Government Debt and Human Capital Formation

Abstract

In this chapter, we focus on another important source of sustained growth, namely human capital formation. Therefore, we study two endogenous growth models with human capital accumulation that is the result of public spending. The government hires teachers and finances additional expenditures for human capital formation. As in the last section, it may run deficits but sets the primary surplus again such that it is a positive function of public debt.

Appendix

5.1.1 Proof of Proposition 29

The balanced growth rate is given by Eq. (5.23) as $g =\dot{ c}/c = -\rho + (1-\tau )(1-\alpha )z^{\alpha }u^{\alpha }$ with z = z ^⋆. Solving Eqs. (5.41), (5.44) and (5.42) with respect to x, p and q leads to x = x(z, ⋅ ),  p = p(z, ⋅ ) and q = q(p(z, ⋅ ), z, ⋅ ), respectively. Inserting that x, p, and q in Eq. (5.43) gives $\dot{z}$ as function of z. Dividing $\dot{z}$ by z leads to the function f ₁ given by

$$\displaystyle\begin{array}{rcl} f_{1}(\cdot )& =& \rho -(1-\tau )(1-\alpha )u^{\alpha }z^{\alpha } + z^{\alpha (1-\beta )-1}\varphi \cdot C_{ 1}^{1-\beta }\cdot {}\\ & &\left (\frac{(1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho } {\xi } \right )^{- \frac{1-\beta } {1-\beta _{h}} }, {}\\ \end{array}$$

with

$$\displaystyle{ C_{1} = \left ( \frac{\alpha (1-\tau )(u - 1) + u\tau } {u^{1-\alpha }(1 - u)^{\beta _{h}/(\beta _{h}-1)}} + \frac{\phi \rho u} {u^{1-\alpha }(1 - u)^{\beta _{h}/(\beta _{h}-1)}(\psi -\rho )}\right ) }$$

On a BGP we must have f ₁(⋅ ) = 0.

Differentiating f ₁(⋅ ) with respect to z leads to

$$\displaystyle\begin{array}{rcl} \frac{\mathit{df }_{1}(\cdot )} {\mathit{dz}} & =& (1-\tau )(\alpha -1)\alpha z^{\alpha -1}u^{\alpha } -\varphi z^{-2}C_{ 1}^{1-\beta }\left (\frac{(1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho } {\xi } \right )^{ \frac{1-\beta } {1-\beta _{h}} } {}\\ & +& \frac{\alpha (\beta -1)\varphi C_{1}^{1-\beta }\left (\frac{(1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho } {\xi } \right )^{- \frac{1-\beta } {1-\beta _{h}} }(\rho (1 -\beta _{h}) +\beta _{h}(1-\tau )(1-\alpha )u^{\alpha }z^{\alpha })} {(1 -\beta _{h})z^{2}((1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho )} {}\\ \end{array}$$

Existence of BGP implies $C_{1}^{1-\beta }((1\,-\,\alpha )(1\,-\,\tau )u^{\alpha }z^{\alpha }/\xi -\rho /\xi )^{- \frac{1-\beta } {1\,-\,\beta _{h}} }\,>\,0,(1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho> 0$ so that $\lim _{z\rightarrow \infty }f_{1}(\cdot ) = -\infty$ and $\lim _{ z\searrow u^{-1}\rho ^{1/\alpha }((1-\alpha )(1-\tau ))^{-1/\alpha }}f_{1}(\cdot ) = +\infty$, where $z \searrow u^{-1}\rho ^{1/\alpha }((1-\alpha )(1-\tau ))^{-1/\alpha }$ means that z approaches that value from above. We can restrict the range of z to $z> u^{-1}\rho ^{1/\alpha }((1-\alpha )(1-\tau ))^{-1/\alpha }$ because otherwise there would be no positive balanced growth rate. Further, for $z \in \left (u^{-1}\rho ^{1/\alpha }((1-\alpha )(1-\tau ))^{-1/\alpha },\infty \right )$ we have df ₁(⋅ )∕dz < 0 so that there is a unique z that solves f ₁(⋅ ) = 0 and, thus, a unique BGP. □ 

