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Optimal capital-income taxation in a model with credit frictions

  • Salem Abo-Zaid EMAIL logo
Veröffentlicht/Copyright: 14. Juni 2014
The B.E. Journal of Macroeconomics
Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 14 Heft 1

Abstract

The optimality of the long-run capital-income tax rate is revisited in a simple neoclassical growth model with credit frictions. Firms pay their factors of production in advance, which requires borrowing at the beginning of the period. Borrowing, in turn, is constrained by the value of collateral that they own at the beginning of the period, leading to inefficiently low amounts of capital and labor. In this environment, the optimal capital-income tax in the steady state is non zero. Specifically, the quantitative analyses show that the capital-income tax is negative and, therefore, the distortions stemming from the credit friction are offset by subsidizing capital. However, when the government cannot distinguish between capital-income and profits, the capital-income tax is positive as the government levies the same tax rate on both sources of income. These results stand in contrast to the celebrated result of zero capital-income taxation of Judd (Judd, K. 1985. “Redistributive Taxation in a Simple Perfect Foresight Model.” Journal of Public Economics 28: 59–83.) and Chamley (Chamley, C. 1986. “Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives.” Econometrica 54: 607–622.).

Keywords: borrowing constraints; credit frictions; optimal capital-income taxation
Corresponding author: Salem Abo-Zaid, Economics, Texas Tech University, 248 Holden Hall, P.O. Box 41014, Lubbock, TX 79409, USA, Phone: +1-806-834-6983, e-mail:

Acknowledgments

I am grateful to the editor and to an anonymous referee for very helpful comments and suggestions.

Appendix

A. The firm’s problem

At the beginning of period t, the firm obtains a loan btf from households, which is repaid at the end of the period at a nominal gross interest rate of Rtf. Borrowing is constrained by the beginning-of-period firm’s collateral. Formally, the firm chooses labor, capital and loans to maximize:

(A.1)ztf(kt, lt)+btf−wtlt−rtkt−Rtfbtf (A.1)

subject to:

(A.2)btf−ϕ(wtlt+rtkt)≥0 (A.2)

and:

(A.3)κqtxt−btf≥0. (A.3)

Letting υt and γt be the Lagrange multipliers on the constraints (A.2) and (A.3), respectively, the optimality condition with respect to btf reads:

(A.4)γt=Rtf+υt−1. (A.4)

Similarly, the first order conditions with respect to lt and kt yield:

(A.5)ztfl,t=(1+ϕυt)wt (A.5)
(A.6)ztfk,t=(1+ϕυt)rt. (A.6)

Recalling that Rtf=1 from the solution to the household’s problem, equation (A.4) becomes:

(A.7)γt=υt (A.7)

and, therefore, the two Lagrange multipliers are equal. By renaming the Lagrange multiplier as μt, we get conditions (11)–(12) in the text.

Alternatively, conditions (A.2) and (A.3) can be combined to get:

(A.8)ϕ(wtlt+rtkt)κqtxt (A.8)

which is condition (10) in the text.

Substituting Rtf=1 in (A.1), the profit function is given by:

(A.9)ztf(kt, lt)−wtlt−rtkt (A.9)

which is condition (9) in the text. Therefore, the optimization problem of the firm is to maximize (A.9) subject to (A.8). Letting μt be the Lagrange multiplier on (A.8), the choices of labor and capital yield conditions (11) and (12) in the text.

B. Efficient allocations

The problem of the social planner is to maximize:

(B.1)max{ct, lt, ht, xt, kt}t=0∞E0∑t=0∞βtu(ct, ht, lt) (B.1)

subject to the sequence of resource constraints:

(B.2)ztf(kt, lt)+(1−δ)kt=ct+kt+1+gt (B.2)

and the market clearing condition of real estate:

(B.3)ht+xt=1. (B.3)

Letting ηt be the Lagrange multiplier associated with condition (B.2), the first-order conditions with respect to ct and kt+1, respectively, read:

(B.4)uc,t=ηt (B.4)
(B.5)ηt=βEt(ηt+1[1−δ+zt+1fk,t+1]) (B.5)

Combining these two conditions gives condition (17) in the text.

C. The present-value implementability constraint

I show here the derivation of the PVIC for the Ramsey problem. Recalling that Rtf=1, the households’ budget constraint becomes:

(C.1)(1−τtl)wtlt+[1−δ+rt−τtk(rt−δ)]kt+(1−τtπ)Πt+qtht+Rt−1bt=ct+kt+1+bt+1+qtht+1. (C.1)

By introducing E0∑t=0∞βtuct to (C.1) and rearranging, we have:

(C.2)E0∑t=0∞βtuct(1−τtl)wtlt+E0∑t=0∞βtuct[1−δ+rt−τtk(rt−δ)]kt+E0∑t=0∞βtuct(1−τtπ)Πt+E0∑t=0∞βtuctqtht+E0∑t=0∞βtuctRt−1bt−E0∑t=0∞βtuctct−E0∑t=0∞βtuctkt+1−E0∑t=0∞βtuctbt+1−E0∑t=0∞βtuctqtht+1=0. (C.2)

Recall that, from the solution to the households’ problem, we have:

(C.3)−ul, tuc, t=(1−τtl)wt (C.3)
(C.4)uc, t=βRtEt(uc, t+1) (C.4)
(C.5)uc, t=βEt(uc, t+1[1−δ+rt+1−τt+1k(rt+1−δ)]) (C.5)
(C.6)qtuc, t=βEt(uh, t+1+qt+1uc, t+1). (C.6)

Substituting (C.3) in the first term of (C.2), (C.4) in the eighth term of (C.2), (C.5) in the seventh term of (C.2) and (C.6) in the last term of (C.2) yield:

