Taxation and human capital accumulation with endogenous mortality

Abstract

We study the impact of taxation on human capital when the mortality of household agents is endogenously determined based on their past levels of consumption. We characterize and solve the dynamic optimization problem facing household agents in the economy. Even though all households in our economy have identical preferences and have access to the same human capital accumulation technology, with endogenous mortality, there is a possibility of multiple equilibria. Households with an initial endowment of human capital above a threshold choose to accumulate human capital, while those below the threshold end up with very low levels of human capital. We use empirical estimates of the income–mortality relationship in Canada, France, and the US to simulate our model. When income has a strong protective effect against mortality, we find that multiple equilibria are empirically plausible. Our simulations suggest that changes in proportional tax rates have more than twice the impact on aggregate human capital in the endogenous mortality model compared to the infinite horizon model.

Appendix

Proof of Lemma 1

The objective of a household is to choose a consumption sequence $\{c_{t}\}_{t=0}^{\infty }$ to maximize

$$\begin{aligned} V=\sum \limits _{t=0}^{\infty }\rho _{t}u(c_{t})\text {,} \end{aligned}$$

where

$$\begin{aligned} \rho _{t+1}=\phi (c_{t},\theta )\rho _{t}\text {,} \end{aligned}$$

and $\rho _{0}=1$. For ease of exposition, let us suppress $\theta$ while denoting the function $\phi (\cdot )$. Consider two feasible consumption sequences ${\mathbb {C}} ^{1}=\left\{ c_{t}^{1}\right\} _{t=0}^{\infty }$ and ${\mathbb {C}} ^{2}=\left\{ c_{t}^{2}\right\} _{t=0}^{\infty }$ such that ${\mathbb {C}} ^{1}\ge {\mathbb {C}} ^{2}$ and ${\mathbb {C}} ^{1}\ne {\mathbb {C}} ^{2}$, i.e., $c_{t}^{1}\ge c_{t}^{2}$ for all t and $c_{t}^{1}>c_{t}^{2}$ for some t. Let $\rho _{t}^{1}$ and $\rho _{t}^{2}$ denote the discount functions generated by sequences ${\mathbb {C}} ^{1}$ and ${\mathbb {C}} ^{2}$ respectively. Since $\phi (\cdot )$ is increasing in $c_{t}$, we have $\rho _{t}^{1}\ge \rho _{t}^{2}$, for all t and $\rho _{t}^{1}>\rho _{t}^{2}$ for some t. From Assumption 1, it follows that

$$\begin{aligned} \mathop {\displaystyle \sum }\limits _{t=0}^{\infty }\rho _{t}^{1}u(c_{t}^{1})=V( {\mathbb {C}} ^{1})>V( {\mathbb {C}} ^{2})=\mathop {\displaystyle \sum }\limits _{t=0}^{\infty }\rho _{t}^{2}u(c_{t}^{2}). \end{aligned}$$

For quasi-concavity it is sufficient to show that for any $j\in (0,1)$, $V(j {\mathbb {C}} ^{1}+(1-j) {\mathbb {C}} ^{2})>V( {\mathbb {C}} ^{2})$. Note that both $u(\cdot )$ and $\phi (\cdot )$ are a strictly increasing and concave in $c_{t}$. Therefore, $u(jc_{t}^{1}+(1-j)c_{t}^{2}) \ge u(c_{t}^{2})$ and $\phi (jc_{t}^{1}+(1-j)c_{t}^{2})\ge \phi (c_{t}^{2})$ with the inequality holding strictly for some t. This implies $V(j {\mathbb {C}} ^{1}+(1-j) {\mathbb {C}} ^{2})>V( {\mathbb {C}} ^{2})$. $\square$

