HISTORY-DEPENDENT PATHS AND TRADE GAINS IN A SMALL OPEN ECONOMY WITH A PUBLIC INTERMEDIATE GOOD*

1. introduction

The supply of public intermediate goods has a great impact on a nation's economic development, welfare, and other important matters. In particular, for a trading country, the supply of public intermediate goods plays an important role in determining trade patterns. For example, during recent decades, the government's role in supporting scientific research and the development of human resources has become important in the rapid economic movement toward globalization. Among trading and well-developed countries, how and on what scale governments should supply these public intermediate goods are recognized as important policies for gaining economic advantage. Therefore, it is imperative to study the role of public intermediate goods in relation to international trade.

The subject itself is not new. In fact, Manning and McMillan (1979) studied trade patterns in a small open Ricardian economy with a public intermediate good. Extending their analysis, Tawada and Abe (1984) and Tawada and Okamoto (1983) examined various traditional trade theorems, including the pattern of trade theorem in the Heckscher–Ohlin type of small open economy with a public intermediate good. Furthermore, Suga and Tawada (2007) recently treated the analysis in a Ricardian two-country model. All these studies are, however, static analyses.

When we consider public intermediate goods such as scientific research, transportation systems, and the development of human resources, they had better be treated as capital stocks. Thus, the analysis should be performed in a dynamic framework. Moreover, among theoretical studies of international trade, in general, the studies incorporating a dynamic aspect have become rigorous in recent years (see, for example, Mountford, 1999; Deardorff, 2001a, 2001b; Hu et al., 2005, 2006). Thus, it is worthwhile and interesting to examine trade theory with public intermediate goods in a dynamic framework.

Among international trade theories featuring a public intermediate good, only McMillan (1978) offers a dynamic analysis investigating trade patterns in a dynamic framework, where a public intermediate good can be accumulated over time as a capital stock. McMillan (1978) considered a small open economy with two private production sectors, one public intermediate good sector, and one primary factor of production. Moreover, two private goods are produced using the primary factor and the public intermediate good, which in turn is produced by the primary factor.² In his model, McMillan showed that the stock of the public intermediate good determines the slope of the production possibility frontier (PPF) and thus determines the pattern of international trade.

However, McMillan's (1978) examination of the dynamic paths and the resulting steady-state solutions was hardly adequate. Thus, in this study, we reexamine his work and present a more precise analysis. In particular, he stated in his Proposition 3 that a unique steady state necessarily exists. We will, however, prove the possibility that two saddle-point stable steady states exist. Thus, our result concerning the patterns of trade along the time path is not necessarily similar to the result he obtained. In particular, we show that if the labor endowment is of moderate size, two saddle-point stable steady states appear, and the patterns of trade heavily depend on the initial stock of the public intermediate good.

In addition to trade patterns, whether trade is gainful is also an important concern. Therefore, we add this into our analysis, since McMillan (1978) did not address it. Concerning the gains from trade in a static economy with public intermediate goods, if a public intermediate good is supplied under the Lindahl rule and if the economy is stable, any small open country necessarily gains from trade. This is because the PPF is unaffected and equilibrium is efficient before and after trade. In a dynamic framework, however, the stock of a public intermediate good may be depreciated; thus, the PPF may shrink in the long run, creating a possibility of trade losses. In fact, we show the possibility of trade losses at a steady-state equilibrium.

This article is organized into the following sections. Section 2 presents the model, and Section 3 explains the dynamic system drawn from the model. Section 4 is devoted to the analysis of specialization patterns. Section 5 considers transition paths from autarky to free trade. Section 6 is allocated to an analysis of gains from trade, and the last section summarizes our results and provides remarks.

2. the model

Following McMillan (1978), we consider a small open economy containing two private sectors, one public production sector, and one primary factor of production, which is assumed to be labor. The two private sectors are assumed to be sectors 1 and 2, where goods 1 and 2 are produced, respectively, under constant returns to scale technology with respect to labor. The public sector produces a public intermediate good with decreasing returns to scale technology with respect to labor. The public intermediate good can be accumulated, and its accumulated stock serves in the production in private sectors as a positive external effect without congestion between sectors. Total labor endowment is assumed to be given and constant over time.

The production function of each private sector is assumed to take the following form: (i= 1, 2):³

where Y_i is the output of good i, R is the stock of the public intermediate good, and L_i is the labor input in sector i. From this formulation, it is clear that labor productivity in private sector i is exhibited by A_i(R) and is dependent on the stock of the public intermediate good R. We make the following assumption for A_i(R):

The production function of the public sector is expressed as G=f(L_R), where L_R is a labor input in the public sector. Concerning f(L_R), we assume that

The accumulation of the public intermediate good is described in the following dynamic equation:⁴

(1)

where β > 0 is the depreciation rate of the stock of the public intermediate good.

