6. Indeterminacy in Open Economies

This specification of production technology is first introduced by Benhabib and Nishimura (1998) who demonstrate that equilibrium intermediacy may hold even in the absence of social increasing returns. Benhabib et al. (2000), Meng (2003), Meng and Velasco (2003, 2004), Mino (2001), and Nishimura and Shimomura (2002a,b) use the same production functions.

See Footnote 11 on this restriction.

Our discussion depends on this assumption. If at least one country completely specializes, the dynamic system of the world economy becomes different from that examined in this section. However, provided that both countries have identical taste and technology, the steady state equilibrium of the world economy is inside the diversification cone where both countries produce both goods. Therefore, our assumption is justified as long as we focus on the local dynamics of the world economy around the steady state equilibrium. To analyze the global behavior of the model, we need to treat the model out of the diversification cone. Atkeson and Kehoe (2000) explored the dynamic behavior of a small country that specializes in producing one of the two goods. Caliendo (2010) presents a detailed analysis of the dynamic behavior of a 2 × 2 × 2 model outside the diversification cone.

The precise expression of \(1/\bar{\sigma }\) is

This conclusion depends on the functional forms of production and utility functions we use as well as on the fact that we restrict our attention to the model behavior near the steady state. As for more general analyses on income and wealth distribution among the countries in the Heckscher-Ohlin setting, see Atkeson and Kehoe (2000) and Bajona and Kehoe (2010). Atkeson and Kehoe treated a small-country model, while Bajona and Kehoe (2010) explored a two-country model.

In the small-country setting, the trade structure assumed here is a type of dependent economy model discussed in open-economy macroeconomics literature. Turnovsky and Sen (1995) treated a small-open economy model with non-tradable capital and Turnovsky (1997, Chapter 7) studied a neoclassical two-country, two-sector model in which capital goods are not traded. Mino (2008) also examined a similar two-country model with external increasing returns. See also Chapter 5 for a brief literature review.

Since \(\widetilde{p} = 1/p\), the precise expression of \(\,\widetilde{r}\,\left (1/\,\widetilde{p}\,\right )\) is \(\widetilde{r}\left (1\,\widetilde{p}\,\right ) = r/p = pa_{1}A_{1}k_{1}\left ( \frac{1} {\widetilde{p}^{}}\right )^{\alpha _{1}-1}\).

The Rybczynski effect of a change in factor endowment depends on the factor-intensity ranking from the private perspective, while the Stolper-Samuelson effect of a price change depends on the factor-intensity ranking from the social perspective. Therefore, if the private and social factor intensity rankings are the same, we obtain the standard results and, hence, equilibrium indeterminacy will not arise.

See, for example, Ghiglino and Olszak-Duquenne (2005).

In the existing literature, Antras and Caballero (2009) introduced financial frictions into the Heckscher-Ohlin mode. Ono and Shibata (2010) and Jin (2012) introduced adjustment costs of investment into 2 × 2 × 2 models.

The structure of the base model is close to the two-country model examined by Turnovsky (1997; Chapter 7). Since Turnovsky (1997) did not assume the presence of external increasing returns, indeterminacy of equilibrium is not the issue in his argument. Baxter and Crucini (1995) used a similar model in their study on international real business cycles.

This assumption is introduced only for notational simplicity.

Denote the steady state values of v _i as v _i ^∗ and v _i ^∗∗ (v _i ^∗ < v _i ^∗∗). It is seen that \(d\dot{v}_{i}/dv_{i} <0\) for v _i = v _i ^∗ and \(d\dot{v}_{i}/dv_{i}> 0\) for v _i = v _i ^∗∗. Since v _i is not a predetermined variable, indeterminacy emerges around v _i = v _i ^∗.

When the dynamic system of \(\left (K,K^{{\ast}},q,q^{{\ast}}\right )\) satisfies equilibrium determinacy under a given level of \(\bar{m}\), then the equilibrium paths of K and K ^∗ are uniquely determined under given levels of K ₀ and K ₀ ^∗. Therefore, the equilibrium path of (6.55) is also uniquely determined. Conversely, if the dynamic system of \(\left (K,K^{{\ast}},q,q^{{\ast}}\right )\) exhibits local indeterminacy, the equilibrium paths of K and K ^∗ cannot be uniquely determined by selecting K ₀ and K ₀ ^∗. This means that (6.55) also holds equilibrium indeterminacy.