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International trade and net investment: theory and evidence


The theory of welfare accounting shows that comprehensive measures of net investment can be used to test whether an economy is following unsustainable paths of consumption. However, the notion of net investment used in most applied studies rules out technological progress and terms-of-trade gains from international trade. This paper considers an augmented expression of net investment derived from a dynamic growth model featuring international trade in different types of resource inputs, exogenous productivity growth in final sectors, and cost-reducing progress in resource extraction. Calculating augmented net investment for the world’s top twenty oil producers, we show that the difference with standard non-augmented measures can be large and may even revert some established conclusions regarding sustainability: prospects are more favorable than previously thought in oil-exporting countries endowed with large reserves like Angola, Azerbaijan, Kuwait, Saudi Arabia and Venezuela. In oil-importing economies, future consumption possibilities are limited by the lack of expected rental incomes from future resource exports.

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  1. Apart from Weitzman’s (1997) calculation of the technological time premium for the United States, and the analysis of ‘natural capital gains’ for Indonesia in Vincent et al. (1997), we do not know of any published work conducting a systematic analysis of augmented measures of net investment in real-world economies.

  2. See Pezzey (1992) for an extensive discussion of sustainability concepts. The notion of sustainability that we employ in this paper corresponds to that of “sustained development” in Pezzey (1992). An alternative definition of sustainable path is that of a development path along which the economy’s level of consumption never exceeds the maximum constant level that could be sustained forever given the available technology, endowments and resource constraints. As noted below, the sustainability properties of the NI indicator remain valid under this alternative definition.

  3. Result (4) underlies most of the results of the theory of welfare accounting but is somewhat neglected in this literature—if not hidden between the lines of several theorems’ proofs—because the vast majority of contributions focus (with the notable exception of Pezzey 2004) on the welfare significance of Net National Product rather than on the predictive power of Net Investment. The proof of Proposition 1 is based on Asheim and Weitzman (2001) but the general result can be attributed to Weitzman (1976) and Dixit et al. (1980).

  4. Proposition 2 is a variant of Pezzey (2004: Proposition 1). Pezzey’s (2004) definition of sustainability is slightly different: a sustainable path is one along which consumption never exceeds the maximum sustainable level. However, the basic property of the NI indicator is unchanged: positive current net investment is necessary but not sufficient for sustainability.

  5. Formally, if we correctly estimate the right hand side of Eq. 2 and we observe \({\rm NI}\left( t\right) <0\), the right hand side of Eq. 4 has to be strictly negative so that there must be an interval of time in the future during which \(\dot{c}\left( v\right) <0\).

  6. The possibility of observing positive net investment in unsustainable economies was first noted by Asheim (1994) and Vellinga and Withagen (1996). Building on these results, Valente (2008) shows that model-specific estimations of the rates of resource regeneration and augmentation may provide an additional criterion for testing sustainability in economies where current genuine savings appear to be positive.

  7. In resource economics, CNNP is called “Green National Product” because it equals Net National Product minus the value of the depletion of the stocks of natural resources and environmental amenities. In this section, we use the term CNNP as it is more generally referred to the frontier of future consumption possibilities.

  8. The re-definition of net investments in the case of population growth is studied in Arrow et al. (2003) and Asheim (2004).

  9. The assumption of perfect foresight is obviously implicit in the optimal paths studied here—defined as paths chosen at time t = 0 by economies that maximize present-value welfare (3).

  10. By well-behaved production we mean that \(\mathcal{F} \left( k,g,m_{h}\right) \) is, with respect to each argument, twice continuously differentiable, strictly increasing, strictly concave, and satisfying the Inada conditions. We also assume that all inputs are essential, i.e., \(\mathcal{F} \left( k,g,m_{h}\right) =0\) if at least one argument is zero. All our results hold for \(\mathcal{F} \left( k,g,m_{h}\right) \) displaying non-increasing returns to scale.

  11. Formally, the weighted present value of future consumption levels is now given by current consumption plus augmented net investment: \(c\left( t\right) + {\rm ANI}\left( t\right) =\int_{t}^{\infty }r\left( v\right) \cdot c\left( v\right) \cdot e^{-\int_{t}^{v}r\left( v^{\prime }\right) dv^{\prime }}dv\).

  12. Clearly, we are implicitly assuming that current prices reflect to a good extent the supporting prices of the optimal path—i.e., the prices that would hold in a welfare-maximizing economy. This assumption is necessary and is in fact made by virtually all studies that calculate net investment on the basis of real data.

