The NATREX stands for NATural Real EXchange rate. It attempts to give a fair value for a currency. It is part of the family of long run equilibrium exchange rate theories (FEER, BEER, and NATREX). Notably the The approach offers an alternative paradigm to the Purchasing Power Parity for equilibrium real exchange rates.

Natrex can be compared to PPP (Purchasing Power Parity) type methods. These are based mainly on an analysis of the goods and services market. However the NATREX also integrates capital flows as long-term determining factors for exchange rates. This makes it easier to consider NATREX as a real equilibrium exchange rate

Stein (2006) has offered the clearest definition of the NATREX-based view of the Equilibrium Real Exchange Rate: The equilibrium value of the real exchange rate NATREX is a sustainable rate that satisfies several criteria. First; it is consistent with internal balance. This is a situation where the rate of capacity utilization is at its longer run stationary mean. Second, it is consistent with external balance. The latter is a situation where, at the given exchange rate, investors are indifferent between holding domestic or foreign assets. At the equilibrium real exchange rate, there is no reason for the exchange rate to appreciate or depreciate. Hence, portfolio balance or external balance implies that real interest rates between the two countries should converge to a stationary mean. As long as there are current account deficits, the foreign debt and associated interest payments rise. If the current account deficit/foreign debt exceeds the growth rate of real GDP, then the ratio of the debt/GDP and the burden of the debt - net interest payments/GDP - will rise. When the debt burden is sufficiently high, devaluation will be required to earn enough foreign exchange through the trade balance to meet the interest payments. The condition for external equilibrium in the longer run is that the ratio of the foreign debt/GDP stabilizes at a tolerable level.

The NATREX is not constant but varies with "fundamentals" Z(t)which are relative productivity and relative "thrift", which is the ratio of relative consumption/GDP ratios. The actual equilibrium real exchange rate converges to the moving "equilibrium" NATREX. The convergence can occur either through variations in the nominal exchange rate or in relative prices.

Schematically the real exchange rate R(t) at any time is a sum involving three terms. R(t) = [R(t) - R [F(t);Z(t)] + [R(F(t);Z(t)] - R [Z(t)] + R [Z(t)] .The medium run equilibrium is R [F(t), Z(t)] where the debt F(t) is given and the real fundamentals Z(t) are given. The long run equilibrium is R [Z(t)] where the debt has converged to its equilibrium value. In the medium run the NATREX exchange rate converges to R [F(t);Z(t)] . In the longer run the debt adjusts and the medium run NATREX converges to R [Z(t)] the longer run NATREX which just depends upon the real fundamentals Z(t) = {relative productivity, relative consumption ratios}. The difference {R(t) - R [F(t);Z(t)] } between the actual real exchange rate and the medium run NATREX results from speculative random behavior and from cyclical factors. In the medium to the longer run, these disturbances have zero means. Thus the NATREX is where the real exchange rate is heading along a trajectory. The PPP model is a special case where the NATREX is constant, and the PPP ignores the NATREX stock-flow dynamics.

In the standard models, an expansionary fiscal policy appreciates the real exchange rate. In NATREX, the expansionary fiscal policy appreciates the medium run NATREX. However, due to the induced accumulation of external debt, the long run NATREX depreciates below its initial level. On the other hands a rise in relative productivity appreciates the medium run NATREX and the long run NATREX appreciates even more.

The underlying techniques of analysis are vector matrix differential equations, and stochastic optimal control/dynamic programming – since the intertemporal optimzation is based upon stochastic processes where the future is unpredictable. The analysis is used to explain the debt crises.

It has been applied to the $US/Euro, the currencies of the Transition Economies in Eastern Europe, Australia, China and to the Latin American currencies by researchers, Central banks and the International Monetary Fund.

