Optimal taxation under equilibrium unemployment and economic profits

Wei Jiang

doi:10.1515/bejm-2017-0226

Published by De Gruyter October 12, 2018

Optimal taxation under equilibrium unemployment and economic profits

Wei Jiang

From the journal The B.E. Journal of Macroeconomics

https://doi.org/10.1515/bejm-2017-0226

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Abstract

This paper develops a heterogeneous agent model with equilibrium unemployment and economic profits due to productive public investment. We find that the presence of profits plays an important role in the determination of long-run optimal tax policy. The Judd-Chamley optimal zero capital tax result can still hold in the model without profits. In this case, the optimal labour wedge is zero in the long run, resulting in welfare gains for all agents and no conflict of interests between agents. But the Benthamite government chooses to subsidise capital income in the long run in the model with economic profits. The resulting labour wedge is non-zero which generates welfare losses of workers despite welfare gains of capitalists. The government also faces a trade-off between efficiency and equity in this case.

Keywords: economic profits; equilibrium unemployment; household heterogeneity; optimal taxation

JEL Classification: E13; E22; E62

Acknowledgements

I would like to thank Konstantinos Angelopoulos, Jagjit Chadha, Jilei Huang, Miguel León-Ledesma, James Malley, and participants and discussants at the University of Glasgow seminar, University of Kent Internal Workshop, China Meeting of the Econometric Society in Beijing and Royal Economic Society Annual Conference at Manchester. I would also like to thank the anonymous referee for helpful comments and suggestions. The remaining errors are mine.

A Appendix

A.1 Optimisation problem of capitalists

The optimisation problem of the capitalist can be expressed mathematically as follows:

max{Ctk,Kt+1k}t=0∞{∑t=0∞βt[μ(Ctk+ωG¯tc)σ−1σ+(1−μ)(1−0)σ−1σ]σσ−1}s.t. Ctk+Kt+1k−(1−δp)Ktk=rtKtk−τtk(rt−δp)Ktk.

The Lagrangian function of the capitalist is then written as:

Lk=∑t=0∞βt{[μ(Ctk+ωG¯tc)σ−1σ+(1−μ)(1−0)σ−1σ]σσ−1+ξt[rtKtk−τtk(rt−δp)Ktk−Ctk−Kt+1k+(1−δp)Ktk]}

where ξ_t is the Lagrangian multiplier on the capitalist’s budget constraint.

The first-order condition (FOC) for Ctk is:

(37)[μ(Ctk+ωG¯tc)σ−1σ+(1−μ)]1σ−1⋅μ(Ctk+ωG¯tc)−1σ=ξt.

The FOC for Kt+1k is:

(38)βξt+1[1+(1−τt+1k)(rt+1−δp)]=ξt.

Consolidating these two FOCs yields the following equation:

(39)(Ctk+ωG¯tc)−1σ[μ(Ctk+ωG¯tc)σ−1σ+(1−μ)]1σ−1=β(Ct+1k+ωG¯t+1c)−1σ[μ(Ct+1k+ωG¯t+1c)σ−1σ+(1−μ)]1σ−1⋅[1+(1−τt+1k)(rt+1−δp)]}.

A.2 Optimisation problem of workers

The optimisation problem for the worker is shown as follows:

maxCtw,Htw{[μ(Ctw+ωG¯tc)σ−1σ+(1−μ)(1−Htw)σ−1σ]σσ−1}s.t. Ctw=(1−τtw)wtHtw+G¯tu(1−Htw).

The Lagrangian function of the worker is written as:

Lw=[μ(Ctw+ωG¯tc)σ−1σ+(1−μ)(1−Htw)σ−1σ]σσ−1+ϕt[(1−τtw)wtHtw+G¯tu(1−Htw)−Ctw]

where ϕ_t is the Lagrangian multiplier on the worker’s budget constraint.