(C.7)E0∑t=0∞βtuct(−ul, tuc, t)lt+E0∑t=0∞βtuct[1−δ+rt−τtk(rt−δ)]kt+E0∑t=0∞βtuct(1−τtπ)Πt+E0∑t=0∞βtuctqtht+E0∑t=0∞βtuctRt−1bt−E0∑t=0∞βtuctct−E0∑t=0∞βtβ(uc, t+1[1−δ+rt+1−τt+1k(rt+1−δ)])kt+1−E0∑t=0∞βtβRt(uc, t+1)bt+1−E0∑t=0∞βtβ(uh, t+1+qt+1uc, t+1)ht+1=0. (C.7)

Combining the second and seventh terms of (C.7) yields:

(C.8)E0∑t=0∞βtuct[1−δ+rt−τtk(rt−δ)]kt−E0∑t=0∞βt+1(uc, t+1[1−δ+rt+1−τt+1k(rt+1−δ)])kt+1=uc, 0[1−δ+r0−τ0k(r0−δ)]k0. (C.8)

Combining the fifth and eighth terms of (C.7) gives:

(C.9)E0∑t=0∞βtuctRt−1bt−E0∑t=0∞βtβRt(uc, t+1)bt+1=uc, 0R−1b0 (C.9)

Combining the fourth and last terms of (C.7) yields:

(C.10)E0∑t=0∞βtuctqtht−E0∑t=0∞βtβ(uh, t+1+qt+1uc, t+1)ht+1=uc, 0q0h0−E0∑t=0∞βt+1uh, t+1ht+1 (C.10)

Also, the first term of (C.7) can be written as:

(C.11)E0∑t=0∞βtuct(−ul, tuc, t)lt=E0∑t=0∞βtultlt. (C.11)

Finally, substituting (C.8)–(C.11) into (C.7) and re-arranging give the PVIC (condition (18) in the text):

(C.12)E0∑t=0∞βt(ctuc,t+ltul, t+βht+1uh, t+1−uc, t(1−τtπ)Πt)=A0 (C.12)

with A0=uc,0R−1b0+uc,0[1−δ+r0−τ0k(r0−δ)]k0+uc,0q0h0.

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Published Online: 2014-6-14
Published in Print: 2014-1-1

©2014 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Advances
  3. Optimal portfolios with wealth-varying risk aversion in the neoclassical growth model
  4. Inventories and the stockout constraint in general equilibrium
  5. Optimal second best taxation of addictive goods in dynamic general equilibrium: a revenue raising perspective
  6. Inflation effects on capital accumulation in a model with residential and non-residential assets
  7. Optimal capital-income taxation in a model with credit frictions
  8. Contributions
  9. Interest rate fluctuations and equilibrium in the housing market
  10. News shocks and learning-by-doing
  11. Capacity utilization and the effects of energy price increases in Japan
  12. Small-scale New Keynesian model features that can reproduce lead, lag and persistence patterns
  13. Optimal policy and Taylor rule cross-checking under parameter uncertainty
  14. The impact of American and British involvement in Afghanistan and Iraq on health spending, military spending and economic growth
  15. Why does natural resource abundance not always lead to better outcomes? Limited financial development versus political impatience
  16. The skill bias of technological change and the evolution of the skill premium in the US since 1970
  17. Aggregate impacts of recent US natural gas trends
  18. Organizational learning and optimal fiscal and monetary policy
  19. Industrial specialization, financial integration and international consumption risk sharing
  20. Leverage, investment, and optimal monetary policy
  21. Public debt in an OLG model with imperfect competition: long-run effects of austerity programs and changes in the growth rate
  22. Temporal aggregation and estimated monetary policy rules
  23. International transmission of productivity shocks with nonzero net foreign debt
  24. Did the euro change the effect of fundamentals on growth and uncertainty?
  25. Topics
  26. Real factor prices and factor-augmenting technical change
  27. Monetary policy and TIPS yields before the crisis
https://doi.org/10.1515/bejm-2013-0112
Schlagwörter für diesen Artikel
borrowing constraints; credit frictions; optimal capital-income taxation

Artikel in diesem Heft

  1. Frontmatter
  2. Advances
  3. Optimal portfolios with wealth-varying risk aversion in the neoclassical growth model
  4. Inventories and the stockout constraint in general equilibrium
  5. Optimal second best taxation of addictive goods in dynamic general equilibrium: a revenue raising perspective
  6. Inflation effects on capital accumulation in a model with residential and non-residential assets
  7. Optimal capital-income taxation in a model with credit frictions
  8. Contributions
  9. Interest rate fluctuations and equilibrium in the housing market
  10. News shocks and learning-by-doing
  11. Capacity utilization and the effects of energy price increases in Japan
  12. Small-scale New Keynesian model features that can reproduce lead, lag and persistence patterns
  13. Optimal policy and Taylor rule cross-checking under parameter uncertainty
  14. The impact of American and British involvement in Afghanistan and Iraq on health spending, military spending and economic growth
  15. Why does natural resource abundance not always lead to better outcomes? Limited financial development versus political impatience
  16. The skill bias of technological change and the evolution of the skill premium in the US since 1970
  17. Aggregate impacts of recent US natural gas trends
  18. Organizational learning and optimal fiscal and monetary policy
  19. Industrial specialization, financial integration and international consumption risk sharing
  20. Leverage, investment, and optimal monetary policy
  21. Public debt in an OLG model with imperfect competition: long-run effects of austerity programs and changes in the growth rate
  22. Temporal aggregation and estimated monetary policy rules
  23. International transmission of productivity shocks with nonzero net foreign debt
  24. Did the euro change the effect of fundamentals on growth and uncertainty?
  25. Topics
  26. Real factor prices and factor-augmenting technical change
  27. Monetary policy and TIPS yields before the crisis
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