Growth, income distribution, and the ‘entrepreneurial state’

Abstract

In this paper, we introduce a twofold role for the public sector in the Goodwin (1967) model of the growth cycle. The government collects income taxes in order to: (a) invest in infrastructure capital, which directly affects the production possibilities of the economy; (b) finance publicly-funded research and development (R&D), which augments the growth rate of labor productivity. We study two versions of the model, with and without induced technical change; that is, with or without a feedback from the labor share to labor productivity growth. In both cases we show that: (i) provided that the output-elasticity of infrastructure is greater than the elasticity of labor productivity growth to public R&D, there exists a tax rate that maximizes the long-run labor share, and it is smaller than the growth-maximizing tax rate; (ii) the long-run share of labor is always increasing in the share of public spending in infrastructure; (iii) different taxation schemes can affect the stability of growth cycles.

Appendices

Appendix A: Proof of propositions 1 and 6

1.1 A.1 The model with public R&D only

Maximizing the labor share is equivalent to minimizing the natural logarithm of its complement (that is, the share of profits) 1 − ω_ss as written in the RHS of Eq. 11. We have that

$$\frac{\partial\ln(1-\omega_{ss})}{\partial\tau}=\frac{1}{1-\tau}-\left( \frac{\eta-\phi}{1-\eta}\right)\frac{1}{\tau}, $$

and

$$\frac{\partial^{2}\ln(1-\omega_{ss})}{\partial\tau^{2}}=\frac{1}{(1-\tau)^{2}}+\left( \frac{\eta-\phi}{1-\eta}\right)\frac{1}{\tau^{2}}>0. $$

Because the steady state profit share is a convex function of τ, the first order condition ∂ ln(1 − ω_ss)/∂τ = 0 is necessary and sufficient for a minimum. It has an interior solution in

$$\tau_{\omega}=\frac{\eta-\phi}{1-\phi}\in(0,1). $$

1.2 A.2 Public R&D and induced technical change

In order to prove the same result in the general model, totally differentiate Eq. 14 with respect to ω and τ to find

$$\left( \frac{(1-\omega_{ss})\beta+(1-\eta)\omega_{ss}}{\omega_{ss}^{1-\beta}(1-\omega_{ss})^{2-\eta}}\right)d\omega=\cfrac{s^{1-\eta}\theta^{\eta}}{\lambda(1-\theta)^{\phi}}\left( \cfrac{\eta-\phi-\tau(1-\phi)}{(1-\tau)^{\eta}\tau^{1-(\eta-\phi)}}\right)d\tau, $$

hence,

$$ \frac{d\omega}{d\tau}=\cfrac{s^{1-\eta}\theta^{\eta}}{\lambda(1-\theta)^{\phi}}\cfrac{\omega_{ss}^{1-\beta}\left( 1-\omega_{ss}\right)^{2-\eta}}{(1-\tau)^{\eta}\tau^{1-(\eta-\phi)}}\cfrac{[\eta-\phi-\tau(1-\phi)]}{[(1-\omega_{ss})\beta+(1-\eta)\omega_{ss}]}. $$

(16)

The denominator is always positive. It follows that $Sign\frac {d\omega }{d\tau }=Sign[\eta -\phi -\tau (1-\phi )]$, which proves that the labor share is maximized by $\tau _{\omega }=\frac {\eta -\phi }{1-\phi }.$

Appendix B: Composition of public expenditure

1.1 B.1 The model with public R&D only

Start with

$$0\leq\frac{\lambda^{\frac{1}{1-\eta}}(1-\theta)^{\frac{\phi}{1-\eta}}}{s(1-\tau)\tau^{\frac{\eta-\phi}{1-\eta}}\theta^{\frac{\eta}{1-\eta}}}\leq1, $$

and re-write the second inequality as

$$\frac{(1-\theta)^{\frac{\phi}{1-\eta}}}{\theta^{\frac{\eta}{1-\eta}}}\equiv m(\theta)\leq(1-\tau)\tau^{\frac{\eta-\phi}{1-\eta}}s/\lambda^{\frac{1}{1-\eta}}\equiv h(\tau). $$

