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This paper analyzes the characteristics of asset bubbles in a small open economy. First, we show that financial globalization relaxes the existence conditions for asset bubbles. This result implies that more countries may experience asset bubbles in a global economy. Second, we show that the effect of asset bubbles in a global economy is larger than in a closed economy. In particular, countries with high financial friction experience a high economic growth rate before a foreign bubble bursts and they are subject ed to more negative influence after that. This conclusion implies that financial globalization may cause large economic movements before and after a bubble bursts.

Asset bubbles are commonly defined as large movements in asset prices that are not explained by their fundamental value. For example, the United States economy experienced a sharp rise and drop in real-estate prices before and after 2007. This movement also occurred in the rest of the world economy around that time. Since the economic conditions or fundamentals do not change so rapidly, such a movement is considered to be asset bubble (i.e., the subprime loan-related bubble).

Some papers have already analyzed the effects of asset bubbles on the economic growth rate. The seminal papers of Samuelson [

In recent years, several papers have introduced the incompleteness of financial markets, which is called “financial friction”, into the analysis. Caballero and Krishnamurthy [

Hirano and Yanagawa [

These papers, however, consider a closed economy case and do not explain the effects of an asset bubble bursting in a large foreign country, like the collapse of Lehman Brothers. This paper analyzes the characteristics of asset bubbles in a small open economy to provide answers to the following two questions. First, do asset bubbles occur more frequently in a global financial market? Second, do foreign asset bubbles have a greater influence on the economic growth rate than internal asset bubbles do?

We extend the models and ideas of Hirano and Yanagawa [

This paper is organized as follows. In Section 2, we introduce the basic setup of the model. In Section 3, we define a competitive equilibrium based on the setup, and derive the economic growth rate in a small open economy. In Section 4, we analyze the effects of financial globalization on the existence conditions for asset bubbles and the economic growth rate. Finally, in Section 5, we summarize the main insights and present some ideas for future research.

The latest asset bubble has one typical characteristic: financial investors outside the United States hold bubble assets in the United States through securitized products. The development of securitization technologies makes it easy for US financial institutions to sell various securitized products to buyers all over the world. We call this phenomenon “financial globalization”. As a result of this financial globalization investors in many parts of the world could indirectly hold foreign assets in the United States whose prices exceed their fundamental values. Thus, the emerging and bursting asset bubbles in the United States (a big country) could affect the economic growth in the rest of the world (small countries).

Such bubble assets which are held by investors in small countries are called “foreign bubbly assets”, and asset bubbles caused by them in small countries are called “foreign asset bubbles”. Conversely, bubble assets in a closed economy, which are generated and held in small countries, are called “internal bubbly assets”, and asset bubbles caused by them are called “internal asset bubbles”. To analyze asset bubbles in a general equilibrium framework, they are introduced into the model as a type of security, and foreign and internal bubbly assets correspond to the portion exceeding the fundamental value.

We construct a model to analyze the effects of asset bubbles in a small open economy by extending the model of Hirano and Yanagawa [

A typical entrepreneur model with financial friction in a discrete-time economy is considered. There is no population growth, and the economy has one homogeneous good and a continuum of entrepreneurs.

A typical entrepreneur has the expected discounted utility function

where i is the index for each entrepreneur, and

Each entrepreneur encounters two types of investment project every period: high productive investment projects (H-projects) and low productive investment projects ( -projects). At the beginning of every period, each entrepreneur encounters H-pro- jects ( -projects) with probability p (probability

where

Each entrepreneur faces the following flow of funds constraint every period:

where

The foreign investment return is assumed to equal or even exceed the marginal productivity of L-projects, because large countries have advanced financial tools and an effective manufacturing sector. Offering a new opportunity for asset management to the entrepreneur in small counties, the interest rate in small counties converges to the foreign investment return

where C is the index for a closed economy and N is the index for an economy with non-bubble assets. The rates ^{1}.