At each moment in time, the economy must face the following full employment constraint on labor:

where L > 0 is labor endowment and is assumed to be given and constant over time. With this constraint and using private production functions, we obtain the PPF for a given L_R and R as

(2)

Now we turn our attention to the behavior of a representative household.⁵ The lifetime utility of a representative household is given by

(3)

where C_i is consumption of good i, ρ > 0 is the rate of time preference, and α∈ (0, 1) is a parameter.

The home government determines to maximize (3) subject to (1), (2), and its balance of payments

(4)

where p is the world price of good 1 relative to that of good 2 and is assumed to be given and constant over time.⁶

The current-value Hamiltonian function is described as

Then, the optimal controls must satisfy

(5)

(6)

(7)

Moreover, the adjoint equation and the transversality condition are, respectively, expressed as⁷

(8)

In view of (4), (5), (6), and (7), the following condition holds along the optimal path:

(9)

3. dynamic system

In this section, we inspect the properties of a dynamic system. Defining the production elasticity of the public intermediate good stock in sector i as , we make the following assumption regarding the impact of the public intermediate good on industries:

Assumption 1 η₁(R) > η₂(R).

The above assumption implies that the public intermediate good serves more significantly in the production in sector 1 than in that of sector 2.

Let be a function satisfying the relation of p=A₂(R)/A₁(R). Then, is decreasing in p under Assumption 1.

Considering (6), we can confirm that, if , the economy is specialized in producing good 2, whereas if , the economy is specialized in producing good 1. Thus, substituting (2), (7), and (9) into (8), we obtain the following Euler equation:

(10)

According to (10), we can verify that the locus of has a “reversed N” shape, as illustrated in the following figures. The dynamic movement of θ according to (10) under a given R is as follows: Above the locus, θ must increase and below the locus, θ must decrease.

Letting L_R=λ (θ, L) express the relation of (9), we have

Replacing L_R with λ (θ, L), we can rewrite (1) as

(11)

According to (11), it is easily verified that the locus of is upward sloping and that the amount of the public good stock must decrease in the upper area and increase in the lower area of the locus. Moreover, the locus is affected by the level of L such that an increase in L shifts the locus downward and vice versa.

In sum, we obtain a dynamic system of θ and R in the small open economy as described by (10) and (11).

4. pattern of specialization

We now investigate the steady state of the dynamic system constructed by (10) and (11). The steady state is a point (R*, θ*) satisfying . Let be a solution of , and consider the following three cases separately:

In Case 1, a unique steady state where the economy specializes in good 1 exists and is saddle-point stable. This case occurs when is sufficiently large and/or p is adequately high. If, in addition, and are sufficiently close to each other, the steady state is globally saddle-point stable. This particular case is displayed in Figure 1, where the gap between points A and B becomes sufficiently small and thus a global saddle path can exist.⁸

In order to see the production pattern on the global saddle path, we suppose in Figure 1 that the initial point is C, where . Thus, starting at R₀, R gradually approaches the steady state E along the saddle path. Then, only good 2 is produced until R reaches . Once R reaches , the economy diversifies in production, and after R exceeds , only good 1 is produced.

In Case 2, where a unique steady state exists, the economy specializes in good 2 and the steady state is saddle-point stable. This case appears if L is sufficiently small and/or p is adequately low. Because this case becomes the reverse of Case 1, we omit the figure. Again, if and are sufficiently close to each other, the steady state becomes globally saddle-point stable.

Concerning the production pattern on the global saddle path, we suppose that the initial stock is given by . Starting at R₀, R gradually approaches the steady state along the saddle path. Then, the production pattern of this economy becomes the opposite of Case 1 with respect to time path, implying that production is first specialized in good 2, then diversified at , and finally specialized again, but in good 1 after R exceeds .

Finally, we consider Case 3. In this case, three steady states, E₁, E₂, and E_d, exist, as shown in Figure 2.⁹E₁ and E₂ are locally saddle-point stable, whereas E_d is unstable.¹⁰ At E₁ and E₂, the economy is specialized in goods 1 and 2, respectively. Both goods are produced at the unstable steady state E_d.

In this case, the initial level of R determines the steady-state specialization pattern: If , the steady state is E₁ and production is specialized in good 1, whereas if , the steady state is E₂ and production is specialized in good 2.

Now we summarize the results obtained in this section as the following theorem:

5. transition from autarky to free trade

We examined the pattern of production at a steady state in a small trading country. Advancing our analysis, we now investigate how the production pattern varies along the time path starting from the autarkic steady-state equilibrium as an initial position. For this purpose, let us derive the autarkic equilibrium of this economy.