m(𝜃) is monotonically decreasing in 𝜃, with $lim_{\theta \rightarrow 0^{+}}=\infty$, and m(1) = 0.h(τ) is hump-shaped, has a maximum in $\tau _{\omega }=\frac {\eta -\phi }{1-\phi }$, with h(0) = h(1) = 0, and $h(\tau _{\omega })=\frac {\eta -\phi }{1-\phi }^{\frac {\eta -\phi }{1-\phi }}\left (\frac {1-\eta }{1-\phi }\right )s/\lambda ^{\frac {1}{1-\eta }}$. We need $m(\theta )\leq h(\tau )\leq \frac {\eta -\phi }{1-\phi }^{\frac {\eta -\phi }{1-\phi }}\left (\frac {1-\eta }{1-\phi }\right )s/\lambda ^{\frac {1}{1-\eta }},$ so that $\theta \geq m^{-1}\left [\frac {\eta -\phi }{1-\phi }^{\frac {\eta -\phi }{1-\phi }}\left (\frac {1-\eta }{1-\phi }\right )s/\lambda ^{\frac {1}{1-\eta }}\right ]\equiv \bar {\theta }$. For any feasible $\theta >\bar {\theta }$ there is a tax rate τ_max(𝜃) such that m(𝜃) = h[τ_max(𝜃)]. Since h(.) is decreasing in the relevant range, it follows that τ ≤ τ_max. Moreover, τ_max(𝜃) is an increasing function because, for $\theta \geq \bar {\theta }$, both m(.) and h(.) are decreasing functions.

1.1.1 B.1.1 Proof of proposition 2

i) Since $\tau _{max}(\bar {\theta })=\tau _{\omega },$ then $\tau _{g}(\bar {\theta })=\tau _{max}(\bar {\theta })=\tau _{\omega }.$ Next, ii) since $\tau _{max}(\theta )\in (\tau _{\omega },1),\forall \theta >\bar {\theta }$, it follows that τ_g(𝜃) = τ_max(𝜃) > τ_ω.

1.1.2 B.1.2 Proof of proposition 3

Start with $\tilde {g}_{A,ss}(\theta )=\lambda \left [(1-\theta )\tau _{g}(\theta )\right ]^{\phi }.$ Let G(𝜃) ≡ (1 − 𝜃)τ_g(𝜃), so that $\tilde {g}_{A,ss}(\theta )=\lambda [G(\theta )]^{\phi }$ and $\tilde {g}_{A,ss}^{\prime }(\theta )=\lambda \phi G^{\prime }(\theta )/[G(\theta )]^{1-\phi }$. We have $sign(\tilde {g}_{A,ss}^{\prime })=sign(G^{\prime }),$ which implies that $\tilde {g}_{A,ss}$ and G have the same stationary points. Let us now analyze $G^{\prime }(\theta )=(1-\theta )\tau _{g}^{\prime }(\theta )-\tau _{g}(\theta ).$ All we know about $\tau _{g}^{\prime }(\theta )$ is $\tau _{g}^{\prime }(\theta )>0$, while we know that τ_g(𝜃) is an increasing function from τ_ω to arbitrarily close to 1. This implies $lim_{\theta \rightarrow 1^{-}}G^{\prime }(\theta )=-1$ so that g_A(𝜃) is decreasing in a left neighborhood of − 1. Starting from − 1, a reduction in 𝜃 increases the growth rate $\tilde {g}_{A,ss}$ as long as G^′(𝜃) < 0. There is no guarantee that G^′(𝜃) will go through 0 as 𝜃 moves from 1 to $\bar {\theta }$. If it does, there is a stationary point 𝜃^∗ that solves $(1-\theta ^{*})=\tau _{g}(\theta ^{*})/\tau _{g}^{\prime }(\theta ^{*})$ and it is a maximum; if it does not, growth is maximized by the lowest admissible composition of public expenditure $\bar {\theta }.$