Proof of Lemma 2

Substituting (6) in (7) we can write the Lagrangian for a household agent’s problem as follows:

$$\begin{aligned} \pounds= & {} \sum \limits _{t=0}^{\infty }\{\rho _{t}u(c_{t})+\widetilde{\lambda }_{t}\left[ B(\nu _{t}h_{t})^{\gamma }x_{t}^{\sigma }+(1-\delta ^{h})h_{t}-h_{t+1}\right] +\mu _{t}[\phi (c_{t},\theta )\rho _{t}-\rho _{t+1}] \nonumber \\&+{\widetilde{\pi }}_{t}[{\widehat{R}}a_{t}+{\widehat{w}}(1-\nu _{t})h_{t}+\Gamma _{t}-c_{t}-x_{t}-a_{t+1}]\text {.} \end{aligned}$$

(20)

The first-order condition for optimal choice of $v_{t}$ is

$$\begin{aligned} \frac{\partial \pounds }{\partial \nu _{t}}=\widetilde{\lambda }_{t}B(\nu _{t}h_{t})^{\gamma -1}x_{t}^{\sigma }\gamma h_{t}-{\widetilde{\pi }}_{t} {\widehat{w}}h_{t}=0\text {. } \end{aligned}$$

Simplifying the above expression and assuming $h_{t}>0$, we get

$$\begin{aligned} {\widetilde{\lambda }}_{t}\gamma B(\nu _{t}h_{t})^{\gamma -1}x_{t}^{\sigma }= {\widetilde{\pi }}_{t}{\widehat{w}}\text {.} \end{aligned}$$

(21)

First-order condition with respect to $x_{t}$ implies

$$\begin{aligned} \frac{\partial \pounds }{\partial x_{t}}=\widetilde{\lambda }_{t}B(\nu _{t}h_{t})^{\gamma }\sigma x_{t}^{\sigma -1}-{\widetilde{\pi }}_{t}=0\text {,} \end{aligned}$$

or

$$\begin{aligned} {\widetilde{\lambda }}_{t}B(\nu _{t}h_{t})^{\gamma }\sigma x_{t}^{\sigma -1}= {\widetilde{\pi }}_{t}\text {.} \end{aligned}$$

(22)

Combining (21) and (22) we get

$$\begin{aligned} \gamma x_{t}=\sigma (\nu _{t}h_{t}){\widehat{w}}\text {.} \end{aligned}$$

(23)

Since the period utility function of the household is strictly increasing in $c_{t}$ Eq. (7) will bind at an optimum. Using the budget constraint to substitute for $x_{t}$, we have

$$\begin{aligned} \gamma [{\widehat{R}}a_{t}+{\widehat{w}}(1-\nu _{t})h_{t}+\Gamma _{t}-c_{t}-a_{t+1}]=\sigma (\nu _{t}h_{t}){\widehat{w}}\text {,} \end{aligned}$$

(24)

or

$$\begin{aligned} \nu _{t}=\frac{\gamma ({\widehat{R}}a_{t}+{\widehat{w}}h_{t}+\Gamma _{t}-c_{t}-a_{t+1})}{(\gamma +\sigma ){\widehat{w}}h_{t}}\text {.} \end{aligned}$$

Substituting the optimal value of $\nu _{t}$ in Eq. (23 ) we get

$$\begin{aligned} x_{t}=\frac{\sigma }{(\gamma +\sigma )}({\widehat{R}}a_{t}+{\widehat{w}} h_{t}+\Gamma _{t}-c_{t}-a_{t+1})\text {.} \end{aligned}$$

$\square$

Derivation of equation (13)

Using (11), we can write

$$\begin{aligned} (1-\nu _{t}){\widehat{w}}h_{t}=\frac{(\gamma +\sigma ){\widehat{w}}h_{t}-\gamma ( {\widehat{R}}a_{t}+{\widehat{w}}h_{t}+\Gamma _{t}-c_{t}-a_{t+1})}{(\gamma +\sigma )} \end{aligned}$$

(25)

