1. The standard versus modified models: an overview

The Feder (1983) model may be simplified without sacrificing its main features. It can then be compared with the basic modified model proposed in this paper. Table 1 provides a summary of the two models, with the Feder-type model labeled as the standard one.

Table 1.

The Standard and Modified Growth Models

Table 1.

The Standard and Modified Growth Models

Standard model	Modified model
Z = (1 - ε)Lf(k,x)	$\begin{array}{ll}\mathrm{Z}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{Lf}\mathrm{(}\mathrm{k}\mathrm{,}\mathrm{x}\mathrm{)}& \mathrm{\left(}\mathrm{1}\mathrm{\right)}\end{array}$
X = εLg(k)	$\begin{array}{ll}\mathrm{X}\mathrm{=}\mathrm{\in }\mathrm{Lg}\mathrm{\left(}\mathrm{k}\mathrm{\right)}& \mathrm{\left(}\mathrm{2}\mathrm{\right)}\end{array}$
L = EN	$\begin{array}{ll}\mathrm{L}\mathrm{=}\mathrm{EN}& \mathrm{\left(}\mathrm{3}\mathrm{\right)}\end{array}$
k = K/L	$\begin{array}{ll}\mathrm{k}\mathrm{=}\mathrm{K}\mathrm{/}\mathrm{L}& \mathrm{\left(}\mathrm{4}\mathrm{\right)}\end{array}$
X = X/L	$\begin{array}{ll}\mathrm{x}\mathrm{=}\mathrm{X}\mathrm{/}\mathrm{L}& \mathrm{\left(}\mathrm{5}\mathrm{\right)}\end{array}$
$\dot{\mathrm{K}}\mathrm{=}\mathrm{Sq}\mathrm{-}\mathrm{\delta k}$	$\begin{array}{ll}\dot{\mathrm{K}}\mathrm{=}\mathrm{Sq}\mathrm{-}\mathrm{\delta k}& \left(6\right)\end{array}$
Ė/E = λ	$\begin{array}{ll}\dot{\mathrm{E}}\mathrm{N}\mathrm{=}\mathrm{h}\mathrm{\left(}\mathrm{X}\mathrm{\right)}\mathrm{+}\mathrm{\lambda L}\mathrm{;}{\mathrm{h}}^{\mathrm{\prime }}\mathrm{>}\mathrm{0}& \left(7\right)\end{array}$
Ṅ = nN	$\begin{array}{ll}\dot{\mathrm{N}}\mathrm{=}\mathrm{nN}& \mathrm{\left(}\mathrm{8}\mathrm{\right)}\end{array}$
Q = Z + X	$\begin{array}{ll}\mathrm{Q}\mathrm{=}\mathrm{Z}\mathrm{+}\mathrm{X}& \mathrm{\left(}\mathrm{9}\mathrm{\right)}\end{array}$
Reduced model
$\dot{\mathrm{k}}\mathrm{/}\mathrm{k}\mathrm{=}{\mathrm{sk}}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{\right\}}\mathrm{-}\mathrm{(}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{+}\mathrm{\delta }\mathrm{)}\mathrm{;}$	$\begin{array}{ll}{\mathrm{sk}}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{\right\}}\mathrm{-}\mathrm{h}\mathrm{[}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{]}\mathrm{-}\mathrm{(}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{+}\mathrm{\delta }\mathrm{)}\mathrm{.}& \mathrm{\left(}\mathrm{10}\mathrm{\right)}\end{array}$
Equilibrium capital-labor ratio (k*)
Root of the equation:
Sk*-1{(1-ε)f[k*,εg(k*)] + εg(k*)} - (λ+n+δ) = 0;	$\begin{array}{ll}\mathrm{sk}{\mathrm{*}}^{\mathrm{-}\mathrm{1}}\mathrm{\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{*}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{\}}\mathrm{-}\mathrm{h}\mathrm{[}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{-}\mathrm{(}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{+}\mathrm{\delta }\mathrm{)}\mathrm{=}\mathrm{0.}& \mathrm{\left(}\mathrm{11}\mathrm{\right)}\end{array}$
Equilibrium growth rate of output $\mathrm{\left(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{\right)}\mathrm{*}$
$\mathrm{\left(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{\right)}\mathrm{*}$	$\begin{array}{ll}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{(}\dot{\mathrm{K}}\mathrm{/}\mathrm{K}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{sk}{\mathrm{*}}^{\mathrm{-}\mathrm{1}}\mathrm{\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{*}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{\}}\mathrm{-}\mathrm{\delta }\mathrm{;}& \mathrm{\left(}\mathrm{12}\mathrm{a}\mathrm{\right)}\end{array}$
$\mathrm{=}\mathrm{(}\dot{\mathrm{K}}\mathrm{/}\mathrm{K}\mathrm{)}\mathrm{*}$
$\mathrm{=}\mathrm{(}\dot{\mathrm{L}}\mathrm{/}\mathrm{L}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{;}$	$\begin{array}{ll}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{(}\dot{\mathrm{L}}\mathrm{/}\mathrm{L}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{h}\mathrm{[}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{+}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{;}& \mathrm{\left(}\mathrm{12}\mathrm{b}\mathrm{\right)}\end{array}$
$\mathrm{or}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{J}\mathrm{(}\mathrm{s}\mathrm{,}\mathrm{\in }\mathrm{,}\mathrm{\delta }\mathrm{,}\mathrm{\lambda }\mathrm{,}\mathrm{n}\mathrm{)}\mathrm{,}$	$\begin{array}{ll}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{J}\mathrm{(}\underset{\mathrm{+}\mathrm{,}}{\mathrm{s}}\mathrm{,}\underset{\mathrm{+}\mathrm{,}}{\mathrm{\in }}\mathrm{,}\underset{\mathrm{-}\mathrm{,}}{\mathrm{\delta }}\mathrm{,}\underset{\mathrm{+}\mathrm{,}}{\mathrm{\lambda }}\mathrm{,}\underset{\mathrm{+}}{\mathrm{n}}\mathrm{)}\mathrm{,}& \mathrm{\left(}\mathrm{12}\mathrm{c}\mathrm{\right)}\end{array}$
with signs: 0,0,0,1,1
Notation:
Q:	total output (GDP), constant dollars
X:	output of the export sector, constant dollars
Z:	output of the non-export sector, constant dollars
L:	labor, in efficiency units, manhours
K:	capital stock, constant dollars
E:	technical-change or productivity multiplier, index number
N:	population
f(·):	production function for non-exports, intensive form
g(·):	production function for exports, intensive form
h(·):	a unit-homogeneous learning function
J(·):	asymptotic (equilibrium) growth function
ε:	rate of employment of capital and labor in the export sector
s:	saving rate
δ:	depreciation rate
λ:	exogenous rate of labor-augmenting technical change
n:	exogenous rate of population growth

