2.1. Price responses for general preference representations

Let $y$ and $o$ indicate the two periods of life. Then, $c^y$ and $c^o$ denote consumption when young and old. Leisure when young is $l$ . A time constraint requires $l\in [0,1]$ . The individual assesses bundles $\left (c^y,l,c^o\right )$ according to a preference relation $\succsim$ on $\left (c^y,l,c^o\right )\in \mathbb{R}_{+}\times [0,1]\times \mathbb{R}_{+}$ . The utility function $U$ represents these preferences, that is, $\left (c^y,l,c^o\right )\mapsto U\left (c^y,l,c^o\right )$ where $U:\mathbb{R}_{+}\times [0,1]\times \mathbb{R}_{+}\to \mathbb{R}$ is twice continuously differentiable, strongly increasing, strictly quasi-concave, and satisfies the Inada conditions at the origin.

Consumption serves as numéraire. With $w$ denoting the real wage, $R$ the real return factor on savings, $s$ , the periodic budget constraints read $c^y+w l+s \leq w$ and $c^o\leq R s$ . Since $U$ is strongly increasing the latter will hold as equalities and can be consolidated. This gives the intertemporal budget constraint in present value terms as $c^y+w l+c^o/R=w$ . Moreover, maximizing $U\left (c^y,l,c^o\right )$ subject to the latter constraint delivers the demand functions:

(2.1) \begin{align} c^y=c^y\left (w,R\right ),\quad l=l\left (w,R\right ),\quad \mbox{and}\quad c^o=c^o\left (w,R\right ). \end{align}

To bring the analysis closer to the remaining parts of this paper I assume henceforth that $U_{13}=U_{23}=0$ , that is, there is time separability between $c^y$ and $c^o$ and between $l$ and $c^o$ . Moreover, let me introduce the following elasticities

(2.2) \begin{align} \eta _1\equiv -c^y\frac{U_{11}}{U_1}\gt 0\quad \mbox{and} \quad \eta _3\equiv -c^o\frac{U_{33}}{U_3}\gt 0, \end{align}

which measure the curvature of $U$ with respect to $c^y$ and $c^o$ at the demands (2.1). Then, the demands for consumption and leisure when young of (2.1) satisfyFootnote ⁶

\begin{align*} \frac{d c^y}{dw}&\gtreqless 0\quad \Leftrightarrow \quad 1 -\frac{U_{12}}{U_1}\frac{1}{\eta _3}\frac{c^o}{Rw}+\left (\frac{-U_{22}}{U_2}+\frac{U_{12}}{U_1}\right )(1-l)\gtreqless 0,\\[3pt] \frac{d l}{dw}&\gtreqless 0\quad \Leftrightarrow \quad -1-\frac{\eta _1}{\eta _3}\frac{c^o}{Rc^y}+\left (\eta _1\frac{w}{c^y}+\frac{U_{21}}{U_1}\right )\left (1-l\right )\gtreqless 0,\\[3pt] \frac{d c^y}{dR}&\gtreqless 0\quad \Leftrightarrow \quad \left (1-\eta _3\right )\left (U_{22}-w U_{12}\right )\gtreqless 0,\\[3pt] \frac{d l}{dR}&\gtreqless 0\quad \Leftrightarrow \quad \left (1-\eta _3\right )\left (w U_{11}-U_{21}\right )\gtreqless 0. \end{align*}

The sign of the comparative statics $d c^y/dw$ and $d l/dw$ hinge on how the substitution effect, represented by the first two terms in the respective expressions, relates to the sum of the ordinary and the endowment income effect, represented by the third term.

Since changing $R$ does not affect the value of the labor endowment, the comparative statics $d c^y/dR$ and $d l/dR$ show only the tension between the substitution and the ordinary income effect. The strength of the latter hinges on the elasticity $\eta _3$ .Footnote ⁷

2.2. Price responses for specific preference representations

With the expressions above it is possible to compare the implications of three frequently used preference representations to the BK-gll representation. Throughout, $\beta \in (0,1)$ is the discount factor.

First, consider a preference representation with periodic utility functions of the King–Plosser–Rebelo log-log type (King, et al. (Reference King, Plosser and Rebelo1988)), henceforth KPR-ll. Then,

(2.3) \begin{align} U\left (c^y,l,c^o\right ) =\ln c^y +\kappa \ln \left (1-\phi \left (1- l\right )\right )+\beta \ln c^o, \end{align}

where $\kappa \gt 0$ and $\phi \gt \left (1+\beta \right )/\left (\kappa +1+\beta \right )\equiv \phi _c$ . Here, $U$ satisfies $\eta _1=\eta _3=1$ and $U_{12}=0$ . This implies that i) the demand for leisure does not hinge on the real wage and ii) neither the demands for consumption when young and leisure nor savings depend on the real return factor. Hence, the joint evolution of increasing real wages and declining hours of work over the long run cannot be replicated with this preference representation.

To generate a demand for leisure that increases in the real wage one may extend (2.3) and allow for subsistence consumption, $\bar{c}\gt 0$ , in both periods of life (see, e.g., Ohanian, et al. (Reference Ohanian, Raffo and Rogerson2008) or Bick, et al. (Reference Bick, Fuchs-Schuendeln and Lagakos2018)), that is,

(2.4) \begin{align} U\left (c^y,l,c^o\right ) =\ln \left (c^y-\bar{c}\right )+\kappa \ln \left (1-\phi \left (1- l\right )\right )+\beta \ln \left (c^o-\bar{c}\right ). \end{align}

Then, $\eta _1=c^y/\left (c^y-\bar{c}\right )\gt 1$ , $\eta _3=c^o/\left (c^o-\bar{c}\right )\gt 1$ , $U_{12}=0$ , and $dl/dw\gt 0$ . Yet, the wage elasticity of the supply of hours worked is not constant. Moreover, the demands for consumption when young and leisure increase whereas savings fall in the real return factor. The latter dependency complicates the derivation of analytical results and may not be consistent with the evidence on the determinants of life cycle savings (see, e.g., Bloom, et al. (Reference Bloom, Canning and Graham2003)).

Finally, consider periodic utility functions of the MaCurdy type (MaCurdy (Reference MaCurdy1981)). Then, the lifetime utility function reads

(2.5) \begin{align} U\left (c^y,l,c^o\right )=\frac{ \left (c^y\right )^{1-\sigma }-1}{1-\sigma }-\frac{(1-l)^{1+\epsilon }}{1+\epsilon }+\beta \frac{\left ( c^o\right )^{1-\sigma }-1}{1-\sigma }, \end{align}

where $\sigma \gt 1$ and $\epsilon \gt 0$ . Here, $\eta _1=\eta _3=\sigma \gt 1$ and $U_{12}=0$ . With the same argument as for subsistence consumption the demand for leisure increases in the real wage. Moreover, the wage elasticities of the supply of hours worked and of savings are constant. Hence, MaCurdy preferences are a special case of Boppart–Krusell preferences (see, Boppart and Krusell (Reference Boppart and Krusell2020), p. 138). Yet, as under (2.4), the demands for consumption when young and leisure increase whereas savings fall in the real return factor.

Table 1 collects the findings derived so far. Its last line states the corresponding properties for a utility representation with periodic BK-gll utility functions that will be derived below. Here, the elasticities (2.2) are constant and differ. From the expressions derived in Section 2.1, it is then immediate that (i) the demand for consumption and leisure when young as well as savings are independent of $R$ since $\eta _3=1$ . Moreover, (ii) the demands for consumption and leisure increase in the wage since $\eta _1\gt 1$ and $U_{12}\gt 0$ . Finally, (iii) the implied supply of hours worked declines in the real wage at a constant proportionate rate. Observe that the BK-gll preference representation is the only functional form that satisfies the properties (i), (ii), and (iii).

