Optimal debt adjustment in a nonlinear endogenous growth model

102 Pages·2013·1.12 MB·English

Optimal debt adjustment
in a nonlinear endogenous growth model
Mário Amorim Lopes
DissertaçãodeMestradoemEconomia
Orientadapor:
Prof. DalilaB.M.M.Fontes
Prof. AnaPaulaRibeiro
2013

About the author
Oscar Wilde once said that “Man is least himself when he talks in his own person. Give
him a mask, and he will tell you the truth”. Since no mask can be put on and the author
standsinlinewithhiswork,nothingbesideshisworkcanserveasafairjudgment.
To help in pursuing his work, the author obtained a MSc in Software Engineering,
is currently submitting a Master Thesis to earn an MSc in Economics and is now a PhD
student. The author has worked abroad for a private multinational company, did research
in computer science and economics, co-founded a software company and travelled the
worldforayearcarryingnothingbutabackpackandabunchofbooksonhisback.
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Acknowledgements
Theses lines wouldn’t be written if it weren’t for the work and assistance of Prof. Fer-
nando A. C. C. Fontes, for which I remain largely (but sustainably) in debt. His seminal
contributionto Optimal Control Theory pavedthe wayto thework performedin endoge-
nous growth theory in the same way that derivates allowed marginal utilities to be cal-
culated. Thousands of words of appreciation would also fall short on thanking for the
guidanceofProf. DalilaB.M.M.Fontes. Finally,Iwouldliketothanktheextremeded-
ication of Prof. Ana Paula Ribeiro in assessing the economic assumptions and providing
a sense of critical rationalism, along with the endless talks we had, with me arguing for a
fallback to sound classical economic principles while she patiently rebutted with Keyne-
sianarguments. ThoseweremomentsofenlightenmentforwhichIamdeeplygrateful.
Butifthesepeoplemadeitpossibleandmotivatedmetowritetheselines,dueappre-
ciation must also be forever paid to the ones that instilled in me a passion for Economics
without which the pages would stay blank. If it weren’t for them, I would be following
a different path, perhaps to a different equilibrium. Destiny, or chance, decided that I
shouldstumbleupontheworksoftheAustrianeconomistsFriedrichvonHayek,Ludwig
von Mises, Henry Hazlitt, along with the insightful work of Adam Smith and, preceding
him, David Hume. It was a life-changing event. It alluded to me the fundamental role
that Economics and the due understanding of simple economic concepts play in society.
It then led me to read Karl Popper and refine my understanding of how philosophy and
economicsweresodeeplyingrained. Theywerethefoundationblocksthatawakenedmy
passion, while contemporary economists like Milton Friedman, Thomas Sowell, Robert
Lucas,RobertBarroandXavierSala-i-Martinsedimentedit.
My intellectual growth was a positive linear function of the work of these men, and
forthatI,butmostlywe,willbeforeverindebt.
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Abstract
This work is a twofold contribution to the economic literature. First, we provide an Op-
timal Control theorem to transform a nonlinear infinite-horizon economic growth model
into an equivalent finite-horizon representation that can then be solved numerically. This
approach has several benefits compared to the methods currently in practice: (a) the non-
linearformofthemodelcanbesolveddirectly,avoidinglog-linearizationqualitativeand
quantitative errors; (b) the dynamic system is solved at each point in time so studying
thetransitionaldynamicsisnowconsiderablyeasier;(c)shockscanhitthesystematany
point in time and do not require the economy to be in a steady-state; (d) more complex
models, once deemed analytically intractable, can now be solved numerically. On top of
this, we developed a framework to exemplify the application of the theorem and demon-
strate its potential use in the exogenous and endogenous economic growth literature by
solvingtheoriginal,nonlineardescriptionoftwostandardmodels: theutility-maximizing
Ramsey-Cass-Koopmans neoclassical growth model and the Lucas endogenous growth
modelwithhumancapital. Weexemplifiedhowtosimulateshockstoasystemthatmight
not be at its steady-state, a strong requirement still in use in the literature, as well as
multiple,sequentialshocks.
