w \)> 0 follows from the normality of second period consumption. The sign of
\( \partial s/\partial r={s}_r \) is ambiguous since the substitution effect and the income effect offset each other, as explained in Chap.
8.
The aggregate macroeconomic production function is
where Y
_{t} is total output, K
_{t} is capital stock, and N
_{t} is labor supply. We assume constant returns to scale technology, so that the production function may be rewritten as
where y
_{t} = Y
_{t}/N
_{t} and k
_{t} = K
_{t}/N
_{t}. y
_{t} is per capita output and k
_{t} is the amount of capital per worker in period t. The production function is well behaved and satisfies the Inada condition:
\( f(0)=0,{f}^{\prime }(0)=\infty, {f}^{\prime}\left(\infty \right)=0. \)
The population grows at the rate of n (> −1). Thus,
Competitive profit maximization and neoclassical technology require that firms hire labor and demand capital in such a way that
Equations (
4.A8) and (
4.A9) imply that the marginal product of capital is equal to the rate of interest and that the marginal product of labor is equal to the wage rate. Constant returns to scale and atomistic competition mean that payments to factors of production exhaust every profitmaximizing producer’s revenue, leaving nothing for profit. Since the markets for renting and purchasing physical capital are competitive, the opportunity cost of owning capital for one period should equal the rental rate.
From Eqs. (
4.A8) and (
4.A9), w
_{t} may be expressed as a function of r
_{t}. Thus,
where w( ) is called the factor price frontier.
In an equilibrium situation, agents can save by holding capital. In this type of economy, equilibrium in the financial market requires
or
We shall assume that the government issues debt b
_{t} to the younger generation in period t. This debt has oneperiod maturity and will be repaid in the next period with interest at the same rate of return as on capital. b can be negative, in which case b means “negative debt”; namely, the government lends b to each individual of the younger generation and will recover this credit with interest.
Let us denote the (percapita) lump sum tax levied on the younger generation and the older generation in period t by T
_{t} ^{1} and T
_{t} ^{2} respectively. Suppose for simplicity that the government does not make any public expenditure. Then, the government budget constraint in period t is
where N
_{t} is the number of people in generation t.
The following cases are of considerable interest.
T
^{2} = 0. The tax collected to finance interest costs minus new debt issuance is a lump sum tax on the younger generation. This debt issue corresponds to Diamond’s internal debt.
T
^{1} = 0. The tax collected to finance interest costs minus new debt issuance is a lump sum tax on the older generation.
b = 0. The government does not issue debt. The government levies the lump sum tax T
^{1} on the younger generation and transfers it to the older generation in the same period. This corresponds to the unfunded payasyougo system.
The private budget constraints of generation t, (
4.A1) and (
4.A2), are rewritten as follows:
Each individual’s lifetime disposable income (
ŵ
_{ t }) is given by
\( \left({\mathrm{w}}_{\mathrm{t}}{\mathrm{T}}_{\mathrm{t}}^1\right) \) and her or his disposable income in the younger period t minus (T
^{2}
_{t+1}/(1 + r
_{t+1})) the present value of the tax in the older period t + 1. Thus, the lifetime budget constraint (
4.A3) is rewritten as
where
\( {\widehat{w}}_t={w}_t{T}_t^1\frac{1}{1+{r}_{t+1}}{T}_{t+1}^2 \).
Considering (
4.A3′), capital accumulation equation (
4.A11) may be rewritten as
Let us define effective taxes by
τ
^{1} and
τ
^{2} are net receipts from the young and old. These two equations (
4.A14.1 and
4.A14.2) are government budget constraints in period t and period t + 1. Thus, dynamic equilibrium can be summarized by the following two equations:
Equation (
4.A15) is the government budget constraint. Equation (
4.A16) comes from Eq. (
4.A13) and is the capital accumulation equation. b, T
^{1}, and T
^{2} do not appear in these two equations.
