Bassam R Awad - s3.amazonaws.coms3.amazonaws.com/zanran_storage/myweb.fsu.edu/ContentPages/... ·...

Bassam R AwadFlorida State UniversityCenter for Economic Forecasting and Analysis3200 Commonwealth Blvd.Tallahassee, FL 32304

Phone: (850) 567-2085

Fax: (850) 645-0191

Email: [email protected]

Homepage: http://myweb.fsu.edu/bawad

Education

Ph.D. Economics, Florida State University, Tallahassee - Florida, 2009.

M.S. Economics, Florida State University, Tallahassee - Florida, 2007.

MBA. Accounting, University of Jordan , Amman - Jordan, 1997.

M.A. Economics, Yarmouk University, Irbid - Jordan, 1994.

B.A. Economics, Yarmouk University, Irbid - Jordan, 1989.

Research Interests

Macroeconomics, Financial Economic and Econometrics.

Experience

Teaching Experience

Mentor. Online course development for Cultural Economics. 2009.

Mentor. Online course in Finance and Banking. 2008.

Teaching assistant. Microeconomics. 2006.

Instructor. Taught Economic Development at a community college in Jordan. 1993.

Research and Teaching Assistant. Microeconomics and Macroeconomics at Yarmouk University. 1990-1992.

PIE Associate. Awarded the Program for Instruction Excellence (PIE) Certificate from Florida StateUniversity. 2008.

Research Experience

Research Associate: The FSU Center for Economic Forecasting and Analysis, 2006 –.

Economist: Central Bank of Jordan, 1994 – 1996.

Internal Auditor: Central Bank of Jordan, 1996 – 2003.

Economist: Central Bank of Jordan, 2003 – 2005.

http://www.unc.edu/

mailto:[email protected]

http://myweb.fsu.edu/bawad//

Bassam R Awad 2

Research

Working Papers

Long-Run Growth versus Welfare: the Importance of Transitional Dynamics When Assessing Alterna-tive Fiscal Policies. Submitted to the Journal of Macroeconomic Dynamics in 07/06/09.

Linearization and Higher-Order Approximations: How Good are They? Results from an EndogenousGrowth Model with Public Capital. Submitted to the Journal of Computational Economics in 09/30/09.

Research in Progress

Optimal Fiscal Policy with Time Invariant Tax Structure: The Importance of a Public Capital External-ity.

Technical Reports

Dollar and Sense: National High Magnetic Field Laboratory and Its Forecasted Impact on the FloridaEconomy. August 2009.

Goliath Grouper: A Survey Analysis of Dive Shop and Charter Boat Operators in Florida. June 2009.

The Tallahassee Economic, Quality of Life and Investment Climate. June 2008.

Business Listing In the Enterprize Zone of Tallahassee. June 2008.

Marketing and Economic Study of CNS Services in Cairo, Georgia. May 2008.

Model Predicts Florida Economy Will Gain By Property Tax Cut Without Changing Sales Tax. May2007.

Assessment of Student Learning in a Laboratory Setting. March 2007

Coastal Training Programs of Florida Needs Assessment: Report and Appendix. December 2006.

Grants

Awarded

August 2009. National High Magnetic Field Laboratory - Florida State University. 2008-2009 EconomicImpact Assessment (Dollar and Sense Report). $10,754

June 2009. Goliath Grouper Study A Survey Analysis of Dive Shop and Charter Boat Operators inFlorida. In partnership with Fish and Wildlife Conservation Commission. $10,000

June 2008. The Tallahassee Economic, Quality of Life and Investment Climate. The Tallahassee/ LeonCounty Economic Development Council. $5,000

March 2008. Database of International Economic Development Council’s of the sixteen county area ofFlorida’s Great Northwest (FGNW). $41,686 for first year, plus $16,379 for annual updates

October 2007. Marketing and Economic Study of CNS Services in Cairo, Georgia. Community NetworkServices, Cairo - GA. $4,795

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1336140




http://www.magnet.fsu.edu/mediacenter/publications/documents/economicimpact-2009.pdf

http://www.magnet.fsu.edu/mediacenter/publications/documents/economicimpact-2009.pdf

http://cefa.fsu.edu/uploaded current projects/GG.pdf

http://cefa.fsu.edu/uploaded current projects/edctemplate.pdf

http://cefa.fsu.edu/uploaded current projects/edctemplatebuslist-Bassam.pdf

http://cefa.fsu.edu/uploaded current projects/Marketing and Economic Study of CNS.pdf

http://www.floridataxwatch.org/resources/pdf/050207FloridaEconomyWillGainByPropertyTaxCut.pdf

http://www.floridataxwatch.org/resources/pdf/050207FloridaEconomyWillGainByPropertyTaxCut.pdf

https://campus.fsu.edu/bbcswebdav/users/bawad/assess.pdf

http://cefa.fsu.edu/uploaded current projects/CTP_report.pdf

http://cefa.fsu.edu/uploaded current projects/appendix.pdf

Bassam R Awad 3

December 2006. Coastal Training Programs of Florida Needs Assessment Report. (A Report funded inpart by the Florida Department of Environmental Protection, Florida Coastal Management Program,pursuant to National Oceanic and Atmospheric Administration Award Number NA05NOS4191074).$45,000

Grant Applications

October 2006. Analytical Services Relating to Property Taxation, Proposal to the Florida Legislature -Office of Demographic Research. $76,800

May 2009. Taxpayer Return on Investment in Florida Public Libraries. $149,800

July 2009. FGNW Targeted Industries Sub-Cluster Analysis. $78,800

Honors

Florida State University Golden Key International Honor Society.(2008-2009).

Yarmouk University Dean’s honor list for undergraduate students in Economics. (1986-1989).

Jordan Social Security Corporation grant for distinguished first-year economics major students. (1986-1989).

Training

Euro Mediterranean partnership, European Union, Brussels-Belgium, April 2005.

Managing Exchange Rate crisis, Arab Planning Institute, Kuwait, March 2005.

Economic Indicators, Arab Monetary Fund, United Arab Emirates, April 2004.

External Sector Policies, International Monetary Fund, Washington, DC, August-September 1997.

Memberships

Society for Computational Economics

American Economic Association.

Econometric Society

Computer Skills

SPSS, STATA, E-views and all Office applications, .

R, MATLAB and Mathematica

Text editing using Latex, WinEdt and Scientific Workplace.

Basic web design abilities.

http://fsugoldenkey.com/photos.php

http://portal.yu.edu.jo/index.php

http://www.ssc.gov.jo/english/

Bassam R Awad 4

References

Dr. Milton Marquis, ProfessorPhD Dissertation AdvisorDepartment of Economics270 Bellamy Building, Florida State University, Tallahassee, FL 32306-2180

Office: (850) 645-1526. Fax: (850) 644-4535. Email: [email protected]

Dr. Julie Harrington, DirectorWork SupervisorCenter for Economic Forecasting and Analysis3200 Commonwealth Blvd. Tallahassee, FL 32304


Dr. Manoj Atolia, Assistant ProfessorPhD Dissertation Committee MemberDepartment of Economics, Florida State University278 Bellamy Building, Florida State University, Tallahassee, FL 32306-2180


Dr. Thomas W. Zuehlke, Associate Professor,Director of Graduate Studies,Department of Economics, Florida State University253A Bellamy Building, Florida State University, Tallahassee, FL 32306-2180


Dr. Carol Bullock, Graduate CoordinatorDepartment of Economics, Florida State University253 Bellamy Building, Florida State University, Tallahassee, FL 32306-2180


13-12-2009

UNOFFICIAL TRANSCRIPT (ALL CREDIT HOURS ON THIS RECORD REFLECTED IN SEMESTER HOURS)

AWAD BASSAM R

Address High School

AMMAN, JO 11191-0000 Residency: Alien

Gender Date of Birth Graduated High School FSU Matriculation Date

Male September 27, 1968 Fall 2005

TEST SCORES CLAST SAT ACT GRE

MAT WRI RDG ESS CR MAT WRI ENG MAT REA SCI COM VER MAT ADV COD

300 740

YARMOUK UNIVERSITY

Term Class Division Major Summer-A 1989

RECEIVED BACHELOR OF ARTS DEGREE

AUGUS 1989 T

YARMOUK UNIVERSITY

Term Class Division Major

Summer-A 1993

RECEIVED MASTER OF ARTS DEGREE

AUGUS 1993 T

UNIVERSITY OF JORDAN


Summer-A 1997

1

Bassam R Awad 1/6

RECEIVED MASTER OF BUSINESS ADMINISTRATION

AUGUST 1997 ADMITTED TO GRADUATE STANDING 09/2005

FLORIDA STATE UNIVERSITY


Fall 2005 Graduate COLLEGE OF

SOCIAL SCIENCES ECONOMICS

Title Type Course Grade Hours

Attempted Hours Earned

GPA Hours

GPA Points

PRO MKT/THEOR OF FRM

ECO5115 A 3.00 3.00 3.00 12.00

MACRO THEORY I ECO5204 A- 3.00 3.00 3.00 11.25

ECONOMETRICS I ECO5416 A- 3.00 3.00 3.00 11.25

MACRO-MICRO WORKSHOP

ECO6938 S 0.00 0.00 0.00 0.00

Term Totals: 9.00 9.00 9.00 34.50



Spring 2006 Graduate COLLEGE OF SOCIAL

SCIENCES ECONOMICS



GPA Hours

GPA Points

COMPTN MKTS DISTRIBN ECO5116 A- 3.00 3.00 3.00 11.25

MACROECON THEORY II ECO5207 A- 3.00 3.00 3.00 11.25

ECONOMETRIC THEORY ECO5423 A- 3.00 3.00 3.00 11.25

PHDWKSHP:QUANTITATIV ECO6938 S 0.00 0.00 0.00 0.00

TEACHING WORKSHOP ECO6939 S 0.00 0.00 0.00 0.00

Term Totals: 9.00 9.00 9.00 33.75



Summer 2006 Graduate COLLEGE OF


2

Bassam R Awad 2/6



GPA Hours

GPA Points

HISTORY ECON THO'T

ECO5305 A 3.00 3.00 3.00 12.00

LIM DEP VAR MODELS

ECO5427 B+ 3.00 3.00 3.00 9.75

Term Tota s:l 6.00 6.00 6.00 21.75



Fall 2006 Graduate COLLEGE OF SOCIAL

SCIENCES ECONOMICS



GPA Hours

GPA Points

FINANCIAL ECON II ECO5282 A 3.00 3.00 3.00 12.00

GENERAL EQUIL MACRO. ECO6209 A 3.00 3.00 3.00 12.00

TPCS:MICROECONOMICS ECO6936 B- 3.00 3.00 3.00 8.25

QUANTITATIVE WRKSHOP

ECO6938 S 0.00 0.00 0.00 0.00

Term Totals: 9.00 9.00 9.00 32.25




SCIENCES ECONOMICS



GPA Hours

GPA Points

TIME SERIES ECO5425 A- 3.00 3.00 3.00 11.25

ECONOMETRICS ECO5936 A- 3.00 3.00 3.00 11.25

MONEY ECO5936 A 3.00 3.00 3.00 12.00

PHDWKSHP:EXPRMNTLECO ECO6938 S 0.00 0.00 0.00 0.00

PRELIM DOCTORAL EXAM ECO8969 P 0.00 0.00 0.00 0.00

Term Totals: 9.00 9.00 9.00 34.50

RECEIVED THE DEGREE

MASTER OF SCIENCE APRIL 28, 2007

3

Bassam R Awad 3/6

PGM : ECONOMICS MAJOR: EC NOMICS O



Summer 2007 Graduate COLLEGE OF SOCIAL

SCIENCES ECONOMICS



GPA Hours

GPA Points

DIS:CMPTTLMETH/GENEQ ECO5907 A 3.00 3.00 3.00 12.00

Term Totals: 3.00 3.00 3.00 12.00



Fall 2007 Graduate COLLEGE OF SOCIAL

SCIENCES ECONOMICS



GPA Hours

GPA Points

PUBLIC FINANCE ECO5505 B 3.00 3.00 3.00 9.00

PHDWKSHP:MACRO/MICRO ECO6938 S 0.00 0.00 0.00 0.00

DISSERTATION ECO6980 S 2.00 2.00 0.00 0.00

GIS LAB GEO5908 S 1.00 1.00 0.00 0.00

GEOG INFO SYS GIS5101 S 3.00 3.00 0.00 0.00

Term Totals: 9.00 9.00 3.00 9.00




SCIENCES ECONOMICS



GPA Hours

GPA Points

DIS:COMPUTATNLMACRO ECO5907 A 2.00 2.00 2.00 8.00

WORKSHOP QUANT ECO6938 S 0.00 0.00 0.00 0.00


4

Bassam R Awad 4/6

Term Totals: 9.00 9.00 2.00 8.00



Summer 2008 Graduate COLLEGE OF




GPA Hours

GPA Points


Term Totals: 6.00 6.00 0.00 0.00



Fall 2008 Graduate COLLEGE OF




GPA Hours

GPA Points


Term Totals: 9.00 9.00 0.00 0.00



Spring 2009 Graduate COLLEGE OF




GPA Hours

GPA Points


Term Totals: 3.00 3.00 0.00 0.00

TRANSFER INFORMATION

Hours Attempted: 0.00 Hours Earned: 0.00

5

Bassam R Awad 5/6

GRADING SYSTEM

USED TO COMPUTE GPA A Excellent

B Good

C Average

D Passing

F Failing

IE Incomplete Expired

GE No Grade Expired NOTE: A grade of "W" is used only to denote that a student was passing a course at the time of withdrawal from the University.

NOT USED TO COMPUTE GPA I Incomplete

S Satisfactory

U Unsatisfactory

EC CLEP Exam Credit/CEEB

P Passed (Graduate Tests)

ED Department Exam Credit

WD Withdrawn with permission of Dean

*W Withdrew

NG No Grade Reported

CR Credit Received

EVALUATION OF TRANSFER (see course type)

D Course duplicated by other work.

E Credit approved toward undergraduate degree.

G Grade below transferable level.

L Course below transferable level.

M Remedial coursework.

N Course not applicable towards degree.

P Credit approved toward graduate degree.

T Vocational or technical terminal course prior to Fall Semester, 1981.

V Vocational or technical terminal course beginning Fall Semester, 1981.

W Gordon Rule Writing Contest.

X Coursework taken while on dismissal.

Y Community college courses taken beyond AA degree.

FORGIVENESS POLICY INDICATORS (see course type)

T Repeated (initial attempt)

R Repeated (last attempted) The information provided is for student use only and can not be given out to a third party

without the student's permission.

6

Bassam R Awad 6/6

Teaching PhilosophyBassam R. Awad

As a graduate student in Jordan, I had the opportunity to teach under-graduate courses in both micro- and macroeconomics. For some, myclass was the first and only economics course they would ever take. Assuch, I understood that this was the only chance that I had to make alasting impression on them. For others, however, my class was the firstof many, and I understood that these students needed to be conditionedfor the more advanced economic courses. Regardless of their careerchoice, I taught my courses in such a way that those students pursuingbachelors or advanced degrees in economics would be prepared, andthose pursuing other degrees would be able to apply economic conceptsto the everyday decisions in their lives . My mission then, and still istoday, is to make sure that every student understands the basic eco-nomic concepts and their importance and usefulness in life.

When teaching, I make every effort to share my enthusiasm of the sub-ject matter with the students in the hopes that they too will becomeexcited about the material. I do this by incorporating examples frommy own research, and their interests into the lectures. One way I amable to gauge their interests is by having them email or bring in articlesthat they have questions about or would like to discuss in class. Whenappropriate, I will post these articles on the class website and orches-trate an interactive classroom discussion. I find that taking the timeto listen to my students and get them involved helps establish a line ofcommunication, wherein the students feel comfortable coming to me ifthey have questions about the material, need academic advisement, orhave general concerns that they would like to express.

Establishing individual relationships with my students is extremelyimportant to me, as I feel that a successful educator is not one whoflourishes solely in the classroom, does not flourish. Instead, I view asuccessful educator as someone who fully integrates themselves intothe community, institution, and classroom. Having been raised abroadI can relate to a diverse student body and try to immerse myself in asmany activities as possible.

While at Florida State University, I worked as a mentor for an online

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course in money and banking. As a mentor, I had no face-to-face contactwith the students I was working with. My teaching mission, however,remained the same. In order to accomplish this mission, I had to adaptmy teaching style to best serve the students of the online environment.This meant taking the time to ensure that my correspondence with thestudents was consistent, clear and concise, encouraging, and thorough.I accomplished this by promptly responding to emails, answering ques-tions on an anonymous message board, and by providing innovativeexamples and practice questions that supplemented the material. Ialso incorporated a wide variety of video clips into the lessons, whichhelp connect various topics to real world applications. Although I didnot develop the on-line course, I feel that my interactions with thiscourse provided me with the skill set necessary to create and maintainmy own distance learning courses.

18-11-2009

2

Research StatementBassam R. Awad

I have worked at the Central Bank of Jordan for 12 years from 1994to 2005. My job duties were varied, but it included mostly the inter-national economic issues, papers and reports related to Jordan. I havebeen employed at the FSU Center for Economic Forecasting and Anal-ysis (CEFA) since August 2006. My primary area of expertise in CEFAis data collection and statistical, econometric analysis, writing reports,literature review and presentations. I have conducted beta testing andprovided feedback on a new software program WITS on economic work-force development in the Big Bend Region in Florida funded by FloridasGreat Northwest (an economic development group including 16 coun-ties in Florida). I have built a good experience in working with variouseconometric and statistical packages such as STATA, Latex, WinEdt,MATLAB, SPSS, Mathematica and Eviews. I have good skills in us-ing other software packages as well, including MS Office, ScientificWorkplace, Advanced SPSS, and REMI and IMPLAN (economic impactanalysis software). I am also the CEFA webpage webmaster. Parallelto that, I have worked as a teaching assistant at the FSU departmentof Economics in different semesters. During that work, I have beenawarded the Certificate of Instruction Excellence. My duties involveddeveloping an online course and mentorship using Blackboard.