View Table

Table 1.

The Standard and Modified Growth Models

Standard model	Modified model
Z = (1 - ε)Lf(k,x)	$\begin{array}{ll}\mathrm{Z}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{Lf}\mathrm{(}\mathrm{k}\mathrm{,}\mathrm{x}\mathrm{)}& \mathrm{\left(}\mathrm{1}\mathrm{\right)}\end{array}$
X = εLg(k)	$\begin{array}{ll}\mathrm{X}\mathrm{=}\mathrm{\in }\mathrm{Lg}\mathrm{\left(}\mathrm{k}\mathrm{\right)}& \mathrm{\left(}\mathrm{2}\mathrm{\right)}\end{array}$
L = EN	$\begin{array}{ll}\mathrm{L}\mathrm{=}\mathrm{EN}& \mathrm{\left(}\mathrm{3}\mathrm{\right)}\end{array}$
k = K/L	$\begin{array}{ll}\mathrm{k}\mathrm{=}\mathrm{K}\mathrm{/}\mathrm{L}& \mathrm{\left(}\mathrm{4}\mathrm{\right)}\end{array}$
X = X/L	$\begin{array}{ll}\mathrm{x}\mathrm{=}\mathrm{X}\mathrm{/}\mathrm{L}& \mathrm{\left(}\mathrm{5}\mathrm{\right)}\end{array}$
$\dot{\mathrm{K}}\mathrm{=}\mathrm{Sq}\mathrm{-}\mathrm{\delta k}$	$\begin{array}{ll}\dot{\mathrm{K}}\mathrm{=}\mathrm{Sq}\mathrm{-}\mathrm{\delta k}& \left(6\right)\end{array}$
Ė/E = λ	$\begin{array}{ll}\dot{\mathrm{E}}\mathrm{N}\mathrm{=}\mathrm{h}\mathrm{\left(}\mathrm{X}\mathrm{\right)}\mathrm{+}\mathrm{\lambda L}\mathrm{;}{\mathrm{h}}^{\mathrm{\prime }}\mathrm{>}\mathrm{0}& \left(7\right)\end{array}$
Ṅ = nN	$\begin{array}{ll}\dot{\mathrm{N}}\mathrm{=}\mathrm{nN}& \mathrm{\left(}\mathrm{8}\mathrm{\right)}\end{array}$
Q = Z + X	$\begin{array}{ll}\mathrm{Q}\mathrm{=}\mathrm{Z}\mathrm{+}\mathrm{X}& \mathrm{\left(}\mathrm{9}\mathrm{\right)}\end{array}$
Reduced model
$\dot{\mathrm{k}}\mathrm{/}\mathrm{k}\mathrm{=}{\mathrm{sk}}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{\right\}}\mathrm{-}\mathrm{(}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{+}\mathrm{\delta }\mathrm{)}\mathrm{;}$	$\begin{array}{ll}{\mathrm{sk}}^{\mathrm{-}\mathrm{1}}\mathrm{\left\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{\right\}}\mathrm{-}\mathrm{h}\mathrm{[}\mathrm{\in }\mathrm{g}\mathrm{\left(}\mathrm{k}\mathrm{\right)}\mathrm{]}\mathrm{-}\mathrm{(}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{+}\mathrm{\delta }\mathrm{)}\mathrm{.}& \mathrm{\left(}\mathrm{10}\mathrm{\right)}\end{array}$
Equilibrium capital-labor ratio (k*)
Root of the equation:
Sk*-1{(1-ε)f[k*,εg(k*)] + εg(k*)} - (λ+n+δ) = 0;	$\begin{array}{ll}\mathrm{sk}{\mathrm{*}}^{\mathrm{-}\mathrm{1}}\mathrm{\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{*}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{\}}\mathrm{-}\mathrm{h}\mathrm{[}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{-}\mathrm{(}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{+}\mathrm{\delta }\mathrm{)}\mathrm{=}\mathrm{0.}& \mathrm{\left(}\mathrm{11}\mathrm{\right)}\end{array}$
Equilibrium growth rate of output $\mathrm{\left(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{\right)}\mathrm{*}$
$\mathrm{\left(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{\right)}\mathrm{*}$	$\begin{array}{ll}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{(}\dot{\mathrm{K}}\mathrm{/}\mathrm{K}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{sk}{\mathrm{*}}^{\mathrm{-}\mathrm{1}}\mathrm{\{}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{\in }\mathrm{)}\mathrm{f}\mathrm{[}\mathrm{k}\mathrm{*}\mathrm{,}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{+}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{\}}\mathrm{-}\mathrm{\delta }\mathrm{;}& \mathrm{\left(}\mathrm{12}\mathrm{a}\mathrm{\right)}\end{array}$