  13. With respect to technical progress in extraction, if the projected parameter γ w is strictly negative—that is, technical progress in the oil sector is actually cost-reducing—the assumption of constant exponential decline in costs is not as optimistic as it may appear at first sight. On the one hand, marginal extraction costs would approach zero in the long run. On the other hand, this would not solve the problem of resource scarcity because sustainability in consumption is far from being guaranteed even when extraction costs are zero at each point in time: as explained in detail by Dasgupta and Heal (1974) and Schulze (1974), the sustainability problem does not arise from extraction costs but from the dynamic productivity loss implied by the use of non-renewable inputs.

  14. If we take the average growth rates of oil imports observed in the past (22 and 8%, respectively) as projected future values for China and India, we have \(r-\gamma _{p_{g}}^{i}-\gamma _{g}^{i}<0\) and thereby an infinite value of the integral representing the present-value loss from terms of trade (that is, insolvency in the long run): see the derivation of equation (17) in Appendix.


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Correspondence to Simone Valente.



Proof of Proposition 1

As shown by Asheim and Weitzman (2001: p. 237, Eq. 9), along the optimal path we have

$$ \frac{d}{dt}{\rm NI}\left( t\right) =r\left( t\right) {\rm NI}\left( t\right) -\dot{c} \left( t\right) $$

in each instant t. Integrating this expression forward and imposing the transversality condition

$$ \lim\limits_{T\rightarrow \infty } {\rm NI}\left( T\right) \cdot e^{-\int_{t}^{T}r\left( v\right) dv}=0, $$

which must hold along the optimal path, we obtain Eq. 4. For further details, see Asheim and Weitzman (2001). The same proof can be equivalently obtained as a special case of Proposition 3 below by excluding trade and technological progress from the model of Section 3.□

Proof of Proposition 2

See the main text. □

Derivation of Eq.  8

Substituting result (4) in definition (7), and integrating by parts, we have

$$\begin{array}{rll}{\rm CNNP}\left( t\right) &=&c\left( t\right) +\int_{t}^{\infty }\dot{c}\left( v\right) \cdot e^{-\int_{t}^{v}r\left( v^{\prime }\right) dv^{\prime }}dv= \\ &=&c\left( t\right) \!+\!\left[\!\lim\limits_{v\rightarrow \infty }c\left( v\right) \cdot e^{-\int_{t}^{v}r\left( v^{\prime }\right) dv^{\prime }}\!\right] -c\left( t\right) \!+\!\int_{t}^{\infty }\!\!r\left( v\right) c\left( v\right) \cdot e^{-\int_{t}^{v}r\left( v^{\prime }\right) dv^{\prime }}dv, \end{array} $$

where the limit in square brackets is zero by the transversality condition that must be satisfied along optimal paths. Hence, the above expression reduces to Eq. 8.

Proof of Proposition 3

Economy Home maximizes welfare (3) subject to Eqs. 1114. The current-value Hamiltonian associated to this problem is

$$\begin{array}{rll} \mathcal{L} &\equiv &u\left( x_{h}+z_{h}\right) +\lambda _{k}\left[ a\cdot \mathcal{F} \left( k,g,m_{h}\right) -x_{h}-x_{f}-w\cdot \left( m_{h}+m_{f}\right) \right] + \\ &&+\lambda _{b}\left[ rb+x_{f}+p_{m}m_{f}-z_{h}-p_{g}g\right] -\lambda _{s} \left[ m_{h}+m_{f}\right] , \end{array} $$

where \(\left\{ \lambda _{k},\lambda _{b},\lambda _{s}\right\} \) are the dynamic multipliers associated to the state variables \(\left\{ k,b,s\right\} \). Maximizing \(\mathcal{L} \) with respect to the control variables \(\left\{ x_{h},x_{f},z_{h},m_{h},m_{f},g\right\} \) we obtain

$$ \lambda _{k} = \lambda _{b}=u^{\prime }\left( x_{h}+z_{h}\right) , $$
$$ \lambda _{s}=\lambda _{k}\cdot \left( a\mathcal{F} _{m_{h}}-w\right) , $$
$$ \lambda _{s} = \lambda _{b}p_{m}-\lambda _{k}w=\lambda _{k}\cdot \left( p_{m}-w\right) , $$
$$ \lambda _{b}p_{g}=\lambda _{k}a\mathcal{F} _{g}. $$