Substituting the equation above and (7) in (1), we get

$$\begin{aligned} h_{t+1}= & {} B\left[ \frac{\gamma ({\widehat{R}}a_{t}+{\widehat{w}}h_{t}+\Gamma _{t}-c_{t}-a_{t+1})}{(\gamma +\sigma ){\widehat{w}}}\right] ^{\gamma }\\&\left[ \frac{\sigma ({\widehat{R}}a_{t}+{\widehat{w}}h_{t}+\Gamma _{t}-c_{t}-a_{t+1})}{ (\gamma +\sigma )}\right] ^{\sigma }+(1-\delta ^{h})h_{t}\text {,} \end{aligned}$$

and after simplifying further we can write

$$\begin{aligned} h_{t+1}= & {} B\left( \frac{\gamma }{\gamma +\sigma }\right) ^{\gamma }\left( \frac{\sigma }{\gamma +\sigma }\right) ^{\sigma }\left( \frac{1}{{\widehat{w}}} \right) ^{\gamma }\\&({\widehat{R}}a_{t}+{\widehat{w}}h_{t}+\Gamma _{t}-c_{t}-a_{t+1})^{\gamma +\sigma }+(1-\delta ^{h})h_{t}\text {,} \end{aligned}$$

or

$$\begin{aligned} h_{t+1}={\widehat{B}}({\widehat{R}}a_{t}+{\widehat{w}}h_{t}+\Gamma _{t}-c_{t}-a_{t+1})^{\gamma +\sigma }+(1-\delta ^{h})h_{t}\text {,} \end{aligned}$$

where ${\widehat{B}}=B\left( \frac{\gamma }{\gamma +\sigma }\right) ^{\gamma }\left( \frac{\sigma }{\gamma +\sigma }\right) ^{\sigma }\left( \frac{1}{{\widehat{w}}}\right) ^{\gamma }$.

Proof of Proposition 1

Since human capital production function satisfies the Inada conditions, for any level of human capital $h_{t}$, $h_{t+1}>0$. To prove this proposition, we will first derive the first-order condition for an interior solution in both $h_{t+1}$ and $a_{t+1}$. From this condition, we will implicitly derive a minimum threshold level of human capital $h_{\max }$ such that if $h_{t}\le h_{\max }$, $a_{t+1}=0$.

Consider the Lagrangian for the household’s maximization problem [Eq. ( 20)]. First-order condition for optimum with respect to $h_{t+1}$ yields

$$\begin{aligned} {\widetilde{\lambda }}_{t}={\widetilde{\lambda }}_{t+1}[\gamma B(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }\nu _{t+1}+(1-\delta ^{h})]+ {\widetilde{\pi }}_{t+1}{\widehat{w}}(1-\nu _{t+1})\text {.} \end{aligned}$$

(26)

We can use Eq. (21) to express ${\widetilde{\pi }}_{t+1}$ as

$$\begin{aligned} {\widetilde{\pi }}_{t+1}=\frac{{\widetilde{\lambda }}_{t+1}\gamma B(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }}{{\widehat{w}}}\text {.} \end{aligned}$$

Using the equation above to substitute for ${\widetilde{\pi }}_{t+1}$ in Eq. (26) we get

$$\begin{aligned} \frac{{\widetilde{\lambda }}_{t}}{{\widetilde{\lambda }}_{t+1}}=\gamma B(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }+(1-\delta ^{h})\text {.} \end{aligned}$$

(27)

Necessary condition for optimum with respect to $a_{t+1}$ implies

$$\begin{aligned} \frac{{\widetilde{\pi }}_{t}}{\widetilde{\pi }_{t+1}}={\widehat{R}}\text {,} \end{aligned}$$

(28)

if $a_{t+1}$

$>0$. Using Eq. (21), we can write

$$\begin{aligned} \frac{{\widetilde{\lambda }}_{t}}{\widetilde{\lambda }_{t+1}}\frac{\gamma B(\nu _{t}h_{t})^{\gamma -1}x_{t}^{\sigma }}{\gamma B(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }}=\frac{{\widetilde{\pi }}_{t}}{ {\widetilde{\pi }}_{t+1}}\text {.} \end{aligned}$$

