Abstract
In this paper, the optimal taxation problem in a small open economy with international trade in capital or investment goods is investigated. The monopolistic power of a small open economy over the terms of trade causes distortions in consumption and investment. The results suggest that due to the external distortion in investment, the long-run optimal capital income tax could be positive under the baseline calibration, and it is increasing in the degree of investment openness and decreasing in the elasticity of substitution between domestic and foreign goods. The long-run optimal labor income tax exhibits the opposite relationships with the openness and elasticity parameters. During the course of business cycles, the fluctuations in the external distortions cause the optimal labor income tax to be more volatile and the optimal capital income tax to be less volatile than their closed-economy counterparts.
Acknowledgement
I would like to thank the editor, Davide Debortoli, and two anonymous referees for helpful comments and suggestions. All errors are mine.
A Mathematical Appendix
A.1 Derivation for Section 3.1
A.1.1 Derivation of (40) and (41)
The government budget constraint (32) can be rewritten as
where the last equality is established using the labor supply condition (24) and the fact that
Rearranging the terms and using the definition that
Multiplying the last equation by UCt yields (41):
The Euler equations (25) and (26) imply
Equations (81) and (80) imply (40):
A.1.2 Derivation of the Lagrangian (43)
The Lagrangian for the Ramsey problem can be written as
The first-order condition with respect to Vt implies
Using this result and defining the pseudo-utility by
we can rewrite the Lagrangian (83) as
This Lagrangian corresponds to (43) with the last term
A.1.3 The first-order conditions
By the price formula (34) and (35), we express the relative prices as functions of the real exchange rate st:
We can derive the elasticities of the above relative prices with respect to st:
Differentiating the Lagrangian (43) with respect to Ct, It, Lt, Kt, bft, and st yields
where
A.1.4 Derivation of (50)
Define
Similarly, defining
Using (85), we can obtain
Using (87) and (88), we can obtain
A.1.5 Derivation of the MRS-to-MRT ratio
As implied by the Armington function (2) and the household optimality condition (6), the MRS is given by
The MRT is given by the ratio of marginal costs (in terms of utility), i.e.
A.2 Derivation for Section 3.2
A.2.1 Derivation of Δ t c Δ t r
We can multiply (45) by pHt and divide the result by νt to obtain
where
Because (34) implies that
Using (94) to eliminate the terms involving bft and ζt, we can rewrite
A.2.2 Derivation of WCt and WLt
The pseudo utility function is
where
where
A.3 Proof to Proposition Proposition 1
For simplicity, we assume that foreign households do not have to pay the portfolio adjustment cost. The foreign household optimality condition for bond holding implies
Differentiating the pseudo-utility function with respect to foreign bonds bf, t and the real exchange rate st yields
Under the assumption that
In the symmetric steady state,
Rearranging the terms yields
If ωc = 0, (108) implies that
If ωc = ωi and ηc = ηi, (108) also implies that
If st = 1 in the deterministic steady state,
Thus, if the steady state is symmetric and Mx is given by (109),
A.4 The Ramsey efficiency conditions
In summary, we can determine the dynamics of 17 variables,
A.5 Data appendix to Section 4.2.2
This section details the sources of the data used in Section 4.2.2. McDaniel’s (2007) data set contains taxes on capital income and investment expenditure for 17 OECD countries from 1978 to 2014, after missing values removed. Because there is no investment tax in our model, the data on capital income tax used in this paper were constructed according to the rule:
where
The degree of investment openness for a country was defined as the trade volume of durable goods relative to the GDP. The data on world trade flows were taken from Feenstra et al. (2005), and the data were only available between 1962 and 2000. The trade flows were categorized into nondurable and durables by using the method in Engel and Wang (2011). The data on GDP were taken from The World Bank (2018).
From the above sources, we obtained balanced panel data on capital income tax and degree of investment openness from 1978 to 2000 for 17 OECD countries, whose summary statistics are shown in Table 4.
