## Abstract

In this study, the optimal value of a tax on capital inflows is estimated so that private agents internalize the social costs of their borrowing decisions in an economy with financial constraints. A key feature of our model is that we provide a theoretical foundation to tax level differentiation by asset volatility. Using Colombian data for the 1996–2011 period (which includes the crisis of 1998–1999), we find the tax would be around 1.2 %.

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## Notes

- 1.
- 2.
There is an alternative interpretation with similar results. To honor their debts, private agents need to sell part of their assets. This fire-sale implies further reductions in asset prices and, consequently, additional sales of assets and so on (see Aghion et al. (2004), Mendoza (2010), Bianchi and Mendoza (2011), and Stein (2012).

- 3.
- 4.
Aghion et al. (2004) draw attention to the fact that economies at an intermediate level of financial development are the most vulnerable to events involving financial instability.

- 5.
For instance, during the last crisis (1999), the value of (devaluation \(-\) inflation)/(1 + inflation) was 24.1 % for Colombia. The value for Indonesia during its crisis period was 118 %.

- 6.
We assume there is only one type of debt \(d_1\)in the model; however, the conclusions remain the same if we introduce different types of debt (with different risk levels). So, the total amount is equal to the sum of all types. Given the concavity of the utility function (risk aversion), the consumer acquires different types of debt only if higher debt volatility implies a lower interest rate (

*r*). For instance, if there are two types of debt, one with high volatility \(\theta _\mathrm{h}\) and the other with low volatility \(\theta _\mathrm{l} ,\theta _\mathrm{h} > \theta _\mathrm{l}\), it is necessary that \(r_\mathrm{h} <r_\mathrm{l}\). In practice, this situation can be observed when an agent borrows in dollars at a low interest rate (but with high foreign exchange risk) or borrows in local currency (avoiding exchange risk) at a higher interest rate. - 7.
Seeing as there is an amplification effect, this condition guarantees the total impact on consumption of changes in initial debt converges to a finite value. See Footnote 10.

- 8.
- 9.
Had we included different types of debt, the equations would be similar, but \(d_1 ({1\!+\!\theta })\) would have to be replaced by \(\sum \nolimits _\mathrm{j} d_1^\mathrm{j} ({1\!+\!\theta _\mathrm{j}})\) where j indicates a specific asset category with a particular level of volatility (\(\theta _\mathrm{j})\).

- 10.
Using \(d_2^\mathrm{{L}} /R=k\left( {\left( {y_\mathrm{{T}} +p_2^\mathrm{{L}} y_\mathrm{{N}} }\right) -d_1 ({1+\theta })}\right) \) and \(c_{\mathrm{{T,}}2}^\mathrm{{L}} =y_\mathrm{{T}} +d_2^\mathrm{{L}} /R-d_1 (1+\theta )\), the initial effect of increasing \(d_1\) (by one unit) on \(c_{\mathrm{{T,}}2}^\mathrm{{L}}\) is \(-(1+\theta )({1+k})\). This effect on consumption reduces \(p_2^\mathrm{{L}} y_\mathrm{N}\) (Eq. (9)) by \(({1-\sigma })/\sigma \) and consumption is reduced again by \(k({1-\sigma })/\sigma \). The final effect on \(c_{\mathrm{{T,}}2}^\mathrm{{L}} \)is \(-({1+\theta })(1+k)({1+k({1-\sigma })/ \sigma +\left[ {k(1-\sigma )/ \sigma } \right] {^2}+\cdots })\), which, on the condition that \(k({1-\sigma })/ \sigma <1\), converges to \(-({1+\theta })({1+k})\sigma /(\sigma -(1-\sigma )k)\)(see Eq. 12).

- 11.
For the analysis with different types of debt included, we would have several first order conditions, one per each debt type and each of them related to its specific volatility.

- 12.
It is not possible to have \(d_{1,\mathrm{{CP}}} >d_1\). Suppose it is. Then, \(c_{\mathrm{{T,}}2,\mathrm{{CP}}}^\mathrm{{L}} <c_{\mathrm{{T,}}2}^\mathrm{{L}} \). From Eqs. (6) and (14), \(\mu _\mathrm{{CP}}^\mathrm{{L}} >\mu ^\mathrm{{L}}\). Using Eqs. (17) and (18), and the fact that marginal utility is decreasing, \(d_{1,\mathrm{{CP}}} <d_1 \), which is a contradiction.

- 13.
The value of \(\tau \) is obtained by solving: \(\frac{\left( {({1+k})\mu ^\mathrm{{L}}-k}\right) (1+\theta )+1-\theta }{1-\tau } =\left( {({1+k})\mu _\mathrm{{CP}}^\mathrm{{L}} -k}\right) (1+\theta )+1-\theta \).

- 14.
In Eq. (19), the expression for \(\lambda ^\mathrm{{L}}/ \mu ^\mathrm{{L}}\) in terms of the primitive parameters is in closed form only for very particular cases (e.g., logarithmic utility functions).