1.2 B.2 Public R&D and induced technical change

1.2.1 B.2.1 Proof of proposition 7

Taking logs in Eq. 5 evaluated at the steady state, we have $\ln g_{A}=\ln \left [\lambda (1-\theta )^{\phi }\tau ^{\phi }\omega (\tau ,\theta )_{ss}^{\beta }\right ]=\ln \lambda +\phi \ln (1-\theta )+\phi \ln \tau +\beta \ln \omega (\tau ,\theta )_{ss}.$ Hence, $\frac {d\ln g_{A}}{d\tau }=\cfrac {\phi }{\tau }+\cfrac {\beta }{\omega }\frac {d\omega }{d\tau }$. Setting $\frac {d\ln g_{A}}{d\tau }= 0,$ while using Eq. 16 we have

$$\cfrac{1}{\tau}\left[\phi+\cfrac{\beta}{\lambda}\cfrac{\left( 1-\omega_{ss}\right)^{2-\eta}}{\omega_{ss}^{\beta}}\cfrac{s^{1-\eta}\theta^{\eta}}{(1-\theta)^{\phi}}\cfrac{\tau_{g}^{\eta-\phi}}{(1-\tau_{g})^{\eta}}\cfrac{[\eta-\phi-\tau_{g}(1-\phi)]}{[(1-\omega_{ss})\beta]+(1-\eta)\omega_{ss}]}\right]= 0, $$

or

$$\phi=\cfrac{\beta}{\lambda}\cfrac{\left( 1-\omega_{ss}\right)^{2-\eta}}{\omega_{ss}^{\beta}}\cfrac{s^{1-\eta}\theta^{\eta}}{(1-\theta)^{\phi}}\cfrac{\tau_{g}^{\eta-\phi}}{(1-\tau_{g})^{\eta}}\cfrac{[\tau_{g}(1-\phi)-(\eta-\phi)]}{[(1-\omega_{ss})\beta+(1-\eta)\omega_{ss}]}, $$

which requires τ_g(1 − ϕ) > η − ϕ, or $\tau _{g}>\frac {\eta -\phi }{1-\phi }.$

1.2.2 B.2.2 Growth-maximizing composition of public expenditure

With respect to the growth-maximizing composition of public expenditure (𝜃^∗), totally differentiate Eq. 14 with respect to ω and τ to find

$$\cfrac{d\omega}{d\theta}=\left( \cfrac{[s(1-\tau)]^{1-\eta}\tau^{\eta-\phi}}{\lambda}\right)\left( \cfrac{\eta-\theta(\eta-\phi)}{\theta^{1-\eta}(1-\theta)^{1+\phi}}\right)\left( \frac{\omega_{ss}^{\beta-1}\left( 1-\omega_{ss}\right)^{2-\eta}}{(1-\omega_{ss})\beta+(1-\eta)\omega_{ss}]}\right). $$

Next, set $\frac {d\ln g_{A}}{d\theta }=\cfrac {-\phi }{1-\theta }+\cfrac {\beta }{\omega }\frac {d\omega }{d\theta }= 0$, to find

$$\cfrac{[s(1-\tau)]^{1-\eta}\tau^{\eta-\phi}}{\lambda}\cfrac{\eta-\theta^{*}(\eta-\phi)}{[(1-\omega_{ss})\beta+(1-\eta)\omega_{ss}]}\frac{\left( 1-\omega_{ss}\right)^{2-\eta}}{\omega_{ss}^{2-\beta}}=\cfrac{\phi}{\beta}\theta^{*(1-\eta)}(1-\theta^{*})^{\phi}. $$

Appendix: C: Uniqueness of the steady state in the model with induced technical change

Rewrite Eq. 14 as

$$\omega_{ss}^{\beta}=\cfrac{\left[s(1-\tau)\right]^{1-\eta}\tau^{\eta-\phi}\theta^{\eta}}{\lambda(1-\theta)^{\phi}}\left( 1-\omega_{ss}\right)^{1-\eta}\equiv{\Gamma}(\tau,\theta)\left( 1-\omega_{ss}\right)^{1-\eta}, $$

where Γ(τ,𝜃) > 0. Raise both sides of the equation to the power 1/β and rearrange to find:

$$\omega_{ss}-\left( {\Gamma}(\tau,\theta)\right)^{\frac{1}{\beta}}\left( 1-\omega_{ss}\right)^{\frac{1-\eta}{\beta}}= 0. $$

