Model

The consumption side of this model is a stylized version of neoclassical growth models in which an infinitely lived household (one in which the generations are assumed to be continuously linked) maximizes an overall utility:

where c is consumption per person and $\rho >0$is the constant rate of time preference. The number of individuals in the household is denoted by L^t which grows at the exogenous rate of n. As in Barra (1974), it is assumed that individuals care about their children’s utility. In order for U(c) to be bounded in the steady state, $\rho >n$is assumed.

The instantaneous utility function is given by

where -θ is the constant elasticity of marginal utility. Households hold assets in the form of “capital” and internal loans. There are no foreign assets in this model. Capital can be envisaged as a broad concept, including both physical capital and human capital. The real rate of return on assets is r. Then, the budget constraint for the household is given by

where a is assets per person and a dot over a variable denotes its differentiation with respect to time. It is assumed that each person supplies one unit of labor. Thus, the wage income per person equals the real wage rate w.

The household maximizes its utility given in equation (1), subject to the budget constraint in equation (3) and to a given stock of initial assets a(0). Thus, the first-order conditions for maximization give the growth rate of consumption:

In this economy, there is a single good produced by a neoclassical production function. It is assumed that domestic production requires foreign inputs—such as raw materials, intermediate goods, and capital goods—in addition to domestic inputs. A small open economy that imports foreign inputs in exchange for the domestic good at given world prices is considered. Since there is no opportunity for foreign borrowing or lending, trade is balanced in every period. Domestic inputs—labor and capital—are assumed to he combined by a Cobb-Douglas process, and, by using imported foreign goods as intermediate inputs, total output is produced by a constant elasticity of substitution production function:

where Q is total output, M is the imported input. K is the “capital” input, and $\hat{L}$ is the “effective” labor input. The effective labor input is assumed to increase over time at the rate of exogenous technological progress x and the population growth rate n:

The production function in equation (5) is assumed to exhibit diminishing marginal returns to each input and constant returns to scale for both inputs.

Define$\hat{q}=Q/\hat{L};\hat{m}=M/\hat{L};\hat{k}=K/\hat{L};$and$z=\hat{m}/{\hat{k}}^{\alpha }\mathrm{.}$The production function is rewritten as

First-order conditions for firms’ profit maximization under perfect competition imply that

where$\mathrm{\Psi }={\gamma }_2^{\sigma },1-\mathrm{\Psi }={\gamma }_1^{\sigma },$and a- denotes the elasticity between domestic inputs and imported inputs, $\sigma =1/(1-\mu )$. The variable V denotes the marginal productivity of capital, and δ denotes the rate of depreciation, $0<\delta <1$. The relative price of the foreign good in terms of the domestic good is denoted by p₁. Under the small-country assumption, p_t is determined by world market conditions. In free trade, the price of the foreign good is assumed to equal the price of domestic output, that is, p =1. Let Ψ denote the share of imported inputs in total output under a free trade regime. Equation (9) can be solved for equilibrium z, and thereby h (z):

By combining the behavior of households and firms, the model is solved for a unique competitive market equilibrium. In equilibrium, the assets per household, a, equal the capital per worker. k. Substituting equations (8) through (12) into equation (3) gives the resource constraint for the economy,

where $\hat{c}={ce}^{-et}$and k(0) is given.

The growth rate of consumption per effective worker is solved by substituting the real interest rate in equation (8) into equation (4).

Equations (13) and (14) and a transversality condition determine the time paths of $\hat{k}$ and $\hat{c}$ in this economy. As is well known, in the steady state of the neoclassical model, consumption, investment, capital stock, and output all grow at the exogenously given rate of technological progress (see Barro and Sala-i-Martin (1991b, chapter 1)). Therefore, the other parameters do not matter for the steady-state growth rates, although they change the steady-state values of $\hat{k}$ and $\hat{c}\mathrm{.}$ With constant p and thereby constant $\hat{z},$ this open economy exhibits the same dynamics as the closed economy does: the economy converges to the steady state at a decreasing rate, and in the steady state all quantities grow at the exogenously given rate of technological progress.