In the second part, we studied optimal debt adjustment by applying the numerical
frameworkanddevelopinganonlinearendogenousgrowthmodelwithpubliccapitaland
public debt in order to study the optimal fiscal policy to curb large levels of public debt.
Given that the model was solved in its nonlinear form, we were able to consider time-
variant tax rates and a quantitative budget rule. It is shown that the optimal fiscal adjust-
mentpolicydependsontheinitiallevelofdebtandtheresultssuggestthereisaninverted
U-shape negative relationship between debt and growth. In the short-tun, the strategy
consists in immediately cutting public expenditure, followed by a tax raise in the capital
income tax. In the long-run, public expenditure resumes, capital income tax converges to
zeroandproductivepublicspendingisthenfullysupportedbyataxonconsumption.
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Resumo
Estetrabalhoapresentaduascontribuçõesparaaliteraturaeconómica. Emprimeirolugar,
disponibilizamos um teorema de Controlo Óptimo que transforma um modelo de cresci-
mento económico não-linear e de horizonte-infinito num problema de horizonte-finito
equivalente que pode então ser resolvido numericamente. Esta abordagem tem vários
benefícios comparativamente com os métodos atualmente usados: (a) a forma não-linear
do modelo pode ser resolvida de forma numérica, evitando os erros qualitativos e quan-
titativos que podem emergir da linearização; (b) o sistema dinâmico é resolvido em cada
instante pelo que analisar a dinâmica de transição torna-se consideravelmente mais fá-
cil; (c) o sistema pode ser sujeito a choques em cada instante e não é necessário que a
economia esteja num ponto de equilíbrio; (d) modelos mais complexos, analiticamente
intratáveis,podemagoraserresolvidosnumericamente. Adicionalmente,desenvolvemos
uma framework para exemplificar a aplicação do teorema e demonstrar o seu uso poten-
cial na literatura de crescimento económico exógeno e endógeno, para tal resolvendo a
versão original, não-linear, de dois modelos standard: o modelo neoclássico de cresci-
mentoeconómico,Ramsey-Cass-Koopmans,eomodelodecrescimentoendógenoecom
capital humano de Lucas. Exemplificamos também como simular choques numa econo-
mia que poderá não estar no seu equilíbrio, uma consideração geralmente assumida na
literatura,assimcomochoquesmúltiplosesequenciais.
Na segunda parte, estudamos o ajustamento óptimo da dívida pública aplicando a
framework numérica e construindo um modelo não-linear de crescimento endógeno com
capitalpúblicoedívidapública,quepermitiráestudarapolíticafiscalóptimaparareduzir
grandesníveisdedívida. Considerandoqueomodeloéresolvidonasuaformanãolinear,
foipossívelconsiderartaxasdeimpostoquepodemvariarcomotempoeaindaumaregra
orçamental quantitativa. Mostramos que a política óptima de ajustamento fiscal depende
no nível inicial de dívida e os resultados sugerem que existe uma relação em U invertido
entre dívida e a taxa de crescimento. No curto-prazo, a estratégia consiste em reduzir
imediatamente o investimento público, seguindo-se um aumento dos impostos sobre o
capital e o trabalho. No longo prazo, ocorre uma retoma da despesa pública, os impostos
sobreocapitaleotrabalhoconvergeparazeroeoinvestimentopúblicoprodutivoéentão
financiadorecorrendounicamenteaimpostosobreoconsumo.