The home government's problem under autarky is to maximize (3) subject to (1), (2), and relevant market clearing conditions

(12)

The current-value Hamiltonian function is described as

Then, the optimality conditions are

Moreover, the adjoint equation is

Using (12) and the optimal conditions, the above adjoint equation can be rewritten as

(13)

where h(R) ≡αη₁(R) + (1 −α)η₂(R).

Thus, the dynamic system of a closed economy is built by dynamic equations (11) and (13). Because the locus of is a convex combination of θ=η₁(R)/[(ρ+β)R] and θ=η₂(R)/[(ρ+β)R], the locus is monotonically downward sloping. As in the case of free trade, the locus of is drawn as a monotonically upward-sloping curve. Therefore, there exists a unique autarkic steady state denoted by .

We consider a transition from autarky to free trade and investigate how a country's welfare changes by this transition. We assume that the economy is initially in a steady state. Then, if (we refer to this case as Case I), the country has a comparative advantage in good 1. This is because the autarkic steady-state price p_a satisfies the condition that , and is a decreasing function of p. Thus, the autarkic steady-state price p_a is lower than the international price p.

Figure 3 illustrates the transition from autarky to free trade in Case I, where E and E_a in the figure are the steady states under free trade and autarky, respectively. As is clear from this figure, Case I occurs when the labor endowment is sufficiently large so that holds.¹²

Likewise, if (we refer to this as Case II), the country has a comparative advantage in good 2. This case occurs when labor endowment is sufficiently small so that holds. The transition from autarky to free trade corresponding to this case can be shown diagrammatically in a fashion similar to Case I, and thus we omit the figure.¹³

In sum, we assert

6. gains from trade

In the previous section, we inspected the autarkic steady-state equilibrium. Using this result, we investigate whether trade is gainful or harmful at the steady state. Let us first present the following lemma:

Proof From (1), dL_R/dR=β/f′ (L_R) holds in the steady state. Using this and (9), we have

In light of (10), which indicates that holds in the steady state, the numerator of the above expression is positive.                   ▪

Now we investigate whether a small open economy gains or loses from trade at the steady state. We separately consider the two cases examined in the previous section.

Case II. .

In this case, we have . Given this and Lemma 1, it is observed that the long-run PPF contracts after trade. This case is illustrated in Figure 4, where the downward straight line closer to the origin exhibits the long-run PPF under free trade, whereas the other downward straight line is that of autarky.

In contrast to Case I, the trading line and the long-run PPF intersect in this case. This means that, depending on preferences, the economy may gain or lose from trade at the steady state. For example, if the autarkic steady-state equilibrium pair of consumption is given by point A₁ in Figure 4, trade enhances welfare at the steady state. On the other hand, if the autarkic steady state is on A₂, trade reduces welfare at the steady state.

Now we establish the following theorem:

7. summary and remarks

In this study, we reexamined McMillan's (1978) analysis of trade patterns and added the gains-from-trade analysis of a small Ricardian economy in a dynamic framework. Our investigation into the patterns of trade presented the following results. If the labor endowment is sufficiently large (small), a small open country specializes in a good whose productivity is more (less) sensitive to the public intermediate good in the long run. In the case where the sensitivity of the productivity to the public intermediate good stock is similar between private industries, a switch in trade patterns along the dynamic path can occur. This possibility of a switch is not properly presented in McMillan (1978), even though the model is essentially the same. Moreover, we demonstrated that, if the labor endowment is of intermediate size, there are two saddle-point stable long-run equilibria, and the initial stock of the public intermediate good determines the long-run trade patterns according to the history-dependent time paths. This aspect was also overlooked in McMillan's analysis.

Concerning trade gains, we obtained the following results. If a country enjoys a comparative advantage in producing a good whose output is more responsive to the stock of a public intermediate good, the economy unambiguously gains from trade at the steady state. In contrast, if a country enjoys a comparative advantage in producing a good whose output is less responsive to the public intermediate good stock, the economy may lose from trade.

We need to extend our present analysis of the Ricardian type to that of the non-Ricardian type. In a two-factor case like the Heckscher–Ohlin model, if we assume that public or private capital can be accumulated, the factor endowment ratio varies from time to time. Therefore, the theorems depending on the factor endowment ratio may be seriously affected in a dynamic trade model. Of another importance is to treat the present issue in a two-country dynamic framework. Thus, we need to handle the two state variables, each of which is a public intermediate good stock specific to the country. Then the model becomes that of a dynamic game. Finally, though we focused on public goods serving private production, it may be interesting to consider the role of public consumption goods in international trade. In fact, Connolly (1970) treated an international trade analysis in a static two-country framework with public consumption good long time ago. However, the dynamic analysis still remains in this case.