3.1. The household sector

The population at $t$ consists of $L_t$ young (cohort $t$ ) and $L_{t-1}$ old individuals (cohort $t-1$ ). Due to birth and other demographic factors the number of young individuals between two adjacent periods grows at rate $g_L\gt (\!-\!1)$ . For short, I shall refer to $g_L$ as the population growth rate.

When young, individuals supply labor, earn wage income, save, and enjoy leisure as well as the consumption good. When old, they retire and consume their wealth.

3.1.1. Preferences, utility, and the optimal plan of cohort $t$

For cohort $t$ , denote consumption when young and old by $c^y_t$ and $c^o_{t+1}$ , and leisure time enjoyed when young by $l_t$ . I normalize the maximum per-period time endowment supplied to the labor market to unity. Then, $1-l_t=h_t$ , where $h_t\in [0,1]$ is hours worked when young. Individuals of all cohorts assess bundles $\left (c^y_t, l_t,c^o_{t+1}\right )$ according to a lifetime utility function:

(3.1) \begin{align} U\left (c^y_t,l_t,c^o_{t+1}\right )=\ln c^y _t+\kappa \ln \left (1-\phi \left (1- l_t\right ) \left (c^y_t\right )^{\frac{\nu }{1-\nu }}\right )+\beta \ln c^o_{t+1}, \end{align}

where $\kappa \in (0,1]$ , $\phi \gt 0$ , and $\nu \in (0,1)$ are parameters to be interpreted below, and $\beta \in (0,1)$ is the discount factor. Let $\mathcal{D}$ denote the domain of $U$ . Since the natural logarithmic function requires a strictly positive argument, $\mathcal{D}$ cannot include the set $\mathcal{B}=\{\left (c_t^y, l_t\right )|1-\phi \left (1- l_t\right ) \left (c_t^y\right )^{\frac{\nu }{1-\nu }}\leq 0\}$ . Hence, $\mathcal{D}=\{\left (c_t^y, l_t,c_{t+1}^o\right )\in \mathbb{R}_{++}\times [0,1]\times \mathbb{R}_{++}\setminus \mathcal{B}\}$ .Footnote ⁸

The function $U$ evaluates consumption and leisure in both periods of life according to a BK-gll utility function. Retirement means that leisure when old, $l^o_{t+1}$ , is equal to unity. Accordingly, the term $\beta \kappa \ln \left (1-\phi \left (1- l^o_{t+1}\right ) \left (c^o_{t+1}\right )^{\frac{\nu }{1-\nu }}\right )$ disappears from $U$ . For ease of notation, I follow Boppart and Krussell and use henceforth

(3.2) \begin{align} x_t\equiv \left (1- l_t\right ) \left (c_t^y\right )^{\frac{\nu }{1-\nu }}. \end{align}

The term $\kappa \ln \left (1-\phi x_t\right )$ reflects the disutility of labor when young. It is more pronounced the greater $\kappa$ or $\phi$ . These parameters may be associated with institutional, cultural, or geographical features of the labor market that affect the disutility of labor in addition to the amount of hours worked and the level of consumption. For instance, in an economy with demanding occupational safety regulations $\kappa$ and $\phi$ may be lower than in an economy without such regulations. Similarly, these parameters should be low if the labor market gives rise to a good matching between individual career aspirations and actual occupations. Alternatively, as suggested respectively by Weber (Reference Weber1930) and Landes (Reference Landes1998), $\kappa$ and $\phi$ may reflect a prevailing work ethic or the climatic conditions under which labor is done. Finally, the parameter $\nu$ determines how the disutility of labor increases with the level of consumption.Footnote ⁹ In the context of Proposition1 below we will see that $\nu \gt 0$ is key for the income effect of a wage hike on the demand for leisure to dominate the substitution effect. In the limit $\nu \to 0$ , $U$ boils down to the KPR-ll preference representation (2.3) for which the income and the substitution effect cancel.

The following lemma summarizes relevant properties of the lifetime utility function (3.1).

Lemma 1. (Properties of $U$ )

The lifetime utility function $U$ of ( 3.1 ) has the following properties:

1. 1. The marginal utility of consumption when young, $U_1$ , satisfies $\lim _{c_t^y\to 0}U_1=\infty$ and is strictly positive for pairs $\left (c_t^y,l_t\right )\in \mathcal{D}$ that also satisfy

   (3.3) \begin{align} \frac{1-\nu }{\phi \left (1-\nu (1-\kappa )\right )} \gt x_t. \end{align}
   Moreover, $U_{11}\lt 0$ .

2. 2. The marginal utility of leisure when young, $U_2$ , satisfies $\lim _{l_t\to 0}U_2\lt \infty$ , and is strictly positive. Moreover, $U_{22}\lt 0$ .

3. 3. $U\left (c_t^y,l_t,c_{t+1}^o\right )$ is strictly concave on $\mathcal{D}$ if

   (3.4) \begin{align} \frac{1-2 \nu +(1-\kappa )\nu ^2}{\phi (1-\nu )\left (1-\nu (1-\kappa )\right )}\gt x_t. \end{align}

Hence, as to consumption when young, $U$ satisfies the Inada condition at the origin. However, the interaction between consumption and leisure implies that $U_1$ is negative if $c_t^y$ becomes too large. As to leisure when young, $U$ is monotonically increasing without satisfying the Inada condition at the origin. Moreover, $U$ is concave in $c_t^y$ and $l_t$ though not necessarily jointly concave in $\left (c_t^y,l_t\right )$ .

Let $w_t\gt 0$ denote the real wage per hour worked and $r_t\gt (\!-\!1)$ the real rental rate per unit of capital. Then, $R_{t+1}\equiv 1+r_{t+1}-\delta \gt (\!-\!1)$ is the perfect foresight real return factor on savings, that is, the total amount of consumption that an investment of one unit of capital at $t-1$ generates at $t$ . I refer to $\left (c^y_t, l_t, c^o_{t+1}, s_t\right )$ as the plan of cohort $t$ . The optimal plan of cohort $t$ solves

(3.5) \begin{align} \max _{\left (c^y_t, l_t, c^o_{t+1}, s_t\right )\in \mathcal{D}\times \mathbb{R}} \ln c^y _t+\kappa \ln \left (1-\phi \left (1- l_t\right ) \left (c^y_t\right )^{\frac{\nu }{1-\nu }}\right )+\beta \ln c^o_{t+1} \end{align}

subject to the per-period budget constraints:

(3.6) \begin{align} c^y_t+s_t\leq w_t (1-l_t) \quad \mbox{and}\quad c^o_{t+1}\leq R_{t+1} s_t. \end{align}

The following assumption assures the existence of a unique optimal plan for all $w_t\gt 0$ and $R_{t+1}\gt (\!-\!1)$ .

Assumption 1. (Upper Bound on $\nu$ )

It holds that

\begin{align*} 0\lt \nu \lt \bar{\nu }\left (\beta, \kappa \right )\equiv \frac{3+\beta -\sqrt{\left (1+\beta \right )^2+4\kappa }}{2\left (2+\beta -\kappa \right )}. \end{align*}

The function $\bar{\nu }:[0,1]^2\to \mathbb{R}_+$ takes on strictly positive values. Moreover, it is monotonically declining in $\beta$ and in $\kappa$ with $\bar{\nu }\left (0,0\right )=1/2$ and $\bar{\nu }\left (1,1\right )=1-1/\sqrt{2}\approx 0.2923$ . In other words, Assumption1 says that $\nu$ must not be too large.