Free Trade Steady-State Equilibrium

Suppose that the price of foreign goods equals the price of domestic goods in the world market as the economy follows a policy of free trade. Equations (11) and (12) simplify to

And equations (13) and (14) are rewritten as

Thus, the free trade steady state has values of capital stock and income per effective worker as follows:

where ${\hat{k}}^{*}$denotes the free trade steady-state capital stock per effective worker, and ${\hat{y}}^{*}$ the free trade steady-state income per effective worker. Note that when the economy is engaged in free trade, the size of imported inputs in total output Ψ does not matter for steady-state income in this model. In other words, what really matters for the steady-state income is free trade policy not the import share itself. For example, a large country that can acquire most of its intermediate inputs domestically, and consequently has a low Ψ, goes to the same steady-state income as does a small country that may depend more on foreign intermediate inputs and consequently will have a high Ψ. This makes sense because there is no technological difference between either domestic or imported inputs. However, once trade is distorted, the size of the import share matters and determines the effects of trade distortions on income. Suppose that this economy cuts off foreign transactions$(M=0)$, then the autarkic output and income become

Not surprisingly, when the imported input is essential in the production of domestic goods $(\sigma <1),$ the autarkic economy exhibits no output growth over time. Unless the domestic and the imported inputs are perfect substitutes $(\sigma =\infty ),$the autarkic steady-state values of the capital stock, consumption, and income per capita are lower than their free trade values. And as Ψ becomes larger, output is lower when the economy is autarkic. The parameter Ψ, which denotes the steady-state share of imports in total output, may depend on the resource endowments of the economy, as will be discussed later. Therefore, implementing a free trade policy may be more important for a smaller country than a larger country.

The steady-state gross saving rate is given by

Thus, import share does not matter for the steady-state saving rate.

Transitional Dynamics and Economic Growth

The steady state in the neoclassical model, where the growth rate is given by the exogenous technological progress x, does not provide any interesting implications for economic growth. The explanation that observed variations in cross-country growth rates are only the result of different rates of exogenous technological progress is not very convincing. As a result, much attention has been given to the process of transitional dynamics.

During the transitional period in which the economy approaches the steady state from a low initial level of capital stock, $\hat{k}$and $\hat{c}$ rise monotonically toward their steady-state values but at decreasing rates (see Barro and Sala-i-Martin (1991b, chapter 1)). In the neoclassical model, the length of the transitional period can be examined by a log-linearized version of equations (13’) and (14’) around the steady state. Following Sala-i-Martin (1990a). the average growth rate of per capita income, y, over a period from initial time 0 to any future time $T>0$is given by

where $\hat{y}*$denotes the steady-state level of income per effective worker. “The speed of convergence” is β, with $\beta >0$. Various empirical studies show that the estimates for the convergence speed, (3, are roughly about 2 percent a year (Barre and Sala-i-Martin (1990a) and Mankiw, Romer, and Weil (1992)). This predicted coefficient implies a lengthy convergence period: about 35 years to move half the distance between the initial per capita income figure and the steady state, and more than 100 years to move 90 percent of the distance. In a special Solow case, in which the saving rate is always constant over time, the convergence speed simplifies to

Then the empirical estimate for θ, roughly 0.02 per year, requires a value for α of around 0.75 with the other parameter values on the order of δ = 0.04, x =0.02, and n= 0.02. Therefore, in these types of neoclassical models, α, the share of “broad capital,” must be well above the conventional value of one-third.

Equation (19) shows that the economy exhibits growth rates exceeding x, the exogenously given rate of technological progress, during the transitional period. If the last term of equation (19) is large, transitional dynamics may explain a significant part of the growth rate, and hence play a larger role in explaining differences in growth rates than that played by the exogenously given rate of technological progress. If this is the case, trade policies, which change the steady-state level of income, may have long-run effects on economic growth in the neoclassical model.

Effects of Trade Distortions

Suppose that the government intervenes in foreign transactions by imposing a tariff τ on the imports of foreign goods so that the price paid by the domestic purchaser is (1+ τ) times the price received by foreign exporters.² Thus,

where p is the relative price of the foreign good in terms of domestic goods.³

Government is assumed to transfer tariff revenues to the public.⁴ The lump-sum transfer directly increases private sector income. Therefore, the tariff has two effects on the economy, namely the distortion of resource allocation and the transfer of revenue. Tariff revenue is given by

where $\hat{G}$ is the tariff revenue per effective worker.