My dissertation title is ”Essays in Fiscal Policy: Computational Accu-racy and Optimal Investments in Public, Private, and Human Capital.”focuses on the growth and welfare effects of alternative tax regimes ina setting where the growth process is driven by investment in humancapital and where public capital enhances the production process. Ihave demonstrated that (i) the policies that induce the highest ratesof economic growth do not always provide the highest welfare. (ii) Thetraditional methods of analyzing the economic: consequences of alter-native tax policy regime can be very large approximation errors, whichdo not occur in the method employed in my analysis. (iii) the short-runimplications of tax reforms can be very different from the longrun con-sequences, and it may take many years for the benefits of tax reform tooffset short-term losses.

1

The second chapter of my dissertation is entitled ”Linearization andHigher-Order Approximations: How Good are They?: Results from anEndogenous Growth Model with Public Capital” I address the accuracyproblem linked to the approximation methods in analyzing the transi-tional dynamics. Where standard procedure for analyzing transitionaldynamics in non-linear macro models has been to employ linear ap-proximations. Recently quadratic approximations have been explored.This chapter examines the accuracy of these and higher-order approx-imations in an endogenous growth model with public capital, therebyextending the work done in the current literature on the neoclassicalgrowth model. We find that significant errors may persist in computedtransition paths and welfare even after resorting to approximations ashigh as fourth order. Moreover, the accuracy of approximations maynot increase monotonically with the increase in the order of approxi-mation. Also, as in the previous literature, we find that achieving ac-ceptable levels of accuracy when computing the welfare consequencesof a policy change typically requires a higher order approximation thanattaining similar levels of accuracy in the computation of the transi-tion path: typically an increase in order of approximation by one issufficient.

The third chapter, Long-Run Growth versus Welfare: The Importance ofTransitional Dynamics When Assessing Alternative Fiscal Policies, uti-lizes the evident superiority of nonlinear solution methods for optimaltransition path. analyzes the effects of distortionary taxes on growthand welfare in an endogenous growth model with a public capital ex-ternality. The model is calibrated to the U.S. economy, and experimentsare run under which the tax regime is shifted from the current mix ofcapital income, labor income, and consumption taxes to a fiscal policyregime with complete reliance on a single source of taxation, includ-ing lump-sum tax. We find that tax policy changes that induce highergrowth rate do not necessarily result in higher welfare due to differenttransitory effects. In fact, a shift to capital income tax while deliver-ing highest long-run growth results in lowest welfare. Furthermore,long-run gains take many years a generation to start getting realized.Among different sources of taxation, we find that, in the long run, com-plete reliance on a consumption tax dominates the current tax regime;however, the current tax regime dominates an exclusive labor income

2

tax, which in turn is less welfare-reducing than an exclusive capitalincome tax. These results are due to the fact that taxes on labor in-come and capital income distort investment decisions in reproduciblecapital, i.e., human capital and physical capital, and therefore have cu-mulative effects that do not result from a tax on consumption. Unlikeprevious studies, we account for the welfare effects of transition usingoptimal decision rules all along the transition path.

The title of the fourth chapter is ”Optimal Consumption Tax Rate in aDynamic Fiscal Policy with Time Invariant Tax Structure: the Impor-tance of Externality” There is a long-standing debate in the literatureon the choice between consumption or expenditure taxes versus capi-tal income taxes that goes back to Thomas Hobbes (1651), Mill (1871)and later Kaldor (1955) who advocated the consumption tax over theincome tax. The advocacy of consumption tax has its solid empiricalevidence as some studies indicated that the tax revenue collected inthe United States includes a relatively small contribution coming fromcapital tax (Roger Gordon, Laura Kalambokidis, Jeffrey Rohaly andJoel Slemrod (2004)). This chapter examines tax policy in an endoge-nous growth model with public capital externality, where human capi-tal serves as the engine of growth. In the previous chapter, this modelwas calibrated to the U.S. economy and experiments were run to cal-culate welfare gains from a shift in the fiscal regime from the currentmix of capital income, labor income, and consumption taxes to com-plete reliance on consumption tax. In those experiments, governmentexpenditures in public capital as a share of output was held fixed. Thechapter showed that the consumption-only tax regime was superior tothe current tax regime and to other tax regimes relying solely on a sin-gle source of taxation. In this chapter, the government tax revenuesas a portion of output are varied in order to find the optimal level ofinvestments in public capital under a consumption-only tax regime. Ifind that in the presence of a significant externality, a modest increasein the consumption tax with a greater investment in public capital canincrease welfare. I also show that a slight shift in taxes from consump-tion to capital income can be welfare improving if the externality ishigh enough.

During my work in the economic research at the Central Bank of Jor-

3

dan and CEFA, I have gained good research skills. These include lit-erature survey, data collection, extensive web-search, mastery of thesoftware requirements, working on different tasks at the same time,committing to deadlines, and continually following-up projects’ com-pletion requirement, and recently, I began applying for grants. Exam-ples of my work are the market survey research for the City of Cairoin Georgia, the preparation of the IEDC tables for the Florida GreatNorthwest, and the economic impact analysis reports. Links to theseare available on my web page under technical reports tab. More infor-mation are available at cefa.fsu.edu

16-11-2009

4

Electronic copy available at: http://ssrn.com/abstract=1336140Electronic copy available at: http://ssrn.com/abstract=1336140

Long-Run Growth versus Welfare: The Importance ofTransitional Dynamics When Assessing Alternative

Fiscal Policies∗

Manoj AtoliaFlorida State University †

Bassam AwadFlorida State University ‡

Milton MarquisFlorida State University §

First Draft: January 2009

∗We are very grateful to Paul Beaumont, Tor Einarsson, Bharat Trehan and other participants of theMacro summer workshop, 2008, at Department of Economics, Florida State University where an earlierversion of this paper was presented. All errors are ours.

†Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 850-644-7088. Email: [email protected].

‡Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 850-567-2085. Email: [email protected].

§Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 850-645-1526. Email: [email protected].

1

Electronic copy available at: http://ssrn.com/abstract=1336140Electronic copy available at: http://ssrn.com/abstract=1336140

.

Abstract

This paper analyzes the effects of distortionary taxes on growth and welfare in anendogenous growth model with a public capital externality. The model is calibratedto the U.S. economy, and experiments are run under which the tax regime is shiftedfrom the current mix of capital income, labor income, and consumption taxes to afiscal policy regime with complete reliance on a single source of taxation, includinglump-sum tax. We find that tax policy changes that induce higher growth rate do notnecessarily result in higher welfare due to different transitory effects. In fact, a shift tocapital income tax while delivering highest long-run growth results in lowest welfare.Furthermore, long-run gains take many years – a generation – to start getting realized.Among different sources of taxation, we find that, in the long run, complete reliance ona consumption tax dominates the current tax regime; however, the current tax regimedominates an exclusive labor income tax, which in turn is less welfare-reducing thanan exclusive capital income tax. These results are due to the fact that taxes on laborincome and capital income distort investment decisions in reproducible capital, i.e.,human capital and physical capital, and therefore have cumulative effects that do notresult from a tax on consumption. Unlike previous studies, we account for the welfareeffects of transition using optimal decision rules all along the transition path.

Keywords: Endogenous growth, tax policy analysis, welfare, public capital, humancapital

JEL Codes: O41, E62, H54

2

1 Introduction

This paper examines the consequences for economic growth and welfare of distortionary

taxes used to finance public capital in an endogenous growth model where public capital

creates a positive production externality as in Barro (1990). The model that we employ

differs from existing literature that either assumes exogenous growth, in which case there

are no long-run distortionary effects on growth, and/or there is no role for public capital in

the production process. We examine these issues from the perspective of shifting taxes from

the status quo, to which the model is calibrated, to alternative tax regimes, thus requiring

a period of transition as the economy adjusts to a new balanced growth path.

Previous analyses of tax distortions that compare alternative tax regimes either ignore the

transitional dynamics (as in King and Rebello, 1990) or use approximations to characterize

the transition paths (such as Mulligan and Sala-i-Martin, 1992).1 In this paper, we employ

a novel reverse-shooting algorithm introduced into economic modeling by Atolia and Buffie

(2009b), in which the decision rules coincide with exact solutions to the saddle-path equilibria

and thus avoid the approximation errors. We highlight the importance of accounting for the

transition when assessing the cumulative welfare consequences of a change in tax policy.2

The theoretical and empirical relevance of human capital in the process of economic

growth is discussed in the survey article by Temple (1999), wherein evidence indicates coun-

tries with high levels of human capital tend to have high levels of income. While investments

in human capital should bolster growth, the proposed mechanisms for increasing the rate of

accumulation of human capital are diverse. They include schooling, parental education, on-

the-job training, and learning-by-doing.3 Romer (1990)-style R&D-based endogenous growth

models also rely on human capital intensive processes to generate growth.4 In this paper, we

choose to use the Lucas (1988)-Uzawa (1965) model of investment in human capital through

the allocation of time that is taken away from either production or leisure. Human capital

is incorporated into the production function as labor-augmenting.

The class of models that treats public capital as generating a production externality relies

1 Also, see Futagmai, Morita and Shibata (1993), Mino (1990), Lee (1992), Einarsson and Marquis (2001),and Greiner (2005).

2 In a companion paper (Atolia, Awad, and Marquis, 2009), we demonstrate that the approximationerrors in a theoretical framework such as the one employed in this paper can be very large.

3Among the vast literature on the subject, see especially Stokey (1988, 1991a, 1991b); Lucas (1988,1993);Barro and Sala-i-Martin (1992); and Barro (1999).

4Other models in this literature include Aghion and Howitt (1992), Jones (1995), and Young (1998),among others.

3

on two empirical facts. First, expenditures on public capital represent a significant share

of GDP. (See Arrow and Kurz, 1969; Shah, 1992; Ratner, 1983; Tatom, 1991; Nadiri and

Mamuneas, 1994; Seitz, 1994; and Romp and de Haan, 2007.) Researchers such as Aschauer

(1989a,b) have offered persuasive arguments that public capital plays an important role in

economic development. Modeling public capital as a production externality, as we do in this

paper, is consistent with the practice of maintaining constant returns to factor inputs that

are endogenously chosen by firms that have no control over either the flow or stock decisions

of public capital.

The complexity of modern theoretical macroeconomic models used to analyze such issues

as distortionary taxes has led researchers to rely heavily on linear approximations of the

nonlinear dynamic system in order to compute approximate values for welfare gains and

losses across tax regimes inclusive of transitional periods. Noting that these errors can be

very large, Schmidt-Grohe and Uribe (2004) have examined the second-order approximations

using a perturbation method and found that significant reductions in approximation errors

are possible. However, the order of approximation required to reduce the errors to acceptable

levels is both model-specific and specific to the exercises that are being carried out for a given

model. We avoid these potential problems by employing the reverse-shooting algorithm of

Atolia and Buffie (2009b)5 which compute exact solutions.

The fiscal policy experiments in this paper begin with a model calibrated to the current

U.S. tax regime of the capital income, labor income, and consumption tax rates. For the

basis of comparison, we first compute the transitional dynamics and the cumulative welfare

gains (losses) associated with a shift to a nondistortionary lump-sum tax regime. We then

conduct similar experiments for shifts from the current status quo to exclusive reliance in

turn on a capital income tax, a labor income tax, and consumption tax, with the results for

each alternative tax regime compared to the shift to the nondistortionary regime.

In the long-run, we find that consumption taxes are preferable to the current tax regime,

which is preferred to an exclusive labor income tax regime, with capital income taxes the

most welfare-reducing. However, in the short-run, cumulative welfare gains and losses are

very different from the long-run consequences for welfare, and it may take many years for

the long-run gains or losses to be fully realized. For example, while a shift to consumption

tax raises overall welfare (and long-run growth), it takes almost a generation (36.7 years)

for the accumulated welfare gains to turn positive. More importantly, the tax regime that

5This algorithm is also used in a fiscal policy experiment by Atolia, Chatterjee, and Turnovsky (2008).

4

produces the highest growth rate is not necessarily the one that coincides with the highest

welfare. In fact, a shift to capital income tax while delivering highest long-run growth results

in lowest welfare. It is even dominated by a shift to labor income tax where the long-run

growth comes to an almost complete halt.

The remaining part of this paper is organized as follows. The model is described in

section 2 whereas details of solving the model are outlined in section 3. Section 4 calibrates

the model to U.S economy. The results for the alternative fiscal policy experiments are

contained in section 5. Section 6 concludes.

2 The Model

We work with continuous-time model with no uncertainty so that the decisions are made

with perfect foresight. The economy is closed and is populated by infinitely-lived homogenous

households.

The representative household derives utility from consumption, c, and leisure, l and is

endowed with one unit of time. It chooses the optimal time paths for consumption, leisure,

time devoted to production (or labor supply) and time devoted to the accumulation of human

capital in order to maximize its lifetime utility. It obtains income from renting his private

physical capital to the firm and receives a wage payment for the time devoted to production.

This income is allocated between consumption and private physical capital accumulation.

Therefore, the only direct form of saving for the household is in the form of private physical

capital accumulation. However, the household can also transfer resources to future indirectly

by accumulating human capital. In addition to the private physical capital and “effective”

units of labor, there is also public physical capital in the model that enters as an externality

in the production function.

The government collects revenue only through taxation; there are no other sources of

revenue. The government levies tax on the income from private physical capital. It also

levies a tax on the labor income of the household. The consumption expenditures are also

taxed. There is also a lump-sum tax available to the government. Tax revenue is used to

finance public capital accumulation and transfers.

5

2.1 Households

The representative household derives its utility from nonnegative streams of consumption,

c, and leisure, l, according to the following per-period utility function, which is logarithmic

in consumption and leisure

u(c, l) = log c + η log l, (1)

where u is defined on <++. It is continuous and strictly increasing in l and c, twice con-

tinuously differentiable and strictly concave. The log specification, which implies elasticity

of intertemporal substitution and coefficient of risk aversion of 1, is used following Marquis

and Einarsson (1999a, 1999b).

The household receives income, rk, from renting physical capital to the firm, where k is

the stock of private physical capital and r is real rental rate. It also receives labor income,

whn, for the time devoted to the production, n, which given the household’s stock of human

capital, h, corresponds to hn effective units of labor that are paid a wage rate of w. In

addition, it receives transfers, T , from the government and profits, π, from the firms in the

form of dividends. After the payment of taxes, net income in excess of consumption is used

to accumulate private physical capital. Accordingly, the household’s budget constraint is

given by

k = π + rk + whn + T − c− δkk − (τkrk + τnwhn + τcc + X), (2)

where k is net investment in private physical capital, δk ∈ (0, 1) is the rate of depreciation

of private physical capital, τk, τn and τc are marginal tax rates on private physical capital,

labor and consumption, and X is the lump-sum tax.

In addition, the household is also constrained in the time it can allocate to leisure,

production, and accumulation of human capital, m, as

l + m + n = 1. (3)

The time devoted to human capital accumulation results in an increase in h according to the

following evolution process:

h = γmh− δhh, (4)

where γ > 0 is a productivity parameter and δh ∈ (0, 1) is rate of depreciation of human

capital. It may be noted that the evolution rule for the human capital is linear in the current

6

state of human capital which generates the endogenous growth in the model. The human

capital evolution equation above was used by Lucas (1988) without a depreciation of human

capital. Accounting for the depreciation of the private human capital is widely recognized

in the literature. See for example Marquis and Einarsson (1999a) and Heckman (1976).

The household maximizes its life time utility by choosing c, l,m, and n

max{c,l,m,n}

∞∫0

e−ρtu(c, l)dt, (5)

where ρ is rate of time preference and e−ρt is the corresponding discount factor. This maxi-

mization is subject to constraints in (2), (3) and (4) and the initial conditions k(0) = ko and

h(0) = ho, where ko and ho are the levels of private physical capital and human capital in

the economy at t = 0.

2.2 Firms

Output, y, is produced using the economy’s resources of public physical capital (kg), private

physical capital, private human capital and labor. The technology is Cobb-Douglas and is

given by

y = f(k, n; h, kg) = Akgα1kα2(hn)1−α2 , (6)

where A, α1, α2 are parameters, with α1, α2, (α1 + α2) ∈ (0, 1), A > 0. Thus, the technology

exhibits constant returns to scale in private capital, which includes the private physical

capital stock and the human capital stock. The parameter α2 is the private physical capital’s

share of output. The parameter A is a scaling parameter.

Furthermore, as seen in (6), output depends on the available stock of government-provided

public capital, which enters as an externality in the production technology and is taken as

given by the firm. The parameter α1 can be interpreted as the elasticity of output with

respect to public physical capital. A higher value of α1 implies a greater externality. The

inclusion of the government spending in the production process was pioneered by Barro

(1990). In his model, it is the flow of government spending that enters the production

function as an external productive input. However, it is more realistic to assume that it is

not the flow of government spending, but the stock of public capital such as infrastructure

which enters the production function as is done here. In this sense, our specification follows

Futagami, Morita and Shibata (1993). (See Atolia, Chatterjee and Turnovsky (2008) for a

7

detailed discussion of the two forms of government spending contribution to output.) It may

be noted again that the firm only chooses k and n and takes the stock of h and kg as given.

