Non-linear pattern of international capital flows

Abstract

In a two-country OLG model with interest rate wedges capturing financial frictions, international capital flows can follow a non-linear pattern, depicted as a U-shaped curve, by first decreasing and then increasing in growth. The turning point of the curve is determined by the world average growth rate and these interest rate wedges. The model developed in this paper can reconcile different theories (i.e, the implications of the neoclassical growth model, up-hill capital flows, and the allocation puzzle) on the pattern of international capital flows.

Appendices

Appendix

1 Definition of equilibrium

Definition 1

The autarky temporal equilibrium.

Given the value determined in the last period $(S_{t-1})$, the exogenous variable $(A_t,N_t)$, and the expected interest rate with perfect foresight $(R^{l}_{t+1})$, the temporary equilibrium is a list of prices $(R^l_t, R^d_t, R_t,w_t)$, allocations $(c^y_t, c^o_t, s_t, I_t)$, and aggregate variables $(K_t, S_t, L_t, Y_t)$ such that:

• 1. Profit is maximized and utility is maximized subjected to the budget constraint.

• 2. The market clearing conditions are satisfied:

  • (a) Labor market: $L_t = N_t$

  • (b) Capital market: $K_{t} = S_{t-1} = N_{t-1} s_{t-1}$

  • (c) Goods market: $N_t c^y_t + N_t s_t + N_{t-1} c^o_t = Y_t$

Definition 2

The autarky inter-temporal equilibrium.

Given an initial lending interest rate $(R^l_{t=0})$, an inter-temporal equilibrium with perfect foresight is a sequence of temporary equilibria that satisfies for all $(t>0)$ the conditions that:

$$\begin{aligned} k(R^l_{t+1},\gamma )=\; & {} s(R^l_t, R^l_{t+1}, \tau , \gamma ) \\ \Leftrightarrow \left(\dfrac{\alpha (1-\gamma )}{R^l_{t+1}}\right)^{1/(1-\alpha )}= & {} \dfrac{(1-\alpha ){(\alpha (1-\gamma ))}^{\alpha /(1-\alpha )}\beta (1-\tau )}{(1+\beta (1-\tau )){(R^l_t)}^{\alpha /(1-\alpha )}} \dfrac{1}{(1+g^A_{t+1})(1+g^N_{t+1})}- \dfrac{\gamma \alpha [\alpha (1-\gamma )]^{\alpha /(1-\alpha )}}{(1+\beta (1-\tau ))(R^l_{t+1})^{1/(1-\alpha )}} \end{aligned}$$

Definition 3

The integration temporal equilibrium.

For country (j) in a two-country economy, given the value determined in the last period $(S^j_{t-1})$, the exogenous variable $(A^j_t,N^j_t)$, and the expected world interest rate with perfect foresight $(R^{l,w}_{t+1})$, the temporary equilibrium is a list of prices $(R^{l,w}_t, R^{d,j}_t, R^j_t,w^j_t)$, allocations $(c^{y,j}_t, c^{o,j}_t, s^j_t, I^j_t)$, and aggregate variables $(K^j_t, S^j_t, L^j_t, Y^j_t)$ in each country j such that:

• 1. Profit is maximized and utility is maximized subjected to the budget constraint.

• 2. The market clearing conditions are satisfied:

  • (a) Labor market: $L^{D,j}_t = N^j_t$

  • (b) World capital market: $\Sigma _j K^j_{t} = \Sigma _j S^j_{t-1} \equiv \Sigma _j N^j_{t-1} s^j_{t-1}$

  • (c) World goods market: $\Sigma _j (N^j_t c^{y,j}_t + N^j_t s^j_t + N^j_{t-1} c^{o,j}_t ) = \Sigma _j Y^j_t$

Definition 4

The integration inter-temporal equilibrium.