The FOC for Ctw is:

(40)[μ(Ctw+ωG¯tc)σ−1σ+(1−μ)(1−Htw)σ−1σ]1σ−1μ(Ctw+ωG¯tc)−1σ=ϕt.

The FOC for Htw is:

(41)[μ(Ctw+ωG¯tc)σ−1σ+(1−μ)(1−Htw)σ−1σ]1σ−1(μ−1)(1−Htw)−1σ+ϕt(1−τtw)wt−ϕtG¯tu=0.

These two FOCs are next combined into one equation as follows:

(1−μ)(1−Htw)−1σ+μG¯tu(Ctw+ωG¯tc)−1σ=μwt(1−τtw)(Ctw+ωG¯tc)−1σ

which can be re-written as:

(42)(1−μ)(1−Htw)−1σ+μr¯twt(Ctw+ωG¯tc)−1σ=μwt(1−τtw)(Ctw+ωG¯tc)−1σ

A.3 Optimisation problem of firms

The optimization problem for the firm can be summarized in the following:

maxKtf,Ntf{Ytf−rtKtf−wtHtf}s.t. Ytf=A(Ktf)α1(Htf)α2.

The Lagrangian function of the firm is written as:

Lf=A(Ktf)α1(Htf)α2−rtKtf−wtHtf.

The FOC for Ktf is:

(43)α1A(Ktf)α1−1(Htf)α2−rt=0rt=α1A(Ktf)α1−1(Htf)α2

The FOC for Htf is:

(44)α2A(Ktf)α1(Htf)α2−1−wt=0wt=α2A(Ktf)α1(Htf)α2−1

A.4 The DCE conditions

The DCE consists of the following conditions:

OCk:(Ctk+ωG¯tc)−1σ[μ(Ctk+ωG¯tc)σ−1σ+(1−μ)]1σ−1=β(Ct+1k+ωG¯t+1c)−1σ[μ(Ct+1k+ωG¯t+1c)σ−1σ+(1−μ)]1σ−1⋅[1+(1−τt+1k)(rt+1−δp)]OCw:(1−μ)(1−Htw)−1σ+μr¯twt(Ctw+ωG¯tc)−1σ=μwt(1−τtw)(Ctw+ωG¯tc)−1σFOk:rt=α1A(Ktf)α1−1(Htf)α2(K¯tg)α3FOl:wt=α2A(Ktf)α1(Htf)α2−1(K¯tg)α3BCk:Ctk+Kt+1k−(1−δp)Ktk=rtKtk−τtk(rt−δp)KtkBCw:Ctw=(1−τtw)wtHtw+r¯twt(1−Htw)BCg:G¯tc+(1−nk)r¯twt(1−Htw)+nkI¯tg=nk[τtk(rt−δp)Ktk+τtkπtk]+(1−nk)τtwwtHtwPCg:K¯t+1g=(1−δg)K¯tg+I¯tgMCk:Ktk=KtfMCl:Htw=nk(1−nk)Htf

A.5 Derivation of ζⁱ

ζⁱ satisfies the follow equation implying that the agent i is as well off in the exogenous policy model as in the Ramsey model.

URi=U¯Ei=[μ(CEi(1+ζi)+ωG¯Ec)σ−1σ+(1−μ)(1−HEi)σ−1σ]σσ−1.

We can solve for ζⁱ in the equation above by taking the following algebra:

(45)(URi)σ−1σ=[μ(CEi(1+ζi)+ωG¯Ec)σ−1σ+(1−μ)(1−HEi)σ−1σ]σσ−1⋅σ−1σ(URi)σ−1σ=μ(CEi(1+ζi)+ωG¯Ec)σ−1σ+(1−μ)(1−HEi)σ−1σ(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ=μ(CEi(1+ζi)+ωG¯Ec)σ−1σ[(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ]σσ−1=[μ(CEi(1+ζi)+ωG¯Ec)σ−1σ]σσ−1[(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ]σσ−1=μσσ−1[CEi(1+ζi)+ωG¯Ec][(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ]σσ−1μσ1−σ=CEi(1+ζi)+ωG¯Ec[(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ]σσ−1μσ1−σ−ωG¯Ec=CEi(1+ζi)[(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ]σσ−1μσ1−σ−ωG¯EcCEi=1+ζi.⇒ζi=[(URi)σ−1σ−(1−μ)(1−HEi)σ−1σ]σσ−1μσ1−σ−ωG¯EcCEi−1