$\mathrm{=}\mathrm{(}\dot{\mathrm{K}}\mathrm{/}\mathrm{K}\mathrm{)}\mathrm{*}$
$\mathrm{=}\mathrm{(}\dot{\mathrm{L}}\mathrm{/}\mathrm{L}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{;}$	$\begin{array}{ll}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{(}\dot{\mathrm{L}}\mathrm{/}\mathrm{L}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{h}\mathrm{[}\mathrm{\in }\mathrm{g}\mathrm{(}\mathrm{k}\mathrm{*}\mathrm{)}\mathrm{]}\mathrm{+}\mathrm{\lambda }\mathrm{+}\mathrm{n}\mathrm{;}& \mathrm{\left(}\mathrm{12}\mathrm{b}\mathrm{\right)}\end{array}$
$\mathrm{or}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{J}\mathrm{(}\mathrm{s}\mathrm{,}\mathrm{\in }\mathrm{,}\mathrm{\delta }\mathrm{,}\mathrm{\lambda }\mathrm{,}\mathrm{n}\mathrm{)}\mathrm{,}$	$\begin{array}{ll}\mathrm{(}\dot{\mathrm{Q}}\mathrm{/}\mathrm{Q}\mathrm{)}\mathrm{*}\mathrm{=}\mathrm{J}\mathrm{(}\underset{\mathrm{+}\mathrm{,}}{\mathrm{s}}\mathrm{,}\underset{\mathrm{+}\mathrm{,}}{\mathrm{\in }}\mathrm{,}\underset{\mathrm{-}\mathrm{,}}{\mathrm{\delta }}\mathrm{,}\underset{\mathrm{+}\mathrm{,}}{\mathrm{\lambda }}\mathrm{,}\underset{\mathrm{+}}{\mathrm{n}}\mathrm{)}\mathrm{,}& \mathrm{\left(}\mathrm{12}\mathrm{c}\mathrm{\right)}\end{array}$
with signs: 0,0,0,1,1
Notation:
Q:	total output (GDP), constant dollars
X:	output of the export sector, constant dollars
Z:	output of the non-export sector, constant dollars
L:	labor, in efficiency units, manhours
K:	capital stock, constant dollars
E:	technical-change or productivity multiplier, index number
N:	population
f(·):	production function for non-exports, intensive form
g(·):	production function for exports, intensive form
h(·):	a unit-homogeneous learning function
J(·):	asymptotic (equilibrium) growth function
ε:	rate of employment of capital and labor in the export sector
s:	saving rate
δ:	depreciation rate
λ:	exogenous rate of labor-augmenting technical change
n:	exogenous rate of population growth

Both standard and modified models have a two-sector structure, involving two neoclassical production functions for exports and non-exports. For simplicity, it is assumed that the export sector employs a constant uniform rate, e, of the total amounts of K and L, with 1-ε being the employment rate prevailing in the non-export sector. ^2/ Both models also assume significant inter-sectoral externalities involving exports and that the marginal productivities in the export sector are higher than those in the non-export sector. Reflecting these assumptions the production functions for non-exports in both the standard and the modified models include exports as a separate input (see equation (1) in Table 1). Both models measure the labor input in efficiency units (equation (3)), ^3/ and employ the neoclassical capital accumulation function (equation (6)), where the net increase in the capital stock is equal to gross saving minus depreciation.