The co-state equations for \(\left\{ k,b,s\right\} \) read

$$ \rho \lambda _{k}-\dot{\lambda}_{k} = \lambda _{k}a\mathcal{F} _{k}, $$
$$ \rho \lambda _{b}-\dot{\lambda}_{b} = \lambda _{b}r, $$
$$ \rho \lambda _{s}-\dot{\lambda}_{s} = 0, $$

and the transversality conditions require

$$ \lim\limits_{t\rightarrow \infty }\lambda _{k}\left( t\right) k\left( t\right) e^{-\rho t}=\lim\limits_{t\rightarrow \infty }\lambda _{b}\left( t\right) b\left( t\right) e^{-\rho t}=\lim\limits_{t\rightarrow \infty }\lambda _{s}\left( t\right) s\left( t\right) e^{-\rho t}=0. $$

Notice that Eqs. 2123 and 2526 imply the following no-arbitrage conditions:

$$ a+ _{k} =r,\quad a\mathcal{F} _{m_{h}}=p_{m},\quad a\mathcal{F} _{g}=p_{g}, $$
$$ \dot{p}_{m}-\dot{w} =\left( p_{m}-w\right) \cdot r, $$

where Eq. 29 establishes the equality between prices and marginal productivities of the inputs in final production, and Eq. 30 is Hotelling’s rule. Also, combining constraints (12)–(14), we obtain

$$ \dot{k}+\dot{b}=a\mathcal{F} \left( k,g,m_{h}\right) -c-wm+rb+p_{m}m_{f}-p_{g}g. $$

Substituting Eq. 31 in Eq. 15, augmented net investment equal

$$ {\rm ANI}=a\mathcal{F} \left( k,g,m_{h}\right) -c-wm+rb+p_{m}m_{f}-p_{g}g+\left( p_{m}-w\right) \cdot \dot{s}+Q. $$

Time-differentiating Eq. 32 we have

$$\begin{array}{rll} {\rm A}\dot{\rm N}{\rm I} &=&\dot{a}\mathcal{F} \left( k,g,m_{h}\right) +a\mathcal{F} _{k} \dot{k}+a\mathcal{F} _{g}\dot{g}+a\mathcal{F} _{m_{h}}\dot{m}_{h}-\dot{c}+ \\ &&-\dot{w}m-w\dot{m}+\dot{r}b+r\dot{b}+\dot{p}_{m}m_{f}+p_{m}\dot{m}_{f}- \dot{p}_{g}g-p_{g}\dot{g}+ \\ &&+\left( \dot{p}_{m}-\dot{w}\right) \cdot \dot{s}+\left( p_{m}-w\right) \cdot \ddot{s}+\dot{Q}. \end{array} $$

Substituting Eq. 29 to eliminate marginal productivities, the Hotelling rule (30) to eliminate \(\left( \dot{p}_{m}-\dot{w} \right) \), and using Eq. 12 to substitute \(\ddot{s}=-\left( \dot{m} _{h}+\dot{m}_{f}\right) \), we obtain

$$\begin{array}{rll} {\rm A}\dot{\rm N}{\rm I} &=&\left[ r\dot{k}+r\dot{b}+r\left( p_{m}-w\right) \cdot \dot{s} \right] -\dot{c}+\dot{Q}+\dot{a}\mathcal{F} \left( k,g,m_{h}\right)\\ &&-\dot{w} m+\dot{r}b+\dot{p}_{m}m_{f}-\dot{p}_{g}g, \\ {\rm A}\dot{\rm N}{\rm I} &=&\left[ r\dot{k}+r\dot{b}+r\left( p_{m}-w\right) \cdot \dot{s} \right] -\dot{c}+\dot{Q}+q, \end{array} $$

where we have used result (16) to obtain the last expression. By definition (15), the term in square brackets equals rANI − rQ, implying

$$ {\rm A}\dot{\rm N}{\rm I}=r{\rm ANI}-\dot{c}+\dot{Q}-rQ+q. $$

By definition (9), the total time-derivative of the time premium is \(\dot{Q}=rQ-q\). As a consequence, the above expression reduces to

$$ {\rm A}\dot{\rm N}{\rm I}=r{\rm ANI}-\dot{c}. $$

Integrating Eq. 31 between over the interval \(\left( t,T\right) \), we have

$$ {\rm ANI}\left( t\right) =\int_{t}^{T}\dot{c}\left( v\right) e^{-\int_{t}^{v}r\left( v^{\prime }\right) dv^{\prime }}dv+{\rm ANI}\left( T\right) \cdot e^{-\int_{t}^{T}r\left( v\right) dv}. $$