In other words, fiscal action is comprehensively summarized by a sequence of effective taxes {τ
_{t} ^{1} } and {τ
_{t} ^{2} }. One of b, T
^{1}, and T
^{2} is redundant in order to attain any fiscal policy. The three cases (a), (b), and (c) are equivalent so long as two of b, T
^{1}, and T
^{2} are adjusted to attain the same {τ
_{t} ^{1} } and {τ
_{t} ^{2} }. In cases (a) and (b), the government budget is not balanced. But in case (c), the government budget is balanced since b = 0. This means that the government deficit is not a useful policy indicator to summarize fiscal action. This is supported by Kotlikoff (
1992), who said that if lump sum taxes are appropriately adjusted, debt policy is not effective and the government deficit is a meaningless policy indicator.
Taxfinanced transfer payments (case (c)) and Diamond’s internal debt (case (a)) have the same effect on competitive equilibrium. In other words, this national debt can be regarded as a device that is used to redistribute income between the younger and older generations. Any intergenerational redistribution that can be supported by debt and taxes can also be supported just with taxes and without debt.
As Auerbach and Kotlikoff (
1987) and Buiter and Kletzer (
1992) stressed, unfunded social security can be easily managed as an explicit government debt policy. The government can label its social security receipts from young workers as either “borrowing” or “taxes.” It can also label benefit payments to retired people as either “principal plus interest payments” with respect to the government’s borrowing or “transfer payments.” The economy’s real behavior is not altered by such relabeling. This makes one wary of relying on official government debt numbers as indicators of the government’s true policy with respect to intergenerational redistribution .
Based on this understanding, Kotlikoff (
1992) proposed a notion of generational accounting. As stressed by Kotlikoff (
1992), generational accounting is a relatively new tool of intergenerational redistribution. It is based on the government’s intertemporal budget constraint, which requires that the government’s bill is paid by current or future generations. Moreover, Fehr and Kotlikoff (
1995) showed how changes in generational accounts relate to the generational incidence of fiscal policy. See Chap.
7 for a further discussion of this issue.
If lump sum taxes are appropriately adjusted among generations, debt policy is meaningless. The government deficit is not a useful policy indicator. This result corresponds to Ricardian debt neutrality. The agent is concerned only with lifetime budget constraint; periodtoperiod budget constraint is meaningless. However, this result does not necessarily deny the effectiveness of fiscal policy with respect to intergenerational redistribution. As shown in (
4.A15) and (
4.A16), changes in {
τ
_{t} ^{1} } and {
τ
_{t} ^{2} } have real effects.
If there is no freedom to adjust lump sum taxes appropriately, then changes in government debt have real effects. This situation has been investigated in terms of debt burden. Let us define the relative burden ratio v with
which is assumed to be constant. When v is exogenously fixed, changes in b have real effects.
Suppose a constant amount of debt per worker (b) is maintained. Considering Eq. (
4.A13), the longrun competitive capital/labor ratio with debt policy,
\( \overset{}{k} \)(b, v), is determined by
where a = s + b. From Eqs. (
4.A12) and (
4.A17), we have
Substituting Eq. (
4.A19) into Eq. (
4.A18) and taking the total derivative of k with respect to b in a steady state, we have
From the assumption of the stability of the system, the denominator is positive; and from the assumption of the normality of the utility function, the numerator is negative. Thus, dk/db is definitely negative. Further, an increase in b reduces k. This result is referred to as the burden of debt, as explained in Sect.
2.3 of this chapter. See also Diamond (
1965), who maintained that an increase in a constant amount of government debt per worker crowds out capital accumulation in the long run .
Barro (
1974) extended the conventional neutrality result (Ricardian neutrality) to the strongest proposition of Barro’s debt neutrality. Under certain conditions, debt policy is meaningless, even if lump sum taxes are not adjusted appropriately among generations.
Barro studied the effect of debt policy in the altruism model of overlapping generations. The altruism model means that households can be represented by the families that act as though they are infinitely lived. He showed that public intergenerational transfer policy becomes ineffective once we incorporate altruistic bequests into the standard overlappinggenerations model. Let us explain intuitively his idea in this section .
A representative individual born at time t has the following budget constraints:
where e
_{t}/(1 + n) is the inheritance received when young and e
_{t+1} is the individual’s bequest that is determined when old.