Summary statistics.
| Investment openness | Capital income tax | |||
|---|---|---|---|---|
| Mean (%) | Standard deviation | Mean (%) | Standard deviation | |
| Australia | 10.46 | 2.02 | 34.91 | 1.72 |
| Austria | 28.34 | 3.35 | 27.02 | 1.33 |
| Belgium | 55.02 | 5.67 | 31.24 | 2.16 |
| Canada | 27.08 | 7.62 | 36.94 | 2.55 |
| Finland | 21.87 | 4.80 | 30.19 | 3.69 |
| France | 17.73 | 2.35 | 26.84 | 2.14 |
| Germany | 10.50 | 12.21 | 25.73 | 1.15 |
| Italy | 16.02 | 2.00 | 22.96 | 4.97 |
| Japan | 10.61 | 1.14 | 25.78 | 2.36 |
| Korea Rep. | 25.45 | 4.69 | 17.80 | 0.98 |
| Netherlands | 31.43 | 5.09 | 26.95 | 2.48 |
| Norway | 19.87 | 1.36 | 33.56 | 3.50 |
| Spain | 14.35 | 4.58 | 19.07 | 4.27 |
| Sweden | 27.54 | 4.11 | 41.06 | 3.49 |
| Switzerland | 29.47 | 2.43 | 17.89 | 0.94 |
| UK | 20.08 | 1.49 | 29.85 | 2.54 |
| USA | 8.78 | 1.85 | 28.41 | 1.82 |
B Optimal policy problem for the case of complete markets
In this section, we briefly describe the optimal policy problem for a small open economy with a complete international asset market in Section 3.4.
B.1 The market equilibrium
Following Auray, de Blas, and Eyquem (2011), we assume that the government budget is always balanced and the domestic monopolistic distortion is eliminated. Thus, the government budget constraint corresponds to (32) with zero profit:
Given the processes of technology shock
B.2 Derivation of (62) and (63)
Define the value of capital by
By the definition Vt and the Euler equation (25),
where the last equality is established using the law of motion for capital (8). The government budget constraint (127) can be rewritten as
where the last equality is established using (128). Rearranging the terms and using the labor supply condition (24), we can rewrite (129) as
where the last equality is established using the fact that
Multiplying (130) by UCt yields (62):
As shown in (128),
B.3 The Lagrangian (65) and the optimality conditions
The Lagrangian for the Ramsey problem can be written as
where Λt, φt, ψt, νt, and ζt are the multipliers associated with the implementability constraint (62), the Euler equation (63), the law of motion of capital (8), the market-clearing condition for home goods (39), and the risk-sharing condition (61). The first-order condition with respect to Vt implies
Using this result and define the pseudo-utility as
we can rewrite the Lagrangian (132) as
This last Lagrangian corresponds to (65) with the term
The first-order conditions with respect to Ct, Lt, It, Kt, and st are given by
where
B.3.1 Derivatives of the pseudo utility
The derivative of the pseudo utility with respect to Ct is given by
where
The derivatives of the pseudo utility with respect to Lt and It are given by
The derivative of the pseudo utility with respect to the real exchange rate is given by:
B.3.2 Derivation of elasticities
Using (85)—(88), we can rewrite
B.3.3 Derivation of (71) and (73)
Multiplying (67) by pHt and dividing the result by νt, we have
where
Using (70) to replace ζt, we can rewrite
where the last equality is established using the fact from the risk-sharing condition (61) that
As in the baseline model, we define the investment externality in the current setting as
where we have used the fact from (35) that
B.3.4 Collection of equilibrium conditions
Given Λ, the following equations constitute a system of
B.3.5 Derivation of the effective capital income tax rate (77)
The steady-state version of the Euler equation (25) implies
By this equation,
Therefore,
C The optimal policy problem for the financial autarky
The financial autarky is equivalent to the baseline model with bft = 0 for all t. The pseudo utility function is
where
where ψt, μt, and νt are multipliers on the law of motion (8), the current-account equation (38), and the market-clearing condition (39). The first-order conditions with respect to Ct, Lt, It, Kt, and st can be written as
D The optimal policy problem for the closed economy
The closed economy is equivalent to the financial autarky with
Thus, the Lagrangian for the Ramsey problem of the closed economy can be written as
where νt is the multiplier on the market-clearing condition (158). The first-order conditions with respect to Ct, Lt, and Kt can be written as
where
E The extended model with nontradable capital goods
In this section, we analyze the optimal policy problem in Section 5.