References

Ahmed, S. 1986. “Temporary and Permanent Government Spending in an Open Economy: Some Evidence for the UK.” Journal of Monetary Economics 17 (2): 197–224.10.1016/0304-3932(86)90028-0Search in Google Scholar

Ai, C., and S. Cassou. 1995. “A Normative Analysis of Public Capital.” Applied Economics 27 (12): 1201–1209.10.1080/00036849500000102Search in Google Scholar

Ambler, S., and A. Paquet. 1996. “Fiscal Spending Shocks, Endogenous Government Spending, and Real Business Cycles.” Journal of Economic Dynamics and Control 20 (1–3): 237–256.10.1016/0165-1889(94)00850-XSearch in Google Scholar

Angelopoulos, K., S. Asimakopoulos, and J. Malley. 2017. “The Optimal Distribution of the Tax Burden Over the Business Cycle.” Macroeconomic Dynamics. DOI: 10.1017/S1365100517000700.Search in Google Scholar

Angelopoulos, K., J. Malley, and A. Philippopoulos. 2011. “Time-Consistent Fiscal Policy Under Heterogeneity: Conflicting or Common Interests?” CESifo Working Paper Series: 3444, CESifo Group Munich.10.2139/ssrn.1844070Search in Google Scholar

Ardagna, S. 2001. “Fiscal Policy Composition, Public Debt, and Economic Activity.” Public Choice 109 (3–4): 301–325.10.1023/A:1013021004195Search in Google Scholar

Ardagna, S. 2007. “Fiscal Policy in Unionized Labor Markets.” Journal of Economic Dynamics and Control 31 (5): 1498–1534.10.1016/j.jedc.2006.05.009Search in Google Scholar

Arsenau, D., and S. Chugh. 2012. “Tax Smoothing in Frictional Labor Markets.” Journal of Political Economy 120 (5): 926–985.10.1086/668837Search in Google Scholar

Aschauer, A. 1985. “Fiscal Policy and Aggregate Demand.” American Economic Review 75 (1): 117–127.Search in Google Scholar

Aschauer, A. 1989. “Is Public Expenditure Productive?” Journal of Monetary Economics 23 (2), 177–200.10.1016/0304-3932(89)90047-0Search in Google Scholar

Aschauer, A., and J. Greenwood. 1985. “Macroeconomic Effects of Fiscal Policy.” Carnegie-Rochester Conference Series on Public Policy 23: 91–138.10.1016/0167-2231(85)90007-7Search in Google Scholar

Asimakopoulos, S., and P. Asimakopoulos. 2017. “Fiscal Policy with Banks and Financial Frictions.” Journal of Financial Stability. DOI: 10.1016/j.jfs.2017.10.01.Search in Google Scholar

Baier, S., and G. Glomm. 2001. “Long-Run Growth and Welfare Effects of Public Policies with Distortionary Taxation.” Journal of Economic Dynamics and Control 25 (12): 2007–2042.10.1016/S0165-1889(00)00017-8Search in Google Scholar

Barro, J. 1981. “Output Effects of Government Purchases.” Journal of Political Economy 89 (6): 1086–1121.10.1086/261024Search in Google Scholar

Barro, J. 1989. “The Neoclassical Approach to Fiscal Policy.” Modern Business Cycle Theory, Cambridge, Massachusetts: Harvard University Press.Search in Google Scholar

Burnside, C., M. Eichenbaum, and J. Fisher. 2003. “Fiscal Shocks and Their Consequences.” Journal of Economic Theory 115 (1): 89–117.10.3386/w9772Search in Google Scholar

