Optimal Life-Cycle Education Decisions of Atomistic Individuals

Abstract

We analyze the optimal life-cycle education decision of a single atomistic individual and show that the standard result of part-time education and part-time work throughout the life-cycle holds only under very special and unrealistic assumptions. Once these assumptions are relaxed, different education strategies become optimal. These range from switching back and forth between work and education (educational leave) to full-time education at the beginning of life and full-time work when older (classical schooling). The resulting strategies for investing in education are better aligned with the observable pattern of educational investments over the life cycle. The particular path chosen by individuals then depends on other aspects, e.g., on the individual’s lifetime horizon or on nonlinearities in human capital accumulation with respect to time invested in education.

This chapter is dedicated to the memory of Arkadii V. Kryazhimskii, who passed away while working on it. He is deeply missed.

Appendices

Appendix 1: Proofs

Proof of Lemma 2.1

Consider an arbitrary admissible control u(t) and its corresponding human capital path h(t) being a solution to Eq. (2.3). Let x(t) and z(t) be solutions to

$$\displaystyle \begin{aligned} \dot x(t) = r(t) x(t) - c(t), \quad x(0) = k_0 + \int_0^{\infty} e^{-r(t,0)} w(t) h(t) u(t) {\,\mathrm d} t {} \end{aligned} $$

(2.18)

and

$$\displaystyle \begin{aligned} \dot z(t) &= r(t) z(t) + w(t) h(t) u(t),\notag\\ z(0) &= -\int_0^{\infty} e^{-r(t,0)} w(t) h(t) u(t) {\,\mathrm d} t. {} \end{aligned} $$

(2.19)

Then, k(t) = x(t) + z(t) for all t ≥ 0. Note that (2.5) and (2.19) imply that

$$\displaystyle \begin{aligned} \lim_{t \to \infty} z(t) = \lim_{t \to \infty} - \int_t^{\infty} e^{-r(s,t)} w(s) h(s) u(s) {\,\mathrm d} s= 0 \end{aligned}$$

such that the No Ponzi Game condition given by Eq. (2.4) leads to

$$\displaystyle \begin{aligned} \lim_{t \to \infty} e^{-r(t,0)} k(t) &= \lim_{t \to \infty} e^{-r(t,0)} x(t) + \lim_{t \to \infty} e^{-r(t,0)} z(t)\notag\\ & = \lim_{t \to \infty} e^{-r(t,0)} x(t) \geq 0. {} \end{aligned} $$

(2.20)

The state variable z(t) does not play any role for the optimization problem with objective function given by Eq. (2.1) and can be omitted in further considerations.

Consider the No Ponzi Game condition in Eq. (2.20). From Eq. (2.18) it follows that

$$\displaystyle \begin{aligned} x(t)=e^{r(t,0)} \left[k_0 + \int_0^{\infty} e^{-r(s,0)} w(s) h(s) u(s) {\,\mathrm d} s - \int_0^t e^{-r(s,0)} c(s) {\,\mathrm d} s\right] \end{aligned}$$

and so the condition given by Eq. (2.20) requires that

$$\displaystyle \begin{aligned} \int_0^{\infty} e^{-r(s,0)} c(s) {\,\mathrm d} s \leq k_0 + \int_0^{\infty} e^{-r(s,0)} w(s) h(s) u(s) {\,\mathrm d} s. {} \end{aligned} $$

(2.21)

This inequality implies that the set of admissible controls c(t) is the largest if the term $\int _0^{\infty } e^{-r(s,0)} w(s) h(s) u(s) {\,\mathrm d} s$ takes the maximal possible value. Thus, maximizing the right hand side of Eq. (2.21) will allow maximizing utility as given by the objective function in Eq. (2.1). Therefore, the original problem given by Eqs. (2.1)–(2.4) is reducible to the two separate problems put forward above. □

