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Chambers and Ting Guo (2009) in their study developed a one-sector endogenous growth model in which renewable natural resources are both a factor of production and measure of environmental quality. Their main objective is to explore the interrelations between the output growth rate and environmental quality along the economy's balanced growth path. The result of their study shown that at a balanced growth, a sustainable economic growth and a non-deteriorating environment are shown to coexist. It was also proven that the steady-state economic growth and natural-resource utilization are positively related.
In their model, the environment is functioning in two roles as:
A provider of factors of production.
A stock of renewable natural resources that accumulates over time to preserve environmental quality as GDP continues to grow.
Chambers and Ting Guo in building their model focusing on the feasibility of a balanced-growth equilibrium with non-deteriorating environmental quality in a one-sector endogenous growth model with renewable natural resources. In their theoretical model, households live forever, provide fixed labor supply and derive utility from consumption goods. On the production side, a continuum of identical, competitive firms produce output using natural resources, which are assumed to regenerate themselves at a constant rate over time, as a factor of production. The economy's aggregate production function displays increasing returns-to-scale because of the presence of productive externalities generated by capital inputs. It was that along the balanced growth path (BGP), output, consumption, and physical capital all grow at a common positive rate, whereas the stock of total natural resources and the level of natural resources allocated to the firms' production process maintain their respective steady-state values.
The Theoretical Model
There is a continuum of identical, competitive firms in the economy, with the total number normalized to one. Each firm produces output Yt using a constant returns-to-scale Cobb-Douglas production function
Yt = Ktα Ht1−αXt, 0 <α< 1, (1)
where Kt and Ht are physical capital and harvested/utilized natural resources (or natural capital), respectively, and Xt represents productive externalities that are taken as given by individual firms. In addition, Xt is postulated to take the form
Xt = AKt1- α, A >0 (2)
where Kt denotes the economy-wide average level of the capital stock. In a symmetric equilibrium, all firms take the same actions such that Kt = Kt. Hence, (2) can be substituted into (1) to obtain the following social production function that displays increasing returns-to-scale:
Yt = AKtHt1−αXt (3)
Under the assumption that factor markets are perfectly competitive, the first-order conditions for the firm's profit maximization problem are given by
rt = α(Yt/Kt) (4)
rt = (1−α)(Yt/Kt) (5)
where rt is the capital rental rate and pt is the real price paid to utilized natural resources.
The economy is populated by a unit measure of identical infinitely-lived households; each has perfect foresight and maximizes a discounted stream of utilities over its lifetime
where Ct is the individual household's consumption, is the subjective discount rate, and is the inverse of the intertemporal elasticity of substitution in consumption.
The budget constraint faced by the representative household is
where is the capital depreciation rate. As is commonly specified in the environmental macroeconomics literature, the economy's ecological process or the law of motion for total renewable resources (as a proxy for environmental quality) Nt is given by
where f(Nt) is the regeneration function that is often assumed to be strictly increasing in Nt. Without loss of any generality, we postulate that the rate of natural regeneration is independent of the environmental state, specifically f(Nt) = . Ht represents not only the extraction of natural resources, but also the disposal of wastes (i.e. pollution) because both activities reduce the environment's absorption capacity represented by f(Nt)Nt.
The first-order conditions for the representative household's dynamic optimization problem are
where Kt and Nt are shadow prices (or utility values) of capital stock and natural resources, respectively. Equation (9) states that the marginal benefit of consumption equals its marginal cost, which is the marginal utility of having an additional unit of physical capital. In addition, (10) and (11) are standard Euler equations that govern the evolution of Kt and Nt over time. Equation (12) shows that the firm utilizes natural resources to the point where the marginal value of more output is equal to the marginal cost of resource depletion. Finally, (13) and (14) are the transversality conditions (TVC).
Balanced Growth Path
In light of the household's CRRA utility formulation (6), together with the linearity of physical capital in the aggregate technology (3), the economy exhibits sustained endogenous growth whereby output, consumption, and physical capital all display a common, positive constant growth rate denoted by g. Moreover, the regeneration/depletion equation (8) implies that in the long run (or in an ecological equilibrium defined as Nt = 0), total and utilized natural resources will reach their respective steady-state levels, N* and H*. This in turn imposes a sustainable long-run environmental quality constraint under exhaustible natural resources, where a constant level of pollution exactly matches the environment's absorption capacity.
To derive a balanced growth path (BGP), we first make the variable transformation Xt = Ct/Kt, and re-express the model's equilibrium conditions as the following autonomous differential equations:
Given the above dynamical system (15)-(17), the balanced-growth equilibrium is characterized by a triplet of positive real numbers that satisfy the condition . It is straightforward to show that our model economy exhibits a unique balanced growth path along which the utilized natural resource maintains its steady-state level
which in turn leads to the expressions for X* and N* as follows:
With (18) and (19), it follows that the common (positive) rate of economic growth g is given by
As a result, the BGP's growth rate ceteris paribus is positively related to the steady-state level of utilized natural resources. That is, a higher (lower) usage of services from the environment in production will raise (reduce) the economy's rate of growth in output, consumption, and physical capital. Moreover, the quantity of utilized natural resources per unit of GDP steadily declines along the economy's balanced growth path.
According to their model, it was found that in the long run, the economy's output growth rate is positively related to the steady-state level of utilized natural resources. In addition, Chambers and Ting Guo have conducted an empirical study to determine the validity of their theoretical model. This led to a panel cross-country growth regression, which includes a broad measure of productive natural resources, which finally provides strong empirical support for this theoretical prediction. The empirical analysis have also shown an estimation results which suggest that the conservation costs are small, and growth strategies based on greater physical capital formation and trade openness outperform those relying on more intensive utilization of the environment.