Corollary 3 Marginal Rates of Substitution of the Tax Rates for the Log- Utility Case) Let the utility function of the representative household be logarith-
4.2 Numerical Solution of the Open Economy Model
At first, an open economy model is solved for the resource-poor country, where there are no exports of natural resources, using NMPC, with different initial conditions of the state variables ofK.0/, andF.0/. The model has these two state variables and
16Potential non-monetary costs of a very high debt can be issues of solvency and political risks as discussed in Blanchard (1983). The penalty term in the objective function, depending on whether the cost of debt is high or low, determines the path of the debt growth. Based on the analysis of marginal cost of debt and implications of different parameters in the penalty term, Blanchard (1983, p. 195) argues that “if a reduction in the growth of debt has to be achieved, it must be done by reducing consumption rather than investment”.
0 5 10 15 20 25 30 1
2 3 4 5 6
7 Capital Stock
0 5 10 15 20 25 30
0.5 0.6 0.7 0.8 0.9 1
Consumption/GDP
0 5 10 15 20 25 30
0 2 4 6 8 10
12 Debt
0 5 10 15 20 25 30
-0.2 0 0.2 0.4 0.6 0.8
Investment/Capital
Fig. 6 Optimal trajectories of state variables and controls in a resource-poor country
two control variables,C;andI. In the NMPC code, control variables are expressed as shares, e.g.,c D C=Y, andi D I=K. The results, with initial conditions of the state variables:K.0/D0:02,F.0/D1:1, and assumed parameter values ofD0:5, ˇ D 0:3,rD 0:04,ı D 0:1, D 0:05as well as with a penalty term included in the preference, where theis assumed to be low, D 0:3, and D 0:0025, is shown in Fig.6.17
For a resource-poor country, funding from external sources would be more limited, as compared with resource-rich countries. Thus, its external debt increases, but at much slower speed. An initial increase in external debt can be associated with a rise in the capital stock. However, there is a decline in capital stock as time passes and debt further increases; thus, the capital stock has an inverted U-shaped curve.
This result is similar to solutions of the closed economy model that was presented earlier. A very large external debt would be associated with a decline in capital stock, as there is less productive investment that would lead to more growth in the future. The excess debt in this case seems to cause consumption to decline after a particular period, when capital stock shows a significant decline and external debt reaches its highest value.
Next, the open economy model for the resource-rich country is solved using NMPC. Since the country is resource abundant, it has a resource constraint in addition to constraints of capital stock and external debt. Thus, there are three control variables,C,I, andX, and three state variables,K,S;andF. A solution to a
17Again, with regard to the choice of values for the parameters and the penalty term, I follow Blanchard (1983), Semmler and Sieveking (2000), and Mittnik and Semmler (2014).
0 5 10 15 20 0.2
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2.2 Capital Stock
0 5 10 15 20
0 1 2 3 4 5 6 7 8 9 10
Stock of Resources
0 5 10 15 20
0 1 2 3 4 5 6 7
8 Debt
Fig. 7 Optimal trajectories of variables in a resource-rich country
variation of the model with penalty term is shown in Fig.7, where control variables are expressed as shares, e.g.,cDC=Y,iDI=K, and the flow of resources as share in total stock,xDX=S. The initial conditions of the state variables areK.0/D1:2, S.0/ D 10, andF.0/ D 0:02and, as before, the parameter values are D 0:5, ˇ D 0:3,r D 0:04,ı D 0:1, D 0:05, D 0:0025anda D 0:5; but now with˛D0:05, and high, e.g.,D 500, because of an increasingly large debt value.18
As expected, the stock of resources declines until it is completely used up. Debt is increasing, but at much slower growth rate; this is because the high penalty term causes the curve to become more concave. Capital stock initially increases sharply, but it then declines significantly. This cyclical pattern repeats over time until it reaches a stable low value; this is in contrast to the capital stock movement in a resource-rich, but closed economy. Typically, this also occurs in a resource-poor open country, where the capital stock rises first, but later declines monotonically, without any cyclical movements. At the beginning, capital stock increases in the economy where large stocks of a resource is available. However, following a sharp fall in the stock of exhaustible resources, capital stock will decline sharply. Initial borrowing from abroad increases the investment which, in turn, would contribute to an increase in capital stock. Nevertheless, further depletion of the resource, as well as excessive unsustainable debt, will lead to an eventual decline in the capital stock.
Thus, the capital stock’s path is affected by both available resources and extent
18Some of these parameter values are taken from Blanchard (1983), Semmler and Sieveking (2000), and Mittnik and Semmler (2014).
F K
2 4 6 8 10 25
20
15
10
5
Fig. 8 Boom-bust cycles
of debt. A positive impact of debt on investment and growth turns negative in the presence of excessive debt.
Bernard and Semmler (2012) have shown that many boom-bust cycles are driven by linkages between borrowing and resource prices, yet borrowing could be over- covered or under-covered by resource value in the marketplace. The collateral value is a rather fuzzy concept based on many speculative elements in the marketplace and on the perceived value of the resource producer. Thus, as shown in Fig.7, a resource boom encourages an overconfidence and overvaluation of assets. This leads over-borrowing and an underestimation of risk. The natural consequence is that too many loans secured by over-valued resources are issued. Finally, the market turns pessimistic and undervaluation ensues. This, in turn, causes the undervaluation of asset prices and a decline in the value of the underlying capital stock.
Technically, there are both absorbing and repelling points in the phase diagram illustration shown in Fig.8, taken from in Bernard and Semmler (2012). In their paper, Bernard and Semmler show how, in the general case, over-borrowing is undertaken, but unless sufficient return is realized on investment, the capital stock will revert to it’s previous value. On the other hand, if sufficient return is realized, a new stability may be achieved. As an example, we can imagine a resource- rich country’s borrowing against its exports to improve its transportation system.
Initially, this increases the value of its capital stock as there is now an efficient method to move resources to export. However, as the resource depletes and/or becomes more difficult to retrieve, the value of this infrastructure declines. A new discovery or extraction technology may re-ignite the boom, but since the infrastructure is already there, the additional returns to investment are more limited.
Obviously the specifics are particular to the situation, but the illustration captures this generality.
5 Open Economy Model with Human Capital and Its Solutions