It turns out that the standard assumption of a direct effect of exports on the output of the non-export sector is not essential to the different equilibrium growth implications of the two models (compare the two expressions for the equilibrium growth rate in (12c) of Table 1). Rather, the critical differences between the standard and modified models lie in their alternative assumptions about the nature of labor-augmenting technical change in relation to the size of the export sector (compare the equilibrium growth equations in (12b)). ^1/ In particular, the standard model assumes that the rate of labor-augmenting technical change is independent of export activity. In contrast, the modified model formalizes the observation made by Keesing (1967) and Feder (1983) that the export sector tends to improve the quality (productivity) of the labor input by providing a valuable learning experience (compare the technical change functions in (7)). In other words, even if the modified model adopts the standard model’s production function for non-exports, in which the export variable appears as another factor input, important differences in the implications of the two models will remain. The assumption about the rate of labor-augmenting technical change turns out to be crucial. The standard model’s hypothesis that h′ = 0 means that labor-augmenting technical change is not affected by export activity, and is taking place at an exogenous constant rate A. In the real world, while a portion of technical change may indeed be exogenous, some technical change is clearly endogenous, partly labor-augmenting, and positively enhanced by the expansion of exports, as argued by Keesing (1967) and Feder (1983) and empirically verified by Romer (1990). ^2/ That is, the hypothesis of this paper that h′ > 0, as against the standard assumption that h′ = 0, appears plausible and merits serious consideration.

Suppose that h′ = 0, as assumed in the standard model. Where does this assumption lead? It is relatively straightforward to show that the equilibrium growth path would be one on which $\dot{\mathrm{Q}}/\mathrm{Q}=\mathrm{n}+\lambda$ (see the standard model’s equation (12b)). Per capita output grows at the exogenous rate of labor-augmenting technical change λ. The saving, factor utilization rates in the export sector, and depreciation rates do not affect the long-run growth rate of per capita output. This is inconsistent with the robust empirical result that saving behavior and export activity do influence the long-term growth rate of output. ^3/ Again, the alternative hypothesis h′ > 0 deserves to be considered. This hypothesis says that labor-augmenting technological innovations are transmitted to the domestic economy partly through the export sector. In the modified model the export variable has two effects on output. First through its role as a factor input in the production of non-exportables, and second through its influence on the rate of labor-augmenting technical progress. The first channel is a level effect (a one-time shift in the production function for non-exports). The second channel is a permanent growth effect. Of these two mechanisms, the second one on technical change is pivotal.

Reflecting this difference in assumptions about the presence or absence of any link between exports and technological progress, the conclusions regarding the long-run growth rate of output are strikingly dissimilar. In the standard model, as in the neoclassical model without exports, the steady-state growth rate of output is fixed by the constant growth rate of population, n, plus the exogenous rate of labor-augmenting technical change, λ. By contrast, in the modified model, in addition to being affected by λ and n, the steady-state growth rate of output can be raised by increasing the employment rate ε in the export sector and the aggregate saving rate s, and by lowering the rate of depreciation of capital δ (compare the signs of the partial derivatives of the growth functions in (12c) of Table 1).

Figure 1 illustrates the workings of the two models. The vertical and horizontal axes measure, respectively, the growth rates of capital and labor (in efficiency units) and the capital-labor ratio. Panels A and B, respectively, depict the standard and modified models. In either model the relationship between the rate of capital accumulation and the capital-labor ratio (the KK curve) is downward-sloping because of the diminishing marginal productivity of capital. In the standard model, the labor growth schedule (the L_SL_S curve) is horizontal with vertical height equal to A + n for all levels of the capital-labor ratio because, by assumption, labor-augmenting technical change is independent of the export-labor ratio and thus of the capital-labor ratio (that is, h′ = 0). In the modified model, the labor growth curve is now upward sloping because the rate of labor-augmenting technological progress is a positive function of output per unit of labor in the export sector and, hence, of the capital-labor ratio. ^1/

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Long-Run Growth Equilibrium

Citation: IMF Working Papers 1993, 041; 10.5089/9781451846027.001.A001

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Now, suppose that the rate of utilization of capital and labor in the export sector, ε, rises for some reason (for example, vigorous implementation of export promotion policies, including competitive exchange rate policy, trade liberalization, and tariff reform). In the standard model (see panel A of Figure 1), the higher value of ε shifts the KK curve to the right, to K′K′, and the new equilibrium is established at point C(k*s1,λ+n), which is characterized by a higher capital-labor ratio but an unchanged growth rate of output. ^2/ The transitional dynamics are traced by the path A-B-C. The growth rate of the capital stock initially jumps to point B, exceeding that of the growth rate of the labor force by an amount equal to AB. This rise in the growth rate of the capital stock and, hence, of output, is only temporary and cannot be sustained over time, since the labor input ultimately becomes a bottleneck in the production process. ^1/ As the capital-labor ratio rises from k*s0, towards k*s1, the marginal productivity of capital declines and firms will slow the rate of investment until the growth rate of the capital stock is brought down to the constant rate of growth of the labor force at point C. Labor growth, being independent of the capital-labor ratio, slides horizontally from A to C. Thus, in the long run the rise in the export employment rate ε raises capital intensity and the level of exports and output, but leaves the growth rate of per capita output unaffected, this being fixed by the exogenous rate of labor-augmenting technical change λ. ^2/

