Following the above discussion of land use change in Latin America and the Caribbean, it is assumed that the rural economy displays land surplus characteristics. In addition, it comprises two separate sectors that exhibit distinctly different patterns of labor, land and natural resource use. One sector consists of commercially oriented activities that convert and exploit available land and natural resources for a variety of traded primary product outputs. Land and other natural resources are sufficiently abundant for use in primary production, but can only be appropriated through employing an increasing amount of labor for this purpose. The other rural sector contains smallholders employing traditional methods to cultivate less favorable agricultural land. For these smallholders, land is also abundant but of extremely poor quality for agricultural production. There is perfect labor mobility throughout the dualistic rural economy.

In addition to the rural economy, there is a modern or leading sector. Following models of structural transformation in developing countries, “the modern sector basically comprises industry along with parts of agriculture and services” (Ocampo et al. 2009, p. 122). Firms in this sector employ capital and labor, innovate through learning-by-doing technological change and generate knowledge spillovers. There is perfect labor mobility between the rural economy and the modern sector.

### Commercial primary production

In this rural sector, production of the primary product (plantation crops, timber, beef, mineral, etc.) depends directly on inputs of land and/or natural resources *N*_{1} and labor *L*_{1}; any capital input is fixed and fully funded out of normal profits. Primary production *Q*_{1} is determined by a function with the normal concave properties and is homogeneous of degree one

$$ Q_{1} = F\left( {N_{1} ,L_{1} } \right),F_{i} > 0,F_{ii} < 0,\quad i = N,L. $$

(1)

The commercial activity can obtain more land or natural resources (hereafter referred to as “resource”) for primary production, but only by employing and allocating more labor for this purpose. It is assumed that increasing *N*_{1} incurs a rising input of *L*_{1}

$$ L_{1} = z\left( {N_{1} } \right),z^{{\prime }} > 0,z^{{\prime \prime }} \ge 0, $$

(2)

where \( z^{\prime}\left( {N_{1} } \right) \) is the marginal labor requirement of obtaining and transforming a unit of the resource input, which is a convex function of the amount of *N*_{1} appropriated.

Letting *p*_{1} be the world price of traded primary products and *w* the wage rate, it follows that total profits are \( \pi = L_{1} \left[ {p_{1} f\left( {n_{1} } \right) - w} \right] = L_{1} \left[ {p_{1} f\left( {{{z^{ - 1} \left( {L_{1} } \right)} \mathord{\left/ {\vphantom {{z^{ - 1} \left( {L_{1} } \right)} {L_{1} }}} \right. \kern-0pt} {L_{1} }}} \right) - w} \right] \), \( n_{1} = {{N_{1} } \mathord{\left/ {\vphantom {{N_{1} } {L_{1} }}} \right. \kern-0pt} {L_{1} }} \), \( f\left( {n_{1} } \right) = F\left( {{{N_{1} } \mathord{\left/ {\vphantom {{N_{1} } {L_{1} ,1}}} \right. \kern-0pt} {L_{1} ,1}}} \right) \). Profit-maximizing, therefore, leads to

$$ f\left( {n_{1} } \right) + f_{N} (n_{1} )\left[ {\frac{1}{{z^{{\prime }} \left( {N_{1} } \right)}} - n_{1} } \right] = f\left( {n_{1} } \right) - f_{N} (n_{1} )n_{1} \left[ {1 - \varepsilon \left( {N_{1} } \right)} \right] = \frac{w}{{p_{1} }},\quad 0 < \varepsilon \left( {N_{1} } \right) < 1, $$

(3)

where \( \varepsilon \left( {N_{1} } \right) \equiv \frac{{\partial N_{1} }}{{\partial L_{1} }}\frac{{L_{1} }}{{N_{1} }} = \frac{1}{{z^{{\prime }} n_{1} }} \) is the elasticity of resource conversion, i.e., the percentage increase in resources appropriated for primary production in response to proportionately more labor devoted to this purpose. It is assumed that the normal case is \( 0 < \varepsilon \left( {N_{1} } \right) < 1 \).^{Footnote 2} Condition (3) indicates that labor will be used in commercial primary production activities until its value marginal productivity equals the wage rate. As labor is used for both appropriating resources and production, the wage rate will be higher if resources are fixed in supply and thus labor was used only for production (\( \varepsilon \left( {N_{1} } \right) = 0 \)). Because in (3) the value marginal productivity of labor declines with respect to *L*_{1}, the wage rate is a decreasing function of labor employed in the primary product sector.