5.1.2 Proof of Lemma 3

From the proof of Proposition 29 we know that f ₁(⋅ ) = 0 must hold on the BGP. Solving (5.41) with respect to x leads to x = x(z, ⋅ ). Inserting that x in Eq. (5.44) and solving the resulting equation with respect to p yields

$$\displaystyle{ p = -u^{\alpha }z^{\alpha }\left ( \frac{\phi } {\psi -\rho }\right ) \leftrightarrow \frac{\phi } {\psi -\rho } = \frac{-p} {u^{\alpha }z^{\alpha }} }$$

Substituting ϕ∕(ψ −ρ) in f ₁(⋅ ) = 0 (from the proof of Proposition 29) leads to

$$\displaystyle\begin{array}{rcl} f_{2}(\cdot )& =& \rho -(1-\tau )(1-\alpha )u^{\alpha }z^{\alpha } +\varphi \cdot \left (\frac{(1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho } {\xi } \right )^{- \frac{1-\beta } {1-\beta _{h}} }. {}\\ & & \left (z^{\alpha -1/(1-\beta )} \frac{\alpha (1-\tau )(u - 1) + u\tau } {u^{1-\alpha }(1 - u)^{\beta _{h}/(\beta _{h}-1)}} + \frac{\rho u} {u(1 - u)^{\beta _{h}/(\beta _{h}-1)}}z^{-1/(1-\beta )}(-p)\right )^{1-\beta } {}\\ \end{array}$$

The function f ₂(⋅ ) is continuous in x and p and this function has the following properties, $\lim _{z\searrow u^{-1}\rho ^{1/\alpha }((1-\alpha )(1-\tau ))^{-1/\alpha }}f_{2}(\cdot ) = +\infty,\ \lim _{z\rightarrow \infty }f_{2}(\cdot ) = -\infty$. From Proposition 29 we know that existence of a BGP implies that it is unique so that $\partial \,f_{2}(\cdot )/\partial \,z <0$ holds for f ₂(⋅ ) = 0. Further, ∂ f ₂(⋅ )∕∂ p < 0 is immediately seen. Implicitly differentiating f ₂(⋅ ) then gives

$$\displaystyle{ \frac{\mathit{dz}} {\mathit{dp}} = -\frac{\partial \,f_{2}(\cdot )/\partial \,p} {\partial \,f_{2}(\cdot )/\partial \,z} <0. }$$

Since the balanced growth rate rises with z the balanced growth rate and the debt ratio are negatively correlated. □ 

5.1.3 Proof of Proposition 31

The balanced growth rate is given by $g =\dot{ c}/c = -\rho + (1-\tau )(1-\alpha )z^{\alpha }u^{\alpha }$. A deficit financed increase in educational spending can be modeled by a decline in ϕ which is immediately seen from Eqs. (5.35) and (5.34). Differentiating g with respect to −ϕ gives

$$\displaystyle{ \frac{\partial \,g} {\partial \,(-\phi )} = (1-\tau )(1-\alpha )u^{\alpha }\alpha z^{\alpha -1} \frac{\partial \,z} {\partial \,(-\phi )}, }$$

with z evaluated on the BGP, that is, at z = z ^⋆.