(29)

Combining with (27) and (28), we have

$$\begin{aligned}{}[\gamma B(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }+(1-\delta ^{h})]\frac{(\nu _{t}h_{t})^{\gamma -1}x_{t}^{\sigma }}{(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }}={\widehat{R}}\text {.} \end{aligned}$$

Let $[\gamma B(\nu _{t+1}h_{t+1})^{\gamma -1}x_{t+1}^{\sigma }+(1-\delta ^{h})]=R_{t+1}^{h}$. This is the marginal return from investing in human capital. Equation (23) implies $\nu _{t}h_{t}=\left( \frac{ \gamma }{\sigma {\widehat{w}}}\right) x_{t}$. Substituting in the equation above we get

$$\begin{aligned} R_{t+1}^{h}\left( \frac{x_{t}}{x_{t+1}}\right) ^{{}^{\gamma +\sigma -1}}= {\widehat{R}}\text {.} \end{aligned}$$

(30)

Equation (30) implies that resource investment in human capital becomes stationary when return from human capital and the risk free are equalized, i.e., if $R_{t+1}^{h}={\widehat{R}}\Longrightarrow$ $x_{t+1}=x_{t}=x$ for all t. It follows that $\nu$ and h will also be stationary, i.e., $\nu _{t}=$ $\nu _{t+1}=\nu$ and $h_{t}=h_{t+1}=h$ for all t.

We will now characterize the minimum level of human capital needed to induce household agents to invest in the financial asset. When $R_{t+1}^{h}= {\widehat{R}}$ Eq. (30) becomes

$$\begin{aligned}{}[\gamma B(\nu h)^{\gamma -1}x^{\sigma }+(1-\delta ^{h})]={\widehat{R}} \text {.} \end{aligned}$$

(31)

While the return from the financial asset is constant, the return from human capital accumulation, $R^{h}$ is high for low values of h and decreasing in h. To characterize $h_{\max }$ we will substitute for x and $\nu$ in (31). Since equation (7) always binds at an optimum, we can write

$$\begin{aligned} x={\widehat{R}}a_{t}+{\widehat{w}}(1-\nu )h+\Gamma _{t}-c_{t}-a_{t+1}\text {.} \end{aligned}$$

(32)

Using Eq. (11) and (25) we can write Eq. (31) as

$$\begin{aligned} (\gamma +\sigma ){\widehat{B}}{\widehat{w}}({\widehat{R}}a_{t}+{\widehat{w}}h+\Gamma _{t}-c_{t}-a_{t+1})^{\gamma +\sigma -1}+(1-\delta ^{h})={\widehat{R}}\text {,} \end{aligned}$$

(33)

where ${\widehat{B}}=$ $B\left( \frac{\gamma }{\gamma +\sigma } \right) ^{\gamma }\left( \frac{\sigma }{\gamma +\sigma }\right) ^{\sigma }\left( \frac{1}{{\widehat{w}}}\right) ^{\gamma }$.

We have shown that if $a_{t+1}>0$, $h_{t+1}=h_{t}=h_{\max }$.

When human capital level becomes stationary equation (13) implies $( {\widehat{R}}a_{t}+{\widehat{w}}h+\Gamma _{t}-c_{t}-a_{t+1})=\left( \frac{\delta ^{h}h}{{\widehat{B}}}\right) ^{\frac{1}{\gamma +\sigma }}$. Substituting in (33), we have

$$\begin{aligned} (\gamma +\sigma ){\widehat{B}}{\widehat{w}}\left( \frac{\delta ^{h}h}{{\widehat{B}} }\right) ^{\frac{\gamma +\sigma -1}{\gamma +\sigma }}+(1-\delta ^{h})= {\widehat{R}}\text {,} \end{aligned}$$

or,

$$\begin{aligned} \left( \frac{\delta ^{h}h}{{\widehat{B}}}\right) ^{\frac{\gamma +\sigma -1}{ \gamma +\sigma }}=\frac{{\widehat{R}}+\delta ^{h}-1}{(\gamma +\sigma ){\widehat{B}}{\widehat{w}}}\text {.} \end{aligned}$$