Turning to the workings of the modified model, the rise in the export employment rate ε shifts the KK curve to the right, to K′K′. However, the L_mL_m, curve also shifts upward to the left, to L_m′L_m′, intersecting the K′K′ at point F. As in the standard model the growth rate of the capital stock initially jumps to point H, exceeding labor growth by EH. But, in contrast to the standard model, the modified model exhibits a new equilibrium that is characterized by a higher growth rate of output (with the growth rate increasing from γ*m0 to γ*m1). ^3/ The main difference between two models lies in the behavior of the growth rate of labor. Referring to panel B, Figure 1, the initial rise in the capital-labor ratio resulting from the increase in the employment rate ε in the export sector leads to an increase in the growth rate of the labor input, instead of remaining constant as in the standard model, for two reasons. First, an increase in ε directly raises exports per unit of labor and thus the rate of labor-augmenting technical change (this is represented by the shift from L_mL_m to L_m′L_m′), an increase shown by the distance DE. Second, as the capital-labor rises a proportion of the increase is used to raise the output of the export sector further, providing an additional boost to the rate of labor-augmenting technological progress (this is represented by the movement along the new L_m′L_m′ curve), an increase traced by E-F. ^1/ While the growth of the capital stock declines after an increase in capital-labor ratio (from H to F, owing to diminishing marginal productivity of capital), the growth rate of the labor input rises from D to E to F. After adjustments are completed, the growth rates of capital and labor converge at some point such as F. Therefore, in the modified model with endogenous technical change the long-run growth rate of per capita output increases when the rate of employment in the export sector is raised.

The available long-run cross-sectional empirical studies reviewed in Khan and Villanueva (1991) find that the saving rate (or investment rate) and some measure of export activity do indeed influence positively and significantly the growth rate of potential output. These findings are consistent with the steady-state behavior of our modified model. They do not support the hypotheses of the standard model that ϵ and s exert no long-run effects on output growth.

To sum up, the restrictive assumption behind most export-cum-growth models is that technical change is given exogenously, typically as a constant rate of labor-augmenting, or Harrod-neutral, technical change λ. The modified model allows for an export-induced endogenous component of technical change ^2/, in addition to an exogenous component. While the inclusion of exports in the standard neoclassical model enriches the transitional growth dynamics, the (asymptotic) long-run growth rate of per capita output remains fixed by the rate λ. ^3/ Thus, the robust empirical result that exports and the growth rate of output are positively correlated in the long run appears to be consistent with our modified model, whereas it is not consistent with those of standard theoretical models formulated to underpin existing empirical work.

2. A two-sector modified model

The foregoing discussion suggests the irrelevance of exports as a third factor of production to the steady-state behavior of the growth rate of output, and the crucial importance of exports as a determinant of the rate of Harrod-neutral technological progress. This subsection develops a modified two-sector neoclassical growth model in which exports enhance the rate of labor-augmenting technical change. This is essentially a formalization of the mechanism identified by earlier authors, notably Keesing (1967). Consider the following model, summarized in Table 2. The sectoral production functions for non-exports and exports, respectively, are given in equations (13) and (14). These functions are assumed to satisfy the Inada (1963) conditions. ^1/ As before, labor is measured in efficiency units. The sectoral resource allocation coefficients, μ and θ, are assumed to be given in (15) and (16), subject to changes resulting from policy shifts and, possibly, by relative profitability in the two sectors. The sectoral stocks of capital and quantities of labor services add up to the economy-wide totals in (17) and (18). Constant proportions of sectoral outputs are saved and invested in (19) and (20), where the net increases in the sectoral capital stocks are equal to sectoral gross saving less depreciation (the same depreciation rate is assumed to apply to the capital stock in each sector). ^2/

Table 2.

The Modified Two-Sector Model

Table 2.