The derivative $\partial \,z/\partial \,(-\varphi )$ is obtained by implicit differentiation of f ₁(⋅ ) (from the proof of Proposition 29) as

$$\displaystyle{ \frac{\partial \,z} {\partial \,(-\phi )} = -\frac{\partial \,f_{1}(\cdot )/\partial \,(-\phi )} {\partial \,f_{1}(\cdot )/\partial \,z} = \frac{z^{\alpha (1-\beta )-1}\varphi (1-\beta )C_{1}^{-\beta }\left (\frac{(1-\alpha )(1-\tau )u^{\alpha }z^{\alpha }-\rho } {\xi } \right )^{- \frac{1-\beta } {1-\beta _{h}} }\rho u^{\alpha }} {(1 - u)^{\beta _{h}/(\beta _{h}-1)}(\psi -\rho )(\partial \,f_{1}(\cdot )/\partial \,z)}. }$$

Because of ∂ f ₁(⋅ )∕∂ z < 0 for all relevant z (see the proof of Proposition 29) we get

$$\displaystyle{ \frac{\partial \,z} {\partial \,(-\phi )}> (<)0\quad \text{for}\ \psi <(>)\rho }$$

 □ 

5.1.4 Proof of Proposition 32

To prove that proposition we note that $\dot{c}/c =\dot{ k}/k$ implies $\dot{k}/k = -\rho + (1-\tau )r$. Using this, the debt to physical capital ratio evolves according to $\dot{p}/p = (\rho -\psi ) -\phi (y/k)p^{-1}$. Thus, the differential equation describing the evolution of public debt relative to physical capital is obtained as

$$\displaystyle{ \dot{p} = p(\rho -\psi ) -\phi \left (\frac{y} {k}\right ), }$$

where $y/k = u^{\alpha }(v/k)^{\alpha }$ is constant if knowledge v and physical k grow at the same rate. Setting $\dot{p} = 0$, the debt to physical capital ratio on the BGP is computed as $p^{\star } = (y/k)\phi /(\rho -\psi )$.

For p ^⋆ > 0 we have ϕ < ( > )0 for ρ < ( > )ψ. In addition, ϕ < ( > )0 gives $\dot{p}> (<)0$ for p = 0. Further, $\partial \,\dot{p}/\partial \,p =\rho -\psi <(>)0$ for ρ < ( > )ψ. This demonstrates that p converges to the unique p ^⋆( > 0) if and only if ρ < ψ.

For p ^⋆ < 0 we have $\varphi> (<)0$ for ρ < ( > )ψ. Further, ϕ > ( < )0 yields $\dot{p} <(>)0$ for p = 0. Again we have $\partial \,\dot{p}/\partial \,p =\rho -\psi <(>)0$ for ρ < ( > )ψ so that p converges to the unique p ^⋆( < 0) if and only if ρ < ψ. □ 

5.1.5 Transition Dynamics

Since the initial conditions with respect to q, z, and p are given, that is, q(0) = q ₀, z(0) = z ₀ and p(0) = p ₀, the economy is saddle point stable if the Jacobian matrix of the differential equation system (5.41)–(5.44) has three negative eigenvalues or three eigenvalues with negative real parts. In that case, there exists a unique value for x(0) such that the economy converges to the BGP.

Table 5.7 gives the balanced growth rate, the debt to capital ratio on the BGP and the signs of the eigenvalues of the Jacobian for different values of ψ and ϕ with the benchmark parameter values used in Sect. 5.2.2.

Table 5.7 suggests that the threshold of the reaction coefficient ψ, which determines whether the economy is stable or unstable, is determined by the subjective discount rate ρ such that for values of ψ larger (smaller) ρ the economy is stable (unstable). It should also be noted that ϕ must not become too small (large) so that a BGP exists in the case of ψ > ( < )ρ. In our example underlying Table 5.7, existence of a BGP is given for ϕ > −0. 04(ϕ < 0. 027) in case of ψ = 0. 05 > ρ(ψ = 0. 025 < ρ).

Table 5.7 Balanced growth rate, g, debt to capital ratio on the BGP, b ^⋆, and the signs of the eigenvalues of the Jacobian for different values of ψ and ϕ

Table 5.8 gives the result for the situation where public debt grows but less than GDP, case (ii) in Definition 9, and the situation of a balanced government budget, case (i) in Definition 9.

Table 5.8 Balanced growth rate, g, growth rate of public debt, g _b , and the signs of the eigenvalues of the Jacobian for ϕ = 0 and for different values of ψ