2.3 Government

The government levies four types of taxes: a tax τk on gross physical capital income, rk; a

tax τn on effective labor income, whn, where hn is quality-adjusted or effective units of labor,

each unit of which earns a real wage, w such that wh is the hourly wage rate; a tax τc on

consumption expenditures, c; and lump-sum tax, X. The first three taxes are distortionary,

whereas the last lump-sum tax is non-distortionary. The government tax revenue, R, is,

therefore, given by

R = τkrk + τnwhn + τcc + X, (7)

where τk, τn ∈ [0, 1) and τc > 0 .

The tax revenue is used to finance the expenditure on accumulation of public capital (G)

and to provide transfers to the households. Therefore, the government’s budget constraint

is

G = R − T. (8)

The public physical capital evolves according to the following rule:

kg = G− δgkg, (9)

where δg ∈ (0, 1) is the rate of depreciation of public capital.

3 Solving the Model

The household’s optimization problem can be solved using the standard optimal control

method. The current-period Hamiltonian for the problem is:

H = u(c, l)+λ1(π + ((1− τk)r − δk)k + (1− τn)whn− (1− τc)c−X + T )+λ2(γ(1−l−n)−δh)h,

and the first-order conditions for the household are:

c : uc(c, l) = (1 + τc)λ1 (10)

8

l : ul(c, l) = λ2γh (11)

n : λ1(1− τn)wh = λ2γh (12)

The left-hand side of (10), uc, is the marginal benefit of consumption and the right-hand

side, (1 + τc)λ1, is the marginal cost, where λ1 is the marginal utility of wealth, and one

unit of consumption costs (1+ τc) units of wealth because of the tax rate τc on consumption.

Equations (11) and (12) show that the optimal allocation equates the marginal benefit across

all uses of time.

The co-state equations for k and h are

λ1

λ1

= ρ + δk − r(1− τk). (13)

λ2

λ2

= ρ + δh − γ(1− l), (14)

The firm works in a perfectly competitive environment taking goods and factor prices

as given. Its maximization problem is essentially static. Every period it rents private phys-

ical capital from the household and uses the household’s labor to maximize profits. Its

optimization problem is therefore

max{k,n}

π = Af(k, n; h, kg)− rk − whn, (15)

where wh is payment per unit of labor time and r is the private physical capital rental. The

first-order conditions for the firm are:

fk = r (16)

fn = wh (17)

Having solved the household’s optimization problem, we now turn to solve for the overall

general equilibrium of the economy. In particular, we solve for the competitive equilibrium.

The variables in the core dynamic system of the economy are c, l, k, kg and h.

The evolution of kg is governed by (9). From (4), using (3), one obtains the differential

equation for h as

h = (γ(1− l − n)− δh)h. (18)

In the perfectly competitive environment, the firm’s maximized profits are zero. Using this

9

result and substituting (7), (8) and (9) in (2), we obtain the economy’s resource constraint

which gives the transition equation for k

k = f(k, n; h, kg)− c−G− δkk. (19)

Note that this shows that the output of the economy is allocated to consumption and gross

investment in private and public physical capital.

The equation for the dynamic path of consumption can be derived from the co-state

equation for the corresponding Lagrange multiplier λ1. In particular, using (10) and (13),

we get

c = ((1− τk)r − (ρ + δk))c. (20)

To derive the l equation, we begin by obtaining the optimality condition for the labor-leisure

choice of the household by eliminating λ1 from (10) using (11) and (12). This yields

uc

(1− τn

1 + τc

)wh = ul. (21)

This equation on log-differentiating gives

l = l

(c

c− h

h− w

w

). (22)

The details of deriving equation for w are standard, and hence, skipped.

The evolution of the core dynamic system is governed by (9), (18), (19), (20) and (22).

The dynamics of the economy is also subject to following transversality conditions:

limt→∞

λ1(t)k(t) = 0, (23)

limt→∞

λ2(t)h(t) = 0. (24)

Before solving the core-dynamic system for the transition dynamics of the model, we

need to solve for the balanced growth path of the economy to which we turn next.

10

3.1 Solving for the Balanced Growth Path

Along the balanced growth path (bgp), all variables of interest (c, h, k, kg, and y) grow at

constant rates (including, possibly some at the zero rate) which may differ across variables. It

is clear from the time allocation constraint in (3), the time allocated to leisure, accumulation

of human capital and labor remains constant along bgp. Further, using standard procedures,

it is possible to show that consumption, output, private physical capital, and public capital

grow at the same rate (denoted φ) along the bgp. On the other hand, human capital grows

at a slower rate ν. These growth rates are related as follows:

φ =1− α2

1− α1 − α2

ν (25)

Furthermore, along the bgp, r is constant and is given by:

r =ρ + δk + φ

(1− τk), (26)

The private capital to output ratio and consumption to output ratio are constant as all these

variables grow at the same ratek

y=

α2

r(27)

c

k=

rα2

(1− τn(1− α2))−Xy−T

yky

− (φ + δk + τkr)

(1 + τc)(28)

Given the tax policy {τk, τn, τc} and X/y and the expenditure policy G/R, the government’s

budget constraint yields

T

y=

(1− G

R

)(α2τk + (1− α2)τn + τc

c

k

k

y+

X

y

)(29)

Finally, other equilibrium conditions yield following relationship among variables that do

not grow along the bgp.

ν = γm− δh (30)

l + m + n = 1 (31)

η =r

α2

(1− τn)(1− α2)l

(1 + τc)ckn

(32)

11

φ = (1− τk)r − (ρ + δk) (33)

Equations (25-33) are nine equations which can be solved for m, l, n, r, c/k, k/y, T/y,

φ, and ν.

3.2 Solving for Transitional Dynamics

Recall that the core dynamics of the economy can be expressed in terms of c, l, k, kg, and h.

However, as the model has endogenous growth, except for l, all these variables experience

constant growth after transitory response to a shock dies out. In order to solve for the

dynamics of the system, therefore, it is necessary to first transform the variables in the

core-dynamic system to the ones that are stationary, i.e., those that ultimately reach a finite

non-zero value.

We perform this transformation by normalizing the variables by the private physical

capital stock. Accordingly, we define following new variables: c = c/k, kg = kg/k, and

h = hφ/ν/k. It is easy to see that c, kg and h are stationary variables. The equations

governing the evolution of stationary variables can be obtained from equations of the original

dynamic system as follows:

˙c =

(c

c− k

k

)c (34)

˙kg =

(kg

kg

− k

k

)kg (35)

˙h =

(φ

ν

h

h− k

k

)h (36)

wherein one can show that

k

k=

r

α2

(1 +

T

y− X

y

)− r

(α2τk + τn(1− α2)

α2

)− (1 + τc)c− δk, (37)

kg

kg

=1

kg

[r

(τkα2 + τn(1− α2)

α2

)− (1 + τc)c +

(X

y− T

y

)y

k− δgkg

]. (38)

The core-dynamic system in normalized variables, c, k, kg and l, consists of (34), (22),

(37) and (38). Having solved this system, we can use (37) to solve for the path of k. The

levels of other variables then follow immediately.

12

3.2.1 Numerical Solution of the Transition Dynamics

In the existing literature, transition dynamics is typically analyzed using linearization. Re-

cently, Schmidt-Grohe and Uribe (2004) have suggested second-order approximation method.

While a number of methods (e.g. projection methods and shooting methods) are now avail-

able to provide more accurate solutions, they are rarely being used.

Recently, in a series of papers Atolia and Buffie (2009a and 2009b) have developed reliable

and user-friendly program to accurately solve dynamic general equilibrium models such as one

of this paper. We use their reverse-shooting program ReverseShoot2D to solve for the global

nonlinear saddlepath of our model. It turns out (and as we show in a companion paper) that

linearization and lower-order Taylor series solutions fail to correctly assess the welfare effect

of the fiscal policy changes that are analyzed in the paper. The failure occurs due to errors in

the solution of the transitional dynamics. By solving for the global nonlinear saddlepath, this

paper avoids these problems and, to our knowledge, for the first time provides an accurate

assessment of the welfare effects of government policy changes in an endogenous growth

model. We present these results in section 5, however, before we can numerically solve the

model, it needs to be calibrated; a task that we undertake in the next section.

4 Calibrating the Model to the Benchmark

There are 22 unknowns in our model: nine endogenous variables (m, l, n, r, c/k, k/y, T/y,

φ, and ν), eight parameters (α1, α2, δk, δh, δg, ρ, η and γ), and five exogenous variables

(X/y and G/R, τk, τn and τc). We use the nine equations to solve for m, l, r, c/k, T/y, ν, ρ,

γ and η and use estimates from the previous literature and empirical data to get the values

φ, α1, α2, δk, δh, δg, τk, τn, τc, n, k/y and G/R. The ratio of lump-sum taxes to output,

X/y, is set to zero in the initial steady-state. Recall, we assume that the entire government

tax revenue in the initial steady-state is collected only from the proportional distortionary

taxes. The calibration results are summarized in Table 1.

Per capita output growth rate (φ). In the last 50 years, (1961-2005), the average long-

run growth rate of output per capita in the United States reached approximately 2.2% as

shown by the World Bank data in Figure 1, which plots of the U.S. growth rate during this

period. The dotted line represents the trend line for the same period. However, in our model

here we set φ to 1.8% based on evidence in Barro (1990). The logic behind our selection is

that Barro’s time series extends for a longer time span from 1870 to 2000.

13

y = -0.0183x + 2.6181

-3

-2

-1

0

1

2

3

4

5

6

7

1961

1963

1965

1967

1969

1971

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

Years

Gro

wth

Rat

e (%

)

Source: World Resources Institute, 2007.

Figure 1: Historical Per Capita Growth Rates in the US (1961-2005)

Elasticity of output with respect to public physical capital (α1). Based on the

evidence provided by Gramlich (1994), the elasticity of output with respect to public physical

capital, or, alternatively, the productivity elasticity of government spending, might range

from 0.1 to 0.2. We set α1 to 0.1 in our model, which is in the range of values used in the

literature. (See Atolia, Chatterjee and Turnovsky, 2008).

Private physical capital share of output (α2). The most frequently assigned value for

α2 in the literature is 0.36. (See for example Kydland and Prescott, 1982; Hansen, 1985; and

Prescott, 1986). A higher value of 0.4 was set by Cooley and Prescott (1995) to account for

the imputed income from public physical capital. However, since public capital is explicitly

considered in our model, a value of α2 between 0.30 and 0.36 may be more appropriate.

Accordingly, a value of 0.337 is assigned for α2 based on a more recent work by Einarsson

and Marquis (1996). They arrived at this value using long-term empirical U.S. data for the

period (1950-1994).

Depreciation rate of private physical capital (δk). Atolia, Chatterjee and Turnovsky

(2008) set this rate to 0.05, whereas Marquis and Einarsson (1999a) set it to 0.0512 in

their calibrated dynamic general equilibrium model of endogenous growth. These are typical

values used in the literature and, following them, we set δk = 0.05.

Depreciation rate of human capital (δh). There is a considerable variation in the choice

of δh in the literature. For example, Haley (1976) uses a value as low as 0.005, whereas

Heckman (1976) sets it to 0.047. In a more recent work, Einarsson and Marquis (2001)

choose a value of 0.05 which is obtained as a rough average of the estimated values from

14

the literature on the labor market. We choose δh = 0.015. A detailed discussion about the

estimation of this rate can be found in Mincer and Ofek (1982). 6

Depreciation rate of public capital (δg). We take the value of 0.035 used by Atolia,

Chatterjee and Turnovsky (2008). Note that the rate of depreciation of public physical

capital is lower than the private physical capital. This captures the fact that public physical

capital is mostly infrastructure which depreciates at a slower pace than plant and machinery.

Time allocated to production (n). Greenwood and Hercowitz (1991) calculated the

average ratio of total hours worked to total nonsleeping hours (16 hours per day) of the

working age population to be .24 while Einarsson and Marquis (2001) use .36. We go with

Einarsson and Marquis and set n = 0.36.

Private physical capital to output ratio (k/y). We set this ratio to 2.17. This value

is obtained from the data for the private physical capital and gross domestic product in

current dollar value in National Income and Product Accounts (NIPA) tables prepared by

the Bureau of Economic Analysis (BEA) at the US Department of Commerce. The data in

the tables covers the post WWII period (1946-2006). Figure 2 shows the time path of k/y

based on this data.

y = 0.0039x + 2.0487R2 = 0.2077

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

1946

1949

1952

1955

1958

1961

1964

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

1997

2000

2003

2006

Years

k/y

US Department of Commerce, 2008.

Figure 2: Private Physical Capital to Output Ratio in the United States (1929-2006)

Gross public investment to total tax receipts ratio (G/R). To estimate this ratio, we

again use the data for the post-WWII period (1946-2006) obtained from the BEA’s NIPA

tables. The average gross public investment to total tax receipts ratio for the period is 19.1%,

6The value of δh is controversial. However, a change in δh will only affect the allocation of non-work timebetween accumulating human capital and depreciation. For example, if δh = 1%, then the values that willchange in Table 1 are l, m, and η. They change as follows: lo = 0.452, mo = 0.188 and η = 0.74.

15

which is the number we use here. The time path of this ratio based on BEA’s data is shown

in Figure 3.

Table 1: Calibration to the Benchmark

Output Shares and Depreciation Rates α1 α2 δk δh δg

0.1 0.337 0.05 0.015 0.035

Data for Calibration k/y G/R n

2.17 0.191 0.36

Government Policy X τc τn τk

0 7% 25% 25%

Deep Parameters of the Model ρ γ η

.0485 .1347 0.67

Consumption and Time Allocations c l m

0.366 0.415 0.225

State Variables h kg y c/y

0.95 0.507 0.461 0.794

Human Capital & Output Growth ν φ

1.53% 1.80%

Tax rates (τk, τn, τc). For the capital income tax rate and labor income tax rate, Turnovsky

(2004) set these both rates to 28%. Here we set the benchmark values to 25%, following the

works of Greenwood and Hercowitz (1991) and Einarsson and Marquis (2001) who discuss

in greater detail the selection of their values. Regarding the consumption expenditure tax

rate, we set this rate as the pure U.S. sales tax rate which is currently 7%.

5 Results

Along the initial bgp, the only sources of government revenue are proportional distortionary

taxes. There is no lump-sum tax. The fiscal policy maintains a budget balance. We begin

by quantifying the distortion caused by the existing tax regime. For this, we consider an ex-

periment where the government replaces the distortionary tax regime with non-distortionary

16

y = -0.0013x + 0.2329

0

0.05

0.1

0.15

0.2

0.25

0.3

1946

1949

1952

1955

1958

1961

1964

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

1997

2000

2003

2006

Years

G/R

(%)


Figure 3: Gross Public Investment to Total Tax Receipts Ratio in the United States (1929-2006)

lump-sum taxes. The ratio of government revenue to GDP along the new bgp is the same as

for the initial bgp. Thus, the policy change is budget neutral. Furthermore, G/R, the ratio of

public physical capital to government revenue remains unchanged as well. In addition, with

the value of G/R determined exogenously and assumed constant, the government budget

constraint implies that the government can manipulate a maximum of three out of the four

policy tools, τc, τk, τn, X/y.

While the evaluation of the current tax regime by considering lump-sum taxation is

insightful, in practice governments do not have access to lump-sum taxes. We, therefore, next

turn attention to tax policies in which only distortionary taxes are available. In particular,

we assess the positive and normative implications of a movement to a tax regime that relies

only on one source of taxation. Comparing the outcome with the lump-sum taxation allows

us to evaluate the distortionary effect of each source of taxation.

Before we present the results, it is instructive to understand the nature of distortionary

effects of taxes on consumption expenditures, labor income, and private physical capital

income relative to financing of government revenue using lump-sum taxation.

The tax on capital is a source of intertemporal distortion which reduces saving. This

reduction in saving lowers the overall productive capacity of the economy as stocks of both

private and public physical capital decline. Consequently, consumption is reduced. Note

that the stock of public physical capital declines as the economy contracts thereby reducing

government revenues. As physical capital and human capital are gross complements in

17

production, a reduction in physical capital also reduces the marginal product of human

capital. The household responds by reducing the time devoted to the accumulation of human

capital and taking more leisure. Thus, the stock of human capital declines as well.

The labor income tax distorts decisions by reducing the return to human capital. The

household again respond by reducing time devoted to human capital accumulation and taking

more leisure. Thus, productive capacity of the economy falls thereby reducing consumption

as the stock of human capital falls. As stated previously, the gross complementarity of human

and physical capital then reduces return to private physical capital reducing incentive to save

and invest. The resulting decrease in physical capital stock further reduces the growth in

the economy.

The tax on consumption results in intra-temporal distortion raising the cost of consump-

tion. The household substitute away from consumption toward leisure. The extra leisure

time comes at the expense of time devoted to human capital accumulation. This reduces the

stock of human capital in the economy. While a consumption tax does not directly cause

intertemporal distortion reducing saving, the reduction in human capital does reduce the

productivity of private physical capital reducing the incentive to save. As a result, the stock

of private physical capital declines as well. The public capital stock falls as the economy as

a whole becomes smaller. Thus, in the end, the growth rate of the economy falls.

5.1 Experiment 1: Shift to Lump-Sum Taxation

In this section, we begin by assessing the extent of the tax distortion in the initial equilibrium.