The left hand side of the above equation is decreasing in h. Define $h_{\max }=\left( \frac{{\widehat{B}}}{\delta ^{h}}\right) \left( \frac{ {\widehat{R}}+\delta ^{h}-1}{(\gamma +\sigma ){\widehat{B}}{\widehat{w}}}\right) ^{ \frac{\gamma +\sigma }{^{\gamma +\sigma -1}}}$. $h_{\max }$ has been defined so that the rate of return from the two assets are equalized. Therefore, it follows that if $h_{t}\le h_{\max }$ $\Longrightarrow$ $R^{h}\ge {\widehat{R}}$ $\Longrightarrow a_{t+1}=0$. We have assumed that $a_{0}=0$ for all households, so only those households with $h_{0}>h_{\max }$ will choose to invest in the financial asset.

Derivation of equation (15) and (16):

To complete our analysis of the household’s optimization problem, let us consider the optimal choice of consumption of a household. First-order condition with respect to $c_{t}$ implies

$$\begin{aligned} \rho _{t}u^{\prime }(c_{t})+\rho _{t}\mu _{t}\phi _{c}(c_{t},\theta )= {\widetilde{\pi }}_{t}\text {,} \end{aligned}$$

if $a_{t+1}>0$. Define $\pi _{t}=\frac{{\widetilde{\pi }}_{t}}{\rho _{t}}$ and $\lambda _{t}=\frac{{\widetilde{\lambda }}_{t}}{\rho _{t}}$. The equation above can be written as

$$\begin{aligned} u^{\prime }(c_{t})+\mu _{t}\phi _{c}(c_{t},\theta )=\pi _{t}\text {,} \end{aligned}$$

(34)

The optimality condition with respect to $\rho _{t+1}$ gives us

$$\begin{aligned} \frac{\partial \pounds }{\partial \rho _{t+1}}=u(c_{t+1})-\mu _{t}+\mu _{t+1}\phi (c_{t+1},\theta )=0\text {.} \end{aligned}$$

(35)

Re-arranging we have

$$\begin{aligned} \mu _{t}=u(c_{t+1})+\mu _{t+1}\phi (c_{t+1},\theta )\text {.} \end{aligned}$$

Substituting one period forward for $\mu _{t+1}$ gives us,

$$\begin{aligned} \mu _{t}=u(c_{t+1})+\phi (c_{t+1},\theta )[u(c_{t+2})+\mu _{t+2}\phi (c_{t+2},\theta )]\text {.} \end{aligned}$$

If we carry out this substitution repeatedly, we find that $\mu _{t}$ is the present discounted value of utility from period $t+1$ onwards, i.e.,

$$\begin{aligned} \mu _{t}=(\rho _{t+1})^{-1}\mathop {\displaystyle \sum }\limits _{s=t+1}^{\infty }\rho _{s}u(c_{s}) \text {.} \end{aligned}$$

Since $u(c_{t}) 0$ and $\rho _{t}>0$, it follows that $\mu _{t}>0$ . Notice, the Lagrange multiplier $\mu _{t}={\widehat{V}}_{t}$, the variable we constructed in proving lemma 1. Using equation (10), we can express Eqs. (28) and (29) as

$$\begin{aligned} \frac{u^{\prime }(c_{t})+\mu _{t}\phi _{c}(c_{t},\theta )}{u^{\prime }(c_{t+1})+\mu _{t+1}\phi _{c}(c_{t+1},\theta )}=\frac{\lambda _{t}}{\lambda _{t+1}}=\frac{\pi _{t}}{\pi _{t+1}}={\widehat{R}}\phi (c_{t},\theta )\text {,} \end{aligned}$$