Proof of Proposition 2.1

The statement of the proposition follows from Theorem 4.1 in Aseev and Veliov (2014): Assumption (A1) is fulfilled by the dynamics of the problem, and Assumption (A2) in Theorem 4.1 is satisfied because the objective function is finite for all admissible controls. For finite objective functions, the notion of weakly overtaking optimality and strong optimality coincide (cf. Definition 2.3 and 2.5 in Aseev and Veliov, 2014). Furthermore, the notion of local weak overtaking optimality is weaker than the property of weak overtaking optimality (cf. footnote 3 in Aseev and Veliov, 2014). Hence, any strongly optimal control is also locally weak overtaking optimal. Thus, the theorem can be applied for this problem and implies that the Maximum Principle holds for the optimal control and the adjoint function can be explicitly stated as in the proposition.

The derivation of functional forms in Proposition 2.1 is purely technical and given in Appendix 2. Note that in the result of Aseev and Veliov (2014) the adjoint multiplier of the objective function in the definition of the Hamiltonian is equal to 1, which simplifies the analysis compared to other results in the literature. □

Proof of Theorem 2.1

We divide the proof into three steps. First, we show that the existence of an optimal balanced growth path with positive human capital growth implies the existence of a singular solution. Then, we derive a necessary condition for an optimal solution to exist. Finally, we show that infinitely many optimal solutions exist.

The assumption of an optimal balanced growth path with positive human capital growth implies that, for some constant d > 0, it holds that $d = \dot h(t)/h(t) = \chi [1-u^*(t)]$. Thus, u ^∗(t) is constant almost everywhere, u ^∗(t) ≡ 1 − d∕χ < 1, for almost all t ∈ [0, ∞). The boundary value u ^∗(t) ≡ 0 has objective value zero in the education problem, which is clearly not optimal because controls that lead to a positive objective value exist. Therefore, 0 < u ^∗(t) < 1 for almost all t ∈ [0, ∞), which proves that a singular optimal solution exists.

Next, we derive the necessary condition for a singular solution presented in Corollary 2.1 by using the optimality conditions of Proposition 2.1. Consider the switching function ζ(t), defined as

$$\displaystyle \begin{aligned} \zeta(t)=e^{-r(t,0)}w(t)-\chi \xi(t). \end{aligned}$$

If a control is singular over the interval [t ₁, t ₂) ⊂ [0, ∞), then the switching function ζ(t) is zero almost everywhere, which implies

$$\displaystyle \begin{aligned} \chi \xi(t) = e^{-r(t,0)} w(t). \end{aligned}$$

Furthermore, $\dot {\zeta }(t)=0$ for t ∈ [t ₁, t ₂). Using the adjoint equation (2.13), the derivative of ζ(t) with respect to time is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot{\zeta}(t) &\displaystyle = &\displaystyle -r(t)e^{-r(t,0)}w(t)+e^{-r(t,0)}\dot{w}(t) -\chi \dot{\xi}(t) = \\ &\displaystyle = &\displaystyle -r(t)e^{-r(t,0)}w(t)+e^{-r(t,0)}\dot{w}(t) \\&\displaystyle &\displaystyle +\chi [1-u^*(t)]e^{-r(t,0)}w(t)+\chi e^{-r(t,0)}w(t)u^*(t) = \\ &\displaystyle = &\displaystyle e^{-r(t,0)} \left[ -r(t)w(t) + \dot{w}(t) + \chi w(t) \right] = \\ &\displaystyle = &\displaystyle e^{-r(t,0)} w(t) \left\{ \frac{ \dot{w}(t)}{w(t)} - [r(t) -\chi] \right\}. \end{array} \end{aligned} $$

The condition $\dot {\zeta }(t)=0$ implies that the first condition in Corollary 2.1 given by Eq. (2.16) is fulfilled.