The following proposition reveals that the optimal plan involves a corner solution $l=0$ if the real wage is lower than the following critical value:

\begin{align*} w_c \equiv \left (\frac{\left (1+\beta \right )\left (1-\nu \right )}{\left (\phi \left (\kappa +\left (1+\beta \right )\left (1-\nu \right )\right )\right )^{1-\nu }\left (1-\nu \left (1+\beta \right )\right )^\nu }\right )^{\frac{1}{\nu }}. \end{align*}

Proposition 1. (Optimal Plan of Cohort $t$ )

Suppose Assumption 1 holds. Then, for cohorts $t=1,2,\ldots, \infty$ , prices $w_t\gt 0$ , and $R_{t+1}\gt (\!-\!1)$ the optimal plan involves continuous, piecewise defined functions:

(3.7) \begin{align} h_t=h\left (w_t\right ),\quad c_t^y=c^y\left (w_t\right ),\quad c_{t+1}^o=c^o\left (w_t, R_{t+1}\right ),\quad \mbox{and}\quad s_t=s\left (w_t\right ). \end{align}

Moreover, there exists a critical wage, $w_c$ , such that

1. 1. if $w_t\leq w_c$ then $l_t=0$ , $h_t=1$ , and $c^y\left (w_t\right )$ is implicitly given by:

   (3.8) \begin{align} c^y_t=\left (\frac{\left (1-\nu \right )\left (1-\phi \left (c^y_t\right )^{\frac{\nu }{1-\nu }}\right )-\nu \kappa \phi \left (c^y_t\right )^{\frac{\nu }{1-\nu }}}{\left (1-\nu \right )\left (1+\beta \right )\left (1-\phi \left (c^y_t\right )^{\frac{\nu }{1-\nu }}\right )-\nu \kappa \phi \left (c^y_t\right )^{\frac{\nu }{1-\nu }}}\right )w_t. \end{align}
   Moreover, $s_t=w_t-c^y\left (w_t\right )= s\left (w_t\right )$ and $c^o_{t+1}=R_{t+1} s\left (w_t\right )= c^o\left (w_t, R_{t+1}\right )$ ;

2. 2. if $w_t\geq w_c$ then $l_t\geq 0$ , $h_t\leq 1$ and

   \begin{align*} h_t= w_c^\nu w_t^{-\nu }, &\quad c^y_t=\frac{1-\nu \left (1+\beta \right )}{\left (1+\beta \right )\left (1-\nu \right )} w_c^\nu w_t^{1-\nu },\\[5pt] s_t=\frac{\beta }{\left (1+\beta \right )\left (1-\nu \right )}w_c^\nu w_t^{1-\nu },&\quad c^o_{t+1}=\frac{\beta R_{t+1}}{\left (1+\beta \right )\left (1-\nu \right )}w_c^\nu w_t^{1-\nu }. \end{align*}

Finally, for members of cohort $0$ , we have $c^o_1=R_{1} s_0\gt 0$ where $s_0\gt 0$ is given.

Proposition1 makes two important points. First, it establishes that the optimal plan hinges on the level of the real wage. If the real wage is below the critical level $w_c$ , then individuals supply their entire time endowment to the labor market. Henceforth, I call this case Regime 0. I refer to Regime 1 if the real wage exceeds $w_c$ . Then individuals supply less than their time endowment to the labor market. Hence, the individual labor supply is indeed piecewise defined, yet, as established in the proposition, continuous at $w=w_c$ . As the real wage increases above its critical level, the supply of hours worked declines at the constant proportionate rate $\nu\in(0,1)$ . This finding suggests that the standard assumption of an inelastic labor supply made in almost all growth models is in fact most plausible for low-wage, i.e., poor economies.

The implied behavioral pattern makes intuitive sense. When wages and incomes are low then the individual demand for consumption goods satisfies basic needs. The demand for leisure is zero since the only way to earn a decent income is by working the maximum of available hours. Rising wages and incomes allow people to adequately satisfy their basic needs, to consume beyond these needs and, eventually, to demand leisure.

The critical wage level, $w_c$ , and, hence, the consumption level $c^y\left (w_c\right )$ above which the demand for leisure becomes positive, reflects an intricate relationship among preference parameters. As mentioned above, these parameters may depend on institutional, cultural, or geographical factors that differ across countries. For instance, $w_c$ and $c^y\left (w_c\right )$ decline in $\kappa$ and $\phi$ . Hence, of two otherwise identical economies the readiness to reduce the labor supply in response to an increasing wage begins at a lower wage level and a correspondingly lower consumption level in the economy with lower occupational safety regulations. Mutatis mutandis, a similar argument can be made for economies that differ in their work ethic or in their climatic conditions.

To understand why the individual labor supply is piecewise defined recall that the utility-maximizing plan involving $\left (c^y_t,l_t,c^o_{t+1}\right )\gg 0$ satisfies the first-order condition $U_2/U_1=w_t$ . However, when the optimal plan involves $l_t=0$ then $U_2/U_1\leq w_t$ . Hence, in Regime 0 it holds that $U_2/U_1\leq w_t \leq w_c$ with equality only if $w_t = w_c$ .Footnote ¹⁰ In other words, $w_c$ has an interpretation as the marginal rate of substitution, $U_2/U_1$ , evaluated at the optimal plan at $w_t=w_c$ .

Second, Proposition1 shows that the individual supply of hours worked, consumption when young, and individual savings are independent of the real return factor on savings. This follows from Section 2. Since the lifetime utility function (3.1) features $\eta _3=1$ the substitution and the income effect in the comparative statics for $c^y_t$ and $l_t$ vanish. Then, the budget constraint when young implies that $s_t$ becomes also independent of $R_{t+1}$ . Observe that the relevant comparative statics are independent of whether $l_t= 0$ or $l_t\gt 0$ . Hence, they apply to the optimal plan under both regimes. An immediate implication of these findings is that consumption and leisure when young are demand complements in Regime 1, that is, $d l/d w\gt 0$ and $d c^y/d w\gt 0$ .Footnote ¹¹

The intertemporal trade-off between consumption when young and consumption when old is governed by the first-order condition $U_1/U_3= R_{t+1}$ . Evaluated at optimal pairs $\left (c^y_t,l_t\right )$ this trade-off delivers the Euler equation in Regime 1 as:

(3.9) \begin{align} \frac{c^o_{t+1}}{c^y_t}=\frac{\beta R_{t+1}}{1-\nu (1+\beta )}. \end{align}

The latter states the desired consumption growth factor of a member of cohort $t$ . The parameter $\nu$ reflects the disutility of consumption when young associated with the labor supply that shows up in the second term of $U$ . Its presence weakens the tendency to smooth consumption over the life cycle.

Regime 1 of Proposition1 exhibits another intuitive property of the optimal plan: $c^y_t$ , $s_t$ , and $c^o_{t+1}$ are proportionate to the wage income, $w_t h_t$ . In particular, one finds that

(3.10) \begin{align} c^y_t=\frac{1-\nu (1+\beta )}{(1+\beta )(1-\nu )}w_t h_t\quad \mbox{and}\quad s_t=\frac{\beta }{\left (1+\beta \right )\left (1-\nu \right )}w_t h_t. \end{align}

Hence, ceteris paribus, the marginal (and average) propensity to consume when young declines in $\nu$ whereas the marginal propensity to save out of wage income increases in $\nu$ .

