Collateral and development

Nicola Amendola; Lorenzo Carbonari; Leo Ferraris

doi:10.1515/bejm-2019-0033

Article

Collateral and development

Nicola Amendola , Lorenzo Carbonari and Leo Ferraris

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 presents a model economy with endogenous credit constraints à la Kiyotaki-Moore and endogenous growth à la Uzawa-Lucas, in which agents face a trade-off between investing resources to improve the pledgeability of collateral assets and the accumulation of human capital. The model generates both growth miracles and stagnant economies.

Keywords: collateral; credit; development; growth; human capital

JEL Classification: G0; O1; O40

Acknowledgments

We thank Alberto Bucci, Guido Cozzi, Alessandra Pelloni, Pasquale Scaramozzino and seminar participants at the “Finance and Economic Growth in the Aftermath of the Crisis” conference held at the Università di Milano, September 11–13, 2017, for their comments and suggestions. We also thank an anonymous referee for the helpful comments. Financial support from the Università di Roma, “Tor Vergata” (Grant Consolidate the Foundations), is gratefully acknowledged. The authors have no conflict of interest. Remaining errors are ours.

Appendix A

A.1 Proofs

Proof of Lemma 1

By equations (9) delayed one period and (12), rearranging we obtain

(1−α)yt+1−wt+1lt+1=(λtλt+11β−αyt+1kt+1−ut)wt+1lt+1ut.

Hence, 1β−αyt+1kt+1−ut>0⇔(1−α)yt+1>wt+1lt+1, which is equivalent to ζt+1>0, by (9). ⊡

Proof of Lemma 2

Suppose these terms are equal. This is consistent with the BCCE system only if (1−α)δ=β, which violates (5). ⊡

Proof of Proposition 1

Substitute (21) and (22) into (20), obtaining one equation in u. The solution is either u¯=min{1+δ2δ,1} in the constrained case, or u~=min{(1−α)δ−βα(1−α)(β+δ)−βα1+δδ,1}, in the unconstrained case, with u~>0 under (5). By Lemma 2 only these two situations are possible. The rest of the system determines uniquely k, c and c^. ⊡

Proof of Proposition 2

a.u¯<1⇔δ>1; b.u~<1⇔β>(1−α)δ(1−α)δ+α. ⊡

Proof of Proposition 3

By (24), the collateral constraint binds at t = 0 iff (1−α)y0≥c^0=k0⇔k0≤h0(1−α)11−α. ⊡

Proof of Proposition 4

To check whether a constrained BCCE with growth becomes unconstrained at some finite date, we need to see whether there exists a finite t > 0, such that

(1−α)y~t=u¯k¯t,

where, on the LHS there is the unconstrained equilibrium value y~t, while on the RHS the constrained equilibrium value u¯k¯t. Therefore, if the equation

(36)(1−α)(β(1−α)(1+δ)(1−α)(β+δ)−βα)th01−αk0α=1+δ2δ(1+δ2)tk0,

has a solution t∗∈(0,∞), the initially constrained CCE with growth becomes unconstrained at ⌈t∗⌉, the closest integer not smaller than t^∗, otherwise the CCE remains indefinitely constrained. Solve (36) for t, obtaining

(37)t=ln⁡((1−α)h01−αk0α−1)−ln⁡1+δ2δln⁡(1+δ2)−ln⁡(β(1−α)(1+δ)(1−α)(β+δ)−βα),

where (1−α)h01−αk0α−1>1 and 1+δ2δ<1, at an initially constrained CCE with growth. Hence, (37), which is finite, is strictly positive iff 1+δ2>β(1−α)(1+δ)(1−α)(β+δ)−βα, which holds under (5). ⊡

Proof of Proposition 5

In a stagnant CCE, u = 1 and k = k₀ always. The CCE is always constrained if k0<h0(1−α)11−α, always unconstrained otherwise. ⊡

Proof of Proposition 6

In the constrained regime, by (16) and (17), effort ut=min{1+δ2δ,1} at all times. The remaining variables are determined as before. ⊡

Proof of Proposition 7

Linearize (28) and (29) around the BCCE, taking first differences, zt+1−zt and ut−ut−1. Solve the characteristic equation of the system,

(Fz−1−ξ)(Gu−1−ξ)−FuGz=0,

where H_υ is the partial derivative of a function H wrt υ evaluated at the BCCE, to find the eigenvalues ξ1=(1−α)δ+β(1−2α)αδ and ξ2=1−ββ, with ξ₁ > 1 and ξ2<1⇔β>12. ⊡

Proof of Proposition 8

In the constrained regime, at the CCE (1−α)yt>c^t=wtlt=−vl(c^t,lt)ltvc(c^t,lt), hence, the CCE allocation violates (31); in the unconstrained regime, the CCE satisfies all the equations (31)–(35), provided μ is chosen appropriately. ⊡

A.2 Figure

The figure illustrates patterns of GDP per-capita for a selection of four countries over the post-WWII period. The break is identified with the Zivot-Andrews test. Data are from the Penn World Tables 9.0. The patterns are in agreement with the theoretical predictions of the model.

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Published Online: 2019-06-29

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Keywords for this article

collateral; credit; development; growth; human capital