Solving the equilibrium z and h(z) by substituting equation (21) into equations (11) and (12), and then using equations (13) and (14), gives the time path of the economy when trade distortions are imposed:

where $\mathrm{\Phi }={\left[(1-\mathrm{\Psi })h\left(\overline{z}\right)\right]}^{1/\sigma }={\left[(1+\pi )-\pi {(1+\tau )}^{1-\sigma }\right]}^{1/(1-\sigma )}$. The parameter π, which is $\mathrm{\Psi }/(1-\mathrm{\Psi }),$denotes the share of imported inputs in the value added in trade. The parameter π, which denotes free trade openness, may depend on structural features of the economy, such as factor endowments and natural trade barriers.

Equation (24) can he solved for the steady-state capital stock per effective worker:

Since ${\mathrm{\Phi }}_{\tau }<0,$the steady-state capital stock decreases with the tariff. Steady-state income is given by

where $\hat{g}*$is the proportion of tariff revenue in private income, that is, income excluding government transfers. Equation (26) shows that the effects of trade distortions on the steady-state income consist of two components: the distortion and the revenue effect. Without the transfer of tariff revenue,$\hat{g}*$will be zero and there is no revenue effect. Equations (25) and (26) imply that the distortionary effect of tariffs always decreases the steady-state levels of the capital stock, output, and thus consumption.

Effects of trade distortions on the steady-state gross saving rate depend entirely on the revenue effect. The steady-state gross saving rate is given by

Trade Distortions and Growth

As shown in equation (19), the effects of the trade distortions depend on the change in steady-state income and the convergence speed. The convergence speed, β, is not changed if a constant saving rate over time is assumed. For the Solow case, in which the saving rate is constant, equation (27) is rewritten as follows:

The log linearization of this equation around the steady state gives the same convergence speed, β as in equation (21). Thus, during the transitional period, the effects of trade distortions depend entirely on the change in the steady-state income. Substituting equation (26) into equation (19) gives the growth rate in the transitional period:⁵

The effects of trade distortions on the growth rate in the transitional period hinge on the last two components—revenue and the distortion effect. If the Cobb-Douglas combination of domestic inputs and foreign goods is assumed, the distortionary effect is simplified to⁶

Thus, the distortionary effects of tariffs on the growth rate evidently hinge on free trade openness: the level of openness magnifies the distortionary effect multiplicatively. Thus, the same trade distortion decreases the growth rate more in an economy that has a high π. Therefore, free trade is more important in a small, resource-scarce economy, which would become more open in the steady state, than in a large, resource- abundant economy.

Expression (30) gives more insights on the effects of trade distortions if one considers a special case in which the saving rate is constant over time and T is small. As T approaches 0, the value of $(1-e^{-\beta T})/T$approaches R. In this case, equation (30), the distortionary effect, is simplified to

Thus, the value of the capital share, α, does not matter for the determination of the tariff effect.

Effects of Trade Distortions in a Model Economy

The effects of trade distortions in a representative economy are now considered. Equation (29) implies that the tariff rate r decreases the growth rate in the transitional period by about 100$(x+n+\delta )\left(\log \mathrm{\Phi }\right)$ percent if the saving rate is constant. Here, the values of plausible parameters are considered to be δ =0.04. n =0.02, and x =0.02. The selection of the parameter value for free trade openness IT is obviously critical in determining the effects of the trade restrictions. For the numerical example, two cases are considered—π = 0.4 and π = 0.2. These figures are chosen by considering the average share of total imports in GDP in a large sample of countries from various data sources.⁷

Table 1. shows the effect of tariffs on growth rates with various elasticities of substitution in production (still assuming a constant saving rate). The results show that trade distortions caused by tariffs decrease the growth rate of per capita income depending on the degree of free trade openness and on the elasticities. As the level of openness rises, the effects of trade distortions increase almost proportionally, given a particular level of substitutability between inputs. Given the openness, the elasticity of substitution between domestic and imported inputs becomes a crucial determinant of the distortionary effects. With a lower elasticity, which implies that imported goods are more essential for production, growth rates decrease more quickly as trade restrictions increase. When both inputs increase in substitutability $(\sigma >1)$, trade distortions are less damaging to the economy. However, for a moderate increase in the tariff rate, the role of substitutability between domestic inputs and foreign goods is less influential than is the openness of trade.