Technical Discussion of Equilibrium Exchange Rates, NATREX and Misalignment

An equilibrium exchange rate is where the exchange rate is heading. The concept and measure of the equilibrium exchange rate depends upon the time horizon and the underlying model. Several reasons have been cited in the literature , why it is important to estimate equilibrium exchange rates. First, there are significant and sustained movements in exchange rates. These movements affect the competitiveness of the economies and their macroeconomic stability. One wants to know whether these movements are ephemeral or whether they are responding to "real fundamentals" . This information is important because the answer has implications for rational macroeconomic policy and for rational investment decisions. If the depreciation of an exchange rate is due to a depreciation of its equilibrium value, then exchange market intervention or a restrictive monetary policy designed to offset the depreciation is counterproductive. Second, in the case of monetary unions such as the Euro area, it is important to know how a potential entrant should select its exchange rate. An "overvalued" rate will depress growth and produce problems such as beset the eastern part of Germany. An "undervalued" rate will generate inflationary pressures. The measure of "over-valuation" and "under-valuation" must contain an explicit measure of an "equilibrium" real exchange rate. Our emphasis is upon the equilibrium real exchange rate. It is defined as the nominal exchange rate times relative prices. In an adjustable peg regime, the nominal exchange rate is fixed and the actual real exchange rate varies due to changes in relative prices. In a floating exchange rate regime, both the nominal exchange rate and relative prices can lead to adjustments in the actual real rate. The only difference from our point of view is that the adjustment of the actual real exchange rate to the equilibrium will be faster when the exchange rate floats, because the nominal exchange rate is more flexible than relative prices. A widely used approach in the literature is to "explain" the exchange rate by the uncovered interest rate parity theory (UIRP) . It states that the anticipated appreciation of the exchange rate is equal to the anticipated interest rate differentials over a period of a given length. There are several limitations of this approach. First: the UIRP equation concerns the change in the exchange rate but does not contain any information concerning where the exchange rate is heading. As Driver and Westaway state, the exchange rate at any given time t will jump around to adjust to any change in either the anticipated exchange rate at some future date t+h or any change in anticipated interest rate differentials over the interval (t, t+h). The UIRP theory per se has no anchor.

Second: for the theory to have significance one must tie down the anchors. One anchor must be the "equilibrium" exchange rate and the second must be the path of the interest rates. This is not done in the UIRP theory.Third: the theory states that the interest rate differential at time t is a good and unbiased predictor of the subsequent change in the exchange rate. The "tests" of the theory generally relate ex-post changes in the exchange rate to the previous interest rate differentials. In general, the results of these tests are not encouraging. The interest rate differential has the incorrect sign and is unsuccessful in predicting exchange rate movements .

For these reasons, authors who are interested in explaining exchange rates focus upon the anchor, the equilibrium exchange rate - where the exchange rate is heading. Then the theory of UIRP has structure. The actual exchange rate at time t is equal to the present value of the equilibrium exchange rate, where the discount factor is the interest rate differential. There are two types of candidates for the equilibrium exchange rate. One is Purchasing Power Parity (PPP), which assumes that the equilibrium real exchange rate is a constant. As mentioned above, this hypothesis is unimpressive as an explanation of the anchor . The other candidate is an equilibrium real exchange rate that depends upon time varying real, measurable "fundamentals" . This has led to the literature of " equilibrium exchange rates", which was given great impetus by John Williamson's influential book (1994). The logic of this approach goes back to Ragnar Nurske's article. The "equilibrium" exchange rate is the exchange rate that is associated with both external and internal balance. Anticipations, speculative capital movements and changes in reserves are excluded from the concept of an equilibrium exchange rate, which is where the exchange rate is heading. The NATREX model of equilibrium exchange rates generalizes the work of Williamson and Nurske. It is a Neoclassical growth model, whose underlying equations are based upon intertemporal optimization by the private sector, but not the government whose decisions are political.

The NATREX explains the fundamental determinants of the medium run equilibrium and the dynamic trajectory to the long run equilibrium. In the medium run equilibrium there are both internal and external balance. In both the medium run and longer run the NATREX equilibrium real exchange rate satisfies equation (1), subject to constraints. The constraints are that there is internal balance, where the rate of capacity utilization is at its longer term mean, and external balance where the real rates of interest at home and abroad are equal, there are neither changes in reserves, nor speculative capital flows based upon anticipations. The equilibrium real exchange rate is the mean of a distribution, which is based upon real fundamentals. The mean will vary over time due to endogenous changes in capital and external debt, as well as changes in the exogenous real fundamentals. Deviations from this mean are produced by speculative factors involving anticipations, by cyclical factors, lags in adjustment, and interest rate differentials. These disequilibrium elements average out to zero. These deviations produce considerable variation but their effects are ephemeral.