In the altruism model, the parent cares about the welfare of her or his offspring instead of the bequest itself. The parent’s utility function is given as
where u
_{t} is the utility from the parent’s own consumption,
\( \mathrm{u}\left({\mathrm{c}}_{\mathrm{t}}^1,\ {\mathrm{c}}_{\mathrm{t}+1}^2\right) \), and
σ
_{ A } is the parent’s marginal benefit of her or his offspring’s utility .
An individual born at time t solves the following problem of maximization:
The optimal conditions with respect to s
_{t} and e
_{t+1} are
Since the first order conditions are independent of government debt, the public intergenerational policy due to debt issuance is completely neutral. It would not affect the real equilibrium. In this regard, Barro (
1974) said that if the altruistic bequest motive is operative, public intergenerational policy is neutral .
Equations (
4.A25.1) and (
4.A25.2) give the longrun rate of interest, r
_{A}, in the altruism model r
_{A}. Thus,
which is independent of b.
Let us define effective bequests with
Recognizing Eqs. (
4.A14.1), (
4.A14.2), (
4.A15), and (
4.A27), Eqs. (
4.A21) and (
4.A22) may be rewritten as
Substituting Eqs. (
4.A21′) and (
4.A22′) into Eqs. (
4.A25.1) and (
4.A25.2), it is easy to see that Eqs. (
4.A25.1) and (
4.A25.2) determine the optimal path of {
e
_{ t } ^{*} }. Public intergenerational transfer through changes between
τ
^{1} and
τ
^{2} (or b, T
^{1}, and T
^{2}) is completely offset by appropriate changes in private transfer, e. When the government changes b, the private sector changes bequests so as to maintain the optimal path of effective bequests, which is determined by Eqs. (
4.A25.1) and (
4.A25.2) .
Say whether the following statements are true or false and explain the reasons.
When a consumer behaves optimally, based on permanent disposable income, public debt does not affect real economic variables.
Debt issuance moves the fiscal burden to future generations and hurts intergenerational equity.
When the fiscal deficit and public debt are large, the conventional Keynesian effect is likely to occur and an increase in taxes may depress private consumption.
Explain the theoretical assumptions for Barro’s neutrality and discuss the plausibility of these assumptions.
$$ {c}_t^1={w}_t{s}_t\kern0.24em \mathrm{and} $$
(4.A1)
$$ {c}_{t+1}^2=\left(1+{r}_{t+1}\right){s}_t. $$
(4.A2)
$$ {c}_t^1+\frac{1}{1+{r}_{t+1}}{c}_{t+1}^2={w}_t. $$
(4.A3)
$$ {u}_t=u\left({c}_t^1,{c}_{t+1}^2\right). $$
(4.A4)
$$ \frac{\partial u}{\partial {c}_t^1}={u}_1\left({c}_t^1,{c}_{t+1}^2\right)>0\kern0.24em for\kern0.24em \left({c}^1,{c}^2\right)>0\kern0.24em \mathrm{and} $$
$$ \frac{\partial u}{\partial {c}_{t+1}^2}={u}_2\left({c}_t^1,{c}_{t+1}^2\right)>0\kern1em for\kern0.24em \left({c}^1,{c}^2\right)>0. $$