This numerical example illustrates that the effects of trade distortions on growth rates are quite significant. For example, in a country where π is 0.4 percent, a tariff rate of 20 percent, which roughly corresponds to the difference between the average tariff rates of the OECD countries and developing countries, reduces the growth rate of per capita income by about 0.5 percent or 0.6 percent depending on the elasticity of substitution in the country concerned.

^{Table 1.}

Effects of Tariffs on Growth Rates in the Transitional Period

(Change of growth rate in percent)

Notes: π denotes the share of imported inputs in value added in free trade, σ is the elasticity of substitution between domestic inputs an d foreign inputs. Pure distortion effects of the tariff and the total effect with a lump-sum transfer of revenue are presented.

^{Table 1.}

Effects of Tariffs on Growth Rates in the Transitional Period

(Change of growth rate in percent)

	σ = 2.0		σ = 1.0		σ = 0.5
Tariff rate in percent	Tariff effect	With transfer	Tariff effect	With transfer	Tariff effect	With transfer
When -π =0.4
0	0.00	0.00	0.00	0.00	0.00	0.00
10	-0.29	-0.22	-0.31	-0.23	-0.32	-0.24
20	-0.52	-0.41	-0.58	-0.45	-0.62	-0.48
30	-0.71	-0.58	-0.84	-0.66	-0.92	-0.71
40	-0.87	-0.72	-1.08	-0.86	-1.22	-0.95
50	-1.00	-0.85	-1.30	-1.05	-1.51	-1.18
60	-1.12	-0.96	-1.50	-1.22	-1.79	-1.41
70	-1.22	-1.06	-1.70	-1.39	-2.07	-1.64
80	-1.31	-1.15	-1.88	-1.55	-2.35	-1.86
90	-1.39	-1.23	-2.05	-1.71	-2.63	-2.09
100	-1.46	-1.30	-2.22	-1.85	-2.90	-2.31
110	-1.52	-1.36	-2.38	-1.99	-3.17	-2.54
120	-1.58	-1.42	-2.52	-2.13	-3.44	-2.76
When π = 0.2
0	0.00	0.00	0.00	0.00	0.00	0.00
10	-0.14	-0.11	-0.15	-0.12	-0.16	-0.12
20	-0.26	-0.21	-0.29	-0.23	-0.31	-0.24
30	-0.36	-0.29	-0.42	-0.33	-0.45	-0.35
40	-0.44	-0.37	-0.54	-0.43	-0.60	-0.46
50	-0.52	-0.43	-0.65	-0.52	-0.74	-0.57
60	-0.58	-0.49	-0.75	-0.61	-0.87	-0.68
70	-0.63	-0.55	-0.85	-0.69	-1.00	-0.79
80	-0.68	-0.59	-0.94	-0.77	-1.13	-0.89
90	-0,72	-0.64	-1.03	-0.85	-1.26	-0.99
100	-0.76	-0.67	-1.11	-0.92	-1.38	-1.10
110	-0.80	-0.71	-1.19	-0.99	-1.51	-1.20
120	-0,83	-0.74	-1.26	-1.05	-1,63	-1.30

Notes: π denotes the share of imported inputs in value added in free trade, σ is the elasticity of substitution between domestic inputs an d foreign inputs. Pure distortion effects of the tariff and the total effect with a lump-sum transfer of revenue are presented.

View Table

^{Table 1.}

Effects of Tariffs on Growth Rates in the Transitional Period

(Change of growth rate in percent)