Each entrepreneur also faces borrowing constraints. He can pledge at most a fraction of future returns from investment to creditors due to financial friction in the economy. Thus, the borrowing constraint is expressed as

where the parameter

Finally, the probability of the bursting of foreign asset bubbles is considered. When it is defined as^{2}. This assumption is consistent with observed facts. For example, Brunnermeier et al. [^{3}.

The previous section provides a basic setup with which to construct a model to analyze the effects of asset bubbles in a small open economy. In this section, we define the competitive equilibrium and derive the economic growth rate in a small open economy. Then we do the same for a closed economy. Comparing these results, we are able to clarify the difference between the existence conditions for asset bubble and the additional effects of foreign bubbly assets on the economic growth rate as a result of financial globalization.

In this subsection, we provide a competitive equilibrium with foreign bubbly assets in a small open economy. The competitive equilibrium is defined as sequences of foreign investment returns

1. Each entrepreneur maximizes their utility under some constraints:

2. The market clearing conditions are

where the aggregate consumption, investment, output, borrowing and purchasing of foreign bubbly assets of each type of entrepreneur at date t are, respectively, designated as

follows:

It is well known that an entrepreneur with the log-linear utility function (1) consumes a fraction

Next we consider the investment function of each entrepreneur to derive the economic growth rate in the equilibrium. An L-entrepreneur prioritizes lending his assets to an H-entrepreneur over investing in L-projects, because the foreign investment return (lending interest rate) equals or even exceeds the marginal productivity of L-projects. An L-entrepreneur lends his assets to H-entrepreneurs up to the limit of the borrowing constraint, and then buys the foreign bubbly assets using residual assets^{4}. An L-entre- preneur, therefore, doesn’t invest in internal projects in his own country. However, an H-entrepreneur borrows assets from L-entrepreneurs and invests all his assets in H-projects, because the marginal productivity of H-projects exceeds the foreign investment return. As a result, H-entrepreneurs are the only entrepreneurs who invest in internal projects in small countries. Combining the budget constraint and the borrowing constraint (7) and (8), the investment function of an H-entrepreneur is

where

vestments to owed capital. We call it the “leverage factor of investments”. Since only H-entrepreneurs invest in internal projects, the investment function of the country is expressed as the aggregate investment of H-entrepreneurs:

The investment function depends on the net worth of H-entrepreneurs at date t. As mentioned before, H-entrepreneurs at date t arise from proportions p of L and H-en- trepreneurs at date

As a result, the investment function (12) is replaced by

Finally, we consider the economic growth rate in a small open economy. The gross income of the country is

In order to characterize the economic growth rate, we define the relative size of foreign bubbly assets (

From Equation (14) and these definitions, the growth rate of aggregate wealth (17) can be expressed as

This equation implies important characteristics of foreign asset bubbles in a small open economy. The first term of the equation corresponds to the leverage factor of investments, and the second term corresponds to the return from investments in foreign bubbly assets. As is clear from this equation, financial globalization, which lets the internal interest rate increase to the foreign investment return, reduces the leverage factor of investments and has a negative effect on the economic growth rate. On the other hand, the emergence of foreign asset bubbles offers the entrepreneur a new investment opportunity and brings him foreign investment income. Since he invests the income on internal projects, capital accumulation is stimulated in the country. These two effects will play a key role when we analyze the effects of foreign bubbly assets in a small open economy.

Combining Equations (8), (10) and the definition of^{5},

Furthermore, from an elementary calculation of the investment function (14), it is clear that the growth rate of the total output (

Theorem 1. The economic growth rate in a small open economy is expressed as a function of the degree of imperfection of the financial market and the foreign investment return of the country. That is, it is given by

The first term of Equation (20) depends on the leverage factor of investments and difference between the marginal investment returns, and the second term corresponds to the return from foreign bubbly assets. Thus, we call these terms “extended investment leverage” and “foreign investment income”, respectively. This second effect is a unique characteristic of a small open economy. If the economy has high financial friction (low

