Indeterminacy in Open Economies

Abstract

In the previous chapters, we restrict our attention to closed-economy models. This chapter examines equilibrium indeterminacy in open economies. The central concern of this chapter is to explore how international transactions affect the dynamic behaviors of macroeconomies. In particular, we focus on the difference in the indeterminacy conditions between open economies and their closed economy counterparts. We first discuss small open economies and then explore the world economy consisting of two large countries.

Appendices

Appendices

6.1.1 Appendix 1: Proof of Proposition6.1

When $\dot{q} =\dot{ q}^{{\ast}} = 0$ in (6.48a) and (6.48b), it holds that

$$\displaystyle{ a_{1}A_{1}k_{1}\left (\,p\right )^{\alpha _{1}-1} = a_{1}A_{1}k_{1}\left (\,p^{{\ast}}\right )^{\alpha _{1}-1} =\rho +\delta. }$$

Hence, by use of (6.15a), we find

$$\displaystyle{ p = p^{{\ast}} = \left (\frac{A_{2}} {A_{1}}\right )\left (\frac{a_{2}} {a_{1}}\right )^{\alpha _{2}}\left (\frac{b_{2}} {b_{1}}\right )^{1-\alpha _{2} }\left ( \frac{\rho +\delta } {a_{1}A_{1}}\right )^{\frac{\alpha _{2}-\alpha _{1}} {\alpha _{1}-1} }. }$$

These conditions show that the steady state levels of p and p ^∗ are uniquely given and it holds that p = p ^∗ in the steady state. The steady state levels of capital stocks satisfying $\dot{K} =\dot{ K}^{{\ast}} = 0$ in (6.47a) and (6.47b) are determined by the following conditions:

$$\displaystyle{ \frac{K - k_{2}\left (\,p\right )} {k_{1}\left (\,p\right ) - k_{2}\left (\,p\right )}A_{1}k_{1}\left (\,p\right )^{\alpha _{1}} =\delta K, }$$

$$\displaystyle{ \frac{K^{{\ast}}- k_{2}\left (\,p^{{\ast}}\right )} {k_{1}\left (\,p^{{\ast}}\right ) - k_{2}\left (\,p^{{\ast}}\right )}A_{1}k_{1}\left (\,p^{{\ast}}\right )^{\alpha _{1} } =\delta K^{{\ast}}. }$$

Using the conditions for $\dot{p} =\dot{ p}^{{\ast}} = 0$ and the fact that p = p ^∗ holds in the steady state, we confirm that the steady state level of capital stock in each county has the same value given by

$$\displaystyle{ K = K^{{\ast}} = \frac{\left (aA_{1}\right )^{ \frac{1} {1-\alpha _{1}} }\left (\rho +\delta \right )^{ \frac{\alpha _{1}} {\alpha _{1}-1} }} {\rho +\delta \left (1 -\delta +\frac{a_{2}b_{1}} {b_{2}} \right )} \left (\frac{a_{2}b_{1}} {a_{1}b_{2}}\right ), }$$

which has a positive value. We also find that the steady state values of labor allocation to the investment good sector are

$$\displaystyle{ L_{1} = L_{1}^{{\ast}} = \frac{a_{1}\delta \left (\frac{a_{2}b_{1}} {a_{1}b_{2}} \right )} {\rho +(1 - a_{1})\delta + a_{1}\delta \left (\frac{a_{2}b_{1}} {a_{1}b_{2}} \right )} \in \left (0,1\right ). }$$

Hence, (6.20) is fulfilled so that both countries specialize imperfectly.

6.1.2 Appendix 2: Proof of Proposition6.2

Since the functional form of $R\left (K,K^{{\ast}},p,p^{{\ast}}\right )$ in (6.55) is complicated, it is simpler to treat a dynamic system with respect to K, K ^∗, q, and q ^∗. We thus focus on the dynamic system consisting of (6.47a), (6.47b), (6.48a), and (6.48b) with $p =\pi \left (K,K^{{\ast}},q,q^{{\ast}};\bar{m}\right )$ and $p^{{\ast}} =\pi ^{{\ast}}\left (K,K^{{\ast}},q,q^{{\ast}};\bar{m}\right )$, where $\bar{m}$ is fixed.¹⁴

To prove Proposition 6.2, the following facts are useful:

Lemma A.1

In the symmetric steady state in which K = K ^∗ and q = q ^∗, the following relations are satisfied:

$$\displaystyle\begin{array}{rcl} & y_{K}^{i}\left (K,p\right ) = y_{K^{{\ast}}}^{i}\left (K^{{\ast}},p^{{\ast}}\right ),\ \ \ i = 1,2, & {}\\ & y_{p}^{i}\left (K,p\right ) = y_{p^{{\ast}}}^{i}\left (K^{{\ast}},p^{{\ast}}\right ),\ \ \ i = 1,2, & {}\\ & \pi _{K}\left (K,K^{{\ast}},q,q^{{\ast}}\right ) =\pi _{ K}^{{\ast}}\left (K,K^{{\ast}},q,q^{{\ast}}\right ) =\pi _{K^{{\ast}}}\left (K,K^{{\ast}},q,q^{{\ast}}\right ) =\pi _{ K^{{\ast}}}^{{\ast}}\left (K,K^{{\ast}},q,q^{{\ast}}\right ),& {}\\ & \pi _{q}\left (K,K^{{\ast}},q,q^{{\ast}}\right ) =\pi _{ q^{{\ast}}}^{{\ast}}\left (K,K^{{\ast}},q,q^{{\ast}}\right ), & {}\\ & \pi _{q^{{\ast}}}\left (K,K^{{\ast}},q,q^{{\ast}}\right ) =\pi _{ q}^{{\ast}}\left (K,K^{{\ast}},q,q^{{\ast}}\right ). & {}\\ \end{array}$$

Proof

By the functional forms of $y_{j}^{i}\left (\cdot \right )\ (i = 1,2,j = K,K^{{\ast}},p,p^{{\ast}})$, it is easy to see that $y_{K}^{i}\left (K,p\right ) = y_{K^{{\ast}}}^{i}\left (K^{{\ast}},p^{{\ast}}\right )$ and $y_{p}^{i}\left (K,p\right ) = y_{p^{{\ast}}}^{i}\left (K^{{\ast}},p^{{\ast}}\right )$ are established when p = p ^∗ and K = K ^∗.As for the rest of the results, we use $p\lambda \left (\cdot \right ) = q$ and $p^{{\ast}}\lambda \left (\cdot \right )\bar{m}^{-\sigma } = q^{{\ast}}$. the total differentiation of $p\lambda \left (\cdot \right ) = q$ and $p^{{\ast}}\lambda \left (\cdot \right )\bar{m}^{-\sigma } = q^{{\ast}}$ yields the following:

$$\displaystyle{ \frac{\partial p} {\partial K} =\pi _{K} = - \frac{p\lambda _{K}} {\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}},\ \ \ \ \frac{\partial p} {\partial K^{{\ast}}} =\pi _{K^{{\ast}}} = - \frac{p\lambda _{K^{{\ast}}}} {\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}}, }$$

$$\displaystyle{ \ \ \ \ \frac{\partial p^{{\ast}}} {\partial K} =\pi _{ K}^{{\ast}} = - \frac{p^{{\ast}}\lambda _{ K}} {\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}},\ \ \ \ \frac{\partial p^{{\ast}}} {\partial K^{{\ast}}} =\pi _{ K^{{\ast}}}^{{\ast}} = - \frac{p^{{\ast}}\lambda _{ K^{{\ast}}}} {\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}}, }$$

$$\displaystyle{ \frac{\partial p} {\partial q} =\pi _{q} = \frac{\lambda +p^{{\ast}}\lambda _{p^{{\ast}}}} {\lambda \left (\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}\right )},\ \ \ \frac{\partial p} {\partial q^{{\ast}}} =\pi _{q^{{\ast}}} = - \frac{p\lambda _{p}} {\lambda \left (\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}\right )}, }$$

$$\displaystyle{ \frac{\partial p^{{\ast}}} {\partial q} =\pi _{ q}^{{\ast}} = - \frac{p^{{\ast}}\lambda _{ p^{{\ast}}}} {\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}},\ \ \ \ \frac{\partial p^{{\ast}}} {\partial q^{{\ast}}} =\pi _{ q^{{\ast}}}^{{\ast}} = \frac{\lambda +p\lambda _{p}} {\lambda +p\lambda _{P} + p^{{\ast}}\lambda _{p^{{\ast}}}}. }$$

Since $\lambda _{K}(\cdot ) =\lambda _{K^{{\ast}}}\left (\cdot \right )$ and $\lambda _{p}\left (\cdot \right ) =\lambda _{p^{{\ast}}}\left (\cdot \right )$ in the steady state where K = K ^∗ and p = p ^∗, we obtain $\pi _{K} =\pi _{ K}^{{\ast}} =\pi _{K^{{\ast}}} =\pi ^{{\ast}},\ \pi _{q} =\pi _{ q^{{\ast}}}^{{\ast}}$ and $\pi _{q^{{\ast}}} =\pi _{ q}^{{\ast}}$. ■