E.1 The model
The representative household owns capital. The stocks of structures Kst, IP products Kip, t, and equipment Ket, evolve according to
We assume that equipment is generated by bundles of home and foreign goods via the CES aggregator (9):
where IeH,t and IeF,t denote the equipment goods produced by Home and Foreign. Bundles IeH,t and IeF,t are generated by the Dixit-Stiglitz aggregator (10):
where ιet(j) is the purchase of good j ∈ [0, 1]. Therefore, by (4) and (6), the investment demand for good j ∈ [0, 1] is given by
As in (6), the demands for home and foreign equipment goods can be derived as
where Pet is the price of equipment goods given by
Analogous to the domestic demand, the foreign demand for home good j ∈ [0, n) is
and the demand for the bundle of home goods is given by
where
Structure capital and IP products are nontradable. Thus, we assume that they generated by the following aggregators:
where ιst(j) and ιip,t(j) are the purchases of domestic good j given by
The household budget constraint at Home can be written as
where
The household optimality condition for capital accumulation implies
where
A typical domestic firm
Given the domestic and foreign demand functions, the world demand for good h is given by
where Yt is given by
The firm optimally chooses the price pt(h) that maximizes the periodic profit flow,
subject to the world demand function and the technology constraint. The optimality condition with respect to pt(h) implies
where
The firm’s optimality conditions for lt(h), kst(h), and ket(h) imply that the factor demands (30) and (31) for the baseline model should be modified as
where
Due to the distinction between tradable and nontradable capital goods, the government budget constraint (32) in the baseline model is modified as
Because only equipment is tradable, the aggregate budget constraint (38) for the baseline model is modified as
where pxt, qmt, X(pxt), and M(qmt) are defined as before. Note that
Due to the above changes, the market-clearing condition for home goods (39) is modified as
where qHt and D(qHt) are defined as before. The term
E.2 The Ramsey problem
Modify the definition of Vt as
where
where
By the definition of Rst and Ret, the Euler equations can be rewritten as
Because the capital income tax is the same for all capital goods, Rst, Rip,t, and Ret are linked by the following equation:
E.3 The Lagrangian
The government attempts to maximize the household lifetime utility subject to the implementability constraint (185), the definition of Vt (184), the laws of motion for structure and equipment (162), (163), and (164), the aggregate budget constraint (182), the market-clearing condition for home goods (183), the Euler equation for foreign bonds (26), the Euler equations for structure and equipment (186) and (187), and the relation (188) that links Rst and Ret.