Next, consider the role of $\nu$ for the response of leisure when young to changes in the real wage. Inspection of the expression for $dl/dw$ from Section 2.1 reveals that leisure is a normal good under the lifetime utility function (3.1) since

\begin{align*} \eta _1=\frac{\kappa \left (1-\nu \left (1+\left (1+\beta \right )\left (1-2\nu \right )\right )\right )+\left (1+\beta \right )^2\left (1-\nu \right )\nu ^2}{\kappa (1-\nu ) (1- \nu (1+\beta ))}\gt 1, \end{align*}

$\eta _3=1$ , and $U_{12}\gt 0$ (see Footnote 9). Using the consolidated budget constraint one readily verifies that

\begin{align*} \frac{d l}{dw}\gtreqless 0\quad \Leftrightarrow \quad -1+\eta _1+\frac{U_{21}}{U_1}\left (1-l\right )\gtreqless 0, \end{align*}

where

\begin{align*} \frac{U_{21}}{U_1}(1-l)=\frac{\nu (1+\beta )(\kappa +(1+\beta )(1-\nu ))}{\kappa \left (1-\nu (1+\beta )\right )}\gt 0. \end{align*}

As $\lim _{\nu \to 0}\eta _1=1$ and $\lim _{\nu \to 0}U_{21}(1-l)/U_{1}=0$ , it is clear that $dl/dw\gt 0$ results since $\nu \gt 0$ implies both $\eta _1\gt 1$ and $U_{21}\gt 0$ . As mentioned above, in the limit $\nu \to 0$ , the lifetime utility function $U$ becomes part of the KPR class and the demand for leisure will no longer respond to changes in the real wage.Footnote ¹²

One readily verifies that the response of the optimal plan to changes in factor prices satisfies

(3.11) \begin{align} h'\left (w_t\right )\leq 0,\quad \left (c^y\right )'\left (w_t\right )\gt 0,\quad s'\left (w_t\right )\gt 0,\notag \\[5pt] c^o_1\left (w_t, R_{t+1}\right )\gt 0,\quad c^o_2\left (w_t, R_{t+1}\right )\gt 0. \end{align}

For Regime 0, this is to be expected. As $h'\left (w_t\right )= 0$ , a higher real wage increases real income one-to-one. Then, consumption smoothing requires that the higher income is used to increase consumption when young and old, hence savings. For Regime 1, a similar intuition holds since

\begin{align*} \frac{d\ln \left (w_t h_t\right )}{d \ln w_t}=1-\nu \gt 0, \end{align*}

that is, the proportionate increase in the wage income induced by a higher wage is still positive even though the labor supply declines. Clearly, only $c^o_{t+1}$ increases in response to a higher $R_{t+1}$ .

Finally, let me turn to the comparative statics of the optimal plan of Proposition1 with respect to the preference parameters $\phi$ and $\beta$ .Footnote ¹³

Corollary 1. (Comparative Statics of the Optimal Plan)

Consider the optimal plan of Proposition 1 at given prices $\left (w_t, R_{t+1}\right )$ .

For Regime 0, it holds that

\begin{align*} \frac{\partial c^y_t}{\partial \phi }\lt 0,\quad \frac{\partial c^o_{t+1}}{\partial \phi }\gt 0, \quad \frac{\partial s_t}{\partial \phi }\gt 0,\\[5pt] \frac{\partial c^y_t}{\partial \beta }\lt 0,\quad \frac{\partial c^o_{t+1}}{\partial \beta }\gt 0,\quad \frac{\partial s_t}{\partial \beta }\gt 0. \end{align*}

For Regime 1, it holds that

\begin{align*} \frac{\partial h_t}{\partial \phi }\lt 0,\quad \frac{\partial c^y_t}{\partial \phi }\lt 0,\quad \frac{\partial c^o_{t+1}}{\partial \phi }\lt 0, \quad \frac{\partial s_t}{\partial \phi }\lt 0,\\[5pt] \frac{\partial h_t}{\partial \beta }\gt 0,\quad \frac{\partial c^y_t}{\partial \beta }\lt 0,\quad \frac{\partial c^o_{t+1}}{\partial \beta }\gt 0,\quad \frac{\partial s_t}{\partial \beta }\gt 0. \end{align*}

Corollary1 shows that the comparative statics properties of the optimal plan hinge on whether the supply of hours worked responds to the respective parameter change or not. First, consider Regime 1. For a greater $\phi$ the disutility of labor is more pronounced. Accordingly, the labor supply falls. Consumption smoothing dictates that the concomitant decline in the wage income reduces consumption in both periods of life, hence, savings decline. Consumption when young is further reduced since the marginal utility of $c_t^y$ falls in $\phi$ . A greater $\beta$ increases the value of consumption when old. Therefore, $c^o_{t+1}$ increases at the expense of the demand for leisure and for consumption when young. Accordingly, the labor supply and savings increase.

In Regime 0 parameter changes do not affect the labor supply. However, a greater $\phi$ reduces the marginal utility of consumption when young whereas a greater $\beta$ increases the value of consumption when old. Hence, unlike in Regime 1, for both parameter changes, $c^y_t$ falls whereas $s_{t}$ and $c^o_{t+1}$ increase.

3.1.2. Some orders of magnitude

The validity of Proposition1 hinges on the parameter restriction of Assumption1. The purpose of this section is to show by example that this assumption can be satisfied for reasonable magnitudes of key parameters of the model. To see this, set $\kappa =1$ and let a period correspond to 30 years. Moreover, suppose that hours worked per worker and the real wage grow at constant annual rates denoted by $g_h$ and $g_w$ . Then, the economy is in Regime 1 and

\begin{align*} \frac{h_{t+1}}{h_t}=\left (1+g_h\right )^{30}\quad \mbox{and}\quad \frac{w_{t+1}}{w_t}=\left (1+g_w\right )^{30}. \end{align*}

According to Proposition1 the growth rates $g_h$ and $g_w$ are linked, that is,

\begin{align*} \frac{h_{t+1}}{h_t}=\left (\frac{w_{t+1}}{w_t}\right )^{-v}\quad \mbox{or}\quad \left (1+g_h\right )^{30}=\left (1+g_w\right )^{-30\nu }. \end{align*}

The latter gives an estimate of $\nu$ as:

\begin{align*} \nu =-\frac{\ln \left (1+g_h\right )}{\ln \left (1+g_w\right )}. \end{align*}

It follows that Assumption1 is satisfied whenever

(3.12) \begin{align} \nu =-\frac{\ln \left (1+g_h\right )}{\ln \left (1+g_w\right )}\lt \bar{\nu }\left (\beta, 1\right ). \end{align}

Boppart and Krusell (Reference Boppart and Krusell2020) estimate that $g_h=-0.57\%$ . Since $\bar{\nu }\left (\beta, 1\right )$ is monotonically declining in $\beta$ , condition (3.12) is easier to satisfy the smaller $\beta$ and the larger $g_w$ . Hence, for $\bar{\nu }\left (1,1\right )= 1-1/\sqrt{2}$ a sufficient condition for (3.12) to hold is

\begin{align*} \nu =-\frac{\ln .9943}{\bar{\nu }\left (1,1\right )}\lt \ln \left (1+g_w\right )\quad \mbox{or}\quad 1.971\%\lt g_w. \end{align*}

If the annual discount factor is equal to $0.96$ as suggested by Prescott (Reference Prescott1986) then $\beta =0.294$ and $\bar{\nu }\left (0.294,1\right )=0.352$ so that (3.12) is satisfied for all $g_w\gt 1.753\%$ .Footnote ¹⁴

Hence, for reasonable values of $\beta$ and $g_w$ sufficiently close to $2\%$ and above, Assumption1 will be satisfied.

3.2. Firms

At all $t$ , the production sector can be represented by a single competitive firm with access to the production function

(3.13) \begin{align} Y_t=\Gamma K_t^\gamma \left (A_t H_t\right )^{1-\gamma },\quad \Gamma \gt 0,\quad 0\lt \gamma \lt 1. \end{align}

Here, $K_t$ is physical capital and $H_t$ the amount of hours of work employed by the firm. Technological knowledge is represented by $A_t$ and advances exogenously at rate $g_A\gt 0$ . Accordingly, $A_t=(1+g_A)^{t-1}A_{1}$ , with $A_1\gt 0$ given. The productivity parameter $\Gamma \gt 0$ may reflect cross-country differences in geography, technical and social infrastructure that affect the transformation of capital and efficient hours worked into the manufactured good.