	σ = 2.0		σ = 1.0		σ = 0.5
Tariff rate in percent	Tariff effect	With transfer	Tariff effect	With transfer	Tariff effect	With transfer
When -π =0.4
0	0.00	0.00	0.00	0.00	0.00	0.00
10	-0.29	-0.22	-0.31	-0.23	-0.32	-0.24
20	-0.52	-0.41	-0.58	-0.45	-0.62	-0.48
30	-0.71	-0.58	-0.84	-0.66	-0.92	-0.71
40	-0.87	-0.72	-1.08	-0.86	-1.22	-0.95
50	-1.00	-0.85	-1.30	-1.05	-1.51	-1.18
60	-1.12	-0.96	-1.50	-1.22	-1.79	-1.41
70	-1.22	-1.06	-1.70	-1.39	-2.07	-1.64
80	-1.31	-1.15	-1.88	-1.55	-2.35	-1.86
90	-1.39	-1.23	-2.05	-1.71	-2.63	-2.09
100	-1.46	-1.30	-2.22	-1.85	-2.90	-2.31
110	-1.52	-1.36	-2.38	-1.99	-3.17	-2.54
120	-1.58	-1.42	-2.52	-2.13	-3.44	-2.76
When π = 0.2
0	0.00	0.00	0.00	0.00	0.00	0.00
10	-0.14	-0.11	-0.15	-0.12	-0.16	-0.12
20	-0.26	-0.21	-0.29	-0.23	-0.31	-0.24
30	-0.36	-0.29	-0.42	-0.33	-0.45	-0.35
40	-0.44	-0.37	-0.54	-0.43	-0.60	-0.46
50	-0.52	-0.43	-0.65	-0.52	-0.74	-0.57
60	-0.58	-0.49	-0.75	-0.61	-0.87	-0.68
70	-0.63	-0.55	-0.85	-0.69	-1.00	-0.79
80	-0.68	-0.59	-0.94	-0.77	-1.13	-0.89
90	-0,72	-0.64	-1.03	-0.85	-1.26	-0.99
100	-0.76	-0.67	-1.11	-0.92	-1.38	-1.10
110	-0.80	-0.71	-1.19	-0.99	-1.51	-1.20
120	-0,83	-0.74	-1.26	-1.05	-1,63	-1.30

Notes: π denotes the share of imported inputs in value added in free trade, σ is the elasticity of substitution between domestic inputs an d foreign inputs. Pure distortion effects of the tariff and the total effect with a lump-sum transfer of revenue are presented.

Table 1. also reports the additional revenue effect of tariffs on the growth rate. The lump-sum transfer of tariff revenue, by increasing steady-state income in the private sector, offsets the negative effects of a tariff, though to a limited degree. The strong distortionary effect with low substitutability is most affected since the lower elasticity of substitution between domestic and imported inputs raises tariff revenue.

Public Investment and Growth

Until now, it has been assumed that government transfers the tariff revenues to the public and that the lump-sum transfer partially offsets the distortionary effect of the tariff by increasing steady-state income in the private sector. This section considers the role of government spending as an input to private production along the lines of Barro (1990). Under this approach, government provides a public input to the production of private goods, which is financed by tariffs. Assume that public inputs are combined with the foreign inputs in a Cobb-Douglas production function for the private good:⁸

Public inputs are interpreted as “publicly provided private goods,” which are “rival” and “excludable.”⁹ Thus, each producer has a property right to the quantity of the public input that is provided by the government. This equation is rewritten in terms of the effective worker:

where a balanced government budget is assumed. The equation shows that the tariff has a positive effect on the production of the private good. Hence, tariff revenue used for the production of private goods diminishes the distortionary effect on the growth rate. If the Cobb-Douglas combination of $\left(k^{\alpha }{\hat{L}}^{1-\alpha }\right)$ and $\left(M^{1-\chi }{G}^{\chi }\right)$ is assumed, the total growth effect of tariffs is simplified to

Thus, the net effect of the tariff on the growth rate is determined as follows:

The expression shows that, even if the government uses all tariff revenue for its most productive use, a high tariff will eventually reduce the growth rate. If one assumes that the share of imported inputs in value added is around 0.3 and the share of total public investment in value added is around 0.033 (which is the 1960-85 average for all countries in Barro (1991)), then tariff rates above 11 percent always lower the growth rate.

The result shows that the relation between tariffs and growth rates is negative except with a very low tariff rate—the distortionary effect of which can be superceded by the stimulating effect of public investment financed by the tariff revenue. This finding that the distortionary effect usually overshadows the productive revenue effect matches the results in Barra (1990) and Easterly (1990). Since the marginal productivity of public investment is very high when the size of public investment is small, the effect of an increase in public input may dominate the distortionary effect of the tariff and create a net positive effect on growth. However, when the tariff rates are high enough, the productivity of public input diminishes; thus, higher tariffs always lead to lower growth rates.