$$ {\mathrm{u}}_1{\mathrm{u}}_{12}>{\mathrm{u}}_2{\mathrm{u}}_{11}\mathrm{f}\mathrm{o}\mathrm{r}\left({\mathrm{c}}^1,\;{\mathrm{c}}^2\right)>0, $$
$$ { \lim}_{c^1\to 0}{u}_1\left({c}^1,{c}^2\right)=\infty\ for\kern0.24em {c}^2>0\kern0.24em \mathrm{and} $$
$$ { \lim}_{c^2\to 0}{u}_2\left({c}^1,{c}^2\right)=\infty\ for\kern0.24em {c}^1>0. $$
$$ {\mathrm{s}}_{\mathrm{t}}=\mathrm{s}\left({\mathrm{w}}_{\mathrm{t}},\;{\mathrm{r}}_{\mathrm{t}+1}\right), $$
(4.A5)
$$ {\mathrm{Y}}_{\mathrm{t}}=\mathrm{F}\left({\mathrm{K}}_{\mathrm{t}},\;{\mathrm{N}}_{\mathrm{t}}\right), $$
$$ {y}_t=f\left({k}_t\right),\kern0.24em {f}^{\prime }>0,\kern0.24em {f}^{{\prime\prime} }<0, $$
(4.A6)
$$ {\mathrm{N}}_{\mathrm{t}}=\left(1+\mathrm{n}\right){\mathrm{N}}_{\mathrm{t}1}. $$
(4.A7)
$$ {f}^{\prime}\left({k}_t\right)={r}_t\kern1em \mathrm{and} $$
(4.A8)
$$ f\left({k}_t\right){f}^{\prime}\left({k}_t\right){k}_t={w}_t. $$
(4.A9)
$$ {\mathrm{w}}_{\mathrm{t}}=\mathrm{w}\left({\mathrm{r}}_{\mathrm{t}}\right),\ {\mathrm{w}}^{\prime}\left({\mathrm{r}}_{\mathrm{t}}\right)={\mathrm{k}}_{\mathrm{t}}<0,\ {\mathrm{w}}^{{\prime\prime} }>0, $$
(4.A10)
$$ {\mathrm{s}}_{\mathrm{t}}{\mathrm{N}}_{\mathrm{t}}={\mathrm{K}}_{\mathrm{t}+1} $$
$$ {\mathrm{s}}_{\mathrm{t}}=\left(1+\mathrm{n}\right){\mathrm{k}}_{\mathrm{t}+1}. $$
(4.A11)
$$ {\mathrm{b}}_{\mathrm{t}1}{\mathrm{N}}_{\mathrm{t}1}\left(1+{\mathrm{r}}_{\mathrm{t}}\right){\mathrm{b}}_{\mathrm{t}}{\mathrm{N}}_{\mathrm{t}}={\mathrm{T}}_{\mathrm{t}}^1{\mathrm{N}}_{\mathrm{t}}+{\mathrm{T}}_{\mathrm{t}}^2{\mathrm{N}}_{\mathrm{t}1} $$
(4.A12)
(a)
T
^{2} = 0. The tax collected to finance interest costs minus new debt issuance is a lump sum tax on the younger generation. This debt issue corresponds to Diamond’s internal debt.
(b)
T
^{1} = 0. The tax collected to finance interest costs minus new debt issuance is a lump sum tax on the older generation.
(c)
b = 0. The government does not issue debt. The government levies the lump sum tax T
^{1} on the younger generation and transfers it to the older generation in the same period. This corresponds to the unfunded payasyougo system.
$$ {c}_t^1={w}_t{s}_t{b}_t{T}_t^1\kern0.24em \mathrm{and} $$
(4.A1′)
$$ {c}_{t+1}^2=\left({s}_t+{b}_t\right)\left(1+{r}_{t+1}\right){T}_{t+1}^2. $$
(4.A2′)
$$ {c}_t^1+\frac{1}{1+{r}_{t+1}}{c}_{t+1}^2={\widehat{w}}_t, $$
(4.A3′)
$$ {w}_t{c}^1\left({\widehat{w}}_t,{r}_{t+1}\right){b}_t{T}_t^1=\left(1+n\right){k}_{t+1}. $$
(4.A13)
$$ {\tau}_t^1={b}_t+{T}_t^1\kern1em \mathrm{and} $$
(4.A14.1)
$$ {\tau}_{t+1}^2=\left(1+{r}_{t+1}\right){b}_t+{T}_{t+1}^2. $$
(4.A14.2)
$$ {\tau}_t^1+\frac{1}{1+n}{\tau}_t^2=0\kern1em \mathrm{and} $$
(4.A15)
$$ w\left({r}_t\right){c}^1\left[w\left({r}_t\right){\tau}_t^1\frac{\tau_{t+1}^2}{1+{r}_{t+1}},{r}_{t+1}\right]{\tau}_t^1=\left(1+n\right){w}^{\prime}\left({r}_{t+1}\right). $$