For this purpose, we consider the experiment where all distortionary taxes are replaced by

non-distortionary lump-sum taxes. Note that the ratio of tax revenue to GDP remains

unchanged and so is the fraction of government revenue devoted to public investment. Figure

4 depicts the response of the main economic variables to this unanticipated change in the

governmental policy. The figures depict the ratios (normalized variables), logarithms, and

growth rates respectively. In each panel, the initial steady-state bgp that prevailed in the

economy before the new policy is shown as the solid line for t < 0. For t ≥ 0, the solid line

reveals the equilibrium path resulting from the new fiscal policy. The dotted line shows the

evolution of variables along the initial bgp in the absence of the change in the tax rates. The

response of various variables across the two balanced growth paths is summarized in Table

2.

As the taxes on physical capital income, labor income, and consumption are removed, the

18

accompanying distortions disappear which, as expected, improves welfare. We follow Atolia,

Chatterjee, and Turnovsky (2008) in using the Lucas’s method to assess the welfare gain in

terms of a proportional increase in consumption, ζ, along the initial bgp that delivers same

utility as the new tax policy. According to this measure, the shift to lump-sum taxation

confers a welfare benefit equivalent to 9.78 per cent increase in consumption.

Table 2: Permanent Shift to Lump-Sum Tax

Initial bgp: τco = 07.0%, τno = 25.0%, τko = 25.0%New bgp: τco = 00.0%, τno = 00.0%, τko = 00.0%

c kg h y i igBenchmark 0.366 0.507 0.949 0.461 0.068 0.027Initial Response Levels 0.319 0.507 0.949 0.487 0.139 0.028

% Change (12.69) - - 5.61 104.10 5.61New bgp Levels 0.300 0.316 0.863 0.417 0.092 0.024

% Change (17.91) (37.71) (9.06) (9.59) 35.19 (9.59)

n m l c/y k/y kg/y

Benchmark 0.360 0.225 0.415 0.794 2.170 1.101Initial Response Levels 0.391 0.348 0.261 0.656 2.055 1.043

% Change 8.58 54.70 (37.08) (17.33) (5.31) (5.31)New bgp Levels 0.360 0.376 0.264 0.721 2.400 0.759

% Change - 67.10 (36.36) (9.20) 10.61 (31.11)

r φ ν ζ

Benchmark 15.5 1.8 1.5 0.0Initial Response Levels 16.4 - - -

% Change 5.61 - - -New bgp Levels 14.0 4.2 3.6 9.78

% Change (9.59) - - -

19

0 10 20 30 40 50Years

0.30

0.32

0.34

0.36

c�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

l

0 10 20 30 40 50Years

0.85

0.90

0.95

1.00

h�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

m

0 10 20 30 40 50Years0.25

0.300.350.400.450.50

kg�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

n

-5 0 5 10 15Years

-1.2-1.0-0.8-0.6-0.4-0.2

0.0

lnc

-5 0 5 10 15Years

0.0

0.2

0.4

0.6

0.8

lnh

-5 0 5 10 15Years-1.0

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lnkg

-5 0 5 10 15Years

0.0

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0.6

0.8

lnk

-5 0 5 10 15Years-1.0

-0.8

-0.6

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0.0

ln y

-5 0 5 10 15Years0.130

0.1350.1400.1450.1500.1550.1600.165

r

-5 0 5 10 15 20 25Years

123456

Φc

-5 0 5 10 15 20 25Years

0.51.01.52.02.53.03.5

Φh

-5 0 5 10 15 20 25Years

2

4

6

8

Φk

Figure 4: Permanent Shift to Lump-Sum Tax

20

In particular, the removal of the tax on private physical capital encourages the accu-

mulation of private physical capital, and the removal of tax on labor income motivates the

accumulation of human capital. As explained earlier, however, a reduction in all taxes ul-

timately gives impetus to the accumulation of both types of private capital either directly

or indirectly which raises the rate of growth of the economy. In fact, φ increases from 1.8%

along the initial bgp (benchmark) to 4.2% along the final bgp. Note that this growth rate is

fueled by the increase in the rate of growth of human capital. As Table 2 shows, ν increases

from 1.5% to 3.6%.

The higher rate of growth of human capital along the bgp implies that the time devoted

to human capital accumulation increases. This fact is evident in Table 2 which shows that m

jumps from 0.225 to 0.376. This increase comes at the expense of time devoted to leisure as

the time devoted to production does not change. Similarly, the higher rate of growth of the

private physical capital increases the investment as share of GDP. Accordingly, consumption

as share of GDP declines from 0.794 to 0.721.7

There are two determinants of return to capital that move in opposite direction as a

result of the change in the tax policy. The elimination of distorting capital tax reduces the

divergence between the private and the social marginal return to private physical capital

which reduces the required return on capital, r, as (26) shows. This boosts the private

physical capital accumulation as referred to earlier. However, as capital accumulation in

the economy (including that of private physical accumulation) rises, the growth rate, φ,

increases, and there is a second round effect that counteracts the fall in the required rate of

return on private physical capital (see (26)). The reason is that the increased growth in the

economy causes the marginal utility of wealth or consumption (λ1) to decline faster making

the household reluctant to save and therefore requiring higher compensation or return on

capital. As the initial effect dominates, r declines from 15.5 per cent to 14.0 per cent which

in turn implies an increase in private physical capital to output ratio (k/y) as Table 2 shows.

The impact effect of the tax policy change on macroeconomic behavior is very similar

to the long-run effect, but there are some differences. First the similarities. There is a

sharp increase in investment in private physical capital accumulation and the time devoted

to human capital accumulation whereas time devoted to leisure falls. However, note that

m increases by a smaller amount on impact than in the long run (Figure 4). On the other

hand, the investment in private capital accumulation overshoots it long-run rate (see path

7Note that the share of public investment to GDP is unchanged.

21

of φk in Figure 4).

The reason that m rises by less in the short run is that the reduction in tax on labor

raises the return on effective labor supply immediately. While investment in human capital

takes advantage of this in the long-run, it cannot do so in the short run as human capital

accumulation takes time. On the other hand, the household can take immediate advantage of

higher return on labor by increasing effective labor supply by increasing the time devoted to

production. Finally, when human capital accumulation has occurred, this labor supply can

be shifted back to accumulation of human capital. This short-run increase in labor supply

also raises the return to private physical capital in the short run, although it falls in the long

run (see Figure 4).

The other difference in the short run is in the response of consumption, which is related

to the incentive for private physical capital accumulation. Although, an increase in time

devoted to production increases economy’s output, it is not enough to meet the increased

demand for investment in private physical capital. The excess investment demand is met

by reduction in current consumption (see Figure 4). The household is willing to sacrifice

current consumption as the current return adequately compensates the household for the

postponement of consumption by providing a high return.

The return on investment is high in the short run and, in fact, overshoots its long-run

value, as private physical capital is relatively scarce initially. This overshooting implies that

not only does the economy’s capital stock grows at a higher rate in the long run but it also

reaches a higher level.

The fact that not only leisure but also consumption falls in short run implies that the

welfare gains are not realized immediately. Figure 5 shows the accumulated welfare gains

over different horizons. Accumulated welfare gain over a time horizon t is calculated as the

proportional increase in consumption (ζ) required over period 0 to t, along the initial bgp, to

deliver same utility over this period as the new policy (see Atolia, Chatterjee, and Turnovsky,

2008). As Figure 5 shows, welfare rises in the long run. However, note that this long-run

increase in welfare comes at a considerable sacrifice in the short run. The short-run welfare

cost amounts to as high as 30 per cent of consumption or more. It takes the economy more

than 50 years (56.1 years) to break even in the sense that it is only after 56.1 years that

utility level exceeds the utility that the household would enjoy along the initial bgp over the

same time horizon.

22

0 50 100 150Years

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2Ζ

Figure 5: Accumulated Welfare Gains From Permanent Shift to Lump-Sum Tax

5.2 Experiment 2: Capital Income Taxation

In this section, we evaluate the distortionary effect of a tax on private physical capital. For

this purpose, we consider a move from the existing tax regime to sole reliance on a capital

income tax. The resulting distortion is assessed by comparing the response of the economy

for a switch to capital income tax versus a switch to lump-sum tax.8 The shift to capital

income tax raises τk to 90.7 per cent. The resulting paths of variables of interest are collected

in Figure 6. The changes across the balanced growth paths are summarized in Table 3.

The removal of tax on labor income and consumption both provide impetus to build

human capital. Accordingly, m rises from 0.225 to 0.302 (see Table 3). On the other hand,

a large increase in taxation of capital substantially reduces the incentive to save and invest.

As a result, the economy’s stock of private capital falls significantly. As Table 3 shows, the

values of all variables in the core dynamic system such as c, h, and kg rises as these variables

are normalized by k. This outcome is exactly opposite of that for shift to lump-sum taxation

as seen from Table 2. As expected, a shift from capital income taxation to lump-sum taxation

would increase the relative availability of private physical capital.

8An alternative plausible experiment would be replace the initial capital income tax with lump-sumtaxation but such an experiment would confound the distortionary effect of capital income taxation with theinteraction effects between capital income tax and other taxes.

23

0 10 20 30 40 50Years0.5

1.01.52.02.53.03.5

c�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

l

0 10 20 30 40 50Years

51015202530

h�

0 10 20 30 40 50Years0.20

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m

0 20 40 60Years

1234567

kg�

0 10 20 30 40 50Years0.20

0.25

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0.40

n

0 20 40 60 80 100Years

-1

0

1

2

lnc

0 10 20 30 40 50Years

0.51.01.52.02.5

lnk

0 20 40 60Years-1.0

-0.5

0.0

0.5

1.0

1.5

lnkg

0 50 100 150Years

-2-1

01234

lnk

0 20 40 60 80 100Years-2

-1

0

1

2

ln y

0 10 20 30Years

0.40.60.81.01.21.4

r

0 5 10 15 20 25Years-10

-8-6-4-2

02

Φc

-5 0 5 10 15 20 25Years

0.51.01.52.02.53.03.5

Φh

-5 0 5 10 15 20 25Years

-30

-20

-10

0

Φk

Figure 6: Permanent Shift to Capital Income Tax

24

Table 3: Permanent Shift to Capital Income Tax


c kg h y i igBenchmark 0.366 0.507 0.949 0.461 0.068 0.027Initial Response Levels 0.522 0.508 0.956 0.362 (0.181) 0.021

% Change 42.77 0.00 (0.00) (21.37) (366.49) (21.37)New bgp Levels 3.778 3.665 32.380 4.098 0.080 0.239

% Change 932.47 622.24 3,312.43 789.17 18.01 789.17

n m l c/y k/y kg/y


% Change (30.41) 69.78 (11.44) 81.56 27.17 27.17New bgp Levels 0.360 0.302 0.338 0.922 0.244 0.895

% Change - 34.35 (18.61) 16.12 (88.75) (18.77)

r φ ν ζ


% Change (21.37) - - -New bgp Levels 138.09 3.02 2.57 -39.44

% Change 789.17 - - -

It is not hard to see that this new tax regime is more distortionary as it relies on taxation

from only one source. In fact, from Table 3 one can see that the new tax policy entails a

welfare loss of 39.44 per cent. However, note that despite a move to a more distortionary

regime and a welfare loss, the growth rate of the economy rises substantially from 1.8 per

cent to 3.02 per cent. This seems puzzling and self-contradictory as one would tend to ex

ante associate higher growth along the new bgp with higher welfare.

Although this appears puzzling at first, the outcome is not very hard to understand. The

engine of growth in the model is human capital. Even though, the new tax policy is more

distortionary, it encourages human capital accumulation as mentioned before. Increased

human capital accumulation raises the growth rate of the economy. However, this increased

growth in the long run (along the new bgp) fails to deliver a welfare gain because increased

25

taxation of income from private physical capital makes it very scarce. As Figure 6 shows,

even though levels of all the variables catch up with their values along the original bgp, they

are considerably lower in the short run due to a strong disincentive to save and invest. In

sum, while growth is higher in the long run, it starts from a much lower level.

While growth accelerates relative to the original bgp, note the economy grows slowly

compared to the case with shift to lump-sum taxation (4.2 per cent vs. 3.02 per cent). The

difference arises due to the absence of the favorable effect of physical capital accumulation

on human capital accumulation (recall, they are gross complements). Also note that, a shift

to lump-sum taxation leads to a short-run increase in the rate of accumulation of k that

exceeds its long-run value whereas the opposite is true for the shift to lump-sum taxation

(see paths of φk in Figures 4 and 6.) This shows that a capital income tax reduces welfare

both by reducing the growth rate of the economy as well as reducing the level of private

physical capital during transition.

0 50 100 150Years

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2Ζ

Figure 7: Accumulated Welfare Gains - Permanent Shift to Lump-Sum Tax Vs. CapitalIncome Tax

The two tax policy shifts (to lump-sum taxation and to capital income taxes only) not

only differ in their long-run effect on welfare but also in the time path of accumulated welfare

as Figure 7 highlights. In particular, note the stark differences in the accumulated welfare

in the short run. With confiscatory tax on capital income, the household refuses to invest

preferring rather to increase consumption (see Figure 6) and takes more leisure (relative to the

case with lump-sum taxation). This clearly raises their welfare in the short-run (the dashed

curve in Figure 7). However, as the stock of physical capital falls consumption ultimately

falls and so does leisure. The resulting loss in welfare gradually pulls the accumulated welfare

26

down to its long-run value.

5.3 Experiment 3: Labor Income Taxation

This section considers a tax policy with only a labor income tax. As before, the share

of government revenue to GDP and of public investment to government revenue is kept

unchanged. This implies a tax rate of 46.1 per cent on labor income. The resulting changes

across the balanced growth paths are collected in Table 4 whereas the paths of variables of

interest are shown in Figure 8.

Table 4: Permanent Shift to Labor Income Tax



% Change (16.72) 0.01 (0.00) 17.90 204.25 17.90New bgp Levels 0.227 0.479 0.434 0.295 0.051 0.017

% Change (38.01) (5.61) (54.25) (35.98) (25.08) (35.98)

n m l c/y k/y kg/y


% Change 28.20 (77.86) 17.73 (29.37) (15.18) (15.18)New bgp Levels 0.360 0.117 0.523 0.769 3.390 1.624

% Change - (47.81) 25.91 (3.16) 56.20 47.44

r φ ν ζ


% Change 17.90 - - -New bgp Levels 9.94 0.09 0.08 -7.54

% Change (35.98) - - -

27

0 10 20 30 40 50Years

0.2

0.3

0.4

0.5

c�

0 10 20 30 40 50Years

0.10.20.30.40.5

l

0 10 20 30 40 50Years0.3

0.40.50.60.70.80.91.0

h�

0 10 20 30 40 50Years

0.10.20.30.40.5

m

0 20 40 60 80 100120Years0.20

0.250.300.350.400.450.500.55

kg�

0 10 20 30 40 50Years

0.10.20.30.40.5

n

-5 0 5 10 15Years

-1.2-1.0-0.8-0.6-0.4-0.2

0.0

lnc

-5 0 5 10 15Years

0.0

0.2

0.4

0.6

0.8

lnh

-5 0 5 10 15Years-1.0

-0.8

-0.6

-0.4

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lnkg

-5 0 5 10 15Years

0.0

0.2

0.4

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0.8

lnk

-5 0 5 10 15Years-1.0

-0.8

-0.6

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-0.2

0.0

ln y

0 5 10 15Years

0.060.080.100.120.140.160.18

r

-5 0 5 10 15 20 25Years

2

4

6

8

Φc

0 5 10 15 20 25Years-2.0

-1.5-1.0-0.5

0.00.51.01.5

Φh

0 5 10 15 20 25Years

5

10

15

Φk

Figure 8: Permanent Shift to Labor Income Tax

28

The increased taxation of labor income or human capital understandably puts a brake

on the growth in the economy as growth is driven by human capital accumulation. The

growth in the economy almost comes to a halt (0.09 per cent) with m falling from 0.225 to

0.117 (see Table 4). On the other hand, elimination of tax on income from physical capital

boosts saving and investment. Both the reduction in τk and lower growth rate reduce the

required return on physical capital from 15.53 per cent to 9.94 per cent per year. As the

economy’s stock of private capital rises, as Table 2.3 shows, the values of all variables in the

core dynamic system such as c, h, and kg fall along the new bgp. This outcome is exactly

the opposite of that for shift to capital income taxation as seen from Table 3.

The tax policy change is again a move to a more distortionary regime given the reliance

on only labor income tax. Indeed, the welfare falls by 7.54 per cent (Table 4). However, note

that the loss in welfare is much less than for shift to capital income taxation where it falls by

39.44 per cent. More importantly, comparison of shifts to a capital income tax and to a labor

income tax shows an exclusive focus on growth rates of the economy along the new balanced

growth path as a measure of welfare can be very misleading. While economy grows at a much

faster rate (in fact growth accelerates) of 3.02 percent with the shift to the capital-income-

tax-only regime, the growth rate is only 0.09 per cent for a shift to labor-income-tax-only

regime; the costs of shifting to a capital-income-tax-only regime are significantly higher.