Real Interest Rate Puzzle

One criticism of the neoclassical model is that it predicts an unreasonably wide range of values for the marginal productivity of capital and hence for real interest rates. This prediction apparently contradicts the observed lack of capital flowing from rich countries to poor ones (King and Rebelo (1989)). The open economy model considered here may provide an answer to this contradiction. With trade distortions, the systematic negative relation between per capita income and the marginal productivity of capital (V) no longer holds. Trade restrictions may lower the marginal productivity of capital, and thus the real interest rate, enough to prevent capital from flowing to poor countries. From equations (25) and (26), V can be written as a function of income and the tariff:

To simplify the discussion, tariff revenues are ignored. The result shows that trade distortions affect V more significantly than the difference in income. It is very plausible, therefore, that V is higher in a high-income, low-distortion country than it is in a low-income, high-distortion country.

Table 2. shows the extent to which V is lower in a tariff-ridden economy than in a free trade one, when both countries have the same income. Assuming the conventional value for the capital share $(\alpha =1/3),$the numerical results show that a tariff rate of about 30 percent reduces V to about two-thirds of its free trade value when π is equal to 0.4. A simple calculation shows that a 30 percent tariff rate offsets about 20 percent of the income differential, thus leaving V the same. Therefore, trade restrictions are crucial in determining real interest rates, and thereby the capital flow.

^{Table 2.}

Marginal Productivity of Capital in a Tariff-Ridden Economy

(Index, zero tariffs equal unity)

Notes: π denotes the share of imported inputs in value added in free trade, σ is the elasticity of substitution between domestic inputs and foreign inputs. Marginal productivity of capital under free trade is normalized to 1.0.

^{Table 2.}

Marginal Productivity of Capital in a Tariff-Ridden Economy

(Index, zero tariffs equal unity)

	π = 0.4			π = 0.2
Tariff rate in percent	σ = 2.0	σ = 1.0	σ = 0.5	σ = 2.0	σ = 1.0	σ = 0.5
0	1.00	1.00	1.00	1.00	1.00	1.00
10	0.90	0.89	0.89	0.95	0.94	0.94
20	0.82	0.80	0.79	0.91	0.90	0.89
30	0.77	0.73	0.70	0.87	0.85	0.84
40	0.72	0.67	0.63	0.85	0.82	0.80
50	0.68	0.61	0.57	0.82	0.78	0.76
60	0.65	0.57	0.51	0.80	0.75	0.72
70	0.63	0.53	0.46	0.79	0.72	0.68
80	0.61	0.49	0.41	0.77	0.70	0.65
90	0.59	0.46	0.37	0.76	0.68	0.62
100	0.58	0.43	0.33	0.75	0.66	0.59
110	0.56	0.41	0.30	0.74	0.64	0.57
120	0.55	0.38	0.27	0.73	0.62	0.54

Notes: π denotes the share of imported inputs in value added in free trade, σ is the elasticity of substitution between domestic inputs and foreign inputs. Marginal productivity of capital under free trade is normalized to 1.0.

View Table

^{Table 2.}

Marginal Productivity of Capital in a Tariff-Ridden Economy

(Index, zero tariffs equal unity)

	π = 0.4			π = 0.2
Tariff rate in percent	σ = 2.0	σ = 1.0	σ = 0.5	σ = 2.0	σ = 1.0	σ = 0.5
0	1.00	1.00	1.00	1.00	1.00	1.00
10	0.90	0.89	0.89	0.95	0.94	0.94
20	0.82	0.80	0.79	0.91	0.90	0.89
30	0.77	0.73	0.70	0.87	0.85	0.84
40	0.72	0.67	0.63	0.85	0.82	0.80
50	0.68	0.61	0.57	0.82	0.78	0.76
60	0.65	0.57	0.51	0.80	0.75	0.72
70	0.63	0.53	0.46	0.79	0.72	0.68
80	0.61	0.49	0.41	0.77	0.70	0.65
90	0.59	0.46	0.37	0.76	0.68	0.62
100	0.58	0.43	0.33	0.75	0.66	0.59
110	0.56	0.41	0.30	0.74	0.64	0.57
120	0.55	0.38	0.27	0.73	0.62	0.54

Notes: π denotes the share of imported inputs in value added in free trade, σ is the elasticity of substitution between domestic inputs and foreign inputs. Marginal productivity of capital under free trade is normalized to 1.0.