Note that the conditions ζ(t) = 0 and $\dot {\zeta }(t)=0$ are independent of the control u(t). If these conditions are fulfilled, every control u(t) satisfies the necessary optimality conditions and therefore infinitely many solutions and infinitely many singular solutions exist. We now show that all of these solutions are optimal. Considering any control u(t), the objective value of the education problem is given by⁵

$$\displaystyle \begin{aligned} \int_0^{\infty} e^{-r(t,0)} w(t) u(t) h(t) {\,\mathrm d} t = \frac{w(0) h(0)}{\chi} \left[ 1- e^{-\chi \lim_{t\to \infty} \int_0^t u(s) {\,\mathrm d} s} \right]. {} \end{aligned} $$

(2.22)

The objective value is thus maximized and equal to w(0)h(0)∕χ for any control u(t) that satisfies Eq. (2.15). This equation is in turn satisfied by infinitely many solutions, which proves the claim of the theorem. By shifting the initial point in time it follows that, along every optimal control, discounted future income is always equal to the current level of human capital multiplied by w(t)∕χ. □

Proof of Theorem 2.2

If r(t) and w(t) are constant, it follows that r(t) ≡ χ for t ∈ [0, ∞), while the wage rate per unit of effective labor is undetermined, w(t) ≡ w > 0.

Consider the maximal achievable income I(t) := χk(t) + wh(t), which is equal to the income at time t under a constant interest rate and a constant wage rate if the agent works full time. Taking the derivative of I(t) with respect to time, we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot I(t) &\displaystyle =&\displaystyle \chi \dot k(t) + w \dot h(t) =\\ &\displaystyle =&\displaystyle \chi [ \chi k(t) + w u(t) h(t) - c(t) ] + w \{ h(t) \chi [1-u(t)] \} =\\ &\displaystyle =&\displaystyle \chi [\chi k(t) + w h(t) - c(t) ] =\\ &\displaystyle =&\displaystyle \chi [ I(t) - c(t)]. \end{array} \end{aligned} $$

Consumption can be rewritten as

$$\displaystyle \begin{aligned} c^*(t) = I(t) \left[ \frac{ \rho - \chi (1-\theta) }{\theta \chi} \right]. \end{aligned} $$

(2.23)

Inserting this expression into the equation for $\dot I(t)$, we obtain that the maximal achievable income grows at a constant rate, independently of the control u(t),

$$\displaystyle \begin{aligned} \frac{\dot I(t)}{I(t)} = \frac{\chi - \rho}{\theta}. \end{aligned}$$

Due to our assumption 0 < (χ − ρ)∕(χθ) < 1, there exists $\tilde u \in (0,1)$ such that $\dot h(t)/h(t) = \chi (1-\tilde u) = (\chi - \rho )/\theta$. Inserting this $\tilde u$ and the expression for c ^∗(t) into the equation for $\dot k(t)$, we obtain

$$\displaystyle \begin{aligned} \frac{\dot k(t)}{k(t)} = \frac{\chi - \rho}{\theta}. \end{aligned}$$

Due to Corollary 2.1, the control $u(t)\equiv \tilde u$ is also optimal, which concludes our proof of the existence of a balanced growth path. □

Appendix 2: Derivations

1.1 Derivation of Eqs. (2.12) and (2.13)

According to Theorem 4.1 in Aseev and Veliov (2014), the adjoint is defined by ξ(t) = Z(t)I(t) with Z(t) being the fundamental matrix solution to the problem

$$\displaystyle \begin{aligned} \dot z(t) = -\chi [1-u(t)] z(t), \end{aligned}$$

and

$$\displaystyle \begin{aligned} I(t) = \int_t^{\infty} Z(s)^{-1} e^{-r(s,0)} w(s) u(s) {\,\mathrm d} s. \end{aligned}$$

The solution to the differential equation with some initial condition z(0) = z ₀ is

$$\displaystyle \begin{aligned} z(t) = e^{-\chi \int_0^t [1-u(\tau)] {\,\mathrm d} \tau} z_0 = Z(t) z_0. \end{aligned}$$

Therefore,

$$\displaystyle \begin{aligned} \xi(t) = Z(t) I(t) = e^{-\chi \int_0^t [1-u(\tau)] {\,\mathrm d} \tau} \int_t^{\infty} e^{\chi \int_0^s [1-u(\tau)] {\,\mathrm d} \tau} e^{-r(s,0)} w(s) u(s) {\,\mathrm d} s. \end{aligned}$$

Simplifying the expression results in Eq. (2.12).

