A. Households

The household section is modeled by an overlapping generations framework as developed by Yaari (1965) and Blanchard (1985). At each instant of time a large cohort of agents is born which face a constant probability of death, β≥0, per unit of time.⁵ The size of the total population at any moment in time is constant and normalized to unity. New generations are not connected to old generations since households do not have a bequest motive. Hence, agents are born without financial assets. In the absence of bequests, households contract actuarially fair (reverse) life insurance with annuities companies. Consumers receive a rate of return on their financial wealth (i.e., β) at each instant of time during their lives, but when they die their entire stock of assets goes to the insurance company.

A representative agent of the generation born at time v≤t derives utility at time τ from private consumption, C(v, τ), and leisure, 1-L(v, τ):⁶

where α and ϵ_c denote, respectively, the constant pure rate of time preference (α≥0) and the consumption share in utility. The logarithmic specification of utility imposes unit intertemporal and intratemporal elasticities of substitution. The agent’s dynamic budget identity in real terms is given by:

where A(v, t) is financial wealth, r(t) is the real rate of interest, W(t) is the age-independent real wage rate, t_L(t) is the tax rate on labor income, and T(t) are net lump-sum transfers which are uniform across all generations. The agent receives r(t)A(v, t) as interest payments on its financial wealth at time t and β A(v, t) is the premium received from the insurance company. A dot above a variable denotes a time derivative, e.g., $\dot{A}(v,t)\equiv {d}A(v,t)/{d}t$. Full consumption, X(v, t), is defined as the sum of the value of goods consumption and the opportunity cost of leisure consumption:

where t_C(t) is a proportional tax rate on private consumption. A No-Ponzi-Game (NPG) solvency condition needs to be imposed to prevent agents from accumulating debt indefinitely:

where the effective rate of interest (i.e., the sum of the real rate of interest and the probability of death) is used as the discount factor.

The representative consumer of vintage v is endowed with perfect foresight and maximizes life-time utility (2.1) subject to the household’s dynamic budget restriction (2.2) and the NPG-condition (2.4). The optimization problem can be solved in two stages which yield the entire time path of consumption and labor supply. In the first stage the consumer chooses the optimal time profile of full consumption. The first-order conditions yield the full consumption Euler equation:

By integrating (2.5) forward and substituting the resulting expression in the household’s budget constraint (which is (2.2) integrated with the NPG condition (2.4) imposed) one derives full consumption for period t as a linear function of total household wealth:

where H(t) is expected lifetime human wealth:

Intuitively, H(t) represents the after-tax market value of the household’s time endowment, i.e., the present discounted value of maximum future after-tax wage income. The constant marginal propensity to consume out of total wealth, α+β, follows from assuming an intertemporal substitution elasticity of unity. Under this assumption the substitution and income effect of a change in the real rate of interest exactly cancel.

In the second stage the representative consumer of cohort v decides about the consumption-leisure choice in each time period. This yields simple expressions relating individual consumption and leisure demand to full consumption:

Due to the constant probability of death, the size of a cohort declines deterministically through time (i.e., in an exponential fashion), although individuals are uncertain about their time of death. Following Blanchard (1985) the household sector can be aggregated by integrating over all currently alive generations. For example, aggregate private consumption, is defined as:

where βe^β(v−t) is the size of a cohort of vintage ν as of time t. Similarly, we can obtain the aggregate values for L(t), A(t), T(t), H(t), and X(t). This yields the following equations describing the behavior of the aggregate household sector:

Aggregate financial wealth accrues at the rate r(t) while individual financial wealth accumulates at a faster rate r(t)+β. Since βA(t) is a transfer from those who die to those who remain alive it therefore does not enter the economy-wide budget constraint. By differentiating household consumption (2.8) with respect to time, using (2.5), the Euler-equation for the growth in individual consumption is obtained:

Equation (2.13) is the familiar Keynes-Ramsey rule which says that the rate of change of individual consumption is positively related to the wedge between the real rate of interest and the pure rate of time preference. An immediate rise in the consumption tax rate (i.e., ${\dot{t}}_C\left(t\right)=0$) does not affect the growth rate of private consumption, while a gradually introduced increase in the consumption tax rate (i.e., ${\dot{t}}_C\left(t\right)>0$) works like a capital income tax and flattens the tilt of the consumption profile (see Auerbach and Kotlikoff, 1987). Aggregating (2.13) yields the Modified Keynes-Ramsey (MKR) rule which describes aggregate consumption growth modified for the existence of overlapping generations:

Aggregate consumption growth differs from consumption growth for individual cohorts because of the turnover of generations. At each instant a cross-section of the existing population dies and is replaced by a new generation. Since average consumption exceeds consumption by newly born agents, the generational turnover effect drags down aggregate consumption growth. See also Section III for a further exposition.

B. Firms

The production sector is characterized by a large number of firms that produce an identical good under perfect competition. Net output,⁷ Y(t), is produced according to a Cobb-Douglas technology with labor, L(t), and physical capital, K(t), as homogeneous factor inputs which are rented from households:

where ϵ_L denotes the share of before-tax wage income in net output. The representative firm maximizes its net present (or equity) value:

with respect to labor and capital subject to the production function (2.15) and the capital accumulation constraint, $\dot{K}(t)=I(t)-\delta K(t)$, where I(t) denotes gross investment, δ is the constant rate of depreciation of capital, and t_K(t) represents a tax on capital income. There are no adjustment costs associated with investment. The first-order conditions imply that the marginal productivity of labor and capital equal the producer costs of these factors:

Combining the expressions in equations (2.17a-b) yields the wage-rental cost ratio as a function of the capital-labor ratio. By substituting (2.17a-b) in (2.16) and using the property of linear homogeneity of the production function, we can derive that the market value of the firm is equal to the replacement value of its capital stock, i.e., V(t) = K(t).

C. The Government and Market Equilibrium

Government expenditure consists of lump-sum transfers to households. The government finances its spending by employing one (or a mix) of the following tax instruments: (i) a capital income tax, (ii) a labor income tax, or (iii) a tax on consumption. The model abstracts from debt policy implying that the government needs to balance its budget in each time period. The dynamic government budget identity is given by:

In a dynamic macroeconomic model with perfect foresight, behavior of households, firms, and the government today depends on future economic conditions. Equilibrium therefore requires that agents’ behavior be consistent not only with current ‘prices’ (i.e., wages, interest rates, and tax rates), but also with the entire path of future prices. Due to the time-separable nature of the utility function (see equation (2.1)), past household decisions only affect current behavior through the household’s accumulated stock of capital.

At each instant of time, factor and goods markets clear instantaneously. In this closed economy households can only accumulate domestic assets. As a result, financial market equilibrium requires that A(t) = K(t). Wage flexibility ensures that the supply of labor by households matches labor demand by firms. Goods market equilibrium is obtained when the supply of goods equals aggregate demand, which consists of private consumption plus net investment: $Y(t)=C(t)+\dot{K}(t)$.

A. Stability

Local stability of the model can be studied by linearizing the nonlinear system of equations around an initial steady state (see Judd (1982)). The main expressions are given in Table 1.

Table 1.

Linearized Version of the Model

Table 1.