0 50 100 150Years

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2Ζ

Figure 9: Accumulated Welfare Gains - Permanent Shift to Lump-Sum Tax Vs. LaborIncome Tax

The reason for the differing welfare implications is that the welfare consequences of the

29

transitional dynamics of the two tax policies are very different; in fact, they are completely

opposite. The shift to an exclusive labor income tax encourages capital accumulation and

time devoted to leisure, while consumption falls (see Figure 8); the reverse is true for a

shift to a capital income tax. As Figures 7 and 9 show, these differences translate into very

different transitory consequences for welfare. Over the short term of 5-10 years, the shift to

the capital-income-tax-only regime encourages consumption which increases instantaneous

felicity but decreases the capital stock. The latter takes a toll in the medium and the long

term. In the case of labor-income-tax-only regime, over the very short term, welfare actually

falls due to reduced consumption, but then recovers in the medium term (5-50 years). The

dashed curve in Figure 9 also shows that, unlike lump-sum taxation, a shift to exclusively

labor income taxation does not involve such a stark intertemporal trade-off in welfare.

The comparison of shifts to capital and labor income tax policies underscores an impor-

tant aspect of the welfare analysis of tax policies. Once cannot assess and rank proposed tax

policies (such as having only capital income tax or only labor income tax) without reference

to the current state of the economy (including current policies.) This is exactly what a

comparison of growth rates along the new bpg does.

5.4 Experiment 4: Consumption Taxation

This section considers our last tax policy experiment, the consumption-tax-only regime, with

the share of government revenue to GDP and the share of public investment to government

revenue remaining unchanged as in earlier experiments. This shift requires a tax rate 41.4

per cent on consumption. The changes across the balanced growth paths are collected in

Table 5, whereas the paths of variables of interest are shown in Figures 10.

The move to a consumption tax results in the growth rate rising to 2.31 per cent, up

from 1.80 per cent. The new growth rate is between that for a shift to capital income tax

and a shift to a labor income tax. This makes intuitive sense. The shift to a consumption

tax removes disincentives for accumulating private physical capital, as does a shift to a labor

income tax. It also removes the factor price distortion as both the tax on labor income and

capital income are reduced to zero, while labor income tax fails to do and in fact causes a

substitution away from human capital in production which ends up causing growth to slow

down.

30

0 10 20 30 40 50Years

0.280.300.320.340.360.380.40

c�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

l

0 10 20 30 40 50Years0.6

0.7

0.8

0.9

1.0

h�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

m

0 10 20 30 40 50Years0.30

0.350.400.450.500.550.600.65

kg�

0 10 20 30 40 50Years0.20

0.25

0.30

0.35

0.40

n

-5 0 5 10 15Years

-1.2-1.0-0.8-0.6-0.4-0.2

0.0

lnc

-5 0 5 10 15Years

0.0

0.2

0.4

0.6

0.8

lnh

-5 0 5 10 15Years-1.0

-0.8

-0.6

-0.4

-0.2

lnkg

-5 0 5 10 15Years

0.0

0.2

0.4

0.6

0.8

lnk

-5 0 5 10 15Years-1.0

-0.8

-0.6

-0.4

-0.2

0.0

ln y

0 5 10 15Years0.10

0.110.120.130.140.150.160.17

r

-5 0 5 10 15 20 25Years

12345

Φc

-5 0 5 10 15 20 25Years

0.5

1.0

1.5

2.0

Φh

0 5 10 15 20 25Years

2468

1012

Φk

Figure 10: Permanent Shift to Consumption Tax

31

The capital income tax causes even higher growth for two reasons. First, instead of just

removing the factor price distortion, it changes the direction of the distortion in favor of

human capital accumulation. Second, it removes the wedge in the intra-temporal labor-

leisure trade-off which exists with a consumption tax. The removal of this wedge, which led

to an inefficient substitution toward leisure, gives an additional impetus for human capital

accumulation, thus raising the rate of growth. As Tables 3 and 5 show, leisure falls to 0.338 in

the capital-income-tax-only regime compared to 0.383 in the consumption-tax-only regime.

Recall, however that while the capital-income-tax-only regime delivers the highest growth

rate, it causes a very significant inter-temporal distortion in saving and capital accumulation.

Table 5: Permanent Shift to Consumption Tax



% Change (13.11) 0.00 (0.00) 9.26 162.50 (73.90)New bgp Levels 0.267 0.362 0.652 0.361 0.073 0.021

% Change (27.13) (28.61) (31.24) (21.70) 7.55 (21.70)

n m l c/y k/y kg/y


% Change 14.29 (4.52) (9.94) (20.47) (8.47) (8.47)New bgp Levels 0.360 0.257 0.383 0.739 2.771 1.004

% Change - 14.39 (7.80) (6.94) 27.71 (8.83)

r φ ν ζ


% Change 9.26 - - -New bgp Levels 12.16 2.31 1.96 4.70

% Change (21.70) - - -

As in other cases, the increase in the growth rate of the economy from 1.80 per cent

to 2.31 per cent requires an increase in m which rises from 0.225 to 0.257. The shift to a

32

consumption tax gives an incentive for the household not only to accumulate human capital,

which is reflected in a higher growth rate, but also to accumulate private physical capital,

as the tax rate on capital income falls to zero. As Table 5 shows, the resulting capital

accumulation lowers the values of all variables in the core dynamic system (c, h, and kg)

along the new bgp. Recall all these variables are normalized by k. The real return to capital

falls from 15.53 per cent to 12.16 per cent, as effect from the elimination of capital income

tax dominates that of an increase in φ (see (26)).

As in the other cases, where tax on capital is removed, the time devoted to leisure

approaches its new long-run value from below. The reason is that the removal of a capital

income tax creates a strong incentive for the household to accumulate physical capital, which

reduces consumption. The resulting increase in the marginal utility of consumption shifts

the labor-leisure trade-off in favor of labor. The reader can verify this from Figure 4, 8, and

10 where n rises in the short run, although it is unchanged in the long run.

5.5 Superiority of Consumption Tax

The tax policy change to consumption-tax-only regime is again a move to a distortionary

regime that places exclusive reliance on one source of taxation. However, it does not lead

to a fall in welfare. Rather, welfare rises by 4.70 per cent. To understand the nature of this

result, first let us note that taxes cause four different types of distortions in this model: (1)

intertemporal distortion caused by capital income tax, (2) intertemporal distortion caused

by labor income tax as it acts as a tax on human capital accumulation, (2) factor-price

distortion caused by differential rates of taxation of capital and labor (or human capital),

and (3) intra-temporal (labor-leisure) distortion caused by the consumption tax and the labor

income tax. As is well known, inter-temporal distortions are generally more severe as they

lead to compounding effect. A move to a consumption tax removes inter-temporal distortions

associated with both the capital income tax and the labor income tax. Furthermore, this shift

also removes the factor-price distortion. While this comes at the cost of increased pressure on

only one–intra-temporal–margin, the gains from the removal of the intertemporal distortions

deliver enough gains to offset the extra losses from the wedge created in the labor-leisure

decision.

This is not to say that consumption taxation is the optimal policy. In fact, it can

be shown that slight deviations from sole reliance on consumption tax improve welfare.

However, it appears that the optimal tax regime will place a disproportionate emphasis on

33

consumption tax. While a consumption-tax-only regime delivers a welfare gain in the long-

run, consumption and leisure both fall in the short run (Figure 10), thus inducing a decline

in felicity in the short run (the dashed curve) and thus this regime involves a sacrifice in the

short run. However, the short-run trade-off is not as severe as lump-sum taxation, as Figure

11 shows.

0 50 100 150Years

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2Ζ

Figure 11: Accumulated Welfare Gains - Permanent Shift to Lump-Sum Tax Vs. Consump-tion Tax

6 Conclusion

This paper examines the consequences of distortionary taxes on welfare and growth in the

framework of an endogenous growth model. It incorporates the role of government capital

in enhancing the productivity of the production process. The model, therefore, accounts for

the roles played by both public capital and human capital in the process of economic growth;

roles which have been widely recognized in both theoretical and empirical literature.

When the model is calibrated to the U.S. economy and the consequences of shifts to

alternative fiscal policies are analyzed, it is found that the implications for growth for various

policy regimes are different from those for welfare. In particular, a fiscal policy delivering the

highest long-run growth is not necessarily the one that maximizes the overall welfare. This

difference arises from the very different welfare outcomes of these fiscal policies during the

transition. An accurate overall welfare assessment of different policies, therefore, requires

solving for the exact nonlinear transitional dynamics of the model which we do using the

reverse-shooting technique.

34

Our experiments suggest that a shift from the current fiscal policy with capital income,

labor income, and consumption taxation to one relying only on a consumption tax would

enhance both the welfare and the long-run growth rate of the economy. This results from

the fact that consumption taxes do not distort the margins that affect the accumulation

of reproducible capital. In contrast, sole reliance on neither capital nor labor income taxes

would result in an increase in both growth and welfare. Surprisingly, reliance exclusively

on a capital income tax increases long-run growth but, it lowers welfare. The labor income

taxation delivers both lower welfare and lower growth rate. Finally, while a shift to con-

sumption tax raises overall welfare (and long-run growth), the welfare gains are not realized

immediately, it takes almost a generation (36.7 years) for the accumulated welfare gains to

turn positive.

In terms of relative ranking, capital taxation delivers highest growth but its benefits are

more than offset by short-run losses in welfare. These short-run losses are so large that,

in terms of welfare, shift to capital taxation fares poorly even when compared to a shift to

labor taxation where the long-run growth comes to an almost complete halt.

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39

Linearization and Higher-Order Approximations:How Good are They?∗†

Results from an Endogenous Growth Model with Public Capital

Manoj AtoliaFlorida State University ‡

Bassam AwadFlorida State University §

Milton MarquisFlorida State University ¶

First Draft: October 2009

∗Based on the second chapter of PhD dissertation of Bassam Awad.†We are very grateful to Paul Beaumont, Tor Einarsson, Bharat Trehan and other participants of the

Macro summer workshop, 2008, at Department of Economics, Florida State University where an earlierversion of this paper was presented. All errors are ours.‡Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 850-644-

7088. Email: [email protected].§Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 850-567-

2085. Email: [email protected].¶Department of Economics, Florida State University, Tallahassee, FL 32306, U.S.A. Telephone: 850-645-

1526. Email: [email protected].

1

.

Abstract

The standard procedure for analyzing transitional dynamics in non-linear macromodels has been to employ linear approximations. Recently quadratic approximationshave been explored. This paper examines the accuracy of these and higher-order ap-proximations in an endogenous growth model with public capital, thereby extendingthe work done in the current literature on the neoclassical growth model. We findthat significant errors may persist in computed transition paths and welfare even af-ter resorting to approximations as high as fourth order. Moreover, the accuracy ofapproximations may not increase monotonically with the increase in the order of ap-proximation. Also, as in the previous literature, we find that achieving acceptablelevels of accuracy when computing the welfare consequences of a policy change typi-cally requires a higher order approximation than attaining similar levels of accuracy inthe computation of the transition path: typically an increase in order of approximationby one is sufficient.

Keywords: Linearization, higher-order approximations, endogenous growth, welfare,public and human capital

JEL Codes: O41, C61, C63

2

1 Introduction

The importance of transitional dynamics for understanding the consequences of various

macroeconomic policies has received much attention in the literature. When the policy

changes are large, a welfare analysis based on comparisons across steady states or balanced

growth paths can be very misleading, often overstating the welfare effects. (See, for example,

Turnovsky (1992,1996) and Einarrson and Marquis (1999,2002).) Until recently, the analysis

of the transition paths and their consequences for the computation of welfare were conducted

using linear – or at best quadratic – approximations to decision rules that govern behavior

along the final (post-policy change) balanced growth path. The fact that optimal decision

rules are not stationary along the transition was ignored.1

The development of computational techniques that could be used to establish time-

varying optimal transition paths appropriate for the analysis of the time path of the econ-

omy and the attendant welfare consequences of once-and-for-all policy changes has been the

subject of work by Becker, Grune, and Semmler (2007), Turnovsky (2000), and Atolia, Chat-

terjee and Turnovsky (forthcoming) which apply the shooting algorithms of Judd (1998) and

Brunner and Strulik (2002), and Atolia and Buffie (2009).

The purpose of this paper is to extend the work of Atolia et al. (forthcoming) who ap-

plied the reverse-shooting algorithm to examine the accuracy of first-order approximation

(linearization) methods in describing the transition path and the welfare consequences of an

economy in which a significant fiscal policy reform is undertaken. In their model, public cap-

ital plays an important role in generating externalities that affect the economy’s transitional

dynamics. We consider two extensions. The public capital externality is examined in the

context of a growth model in which growth depends on the investment in human capital as

in Uzawa (1966) and Lucas (1988).2 This change introduces an important new dimension

to the optimal transition path. Second, we do not limit our comparisons of the accuracy of

approximate decision rules to first-order, but consider up to fourth-order approximations.

We obtain several results in this model that appear to be quite general, in that they are

likely to be present in other models where transitional dynamics play an important role in

1We use the term ‘optimal’ to mean welfare-maximizing decision rules in an economy with distortions,which of course is not Pareto optimal.

2We note that Turnovsky (1996) examined fiscal policy issues in an AK model, whose transitional dy-namics differs considerably from an Uzawa-Lucas type model. The work of Turnovsky has motivated sev-eral later works analyzing different aspects of alternative fiscal policies. See for example Petrucci (2009),Gokan(2008a,b), Baier (2001), Fung(2000), Turnovsky(1997), and Palivos (1995).

3

the analysis:

• The order of approximation required to obtain a tolerable level of accuracy in terms of

characterizing the transition path may not be the same as the order of approximation

required to obtain acceptable accuracy in the computation of welfare, which generally

requires a higher-order approximation.

• The order of approximation required to obtain acceptable accuracy varies with the

manner in which the tax distortions in the economy are varied under a fiscal policy

reform.

• While higher-order approximations tend to improve the accuracy of the welfare cal-

culations, these improvements are not universally monotonic with increasing order.

For some parameterizations of the model, second-order approximations do not yield

transitional patterns or welfare calculations that are superior to first-order.

• Finally, even fourth-order approximations can be insufficient to obtain acceptable levels

of accuracy.

The paper proceeds with a description of the model in Section 2, the solution of the

economy’s balanced growth path and the calibration exercise in Sections 3, the solution of

the transitional dynamics in Section 4, the fiscal policy exercises in Section 5, and sensitivity

analysis in Section 6. Section 7 concludes.

2 The Model

In this perfect foresight representative agent economy, households choose time allocations

between labor, leisure, and human capital accumulation. The last of these ultimately de-

termines the rate of growth of the econmoy. Public capital introduces a positive externality

in production and is funded by a combination of consumption, labor income, and capital

income taxes, while the government runs a balanced budget.

2.1 Household’s Optimization

The household maximizes lifetime utility by choosing time paths for consumption, c, and its

allocation of time between leisure, l, labor, n, and human capital accumulation, m.

4

max{c,l,m,n}

∞∫0

e−ρtu(c, l)dt, (1)

where ρ is rate of time preference and e−ρt is the corresponding discount factor, and the CES

felicity function is given by:

u(c, l) =

11−µ

[(clη)1−µ − 1

]if µ > 1,

log c+ η log l if µ = 1.(2)

where µ−1 is the intertemporal elasticity of substitution.

The household faces a budget constraint that determines its gross investment in private

physical capital, k, as its income less its consumption, its taxes, and the depreciation loss in

capital, δkk, where δk ∈ (0, 1) is the depreciation rate. It receives income from labor, whn,

where w is the wage rate paid on effective units of labor, hn, with h equal to the household’s

stock of human capital, its rental income from capital, rk, where r is the real rental rate,

dividends from the (aggregate) firm, π, which are a per capita share of the firm’s profits, and

a lump-sum transfer from the government, T . In addition to a lump-sum tax, X, (which

may be zero), the household faces tax rates of τc on consumption, τn on labor income, and

τk on capital income.

k = π + rk + whn+ T − c− δkk − (τkrk + τnwhn+ τcc+X), (3)

The technology available to the household to build human capital is given by:

h = γmh− δhh, (4)

where γ > 0 is a productivity parameter and δh ∈ (0, 1) is rate of depreciation of human

capital.3

The time resource constraint is:

l +m+ n = 1 (5)

3It may be noted that the evolution rule for the human capital is linear in the current state of humancapital which generates the endogenous growth in the model. The human capital evolution equation abovewas used by Lucas (1988) without a depreciation of human capital. Accounting for the depreciation of theprivate human capital is widely recognized in the literature. See, for example, Heckman (1976) and Marquisand Einarsson (1990).

5

This maximization of (1) is then subject to constraints (3), (4) and (5), the initial con-

ditions k(0) = ko, h(0) = ho, kg(0) = kgo, and the transversality conditions:

limt→∞

λ1(t)k(t) = 0 (6)

limt→∞

λ2(t)h(t) = 0 (7)

where ko, ho, kgo are the levels of private physical capital stock, human capital stock, and

public physical capital stock in the initial state of the economy and λ1 and λ2 are the

Lagrange multipliers on (3) and (4).

The maximization yields the following Euler equation for the labor-leisure decision and

co-state equations for investments in private physical capital and human capital:

uc(1− τn)wh = ul(1 + τc) (8)

λ1

λ1

= ρ− r(1− τk) + δk (9)

λ2

λ2

= ρ− γ(m+ n) + δh (10)

where

λ1 = uc/(1 + τc) (11)

and

λ2 = ul/γh (12)

with uc and ul being the marginal utilities of consumption and leisure.

2.2 Firm’s Optimization

Output per capita, y, is determined by a technology that transforms the economy’s resources

of public and private physical capital, private human capital and labor into desired output

goods. The technology assumes the following constant elasticity of substitution (CES) form:

6

y = f(k, n;h, kg) =

A[(1− α2)(kg

θhn)(1− 1σ

) + α2k(1− 1

σ)]( 1

1− 1σ

)if σ 6= 1;

Akgα1kα2(hn)1−α2 if σ = 1

(13)

where A,α1, α2 are parameters, with α2 ∈ (0, 1), A > 0, and θ = α1

1−α2.