Let Λt, φt, ψst, ψip, t, ψet, μt, νt, ζbt, ζst, ζet, and ζrt be the multipliers associated with (26), (162), (163), (164), (182), (183), (184), (185), (186), (187), and (188). Define the pseudo-utility by
where
The Lagrangian for the Ramsey problem can be written as
where the rental rates
E.4 The first-order conditions
Differentiating the Lagrangian (43) with respect to Vt, Ct, Iet, Iip, t, Ist, Lt, Ket, Kip, t, Kst, bft, st, Ret, Rst, Rip,t, and χt yields
where
References
Adolfson, M., S. Laséen, J. Lindé, and M. Villani. 2007. “Bayesian Estimation of an Open Economy DSGE Model with Incomplete Pass-Through.” Journal of International Economics 72: 481–511.10.1016/j.jinteco.2007.01.003Search in Google Scholar
Adolfson, M., S. Laséen, J. Lindé, and M. Villani. 2008. “Evaluating an Estimated new Keynesian Small Open Economy Model.” Journal of Economic Dynamics and Control 32: 2690–2721.10.1016/j.jedc.2007.09.012Search in Google Scholar
Aiyagari, S. R. 1995. “Optimal Capital Income Taxation with Incomplete Markets, Borrowing Constraints, and Constant Discounting.” Journal of Political Economy 103: 1158–1175.10.1086/601445Search in Google Scholar
Armington, P. S. 1969. “A Theory of Demand for Products Distinguished by Place of Production.” IMF Staff Papers 16: 159–178.10.2307/3866403Search in Google Scholar
Atkeson, A., V. V. Chari, and P. J. Kehoe. 1999. “Taxing Capital Income: A Bad Idea.” Federal Reserve Bank of Mineapolis Quarterly Review 23: 3–17.Search in Google Scholar
Auray, S., B. de Blas, and A. Eyquem. 2011. “Ramsey Policies in a Small Open Economy with Sticky Prices and Capital.” Journal of Economic Dynamics and Control 35: 1531–1546.10.1016/j.jedc.2011.03.013Search in Google Scholar
Benigno, G., and P. Benigno. 2003. “Price Stability in Open Economies.” Review of Economic Studies 70: 743–764.10.1111/1467-937X.00265Search in Google Scholar
Benigno, P., and M. Woodford. 2006. “Optimal Taxation in an RBC Model: A Linear-Quadratic Approach.” Journal of Economic Dynamics and Control 30: 1445–1489.10.1016/j.jedc.2005.08.009Search in Google Scholar
Benigno, G., and B. De Paoli. 2010. “On the International Dimension of Fiscal Policy.” Journal of Money, Credit, and Banking 42: 1523–1542.10.1111/j.1538-4616.2010.00352.xSearch in Google Scholar
Benigno, P., and M. Woodford. 2012. “Linear-Quadratic Approximation of Optimal Policy Problems.” Journal of Economic Theory 147: 1–42.10.1016/j.jet.2011.10.012Search in Google Scholar
Bull, N. 1993. “When all the Optimal Dynamic Taxes are Zero.” Working Paper Series / Economic Activity Section 137, Board of Governors of the Federal Reserve System (U.S.).Search in Google Scholar
Cavallo, M., and A. Landry. 2018. “Capital-goods imports and US growth.” Staff Working Papers 18-1, Bank of Canada.Search in Google Scholar
Chamley, C. 1986. “Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives.” Econometrica 54: 607–622.10.2307/1911310Search in Google Scholar
Chari, V. V., and P. J. Kehoe. 1999. “Optimal Fiscal and Monetary Policy.” In Handbook of Macroeconomics, edited by J. B. Taylor and M. Woodford, Vol. 1C, 1671–1745. Amsterdam: Elsevier.10.3386/w6891Search in Google Scholar
Conesa, J. C., S. Kitao, and D. Krueger. 2009. “Taxing Capital? Not a Bad Idea after All!” American Economic Review 99: 25–48.10.1257/aer.99.1.25Search in Google Scholar
Correia, I. H. 1996. “Dynamic Optimal Taxation in a Small Open Economy.” Journal of Economic Dynamics and Control 20: 691–708.10.1016/0165-1889(95)00871-3Search in Google Scholar
Corsetti, G., and P. Pesenti. 2001. “Welfare and Macroeconomic Interdependence.” Quarterly Journal of Economics 116: 421–445.10.1162/00335530151144069Search in Google Scholar
Corsetti, G., and P. Pesenti. 2005. “International Dimension of Optimal Monetary Policy.” Journal of Monetary Economics 52: 281–305.10.1016/j.jmoneco.2004.06.002Search in Google Scholar