The Modified Two-Sector Model

$\begin{array}{cc}\mathrm{Z}=\mathrm{F}({\mathrm{K}}_{\mathrm{z}},{\mathrm{E}}_{\mathrm{z}}{\mathrm{N}}_{\mathrm{z}})& \left(13\right)\end{array}$
$\begin{array}{cc}\mathrm{X}=\mathrm{G}({\mathrm{K}}_{\mathrm{x}},{\mathrm{E}}_{\mathrm{x}}{\mathrm{N}}_{\mathrm{x}})& \left(14\right)\end{array}$
$\begin{array}{cc}{\mathrm{K}}_{\mathrm{x}}/{\mathrm{K}}_{\mathrm{z}}=\mu & \left(15\right)\end{array}$
$\begin{array}{cc}{\mathrm{N}}_{\mathrm{x}}/{\mathrm{N}}_{\mathrm{z}}=\theta & \left(16\right)\end{array}$
$\begin{array}{cc}{\mathrm{K}=\mathrm{K}}_{\mathrm{x}}+{\mathrm{K}}_{\mathrm{z}}& \left(17\right)\end{array}$
$\begin{array}{cc}{\mathrm{N}=\mathrm{N}}_{\mathrm{x}}+{\mathrm{N}}_{\mathrm{z}}& \left(18\right)\end{array}$
$\begin{array}{cc}{\dot{\mathrm{K}}}_{\mathrm{x}}={\chi }^{\mathrm{X}}-\delta {\mathrm{K}}_{\mathrm{x}}& \left(19\right)\end{array}$
$\begin{array}{cc}{\dot{\mathrm{K}}}_{\mathrm{z}}={\eta }^{\mathrm{Z}}-\delta {\mathrm{K}}_{\mathrm{z}}& \left(20\right)\end{array}$
$\begin{array}{cc}{\mathrm{E}}_{\mathrm{x}}/{\mathrm{E}}_{\mathrm{z}}=1+\alpha ;\alpha \ge 0& \left(21\right)\end{array}$
$\begin{array}{cc}{\dot{\mathrm{E}}}_{\mathrm{z}}{\mathrm{N}}_{\mathrm{z}}=\mathrm{h}(\mathrm{X})+\lambda {\mathrm{E}}_{\mathrm{z}}{\mathrm{N}}_{\mathrm{z}};{\mathrm{h}}^{\prime }>0& \left(22\right)\end{array}$
$\begin{array}{cc}\dot{\mathrm{N}}=\mathrm{nN}& \left(23\right)\end{array}$
$\begin{array}{cc}{\mathrm{k}}^{\prime }=\mathrm{k}/{\mathrm{E}}_{\mathrm{z}}\mathrm{N}& \left(24\right)\end{array}$
$\begin{array}{cc}\mathrm{Q}=\mathrm{X}+\mathrm{Z}& \left(25\right)\end{array}$
Reduced model
$\begin{array}{ccc}\hfill {\dot{\mathrm{k}}}^{\prime }/{\mathrm{k}}^{\prime }& =\chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }]+\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }]\hfill & \hfill \\ \hfill & -\mathrm{h}(\mathrm{G}[(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime },(1+\alpha )\theta ]-(\lambda +\mathrm{n}+\delta )\hfill & \left(26\right)\hfill \end{array}$
Equilibrium capital-iabor ratio (k′*)
Root of the equation:
$\begin{array}{cc}\chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }*]+\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }*]& \\ -\mathrm{h}(\mathrm{G}[(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime }*,(1+\alpha )\theta ]-(\lambda +\mathrm{n}+\delta )=0& \left(27\right)\end{array}$
Equilibrium growth rate of output (Q/Q)*
$\begin{array}{ccc}[(\mathrm{Q})/\mathrm{Q}]*=(\dot{\mathrm{K}}/\mathrm{K})*=& \chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }*]\hfill & \\ & +\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }*]-\delta \hfill & \left(28\mathrm{a}\right)\end{array}$
$\begin{array}{cc}[\left(\mathrm{Q}\right)/\mathrm{Q}]*=({\dot{\mathrm{E}}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{z}})*+(\dot{\mathrm{N}}/\mathrm{N})*=\mathrm{h}(\mathrm{G}\left[\right(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime }*(1+\alpha )\theta ]+\lambda +\mathrm{n}& \left(28\mathrm{b}\right)\end{array}$
Notation:
The notation is identical for the same variables appearing in Table 1. The subscripts x and z refer to exports and non-exports, respectively. Other variables and parameters are defined as follows.
	k′: ratio of K to EZN
	χ: saving rate of the export sector
	η: saving rate of the non-export sector
	μ: ratio of sectoral capital stocks
	θ: ratio of sectoral labor services
	α: a proportional factor by which Ex exceeds Ez

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Table 2.