(36)

if $a_{t+1}=0$. If $h_{t}\le h_{\max }$ the household will only invest in human capital, i.e., $a_{t+1}=0$. Equation (36) becomes

$$\begin{aligned} \frac{u^{\prime }(c_{t})+\mu _{t}\phi _{c}(c_{t},\theta )}{u^{\prime }(c_{t+1})+\mu _{t+1}\phi _{c}(c_{t+1},\theta )}=\frac{\lambda _{t}}{\lambda _{t+1}}=R_{t+1}^{h}\phi (c_{t},\theta )\text {,} \end{aligned}$$

(37)

where $R_{t+1}^{h}=[(\gamma +\sigma ){\widehat{B}}{\widehat{w}}( {\widehat{w}}h_{t+1}+\Gamma _{t+1}-c_{t+1})^{\gamma +\sigma -1}+(1-\delta ^{h})]$. We can combine Eqs. (36) and (37) and write it more compactly as

$$\begin{aligned} \frac{u^{\prime }(c_{t})+\mu _{t}\phi _{c}(c_{t},\theta )}{u^{\prime }(c_{t+1})+\mu _{t+1}\phi _{c}(c_{t+1},\theta )}=R_{t+1}^{h}\phi (c_{t},\theta )\ge {\widehat{R}}\phi (c_{t},\theta )\text {,} \end{aligned}$$

with the equality between $R_{t+1}^{h}$ and ${\widehat{R}}$ holding if $h_{t}>h_{\max }$. $\square$

Proof of Proposition 2

Consider the function $\Psi (h)=\phi \left[ \eta (h),\theta \right] R^{h}(h)-1$. We want to show that there exists at least one point in the interval $(0,h_{\max })$ where $\Psi (h)$ intersects the horizontal axis. When $h=0$, $\Psi (0)=\phi (0,\theta )R^{h}(0)-1$. The effective discount factor of the household $\phi [0,\theta ]$ is bounded between 0 and 1. Since the human capital production function satisfies the Inada conditions, it follows that $\Psi (0)>0$. From the definition of $h_{\max }$, we know that when $h=h_{\max }$, $R^{h}={\widehat{R}}=(1+{\widehat{r}})$. From assumption 3, we have $\beta (1+r)-1=0$. Therefore, $\Psi (h_{\max })=\phi \left[ \eta (h_{\max }),\theta \right] (1+{\widehat{r}})<\beta (1+r)-1=0$. We know that sum, product and composition of continuous functions are continuous. The functions $\eta (h)$, $R^{h}(h)$ and $\phi (\cdot )$ are all continuous functions. This would imply that $\Psi (h)$ is also continuous. Since $\Psi (h)$ is a continuous function of h there will be a $h^{1}\in (0,h_{\max })$ such that $\Psi (h^{1})=0$ and $\Psi (h)<0$ for some $h\in (h^{1},h_{\max })$. There must be at least one locally unique steady-state solution. Suppose there are two locally unique steady-state solutions to the household’s maximization problem. It means there are two values of h, say $0<h^{1}<$ $h^{2}<h_{\max }$ such that $\Psi (h^{1})=\Psi (h^{2})=0$ and $\Psi (h)>0$ for some $h\in (h^{2},h_{\max })$. Since $\Psi (h_{\max })<0$, continuity of $\Psi (h)$ would ensure the existence of another value of h, say $h^{3}\in (h^{2},h_{\max })$ such that $\Psi (h^{3})=0$ and $\Psi (h)<0$ for some $h\in (h^{3},h_{\max })$. This argument can be generalized to prove that the number of locally unique steady-state solutions is odd. $\square$