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Contents
1 Introduction 13
2 Anumericalframeworkforeconomicgrowthmodels 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Modelsofeconomicgrowth . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Neoclassicalgrowthmodel . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Uzawa-Lucasendogenousgrowthmodel . . . . . . . . . . . . . 23
2.3 Aframeworkforinfinite-horizonmodels . . . . . . . . . . . . . . . . . . 24
2.3.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Evaluationofnumericalresults . . . . . . . . . . . . . . . . . . 30
2.4 Transitionaldynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Expectedshocks . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 Non-steadystateshocks . . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Multiple,sequentialshocks . . . . . . . . . . . . . . . . . . . . . 38
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Optimaldebtadjustmentinanendogenousgrowthmodel 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Backgroundonpublicdebt . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Empiricaldata . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Debtandgrowth . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.3 Debtdynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Debtadjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 Optimaldebtadjustment . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1 Themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Numericalsolution . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.3 Transitionaldynamics . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.4 Debtandthegrowth . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4 Remarksandpolicyimplications . . . . . . . . . . . . . . . . . . . . . . 91
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CONTENTS
3.5 Conclusionandfuturework . . . . . . . . . . . . . . . . . . . . . . . . . 93
A Equilibriumindecentralizedmarkets 101
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List of Figures
2.1 NumericaltrajectoriesfortheRamsey-Cass-Koopmansmodel. . . . . . . 28
2.2 NumericaloptimizationoftheUzawa-Lucasendogenousgrowthmodel. . 31
2.3 log oftheresidualoftheconsumptiontrajectory. . . . . . . . . . . . . 32
10
2.4 Impulseresponsestoananticipatedincreaseinτ att =20. . . . . . . . 35
c
2.5 NumericaloptimizationoftheUzawa-Lucasendogenousgrowthmodel. . 36
2.7 Responsestomultiplesequentialshocks,bothexpected. . . . . . . . . . . 38
2.6 Impulseresponsestoananticipatedincreaseinτ . . . . . . . . . . . . . . 40
c
3.1 Publicdebt-to-GDPratiooftheUnitedStatesofAmerica,1790-2010. . . 46
3.2 Publicdebt-to-GDPratioofsomeEuropeancountries . . . . . . . . . . . 47
3.3 LevelofthePortuguesepublicdebtbetweentheperiodof1999-2012 . . 48
3.4 OutstandingpublicdebtissuedbythePortuguesegovernment . . . . . . 49
3.5 ShareofthepublicdebtoftheUSAheldinTIPS . . . . . . . . . . . . . 50
3.6 GrowthrateandstatisticaltrendlineoftheGDPofPortugal . . . . . . . 51
3.7 Empiricalevidenceongrowthandthedebt-to-GDPratio . . . . . . . . . 53
3.8 Optimaltrajectoriesforthereductionofpublicdebt . . . . . . . . . . . . 81
3.9 Growth rates of private and public capital, debt and of the output of the
economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.10 Optimalincome/capitalandconsumptiontaxratesduringtheadjustment. 83
3.11 Optimalpublicexpenditureinpubliccapitalduringtheadjustment. . . . 85
3.12 Theevolutionoftheprimarybudget. . . . . . . . . . . . . . . . . . . . . 86
3.13 Theevolutionofprivatecapital . . . . . . . . . . . . . . . . . . . . . . . 86
3.14 Theevolutionofconsumption . . . . . . . . . . . . . . . . . . . . . . . 87
3.15 Theevolutionofinvestment . . . . . . . . . . . . . . . . . . . . . . . . 87
3.16 Simulationfordifferentinitialvaluesofpublicdebt. . . . . . . . . . . . . 88
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List of Tables
2.1 CalibratedparametersforthenumericalsimulationoftheRCKmodel. . . 28
2.2 CalibratedparametersfortheUzawa-Lucasmodel. . . . . . . . . . . . . 30
2.3 CalibratedparametersassetbyBrunnerandStrulik(2002). . . . . . . . . 32
2.4 Maximumabsoluteerrorsandrelativeerrors. . . . . . . . . . . . . . . . 33
2.5 Calibratedparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Welfareanalysisforthreedistinctinitialvaluesfork . . . . . . . . . . . . 38
0
3.1 Structuralparametervaluesandtheinitiallevelsfortheendogenousstate
variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2 Long-rungrowthratesfordifferentvaluesoftheinitialdebt-to-GDPratios. 88
10