Chamley, C. 1986. “Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives.” Econometrica 54 (3): 607–622.10.2307/1911310Search in Google Scholar

Chari, V., C. Lawrence, and P. Kehoe. 1994. “Optimal Fiscal Policy in a Business Cycle Model.” Journal of Political Economy 102 (4): 617–652.10.1086/261949Search in Google Scholar

Chari, V., and P. Kehoe. 1999. “Optimal Fiscal and Monetary Policy.” In Handbook of Macroeconomics, edited by J.B. Taylor and M. Woodford, Vol. 1, 1671–1745. Elsevier.10.1016/S1574-0048(99)10039-9Search in Google Scholar

Chari, V., P. Kehoe, and E. McGrattan. 2002. “Accounting for the Great Depression.” American Economic Review 92 (2): 22–27.10.1257/000282802320188934Search in Google Scholar

Chari, V., P. Kehoe, and E. McGrattan. 2007. “Business Cycle Accounting.” Econometrica 75 (3): 781–836.10.1111/j.1468-0262.2007.00768.xSearch in Google Scholar

Conesa, J., S. Kitao, and D. Krueger. 2009. “Taxing Capital? not a Bad Idea After All!” American Economic Review 99 (1): 25–48.10.1257/aer.99.1.25Search in Google Scholar

Cooley, T., and G. Hansen. 1992. “Tax Distortions in a Neoclassical Monetary Economy.” Journal of Economic Theory 58 (2): 290–316.10.1016/0022-0531(92)90056-NSearch in Google Scholar

Correia, I. 1996. “Should Capital Income be Taxed in the Steady State?” Journal of Public Economics 60 (1): 147–151.10.1016/0047-2727(95)01514-0Search in Google Scholar

Daveri, F., and M. Maffezzoli. 2001. “A Numerical Approach to Fiscal Policy, Unemployment, and Growth in Europe.” Econometrics and Applied Economics Working Paper, Istituto di Economia Politica, Bocconi University, No. 4.Search in Google Scholar

Domeij, D. 2005. “Optimal Capital Taxation and Labour Market Search.” Review of Economic Dynamics 8 (3): 623–650.10.1016/j.red.2005.01.011Search in Google Scholar

Finn, M. 1998. “Cyclical Effects of Government’s Employment and Goods Purchases.” International Economic Review 39 (3): 635–657.10.2307/2527394Search in Google Scholar

Gali, J., J. López-Salido, and J. Vallés. 2007. “Understanding the Effects of Government Spending on Consumption.” Journal of the European Economic Association 5 (1): 227–270.10.1162/JEEA.2007.5.1.227Search in Google Scholar

Guo, J., and K. Lansing. 1999. “Optimal Taxation of Capital Income with Impefectly Competitive Product Markets.” Journal of Economic Dynamics and Control 23 (7): 967–995.10.1016/S0165-1889(98)00047-5Search in Google Scholar

Judd, K. 1985. “Redistributive Taxation in a Simple Perfect Foresight Model.” Journal of Public Economics 28 (1): 59–83.10.1016/0047-2727(85)90020-9Search in Google Scholar

Judd, K. 1997. “The Optimal Tax on Capital Income is Negative.” NBER Working Paper, No. 6004.10.3386/w6004Search in Google Scholar

Judd, K. 1999. “Optimal Taxation and Spending in General Competitive Growth Models.” Journal of Public Economics 71 (1): 1–26.10.1016/S0047-2727(98)00054-1Search in Google Scholar

Judd, K. 2002. “Capital Income Taxation with Imperfect Competition.” American Economic Review 92 (2): 417–421.10.1257/000282802320191723Search in Google Scholar

Klein, P. 2000. “Using the Generalized Schur Form to Solve a Multivariate Linear Rational Expectations Model.” Journal of Economic Dynamics and Control 24 (10): 1405–1423.10.1016/S0165-1889(99)00045-7Search in Google Scholar

