Financial deepening in a two-sector endogenous growth model with productivity heterogeneity

Quoc Hung Nguyen

doi:10.1515/bejm-2019-0039

Article

Financial deepening in a two-sector endogenous growth model with productivity heterogeneity

Quoc Hung Nguyen

Published/Copyright: June 29, 2019

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From the journal The B.E. Journal of Macroeconomics Volume 20 Issue 1

Abstract

This paper studies the effects of financial deepening and fiscal policy on human capital formation, working hours and growth in a model with financial frictions and productivity heterogeneity. The paper first shows that in the range of capital tax rates that attains a balanced growth path, taxation exerts inverted U-shaped effects on growth. The paper then analytically derives and shows that the growth maximizing tax rate and the corresponding growth are increasing concave functions of the financial deepening level. Finally, it is shown that theoretical predictions of the model are in line with data from OECD countries.

Keywords: financial deepening; heterogeneity; taxation

JEL Classification: E10; E22; E44; O16

Acknowledgments

I would like to thank two anonymous referees for constructive comments and suggestions. All errors are still solely mine.

Appendix A

Data-set description

Data for years of total schooling the age group beyond 15 is taken from Barro-Lee Education Attainment Data-set: Barro and Lee 2013Barro and Lee. 2013, “A New Data Set of Educational Attainment in the World, 1950–2010.” Journal of Development Economics, vol 104, pp. 184–198, available for download at http://www.barrolee.com/. The Years of Schooling variable is obtained by taking the average of years of total schooling from 1995 to 2004.
Financial data is taken from Financial Development and Structure Data-set (Updated Nov. 2013): Thorsten Beck, Asli Demirguc-Kunt, Ross Eric Levine, Martin Cihak and Erik H.B. Feyen, available for download at http://siteresources.worldbank.org/INTRES/Resources/469232-1107449512766/FinStructure_November_2013.xlsx. The Private Credit Indicator that reflects financial deepening level is then computed by taking the average of the ratio of total credit issued by depository and other financial institutions to the private sector over GDP from 1985 to 2004.
Data for all other variables is obtained from The Penn World Table 9.0 (PWT 9.0): Feenstra, Inklaar, and Timmer (2015), “The Next Generation of the Penn World Table” American Economic Review, 105(10), 3150–3182, available for download at www.ggdc.net/pwt. In particular, growth rate is computed from rgdpo, which is output-side real GDP at chained PPPs in mil. 2011US$; TFP level, ctfp, is at current PPPs where USA is equal to one; human capital index, hc, is computed based on years of schooling and returns to education; average working hours, avh, is the average annual hours worked by persons engaged. All indicators in this paper are then computed by taking the average of the corresponding variables from 1985 to 2004.
Data for public spending on eduction and health is the annual government expenditure on education and health as a percentage of GDP and is taken from UNESCO and OECD statistics.

Appendix B

Workers and firms’ optimization

The Representative Worker: The representative worker optimizes the following sum of discounted utilities from consumption

∫0∞e−ρtlog⁡cw(t)dt

subject to

h˙=G(h,ge)(1−u)cw(t)=u(t)h(t)w(t)

After substituting the budget constraint into the instantaneous utility function and denoting the current-value costate variable by μ(t), the current-value Hamiltonian for the representative worker’ optimization problem can be expressed as:

H(u,h,λ)=log⁡[wuh]+μ[G(h,ge)(1−u)]

The F.O.Cs of this optimization state:

1u=μG(h,ge)1h+μ(1−u)Gh(h,ge)=ρμ−μ˙limt→∞e−ρtμ(t)h(t)=0

These F.O.Cs together with the evolution equation for h imply that optimal allocating time to work, u(t), will obey the following differential equations:

u˙u−G(h,ge)hu+ρ=0

Firms’ Problem: Each entrepreneur maximizes the profit of his private firm subject to the technology (6) and his borrowing constraint (9), hence the profit of an entrepreneur can be expressed as follows:

Π(a,z)=maxk,n{f(z,k,n)−wn−rk}s.t.k≤λa

The first order condition of this problem with respect to labor states:

(1−α)(zk)αn−α=w

The implied labor demand then is, nd=(1−αw)1α(zk). After substituting the optimal labor demand into the technology equation (6) we then obtain the production function that is linear in individual entrepreneur’s capital input as follows:

F(z,k)=(1−αw)1−αα(zk)

Consequently, an entrepreneur’s profits can be expressed as:

Π(a,z)=maxk{α(1−αw)1−ααzk−rk}s.t.k≤λa

This entrepreneur’s profit is linear in capital input, hence implying capital demand, labor demand and the productivity cutoff as in (10).