For country j in a two-country economy, given an initial world lending interest rate $(R^{l,w}_{t=0})$, an inter-temporal equilibrium with perfect foresight is a sequence of temporary equilibria that satisfies for all $t>0$ the conditions:

$$\begin{aligned} \Sigma _j \dfrac{A^j_{t} N^j_{t}}{\Sigma _jA^j_{t} N^j_{t}} k(R^{l,w}_{t+1},\gamma ^j)=\; & {} \Sigma _j \dfrac{A^j_{t} N^j_{t}}{\Sigma _jA^j_{t} N^j_{t}} s(R^{l,w}_t, R^{l,w}_{t+1}, \tau ^j, \gamma ^j) \end{aligned}$$

where $k(R^{l,w}_{t+1},\gamma ^j)\equiv \dfrac{K^j_{t+1}}{A^j_{t} N^j_{t}};\; s(R^{l,w}_t, R^{l,w}_{t+1}, \tau ^j, \gamma ^j)\equiv \dfrac{S^j_t}{A^j_{t}N^j_{t}}$.

2 Proofs

Theorem 2.2.1

Proof

We define the function $\Delta (R^l_{t+1}, R^l_t)$ as follows:

$$\begin{aligned}&\Delta (R^l_{t+1}, R^l_t) \equiv k(R^l_{t+1},\gamma ) - s(R^l_{t+1},R^l_t,\tau ,\gamma ) \\&\quad= \left[ (\alpha (1-\gamma ))^{1/(1-\alpha )}+\dfrac{\gamma \alpha [\alpha (1-\gamma )]^{\alpha /(1-\alpha )}}{1+\beta (1-\tau )}\right] \dfrac{(1+g^A_{t+1})(1+g^N_{t+1})}{(R^l_{t+1})^{1/(1-\alpha )}}- \dfrac{(1-\alpha ){(\alpha (1-\gamma ))}^{\alpha /(1-\alpha )}\beta (1-\tau )}{1+\beta (1-\tau )}\dfrac{1}{{(R^l_t)}^{\alpha /(1-\alpha )}} \end{aligned}$$

• 1. Existence of steady state

  $$\begin{aligned} \lim _{R^l_{t+1} \rightarrow 0} \Delta (R^l_{t+1}, R^l_t) > 0 \hbox { and} \lim _{R^l_{t+1} \rightarrow \infty } \Delta (R^l_{t+1}, R^l_t) < 0 \end{aligned}$$

• 2. Uniqueness and global stability.

Since $\dfrac{\partial \Delta (R^l_{t+1}, R^l_t)}{\partial R^l_{t+1}} \ne 0$, $R^l_{t+1}$ is an implicit function of $R^l_t$, denoted by: $R^l_{t+1} = h(R^l_t)$. By the implicit function theorem, $\dfrac{\partial R^l_{t+1}}{\partial R^l_t}>0$, since:

$$\begin{aligned}&\dfrac{\partial \Delta }{\partial R^l_{t+1}} = - \left[ (\alpha (1-\gamma ))^{1/(1-\alpha )}+\dfrac{\gamma \alpha [\alpha (1-\gamma )]^{\alpha /(1-\alpha )}}{1+\beta (1-\tau )}\right] \dfrac{(1+g^A_{t+1})(1+g^N_{t+1})}{(1-\alpha )(R^l_{t+1})^{(2-\alpha )/(1-\alpha )}}<0\\&\dfrac{\partial \Delta }{\partial R^l_{t}} = \dfrac{\alpha {(\alpha (1-\gamma ))}^{\alpha /(1-\alpha )}\beta (1-\tau )}{1+\beta (1-\tau )}\dfrac{1}{{(R^l_t)}^{1/(1-\alpha )}}>0 \end{aligned}$$

Therefore, $h(R^l_t)$ is a monotonically increasing function and converges to a positive value, if any. By l’Hôpital’s rule, the graph of $h(R^l_t)$ would be below the 45 degree line for a large enough value of $R^l_t$.

$$\begin{aligned} \lim _{R^l_t \rightarrow \infty } \dfrac{R^l_{t+1}}{R^l_t} = 0 <1 \end{aligned}$$

In sum, $R^l_{t+1} = h(R^l_t)$ is both a monotonically increasing and concave function. Then, it would converge to a finite positive value $0<R^l<\infty$. The comparative statics are easy to derive since we have a closed-form solution for the interest rate for $\sigma =1$. $\square$