The terms in square brackets are that investment investment less saving [I(t) - S(t)] plus the current account is equal to zero. The current account [B(t )- r(t)F(t)] is the trade balance B(t) less transfers of interest and dividends rtFt. The net external debt is F(t) and r(t) is the "interest/dividend" rate. The international investment position consists of equity, portfolio investment and direct investment. The debt F(t) is the negative of the net international investment position. Measure investment, saving and the debt as fractions of the GDP.

(1) [I(t) - S(t)] + [B(t) - r(t)F(t)] = 0

All of the authors who take the equilibrium real exchange rate approach use equation (1) to determine the exchange rate. The main differences among them concern their treatment of the two terms. Some work with a concept of what is a "sustainable" current account such that the debt does not "explode". As is discussed in Stein (2006) chapter 9 on the United States current account deficits, their estimates are subjective, so their equilibrium exchange rate is a "normative" concept. The NATREX approach is quite different in several respects, primarily because the endogenous current account generates an evolving external debt, which feeds back into the medium run equation (1). A trajectory to longer run equilibrium is generated. The other difference is that the underlying equations are derived from inter-temporal optimization by the private sector. The dynamics of the debt/GDP ratio F(t) is equation (2), where g is the growth rate. The current account deficit is the change in the external debt. The real exchange rate affects the trade balance B in equation (1), and the trade balance affects the evolution of the actual debt ratio in equation (2). There is a dynamic interaction between the endogenous real exchange rate and debt ratio.

(2)dF(t)/dt = (I(t) - S(t)) - g(t)F(t) = [r(t)F(t) - B(t)] - g(t)F(t)] = (r(t)-g(t)F(t) - B(t)

In longer run equilibrium, the debt ratio stabilizes at F*(t) at a value that satisfies equation (3). The trade balance B(t) is sufficient to finance the interest plus dividend transfer on the debt net of growth [r(t) - g(t)] F(t). A negative debt is net foreign assets.and the exchange rate at R*(t). A negative debt is net foreign assets.

(3) [r(t)-g(t)] F(t)- B(t) = 0.

The longer-run equilibrium real exchange rate R(t)* and debt/GDP ratio F(t)* are endogenous variables that satisfy both equations (1) and (3). They are written as (4) and (5) to indicate that they both depend upon the real fundamentals Z(t).

(4) R(t)* = R(Z(t))(5) F(t)* = F(Zt))

We call dynamic stock-flow model equations (1) - (3) the NATREX model, which is an acronym for the Natural Real Exchange Rate . This is a model of positive economics. The literature associated with Williamson's FEER uses equation (1) and does not contain the dynamic interactions, equations (2) and (3). The NATREX model derives the private saving, private investment and trade balance equations from optimization criteria. There is no presumption that the government saving and investment decisions are optimal, since they are based upon political considerations not upon social welfare.

Populist and Growth Scenarios

The NATREX model is a technique of analysis . The purpose of the model is to understand the effects of policies and external disturbances upon the trajectories of the equilibrium real exchange rate R(t) and equilibrium debt ratio F(t) ,which depend upon the vector of fundamentals Z(t). Insofar as the fundamentals vary over time, the equilibrium real exchange rate and debt ratio will vary over time, as indicated in equations (4) and (5). The logic and insights of the NATREX model can be summarized in two scenarios. Each scenario concerns different elements in the vector Z(t) of the fundamentals, and has different effects upon the equilibrium trajectories of the real exchange rate NATREX and of the external debt. NATREX analysis concerns the equilibrium real exchange rate and it is neither the actual real exchange rate nor the optimal exchange rate that would lead to the optimal debt ratio.

