Financing Sustainable Growth Through Energy Exports and Implications for Human Capital Investment

Abstract

This paper examines the impact of energy resources financing on investment in human capital through the mechanism of growth dynamics. This is done within a context that includes global financial markets and exports of non-renewable energy. These are frequently related to issues of debt accumulation, which naturally raises questions relating to sustainability and welfare—both present and future. Energy export’s contribution to economic growth is emphasized and the distinction between resource-rich and resource-poor countries is highlighted. Major external disturbances for sustained resource-driven development, which can make a country more vulnerable to economic shocks, are discussed. Numerical analysis using Nonlinear Model Predictive Control confirms the empirically observed long-run patterns when non-renewable resources decline monotonically and become depleted. The solutions also confirm typical boom/bust cycle phenomena, where excessive debt may effectively strangle growth. In addition, the implications of investment in human capital for inequality are discussed.

Appendices

Appendix 1

The basic growth model is solved with the current-value Hamiltonian with two constraints as follows:

$$\displaystyle{ H = U(C) + q_{1}\left (Q(K,X) - C\right ) + q_{2}(-X) }$$

(23)

where q ₁ and q ₂ are co-state variables or shadow prices of capital accumulation and resource constraint respectively. The necessary optimality conditions are obtained as follows:

$$\displaystyle{ \frac{\partial H} {\partial C} = U^{{\prime}}(C) - q_{ 1} = 0 }$$

(24)

$$\displaystyle{ \frac{\partial H} {\partial X} = q_{1}Q_{X} - q_{2} = 0 }$$

(25)

$$\displaystyle{ \dot{q_{1}} = rq_{1} -\frac{\partial H} {\partial K} = q_{1}(r - Q_{K}) }$$

(26)

$$\displaystyle{ \dot{q_{2}} = rq_{2} -\frac{\partial H} {\partial S} = rq_{2} }$$

(27)

with $Q_{X} = \frac{\partial Q(K,X)} {\partial X}$, $Q_{K} = \frac{\partial Q(K,X)} {\partial K}$, $\frac{dq_{1}} {dt} =\dot{ q_{1}}$ and $\frac{dq_{2}} {dt} =\dot{ q_{2}}$.

As shown in Dasgupta and Heal (1974, p. 11) and in Greiner and Semmler (2008, pp. 165–166), differentiating Eq. (24) with respect to time and substituting Eq. (26) yields the following consumption rate along an optimal path:

$$\displaystyle{ \frac{\dot{C}} {C} = \frac{Q_{K} - r} {\eta (C)} }$$

(28)

with $\eta (C) = -\frac{CU^{{\prime\prime}}(C)} {U^{{\prime}}(C)}$ as an elasticity of marginal utility (Dasgupta and Heal 1974, p. 5). Greiner and Semmler (2008) note that from the optimal path of consumption we can observe that higher discount rate is associated with further fall of the rate of consumption over time.

Appendix 2

The current-value Hamiltonian of the open economy growth model with three constraints is as follows:

$$\displaystyle{ H = U(C) + q_{3}(I -\delta K - aX) + q_{4}(-X) + q_{5}\left (\theta F -\left (Y - C - I -\varphi (I,K)\right )\right ) }$$

(29)

The optimality conditions are:

$$\displaystyle{ \frac{\partial H} {\partial C} = U^{{\prime}}(C) + q_{ 5} = 0 }$$

(30)

$$\displaystyle{ \frac{\partial H} {\partial I} = q_{3} + q_{5}\left (1 +\varphi _{I}\right ) = 0 }$$

(31)

$$\displaystyle{ \frac{\partial H} {\partial X} = -q_{3}a - q_{4} - q_{5}Q_{X} = 0 }$$

(32)

$$\displaystyle{ \dot{q_{3}} = rq_{3} -\frac{\partial H} {\partial K} = (r+\delta )q_{3} + (Q -\varphi _{K})q_{5} }$$

(33)

$$\displaystyle{ \dot{q_{4}} = rq_{4} -\frac{\partial H} {\partial S} = rq_{4} }$$

(34)

$$\displaystyle{ \dot{q_{5}} = rq_{5} -\frac{\partial H} {\partial F} = (r-\theta )q_{5} }$$

(35)

with $\varphi _{I} = \frac{\partial \varphi (I,K)} {\partial I}$, $\varphi _{K} = \frac{\partial \varphi (I,K)} {\partial K}$, $Q_{X} = \frac{\partial Q(K,X)} {\partial X}$, $Q_{K} = \frac{\partial Q(K,X)} {\partial K}$ and $Q_{S} = \frac{\partial Q(K,X)} {\partial S}$ and q ₃, q ₄, and q ₅ are co-state variables or shadow prices of capital accumulation, resource constraint, and external debt respectively. Here, initial values for all the state variables K(0), R(0) and F(0) are given. The path of consumption is:

$$\displaystyle{ \frac{\dot{C}} {C} = \frac{r-\theta } {\eta (C)} }$$

(36)

with $\eta (C) = -\frac{CU^{{\prime\prime}}(C)} {U^{{\prime}}(C)}$ as an elasticity of marginal utility.