Entrepreneur’s Consumption Rule: Let V(a, z, t) be the value function of this optimality problem, then the Hamilton-Jacobi-Bellman equation is set as follows,

ρV(a,z,t)=maxce(t){log⁡(ce(t))+1dtEa,z[dV(a,z,t)]s.t.(14)}

This optimal rule can be confirmed as follows. First guess that the value function takes the form V(a,z,t)=B[v(z,t)+log⁡a], where B is an undermined constant. Substitute this form back to the above Hamilton-Jacobi-Bellman equation and take first order condition to obtain c = a/B. Substitute back in and then apply the envelop theorem to obtain B = 1/ρ.

Appendix C

Proof of the proposition Proposition 1

In this economy, the aggregate output denoted by y can be obtained by summing the amounts of homogenous final goods produced by all active entrepreneurs, i.e. entrepreneurs with idiosyncratic productivity higher than the cut-off z:

(48)y=∫f(z,k,n)dΦt(a,z)=∬(1−αw)1−ααzλag(z)ψ(a)dadz=παλ∫aψ(a)da∫zg(z)dz=παλk∫z_∞zg(z)dz=παλXk,whereX≡∫z_∞zg(z)dz

Substituting the optimal labor demand (11) into the labor market clearing condition (18), we obtain

uh=∫nd(a,z)dΦ(a,z)=(πα)11−αλXk

which then implies that:

(49)π=α(λX)α−1kα−1(uh)1−α

Plugging this equation back to (48) we obtain the aggregate output as follows:

y=(λX)αu1−αk[1+α−1](h)1−α=Akα(uh)1−α

where A is the endogenous measured TFP

A(t)≡(λX)α=(∫z_∞zg(z)dz(1−G(z_)))α=E[z|z≥z_]α

Substituting capital demand from (10) into (17) and recall that z is i.i.d we obtain the following capital market equilibrium equation

(50)1=λ(1−G(z_))

which in turn determines the productivity cut-off z_=λ1φ.

The wage rate, w, can be obtained by substituting (49) into the definition of π (13):

w=(1−α)Akα(uh)−α

Similarly, the capital return, r, can be expressed as:

r=αz_E[z|z≥z_]Akα−1(uh)1−α

The budget constraint of the government becomes

ge=∫τadΦt(a,z)=∬τag(z)ψ(a)dadz=kτ

We then derive the dynamic equation of the aggregate capital stock, k, by first aggregating wealth of all entrepreneurs

k˙k=1k∫a˙dΦ(a,z)=1k∬a˙g(z)ψ(a)dadz=1k∬(λmax{zπ−r,0}+r−τ−ρ)ag(z)ψ(a)dadz=∫0∞(λmax{zπ−r,0}+r−τ−ρ)g(z)dz

and then dividing entrepreneurs into the inactive group (z<z_) and the active group (z≥z_), therefore

k˙k=r−ρ−τ+∫z_∞λ{zπ−r}g(z)dz=r−ρ−τ+πλ∫z_∞zg(z)dz−rλ∫z_∞g(z)dz=πλX+r−ρ−τ−r=α(λX)α−1kα−1λX−ρ=αAkα−1(uh)1−α−ρ−τ

where the third and forth equal signs are implied by the definition of X in (19), z_ in (12), and π in (49).

Appendix D

Proof of the proposition Proposition 2

Rewrite the system of two non-linear differential equations (41) and (42) as:

(51)κ˙=H(κ,u)u˙=U(κ,u)

where H(κ,u)≡[αAκα−1u1−α−bτ1−ϕκ1−ϕ(1−u)−ρ−τ]κ, U(κ,u)≡[bτ1−ϕκ1−ϕu−ρ]u

The Jacobian matrix of the non-linear system (51) evaluated at the equilibrium obtained from the Proposition 1 is:

(52)JE≡[HκHuUκUu]

The trace of this Jacobian matrix is equal to:

(53)trace(JE)=α(α−1)Aκα−1u1−α−b2τ1−ϕκ1−ϕ[(1−ϕ)(1−u)−u]

Since u ∈ (0, 1) and ϕ ∈ (0, 1), the value of trace(JE) can be positive, negative and equal to zero. In the meantime, the determinant of this Jacobian matrix is equal to:

(54)|JE|=2α(α−1)Abτ1−ϕκα−ϕu2−ϕ−(1−ϕ)b2τ2(1−ϕ)κ2(1−ϕ)u

Since α ∈ (0, 1) and ϕ ∈ (0, 1), it is straightforward that the value of this determinant of this Jacobian matrix is always negative. As a result, the reduced linearization of the system of two non-linear differential equations (41) and (42) has two roots with opposite signs. Therefore, the equilibrium obtained the Proposition 1 is saddle point in the local sense.

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Published Online: 2019-06-29

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https://doi.org/10.1515/bejm-2019-0039

Keywords for this article

financial deepening; heterogeneity; taxation