As seen in (13), output depends on the available stock of government-provided public

capital, kg, which enters as an externality in the production technology and is taken by the

firm as given. The parameter α1 can be interpreted as the elasticity of output with respect

to public physical capital. A higher value of α1 implies a greater externality.4

The firm operates in a perfectly competitive environment taking goods and factor prices

as given. Its profit-maximization problem is essentially static, choosing the quantity of

private physical capital to rent from the household and the amount of labor to employ.

max{k,n}

π = Af(k, n;h, kg)− rk − whn, (14)

where whn is the firm’s wage bill with wh the unit labor cost, and rk is the rental

payment on private physical capital. The first-order conditions for the firm are:

fn = wh (15)

fk = r (16)

where fn and fk are the marginal products of labor and capital.

2.3 Government

The government levies four types of taxes: a tax τk on gross physical capital income, rk; a tax

τn on effective labor income, whn; a tax τc on consumption expenditures, c; and lump-sum,

non-distortionary tax, X. Total tax revenue for the government tax can be written as:

R +X = τkrk + τnwhn+ τcc+X, (17)

4The inclusion of the government spending in the production process was pioneered by Barro (1990). Inthis model, we assume that it is not the flow of government spending, but the stock of public capital, such asinfrastructure, that enters the production function. In this regard, we follow Futagami, Morita and Shibata(1993). See Atolia, Chatterjee and Turnovsky (2008) for a detailed discussion of the two forms of governmentspending contribution to output.

7

where τk, τn ∈ [0, 1), τc > 0. R represents the total tax revenue raised from distortionary

taxes.

This revenue is used to finance government spending on public capital (G) and to provide

transfers to the households. The government observes policy rules governing gross investment

as a share of output and lump-sum taxes as a share of output, which we write as:

G = G(y) (18)

and

X = X(y). (19)

Subject to these policy rules, the government runs a balanced-budget:

G = R +X − T (20)

Net investment in public physical capital is given by the government’s gross investment

less capital depreciation, or

kg = G− δgkg, (21)

where δg ∈ (0, 1) is the rate of depreciation of public capital.

2.4 Competitive Equilibrium

Equilibrium in the goods market is given by:

f(k, n;h, kg) = c+ k + δkk + kg + δgkg (22)

where output is allocated between consumption, gross investment in private capital, and

gross investment in public physical capital.

The dynamic path for the economy is found as the solution to the set of equations: (4),

(5), (8) – (22), given the economy’s initial state, k(0) = ko, h(0) = ho, kg(0) = kgo, which

satisfy the transversality conditions (6) and (7).

The model can reduced to a core dynamic system consisting of h, λ1, λ2, kg, and k that

are represented in the five differential equations (3), (4), (9), (10) and (21). It includes the

8

following endogenous variables r, w, y, π, G, X, T , R, c, l, m, and n which can be eliminated

from the system by using (5), (8), and (11) – (20). We then choose to replace the dynamic

shadow prices λ1 and λ2 in the core dynamic system by the control variables c and l using

(11) and (12), allowing us to solve the core system for the dynamic paths of c, l, k, kg, and

h.

3 Calibrating the Model for the Benchmark Balanced

Growth Path

Along the balanced growth path (bgp), the set of variables (c, k, kg, h) in the core system

are growing at constant, but not necessarily identical rates, while l is a constant. Of the

remaining variables, (G,X, T,R, π, w, y) are also growing at constant rates,5 while (r,m, n, l)

are constant.

As shown in the Appendix:

c

c=kgkg

=k

k=y

y=G

G=X

X=T

T= φ (23)

h

h= ν = (1− θ)φ (24)

w

w= φ− ν (25)

Note that equation (4) makes clear that the engine of growth is human capital; it drives

output growth. A lesser investment of time, m, in human capital will retard the economy’s

growth rate. However, the presence of public capital amplifies the effect of human capital on

output. That is, φ > ν when α1 > 0, or the higher the share of public capital in production,

α1, the wider is the gap between the growth rate of private physical capital and human

capital.6

5We note again that lump-sum taxes, X, may be zero.6We note that φ does not have an analytical solution in this model, but can be approximated arbitrarily

close with numerical methods.

9

3.1 Normalization

To calibrate the model, a normalization must be chosen that renders the model stationary.

We choose to normalize by the private physical capital stock, k. Given the different growth

rates for the system’s core variables, the normalization may also differ for the variables in the

model. Begin with c, kg, y, G, X, and T , which grow by the same rate, φ, as the normalizing

variable k, and define the normalized variables to be:

c =c

k, kg =

kgk, y =

y

k, G =

G

k, X =

X

k, R =

R

k, and T =

T

k. (26)

Human capital, h, grows at a rate ν. Therefore, a stationary normalization can be chosen

as:

h =h

1(1−θ)

k, (27)

The wage rate, w, grows at a rate (φ − ν), which suggests a stationary normalization

given by:

w =w

1θ

k(28)

We can now proceed with the calibration.

3.2 Calibration

The normalized model consists of fifteen equations: five from the household’s problem, con-

straints (4), the static Euler equation (5), and the co-state equations (8) – (10) (with the

multipliers eliminated); the output and profit functions (13) and (14) and two first-order

conditions from the firm’s problem, (15) and (16); four equations from the government sec-

tor, the accounting identity (17), two policy rules (18) and (19), and the balanced budget

(20); and goods market equilibrium (22). The fifteen endogenous variables are: m, l, n, r,

w, c, y, kg, h, T , X, G, R, φ, and ν. The model contains fourteen parameters: θ, µ, σ, α2,

δk, δh, δg, ρ, η, A, γ, and the tax rates τk, τn, and τc.

To solve the normalized model in steady state, we need fourteen constraints. We use

estimates from the previous literature and empirical data to get the values φ, α1, α2, δk, δh,

δg, τk, τn, τc, n, y and GR

. The normalized lump-sum tax, X, is set to zero and the scale

variable A is normalized to one. The solution determines the three deep parameters ρ, γ

10

and η that are consistent with the benchmark bgp. The discussion of the choice of unknown

parameters and key ratios follows.

Per capita output growth rate (φ). Over the 54-year sample period 1961-2005, the

average long-run growth rate of output per capita in the United States was approximately

2.2 percent. Over a longer time period from 1870 to 2000, Barro (2003) estimates a slightly

lower figure of 1.8 percent. We chose to use the Barro figure for our calibration.

Private physical capital share of output (α2). The most frequently assigned value for

α2 in the literature is 0.36. (See for example Kydland and Prescott (1982), Hansen (1985),

Cooley and Prescott (1995). A higher value of 0.4 was set by Cooley (1995) to account for

the imputed income from public physical capital. However, since the government is explicitly

set in our model, a value of α2 between 0.30 and 0.36 may be more appropriate. Accordingly,

a value of 0.337 is assigned for α2 based on a more recent work by Einarsson and Marquis

(1996). They arrived at this value using long-term empirical U.S. data for the period (1950-

1994) treating proprietor’s income as having the same share of labor income as the public

corporations.

Elasticity of output with respect to public physical capital [θ(1 − α2)]. Based on

the evidence provided by Gramlich, the elasticity of output with respect to public physical

capital, or, alternatively, the productivity elasticity of government spending, might range

from 0.1 to 0.2. We set θ(1 − α2) to 0.1 in our model. Which is essentially in the range

obtained in the empirical investigation. See Atolia, Chatterjee and Turnovsky (forthcoming).

And since α2 = 0.337, then the value of θ = 0.151.

Depreciation rate of private physical capital (δk). Atolia, Chatterjee and Turnovsky

(forthcoming) set this rate to 0.05, whereas Marquis and Einarsson (1996) set it to 0.0512 in

their calibrated dynamic general equilibrium model of endogenous growth. These are typical

values used in the literature and, following them, we set δk = 0.05.

Depreciation rate of human capital (δh). There is a considerable variation in the choice

of δh in the literature. For example, Haley (1976) uses a value as low as 0.005, whereas

Heckman (1976) sets it to 0.047. In a more recent work, Einarsson and Marquis (1996)

choose a value of 0.05 which is obtained as a rough average of the estimated values from

the literature on the labor market. We choose δh = 0.015. A detailed discussion about the

11

estimation of this rate can be found in Mincer and Ofek(1982). 7

Depreciation rate of public capital (δg). We take the value of 0.035 used by Atolia,

Chatterjee and Turnovsky (forthcoming). Note that the rate of depreciation of public phys-

ical capital is lower than the private physical capital. This captures the fact that public

physical capital is mostly infrastructure which depreciates at a slower pace than plant and

machinery.

Time allocated to production (n). The proportion of time allocated to work is often

computed in the literature to be the ratio of hours worked to total non-sleep time, which

is 0.357 for a 40-hour workweek, assuming total non-sleep time per day is 16 hours. See

Greenwood and Hercowitz (1991) and Einarsson and Marquis (2001). We set the steady-

state fraction of time allocated to work at n = 0.36.

y = 0.0039x + 2.0487R2 = 0.2077

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

1946

1949

1952

1955

1958

1961

1964

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

1997

2000

2003

2006

Years

k/y


Figure 1: Private Physical Capital to Output Ratio in the United States (1929-2006)

Private physical capital to output ratio (y). We set the steady-state value of y equal

to the inverse of the capital-labor ratio of 2.17 (or y = 0.461) This value is obtained from the

data for the private physical capital and gross domestic product in current dollar value in

National Income and Product Accounts (NIPA) tables prepared by the Bureau of Economic

Analysis (BEA) at the US Department of Commerce. The data in the tables covers the post

7We note that a change in δh will only affect the allocation of non-work time between accumulatinghuman capital and leisure. For example, if δh = 1%, then the values that will change in Table 1 are l, m,and η. They change as follows: lo = 0.452, mo = 0.188 and η = 0.74.

12

y = -0.0013x + 0.2329

0

0.05

0.1

0.15

0.2

0.25

0.3

1946

1949

1952

1955

1958

1961

1964

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

1997

2000

2003

2006

Years

G/R

(%)


Figure 2: Gross Public Investment to Total Tax Receipts Ratio in the United States (1929-2006)

WWII period (1946-2006). The actual data for private physical capital and GDP is shown

in Figure 1, and is seen to have experienced a modest upward trend.

Gross public investment to total tax receipts ratio (G/R). To estimate this ratio, we

again use the data for the post-WWII period (1946-2006) obtained from the BEA’s NIPA.

The average gross public investment to total tax receipts ratio for the period is 19.1%,

which is the number we use here. The actual data for the government’s current receipts and

expenditures is shown in Figure 2, where it is seen to be relatively absent of a trend in recent

years.

Tax rates (τk, τn, τc). For the gross tax rate on capital rental income and labor income,

Turnovsky (2000) set both rates to 28%. Here we set the benchmark values to 25%, following

the works of Greenwood and Hercowitz (1991) and Einarsson and Marquis (2001) who discuss

in greater detail the selection of their values. Regarding the consumption expenditure tax

rate, we set this rate as the pure U.S. sales tax rate currently, which is 7%.8

The calibration results are summarized in Table 1.

8Note that the τk is the tax rate on gross capital income. The tax rate on net capital income τk′ ismathematically given by (rk− δkk)τk′ = τkrk. Hence, τk′ = r

r−δkτk. Substituting for the steady-state value

of r in the above equation, we get τk′ = ρ+δk+µφρ+δk+µφ−δk(1−τk)τk. In our model, τk′ ranged from 34.5%-42.9%

depending on the value of σ, with an average of 38.1%. This number is close to empirical literature. Forexample, Mendoza et al. (1994) estimated it to be 41.5%. Since then, the tax rates on capital in the U.S.has fallen.

13

Table 1: Calibration to the Benchmark

Output Shares and Depreciation Rates α2 θ δk δh δg0.337 0.151 0.05 0.015 0.035

Data for Calibration k/y G/R n2.17 0.191 0.36

Government Policy RX τc τn τk0 7% 25% 25%

Deep Parameters of the Model ρ γ η4.85% 13.47% 0.67

Consumption and Time Allocations c l m0.366 0.415 0.225

State Variables h kg y c/y0.95 0.507 0.461 0.794

Human Capital & Output Growth ν φ1.53% 1.80%

4 Solving for Transitional Dynamics

Recall that the core dynamic system consists of c, l, k, kg, h. Except for l, all these variables

experience constant growth. In order to solve the model for the transition, it is necessary to

rewrite the dynamic system in terms of the normalized variables to ensure that the system

is stationary.

4.1 The Core Dynamic System

The equations governing the evolution of stationary variables can be obtained from equations

of the original dynamic system as follows:

˙c =

(c

c− k

k

)c (29)

14

˙kg =

(kgkg− k

k

)kg (30)

˙h =

(φ

ν

h

h− k

k

)h (31)

Using the functional form for utility (2) and equations (4), (5), (8) –(12), the dynamic

system can be rewritten as:

˙c =

[η(1− µ)(ρ+ δh − γ(1− l)) + η(1− µ)(γm− δh)− (η(1− µ)− 1) (ρ+ δk − (1− τk)r)

η(1− µ)2 + µ(η(1− µ)− 1)− k

k

]c

(32)

l

l=

(1− µ)(ρ+ δk − (1− τk)r) + µ(ρ+ δh − γ(1− l)) + µ(γ(1− l − n)− δh)η(1− µ)2 + µ(η(1− µ)− 1)

(33)

˙kg =

[1

kg(τkr + τnw

θh1−θn+ τcc+ X + T − δgkg)−k

k

]kg (34)

˙h =

[γ(1− l − n)− δh −

k

k

]h (35)

where (i) normalizing the household’s budget constraint (16) yields:

k

k= y − c− δk − τkr − τnwθh1−θn− τcc− X + T (36)

(ii) the firm’s normalized first-order conditions are:

r = Aα2

[(1− α2)(kg

θhn)(σ−1

σ) + α2

]( 1σ−1)

, (37)

w = A(1− α2)kgθ(σ−1

σ)h−

1σn

−1σ

[(1− α2)(kg

θhn)(σ−1

σ) + α2

]( 1σ−1)

, (38)

15

and (iii) the normalized goods market equilibrium condition yields:

n = α2

σσ−1

1+τc

1−τnηcl

A(1− α2)(kgθh1−θ

)σ−1σ

σ−1σ

− (1− α2)(kgθh1−θ

)σ−1σ

σ−1σ

. (39)

4.2 Analysis of the Accuracy of Approximation Methods

The common approach in the literature to solving models in which a lengthy transition from

one balanced growth path to another plays a critical role in the analysis is to approximate the

decision rules of the model with Taylor series expansions around the final balanced growth

path and run the transition from an initial point on the original balanced growth path.

This approximation introduces errors in estimated dynamic paths taken by the economy and

affects any welfare calculations that are usually important aspects of the analysis.

In this paper we wish to examine properties of the approximation within the context of

a calibrated model in which we alter the tax regime and estimate the dynamic path of the

model economy from one balanced growth path to another. We perform the analysis by first

solving the dynamic system for the exact transition path using the reverse-shooting algorithm

developed by Atolia and Buffie (2009). We then estimate the model using first- and higher-

order approximations to the decision rules and examine the errors present in the dynamic

paths of the economy’s key macroeconomic variables. These approximations are also solved

using reverse-shooting algorithm of Atolia and Buffie. We also compute the welfare gains

(losses) associated with the change in the tax regime for both the exact solution and the

approximations in order to gauge the magnitude of the approximation errors. A sensitivity

analysis is also conducted to examine the robustness of our results to reasonable variations

in the parameters of the model.9

5 Results

This section investigates the errors introduced by Taylor’s series approximations in the com-

putation of transition paths. The exact solution to the nonlinear system of equations that

comprise the core dynamic system is first obtained. This exact solution is then approximated

with first-, second-, third-, and fourth-order approximations of the decision rules. In the next

9Details of the solution algorithms used are available from the authors.

16

section, the sensitivity analysis examines 68 different cases in which various combinations

of key parameters of the model that have a range of reported values in the literature are

examined. These key parameters are the output elasticity of public physical capital, which

assumes values of α1 ∈ {0.01, 0.1, 0.2}, the elasticity of substitution between labor and pri-

vate physical capital in production, with values of σ ∈ {0.75, 1, 1.25}, and the intertemporal

elasticity of substitution, which is assumed to be µ−1 ∈ {0.5, 1}.

5.1 The Standard Case: α1 = 0.1, σ = 1, µ = 1

We begin the analysis by studying what we refer to as the standard case, which is identified

with the following parameterization: α1 = 0.1, σ = 1, µ = 1. This corresponds to the

midpoints of the ranges for α1 and σ and to the commonly used log-linear utility function.

The exercises begin with the economy on the benchmark balanced growth (bgp). The

benchmark tax code sets: τk = τn = 0.25, τc = 0.07 and X = 0. A new tax regime is then

implemented and a new balanced growth path is computed, along with the exact transition

path. Comparisons are made of the initial and final bgps and then welfare comparisons are

made that incorporate the transition dynamics. Four alternative tax regimes are studied,

where the federal tax revenues are shifted from the benchmark mix of labor income, capital

income, and consumption taxes to a single tax source of: lump-sum, consumption, labor

income, or capital income, where the ratio of government investment in public physical

capital to output is held fixed. Table 2 shows the new steady-state results of the economy

under the alternative tax regimes. These can be compared with the benchmark steady state

displayed in Table 1.