1.2 Derivation of Eq. (2.22)

For a given control path u(t), human capital at time t is given by

$$\displaystyle \begin{aligned} h(t) = h(0) e^{\int_0^t \chi [1-u(s)] {\,\mathrm d} s}. \end{aligned}$$

Using this expression, we obtain

$$\displaystyle \begin{aligned} \int_0^{\infty} e^{-r(t,0)} w(t) u(t) h(t) {\,\mathrm d} t = \int_0^{\infty} \left[ e^{-r(t,0)} e^{\chi t} w(t) \right] \left[ h(0) u(t) e^{-\chi \int_0^t u(s) {\,\mathrm d} s} \right] {\,\mathrm d} t. \end{aligned}$$

Integrating by parts, where the first square bracket is differentiated, while the second is integrated, results in

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle e^{-r(t,0)} e^{\chi t} w(t) \frac{h(0)}{\chi} e^{-\chi \int_0^t u(s) {\,\mathrm d} s} \Big|{}_{t=0}^{\infty} \\&\displaystyle &\displaystyle - \int_0^{\infty} [\dot w(t) + \chi w(t) - r(t) w(t)] e^{-r(t,0)} e^{\chi t} h(0) e^{-\chi \int_0^t u(s) {\,\mathrm d} s} {\,\mathrm d} t. \end{array} \end{aligned} $$

Since the wage satisfies Eq. (2.16), the integrand of the second term is zero. Equation (2.16) further implies that w(t) = w(0)e ^{r(t, 0)}e ^−χt, which leads to the first term being equal to

$$\displaystyle \begin{aligned} \frac{w(0) h(0)}{\chi} \left[ 1- e^{-\chi \lim_{t\to \infty} \int_0^t u(s) {\,\mathrm d} s} \right], \end{aligned}$$

which is Eq. (2.22).

1.3 Derivation of Eq. (2.23)

Equation (2.23) is a representation of optimal consumption—using the notion of “maximal achievable income” I(t)—which is not common. Therefore, we sketch how to obtain this formulation from the known equation (2.11).

We start by applying a time shift: exchanging in Eq. (2.11) for $c^*_0$ the integration variable t by s and then the 0 by t, we obtain

$$\displaystyle \begin{aligned} c^*(t) = c^*_t, \qquad c^*_t = \frac{ x(t)}{ \int_t^{\infty} e^{-r(s,t)} e^{\frac{r(s,t) - \rho (s-t)}{\theta}} {\,\mathrm d} s}. \end{aligned}$$

Since r(t) ≡ χ for all t ∈ [0, ∞), it follows that r(s, t) = χ(s − t) and the integral in the denominator can be solved:

$$\displaystyle \begin{aligned} \int_t^{\infty} e^{-r(s,t)} e^{\frac{r(s,t) - \rho (s-t)}{\theta}} {\,\mathrm d} s = \left[ \frac{\theta}{ \rho- \chi(1-\theta)} \right]. \end{aligned}$$

The numerator x(t) is by definition equal to the future discounted income plus today’s assets,

$$\displaystyle \begin{aligned} x(t) = k(t) + \int_t^{\infty} e^{-r(s,t)} w(s) u(s) h(s) {\,\mathrm d} s, \end{aligned}$$

where the integral is, for any optimal control, equal to w(t)h(t)∕χ, see Eq. (2.17) and the first paragraph of Appendix 2. Combining the equations above, we obtain Eq. (2.23):

$$\displaystyle \begin{aligned} \begin{array}{rcl} c^*(t) = c^*_t;\quad \; c^*_t &\displaystyle =&\displaystyle \left[ \chi k(t) + w h(t) \right] \left[ \frac{ \rho - \chi (1-\theta)}{\theta \chi} \right] \\&\displaystyle =&\displaystyle I(t) \left[ \frac{ \rho - \chi (1-\theta)}{\theta \chi} \right]. \end{array} \end{aligned} $$