The first scenario, called the Populist scenario, involves a decline in the ratio of social saving/GDP. This could occur when the government incurs high-employment budget deficits, lowers tax rates that raise consumption, or offers loan guarantees/subsidies for projects with low social returns. This represents rise in the consumption ratio/a decline in the saving ratio, a shift in the S function in equations (1) and (2). These Populist expenditures are designed to raise the standards of consumption/quality of life for the present generation. The second scenario, called the Growth scenario, involves policies designed to raise the productivity of capital. Policies that come to mind involve the liberalization of the economy, increased competition, wage and price flexibility, the deregulation of financial markets, improved intermediation process between savers and investors, and an honest and objective judicial system that enforces contracts. Growth policies improve the allocation of resources and bring the economy closer to the boundary of an expanding production possibility curve. Table 1 summarizes the differences between the two scenarios in the medium and the long run. The stories behind the dynamics are as follows.The Populist scenario involves increases in social (public plus private) consumption relative to the GDP. External borrowing must finance the difference between investment and saving. The capital inflow appreciates the real exchange rate from initial level R(0) to medium run equilibrium R(1), where T = 1 denotes medium run equilibrium. The current account deficit is balanced by the capital inflow. The debt rises, since the current account deficit is the rate of change of the debt - equation (2). Current account deficits lead to growing transfer payments rtFt. This Populist scenario is potentially dynamically unstable because the increased debt raises the current account deficit, which then increases the debt further. The exchange rate depreciates, and the debt rises, steadily. Stability can only occur if the rise in the debt, which lowers net worth equal to capital less debt, reduces social consumption/raises social saving. For example, the growing debt and depreciating exchange rate force the government to decrease the high employment budget deficit. Thereby, saving less investment rises. Long-run equilibrium (denoted by T = 2) is reached at a higher debt F(2) > F(0) and a depreciated real exchange rate R(2) < R(0). The longer-run depreciation of the exchange rate R(2) < R(0) can be understood from equation (3). The debt is higher than initially. Therefore, the trade balance B(2) must be higher than initially to generate the foreign exchange to service the higher transfers r(t)F(2). The real exchange rate must depreciate to R(2) < R(0) in order to raise the trade balance to B(2).

Table 1NATREX dynamics of exchange rate and external debt: Two Basic ScenariosScenariosR = real exchange rate (rise is appreciation), F = external debt/GDP; initial period T = 0, medium run T=1, long-run T=2. Medium-run, T = 1 Longer-run T = 2

Populist: Rise in social in social consumption (discount rate, time preference), rise in high employment government budget deficit, decline social saving R(1) > R(0) appreciationDebt risesF(1) > F(0) R(2) < R(0) < R(1) depreciationDebt risesF(2) > F(1) > F(0)

Growth oriented: Rise in productivity of investment, expansion of production possibility set. Rise in growth, rise in competitiveness appreciationR(1) > R(0)Debt risesF(1) > F(0) appreciationR(2) > R(1) > R(0)Debt declinesF(2) < F(0) < F(1)

The Growth scenario is summarized in the lower half of table 1. The perturbation is a rise in the productivity of investment and an expansion of the production possibility set. Investment rises because of the rise in the rate of return. The difference between investment and saving is financed by a capital inflow. The exchange rate appreciates to R(1) > R(0) which reduces the trade balance and produces a current account deficit. The initial current account deficit equal to [I(0) - S(0)] raises the debt. The trade deficit provides the resources to finance capital formation, which raises the growth rate and the competitiveness of the economy. It does not matter much where the rise in the return on investment occurred or what factors led to an expansion of the production possibility set. If they are in the traditional export or import competing sectors, the trade balance function B = B(R;Z) increases. The B function, which relates the real value of the trade balance to the real exchange rate R, increases with a rise in the overall productivity of the economy. For example, the reallocation of resources leads to the production of higher quality/value goods that can compete in the world market. If the rate of return on investment and productivity increase in the sectors that are not highly involved in international trade, resources can then be released for use in the more traditional "tradable" sectors. Again, the B function supply curve increases. The trajectory to longer-run equilibrium differs from that in the Populist scenario. The crucial aspect implied by the Growth Scenario is that, at medium run equilibrium exchange rate R(1), the trade balance function increases. The real exchange rate appreciates and there are now current account surpluses, excess of saving over investment. As a result, the debt then declines to a new equilibrium F(2) < F(0). The trajectory of the debt is not monotonic. The net effect in the longer-run can be understood from equation (3). The debt is lower, the growth rate is higher and the trade balance function B has shifted to the right. The long-run equilibrium exchange rate must appreciate to reduce B to equal the lower value of (r-g)F*. The dynamic process in the Growth scenario is summarized in the lower half of table 2. The real exchange rate appreciates steadily to a higher level R(2) > R(1) > R(0). The external debt reaches a maximum and then declines to F(2) < F(0) < F(1).

References: Stein, Jerome L. "Stochastic Optimal Control, International Finance and Debt Crises", Oxford University Press, 2006; Independent Evaluation Office, (2007) International Monetary Fund, "The IMF Exchange Rate Policy Advice 1999-2005", Background Document III.

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