Linearized Version of the Model

$\begin{array}{ccc}\dot{\tilde{K}}(t)=\left(\frac{r}{(1-t_K)(1-{{ϵ}}_L)}\right)\left[\tilde{Y}(t)-\tilde{C}(t)\right]& & ({T}.1)\end{array}$
$\begin{array}{ccc}\dot{\tilde{C}}(t)=(r-{\alpha })\left[\tilde{C}(t)+{\tilde{t}}_C(t)-\tilde{K}(t)\right]+r\tilde{r}(t)-{\dot{\tilde{t}}}_C(t)& & ({T}.2)\end{array}$
$\begin{array}{ccc}\begin{array}{l}\tilde{T}(t)=(1+t_C)\left[{\tilde{t}}_C(t)+\left(\frac{t_C}{1+t_C}\right)\tilde{C}(t)\right]\\ +{{ϵ}}_L(1-t_L)\left[{\tilde{t}}_L(t)+\left(\frac{t_L}{1-t_L}\right)\tilde{Y}(t)\right]+(1-{{ϵ}}_L)(1-t_K)\left[{\tilde{t}}_K(t)+\left(\frac{t_K}{1-t_K}\right)\tilde{Y}(t)\right]\end{array}& & ({T}.3)\end{array}$
$\begin{array}{ccc}\tilde{Y}(t)-\tilde{L}(t)=\tilde{W}(t)& & ({T}.4)\end{array}$
$\begin{array}{ccc}\tilde{Y}(t)-\tilde{K}(t)={\tilde{t}}_K(t)+\tilde{r}(t)& & ({T}.5)\end{array}$
$\begin{array}{ccc}\tilde{L}(t)={{\omega }}_{\mathit{\text{LL}}}[\tilde{W}(t)-{\tilde{t}}_L(t)-\tilde{C}(t)-{\tilde{t}}_C(t)]& & ({T}.6)\end{array}$
$\begin{array}{ccc}\tilde{Y}(t)={{ϵ}}_L\tilde{L}(t)+(1-{{ϵ}}_L)\tilde{K}(t)& & ({T}.7)\end{array}$
Shares and parameters:
ϵL ≡WL/Y	Share of before-tax wage income in real net output
ωLL≡(1-L)/L	Ratio of leisure to labor
α	Pure rate of time preference
β	Probability of death

View Table

Table 1.

Linearized Version of the Model

$\begin{array}{ccc}\dot{\tilde{K}}(t)=\left(\frac{r}{(1-t_K)(1-{{ϵ}}_L)}\right)\left[\tilde{Y}(t)-\tilde{C}(t)\right]& & ({T}.1)\end{array}$
$\begin{array}{ccc}\dot{\tilde{C}}(t)=(r-{\alpha })\left[\tilde{C}(t)+{\tilde{t}}_C(t)-\tilde{K}(t)\right]+r\tilde{r}(t)-{\dot{\tilde{t}}}_C(t)& & ({T}.2)\end{array}$
$\begin{array}{ccc}\begin{array}{l}\tilde{T}(t)=(1+t_C)\left[{\tilde{t}}_C(t)+\left(\frac{t_C}{1+t_C}\right)\tilde{C}(t)\right]\\ +{{ϵ}}_L(1-t_L)\left[{\tilde{t}}_L(t)+\left(\frac{t_L}{1-t_L}\right)\tilde{Y}(t)\right]+(1-{{ϵ}}_L)(1-t_K)\left[{\tilde{t}}_K(t)+\left(\frac{t_K}{1-t_K}\right)\tilde{Y}(t)\right]\end{array}& & ({T}.3)\end{array}$
$\begin{array}{ccc}\tilde{Y}(t)-\tilde{L}(t)=\tilde{W}(t)& & ({T}.4)\end{array}$
$\begin{array}{ccc}\tilde{Y}(t)-\tilde{K}(t)={\tilde{t}}_K(t)+\tilde{r}(t)& & ({T}.5)\end{array}$
$\begin{array}{ccc}\tilde{L}(t)={{\omega }}_{\mathit{\text{LL}}}[\tilde{W}(t)-{\tilde{t}}_L(t)-\tilde{C}(t)-{\tilde{t}}_C(t)]& & ({T}.6)\end{array}$
$\begin{array}{ccc}\tilde{Y}(t)={{ϵ}}_L\tilde{L}(t)+(1-{{ϵ}}_L)\tilde{K}(t)& & ({T}.7)\end{array}$
Shares and parameters:
ϵL ≡WL/Y	Share of before-tax wage income in real net output
ωLL≡(1-L)/L	Ratio of leisure to labor
α	Pure rate of time preference
β	Probability of death