In each period, the firm chooses the amounts of capital, $K_t$ , and of hours of work, $H_t$ , to maximize the net-present value of profits. Doing so, it takes the evolution of $A_t$ as given. Void of intertemporal considerations, the respective first-order conditions read

(3.14) \begin{align} w_t=\Gamma (1-\gamma ) K_t^\gamma A_t^{1-\gamma } H_t^{-\gamma }\quad \mbox{and}\quad r_t=\Gamma \gamma K_t^{\gamma -1} \left (A_t H_t\right )^{1-\gamma }, \end{align}

where $r_t$ is the real rental rate of capital at $t$ .

4.1. Definition, existence, and uniqueness

A price system corresponds to a sequence $\left \{w_t,r_t\right \}_{t=1}^\infty$ . An allocation is a sequence $\{c^y_t,l_t,c^o_t,s_t,Y_t,H_t,K_t\}_{t=1}^\infty$ that comprises a plan $\{c^y_t,l_t,c^o_{t+1},s_t\}_{t=1}^\infty$ for all cohorts, consumption of the old at $t=1$ , $c_1^o$ , and a plan for the production sector $\{Y_t,H_t,K_t\}_{t=1}^\infty$ .

For an exogenous evolution of the labor force, $L_{t}=L_1\left (1+g_L\right )^{t-1}$ with $L_1\gt 0$ , an exogenous evolution of technological knowledge, $A_t=A_1(1+g_A)^{t-1}$ with $A_1\gt 0$ , and a given initial level of capital, $K_1\gt 0$ , an intertemporal general equilibrium with perfect foresight corresponds to a price system and an allocation that satisfy the following conditions for all $t=1,2,\ldots, \infty$ :

1. (E1) The plan of each cohort satisfies Proposition1.

2. (E2) The production sector satisfies (3.14).

3. (E3) The market for the manufactured good clears, that is,

   (4.1) \begin{align} L_{t-1}c_t^o+L_{t}c_t^y+I_t=Y_t+\left (1-\delta \right )K_t, \end{align}
   where $I_t$ is aggregate capital investment.

4. (E4) There is full employment of labor, that is,

   (4.2) \begin{align} H_t= L_t h_t. \end{align}

(E1) guarantees the optimal behavior of the household sector under perfect foresight. Since the old own the capital stock, their consumption at $t=1$ is $L_0c^o_1=R_1 K_1=\left (1+r_1-\delta \right )K_1$ and $s_0=K_1/L_0$ . (E2) assures the optimal behavior of the production sector and zero profits. (E3) states that the aggregate demand for the manufactured good at $t$ is equal to its produced output plus the non-depreciated capital stock. According to (E4) the demand for hours worked must be equal to the supply.

The labor market requires a special treatment. Since both the aggregate demand for hours worked and the aggregate supply of hours worked are decreasing in the real wage there may be none, one, or multiple wage levels at which demand is equal to supply. To address this issue let me refer to the first condition of (3.14) as the firms’ aggregate demand for hours worked at $t$ and restate it as:

(4.3) \begin{align} H_t^d=K_t A_t^{\frac{1-\gamma }{\gamma }}\left (\frac{\Gamma (1-\gamma )}{w_t}\right )^{\frac{1}{\gamma }}\equiv H_t^d\left (w_t\right ). \end{align}

Let $H_t^s=L_t h_t$ denote the aggregate supply of hours worked at $t$ . Using Proposition1 gives

(4.4) \begin{align} H_t^s=\left \{\begin{array}{cc} L_t w_c^\nu w_t^{-\nu }& \mbox{if $w_t\geq w_c$} \\[5pt] L_t \cdot 1 & \mbox{if $0\lt w_t\leq w_c$} \end{array}\right \}\equiv H_t^s\left (w_t\right ). \end{align}

Then, the labor market equilibrium, $\left (\hat{w}_t,\hat{H}_t\right )$ , satisfies $\hat{H}_t=H_t^d\left (\hat{w}_t\right )=H_t^s\left (\hat{w}_t\right )$ . To simplify the notation let $k_t\equiv K_t/\left (A_t^{1-\nu }L_t\right )$ and define the time-varying critical value:

(4.5) \begin{align} k_{c,t}\equiv \left [\frac{w_c}{\Gamma (1-\gamma )A_t^{1-\gamma \nu }}\right ]^{\frac{1}{\gamma }}. \end{align}

Then, one readily verifies that at all $t$ there is a unique labor market equilibrium where the equilibrium real wage is given by:

(4.6) \begin{align} \hat{w}_t=\left \{\begin{array}{cc} A_t \cdot \left (\frac{\Gamma (1-\gamma )}{w_c^{\gamma \nu }}\right )^{\frac{1}{1-\gamma \nu }}\cdot k_t^{\frac{\gamma }{1-\gamma \nu }}&\mbox{ if $k_t \geq k_{c,t}$}, \\[5pt] A_t^{1-\nu \gamma } \cdot \Gamma (1-\gamma )\cdot k^\gamma _t & \mbox{ if $k_t\leq k_{c,t}$,} \end{array}\right . \end{align}

and the equilibrium amount of hours worked is

(4.7) \begin{align} \hat{H}_t=\left \{\begin{array}{cc} \frac{L_t}{A_t^\nu }\cdot \left (\frac{w_c}{\Gamma (1-\gamma )}\right )^{\frac{\nu }{1-\gamma \nu }}\cdot k_t^{\frac{-\gamma \nu }{1-\gamma \nu }} & \mbox{if $k_t\geq k_{c,t}$}, \\[5pt] L_t \cdot 1& \mbox{ if $k_t\leq k_{c,t}$.} \end{array}\right . \end{align}

Existence and uniqueness follow from two properties of the labor market that are illustrated in Figure 4.1. First, since $\nu$ is quite small the individual and the aggregate supply of hours worked, $h\left (w_t\right )$ and $H_t^s\left (w_t\right )$ , is fairly flat. Second, the image of the aggregate demand for hours worked, $H_t^d\left (w_t\right )$ , is $\mathbb{R}_{++}$ since the aggregate production function satisfies both Inada conditions.

If $k_t\geq k_{c,t}$ , then $\hat{w}_1\geq w_c$ and the labor market equilibrium is in Regime 1. Intuitively, for a given aggregate supply of hours worked this is the case if the aggregate demand for hours worked is large (see $\left (\hat{w}_t^1, \hat{H}_t^1\right )$ in Figure 4.1). From (4.3) the latter is more likely the greater $K_t$ , $A_t$ , or $\Gamma$ . Intuitively, modern industrialized economies should possess these features. Conversely, economies with a low demand for hours worked would find their labor market equilibrium in Regime 0 (see $\left (\hat{w}_t^0, \hat{H}_t^0\right )$ where $\hat{H}_t^0=L_t\cdot 1$ in Figure 4.1).

For a given aggregate demand for hours worked the labor market equilibrium is more likely to be in Regime 1 the smaller the total amount of workers, that is, the smaller $L_t$ . Intuitively, when $L_t$ falls then the aggregate supply of hours worked shifts downward. Labor becomes scarcer so that the equilibrium wage increases. Then, even for a low aggregate demand of hours worked such as $H_t^{d0}$ an equilibrium wage in Regime 1 is possible.

Figure 4.1. The labor market equilibrium of period $t$ . Note: If the aggregate demand for hours worked is $H^{d1}_t$ then the labor market equilibrium is $\left (\hat{w}_t^1, \hat{H}_t^1\right )$ . The individual supply of hours worked is in Regime 1, that is, it falls in $w_t$ , and aggregate demand for hours worked is high. If the aggregate demand for hours worked is $H^{d0}_t$ then the labor market equilibrium is $\left (\hat{w}_t^0, \hat{H}_t^0\right )$ where $\hat{H}_t^0=L_t\cdot 1$ . The individual supply of hours worked is in Regime 0, that is, it does not hinge on $w_t$ , and the aggregate demand for hours worked is low.

