The Dutch disease revisited: absorption constraint and learning by doing

Abstract

This paper revisits the Dutch disease by analyzing the general equilibrium effects of a resource shock on a dependent economy, both in a static and dynamic setting. The novel aspect of this study is to incorporate in one coherent framework two distinct features of the Dutch disease literature that have previously been analyzed in isolation: capital accumulation with absorption constraint, and productivity growth induced by learning-by-doing. The result of long run exchange rate appreciation is maintained in line with part of the Dutch Disease literature. In addition, a permanent change in the employment shares occurs after the resource windfall, in favor of the non-traded sector and away from the traded sector growth engine of the economy. In other words, in the long run both of the classic symptoms of the Dutch Disease remain in place.

Appendix: A

1.1 A1

The representative household endowed with Cobb-Douglas utility function maximizes the static utility $u(C_{N},C_{T})=C_{T}^{\delta }C_{N}^{1-\delta }$ subject to the static version of the aggregate income constraint given in (7):

$$\begin{array}{@{}rcl@{}} &\max\limits_{\left( C_{N},C_{T}\right) }C_{T}^{\delta }C_{N}^{1-\delta }, \\ s.t.\qquad& PC_{N}+C_{T}=AR+X-PI_{N}. \end{array} $$

Setting the Lagrangian Γ and computing the first order conditions, the solution to this static problem is:

$$\begin{array}{@{}rcl@{}} &&{\Gamma} =\delta \log C_{T}+\left( 1-\delta \right) \log C_{N}-\lambda (PC_{N}+C_{T}), \\ &&\lbrack C_{T}]\qquad \frac{\delta }{C_{T}}=\lambda \quad \qquad \lbrack C_{N}]\qquad \frac{1-\delta }{C_{N}}=P\lambda ,\\ &&C_{N}=\left( \frac{1-\delta }{P}\right) C,\qquad C_{T}=\delta C. \end{array} $$

1.2 A2

Let us give a closer look at the overall sign of the derivatives in (20, 23):

$$\begin{array}{@{}rcl@{}} \frac{\partial \eta }{\partial A} &=&\frac{\frac{(1-\alpha )\eta }{A}\left[ \varphi^{\prime }\alpha -\varphi \left( \cdot \right) \left( \frac{K_{N}}{ A\eta }\right)^{1-\alpha }\right] }{1+\frac{\left( 1-\alpha \right) \left( 1-\delta \right) }{\delta }-\alpha \left[ \varphi \left( \cdot \right) \left( \frac{K_{N}}{A\eta }\right)^{1-\alpha }+\varphi^{\prime }(1-\alpha ) \right] }, \\ \frac{\partial \eta }{\partial K_{N}} &=&\frac{\frac{(1-\alpha )\eta }{K_{N}} \left[ \varphi \left( \cdot \right) \left( \frac{K_{N}}{A\eta }\right)^{1-\alpha }-\varphi^{\prime }\alpha \right] }{1+\frac{\left( 1-\alpha \right) \left( 1-\delta \right) }{\delta }-\alpha \left[ \varphi \left( \cdot \right) \left( \frac{K_{N}}{A\eta }\right)^{1-\alpha }+\varphi^{\prime }(1-\alpha )\right] }. \end{array} $$

The common denominator of both derivatives is always positive since:

$$\begin{array}{@{}rcl@{}} &&\varphi \left( \cdot \right) \left( \frac{K_{N}}{A\eta }\right)^{1-\alpha }+\varphi^{\prime }(1-\alpha )<0, \\ &&\varphi^{\prime }>-\frac{\varphi \left( \cdot \right) }{(1-\alpha )}\left( \frac{K_{N}}{A\eta }\right)^{1-\alpha }, \end{array} $$

which is always true since by definition φ ^′ > 0. As regards the numerator, we observe that as long as the following condition holds:

$$ \varphi^{\prime }>\frac{\varphi (\cdot )}{\alpha }\left( \frac{K_{N}}{A\eta }\right)^{1-\alpha }, $$

(42)

we can determine the overall signs of both derivatives and conclude that $\frac {\partial \eta }{\partial A}>0$ and $\frac {\partial \eta }{\partial K_{N}}<0$. Notice that this condition is not inconsistent with the condition given in Eq. 16.