5.1.1 Across Steady States

Shifting the economy from distortionary taxes to a nondistortionary lump-sum tax regime

leads to slight changes in the new state of the core dynamic system variables relative to the

benchmark. Normalized consumption, c, drops from 0.366 to 0.300, or by 18% as a direct

effect of the 7% cut in consumption taxes, hence raising consumption, and as a result of

the increase in private physical capital due to the cut in capital income taxes by 25%. The

elimination of the 25% tax on labor income also affects c indirectly as there is an increase

in savings in the form of private physical capital. The lifting of income taxes encourages

the household not only to increase time allocated to labor, but also to increase the time

17

Table 2: New Steady State: Standard Cases

Lump-Sum Tax(30.6%)Consumption and Time Allocations c l m k/y

0.300 0.264 0.376 2.40


Human Capital & Output Growth ν φ ∆ζ3.6% 4.2% 9.8%

Capital Income Tax(90.6%)Consumption and Time Allocations c l m k/y

3.778 0.338 0.302 0.244


Human Capital & Output Growth ν φ ∆ζ2.57% 3.02% -39.4%

Labor Income Tax(46.1%)Consumption and Time Allocations c l m k/y

0.227 0.523 0.117 3.39


Human Capital & Output Growth ν φ ∆ζ0.08% 0.09% -7.5%

Consumption Tax(41.4%)Consumption and Time Allocations c l m k/y

0.267 0.383 0.257 2.771

State Variables h kg y c/y0.652 0.362 0.0.361 0.739

Human Capital & Output Growth ν φ ∆ζ1.96% 2.31% 4.7%

18

allocated toward the accumulation of human capital, both coming at the expense of leisure,

which drops by 36.5%. As a result, the growth rate of human capital growth more than

doubles from 1.54% to 3.6% along the new balanced growth path.

The comparison of the benchmark tax regime with the lump-sum tax regime facilitates the

analysis of distortions that can arise when taxes are shifted to a single source of tax revenue.

The welfare costs of the current regime, measured as the percentage gain in consumption

associated with the removal of the tax distortions is shown in Table 2 to be ∆ζ = 9.8%.

When all taxes are raised from capital income, dramatic shifts in the decision rules of

households significantly alter the core dynamic system variables relative to the benchmark

tax regime. In this case, maintaining the benchmark ratio of public physical capital invest-

ment to output (G/y) requires a 90.6% marginal tax rate on capital income. As a result

of both the higher capital income tax and the elimination of the consumption tax, there is

a sharp decline in personal savings, and normalized consumption, c increases by more than

11 times the benchmark value from 0.366 to 3.778. The household also allocates more time

to human capital accumulation at the expense of leisure, which falls by 18.6%. The human

capital growth rate increases to 2.57% thereby inducing an increase in the growth rate of

output to 3.02%. However, the dramatic decline in private physical capital results in a steep

drop in welfare. Welfare losses are measured to be ∆ζ = −39.4%.

Replacing the current tax regime is with a labor-income-tax-only regime necessitates a

marginal tax rate on labor income of 46.1%. In this case, c decreases by 32.4% and the

time allocated to the accumulation of human capital is nearly cut in half from 0.225 to

0.117 (48%). This decline in time devoted to building human capital is enough to nearly

stagnate the economy. The growth rate of human capital falls to only 0.08% and output

growth declines to 0.09%. The normalized stock of human capital, h, falls sharply with a

decline of 54.3% along the new balanced growth path compared to the benchmark. The

private physical capital to output ratio, k/y, increases by 56.2% due to the sharp decline in

output. The subsequent decline in time allocation to labor and human capital accumulation

induces a large increase in leisure (52.3%). A shift to the labor-income-tax-only regime also

generates welfare losses of ∆ζ = −7.5%.

The final fiscal policy scenario that we examine is a shift to a consumption-tax-only

regime. The marginal tax rate on consumption is increased from 7% in the benchmark to

41.4%, or about 6 times. Taxing consumption does not adversely affect the growth rate

adversely as was the case with the labor tax; the household is encouraged to accumulate

19

savings and allocate less time to leisure and more time to accumulating human capital. In

this case, economic growth increases 2.31% from the benchmark value of 1.80%. Though

not shown in the two states of our normalized model, the size of the economy is expected

to be the biggest when the government punishes consumption than when it punishes the

accumulation of saving and human capital. c decreases by 20.5% in the new state, and the

time allocated to leisure decrease by 7.7% and that allocated to human capital accumulation

increases by 14.2%. The result is an improvement in welfare relative to the current tax

regime, with ∆ζ = 4.7%.

The question of whether linearization and other higher-order approximations are close

to the actual solution is irrelevant when comparing balanced growth paths. However, the

approximations may become important in two situations: explaining the response of the

household to the new balanced growth path during the transition, and in the computation

of cumulative welfare gains or losses along the transition paths. These issues are explored

next.

5.1.2 Transitional Dynamics: the Exact Solution

The welfare consequences of any policy change is affected by the transitional dynamics.

Approximation errors in establishing the transition paths may result in seriously misrepre-

senting the welfare consequences of the policy change. The greater the policy change, the

more significant the transition paths are likely to be, and the greater the welfare compu-

tations may be in error. We illustrate these problems with the standard parameterization

of the model by computing the actual (exact) transition path using the reverse-shooting

algorithm and comparing the results with transitions computed using first-, second-, third-,

and fourth-order approximations. We report these results in Table 3.

The transition to a new balanced growth path following a policy change can be quite

lengthy. Policy changes that are beneficial in the long run may yield welfare losses in the

short run. We illustrate this phenomenon in the first column of Table 3 labeled “Actual.”

The symbol ζ represents the cumulative (discounted) welfare gains for ζ > 1 or losses ζ < 1

at various time horizons following the switch from the benchmark tax regime to the four

alternative tax regimes described above labeled LS (lump-sum), τk (capital-income-tax only),

τn (labor-income-tax only), and τc (consumption-tax only).

In the initial year following the policy change, welfare losses accrue for the lump-sum,

labor-income-only, and consumption-only tax regimes, while welfare gains accompany the

20

capital-income-only tax regime. The reason for these results is the strong substitution effects

in both the consumption-savings decision and the time allocation decisions. Removing the

disincentives to save, work, and invest in human capital, as is true of the lump-sum and

consumption-only cases lowers consumption and leisure in the current period, thus reducing

welfare early on. The effect is slightly more pronounced in the lump-sum case which lacks

the disincentive of the consumption tax. In the capital-only tax regime the disincentives for

saving, working, and investing in human capital are strengthened and the household chooses

instead to increase consumption and leisure immediately, thus increasing welfare in the early

stages of the transition. The labor-income-only tax also produces welfare losses in the early

years of the tax regime change despite the fact that by punishing work effort and investment

in human capital, the household can increase leisure to enhance welfare. The reason is the

sharp decline in consumption. The decline in output due to the decline in labor that tends

to reduce consumption is exacerbated by the labor income tax distortion that induces a

substitution of capital for labor in production, further creating a cutback of consumption.

These patterns shift remarkably over time. The last panel of Table 3 indicates the

long-run welfare gains and losses. The lump-sum tax regime realizes the full benefit of

removal of the tax distortions and has the highest welfare gains. The consumption-only tax

also produces welfare gains, as the incentives for saving and investing in human capital are

alleviated. The most punitive tax alternative is the capital-income-only tax where the early

welfare gains associated with a short-term consumption boom are more than offset in the

long-run due to a lack of savings. The labor-income-only tax also produces welfare losses,

due to the disincentive to invest in human capital, however the losses are not as great as

those resulting from the capital-income-only tax.

5.2 Transitional Dynamics: Higher-Order Approximations

We are now ready to examine the magnitude of the approximation errors in computing the

welfare effects of a fiscal policy change. The welfare gains and losses are computed for the

first-, second-, third-, and fourth-order approximations of the core dynamic system. Using the

same metric as in in the previous subsection, the cumulative (discounted) values at various

time horizons are reported in Table 3 as ζ1, ζ2, ζ3, and ζ4. The errors are then compiled as

a percentage relative to the actual numbers found under the exact solution. These errors

are labeled ε1, ε2, ε3, and ε4 in the Table 3. The entire time-horizon calculations for each of

these exercises are also displayed in Figures 3-6. Note that the horizontal dotted grid points

21

demarcate the long-run welfare gains and losses to which the model solutions converge for

various orders.

We begin by noting the lack of monotonicity in reducing errors as you work either down

the columns of Table 3, thus approaching the computed long-run welfare gains or loses, or

across the rows of Table 3, thus increasing the order of the approximation. The first of these

observations is not too surprising, since the approximation errors could be switching from

positive to negative or vice versa as the time horizon lengthens, thus tending to cancel out

over some periods and accumulate over others. For the three time horizons reported in Table

3, only consumption taxes show a uniform increase in the percentage errors in the computed

cumulative welfare gains or losses as the time horizon lengthens. In contrast, the errors

computed in capital-income-tax-only case are highest in the initial period (first panel) for

the first-order approximation, in the long-run (third panel) for the second- and third-order

approximations, and after 50 years (middle panel) for the fourth-order approximation.

Table 3: Welfare Results for Different Taxation Policies - Standard Cases

Actual 1st Order 2nd Order 3rd Order 4th Order

τ ζ ζ1 ε1 ζ2 ε2 ζ3 ε3 ζ4 ε4Initial Response

LS 0.6387 0.6412 0.39 0.6382 0.07 0.6388 0.02 0.6386 0.01τk 1.3153 3.5185 167.49 2.7574 109.63 2.3770 80.71 2.1334 62.19τn 0.9297 0.9506 2.25 0.9340 0.46 0.9259 0.42 0.9311 0.15τc 0.8097 0.8159 0.76 0.8099 0.02 0.8095 0.02 0.8097 0.00

50 YearsLS 0.9791 0.9262 5.41 0.9862 0.72 0.9771 0.21 0.9798 0.07τk 0.5981 0.3047 49.04 3.9387 558.59 3.0487 409.78 2.4866 315.79τn 0.9914 0.6737 32.05 0.9286 6.34 1.0088 1.75 0.9906 0.09τc 1.0173 1.0025 1.45 1.0094 0.77 1.0180 0.07 1.0173 0.00

750 YearsLS 1.0978 1.0350 5.72 1.1061 0.75 1.0971 0.06 1.1003 0.22τk 0.6056 0.3131 48.30 4.1680 588.25 3.2161 431.07 1.4098 132.79τn 0.9246 0.6114 33.87 0.8635 6.60 0.9445 2.16 0.9274 0.31τc 1.0470 1.0308 1.55 1.0387 0.79 1.0504 0.32 1.0496 0.25

22

0 50 100 150Years

-0.4

-0.2

0.0

0.2

Ζ

Figure 3: The Transition Path of ζ for a Shift to Lump-Sum Taxation

Perhaps more surprising, increasing the order of the approximation does not necessarily

improve the accuracy of the welfare calculations. For example, the largest approximation

errors in the capital-income-tax- only case were for the second-order approximation, not the

first-order. While a sufficiently high-order approximation may eventually yield satisfactory

error limits, each exercise is likely to require a different order. Table 3 suggests that for this

model an accuracy of within one percent error in the long-run calculation (bottom panel)

requires at least a second-order approximation for the lump-sum-tax and consumption-tax-

only regimes, while the labor-income-tax-only regime requires a fourth-order approximation,

and the capital-income-tax-only regime could not meet the criterion even with a fourth-order

approximation, which produced an errror of 133 percent.

23

0 50 100 150Years

0

1

2

3

4Ζ

Figure 4: The Transition Path of ζ for a Shift to Capital Income Taxation

0 50 100 150Years

-0.4

-0.2

0.0

0.2

Ζ

Figure 5: The Transition Path of ζ for a Shift to Labor Income Taxation

24

0 50 100 150Years

-0.4

-0.2

0.0

0.2

Ζ

Figure 6: The Transition Path of ζ for a Shift to Consumption Expenditure Taxation

6 Sensitivity Analysis

This section examines how the approximation errors discussed in the ”standard model” vary

with changes in key deep parameters. We vary the output elasticity of public capital over

the range, α1 ∈ {0.01, 0.1, 0.2}, the elasticity of substitution between labor and private

physical capital over the range of σ ∈ {0.75, 1, 1.25}, and the intertemporal elasticity of

substitution in consumption between µ−1 ∈ {1/2, 1}. Changes in these deep parameters

require a recalibration of the model which results in corresponding adjustments in the values

of other deep parameters. The calibration also alters some characteristics of the initial

balanced growth path.

6.1 Calibrations and New Tax Rates under Alternative Tax Regimes

For these exercises, we maintain the current set of marginal tax rates: 25 percent for both

capital and labor income and 7 percent on consumption expenditures. A change in either of

the deep parameters σ and µ requires a new calibrated value for the household’s discount rate,

ρ. These values are displayed in Table 4 for each of the 18 combinations of deep parameters

examined. In general, an increase in the elasticity of substitution between private physical

25

capital and labor increases the marginal product of capital and thus the real rental rate.10

This higher real rental rate necessitates an increase in the calibrated value of ρ. Similarly, a

decrease in the intertemporal elasticity of substitution in consumption, or a rise in µ, requires

a higher calibrated value for ρ, given the real rental rate on private physical capital.

For the new tax regimes, the growth rate of the economy, φ, tends to be higher when there

is greater flexibility in production with a higher elasticity of substitution between private

physical capital and labor, σ, and a more pronounced public capital externality, or a higher

α1. Likewise, an increase in the willingness of household to substitute utility through time,

or a lower µ, increases the growth rates under the new tax regimes. We note that the effect

on growth of shifting to the alternative tax regimes under the various parameterizations is

consistent with the results reported in Table 2 for standard cases. For example, the average

growth rates across all parameterization under the alternative tax regimes, reported at the

bottom of Table 4, indicate that the regimes can be ranked by growth rates from high to low

as: lump-sum tax (φ = 3.23%), capital-income-tax-only (φ = 3.02%), consumption-tax-only

(φ = 2.03%), and labor-income-tax-only (φ = 0.94%).

10See Klump and De La Grandville(2000) and Papageorgiou and Saam(2005) for a full discussion of thisissue.

26

Table 4: Tax Rates For All Taxation Instruments Scenarios

Parameters Lump-Sum Capital Tax Labor Income Tax Consumption Taxσ α1 µ ρ φ φ τk φ τn φ τc

0.75 0.01 1 2.20 2.02 1.95 70.70 1.74 40.31 1.83 40.3190.75 0.10 1 2.20 2.16 2.04 70.37 1.70 40.29 1.89 40.3390.75 0.20 1 2.20 2.37 2.18 69.87 1.64 40.27 1.94 40.3700.75 0.01 2 0.40 1.90 1.87 70.76 1.77 40.31 1.82 40.2820.75 0.10 2 0.40 1.94 1.89 70.56 1.76 40.30 1.83 40.2800.75 0.20 2 0.40 1.99 1.92 70.33 1.75 40.29 1.84 40.2781.00 0.01 1 4.85 3.76 3.27 57.27 0.44 46.09 2.22 41.2941.00 0.10 1 4.85 4.19 3.60 56.91 0.09 46.09 2.22 41.2941.00 0.20 1 4.85 4.89 4.11 56.38 (0.52) 46.09 2.22 41.2941.00 0.01 2 3.05 2.47 2.30 57.83 1.55 46.09 1.93 40.9911.00 0.10 2 3.05 2.53 2.34 57.73 1.53 46.09 1.94 40.9881.00 0.20 2 3.05 2.60 2.39 57.60 1.52 46.09 1.95 40.9851.25 0.01 1 6.80 5.14 4.67 52.79 1.15 53.42 2.24 42.1321.25 0.10 1 6.80 5.87 5.26 52.90 0.10 54.51 2.35 42.1991.25 0.20 1 6.80 7.04 6.22 53.13 (0.95) 55.30 2.55 42.3151.25 0.01 2 5.00 2.83 2.74 52.62 0.61 54.93 1.91 41.7991.25 0.10 2 5.00 2.90 2.79 52.66 0.54 55.07 1.92 41.7961.25 0.20 2 5.00 2.99 2.86 52.68 0.45 55.24 1.93 41.792Average 3.71 3.31 3.02 60.17 0.94 47.04 2.03 41.153

27

Table 5: Average Approximation Errors at Various Time Horizons

Time Horizon 1st-Order 2nd-Order 3rd-Order 4th-Order0 2.81 0.42 0.25 0.04

(2.83) (0.88) (1.06) (0.12)50 5.96 3.04 1.19 0.49

(8.05) (7.62) (4.72) (2.58)100 6.10 3.14 1.22 0.52

(8.26) (7.84) (4.83) (2.72)200 6.14 3.14 1.19 0.54

(8.28) (7.83) (4.81) (2.74)750 6.14 3.15 1.19 0.54

(8.28) (7.83) (4.81) (2.74)

Average 5.43 2.58 1.01 0.42(7.53) (7.01) (4.30) (2.41)

6.2 Fiscal Policy Instruments

Due to the large number of cases examined (68) in this sensitivity analysis, we report the

results in the form of summary statistics.11 We begin by computing the average of the

absolute values of the errors in the calculation of welfare gains (losses) across all cases for each

order of the approximation using the same metric as in Section 5.2. The results at various

time horizons during the transitions to the new balanced growth paths are reported in Table

5. The number in parenthesis is the standard deviation of the mean errors. We note two

broad patterns that emerge in these statistics. First, the higher is the order of approximation,

the lower tends to be the errors, with fourth-order approximations producing error estimates

whose average is roughly one-half of one percent or less at all time horizons. As we know

from the previous Section 5, these results are not uniform across all tax regime changes,

with the capital-income-tax-only cases contributing disproportionately to these estimates.