Theorem 2.2.2

Proof

For country j in a two-country economy, we define the function $\Delta (R^{l,w}_{t+1}, R^{l,w}_t)$ as follows:

$$\begin{aligned}&\Delta (R^{l,w}_{t+1}, R^{l,w}_t) \equiv \Sigma _j \dfrac{A^j_{t} N^j_{t}}{\Sigma _jA^j_{t} N^j_{t}} k(R^{l,w}_{t+1},\gamma ^j) - \Sigma _j \dfrac{A^j_{t} N^j_{t}}{\Sigma _jA^j_{t} N^j_{t}} s(R^{l,w}_t, R^{l,w}_{t+1}, \tau ^j, \gamma ^j) \\&=\left( \dfrac{1}{R^{l,w}_{t+1}}\right) ^{1/(1-\alpha )} \Sigma _j \dfrac{A^j_{t} N^j_{t}}{\Sigma _jA^j_{t} N^j_{t}} \left[ (\alpha (1-\gamma ^j))^{1/(1-\alpha )}+\dfrac{\gamma ^j\alpha [\alpha (1-\gamma ^j)]^{\alpha /(1-\alpha )}}{1+\beta (1-\tau ^j)}\right] (1+g^{A,j}_{t+1}) (1+g^{N,j}_{t+1}) \\&\quad - \left( \dfrac{1}{R^{l,w}_t}\right) ^{\alpha /(1-\alpha )} \Sigma _j \dfrac{A^j_{t} N^j_{t}}{\Sigma _jA^j_{t} N^j_{t}} \dfrac{(1-\alpha ){(\alpha (1-\gamma ))}^{\alpha /(1-\alpha )}\beta (1-\tau ^j)}{(1+\beta (1-\tau ^j))} \end{aligned}$$

• 1. Existence of steady state

  $$\begin{aligned} \lim _{R^{l,w}_{t+1} \rightarrow 0} \Delta (R^{l,w}_{t+1}, R^{l,w}_t) > 0 \hbox { and} \lim _{R^{l,w}_{t+1} \rightarrow \infty } \Delta (R^{l,w}_{t+1}, R^{l,w}_t) < 0 \end{aligned}$$

• 2. Uniqueness and global stability

Since $\dfrac{\partial \Delta (R^{l,w}_{t+1}, R^{l,w}_t)}{\partial R^{l,w}_{t+1}} \ne 0$, then $R^{l,w}_{t+1}$ is a function of $R^{l,w}_t: R^{l,w}_{t+1} \equiv h(R^{l,w}_t)$.

By the implicit function theorem, $\dfrac{\partial \Delta }{\partial R^{l,w}_{t+1}} < 0$ and $\dfrac{\partial \Delta }{\partial R^{l,w}_{t}} > 0$. Then, $\dfrac{\partial R^{l,w}_{t+1}}{\partial R^{l,w}_t}>0$. Therefore, $h(R^{l,w}_t)$ is a monotonically increasing function, and it would converge to a positive value, if any.

Moreover, by l’Hôpital’s rule,

$$\begin{aligned} \lim _{R^{l,w}_t \rightarrow \infty } \dfrac{R^{l,w}_{t+1}}{R^{l,w}_t} = 0 <1 \end{aligned}$$

In sum, $R^{l,w}_{t+1} = h(R^{l,w}_t)$ is both a monotonically increasing and concave function since its graph would be below the 45 degree line for a large enough value of $R^{l,w}_t$. Then, it would converge to a finite positive value $0<R^{l,w}<\infty$.

For the case of a symmetric growth rate across countries, $\lambda ^j_t\equiv \dfrac{Y^j_t}{Y^w_t}, \theta ^j_t\equiv \dfrac{\lambda ^j_t \vartheta ^j}{\bar{\vartheta }}$,