Notice that, using definition (15) and the Hotelling rule (30), the last term in Eq. 35 can be written as

$$\begin{array}{rll} {\rm ANI}\left( T\right) \cdot e^{-\int_{t}^{T}r\left( v\right) dv}&=&\left[ \dot{k} \left( T\right) +\dot{b}\left( T\right) +Q\left( T\right) \right] \cdot e^{-\int_{t}^{T}r\left( v\right) dv}\\ &&+\left( p_{m}\left( t\right) -w\left( t\right) \right) \cdot \dot{s}\left( T\right) . \end{array} $$

Using the co-state equations (25)–(27), the transversality conditions (28) imply

$$ \lim\limits_{T\rightarrow \infty }k\left( T\right) e^{-\int_{t}^{T}r\left( v\right) dv}=\lim\limits_{T\rightarrow \infty }b\left( T\right) e^{-\int_{t}^{T}r\left( v\right) dv}=0\text{ and }\lim\limits_{T\rightarrow \infty }s\left( T\right) =0. $$

As a consequence,

$$ \lim\limits_{T\rightarrow \infty }\dot{k}\left( T\right) e^{-\int_{t}^{T}r\left( v\right) dv}=\lim\limits_{T\rightarrow \infty }\dot{b}\left( T\right) e^{-\int_{t}^{T}r\left( v\right) dv}=\lim\limits_{T\rightarrow \infty }\dot{s} \left( T\right) =0. $$

Moreover, \(\dot{Q}=rQ-q\) (which is well defined only if q = 0 because Q = 0 otherwise) implies

$$ \lim\limits_{T\rightarrow \infty }Q\left( T\right) e^{-\int_{t}^{T}r\left( v\right) dv}=0. $$

Results (37) and (38) imply that taking the limit as T → ∞ in Eq. 36, we have

$$ \lim\limits_{T\rightarrow \infty }{\rm ANI}\left( T\right) \cdot e^{-\int_{t}^{T}r\left( v\right) dv}=0. $$

As a consequence, taking the limit as T → ∞ in Eq. 35, we obtain \({\rm ANI}\left( t\right) =\int_{t}^{T}\dot{c}\left( v\right) e^{-\int_{t}^{v}r\left( v^{\prime }\right) dv^{\prime }}dv\), as stated in Proposition 3. The fact that \({\rm ANI}\left( t\right) <0\) implies unsustainability follows by analogy with Proposition 2.□

Derivation of Eq.  17

By definitions (9) and (16), the value of time is given by

$$\begin{array}{rll} Q\left( t\right) &=&\int_{t}^{\infty }\frac{\dot{a}\left( v\right) }{a\left( v\right) }\cdot a\left( v\right) \mathcal{F} \left( k\left( v\right) ,g\left( v\right) ,m_{h}\left( v\right) \right) \cdot e^{-\bar{r}\left( v-t\right) }+ \\ &&+\int_{t}^{\infty }\dot{p}_{m}\left( v\right) m_{f}\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv-\int_{t}^{\infty }\dot{w}\left( v\right) m\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv+ \\ &&-\int_{t}^{\infty }\frac{\dot{p}_{g}\left( v\right) }{p_{g}\left( v\right) }\cdot p_{g}\left( v\right) g\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv+ \\ &&+\int_{t}^{\infty }\dot{r}\left( v\right) b\left( v\right) \cdot e^{-\bar{r }\left( v-t\right) }dv. \end{array} $$

Considering the first line in Eq. 39, we substitute \(\dot{a} \left( v\right) /a\left( v\right) \approx \gamma _{a}\) inside the integral and, defining the average future growth rate of output as \(\gamma _{x}\equiv \frac{1}{v-t}\cdot \int_{t}^{v}\dot{x}\left( \tau \right) /x\left( \tau \right) d\tau \), obtain

$$ Q_{1}\!=\!\gamma _{a}\cdot\!\!\int_{t}^{\infty}\!\!\!a\left( v\right) \mathcal{F} \left( k\left( v\right) ,g\left( v\right) ,m_{h}\left( v\right) \right) \cdot e^{- \bar{r}\left( v-t\right) }dv\!=\!\gamma _{a}\cdot x\left( t\right) \cdot \!\!\int_{t}^{\infty}\!\!\!e^{-\left( \bar{r}-\gamma _{x}\right) \left( v-t\right) }dv. $$