Proof of Proposition 3

The inverse of the discount factor, $\frac{1}{\phi (\eta (h_{0}),\theta )}$, is called the rate of time preference. We need to show that if the return from human capital accumulation at the initial endowment of human capital exceeds the rate of time preference then $\{h_{t}=h_{0}\}_{t=0}^{\infty }$ and $\{c_{t}=\eta (h_{0},{\widehat{w}},\delta ^{h})={\widehat{w}}h_{0}-(G^{-1})(\delta ^{h}h_{0})\}_{t=0}^{\infty }$ cannot be an optimum for an agent. Furthermore $h_{1}>h_{0}$. Suppose

$$\begin{aligned} {\widehat{w}}G^{\prime }({\widehat{w}}h_{0}-\eta (h_{0}))+(1-\delta ^{h})>\frac{1 }{\phi (\eta (h_{0}),\theta )}\text {.} \end{aligned}$$

(38)

If we reduce consumption for only one period by $\epsilon$, where $\epsilon$ is close to zero the resultant loss in utility will be $[u^{\prime }(c_{0})+\mu _{0}\phi _{c}(c_{0},\theta )]$, where $\mu _{0}=\frac{u(c_{0})}{ 1-\phi (c_{0},\theta )}$. The saved amount can be used to accumulate human capital which will provide higher wage income in the next period. If that increased income were to be consumed the discounted value of its utility will be $\phi (c_{0},\theta )[{\widehat{w}}G^{\prime }({\widehat{w}} h_{0}-c_{0})+(1-\delta ^{h})][u^{\prime }(c_{0})+\mu _{0}\phi _{c}(c_{0},\theta )]$. Since $\phi (\eta (h_{0}),\theta )[{\widehat{w}} G^{\prime }({\widehat{w}}h_{0}-\eta (h_{0}))+(1-\delta ^{h})]>1$, from the condition above it follows that the gain in utility will outweigh the loss. Hence $\{h_{t}=h_{0}\}_{t=0}^{\infty }$ and $\{c_{t}=\eta (h_{0},w,\delta ^{h})=wh_{0}-(G^{-1})(\delta ^{h}h_{0})\}_{t=0}^{\infty }$ cannot be an optimal sequence for an agent and the agent will want to accumulate human capital, i.e., $h_{1}>h_{0}$. We can use a similar argument to show that if

$$\begin{aligned} {\widehat{w}}G^{\prime }({\widehat{w}}h_{0}-\eta (h_{0}))+(1-\delta ^{h})<\frac{1 }{\phi (\eta (h_{0}),\theta )} \end{aligned}$$

(39)

then human capital must be strictly decreasing. $\square$

Proof of Proposition 4

By definition $\Psi ({\widehat{h}})=0$. Consider $\Psi (h)=\phi (\eta (h,\cdot ),\theta )[{\widehat{w}}G^{\prime }({\widehat{w}}h-\eta (h,\cdot ))+(1-\delta ^{h})]-1$. $\frac{\partial }{\partial \theta }\Psi (h,\cdot )=\phi _{\theta }(\cdot ,\theta )[{\widehat{w}}G^{\prime }({\widehat{w}}h-\eta (h,\cdot ))+(1-\delta ^{h})]>0$. The entire $\Psi (h)$ curve shifts upward as $\theta$ increases. The proportion of households converging to $h_{\rm low}$ will fall as $\theta$ increases.$\square$

Proof of Proposition 5

We will prove this proposition by showing that if either $\tau ^{K}$ or $\tau ^{L}$ increases the entire $\Psi (h)$ will shift downwards. Recall, ${\widehat{w}}=(1-\tau ^{L})w(r,\tau ^{K})$ where $w(\cdot )$ is a decreasing function of $\tau ^{K}$. Therefore, ${\widehat{w}}$ decreases if either $\tau ^{K}$ or $\tau ^{L}$ increase. It is easy to see that the function $\Psi (h)=\phi [\eta (h,{\widehat{w}},\delta ^{h}),\theta ]\{ {\widehat{w}}G^{\prime }[(G^{-1})(\delta ^{h}h)]+(1-\delta ^{h})\}-1$ will shift downwards if ${\widehat{w}}$ decreases. In case there are two stable steady-state solutions to a household’s problem a downward shift of $\Psi (h)$ implies that a higher threshold is needed to induce a household to accumulate high level of human capital in steady state. $\square$