In this subsection, we derive the equilibrium interest rate and the economic growth rate in a closed economy. These cases have been already derived in Hirano and Yanagawa [

As mentioned above, if internal bubbles emerge in a closed economy, an entrepreneur has an opportunity to buy internal bubbly assets instead of foreign bubbly assets. The internal bubble equilibrium, therefore, is defined by introducing a domestic interest rate instead of the foreign investment return. Based on the same idea as Theorem1, the economic growth rate is expressed as

where C is the index for a closed economy, and

In an economy with no bubble assets, an L-entrepreneur invests his residual assets in L-projects. He, however, is able to lend all his assets to an H-entrepreneur, if the level of financial friction is not so severe. Thus, the equilibrium economic growth rate depends on the conditions

where

Here,

where N is an index indicating a no-bubble economy^{7}.

In this subsection, we analyze the effects of financial globalization on the existence conditions for asset bubbles. We provide an answer to the question of whether asset bubbles occur more frequently in a global economy. Comparing the range of the existence conditions in a small open economy with that in a closed economy, we can clarify the effects of globalization. First, we derive the existence conditions in a small open economy, and then a closed economy is considered.

In a small open economy, the following conditions need to be satisfied to sustain foreign bubbly assets:

Equation (26) corresponds to the condition that the entrepreneur invests in foreign bubbly assets in the equilibrium. Equation (27) corresponds to the condition that the foreign investment return equals or even exceeds the equilibrium interest rate and does not exceed the marginal productivity of -projects. Equation (28) is a condition specific to a small open economy, which we mentioned before. As a result, we have the following existence conditions for foreign bubbly assets:

On the other hand, the closed economy case is derived using the same ideas as the small open economy case. Taking

It is clear from

Theorem 2. Financial globalization relaxes the existence conditions for asset bubbles.

Here, we discuss the reason why asset bubbles may emerge in countries with higher financial friction. L-entrepreneurs have a lot of residual assets in both closed and open economies. In a closed economy, however, the amount of internal bubbly assets is limited. As a result, the expected return from internal bubbly assets dips below the marginal productivity of investments in L-projects. Thus, there is the floor (

In this subsection, we analyze the effects of financial globalization on the economic growth rate. We provide an answer to the question of whether foreign asset bubbles have a greater influence on the economic growth rate than internal asset bubbles.

First, to understand the effects of financial globalization, we compare the economic growth rate in a foreign asset bubble equilibrium with that in a no-bubble economy. Financial globalization turns the interest rate in a small country into the foreign investment return. As mentioned before, that increase of the equilibrium interest rate has two effects on the economic growth rate in the country. First of all, it improves the investment return of L-entrepreneurs. On the other hand, it decreases the leverage factor of investments, because budget constraints become tighter as a result of the upturn in interest rate.

Combining equations (20) and (24), we derive the interest rates that make economic growth rates equal in both economies (see Appendix C):

where

Theorem 3. Financial globalization enhances the economic growth rate in small countries with high financial friction, if the foreign investment return satisfies

Theorem 3 shows the following things. In a country with an advanced financial market (

Next, we compare the economic growth rate in a foreign asset bubble equilibrium with that in an internal asset bubble equilibrium. Because foreign bubbly assets offer a higher investment return to an L-entrepreneur than internal bubbly assets do, L-entrepre- neurs invest residual assets in foreign bubbly assets instead. That upturn of the equilibrium interest rate has the same two effects on the economic growth rate as in a no- bubble economy. Combining Equations (20) and (22), the interest rates when the economic growth rates in both cases become equal are (see Appendix D):

where

As a result, we have the following theorem.