Finally, observe that the equilibrium conditions (E1) – (E4) imply for all $t$ and both regimes that aggregate saving equals capital investment, that is,

(4.8) \begin{align} s_t L_t=I_t = K_{t+1}. \end{align}

The following proposition states and proves the existence and the uniqueness of the intertemporal general equilibrium.

4.2. Dynamical system for Regime 1 and steady-state analysis

Using Proposition1, the labor market equilibrium, and the capital market equilibrium (4.8) reveals that in Regime 1 the intertemporal general equilibrium may be studied by means of the sequence $\{k_t\}_{t=1}^\infty$ . This state variable will be constant in steady state and equal to

(4.9) \begin{align} k^*\equiv w_c^\nu \left [\frac{\beta \left (\Gamma (1-\gamma )\right )^{\frac{1-\nu }{1-\gamma \nu }}}{\left (1+\beta \right )\left (1-\nu \right )\left (1+g_L\right )\left (1+g_A\right )^{1-\nu }}\right ]^{\frac{1-\gamma \nu }{1-\gamma }}. \end{align}

The following assumption serves the purpose of this section.

Assumption 2. The initial conditions $\left (K_1, L_1,A_1\right )$ satisfy

\begin{align*} A_1\left (\frac{K_1}{A_1 L_1}\right )^\gamma \Gamma (1-\gamma )\geq w_c \end{align*}

and

\begin{align*} A_1\left [\frac{\beta \left [\Gamma \left (1-\gamma \right )\right ]^{\frac{1}{\gamma }}}{\left (1+\beta \right )\left (1-\nu \right )\left (1+g_L\right ) \left (1+g_A\right )^{1-\nu }}\right ]^{\frac{\gamma }{1-\gamma }}\gt w_c. \end{align*}

The first inequality guarantees that the equilibrium wage in $t=1$ satisfies $\hat{w}_1\geq w_c$ . Hence, the economy starts in Regime 1. In terms of the state variable, $k_t$ , and its critical value, $k_{c,t}$ , defined in (4.5) this means that $k_1\geq k_{c,1}$ . The second inequality ensures $k^*\gt k_{c,1}$ , that is, over time the sequence $\{k_t\}_{t=1}^\infty$ remains in Regime 1.Footnote ^{15 ,} Footnote ¹⁶

Proposition 3. (Dynamical System - Regime 1)

Suppose the initial conditions $\left (K_1, L_1,A_1\right )$ are such that Assumption 2 holds. Then, the transitional dynamics of the intertemporal general equilibrium is given by a unique and monotonous sequence $\{k_t\}_{t=1}^\infty$ , generated by the difference equation:

(4.10) \begin{align} k_{t+1}=\frac{\beta \left [w_c^{\nu (1-\gamma )} \left (\Gamma (1-\gamma )\right )^{1-\nu }\right ]^{\frac{1}{1-\gamma \nu }}}{(1+\beta )(1-\nu )(1+g_L)(1+g_A)^{1-\nu }} \cdot k_t^{\frac{\gamma (1-\nu )}{1-\gamma \nu }} \end{align}

with

(4.11) \begin{align} \lim _{t\to \infty }k_t=k^*. \end{align}

Hence, in Regime 1, the evolution of the state variable is governed by the difference equation (4.10). Since $\gamma (1-\nu )/\left (1-\gamma \nu \right )\lt 1$ the sequence generated by this equation is monotonous and the steady state is stable. This is illustrated in Figure 4.2.

Figure 4.2. The dynamical system under Regime 1. Note: Under Assumption2 the steady state of the equilibrium difference equation (4.10), $k^*$ , is unique and stable for $k_1\gt k_{c,1}$ .

The key parameter in the difference equation (4.10) is $\nu$ . It affects the sequence $\{k_t\}_{t=1}^\infty$ through four channels. To see this let me write (4.8) using (3.10) and Proposition1 as:

(4.12) \begin{align} \frac{\beta }{\left (1+\beta \right )\left (1-\nu \right )}\hat{w}_t h\left (\hat{w}_t\right )L_t=K_{t+1}, \end{align}

where $\hat{w}_t$ is the equilibrium wage for Regime 1. Hence, the factor $(1-\nu )$ in the denominator of (4.10) shows the effect of $\nu$ on the marginal propensity to save. A greater $\nu$ increases the fraction of the wage income that is saved and invested, hence, $K_{t+1}$ increases. This is the first channel.

Next, observe that Proposition1 and (4.6) imply that the equilibrium individual wage income can be expressed as:

(4.13) \begin{align} \hat{w}_t h\left (\hat{w}_t\right )=A_t^{1-\nu }\left [w_c^{\nu (1-\gamma )} \left (\Gamma (1-\gamma )\right )^{1-\nu }\right ]^{\frac{1}{1-\gamma \nu }}k_t^{\frac{\gamma (1-\nu )}{1-\gamma \nu }}. \end{align}

The remaining three channels show how $\nu$ affects the difference equation (4.10) through this expression.

The second channel is related to the factor $A_t^{1-\nu }$ . It captures that, given $k_t$ , a greater $A_t$ increases the equilibrium wage and reduces the individual supply of hours worked. This channel shows up as $\left (1+g_A\right )^{1-\nu }$ in the denominator of (4.10) since the latter equation expresses (4.12) in units of $A^{1-\nu }_{t+1}L_{t+1}$ . As $g_A\gt 0$ a greater $\nu$ implies a greater $k_{t+1}$ .

The third channel reflects all effects of $\nu$ related to the bracketed term in the numerator of (4.10). From (4.13) it is clear that this term shows how preference and technology parameters affect the equilibrium individual wage income given $\left (K_t,A_t,L_t\right )$ .

Finally, $\nu$ impacts on how the state variable affects the equilibrium individual wage income. Increasing $\nu$ augments the exponent of $k_t$ on the right-hand side of (4.10) and accelerates the process of convergence toward the steady state. Indeed, the speed of convergence defined as

\begin{align*} -\frac{\partial \ln \left (\frac{k_{t+1}}{k_t}\right )}{\partial \ln k_t}=\frac{1-\gamma }{1-\gamma \nu }\gt 0 \end{align*}

increases in $\nu$ .

The following proposition characterizes the steady state of Regime 1.

Proposition 4. (Properties of the Steady State - Regime 1)

Along the steady-state path the growth rate of the real wage is $g_w=g_A\gt 0$ , and the real rental rate of capital is constant, that is, $\hat{r}_t=\hat{r}$ . Moreover, it holds that

\begin{align*} a)\quad &\frac{h_{t+1}}{h_{t}}=\left (1+g_A\right )^{-\nu },\quad \frac{\hat{H}_{t+1}}{\hat{H}_{t}}=\left (1+g_A\right )^{-\nu }\left (1+g_L\right ),\\[5pt] b)\quad & \frac{c_{t+1}^y}{c_{t}^y}=\frac{c_{t+1}^o}{c^o_{t}}=\frac{s_{t+1}}{s_t}=\left (1+g_A\right )^{1-\nu },\\[5pt] c)\quad &\frac{Y_{t+1}}{Y_t}=\frac{K_{t+1}}{K_{t}}=\left (1+g_A\right )^{1-\nu }\left (1+g_L\right ),\\[5pt] d)\quad &\frac{\partial \left (1+g_A\right )^{1-\nu }}{\partial \nu }\lt 0. \end{align*}

Hence, in steady state the individual supply of hours worked declines at an approximate rate $\nu g_A$ since $(-\nu )$ is the wage elasticity of $h(w_t)$ . The steady-state growth rate of the aggregate supply of hours worked is approximately equal to $-\nu g_A+g_L$ . It reflects the intensive and the extensive margin of the labor supply. Depending on which margin dominates it may be positive or negative. The growth rates under b) follow from Proposition1 as the wage elasticity of $c^y_t$ , $c^o_{t+1}$ , and $s_t$ is $1-\nu$ .