Substituting the approximation based on the Keynes-Ramsey rule, \(\rho \approx \bar{r}-\gamma _{x}\) with ρ > 0 constant, direct integration yields \(Q_{1}=\gamma _{a}x\left( t\right) /\rho \). Exploiting m = m h  + m f , the second line of Eq. 39 can be re-written as

$$\begin{array}{rll} Q_{2}-Q_{4}&=&\int_{t}^{\infty }\left[ \dot{p}_{m}\left( v\right) -\dot{w} \left( v\right) \right] m_{f}\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv\\&&-\int_{t}^{\infty }\frac{\dot{w}\left( v\right) }{w\left( v\right) }w\left( v\right) m_{h}\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv \end{array} $$

where we can substitute \(\dot{w}\left( v\right) /w\left( v\right) \approx \gamma _{w}\) and the Hotelling rule \(\left[ \dot{p}_{m}\left( v\right) -\right.\) \(\left.\dot{ w}\left( v\right) \right] =\bar{r}\cdot \left[ p_{m}\left( v\right) -w\left( v\right) \right] \) to obtain

$$\begin{array}{rll} Q_{2}-Q_{4}&=&\bar{r}\cdot \int_{t}^{\infty }\left[ p_{m}\left( v\right) -w\left( v\right) \right] \cdot m_{f}\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv\\&&-\gamma _{w}\int_{t}^{\infty }w\left( v\right) m_{h}\left( v\right) \cdot e^{-\bar{r}\left( v-t\right) }dv. \end{array}$$

Further substitute \(w\left( v\right) \approx w\left( t\right) e^{\gamma _{w}\left( v-t\right) }\), \(m_{h}\left( v\right) \approx m_{h}\left( t\right) e^{\gamma _{m_{h}}\left( v-t\right) }\) and \(p_{m}\left( v\right) -w\left( v\right) =\left[ p_{m}\left( t\right) -w\left( t\right) \right] \cdot e^{ \bar{r}\left( v-t\right) }\) yields

$$ Q_{2}-Q_{4}=\bar{r}\cdot \left( p_{m}\left( t\right) -w\left( t\right) \right) \cdot \int_{t}^{\infty }m_{f}\left( v\right) dv-\frac{\gamma _{w}}{ \bar{r}-\gamma _{w}-\gamma _{m_{h}}}\cdot w\left( t\right) m_{h}\left( t\right) . $$

Considering the third line in Eq. 39, substituting \(\dot{p} _{g}\left( v\right) /p_{g}\left( v\right) \approx \gamma _{p_{g}}\), \( p_{g}\left( v\right) \approx p_{g}\left( t\right) e^{\gamma _{p_{g}}\left( v-t\right) }\) and \(g\left( v\right) \approx g\left( t\right) e^{\gamma _{g}\left( v-t\right) }\) we obtain

$$ Q_{3}\left( t\right) =\frac{\gamma _{p_{g}}}{\bar{r}-\gamma _{p_{g}}-\gamma _{g}}\cdot p_{g}\left( t\right) g\left( t\right) . $$

Finally, the assumption \(r\left( v\right) \approx \bar{r}\) implies \(\dot{r} \left( v\right) \approx 0\) so that the last line in Eq. 39 is equal to zero. Notice that the integrals yielding Q 3 and Q 4 are bounded provided that \(\bar{r}>\gamma _{w}+\gamma _{m_{h}}\) and \(\bar{r} >\gamma _{p_{g}}+\gamma _{g}\). Both these inequalities can be shown to hold necessarily along an optimal path in order to fulfill the various transversality conditions associated to the state variables. In particular, if \(\bar{r}<\gamma _{p_{g}}+\gamma _{g}\), the integral yielding \(Q_{3}\left( t\right) \) becomes unbounded and does not fulfill intertemporal solvency with the rest of the world: the sequence of future trade deficits explodes at a rate that exceeds the interest on foreign debt.

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Bretschger, L., Valente, S. International trade and net investment: theory and evidence. Int Econ Econ Policy 8, 197–224 (2011).

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  • International trade
  • Natural resources
  • Net investment
  • Sustainability
  • Technological progress

JEL Classification

  • E22
  • F11
  • O11