The Modified Two-Sector Model

$\begin{array}{cc}\mathrm{Z}=\mathrm{F}({\mathrm{K}}_{\mathrm{z}},{\mathrm{E}}_{\mathrm{z}}{\mathrm{N}}_{\mathrm{z}})& \left(13\right)\end{array}$
$\begin{array}{cc}\mathrm{X}=\mathrm{G}({\mathrm{K}}_{\mathrm{x}},{\mathrm{E}}_{\mathrm{x}}{\mathrm{N}}_{\mathrm{x}})& \left(14\right)\end{array}$
$\begin{array}{cc}{\mathrm{K}}_{\mathrm{x}}/{\mathrm{K}}_{\mathrm{z}}=\mu & \left(15\right)\end{array}$
$\begin{array}{cc}{\mathrm{N}}_{\mathrm{x}}/{\mathrm{N}}_{\mathrm{z}}=\theta & \left(16\right)\end{array}$
$\begin{array}{cc}{\mathrm{K}=\mathrm{K}}_{\mathrm{x}}+{\mathrm{K}}_{\mathrm{z}}& \left(17\right)\end{array}$
$\begin{array}{cc}{\mathrm{N}=\mathrm{N}}_{\mathrm{x}}+{\mathrm{N}}_{\mathrm{z}}& \left(18\right)\end{array}$
$\begin{array}{cc}{\dot{\mathrm{K}}}_{\mathrm{x}}={\chi }^{\mathrm{X}}-\delta {\mathrm{K}}_{\mathrm{x}}& \left(19\right)\end{array}$
$\begin{array}{cc}{\dot{\mathrm{K}}}_{\mathrm{z}}={\eta }^{\mathrm{Z}}-\delta {\mathrm{K}}_{\mathrm{z}}& \left(20\right)\end{array}$
$\begin{array}{cc}{\mathrm{E}}_{\mathrm{x}}/{\mathrm{E}}_{\mathrm{z}}=1+\alpha ;\alpha \ge 0& \left(21\right)\end{array}$
$\begin{array}{cc}{\dot{\mathrm{E}}}_{\mathrm{z}}{\mathrm{N}}_{\mathrm{z}}=\mathrm{h}(\mathrm{X})+\lambda {\mathrm{E}}_{\mathrm{z}}{\mathrm{N}}_{\mathrm{z}};{\mathrm{h}}^{\prime }>0& \left(22\right)\end{array}$
$\begin{array}{cc}\dot{\mathrm{N}}=\mathrm{nN}& \left(23\right)\end{array}$
$\begin{array}{cc}{\mathrm{k}}^{\prime }=\mathrm{k}/{\mathrm{E}}_{\mathrm{z}}\mathrm{N}& \left(24\right)\end{array}$
$\begin{array}{cc}\mathrm{Q}=\mathrm{X}+\mathrm{Z}& \left(25\right)\end{array}$
Reduced model
$\begin{array}{ccc}\hfill {\dot{\mathrm{k}}}^{\prime }/{\mathrm{k}}^{\prime }& =\chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }]+\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }]\hfill & \hfill \\ \hfill & -\mathrm{h}(\mathrm{G}[(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime },(1+\alpha )\theta ]-(\lambda +\mathrm{n}+\delta )\hfill & \left(26\right)\hfill \end{array}$
Equilibrium capital-iabor ratio (k′*)
Root of the equation:
$\begin{array}{cc}\chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }*]+\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }*]& \\ -\mathrm{h}(\mathrm{G}[(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime }*,(1+\alpha )\theta ]-(\lambda +\mathrm{n}+\delta )=0& \left(27\right)\end{array}$
Equilibrium growth rate of output (Q/Q)*
$\begin{array}{ccc}[(\mathrm{Q})/\mathrm{Q}]*=(\dot{\mathrm{K}}/\mathrm{K})*=& \chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }*]\hfill & \\ & +\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }*]-\delta \hfill & \left(28\mathrm{a}\right)\end{array}$
$\begin{array}{cc}[\left(\mathrm{Q}\right)/\mathrm{Q}]*=({\dot{\mathrm{E}}}_{\mathrm{z}}/{\mathrm{E}}_{\mathrm{z}})*+(\dot{\mathrm{N}}/\mathrm{N})*=\mathrm{h}(\mathrm{G}\left[\right(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime }*(1+\alpha )\theta ]+\lambda +\mathrm{n}& \left(28\mathrm{b}\right)\end{array}$
Notation:
The notation is identical for the same variables appearing in Table 1. The subscripts x and z refer to exports and non-exports, respectively. Other variables and parameters are defined as follows.
	k′: ratio of K to EZN
	χ: saving rate of the export sector
	η: saving rate of the non-export sector
	μ: ratio of sectoral capital stocks
	θ: ratio of sectoral labor services
	α: a proportional factor by which Ex exceeds Ez

Equation (21) says that the export sector is at least as technologically advanced as the non-export sector, such that the labor-augmenting technical change multiplier E_x is at least as large as E_z. This is a very plausible assumption.

Equation (22) is the most important relationship in the modified model. It hypothesizes that, as a form of inter-sectoral externality, the productivity of labor employed in the non-export sector is influenced positively by export activity (h′ > 0), and also by exogenous factors. The modified model collapses to the standard Feder-type model if it is assumed that h′ - 0; that is, assuming that labor productivity in the non-export sector is independent of export activity. As demonstrated earlier, inter-sectoral externalities that assign exports the role of an additional input in the production function of the non-export sector (à la Feder) make no difference to the asymptotic behavior of the growth model. Finally, equation (24) defines a new variable k′ as the ratio of K to E_ZN. Equations (23) and (25) are the same as in Table 1.

a. Equilibrium behavior

The growth rate of the effective capital stock, denoted ωj(k′), may be derived by differentiating equation (17) with respect to time and substituting (13)-(16), (19)-(21), and (24):

$\begin{array}{cc}\omega ({\mathrm{k}}^{\prime })=\chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }]+\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }]-\delta & \left(\begin{array}{c}29\end{array}\right)\end{array}$

Similarly, the growth rate of the effective labor input, denoted ψ(k′), may be derived by differentiating equation (18) with respect to time and substituting (13), (15)-(16), and (21)-(24):