Under a given level of $\bar{m}$, let us linearize the dynamic system of (6.47a), (6.47b), (6.48a), and (6.48b) at the steady state. The coefficient matrix of the linearized system is given by

$$\displaystyle{ J = \left [\begin{array}{cccc} y_{K}^{1} -\delta +y_{p}^{1}\pi _{K}& y_{p}^{1}\pi _{K^{{\ast}}} & y_{p}^{1}\pi _{q} & y_{p}^{1}\pi _{q^{{\ast}}} \\ y_{p^{{\ast}}}^{1}\pi _{K}^{{\ast}} &y_{K^{{\ast}}}^{1} -\delta +y_{p^{{\ast}}}^{1}\pi _{K^{{\ast}}}^{{\ast}}& y_{p^{{\ast}}}^{1}\pi _{q}^{{\ast}} & y_{p^{{\ast}}}^{1}\pi _{q^{{\ast}}}^{{\ast}} \\ - q\hat{r}^{{\prime}}\pi _{K} & - q\hat{r}^{{\prime}}\pi _{K^{{\ast}}} & - q\hat{r}^{{\prime}}\pi _{q} & - q\hat{r}^{{\prime}}\pi _{q^{{\ast}}} \\ - q\hat{r}^{{\prime}}\pi _{K}^{{\ast}} & - q\hat{r}^{{\prime}}\pi _{K^{{\ast}}}^{{\ast}} &- q\hat{r}^{{\prime}}\pi _{q}^{{\ast}}&- q\hat{r}^{{\prime}}\pi _{q^{{\ast}}}^{{\ast}} \end{array} \right ]. }$$

By use of Lemma A.1, we see that the characteristic equation of J is written as

$$\displaystyle\begin{array}{rcl} \Gamma \left (\eta \right )& =& \det \left [\eta I - J\right ] {}\\ & =& \det \left [\begin{array}{cccc} \eta - (y_{K}^{1} -\delta +y_{p}^{1}\pi _{K})& - y_{p}^{1}\pi _{K} & - y_{p}^{1}\pi _{q} & - y_{p}^{1}\pi _{q^{{\ast}}} \\ - y_{p}^{1}\pi _{K} &\eta - (y_{K}^{1} -\delta +y_{p}^{1}\pi _{K})& - y_{p}^{1}\pi _{q^{{\ast}}}& - y_{p}^{1}\pi _{q} \\ q\hat{r}^{{\prime}}\pi _{K} & q\hat{r}^{{\prime}}\pi _{K} & \eta + q\hat{r}^{{\prime}}\pi _{q} & q\hat{r}^{{\prime}}\pi _{q^{{\ast}}} \\ q\hat{r}^{{\prime}}\pi _{K} & q\hat{r}^{{\prime}}\pi _{K} & q\hat{r}^{{\prime}}\pi _{q} & \eta + q\hat{r}^{{\prime}}\pi _{q} \end{array} \right ] {}\\ & =& \det \left [\begin{array}{cccc} \eta -\left (y_{K}^{1}-\delta \right )& 0 & \eta & 0 \\ 0 &\eta - (y_{K}^{1}-\delta )& 0 & \eta \\ q\hat{r}^{{\prime}}\pi _{K} & q\hat{r}^{{\prime}}\pi _{K} &\eta + q\hat{r}^{{\prime}}\pi _{q}& q\hat{r}^{{\prime}}\pi _{q^{{\ast}}} \\ q\hat{r}^{{\prime}}\pi _{K} & q\hat{r}^{{\prime}}\pi _{K} & q\hat{r}^{{\prime}}\pi _{q} &\eta + q\hat{r}^{{\prime}}\pi _{q} \end{array} \right ] {}\\ & =& \left [\eta -\left (y_{K}^{1}-\delta \right )\right ]\left [\eta +q\hat{r}^{{\prime}}(\pi _{ q} -\pi _{q^{{\ast}}})\right ]\xi \left (\eta \right )\text{,} {}\\ \end{array}$$

where η denotes the characteristic root of J and

$$\displaystyle{ \xi \left (\eta \right ) \equiv \eta ^{2} + \left [q\hat{r}^{{\prime}}\left (\pi _{ q} +\pi _{q^{{\ast}}}\right ) -\left (y_{K}^{1}-\delta \right ) - 2y_{ p}^{1}\pi _{ K}\right ]\eta - q\hat{r}^{{\prime}}\left (y_{ K}^{1}-\delta \right )\left (\pi _{ q} +\pi _{q^{{\ast}}}\right ). }$$