### Traditional agriculture on marginal land

Production of non-traded agricultural output on less favored, or marginal, land also involves two inputs, land \( \left( {N^{m} } \right) \) and labor \( (L^{m} ) \); any capital input is fixed and fully funded out of normal profits. Both land and labor are required for traditional agricultural production, \( Q^{m} \), which is determined by the following linearly homogeneous function

$$ Q^{m} = G\left( {N^{m} ,L^{m} } \right),G_{i} \ge 0,G_{ii} < 0,\quad i = N,L $$

(4)

Note that the marginal productivity of land is not necessarily positive, which is the case for all less favored agricultural land. This Ricardian surplus land condition follows from the assumption that poor quality marginal land is unproductive in cultivation (Hansen 1979). That is, for traditional agriculture on less favored land \( G_{N} = 0 \), and because (3) does not apply to marginal land conversion, equilibrium is determined by

$$ g_{N} \left( {n^{m} } \right) = 0, \quad q^{m} = g\left( {n^{m} } \right) = \frac{w}{{p^{m} }}, \quad n^{m} = \frac{{N^{m} }}{{L^{m} }}, \quad q^{m} = \frac{{Q^{m} }}{{L^{m} }} = G\left( {{{N^{m} } \mathord{\left/ {\vphantom {{N^{m} } {L^{m} ,1}}} \right. \kern-0pt} {L^{m} ,1}}} \right). $$

(5)

The result is that there are no diminishing returns to labor in the use of less favored land for agricultural production. Real wages are invariant to rural employment and determined by the average product of labor. Moreover, the condition of zero marginal productivity fixes the land–labor ratio on less favored agricultural land, which can be designated as \( n^{m} \). Given the average product of labor relationship in (5), the fixed land–labor ratio will determine the nominal wage rate *w* for the predetermined output price *p*^{m}. Thus, the best that rural households on less favored agricultural land can do is either to sell their labor to each other and obtain an equilibrium real wage \( {w \mathord{\left/ {\vphantom {w p}} \right. \kern-0pt} p}^{m} \), or alternatively, farm their own plots of land and earn the same real wage. Since there is little advantage in selling their labor, households will tend to use their labor to farm their own land. Hence, under this marginal land condition, small family farms consuming their own production will predominate. Unless the population increases, no more land will be brought into production and there will be surplus of unfarmed less favored land.

### Modern sector

The modern sector, which includes industry but also technically advanced agriculture and services, has labor-augmenting technology that benefits from learning-by-doing and knowledge spillovers. Production in the sector depends on both unskilled labor and capital, which could also comprise human capital (skills). For the representative firm, an increase in the firm’s capital stock leads to a parallel increase in its stock of knowledge. Each firm’s knowledge is a public good, however, that any other firm can access at zero cost.

For the representative *i*th firm, output *Q*_{2i} is produced by hiring capital *K*_{2i} and labor *L*_{2i}, and *A*_{2i} is the amount of labor-augmenting technology available to the firm. But, with the presence of learning-by-doing and knowledge spillovers \( A_{2i} = K_{2} = \sum\limits_{i} {K_{2i} } \) and the representative firm’s production function is

$$ Q_{2i} = H\left( {K_{2i} ,A_{2i} L_{2i} } \right) = H\left( {K_{2i} ,K_{2} L_{2i} } \right),H_{j} > 0,H_{jj} < 0,\quad j = K,L. $$

(6)

Production of the firm displays diminishing returns to its own stock of capital *K*_{2i}, provided that *K*_{2} and *L*_{2i} are constant. However, if each producer in the sector expands its own capital, then *K*_{2} will rise and produce a spillover benefit that increases the productivity of all firms, which is the increasing returns effect. Each firm’s production is nonetheless homogeneous of degree one with respect to its own capital *K*_{2i} and labor *L*_{2i}, and if *K*_{2i} and *K*_{2} expand together by the same amount while *L*_{2i} is fixed, production also displays constant returns to scale.