1.3 A3

Dynamic stability analysis. At first, by totally differentiating (33, 34) and exploiting the convenient result that (∂ g _t /∂ R _t ) = 0, the dynamic system can be rewritten as:

$$\begin{array}{@{}rcl@{}} -\frac{\partial \eta_{t}}{\partial A_{t}}-\frac{\partial \eta_{t}}{ \partial K_{Nt}} &=&0, \\ \left[ \eta_{t}+A_{t}\frac{\partial \eta_{t}}{\partial A_{t}}\right] + \left[ A_{t}\frac{\partial \eta_{t}}{\partial K_{Nt}}-\frac{A_{t}\eta_{t}}{ K_{Nt}}\right] &=&0. \end{array} $$

Let us then insert these derivatives into the Jacobian J and evaluate it at the dynamic steady-state (26) (for any given g _t ):

$$J= \left|\begin{array}{cc} -\frac{\partial \eta_{t}}{\partial A_{t}} & -\frac{\partial \eta_{t}}{ \partial K_{Nt}} \\ \eta_{t}+A_{t}\frac{\partial \eta_{t}}{\partial A_{t}} & A_{t}\frac{ \partial \eta_{t}}{\partial K_{Nt}}-\frac{A_{t}\eta_{t}}{K_{Nt}} \end{array}\right| $$

By recalling from the static model and from the Section A2 of the Appendix that $\frac {\partial \eta }{\partial A}>0$ and $\frac {\partial \eta }{ \partial K_{N}}<0$ we can immediately evaluate that:

$$tr(J)=-\frac{\partial \eta_{t}}{\partial A_{t}}+\left[ A_{t}\frac{\partial \eta_{t}}{\partial K_{Nt}}-\frac{A_{t}\eta_{t}}{K_{Nt}}\right] <0. $$

The trace is unambiguously negative. The determinant is instead given by:

$$\begin{array}{@{}rcl@{}} \det (J)&=&-\frac{\partial \eta_{t}}{\partial A_{t}}\left[ A_{t}\frac{ \partial \eta_{t}}{\partial K_{Nt}}-\frac{A_{t}\eta_{t}}{K_{Nt}}\right] + \frac{\partial \eta_{t}}{\partial K_{N}}\left[ \eta_{t}+A_{t}\frac{ \partial \eta_{t}}{\partial A_{t}}\right] , \\ \det (J)&=&\eta_{t}\left[ \frac{A_{t}}{K_{Nt}}\frac{\partial \eta_{t}}{ \partial A_{t}}+\frac{\partial \eta_{t}}{\partial K_{N}}\right] . \end{array} $$

Let us now substitute for the analytical expression of the two derivatives (20, 23). Redefine for convenience the positive common denominator as ${\Psi } =1+\frac {\left (1-\alpha \right ) \left (1-\delta \right ) }{\delta }-\alpha \left [ \varphi \left (\cdot \right ) \left (\frac {K_{N}}{A\eta }\right )^{1-\alpha }+\varphi ^{\prime }(\cdot )(1-\alpha )\right ]$ and rewrite det(J) as:

$$\begin{array}{@{}rcl@{}} \det (J) &=&\frac{\eta_{t}}{\Psi }\left\{ \begin{array}{c} \frac{A_{t}}{K_{Nt}}\frac{(1-\alpha )\eta_{t}}{A_{t}}\left[ \varphi^{\prime }(\cdot )\alpha -\varphi \left( \cdot \right) \left( \frac{K_{Nt}}{ A_{t}\eta_{t}}\right)^{1-\alpha }\right] + \\ \frac{(1-\alpha )\eta_{t}}{K_{Nt}}\left[ \varphi \left( \cdot \right) \left( \frac{K_{Nt}}{A_{t}\eta_{t}}\right)^{1-\alpha }-\varphi^{\prime }(\cdot )\alpha \right] \end{array} \right\} , \\ \det (J) &=&\frac{(1-\alpha ){\eta_{t}^{2}}}{\Psi K_{Nt}}\left\{ \left[ \varphi^{\prime }(\cdot )\alpha -\varphi \left( \cdot \right) \left( \frac{ K_{Nt}}{A_{t}\eta_{t}}\right)^{1-\alpha }\right] +\left[ \varphi \left( \cdot \right) \left( \frac{K_{Nt}}{A_{t}\eta_{t}}\right)^{1-\alpha }-\varphi^{\prime }(\cdot )\alpha \right] \right\} =0. \end{array} $$

1.4 A4

Let us analyze closely the change in the ratio of the productivity level with respect to the non-traded capital stock level Φ _t , after the resource shock. As it can be seen from Fig. 4 in the paper and the Fig. 6 below, the information at our disposal is that the new dynamic equilibrium F will in any case lay in the area down to the right of the two initial isoclines. The border of this area is marked by thicker isoclines:

Fig. 6

Productivity and capital stock in the new dynamic equilibrium

Let us redefine for convenience, in the most general case, the two isoclines as such (with A > 0,K _N > 0):

$$\begin{array}{@{}rcl@{}} A &=&\alpha K_{N}+m,\qquad \alpha >0, \\ A &=&\beta K_{N}+q,\qquad \beta >0, \\ \alpha &>&\beta ,\quad q>m. \end{array} $$

This allows me to start computing the ratio at the initial dynamic equilibrium in E:

$${\Phi}_{E}=\frac{\alpha K_{N}+m}{K_{N}}=\frac{\beta K_{N}+q}{K_{N}}. $$

In order to cover all the possible outcomes for the new ratio between productivity and the capital level, let us consider the two following “corner solutions”, in which all the possible new equilibriums lay to the right of (but infinitely close to) the thicker parts of the isoclines:

$$\begin{array}{@{}rcl@{}} &&{\Phi}_{F}^{^{\prime }}(A^{\prime },K_{N}^{\prime })=\left( A^{\prime }=\alpha K_{N}+m;\quad K_{N}^{\prime }=K_{N}+\varepsilon \right) , \\ &&{\Phi}_{F}^{^{\prime \prime }}(A^{^{\prime \prime }},K_{N}^{^{\prime \prime }})=\left( A^{^{\prime \prime }}=\beta K_{N}+q-\varepsilon ;\quad K_{N}^{^{\prime \prime }}=K_{N}\right) , \end{array} $$

where ε > 0 is infinitely small. It is easy to show that, for both of these cases:

$$\begin{array}{@{}rcl@{}} {\Phi}_{F}^{^{\prime }}&=&\frac{\alpha K_{N}+m}{K_{N}+\varepsilon }=\frac{\beta K_{N}+q}{K_{N}\left( 1+\frac{\varepsilon }{K_{N}}\right) }={\Phi}_{E}\left( \frac{1}{1+\frac{\varepsilon }{K_{N}}}\right) <{\Phi}_{E}, \\ {\Phi}_{F}^{^{\prime \prime }}&=&\frac{\beta K_{N}+q-\varepsilon }{K_{N}}=\frac{ \left( \beta K_{N}+q\right) \left( 1-\frac{\varepsilon }{\beta K_{N}+q} \right) }{K_{N}}={\Phi}_{E}\left( 1-\frac{\varepsilon }{\beta K_{N}+q}\right) <{\Phi}_{E}, \end{array} $$

which completes the proof. The ratio between productivity and capital stock in the new dynamic equilibrium has in any case decreased after the resource shock, with respect to the initial dynamic equilibrium [Φ _F <Φ _E ].