Appendix 3

Nonlinear Model Predictive Control (NMPC) is a methodological paradigm for the analysis of nonlinear feedback control systems. Although it was developed in the early 1960s, the computational complexity of the algorithms created rendered it impractical until relatively recently. However, NMPC algorithms developed by Grüne and Pannek (2011), as I have written previously in Nyambuu and Semmler (2014), can now be implemented on modern computers and used for the analysis of the dynamics of our closed economy model with exhaustible resources. The extension of the model in an open economy with external debt constraint is also solved with NMPC.

NMPC is an optimization-based method of feedback for control of a nonlinear system. For example, consider a controlled and predicted process where at each instant of time a control input, w(n), can be chosen. The control inputs affect the future behavior of the state of the system variable, v(n), which is measured at discrete time instants. The control inputs, w(n), are determined to track the state of the system, v(n), so that it is as close as possible to a given reference, v ^ref(n) (Grüne and Pannek 2011). When the reference is constant and equal to zero, as shown in Eq. (37), the tracking problem reduces to a stabilization problem of the following type²⁰:

$$\displaystyle{ v^{ref}(n) = v^{{\ast}} = 0 }$$

(37)

Note, however, the procedure also works—and thus is the novelty of the Grüne-Pannek procedure—if the reference value, for example the steady state as terminal condition, is not known. Greiner et al. (2013) point out that in the case of a very long decision horizon, NMPC can approximate the infinite time horizon solution well. Even with a short decision horizon (N = 10), one can still investigate important issues raised in the context of the model.

Following Greiner et al. (2013, pp. 16–18), suppose the optimal decision problem is given as follows:

$$\displaystyle{ max\mathop{\int }\nolimits _{0}^{\infty }e^{-\delta t}g(v(t),w(t))dt }$$

(38)

with $\dot{v}(t) = f(v(t),w(t))$, v(0) = v ₀. 

An approximate discrete time problem can be written as:

$$\displaystyle{ max\sum _{i=0}^{\infty }\beta ^{i}g(v_{ i},w_{i}) }$$

(39)

where $v_{i+1} = \Phi (h,v_{i},w_{i})$ and β = e ^{−δ h} with h > 0 discretization time step (Greiner et al. 2013, p. 17). Accordingly, instead of the infinite horizon of Eq. (39) we can use a finite horizon functional as shown in Eq. (40).

$$\displaystyle{ max\sum _{k=0}^{N}\beta ^{i}g(v_{ k,i},w_{k,i}) }$$

(40)

where the index, i, is the number of the iteration. Here, the decision horizon is “truncated” and is represented by a “finite” horizon, N, where $v_{k+1,i} = \Phi (h,v_{k,i},w_{k,i})$ after the decision has been made. The converted static nonlinear optimization problem can be solved numerically using the Matlab f mincon solver (Greiner et al. 2013, pp. 16–18).

The solution (v _i , w _i ) of the problem converges to the accurate solution of Eq. (39) as N → ∞. This convergence is ensured under the assumptions such that an optimal equilibrium for the infinite horizon problem in Eq. (39) exists (Greiner et al. 2013, p. 18). Thus, in accordance with the recent studies already mentioned, I also assume that the solution converges to the optimal solution even without knowing the equilibrium values (Grüne 2013).

Appendix 4

I solve the optimal control models described in this paper by using NMPC for different initial values of state variables and various parameters. In Sect. 3 of the paper, I showed the optimal trajectories of the variables for initial conditions of K(0) = 1 and S(0) = 4 and parameter values of η = 0. 5, β = 0. 3, r = 0. 03 for the resource intensive production function. The results of the evolutions of state variables are illustrated in Fig. 5. Now, I present the optimal trajectories of the state and control variables for capital intensive economy with β = 0. 7 and other parameter values as before: η = 0. 5, r = 0. 03 and initial conditions of K(0) = 1 and S(0) = 4. The result is shown in Fig. 10.

Fig. 10

Optimal trajectories of K and S in the capital intensive economy

Fig. 11

Evolution of consumption, utility, output and flow of resources in the capital intensive economy

Fig. 12

Optimal trajectories of K and S with different initial conditions of the state variables

Similar to the previous case, we observe an inverted U-shaped movement of capital stock with monotonically decreasing stock of the remainder of exhaustible resources. However, it has taken a longer period for the capital stock to start declining. This indicates that the stock of the capital has begun to fall when the resources had reached a much lower level in comparison to the level in previous result. In addition, corresponding optimal paths of consumption level, flows of exhaustible resources, utility level and output, depicted in Fig. 11, show similar movements as we saw in the first scenario described in Sect. 3. Thus, after reviewing numerous variations and their numerical solution, we generally observe an inverted U-shaped capital accumulation movement and monotonically declining stock of non-renewable resources when initial condition of capital stock is not high. Besides the different values of the parameters, we can be interested in the solutions with the different initial conditions of the state variables. These are shown in Fig. 12.