Our assumptions mean that $\frac{a_{1}} {b_{1}} -\frac{a_{2}} {b_{2}} < 0\$ and α ₁ −α ₂ > 0. Thus from (6.21a) we see that y _K ¹ −δ < 0. In addition, as to the partial derivatives of  $\pi \left (.\right )$ function displayed above, we see that $\pi _{q} -\pi _{q^{{\ast}}} = 1/\lambda \left (> 0\right )$. Hence, using $\hat{r}\left (\,p\right ) \equiv a_{1}A_{1}k_{1}\left (\,p\right )^{\alpha _{1}-1}$, we obtain

$$\displaystyle{ \hat{r}^{{\prime}}\left (\pi _{ q} -\pi _{q^{{\ast}}}\right ) = a_{1}\left (a_{1} - 1\right )A_{1}\left (k_{1}\left (\,p\right )\right )^{a_{1}-2}\frac{k_{1}^{{\prime}}\left (\,p\right )} {\lambda }> 0. }$$

As a consequence, at least two roots of $\Gamma \left (\eta \right ) = 0$ have negative real parts. Similarly, we find that

$$\displaystyle{ \pi _{q} +\pi _{q^{{\ast}}} = \frac{1} {\lambda +2p\lambda _{p}}, }$$

where

$$\displaystyle\begin{array}{rcl} \lambda _{p}& =& \frac{\partial } {\partial p}\left (1 +\bar{ m}\right )^{\frac{1} {\sigma } }\left [y^{2}\left (K,p\right ) + y^{2}\left (K^{{\ast}},p^{{\ast}}\right )\right ]^{-\frac{1} {\sigma } } {}\\ & =& -\frac{y_{p}^{2}} {\sigma } \left (1 +\bar{ m}\right )^{\frac{1} {\sigma } }\left [y^{2}\left (K,p\right ) + y^{2}\left (K^{{\ast}},p^{{\ast}}\right )\right ]^{-\frac{1} {\sigma } -1} <0. {}\\ \end{array}$$

Therefore, in the steady state equilibrium. the following holds:

$$\displaystyle{ \lambda +2p\lambda _{p} = \frac{1} {\sigma } \left [\sigma -\frac{py_{p}^{2}\left (K,p\right )} {y^{2}\left (K,p\right )} \right ]. }$$

Note that under our assumptions, it holds that $y_{p}^{2}\left (K,p\right )> 0$. Suppose that σ is small enough to satisfy σ < py _p ²∕y ². Then, λ _p + 2pλ _p > 0 so that $\pi _{q} +\pi _{q^{{\ast}}} <0$, which leads to

$$\displaystyle{ -q\hat{r}^{{\prime}}\left (y_{ K}^{1}-\delta \right )\left (\pi _{ q} +\pi _{q^{{\ast}}}\right ) <0. }$$

This means that $\xi \left (\eta \right ) = 0$ has one positive and one negative root. As a result, $\Gamma \left (\eta \right ) = 0$ has three stable roots. Hence, if σ is smaller than the price elasticity of supply function of consumption goods, then there locally exists a continuum of equilibrium paths converging to the steady state.

Now suppose that σ is larger than py _p ²∕y ². Then, we obtain $\pi _{q} +\pi _{q^{{\ast}}}> 0$. Furthermore, it holds that

$$\displaystyle\begin{array}{rcl} -2y_{p}^{1}\pi _{ K}& =& -2y_{p}^{1}\left (- \frac{p\lambda _{K}} {\lambda +2p\lambda _{p}}\right ) {}\\ & =& -\frac{2py_{p}^{1}} {\lambda +2p\lambda _{p}} y_{K}^{2}\left [\frac{(1 +\bar{ m})^{\sigma ^{-1} }} {\sigma } \right ]\left (2y^{2}\right )^{-\sigma ^{-1}-1 }> 0, {}\\ \end{array}$$

because y _p ¹ < 0 and y _K ² > 0 under our assumptions. Consequently, the following inequalities are established:

$$\displaystyle\begin{array}{rcl} & -q\hat{r}^{{\prime}}\left (y_{K}^{1}-\delta \right )\left (\pi _{q} +\pi _{q^{{\ast}}}\right )> 0, & {}\\ & q\hat{r}^{{\prime}}\left (\pi _{q} +\pi _{q^{{\ast}}}\right ) -\left (y_{K}^{1}-\delta \right ) - 2y_{K}^{1}\pi _{K}> 0.& {}\\ \end{array}$$

These conditions mean that $\xi \left (\eta \right ) = 0$ has two roots with negative real parts and, hence, all the roots of $\Gamma \left (\eta \right ) = 0$ are stable ones. In sum, if $\frac{a_{1}} {b_{1}} -\frac{a_{2}} {b_{2}} <0$ and α ₁ −α ₂ > 0, then the characteristic equation of the linearized system involves at least three stable roots, meaning that the converging path towards the steady state is locally indeterminate.