The left hand side is the difference between two continuous functions of ω, with ω ∈ [0, 1]. The first one increases linearly from 0 to 1; the second one is a power function decreasing from $\left ({\Gamma }(\tau ,\theta )\right )^{\frac {1}{\beta }}$to 0. Therefore, the left hand side is a continuous function that starts from $-\left ({\Gamma }(\tau ,\theta )\right )^{\frac {1}{\beta }}$and increases monotonically to 1: it crosses the horizontal axis once and only once. By Eqs. 12 and 13, both χ_ss and e_ss are monotonic functions of the labor share: if the steady state value of the latter is unique, so are the former.

Appendix D: Stability analysis

1.1 D.1 Proof of proposition 8 - income tax with induced technical change

Linearization of the system formed by Eqs. 6, 7 and 8 around its steady state position, when β ∈ (0, 1), yields the following Jacobian matrix:

$$\begin{array}{ll} J(\chi_{ss},e_{ss},\omega_{ss}) & =\left[\begin{array}{ccc} J_{11} & 0 & J_{13}\\ 0 & 0 & J_{23}\\ 0 & J_{32} & J_{33} \end{array}\right], \end{array} $$

with

$$\begin{array}{ll} J_{11} & =-\tau\theta\chi_{ss}^{\eta-1}<0;\\ J_{13} & =s(1-\tau)\chi_{ss}^{1+\eta}>0;\\ J_{23} & =-e_{ss}\left\{ (1-\eta)s(1-\tau)\chi_{ss}^{\eta}+\lambda\beta[(1-\theta)\tau]^{\phi}\omega^{\beta-1}\right\} <0;\\ J_{32} & =\delta^{-1}e_{ss}^{\frac{1-\delta}{\delta}}\omega_{ss}>0;\\ J_{33} & =-\lambda\beta\left( (1-\theta)\tau\right)^{\phi}\omega_{ss}^{\beta}<0. \end{array} $$

The Routh-Hurwitz necessary and sufficient conditions for stability of the steady state require that:

• 1. TrJ < 0. We have that TrJ = J₁₁ + J₃₃ < 0 as required.

• 2. DetJ < 0. We have that DetJ = J₁₁ × (−J₂₃J₃₂) < 0 as required.

• 3. PmJ > 0, where PmJ denotes the sum of the principal minors of J. In fact, PmJ = −J₂₃J₃₂ + J₁₁J₃₃ > 0 as required.

• 4. Finally, we need to check that − PmJ + DetJ/TrJ < 0. Since TrJ < 0, the condition can be rewritten as DetJ > TrJ(PmJ). We have $-J_{11}J_{23}J_{32}>\left (J_{11}+J_{33}\right )\left [-J_{23}J_{32}+J_{11}J_{33}\right ]=-J_{11}J_{23}J_{32}-J_{33}J_{23}J_{32}+J_{11}^{2}J_{33}+J_{11}J_{33}^{2}$, ⇔$0>J_{33}\left (-J_{23}J_{32}+J_{11}^{2}+J_{11}J_{33}\right ),$ which is always true.

Notice that the stability conditions hold for any value of the main parameters of the model s ∈ (0, 1],τ ∈ (0, 1),𝜃 ∈ (0, 1],η ∈ [0, 1).

1.2 D.2 Proof of proposition 4 - income tax with public R&D only

Linearization of the system formed by Eqs. 6, 7 and 8 around its steady state position, evaluated at β = 0, yields the following Jacobian matrix:

$$\begin{array}{ll} J(\chi_{ss},e_{ss},\omega_{ss}) & =\left[\begin{array}{ccc} J_{11} & 0 & J_{13}\\ 0 & 0 & J_{23}\\ 0 & J_{32} & 0 \end{array}\right], \end{array} $$

with

$$\begin{array}{ll} J_{11} & =-\tau\theta\chi_{ss}^{\eta-1}<0;\\ J_{13} & =s(1-\tau)\chi_{ss}^{1+\eta}>0;\\ J_{23} & =-(1-\eta)s(1-\tau)\chi_{ss}^{\eta}e_{ss}<0;\\ J_{32} & =\delta^{-1}e_{ss}^{\frac{1-\delta}{\delta}}\omega_{ss}>0. \end{array} $$

The Routh-Hurwitz necessary and sufficient conditions for stability of the steady state require that:

• 1. TrJ < 0. We have that TrJ = J₁₁ < 0 as required.