De Paoli, B. 2009. “Monetary Policy Under Alternative Asset Market Structures: The Case of a Small Open Economy.” Journal of Money, Credit, and Banking 41: 1301–1330.10.1111/j.1538-4616.2009.00257.xSearch in Google Scholar
Eaton, J., and S. Kortum. 2001. “Trade in Capital Goods.” European Economic Review 45: 1195–1235.10.1016/S0014-2921(00)00103-3Search in Google Scholar
Engel, C., and J. Wang. 2011. “International Trade in Durable Goods: Understanding Volatility, Cyclicality, and Elasticities.” Journal of International Economics 83: 37–52.10.1016/j.jinteco.2010.08.007Search in Google Scholar
Erosa, A., and M. Gervais. 2002. “Optimal Taxation in Life-Cycle Economies.” Journal of Economic Theory 105: 338–369.10.1006/jeth.2001.2877Search in Google Scholar
Faia, E., and T. Monacelli. 2008. “Optimal Monetary Policy in a Small Open Economy with Home Bias.” Journal of Money, Credit, and Banking 40: 721–750.10.1111/j.1538-4616.2008.00133.xSearch in Google Scholar
Feenstra, R. C., R. E. Lipsey, H. Deng, A. C. Ma, and H. Mo. 2005. “World Trade Flows: 1962–2000.” NBER Working Paper 11040.10.3386/w11040Search in Google Scholar
Guo, J.-T., and K. J. Lansing. 1999. “Optimal Taxation of Capital Income with Imperfectly Competitive Product Markets.” Journal of Economic Dynamics and Control 23: 967–995.10.1016/S0165-1889(98)00047-5Search in Google Scholar
Jones, L. E., R. E. Manuelli, and P. E. Rossi. 1993. “Optimal Taxation in Models of Endogenous Growth.” Journal of Political Economy 101: 485–517.10.1086/261884Search in Google Scholar
Jones, L. E., R. E. Manuelli, and P. E. Rossi. 1997. “On the Optimal Taxation of Capital Income.” Journal of Economic Theory 73: 93–117.10.1006/jeth.1996.2238Search in Google Scholar
Judd, K. L. 1985. “Redistributive Taxation in a Simple Perfect Foresight Model.” Journal of Public Economics 28: 59–83.10.1016/0047-2727(85)90020-9Search in Google Scholar
Judd, K. L. 1997. “The Optimal Tax Rate for Capital Income is Negative.” NBER Working Papers 6004, National Bureau of Economic Research, Inc.10.3386/w6004Search in Google Scholar
Judd, K. L. 2002. “Capital Income Taxation with Imperfect Competition.” American Economic Review 92: 417–421.10.1257/000282802320191723Search in Google Scholar
Lee, J.-W. 1995. “Capital Goods Import and Long-Run Growth.” Journal of Development Economics 48: 91–110.10.1016/0304-3878(95)00015-1Search in Google Scholar
Lucas Jr., R. E. 1990. “Supply-Side Economics: An Analytical Review.” Oxford Economic Papers 42: 293–316.10.1093/oxfordjournals.oep.a041948Search in Google Scholar
McDaniel, C. 2007. Average Tax Rates on Consumption, Investment, Labor and Capital in the OECD 1950–2003. Arizon State University. URL: http://www.caramcdaniel.com/researchpapers.Search in Google Scholar
Monacelli, T. 2013. “Is Monetary Policy in an Open Economy Fundamentally Different?” IMF Economic Review 61: 6–21.10.1057/imfer.2013.3Search in Google Scholar
Obstfeld, M., and K. Rogoff. 1998. “Risk and Exchanges Rates.” NBER Working Paper No. 6694.10.3386/w6694Search in Google Scholar
Ramsey, F. 1927. “A Contribution to the Theory of Taxation.” Economic Journal 37: 47–61.10.2307/2222721Search in Google Scholar
Schmitt-Grohé, S., and M. Uribe. 2003. “Closing Small Open Economy Models.” Journal of International Economics 61: 163–185.10.1016/S0022-1996(02)00056-9Search in Google Scholar
The World Bank. 2018. “World Development Indicators.” URL: http://data.worldbank.org/data-catalog/world-development-indicators.Search in Google Scholar
Tille, C. 2001. “The Role of Consumption Substitutability in the International Transmission of Monetary Shocks.” Journal of International Economics 53: 421–444.10.1016/S0022-1996(00)00071-4Search in Google Scholar
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