The findings under a) and b) highlight why the optimal plan of Proposition1 is consistent with a steady state equilibrium. The steady-state growth factor of individual hours worked is $\left (1+g_h\right )=\left (1+g_A\right )^{-\nu }$ , the one of $c^y_t$ , $c^o_{t+1}$ , and $s_t$ is $\left (1+g_A\right )^{1-\nu }$ . In steady state, individual wage income, $w_t h_t$ , grows at a factor $\left (1+g_h\right )\left (1+g_w\right )=\left (1+g_A\right )^{1-\nu }$ that coincides with the growth factor of $c^y_t$ , $c^o_{t+1}$ , and $s_t$ . Therefore, these growth patterns are consistent with the individual and the economy-wide budget constraints. As to c), we obtain from (4.8) that in steady state $\left (1+g_K\right )=\left (1+g_A\right )^{1-\nu }\left (1+g_L\right )$ . Then, the production function delivers $g_Y=g_K$ .

Overall, the rule is that the steady-state growth factor of economic aggregates like $Y_t$ , $K_t$ , or aggregate consumption, $L_tc^y_t+L_{t-1}c^o_{t}$ , is the growth factor of aggregate efficient hours worked, $A_t H_t=A_t L_t h_t$ . The growth factor of per-capita variables like $c^y_t$ , $c^o_{t+1}$ , $s_t$ , or output per worker, $Y_t/L_t$ , is the one of efficient individual hours worked, $A_t H_t/L_t=A_t h_t$ . The latter growth factor is $\left (1+g_A\right )^{1-\nu }$ and reflects the attenuating effect of a declining individual supply of hours worked on the growth rate of per-capita variables. Hence, all growth factors under a) – c) are endogenous.

According to d), the attenuation of the growth factor is more pronounced the greater is $\nu$ . Hence, the growth rate of per-capita variables declines in $\nu$ , and, ceteris paribus, an economy with a greater $\nu$ is predicted to grow slower in per-capita terms.

Observe that for all adjacent periods $t$ and $t+1$ hours worked per worker and hours worked per capita grow at the same rate. To see this, denote the population at $t$ by $N_t=L_{t}+L_{t-1}$ . Then, hours worked per capita at $t$ is the product of hours worked per worker and the labor market participation rate, that is,

\begin{align*} \frac{H_t}{N_t}=h_t\times \frac{L_t}{L_{t}+L_{t-1}}=h_t\times \frac{1+g_L}{2+g_L}. \end{align*}

Hence, in line with the cross-country evidence over the long run the participation rate is constant (Boppart and Krusell (Reference Boppart and Krusell2020)). Moreover, the growth factor of hours worked is $h_{t+1}/h_t$ and, in steady state, equal to $\left (1+g_A\right )^{-\nu }$ .

Define a Boppart–Krusell Balanced Growth Path, as an allocation that satisfies Kaldor’s growth facts (Kaldor (Reference Kaldor and Kaldor1961)), that is, the capital-output ratio, the real rental rate of capital, and factor shares remain constant whereas output per worker grows at a constant rate, and has the supply of hours worked declining at a constant rate.

Corollary2 follows as the allocation described by Proposition4 implies indeed that the labor share, $\hat{w}_t \hat{H}_t/Y_t$ , and the capital share, $\hat{r} K_t/Y_t$ , are time-invariant. Hence, this path is consistent with the stylized facts discussed in the Introduction.

Before closing this section two remarks are in order. First, observe that Proposition3 and 4 do not hinge on the depreciation rate, that is, they hold for any $\delta \in [0,1]$ . This contrasts with the discrete-time neoclassical growth model where a closed-form solution requires $\delta =1$ (Boppart and Krusell (Reference Boppart and Krusell2020), Appendix B.2). This difference is due to the fact that savings of cohort $t$ do not hinge on $R_{t+1}$ .

Second, consider the limit $\nu \to 0$ in which the lifetime utility function (3.1) converges to (2.3). Then, $\lim _{\nu \to 0} w_c=0$ implies $\lim _{\nu \to 0} k_{c,1}=0$ . Moreover, the shape of the difference equation (4.10) depends on whether $\phi \gt \phi _c$ or $\phi =\phi _c$ . In the former case, the demand for leisure is strictly positive since $\lim _{\nu \to 0}h(w_t)=\lim _{\nu \to 0} w_c^\nu =\left (1+\beta \right )/\left (\phi (\kappa +1+\beta )\right )\lt 1$ . Accordingly, (4.10) becomes

\begin{align*} k_{t+1}=\frac{\beta \left (\frac{1+\beta }{\phi (\kappa +1+\beta )}\right )^{1-\gamma } \Gamma (1-\gamma )}{(1+\beta )(1+g_L)(1+g_A)} \cdot k_t^\gamma . \end{align*}

If $\phi =\phi _c$ then the demand for leisure vanishes since $\lim _{\nu \to 0}h(w_t)=\lim _{\nu \to 0} w_c^\nu =\left (1+\beta \right )/\left (\phi _c(\kappa +1+\beta )\right )=1$ and (4.10) boils down to

\begin{align*} k_{t+1}=\frac{\beta \Gamma (1-\gamma )}{(1+\beta )(1+g_L)(1+g_A)} \cdot k_t^\gamma . \end{align*}

The latter coincides with the difference equation of the canonical OLG model with ( $g_A\gt 0$ ) or without ( $g_A=0$ ) technological progress.

4.3. Global dynamics: technological progress as an engine of liberation

This section studies the global dynamics of the economy of Section 3. The analysis reveals that sustained technological progress is the main cause for why workers have enjoyed more and more leisure over time. It liberated poor individuals from the necessity to supply long hours of work to assure a subsistence income. In this sense, technological progress has been an engine of liberation.Footnote ¹⁷

On the supply side, technological progress increases the marginal product of total hours worked. Accordingly, equilibrium real wages increase. During the transition to the steady state the growth rate of real equilibrium wages also reflects the growth rates of the physical capital stock and of the total supply of hours worked. However, in the long run it is technological progress alone that determines the growth rate of equilibrium wages. In addition, with technological progress aggregate output of the manufactured good increases.

On the household side, individuals who see their real income increase want to buy more of the consumption good. Additional purchases of the consumption good become feasible since technological progress allows for the total output of the consumption good to increase. Preferences exhibit a latent desire to work less. As consumption per capita increases the valuation of leisure increases. Eventually, individuals decide to enjoy more and more leisure and to supply less labor.

Section 4.3.1 starts out with the analysis of an economy where capital accumulation is the only source of economic growth. There is no technological progress. Initially, the economy is in Regime 0. Hence, real wages are low and individuals are poor. As a consequence, they supply their entire time endowment to the labor market. For the chosen parameter values, I establish that this economy converges toward a steady state with a constant real wage below $w_c$ . Hence, while real wages may grow over time due to capital accumulation individuals remain poor and supply their entire time endowment to the labor market.Footnote ¹⁸ Section 4.3.2 adds sustained technological progress to an otherwise identical economy. The initial state of the economy is again in Regime 0. However, due to technological progress the economy evolves in finite time from Regime 0 into Regime 1 where the supply of individual hours worked continuously declines. Eventually, there is convergence to the steady state of Proposition3.