Second, we find that the average error estimates tend to accumulate over time, with rough

convergence for each order of approximation achieved after about 100 periods. These results

11We note that four of the labor-income-tax-only regime cases reported in Table 4 are extreme, withnear zero or negative growth rates. These cases are deleted from the subsequent sensitivity analysis. Morecomplete results are available from the authors on request.

28

are also consistent with the lengthy transition periods required to absorb the dramatic fiscal

policy changes examined in this model.

To examine how these results likely vary across the alternative fiscal policy regimes, we

compute for each of the alternative tax regimes, the average approximation errors across all

cases after convergence to the new long-run balanced growth path is achieved. The results

(with standard deviations in parentheses) are reported in Table 6. Two broad patterns

emerge. The first is the general tendency of the errors to fall as the order of the approximation

increases. However, a monotonic decline in the long-run approximation errors with the order

of approximation does not always occur. For the capital-income-tax-only case, we find that

the second-order approximation results in larger long-run errors in estimating the welfare

gains (or losses) than the first-order approximation. This pattern had previously revealed

itself in the standard case, as reported in Table 3.

Table 6: Approximation Errors by Tax Regime

1st-Order 2nd-Order 3rd-Order 4th-OrderPolicy instrument (LS, τk, τn, τc)

LS 7.34 1.27 0.19 0.13(11.51) (2.82) (0.34) (0.25)

τk 6.93 8.49 3.80 1.81(7.47) (13.73) (8.98) (5.22)

τn 7.00 1.58 0.50 0.08(9.10) (1.96) (0.83) (0.14)

τc 3.48 0.90 0.13 0.05(2.80) (0.69) (0.11) (0.08)

Average 6.14 3.15 1.19 0.54(8.28) (7.83) (4.81) (2.74)

The second pattern that is displayed in Table 6 is that order of approximation needed to

reduce the errors to within acceptable limits varies with tax regime. For example, to reduce

the average approximation errors to less than one percent in this model requires second-

order approximations for the consumption tax regime, third-order for the lump-sum and

labor-income-tax-only regimes, and even fourth-order is insufficient for the capital-tax-only

regime. These results reinforce the conclusion in Section 5 from the standard cases that

the order or approximation required to obtain an acceptable level of accuracy is likely to be

29

model- and experiment-specific.

We next examine how the approximation errors vary with changes in the deep parameters.

6.3 Deep Parameters

In this section we alter the three key deep parameters that are likely to be important to

welfare analysis – the elasticity of substitution between private physical capital and labor in

production, σ, the public capital externality, α1, and intertemporal elasticity of substitution,

µ−1, – and examine the impact they have on the average approximation errors across all

regime changes.

6.3.1 Elasticity of Substitution between Private Physical Capital and Labor

We begin with the elasticity of substitution between private physical capital and labor, which

takes on values of σ ∈ {0.75, 1.00, 1.25}. Table 7 reports the mean and standard deviation

(in parentheses) of the approximation errors in the computation of the welfare gains (and

losses) by order of approximation.

Table 7: Approximation Errors by the Elasticity of Substitution between Private PhysicalCapital and Labor (σ)

σ 1st-Order 2nd-Order 3rd-Order 4th-Order0.75 1.91 0.73 0.13 0.03

(1.12) (0.80) (0.20) (0.05)1.00 6.66 2.24 0.57 0.23

(8.16) (3.49) (1.06) (0.53)1.25 10.41 6.90 3.08 1.46

(10.67) (12.97) (8.41) (4.85)Average 6.14 3.15 1.19 0.54

(8.28) (7.83) (4.81) (2.74)

The results indicate that the greater the substitution between private physical capital

and labor, the larger will be the approximation errors. This finding is consistent with the

suggestion that greater flexibility in the production process is likely to induce greater re-

sponses of factor demands to given policy changes that increase the likelihood of errors in

30

approximating the dynamic response of the economy to those changes, thus resulting in more

significant computational errors.

As the table shows, these errors tend to fall with the order of the approximation. How-

ever, the minimum order of approximation varies with the choice of σ. With reference to a

maximum error tolerance in the computation of welfare gains (losses) of one percent, a low

σ of 0.75 requires a second-order approximation, a σ of one requires a third-order approx-

imation, and the higher σ of 1.25 implies a failure to satisfy the minimum error tolerance

on average for even fourth-order approximations. This is clearly a parameter that can con-

tribute significantly to misrepresenting the welfare consequences of policy experiments where

factor demand decisions play an important role in the transitional dynamics.

6.3.2 Public Capital Externality

Table 8 displays the results of the sensitivity analysis that varies the public capital externality

over the range α1 ∈ {0.01, 0.2, 0.2}. The results resemble those for the σ in that an increase in

the externality that has a greater effect on the factor employment decisions tends to increase

the magnitude of the approximation errors. Again, the order of approximation that would

satisfy on average the one percent maximum error varies with the parameter value. For lower

values of α1 of 0.01 or 0.1, third-order approximation is required. For the higher value of α1

equal to 0.2, even fourth-order is insufficient. These results suggest that researchers should

exercise caution when interpreting the welfare analysis of fiscal policy reforms in models in

which a public capital externality is present.

Table 8: Approximation Errors by Public Physical Capital Externality, α1

α1 1st-Order 2nd-Order 3rd-Order 4th-Order0.01 4.31 1.82 0.42 0.07

(4.23) (3.76) (1.01) (0.08)0.10 5.62 2.97 0.94 0.33

(7.15) (6.55) (2.63) (1.03)0.20 8.61 4.72 2.27 1.25

(11.66) (11.45) (7.98) (4.69)Average 6.14 3.15 1.19 0.54

(8.28) (7.83) (4.81) (2.74)

31

6.3.3 Intertemporal Elasticity of Substitution

The final parameter over which we conduct a sensitivity analysis is the intertemporal elas-

ticity of substitution, which is varied from µ−1 ∈ {0.5, 1}. The results are displayed in Table

9.

Table 9: Sensitivity Analysis: Intertemporal Elasticity of Substitution

µ−1 1st −Order 2nd −Order 3rd −Order 4th −Order0.50 4.76 0.94 0.26 0.06

(3.47) (0.98) (0.48) (0.12)1.00 7.70 5.63 2.24 1.08

(11.39) (10.92) (6.90) (3.96)Average 6.14 3.15 1.19 0.54

(8.28) (7.83) (4.81) (2.74)

When the intertemporal elasticity of substitution is very low, households respond less in

their resource allocation decisions to changes in the tax distortions. This factor mitigates

the changes from one balanced growth path to another and lessens the role of transitional

dynamics in computing welfare gains (losses). For example, as the Table 9 suggests, a

relatively low value for the intertemporal elasticity of substitution of µ−1 = 0.5, requires a

second-order approximation for the average error in computing the welfare gains (losses) in

this model to be within a one percent maximum. In contrast, logarithmic utility (µ = 1)

requires a fourth-order approximation. This imprecisely measured preference parameter

can significantly alter the inferences one might draw from policy experiments of the type

described in this paper.

7 Conclusions

This paper examines the accuracy of computing transitional dynamics associated with fiscal

policy reforms with linear and higher-order Taylor’s series expansions of a core economic

model in which there exists a public capital externality and growth that is determined en-

dogenously as in Lucas (1988) and Uzawa (1966). By comparing the approximations of

the post-policy reform transition paths and the welfare implications that they engender

32

with those obtained from exact characterizations of the transitional dynamics found by the

reverse-shooting methods described in Atolia and Buffie (2007), we find that the traditional

approximation methods employed in performing policy analysis may induce significant errors

that could lead to erroneous conclusions regarding the impact of fiscal policy reforms.

Moreover, these problems may not be easily minimized by resorting to ever-higher ap-

proximations. The order of approximation required to obtain approximation errors that

may be seen as acceptable (even when this minimum error in the estimation of welfare gains

(losses) as large as one percent) varies widely with the type of fiscal policy reform under-

taken, the time horizon of the analysis, and the setting of deep parameters about which little

may be known. In some cases, we find that even a fourth-order approximation of the core

dynamic system is insufficient to come within the error bounds that any researcher would

likely deem to be acceptable.

While these results were obtained for a particular model economy, there is no reason to

believe that similar problems will not be present in other models. We believe that these

results we report in this paper strongly encourage the exercise of caution when undertak-

ing an analysis whose conclusions require computations that entail the approximation of a

significant set of transitional dynamics.

33

A SOLVING FOR THE BALANCED GROWTH PATH

Along the balanced growth path (bgp), all variables of interest (c, h, k, kg, y and w) grow

at constant rates which may differ across variables.

In addition, as it appears from the time allocation constraint in (5), the time allocated

to leisure, accumulation of human capital and labor remains constant along bgp, i.e.,

l

l=m

m=n

n= 0 (A.1)

Let the growth rate of consumption be φ, i.e.,

c

c= φ, (A.2)

and the growth rate of human capital accumulation be ν, i.e.,

h

h= ν (A.3)

We begin by showing that interest rate is constant along the bgp. We proceed in steps.

After substituting the functional form of u(.) given in (2) into (8), we get

(clη)−µ lη = (1 + τc)λ1 (A.4)

Log differentiating (A.4) and using (A.1), we obtain

λ1

λ1

= −µcc

= −µφ (A.5)

Substituting (A.5) in the costate equation of the private physical capital (9) we get:

r =ρ+ δk + µφ

(1− τk), (A.6)

which implies that r is constant along the bgp, or that

r

r= 0 (A.7)

34

To solve for bgp and the growth rate of other variables, we begin by obtaining the opti-

mality condition for the labor-leisure choice of the household by rearranging (8):

uc

(1− τn1 + τc

)wh = ul. (A.8)

The left-hand side of equation(A.8) is the marginal utility of extra consumption that the

household can enjoy or obtain by working more. To see that, note that by working for

an extra hour, the household earns a net wage of (1 − τn)wh. Which allows it to have

(1− τn)wh/(1 + τc) units of consumption which yields an extra utility of uc(1− τn)wh/(1 + τc).

The right-hand side of (A.8) is the marginal utility cost of reduction in leisure, ul. The opti-

mality of the household decision requires that the marginal cost equal the marginal benefit

of the decision.

Substituting for the functional form in (A.8), we obtain:(1− τn1 + τc

)whl = ηc (A.9)

Log differentiating (A.9), we get:

c

c=w

w+h

h+l

l. (A.10)

Using equations (A.1), (A.2) and (A.3) in equation (A.10), we get the growth rate of w as:

w

w= φ− ν. (A.11)

To compute the growth of the remaining variables, the first-order conditions of the firm

are used. We begin by substituting the functional form of the production function (13) in

(15) to obtain:

wh = A(1− α2)k( θσ−θ

σ)

g h(σ−1σ

)n−1σ

[(1− α2)(kg

θhn)(1− 1σ

) + α2k(1− 1

σ)]( 1

σ−1), (A.12)

logw = log (1− α2) + (θσ − θσ

) log kg −1

σlog h− 1

σlog n+

1

σlog y (A.13)

Differentiating and using equations (A.1), (A.3) and (A.11)

φ− ν =1

σ

y

y+ (

θσ − θσ

)kgkg− ν

σ(A.14)

35

To find the growth rate of public capital along the bgp, substitute out G from (20) using

(21) to obtain:

kg = R− T − δgkg. (A.15)

Rearranging and dividing (A.15) by y gives:

kgy

(kgkg

+ δg

)=R− Ty

. (A.16)

The terms in the parenthesis in the left-hand side of equation (A.16) and the right-hand side

are both constant. Therefore, kg grow at the same rate as k and y along the bgp, i.e.,

kgkg

=y

y= ψ (A.17)

Using equation (A.17) in equation (A.14), we get the growth rate of output as:

y

y=σφ− σν + ν

1− θ + σθ= ψ (A.18)

which implies that output grows at the same rate as consumption only if both the elasticity

of substitution and intertemporal elasticity of substitution equal one.12

Similarly, substituting the functional form of the production function (13) in (16), we

get:

r = Aα2k−1σ

[(1− α2)(kg

θhn)(1− 1σ

) + α2k(1− 1

σ)]( 1

σ−1)(A.19)

log r = logα2 −1

σlog k − 1

σlog y (A.20)

Therefore:k

k=y

y=kgkg

= ψ. (A.21)

Clearly, the private physical capital grows at the same rate as output and public physical

capital. Using the budget constraint in (3), it could be easily shown that consumption also

grows at the same rate and the private physical capital. Therefore:

ψ = φ (A.22)

12The relation between φ, ν and ψ will be explored shortly

36

Using this fact in (A.18) and (A.22), we obtain the relationship between the growth rate of

human capital and that of output as:

ν = (1− θ)φ (A.23)

After determining the growth rates of all the variables in the model, the model needs to

be normalized before calibrating it to the benchmark. Before this, we determine the growth

rates of λ2 and λ3. Using the first order condition for l in (2) and using (A.1), the growth

rate of λ3 is:λ3

λ3

= (1− µ)φ (A.24)

This equation implies that the growth rate of λ3 may be zero, less or greater than that of

the consumption, though in an opposite direction. 13 Similarly, to obtain the growth rate of

λ2, we make use of the first household first order condition of m in (2) in addition to (A.3)

and (A.24):

λ2

λ2

= (1− µ)φ− ν (A.25)

13Note that for µ = 1, which reflects the logarithmic utility function, λ3 does not grow.

37

B CORE DYNAMIC SYSTEM

Recall from the first order conditions that the functional forms of the equations are as follows:

(clη)−µ lη = (1 + τc)λ1 (B.1)

(clη)−µ ηclη−1 = λ2γh (B.2)

Rearranging, we obtain:

c−µlη(1−µ) = (1 + τc)λ1 (B.3)

ηc(1−µ)l(η(1−µ)−1) = λ2γh (B.4)

Taking log and differentiating we get:

− µcc

+ η(1− µ)l

l=λ1

λ1

(B.5)

(1− µ)c

c+ (η(1− µ)− 1)

l

l=λ2

λ2

h

h(B.6)

Multiplying (B.5) by (1− µ) and (B.6) by µ and summing the two equations to eliminate c

we get: [µ(η(1− µ)− 1) + η(1− µ)2] l

l= µ

λ2

λ2

h

h+ (1− µ)

λ1

λ1

(B.7)

Substituting for the values of hh, λ1

λ1and λ2

λ2from equations (4), (9) and (10) respectively, we

get the transitional dynamics equation for l as follows:

l =

[(1− µ)(ρ+ δk − (1− τk)r) + µ(ρ+ δh − γ(1− l)) + µ(γ(1− l − n)− δh)

η(1− µ)2 + µ(η(1− µ)− 1)

]l (B.8)

To get the transitional dynamic equation for c we follow a similar logic. First multiply (B.5)

by (η(1 − µ) − 1) and (B.6) by η(1 − µ) and summing the two equations to eliminate c we

get: [µ(η(1− µ)− 1) + η(1− µ)2] c

c= η(1− µ)

λ2

λ2

h

h+ (η(1− µ)− 1)

λ1

λ1

(B.9)

38

To get the transitional dynamics equation for c, we proceed same as l above. Substituting

for the values of hh, λ1

λ1and λ2

λ2from equations (4), (9) and (10).

c

c=η(1− µ)(ρ+ δh − γ(1− l)) + η(1− µ)(γm− δh)− (η(1− µ)− 1) (ρ+ δk − (1− τk)r)

η(1− µ)2 + µ(η(1− µ)− 1)(B.10)

˙c =c

cc (B.11)

The other two core system equations is straightforward. For the public capital, kg, it is simply

the government investment equation normalized the private physical capital. Whereas the

human capital transitional dynamics was in our model setup.

˙kg =

[1

kg(τkr + τnw

θh1−θn+ τcc+ X + T − δgkg)−k

k

]kg (B.12)

˙h =

[γ(1− l − n)− δh −

k

k

]h (B.13)

39

.

Abstract

There is a long-standing debate in the literature on the choice between consumptionor expenditure taxes versus capital income taxes that goes back to Thomas Hobbes(1651), Mill (1871) and later Kaldor (1955) who advocated the consumption tax overthe income tax. The advocacy of consumption tax has its solid empirical evidence assome studies indicated that the tax revenue collected in the United States includes arelatively small contribution coming from capital tax (Roger Gordon, Laura Kalam-bokidis, Jeffrey Rohaly and Joel Slemrod (2004)). This paper examines tax policyin an endogenous growth model with public capital externality, where human capitalserves as the engine of growth. In a companion paper, this model was calibrated tothe U.S. economy and experiments were run to calculate welfare gains from a shift inthe fiscal regime from the current mix of capital income, labor income, and consump-tion taxes to complete reliance on consumption tax. In those experiments, governmentexpenditures in public capital as a share of output was held fixed. The paper showedthat the consumption-only tax regime was superior to the current tax regime and toother tax regimes relying solely on a single source of taxation. In this paper, the gov-ernment tax revenues as a portion of output are varied in order to find the optimallevel of investments in public capital under a consumption-only tax regime. I find thatin the presence of a significant externality, a modest increase in the consumption taxwith a greater investment in public capital can increase welfare. I also show that aslight shift in taxes from consumption to capital income can be welfare improving ifthe externality is high enough.

Keywords: Consumption tax, public capital externality, endogenous growth, welfare,public and human capital

JEL Codes: O41, C61, C63

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