For each firm, the total capital stock *K*_{2} of the modern sector is exogenously determined. In addition, assume that the output of each firm is a homogenous product with a given price *p*_{2}. If all firms make the same choices so that *k*_{2i} = *k*_{2} and *K*_{2} = *k*_{2}*L*_{2}, then profit-maximizing by each firm yields

$$ p_{2} h_{K} \left( {k_{2} ,K_{2} } \right) = p_{2} \left[ {\tilde{h}\left( {L_{2} } \right) - L_{2} \tilde{h}^{{\prime }} \left( {L_{2} } \right)} \right] = r,\quad q_{2} = \frac{{Q_{2} }}{{L_{2} }} = h\left( {k_{2} ,K_{2} } \right) $$

(7)

$$ h\left( {k_{2} ,K_{2} } \right) - k_{2} h_{K} \left( {k_{2} ,K_{2} } \right) = K_{2} \tilde{h}^{{\prime }} \left( {L_{2} } \right) = \frac{w}{{p_{2} }} $$

(8)

where use is made of the following expressions for the average product of capital \( \tilde{h}\left( {L_{2} } \right) \) and private marginal product of capital \( h_{K} \left( {k_{2} ,K_{2} } \right) \), respectively.

$$ \frac{{h\left( {k_{2} ,K_{2} } \right)}}{{k_{2} }} = \tilde{h}\left( {\frac{{K_{2} }}{{k_{2} }}} \right) = \tilde{h}\left( {L_{2} } \right),h_{K} \left( {k_{2} ,K_{2} } \right) = \tilde{h}\left( {L_{2} } \right) - L_{2} \tilde{h}^{{\prime }} \left( {L_{2} } \right),\tilde{h}^{{\prime }} \left( {L_{2} } \right) > 0,\tilde{h}^{{\prime \prime }} \left( {L_{2} } \right) < 0. $$

(9)

Condition (7) indicates that the value marginal productivity of capital for a modern sector firm equals the interest rate, *r*. Condition (8) indicates that the value marginal productivity of labor employed by a modern sector firm equals the real wage rate.

Both the private marginal product of capital and average product of capital are invariant with respect to the capital–labor ratio because learning-by-doing and spillovers eliminate diminishing returns to capital. As (9) indicates, the private marginal product of capital is less than the average marginal product of capital. The private marginal product of capital is increasing in *L*_{2}, given \( \tilde{h}^{{\prime \prime }} \left( {L_{2} } \right) < 0 \). These results (7)–(9) for production and input use involving learning-by-doing and knowledge spillovers are standard for these types of relationships (Barro and Sala-I-Martin 2004).

### Primary production trade, growth dynamics and labor market equilibrium

Because condition (5) indicates that the fixed land–labor ratio on less favored land *n*^{m} determines the nominal wage rate *w*, the rural economy is recursive with respect to resource use, labor and output in the primary production sector. If the elasticity of resource conversion \( \varepsilon \left( {N_{1} } \right) \) is constant, *p*_{1} given and *w* known, then (3) yields the resource–land ratio *n*_{1} for primary production. With *n*_{1} determined, the relationship \( \varepsilon = {1 \mathord{\left/ {\vphantom {1 {z^{{\prime }} n_{1} }}} \right. \kern-0pt} {z^{{\prime }} n_{1} }} \) can be solved for resource conversion and use *N*_{1}. Employment *L*_{1}, and from (1), primary production *Q*_{1} can then be found.

Primary products are exported, and *p*_{1} is the given world price for these commodities. These products are exchanged for imports *M*, which are substitutes for consumption of domestic output from the modern sector. The balance of trade is

$$ pQ_{1} = M,\quad p = \frac{{p_{1} }}{{p_{2} }}, $$

(10)

where *p* is the terms of trade, expressed in terms of modern sector commodities as the numeraire. Note that, because *p* is given and *Q*_{1} known, imports to the small economy are recursively determined.

If there is no population growth, the representative household seeks to maximize its discounted flow of welfare over time as given by \( U = \int_{0}^{\infty } {\left[ {\frac{{\left( {c^{1 - \theta } + m^{1 - \theta } } \right) - 1}}{1 - \theta }} \right]e^{ - \rho t} {\text{d}}t} \) subject to the budget constraint \( \dot{a} = {ra} + w - c \), where *m* is per capita imports, *a* is the household’s assets per person, *r* is the interest rate, *w* the wage rate, ρ is the rate of time preference, and θ is the intertemporal elasticity of substitution.

However, as imports are determined by the primary products balance of trade condition (6), the household is free to choose only its per capita consumption. As shown in the “Appendix”, the growth dynamics of the modern sector and thus the economy are governed by

$$ \dot{k}_{2} = p_{2} \tilde{h}\left( {L_{2} } \right)k_{2} - c,\quad k_{2} \left( 0 \right) = k_{20} $$

(11)

$$ c\left( t \right) = \varphi k_{2} \left( t \right),\quad \varphi = p_{2} \tilde{h}\left( {L_{2} } \right) - \gamma $$