The dynamic part of the model consists of the aggregate capital stock (which is a predetermined variable) and aggregate consumption (which is a forward-looking or jump variable). By using labor demand (T.4), labor supply (T.6), and the aggregate production function (T.7), a useful ‘quasi-reduced form’ expression for aggregate output is obtained:

where $\tilde{Y}(t)\equiv {d}Y(t)/Y$, $\tilde{K}(t)\equiv {d}K(t)/K$, $\tilde{C}(t)\equiv {d}C(t)/C$, ${\tilde{t}}_L(t)\equiv {d}{t}_L(t)/(1-t_L)$, ${\tilde{t}}_C(t)\equiv {d}{t}_C(t)/(1+t_C)$, and ϕ is a crucial parameter representing the intertemporal labor supply effect:

where ω_LL(≡(1-L)/L≥0) is the leisure to labor ratio, which also represents the intertemporal substitution elasticity of labor supply. Note that ϕ = 1 if labor supply is exogenous (since L = 1, which implies that ω_LL = 0). Higher values of ω_LL correspond to a larger intertemporal labor supply effect. By using (3.1) and (T.5) in (T. 1)-(T.2), the dynamical system characterizing the evolution of the economy can be written in terms of $\tilde{K}$ and $\tilde{C}$:

where $\dot{\tilde{K}}(t)\equiv {d}\dot{K}(t)/K$, $\dot{\tilde{C}}(t)\equiv {d}\dot{C}(t)/C$ and the square matrix on the right-hand side of (3.3) is the Jacobian, J, evaluated at the steady state. The (potentially time-varying) forcing terms γ_k(t) and γ_C(t) are given by:

The following notational conventions are adopted. A deviation of a variable relative to the initial steady state is denoted by a tilde (∼), e.g., $\tilde{K}(t)\equiv {d}K(t)/K$. For the tax tildes are defined as follows: ${\tilde{t}}_i(t)\equiv {d}{t}_i(t)/(1-t_i)$, for i = K, L, t_C(t) ≡ dt(t)_C(t)/(1+t_C) and $\tilde{T}(t)\equiv {d}T(t)/Y$. For the time rate of change of a variable we use a tilde and a dot: $\dot{\tilde{K}}(t)\equiv {d}\dot{K}(t)/K=\dot{K}(t)/K$, since $\dot{K}(0)=0$ in the initial steady state, except ${\dot{\tilde{t}}}_C(t)\equiv (t){d}{\dot{t}}_C(t)/(1+t_C)$.

The equilibrium is a saddle point. Saddle-point stability holds because the determinant of the Jacobian matrix is negative:

where the characteristic roots are denoted by -λ₁ < 0 and λ₂ >0. The fact is used that in the general case of finite horizons (β > 0) the steady-state rate of interest exceeds the pure rate of time preference, i.e., r>α. Proposition 1 summarizes some stability results for the model that will prove useful in the discussion of tax policy changes.

B. Graphical Apparatus

In order to facilitate the discussion, the model is first summarized graphically by means of two schedules used in Figures 2 to 4 below. The CSE curve represents all points for which the goods market is in equilibrium with a constant capital stock $(\dot{\tilde{K}}(t)=0)$. The MKR curve is the MKR rule, which represents the steady-state aggregate Euler equation augmented for the turnover of generations $(\dot{\tilde{C}}(t)=0)$. The CSE curve is obtained by rewriting the first equation in (3.3) in steady-state form:

The Markets for Capital and Labor

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The Markets for Capital and Labor

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The Markets for Capital and Labor

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The Effects of the Capital Income Tax

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The Effects of the Capital Income Tax

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The Effects of the Capital Income Tax

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Exogenous Labor Supply and Finite Lives

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Exogenous Labor Supply and Finite Lives

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Exogenous Labor Supply and Finite Lives

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The Effects of the Labor Income and Consumption Tax

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The Effects of the Labor Income and Consumption Tax

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The Effects of the Labor Income and Consumption Tax

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which is unambiguously upward sloping in $(\tilde{C}(t),\tilde{K}(t))$ space. The dynamic forces operating on the economy off the CSE curve are obtained from the first equation of (3.3). Points above the CSE curve are associated with a falling capital stock over time because both goods consumption is too high and labor supply (and hence production) is too low. The opposite is the case for points below the CSE curve.

The MKR curve is obtained by using the steady-state version of the second equation of (3.3):

The term in square brackets on the right-hand side of (3.7) is positive (see (3.2)) but the sign of the term in square brackets on the left-hand side is ambiguous. The slope of the MKR curve is thus ambiguous because it is determined by two effects which work in opposite directions, viz. the generational turnover (GT) effect and the labor supply (LS) effect. The intuition behind these two effects can best be explained by looking at the two polar cases separately.