Kormendi, R. 1983. “Government Debt, Government Spending, and Private Sector Behavior.” American Economics Review 73 (5): 994–1010.Search in Google Scholar

Koskela, E., and L. von Thadden. 2008. “Optimal Factor Taxation Under Wage Bargaining: a Dynamic Perspective.” German Economic Review 9 (2): 135–159.10.1111/j.1468-0475.2008.00428.xSearch in Google Scholar

Krusell, P. 2002. “Time-Consistent Redistribution.” European Economic Review 46 (4–5): 755–769.10.1016/S0014-2921(01)00211-2Search in Google Scholar

Lansing, K. 1999. “Optimal Redistributive Capital Taxation in a Neoclassical Growth Model.” Journal of Public Economics 73 (3): 423–453.10.1016/S0047-2727(99)00016-XSearch in Google Scholar

Ljungqvist, L., and T. Sargent. 2006. “Do Taxes Explain European Employment? Indivisible Labor, Human capital, Lotteries, and Savings.” NBER Macroeconomics Annual 21: 181–246.10.1086/ma.21.25554955Search in Google Scholar

Lucas, R. 1990. “Supply-Side Economics: An Analytical Review.” Oxford Economic Papers 42 (2): 293–316.10.1093/oxfordjournals.oep.a041948Search in Google Scholar

Malley, J., A. Philippopoulos, and U. Woitek. 2009. “To React or not? Technology Shocks, Fiscal Policy and Welfare in the EU-3.” European Economic Review 53 (6): 689–714.10.1016/j.euroecorev.2009.01.005Search in Google Scholar

Mortensen, D., and C. Pissarides. 1994. “Job Creation and Job Destruction in the Theory of Unemployment.” Review of Economic Studies 61 (3): 397–415.10.2307/2297896Search in Google Scholar

Munnell, A. 1990. “Why has Productivity Growth Declined? Productivity and Public Investment.” Federal Reserve Bank of Boston, New England Economic Review, January/February, 3–22.Search in Google Scholar

Ohanian, L. 1997. “The Macroeconomic Effects of War Finance in the United States: World War II and the Korean War.” American Economic Review 871: 23–40.Search in Google Scholar

Pissarides, C. 1998. “The Impact of Employment Tax Cuts on Unemployment and Wages; The Role of Unemployment Benefits and Tax Structure.” European Economic Review 42 (1): 155–183.10.1016/S0014-2921(97)00090-1Search in Google Scholar

Prescott, E. 2002. “Prosperity and Depression.” American Economic Review 92 (2): 1–15.10.1257/000282802320188916Search in Google Scholar

Prescott, E. 2004. “Why do Americans Work so Much More Than Europeans?” Quarterly Review, Federal Reserve Bank of Minneapolis, issue July, 2–13.10.3386/w10316Search in Google Scholar

Shimer, R. 2009. “Convergence in Macroeconomics: The Labor Wedge.” American Economic Journal: Macroeconomics 1 (1): 280–297.10.1257/mac.1.1.280Search in Google Scholar

Werning, I. 2007. “Optimal Fiscal Policy with Redistribution.” The Quarterly Journal of Economics 122 (3): 925–967.10.1162/qjec.122.3.925Search in Google Scholar

Published Online: 2018-10-12

Optimal taxation under equilibrium unemployment and economic profits

Abstract

Acknowledgements

A Appendix

A.1 Optimisation problem of capitalists

A.2 Optimisation problem of workers

A.3 Optimisation problem of firms

A.4 The DCE conditions

A.5 Derivation of ζⁱ

References

Journal and Issue

Articles in the same Issue

Optimal taxation under equilibrium unemployment and economic profits

Abstract

Acknowledgements

A Appendix

A.1 Optimisation problem of capitalists

A.2 Optimisation problem of workers

A.3 Optimisation problem of firms

A.4 The DCE conditions

A.5 Derivation of ζi

References

Journal and Issue

Articles in the same Issue

A.5 Derivation of ζⁱ