(4.A16)
$$ {v}_t=\frac{T_t^2}{T_t^1\left(1+n\right)}, $$
(4.A17)
$$ a\left(\widehat{w},r\right)=\left(1+n\right)k+b, $$
(4.A18)
$$ {\widehat{w}}_t={w}_t\left\{\frac{r_tn}{\left(1+v\right)\left(1+n\right)}+\frac{v\left({r}_{t+1}n\right)}{\left(1+{r}_{t+1}\right)\left(1+v\right)}\right\}b. $$
(4.A19)
$$ \frac{dk}{db}=\frac{\frac{\partial a}{\partial b}1}{1+n\frac{\partial a}{\partial k}}. $$
(4.A20)
$$ {c}_t^1={w}_t{s}_t{b}_t{T}_t^1+\frac{e_t}{1+n}\kern0.36em \mathrm{and} $$
(4.A21)
$$ {c}_{t+1}^2=\left(1+{r}_{t+1}\right)\left({s}_t+{b}_t\right){e}_{t+1}{T}_{t+1}^2, $$
(4.A22)
$$ {U}_t={u}_t+{\sigma}_A{U}_{t+1}, $$
(4.A23)
$$ \begin{array}{l}{W}_t=u\left[w\right({r}_t\left){s}_t{b}_t{T}_t^1+\frac{e_t}{1+n},\left(1+{r}_{t+1}\right)\left({s}_t+{b}_t\right){T}_{t+1}^2{e}_{t+1}\right]+\\ {}{\sigma}_A\Big\{u\left[w\left({r}_{t+1}\right){s}_{t+1}{b}_{t+1}{T}_{t+1}^1+\frac{e_{t+1}}{1+n},\left(1+{r}_{t+2}\right)\left({s}_{t+1}+{b}_{t+1}\right){T}_{t+2}^2{e}_{t+2}\right]\\ {} + {\sigma}_A{U}_{t+2}\Big\}.\end{array} $$
(4.A24)
$$ \frac{\partial u}{\partial {c}_t^1}=\left(1+{r}_{t+1}\right)\frac{\partial u}{\partial {c}_{t+1}^2}\kern0.24em \mathrm{and} $$
(4.A25.1)
$$ \left(1+n\right)\frac{\partial u}{\partial {c}_{t+1}^2}={\sigma}_A\frac{\partial u}{\partial {c}_{t+1}^1}. $$
(4.A25.2)
$$ n={\sigma}_A\left(1+{r}_A\right)1. $$
(4.A26)
$$ {e}_t^{*}={\tau}_t^2+{e}_t. $$
(4.A27)
$$ {c}_t^1={w}_t{s}_t+\frac{1}{1+n}{e}_t^{*}\kern0.24em \mathrm{and} $$
(4.A21′)
$$ {c}_{t+1}^2=\left(1+{r}_{t+1}\right){s}_t{e}_{t+1}^{*}. $$
(4.A22′)
4.1
Say whether the following statements are true or false and explain the reasons.
(a)
When a consumer behaves optimally, based on permanent disposable income, public debt does not affect real economic variables.
(b)
Debt issuance moves the fiscal burden to future generations and hurts intergenerational equity.
(c)
When the fiscal deficit and public debt are large, the conventional Keynesian effect is likely to occur and an increase in taxes may depress private consumption.
4.2
Explain the theoretical assumptions for Barro’s neutrality and discuss the plausibility of these assumptions.
(a)
When a consumer behaves optimally, based on permanent disposable income, public debt does not affect real economic variables.
(b)
Debt issuance moves the fiscal burden to future generations and hurts intergenerational equity.
(c)
When the fiscal deficit and public debt are large, the conventional Keynesian effect is likely to occur and an increase in taxes may depress private consumption.
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 Titel
 Public Debt
 DOI
 https://doi.org/10.1007/9789811023897_4
 Autor:

Toshihiro Ihori
 Verlag
 Springer Singapore
 Sequenznummer
 4
 Kapitelnummer
 4