$\begin{array}{cc}\psi ({\mathrm{k}}^{\prime })=\mathrm{h}(\mathrm{G}[(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime },(1+\alpha )\theta ]+\lambda +\mathrm{n}& \left(30\right)\end{array}$

The growth rate of the capital-labor ratio, k′, is thus equal to (see equation (26), Table 2):

$\begin{array}{cc}{\dot{\mathrm{k}}}^{\prime }/{\mathrm{k}}^{\prime }=\omega ({\mathrm{k}}^{\prime })-\psi ({\mathrm{k}}^{\prime })& \left(31\right)\end{array}$

The reduced model, equation (31), is a single equation involving the variables ${\dot{\mathrm{k}}}^{\prime }/{\mathrm{k}}^{\prime }$ and k′ alone. Given the assumed properties of the neoclassical production function (the Inada (1963) conditions), equation (31) graphs as in Figure 2, Panel A. The downward slope of the ${\dot{\mathrm{k}}}^{\prime }/{\mathrm{k}}^{\prime }$ equation follows from the assumption of positive but diminishing marginal productivities of capital in the two sectors. The reasons why the curve representing equation (31) lies partly in the first quadrant and partly in the fourth quadrant are given by the other Inada (1963) conditions--for some initial values of the capital-labor ratio it is possible for capital to grow either faster or slower than labor.

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The Two-Sector Model

Citation: IMF Working Papers 1993, 041; 10.5089/9781451846027.001.A001

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It is obvious by inspection that, at any point on the ${\dot{\mathrm{k}}}^{\prime }/{\mathrm{k}}^{\prime }$ curve, the economic system would move in the direction indicated by the arrows. Thus, k′ tends to settle at an equilibrium value k′*. At this point, K and E_ZN would grow at the same rate and, by the constant returns assumption, output Q also would grow at this rate. Indicating this equilibrium growth rate γ:

$\begin{array}{cc}\gamma ({\mathrm{k}}^{\prime }*)=\chi \mathrm{G}[(\mu /(1+\mu )),(1+\alpha )/({1+\theta }^{-1}){\mathrm{k}}^{\prime }*]+\eta \mathrm{F}[(1/(1+\mu )),1/(1+\theta ){\mathrm{k}}^{\prime }*]-\delta & \left(28\mathrm{a}\right)\end{array}$

$\begin{array}{cc}\gamma ({\mathrm{k}}^{\prime }*)=\mathrm{h}(\mathrm{G}[(\mu /(1+\mu ))(1+\theta ){\mathrm{k}}^{\prime }*,(1+\alpha )\theta ]+\lambda +\mathrm{n}& \left(28\mathrm{b}\right)\end{array}$

Given the production functions F, G, and the learning function h, and since k′* is a function of the structural parameters of the model, γ is in general a function of χ, η, μ, θ, α, δ, n, and λ. Note that this function γ = γ (χ, η, χ, θ, α, δ, n, λ|G, F, h) has partial derivatives with signs +, +, +, +, +, -, +, +. The exact form of the growth rate γ-function is implied by the production functions G, F and the learning function h. However, even for a simple Cobb-Douglas production function and a linear learning function, an explicit solution for the γ-function is generally difficult, if at all possible.

3. Some extensions

The basic growth model (see the modified model, Table 1) can be extended in several directions. Maizels (1968) has argued that the marginal propensity to save in the export sector could be larger than elsewhere, in which case the overall saving-income ratio, s, would increase with an expanding export sector. This hypothesis can be incorporated in the model by assuming that s = s(χ), with s′ > 0. The growth effects of export expansion would be magnified by this extension, because of an additional chapnel (via a higher overall saving-income ratio) through which increased export activity raises the growth rate of the capital stock.

The model can also be extended to incorporate fiscal variables by disaggregating total domestic saving into private and government saving: S = S^P + τ^Q - C^g, where τ is the average income tax rate, and C^g is government real current expenditure on goods and services. Allowing for a degree of debt neutrality or incomplete Ricardian equivalence ^1/, private saving can be assumed to be a constant fraction of disposable income: S^P = σ(1-βτ)Q - (l-β) (τQ - C^g), where σ is the private saving ratio, 1-β is the proportion of a change in government saving offset by an opposite change in private saving (β = 0 means full debt neutrality or complete Ricardian equivalence). ^2/ Assuming that C^g = ΓQ, where Γ is a policy parameter, the aggregate saving ratio now becomes: s = [σ(χ) (1-γτ)+(τ -Γ)β], with σ′(χ) > 0. As explained in the preceding paragraph, an increase in χ would have a magnified growth effect through an increase in the private saving rate σ(χ). The growth effects of changes in the average income tax rate τ and the current expenditure ratio Γ can also be analyzed in this extended model. As long as Ricardian equivalence is incomplete, that is, γ is non-zero, an increase in the tax-income ratio would raise the overall domestic saving ratio and thus the equilibrium growth rate of output. ^3/ The opposite effects would be brought about by an increase in the current government expenditure rate Γ.