Theorem 4. Foreign asset bubbles enhance the economic growth rate in small countries with high financial friction much more than internal asset bubbles do:

Theorem 4 shows that financial globalization brings an additional investment income to the entrepreneur and he invests the income in internal projects in the future which stimulates capital accumulation in the country (

In this subsection, we analyze the effects of bubbles bursting in a large foreign country. The value of foreign bubbly assets becomes zero after the bubble bursts. Combining Equation (20) and the condition on

where A is an index indicating an economy after the bubble bursts. As is clear from this equation, the effect of bubbles bursting in a large foreign country is much larger than it is in countries with high financial friction (

We have analyzed the characteristics of asset bubbles in a global economy. We introduced foreign bubbly assets into the model of Hirano and Yanagawa [

This paper has made several contributions to the literature. First, we have shown that financial globalization relaxes the existence conditions for asset bubbles. Small countries with financial friction have some residual assets and invest them in foreign bubbly assets. As a result, foreign asset bubbles are introduced into the small countries. This means that more countries have the potential to experience asset bubbles in a global economy. This result is consistent with observed facts before and after the collapse of Lehman Brothers.

Second, we showed that financial globalization enhances the economic growth rate in small countries with high financial friction before the bursting of a foreign asset bubble. They are also subject to a greater negative impact after that. That is because countries with high financial friction invest much more in foreign bubbly assets than countries with advanced financial markets do. As a result, they gain a much greater investment return from foreign bubbly assets before the bubble bursts, but they lose much more afterwards.

We have also shown that the effect of asset bubbles in a global economy becomes larger than it is in closed economy. That is because the investment return from foreign bubbly assets is higher than it is from internal bubbly assets. This conclusion implies that financial globalization may cause large movements in the world economy. These results are different from those of Olivier [

This paper leaves some promising areas for a future research. The introduction of capital accumulation would enable the study of a negative growth rate after a bubble bursts. Second, the prior distribution of risk is important for the construction of an asset portfolio. A very interesting topic for analysis would be the effects of a change in risk distribution on economic growth. I look forward to continuing my research in this field.

My heartfelt appreciation goes to attendees for a research seminar held in Development Bank of Japan Inc., whose comments and suggestions were of inestimable value for my study.

Motohashi, A. (2016) Economic Growth with Asset Bubbles in a Small Open Economy. Theoretical Economics Letters, 6, 942-961. http://dx.doi.org/10.4236/tel.2016.65097

To clarify the meaning of financial friction, we provide a micro-foundation for

Here, we can redefine the parameter

1. Necessary condition

If an economy has a stable internal bubble equilibrium, the relative size of foreign bubbly assets is required to be constant (

As a result,

2. Sufficient condition

The transversality condition in an internal bubble equilibrium is

At date

Considering that an entrepreneur with the log-linear utility function (1) consumes a fraction

As a result,

(Q.E.D)

We first consider the effects of financial globalization on the economic growth rate in countries with high financial friction, and then the case of low financial friction is analyzed. Countries with low financial friction realize the same economic growth rate as a perfect financial market, because the borrowing constraint does not bite. However, countries with high financial friction do not realize such a high economic growth rate. Thus, we consider following two cases.

1. Countries with high financial friction (

The foreign investment return that equalizes the economic growth rate in a foreign bubble equilibrium with that in a no-bubble equilibrium is derived as the solution of following equation:

The solutions of the above quadratic function satisfy the conditions

It is clear that the economic growth rate in both countries becomes equal when the investment interest rate becomes equal to internal equilibrium rate. Thus,

Considering the shape of a quadratic function, we have

2. Countries with low financial friction (

The foreign investment return that equalizes the economic growth rate in foreign bubble equilibrium with that in an internal bubble equilibrium is derived from the following equation:

This has the solution

In this case, it is clear that

(Q.E.D)

The foreign investment return that equalizes the economic growth rate in a foreign bubble equilibrium with that in an internal bubble equilibrium is derived as the solution of the following equation:

This equation can be rewritten as follows:

where

It is clear that the economic growth rate in both countries becomes equal when the investment interest rate becomes equal to the internal equilibrium rate. Thus,

Considering the shape of a quadratic function, we have

The magnitude relationship of these solutions depends on the level of financial friction. The branch point in the level of financial friction is

where

(Q.E.D)

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