(12)

$$ \frac{{\dot{q}_{2} }}{{q_{2} }} = \frac{{\dot{k}_{2} }}{{k_{2} }} = \frac{{\dot{c}}}{c} = \gamma ,\quad \gamma = \frac{1}{\theta }\left( {p_{2} \left[ {\tilde{h}\left( {L_{2} } \right) - L_{2} \tilde{h}^{{\prime }} \left( {L_{2} } \right)} \right] - \rho } \right) $$

(13)

Equation (11) is the usual condition for capital accumulation in an economy. If output per capita, valued at the price *p*_{2}, exceeds consumption, and will increase capital per person. Condition (12) indicates that per capita consumption is proportional to capital per person. Consequently, as (13) depicts, capital and output per worker in the modern sector grow at the same (constant) rate as consumption per capita. The per capita growth rate, *γ*, is determined by the total number of workers employed in the sector, *L*_{2}. An expansion (contraction) in the aggregate modern sector labor force, *L*_{2}, therefore, increases (decreases) per capita growth in this sector.

With the nominal wage determined by the fixed land–labor ratio on less favored agricultural land, the value marginal productivity condition (8) for the modern sector must equal *w*. However, suppose that initially capital in the sector is some given level *K*_{20}. Equilibrium employment must, therefore, be the unique solution to \( \tilde{h}^{{\prime }} \left( {L_{2} } \right) = {w \mathord{\left/ {\vphantom {w {p_{2} }}} \right. \kern-0pt} {p_{2} }}K_{20} \). It is possible that this level of employment is large enough so that growth of the modern sector is positive, i.e., \( \gamma > 0 \). But this requires a relatively large initial stock of aggregate capital for the modern sector, as the equilibrium employment condition implies that more *L*_{2} requires a higher *K*_{20}. For most LAC economies, the initial stock of aggregate capital in the modern sector is likely to be relatively small rather than large. Thus, it follows that employment *L*_{2} will also be small, and if this is the case, it is more likely that (13) will yield \( \gamma \le 0 \). If it turns out that \( \gamma < 0 \), then the capital–labor ratio and aggregate capital will decline, employment will fall and the modern sector will contract.

It is also possible that the modern sector neither contracts nor declines. For example, with *w* predetermined and for a given *K*_{20}, the equilibrium *L*_{2} that satisfies (8) is just sufficient to ensure \( \gamma = 0 \) in (13). This outcome ensures constant employment and aggregate capital in the modern sector, and thus an equilibrium output level *Q*_{2}. Such a steady-state result is depicted in Fig. 3.

The total labor force in the developing economy is \( L = L_{1} + L_{2} + L^{m} \). With employment in primary production and the modern sector known, the residual labor on marginal land *L*^{m} can be found. As *n*^{m} is already known, the total marginal land used in traditional agriculture *N*^{m} is determined. Thus, the full labor market equilibrium corresponds to

$$ p_{1} \left[ {f\left( {n_{1} } \right) - f_{N} \left( {n_{1} } \right)n_{1} \left( {1 - \varepsilon } \right)} \right] = p_{2} K_{2} \tilde{h}^{{\prime }} \left( {L_{2} } \right) = p^{m} g\left( {n^{m} } \right) = w. $$

(14)

As described previously, the average productivity of labor on marginal land determines the equilibrium wage rate in the economy, and employment in both the primary production and modern sectors equates their respective marginal productivities with *w*. In addition, the amount of labor employed in the modern sector *L*_{2} must correspond to \( \gamma = 0 \). The solid lines in Fig. 3 depict the labor market equilibrium for the economy and the corresponding zero growth rate for the modern sector.

The equilibrium outcome indicated in Fig. 3 is not very optimistic. Although some labor may be employed in the modern sector, a constant capital stock eliminates any productivity gains from spillover and learning-by-doing in the sector. As a consequence, the modern sector competes with the commercial primary production sector for available labor, with marginal land absorbing the residual. Without the dynamic productivity effects of positive growth, the modern sector does not generate a self-reinforcing labor absorption process that leads workers to shift from the rural economy to this sector. The economy remains fundamentally dualistic; commercial primary production and a static modern sector are the two principal sectors, with less favored agricultural land absorbing the remaining rural households. This latter process is a key structural feature of the land surplus rural economy. The concentration of the rural populations on marginal land is essentially a barometer of economy-wide development. As long as there is abundant less favored land for cultivation, it absorbs rural migrants, population increases and displaced unskilled labor from elsewhere in the economy.

On the other hand, the rural populations on less favored agricultural land can also be thought of as a large pool of unskilled surplus labor that, under the right conditions, could potentially be absorbed by the commercial primary production and modern sectors. These conditions are explored in turn.