- 15.
For example, in this paper, calculating the optimal tax by estimating the primitive parameters would necessitate, among other things, estimating the value of interest rates and the volatility of every single asset relevant to the entry of capital flows into Colombia. This data requirement comes from the fact that \(\lambda ^\mathrm{{L}}/ \mu ^\mathrm{{L}}\) incorporates information on \(c_{\mathrm{{T}},2}^\mathrm{{L}}\) and \(d_2^\mathrm{{L}}\), and estimating them calls for information on the aforementioned assets (see Footnote 9).

- 16.
For example, it is not appropriate to analyze how the optimal tax value changes in response to changes in

*k*, using Eq. (19), if the fact that k affects the value of \(\lambda ^\mathrm{{L}}/ \mu ^\mathrm{{L}}\)is not taken into account. - 17.
The Korinek model (2011b) requires the estimation of 9 parameters, 10 in that of Mendoza (2010), 12 in that of Bianchi and Mendoza (2011), and 15 in that of Bianchi (2011). None of these models differentiates the value of the tax by the type of capital flows. However, some of them stochastically model income flows (e.g., 8 out of the 15 parameters estimated by Bianchi (2011) are required to that end).

- 18.
The calibration of this ratio is based on a second order Taylor approximation of the Euler equation in Korinek (2010), which is similar to the one in this paper (Eq. (8)), but more general because it corresponds to a model with infinite periods. In Korinek (2010), the equation is \(\lambda /\mu =1-u^{\prime }(c_{t+1})/ u^{\prime }(c_t)\). The second order Taylor approximation on \(c_{t+1} =c_t \) results in \(\lambda / \mu \approx -\gamma ({\Delta c_{t+1} / c_t})+\gamma ({1+\gamma })({\Delta c_{t+1} / c_t})^2/2\). Korinek (2010) resorts to a first order approximation. However, in the present paper, the second order effects are significant for the estimate at the second-decimal level.

- 19.
The value is highly stable throughout the period, reaching a minimum of 69 % and a maximum of 73 %. It also is similar to the one calibrated by Bianchi (2011) for Argentina (69 %).

- 20.
For its numerical examples, Korinek (2010) takes the maximum value of debt over GDP estimated by Reinhart et al. (2003) at 50 %. However, maximum values could be reached at times when the economy is not in crisis (e.g., in times of high liquidity and international financial boom). Therefore, because the economy is not financially constrained, those values would not be a good approximation of

*k*. - 21.
In solving Eq. (5) for

*k*(when the restriction is binding), the result suggests this parameter should be estimated as the ratio of total debt to net income. However, the simplifications in the model do not make that approach necessarily the most precise one. For example, in the model, debt is paid back in one period; however, in reality, it could be amortized within different time frames. For this reason, a small modification is made, and the borrowing capacity of a constrained economy is measured in a more standard way similar to what is found in other documents (e.g., Korinek (2010)), using the debt level as a proportion of GDP. - 22.
The obtained value is consistent with those estimated by Bianchi (2011) (32 %) and by Bianchi and Mendoza (2011) (36 %), who calibrated the value of

*k*so their models would reproduce the likelihood of a crisis in Argentina and the United States, respectively. In a similar exercise, Mendoza (2010) estimates a value of 20 % for Mexico. - 23.
In this paper, using Eq. (19) and bearing in mind that a is the asset with higher risk and b, the asset with lower risk (\(\theta _\mathrm{b} =0)\), the result is:

$$\begin{aligned} \tau _{\theta _\mathrm{a} } =\frac{x_1}{x_1 +x_2}( {1+\theta _\mathrm{a}})\tau _{\theta _\mathrm{b}}, \end{aligned}$$where \(x_1 =( {1+k})\sigma ({2-\lambda ^\mathrm{L}/\mu ^\mathrm{L}})-2k({1-\lambda ^\mathrm{L}/ \mu ^\mathrm{L}})\) and \(x_2 =\sigma ({1+k})({\lambda ^\mathrm{L}/ \mu ^\mathrm{L}})\theta _\mathrm{a}\). Using the values estimated herein, \(x_1 /({x_1 +x_2})=0.97\). Furthermore, the ratio \(x_1 /({x_1 +x_2 })\) is decreasing in the volatility level.

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## Acknowledgments

The authors wish to thank the editor, Eckhard Janeba, and two anonymous referees for helpful comments. We also thank J. Bejarano, J. E. Gomez, J. Ojeda, H. Rincon, H. Vargas and A. Velasco for their valuable suggestions; A. Korinek for clarifications on the empirical analysis of his document (Korinek 2010). The opinions expressed herein are solely the responsibility of the authors and do not necessarily reflect those of the Central Bank of Colombia or its Board of Directors.

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Parra-Polania, J.A., Vargas, C.O. Optimal tax on capital inflows discriminated by debt-risk profile.
*Int Tax Public Finance* **22, **102–119 (2015). https://doi.org/10.1007/s10797-013-9300-1

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### Keywords

- Optimal tax
- Capital flows
- Externalities
- Financial constraint

### JEL Classification

- H23
- D62
- F34