• 2. DetJ < 0. We have that DetJ = J₁₁ × (−J₂₃J₃₂) < 0 as required.

• 3. PmJ > 0, where PmJ denotes the sum of the principal minors of J. It is easy to check that, in fact, PmJ = −J₂₃J₃₂ > 0 as required.

• 4. Finally, we need to check that − PmJ + DetJ/TrJ < 0. This condition is violated. In fact, DetJ/Tr_J = Pm₁J = PmJ, so we have − PmJ + PmJ = 0. As illustrated by Julius (2006), when the fourth condition goes from negative (see the previous appendix) through zero, the Hopf bifurcation theorem implies that the system has a family of closed orbits in a neighborhood of the steady state. This is happening as β goes from positive to zero.

Notice that, as long as β = 0, the above properties are not sensitive to alternative parametric specifications: the entries of the Jacobian matrix will not change signs as long as s ∈ (0, 1],τ ∈ (0, 1),𝜃 ∈ (0, 1],η ∈ [0, 1). Thus, the limit cycle is robust in the parameter space.

1.3 D.3 Proof of proposition 9 - profit tax

The Jacobian matrix evaluated at the steady state has the following structure:

$$\begin{array}{ll} J(\chi_{ss},e_{ss},\omega_{ss}) & =\left[\begin{array}{ccc} J_{11} & 0 & 0\\ 0 & 0 & J_{23}\\ 0 & J_{32} & J_{33} \end{array}\right], \end{array} $$

with

$$\begin{array}{ll} J_{11} & =-\tau\theta(1-\omega)\chi_{ss}^{\eta-1}<0;\\ J_{23} & =-\frac{e_{ss}}{\omega_{ss}(1-\omega)_{ss}}g_{A,ss}[\omega_{ss}(1-\phi)+\beta(1-\omega_{ss})]<0;\\ J_{32} & =\delta^{-1}e_{ss}^{\frac{1-\delta}{\delta}}\omega_{ss}>0;\\ J_{33} & =-\frac{g_{A,ss}}{1-\omega_{ss}}[\beta(1-\omega_{ss})-\phi\omega_{ss}]. \end{array} $$

With $\beta \in [0,\frac {\phi \omega _{ss}}{1-\omega _{ss}})$, J₃₃ > 0 and the steady state is unstable. With $\beta =\frac {\phi \omega _{ss}}{1-\omega _{ss}},J_{33}= 0$, which is the limit cycle case. With $\beta >\frac {\phi \omega _{ss}}{1-\omega _{ss}},J_{33}<0$, and the steady state is stable.

1.4 D.4 Wage tax with public R&D only

The Jacobian matrix evaluated at the steady state has the following structure:

$$\begin{array}{ll} J(\chi_{ss},e_{ss},\omega_{ss}) & =\left[\begin{array}{ccc} J_{11} & 0 & J_{13}\\ 0 & 0 & J_{23}\\ 0 & J_{32} & J_{33} \end{array}\right], \end{array} $$

with

$$\begin{array}{ll} J_{11} & =-\tau\theta\omega\chi_{ss}^{\eta-1}<0;\\ J_{23} & =\frac{e_{ss}}{\omega_{ss}(1-\omega)_{ss}}g_{A,ss}[\eta-\omega_{ss}-(1-\omega_{ss})(\phi+\beta)];\\ J_{32} & =\delta^{-1}e_{ss}^{\frac{1-\delta}{\delta}}\omega_{ss}>0;\\ J_{33} & =-\frac{g_{A,ss}}{\omega}[\beta+\phi]<0. \end{array} $$

The condition η < ω_ss is sufficient for J₂₃ < 0, which ensures local stability. This condition is verified under our parameterization: as mentioned in Section 3.3, estimates for η are around 15%, way below the long-run value of the labor share.