To straighten the presentation I choose particular parameter values and make the following simplifying assumptions. On the household side, I set

(4.14) \begin{align} \nu =\frac{1}{4},\quad \beta =\frac{1}{3},\quad \kappa =1,\quad \mbox{and}\quad \phi =\frac{1}{2} \left (\frac{3}{2}\right )^{\frac{1}{3}}. \end{align}

While the values for $\nu$ and $\beta$ are not far away from those discussed in Section 3.1.2, this calibration involves a judicious choice of $\kappa$ and $\phi$ so that $w_c=1$ . Assumption1 is satisfied since $1/4\lt 0.352$ , and the optimal plan for Regime 1 involves

(4.15) \begin{align} h_t= w_t^{-\frac{1}{4}},\quad c_t^y=\frac{2}{3}w_t^{\frac{3}{4}},\quad c_{t+1}^o=\frac{R_{t+1}}{3}w_t^{\frac{3}{4}},\quad \mbox{and}\quad s_t=\frac{w_t^{\frac{3}{4}}}{3}. \end{align}

Without loss of generality for my qualitative results, I simplify further and set $L_t=1$ for all $t$ . Then, the evolution of the capital stock (4.8) becomes

(4.16) \begin{align} s\left (w_t\right )=K_{t+1}. \end{align}

On the production side, let $\Gamma =3/2$ and $\gamma =1/3$ . Then, from (3.14) the inverse aggregate demand for hours worked is

(4.17) \begin{align} w_t=A_{t}^{\frac{2}{3}}\left (\frac{K_t}{H^d_t}\right )^{\frac{1}{3}}. \end{align}

Throughout, it proves convenient to describe the evolution of the economy in terms of its equilibrium real wage, $\hat{w}_t$ .

4.3.1. No technological progress: the equilibrium dynamics in Regime 0

Consider the intertemporal general equilibrium of Section 4 void of technological progress, that is, $A_t=1$ for all $t$ . The initial state of the economy is in Regime 0. Hence, the aggregate supply of hours worked is $H^s_t=1 \cdot 1$ . Then, from (4.17) the equilibrium real wage is $\hat{w}_t=K_t^{1/3}$ . Combining the latter with (4.16) delivers the evolution of the equilibrium real wage in Regime 0 as

(4.18) \begin{align} \hat{w}_{t+1}=\left [s\left (\hat{w}_t\right )\right ]^{\frac{1}{3}}\quad \mbox{for}\quad \hat{w}_t\leq s^{-1}\left (w_c^3\right ), \end{align}

where the latter inequality assures that $\hat{w}_{t+1}\leq w_c$ .

The following proposition characterizes the steady state and the transitional dynamics of Regime 0.

Proposition 5. (Dynamical System - Regime 0)

The difference equation ( 4.18 ) gives rise to a unique, strictly positive steady-state equilibrium real wage $\hat{w}^{**}\lt w_c$ given by:

(4.19) \begin{align} \hat{w}^{**}=\left [s\left (\hat{w}^{**}\right )\right ]^{\frac{1}{3}}. \end{align}

Suppose $0\lt \hat{w}_1\lt w_c$ then the sequence $\{\hat{w}_t\}_{t=1}^\infty$ generated by ( 4.18 ) converges monotonically with $\lim _{t\to \infty } \hat{w}_t=\hat{w}^{**}$ .

The point of Proposition5 is that a poor economy may not escape from poverty without technological progress but remain forever stuck in Regime 0. The reason is that wage growth is driven by the process of capital accumulation alone. Due to a declining impact of additional capital on equilibrium wages the latter eventually peters out and the growth of wages comes to a halt. This tendency cannot be outweighed by the utility interaction between consumption and leisure when young that reduces the marginal utility of consumption when young and, thus, implies higher savings per worker.Footnote ¹⁹ The following section shows that sustained technological progress annihilates the possibility of a steady state involving a stationary real wage.

Figure 4.3. The switch between Regime 0 and 1. Note: At $t_c$ the equilibrium in the labor market is $(\hat{w}_{t_c}, 1)$ . Then, the difference equation (4.20) delivers $w_{t_c+1}\gt w_c$ which is the intersection between $H^{d}_{t_c+1}$ and $H^{s}_{t_c}=1$ . However, for wage levels greater than $w_c$ the equilibrium expression for the aggregate supply of hours worked is $H^{s}_{t_c+1}$ . Accordingly, the labor market equilibrium in period $t_c+1$ is $(\hat{w}_{t_c+1}, \hat{H}_{t_c+1})$ where $\hat{w}_{t_c+1}\gt w_{t_c+1}$ .

4.3.2. Global dynamics with sustained exogenous technological progress

Sustained technological progress means that $A_t$ grows over time at a constant rate $g_A\gt 0$ . Let the economy start in Regime 0 with an equilibrium real wage $\hat{w}_1\lt \hat{w}^{**}\lt w_c$ . Then, equations (4.16) and (4.17) deliver the evolution of the equilibrium real wage in Regime 0 as:

(4.20) \begin{align} \hat{w}_{t+1}=A_{t}^{\frac{2}{3}}\left [s\left (\hat{w}_t\right )\right ]^{\frac{1}{3}}\quad \mbox{for}\quad \hat{w}_t \leq s^{-1}\left (\frac{w_c^3}{A_t^2}\right ), \end{align}

where the latter inequality assures that $\hat{w}_{t+1}\leq w_c$ .

As seen above, $\hat{w}_t$ increases over time in Regime 0 even without technological progress. However, technological progress prevents the economy from converging to the steady state of Proposition5 since the right-hand side of the difference equation (4.20) shifts up by a factor $\left (1+ g_A\right )^{2/3}$ between any pair of periods $t$ and $t+1$ .

Instead, there is a finite $t_c$ at which $\hat{w}_{t_c} \gt s^{-1}\left (w_c^3/A_{t_c}^2\right )$ so that (4.20) prescribes a real wage $w_{t_c+1}\gt w_c$ . This is illustrated in Figure 4.3. However, since $w_{t_c+1}$ falls into Regime 1 it is not the equilibrium wage of period $t_c+1$ . Intuitively, at $t_c+1$ individuals realize that the real wage is so high that they want to reduce their supply of hours worked. Accordingly, the aggregate supply of hours worked becomes $H_{t_c+1}^s$ of (4.4) for $w_{t_c+1}\gt w_c$ . Equating the latter with the aggregate demand for hours worked, $H_{t_c+1}^d$ of (4.3), delivers the labor market equilibrium $\left (\hat{w}_{t_c+1},\hat{H}_{t_c+1}\right )$ . Here, $\hat{w}_{t_c+1}\gt w_{t_c+1}$ since the decline in the supply of hours worked increases the equilibrium wage.

At $t_c+1$ the capital stock and the level of technological knowledge will be $K_{t_c+1}$ and $A_{t_c+1}$ , respectively. Hence, $k_{t_c+1}=K_{t_c+1}/\left (A^{1-\nu }_{t_c+1}\right )$ as $L_t=1$ for all $t$ . It satisfies $k_{t_c+1}\gt k_{c,t_c+1}$ and serves as the initial condition for the dynamical system of Regime 1 as outlined in Proposition3. Using (4.6) the latter can be expressed as:

(4.21) \begin{align} \frac{\hat{w}_{t+1}}{A_{t+1}}=\left (\frac{1}{3(1+g_A)^{3/4}}\right )^{\frac{4}{11}}\cdot \left (\frac{\hat{w}_{t}}{A_{t}}\right )^{\frac{3}{11}}, \quad t= t_c+1, t_c+2, t_c+3 \ldots \end{align}

Proposition 6. (Dynamical System with Exogenous Technological Progress)

Consider the intertemporal general equilibrium of Section 4 under ( 4.14 ) - ( 4.17 ). If the economy starts in Regime 0 with initial conditions such that $\hat{w}_1\lt \hat{w}^{**}\lt w_c$ then it switches in finite time into Regime 1 and remains there. The sequence $\{\hat{w}_t/A_t\}_{t=t_c+1}^\infty$ generated by ( 4.21 ) converges monotonically with

(4.22) \begin{align} \lim _{t\to \infty } \left (\frac{\hat{w}_t}{A_t}\right )=\frac{1}{\sqrt{2\left (1+g_A\right )^{3/4}}}. \end{align}

Hence, technological progress drives the economy out of the poverty Regime 0. The advantages of productivity growth are not confined to the possibility to buy larger amounts of the consumption good. They also open the opportunity to enjoy more leisure.