$$\begin{aligned} R^{l,w}_t=\; & {} \dfrac{\alpha }{(1-\alpha )} \dfrac{Y^w_t}{Y^w_{t-1}} \left( \dfrac{\Sigma _j \lambda ^j_t \left[ \mu (\gamma ^j,\tau ^j)+1-\gamma ^j\right] }{\bar{\vartheta }}\right) = \dfrac{\alpha }{(1-\alpha )} \dfrac{Y^w_t}{Y^w_{t-1}} \Sigma _j \lambda ^j_t \dfrac{\vartheta ^j}{\bar{\vartheta }} \left( \dfrac{\left[ \mu (\gamma ^j,\tau ^j)+1-\gamma ^j\right] }{\vartheta ^j}\right) \\=\; & {} \dfrac{\alpha }{(1-\alpha )} \dfrac{Y^w_t}{Y^w_{t-1}} \Sigma _j \theta ^j_t \left( \dfrac{\left[ \mu (\gamma ^j,\tau ^j)+1-\gamma ^j\right] }{\vartheta ^j}\right) =\Sigma _j \theta ^j_t R^{l,j}_t \end{aligned}$$

Therefore, $min_j R^{l,j}_t< R^{l,w}_t < max_j R^{l,j}_t$. $\square$

Appendix 3: CRRA utility

We show that, with a general utility function, the main mechanism whereby the equilibrium interest rate is increasing in the saving wedge still holds. Therefore, the use of log-utility in the main text is valid.

In particular, we employ the standard isoelastic preferences $u(c) = (c^{1-1/\sigma } - 1)/(1-1/\sigma )$, where $\sigma$ is the intertemporal elasticity of substitution coefficient. The Euler equation yields the saving function for the young agent.

$$\begin{aligned} \dfrac{u'(c^y_t)}{u'(c^o_{t+1})} = \beta (1-\tau ) R^l_{t+1} \Rightarrow s_t = \dfrac{w_t - (\beta (1-\tau ) R^l_{t+1})^{-\sigma } z_{t+1}}{1+\beta ^{-\sigma } ((1-\tau ) R^l_{t+1})^{(1-\sigma )}} \end{aligned}$$

(21)

Using the income transfer (6), the shares of output allocated to labor income $(w_t N_t = (1-\alpha )Y_t)$ and capital income $(R_{t+1} I_{t+1} =R_{t+1} K_{t+1} = \alpha Y_{t+1})$, we obtain the total savings in equilibrium.

$$\begin{aligned} S_t = N_t s_t=\dfrac{(1-\alpha ) Y_{t} - [\beta (1-\tau ) R^l_{t+1}]^{-\sigma } \gamma \alpha Y_{t+1}}{1+[\beta (1-\tau )]^{-\sigma } (R^l_{t+1})^{(1-\sigma )}} \end{aligned}$$

With an income transfer by the government, the saving wedge enters the saving-output ratio $(S_t/Y_t)$ with the power $(-\sigma )$. Therefore, given the output growth rate $(Y_{t+1}/Y_t)$, the saving rate is always decreasing in the level of the saving wedge. As a result, a country with a smaller saving wedge would have a higher saving rate. Since the saving-output ratio is decreasing in the saving wedge for all values of the inter-temporal elasticity of substitution coefficient while the investment-output ratio is independent of the inter-temporal elasticity of substitution coefficient, we can focus on the case of log utility $(\sigma =1)$ without losing generality.

The positive impact of the saving wedge on the lending interest rate relies on the modeling of capital taxation. In the set-up where the income taxation is a transfer to the households, a larger saving wedge reduces savings, then raises the lending rate, even in the case of a more general isoelastic utility function. Indeed, by Eq. (21) in partial equilibrium, the wedge affects the deposit interest rate and, in turn, saving through the substitution and income effects. A larger wedge decreases the deposit rate, which, in turn, reduces the saving by the substitution effect but raises the saving by the income effect. Therefore, the role of the wedge in saving depends on the value of the inter-temporal substitution coefficient $(\sigma )$. In general equilibrium with the taxation income being transferred to households, the income effect disappears. Then, only the substitution effect works: a larger wedge curtails saving and raises the lending rate, given the investment demand.

However, when the taxation income is used to finance public expenditure, the relationship between saving and taxation depends on the value of $(\sigma )$. In the online Appendix, we show that with public expenditure and for $0<\sigma <1$, the income effect dominates the substitution effect. In that case, the lending rate is decreasing in the saving wedge because a larger saving wedge raises the supply of savings, which reduces the autarky interest rate, given the investment demand.