Generalized Economics of Exchange Rates and Policy Options:
Empirical Evidence
Hsieh Hsih-chia and Hsieh Pei-gin
Providence University
9,Alley 24, Lane 52
Shih-wey Rd.
Taipei,Taiwan
Fax 886-2-27056893
Case Western Reserve University,
Department of Accountancy,
Cleveland, Ohio 44106,
U.S.A.
pxh29@po.cwru.edu
Abstract
The minmax solution of foreign exchange rates is a consistent efficiency test for
micro and macro efficiency. Deviations from the competitive equilibrium lead to
persistent devaluation and output declines. The optimal fiscal and monetary policy is
estimated with a minimum and shrinking variance of errors. The necessary and
sufficient Lagrange multiplier, nonlinear stability, and convergence probability are
defined. A family of evolution equations yields the minmax solution, which is
equilibrium level of outcomes to be evaluated, when the response coefficient of
strategies alternates in sign around the equilibrium. The larger the conditional
convergence probability is, the steeper the wave curvature of the objective welfare
and asset allocation would be, and the faster the convergence time is. Beyond the
equilibrium, the positive sign of coefficient synergy tends to blow up and becomes
transformed into a negative response. Thus we solve the optimization problem at all
horizons and times. The Lagrange multiplier is a coordinator and indicates the
evolution direction and the convergence probability towards the stable equilibrium.
The generalized solution is found consistent with real data and satisfies the equality
and inequality constraints of resources, when resources are potential rather than
constant.
Keywords: Efficiency test of real exchange rates; Second-order and fourth-order
dimensions. Micro-macro; General equilibrium in a continuous sense; conditional
convergence probability; time-consistent strategy; wave and evolution equations.
JEL Classification F31;D82; E4; E5.
AMS subject classification. primary: 53C25; secondary: 65L06, 65M20
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1. Introduction
In this paper, we resolve contradictory theories. Devaluation promotes exports
but can either reduce or raise output growth. Similarly, foreign debt is capital but also
may lead to currency crisis(Hausmann, et al. (2003). The Keynesian and monetarists
find a positive correlation between output and price levels (Almonacid, et al. 2003).
We find the competitive equilibrium whereby the output growth and inflation rate turn
from a positive into a negative relationship. In the autoregression of exchange rates
and output, the second-order autoregressive coefficients changes in sign over time
around the equilibrium. Suppose that the cost function is convex-up, and that the
marginal revenue is declining. The price-takers of the firms perform better than its
rivals, when the observations converge toward the maximum equilibrium profits such
that the price equals the marginal costs. The efficiency test implies that the real
exchange rate is equal to the purchasing power parity.
In macroeconomics, when the output growth serves as the dependent variable,
the cross-section data show that education spending has a positive impact upon output
growth, while the time-series data show that education spending has a negative
impact(Yang, 2003). Although an increase in government spending can raise output,
the increase in tax revenue tends to reduce output. The Keynesian counter-cyclical
policy is assumed to have a positive stabilization impact upon output and effective
demand. A large variance of government spending, however, has negative impacts
upon output growth(Fatas, et al. 2003). We find that the optimal policy is the ratio of
government spending to output and denotes the minmax solution, where the
equilibrium is stable and has the minimum and shrinking variance. Thus, output
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growth and profits are maximized and the variance of disturbances is minimized.
Empirically, international assistance to the developing countries is focused to debt
concessions, food, drugs, and cloths for short periods of survival. Many African
people, however, do not know how to use the movie for mass education and to utilize
the machinery for self subsistance. Lack of education, infrastructure, democratic
voting or referendum leads to persistence of wars and strife. When the volatility of
policy deviates from the maximum equilibrium, it has a negative impact upon output;
and vice versa. We solve a welfare function as the minmax solution, which is
equilibrium level of outcomes to be evaluated, when the response coefficient of
strategies alternates in sign around the equilibrium. The minmax solution serves a
consistent efficiency test for micro and macro efficiency.
Section 2 presents a general model of foreign exchange rate economics.
Section 3 illustrates examples. Section 4 concludes with remarks.
2. The Model
Following Hsieh(2003), suppose x(t)=F(x(t-1), u(t-1), t) is the welfare function
of exchange rates. x is the state variable or exchange rates; u is the control policy; t is
time. The generalized solution of the law of motion of exchange rate is
Step 1: Estimate the equilibrium x* and the optimal policy u**:
(2.12)
min max
 
x

n
t 1
v2(t)
p(x)=exp(dx/dt)=  0 + 1 x(t-1)+  2 x2 (t-1) +  3 u(t-1)+  4 u2 (t-1)+v(t)
p(x) is the convergence probability and denotes the value of policy options.
x*=max(log Y*(t)-logY(t), 0)
where x* is the value of
optimal policy options and outcomes or competitive
equilibrium. Y(t) is the market output; Y* is the strike price or the exercise price,
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which is the maximum likelihood estimator and the almost uniformly most powerful
unbiased estimator as well as the minmax solution, when the welfare function F(x,u,t)
is nonsmooth and nondifferentiable . u* is optimal policy or the incentive.
Step 2: The heteroschedascity of errors is reduced into homoschedasticity:
p(x)= exp(  x(t))= 
t*
t 0
f ( x(t ), u(t ) )dt
=  2 (x(t-1)-x*)2 +  4 (u(t-1)-u**)2 +  5 (x(t-1)-x*)(u(t-1)-u**)2
+  6 (  x(t-1)) +  7  x(t-2)+v(t)
where 1 ,  2 ,  3 ,  4 , … are parameters and can be trqnsformed into Lagrangian
multipliers of constraints, such as  2 (x(t-1)-x*)2=0. The equilibrium is x*=- 1 /2  2 ;
the optimal policy is u**=-  3 /2  4 . The interactive response  5 may be negatively or
positively switching and stabilize the equilibrium x*.  6 and  7 denote time-delay
responses of the state variable.
3. Examples
Example 1 on convergence probability to the competitive equilibrium at infinite
horizons. From the generalized solution, we detect the distorted magnified impact of
misleading policy upon the welfare and firms’ value, lim F(x(u(n),n))= lim
n
n

n
1
F(x(t), t)exp(xt). Non-uniqueness implies that multiple equilibria comprise the upper
and the lower bound of output and foreign exchange rates.
To maximize the welfare, Hsieh(2003) solves the puzzle of the shadow cost,
which is the unique equilibrium. When the minmax solution for country A is equal to
the maxmin solution for country B. The maximum solution is determined by the
non-Ricardian fiscal budget inequality and by the Ricardian monetary equality of
budget constraints.
4
Fig. 1 The maxmin solution in Concave Functions
demand curve
Output Growth
supply curve
x*,q*=1
momentum
contrarian
b1  0
t  t*
0
 b2  0
t*=t**
t  t*
u**
Optimal policy
 P/P( t * )
 M(t*)/M(t*)
q*=1
price inflation
Growth of money supply or cash flow
real exchange rates
equal the purchasing power parity
In the short-run
In the long-run
As in Figure 1, x is output growth; u is control policies, such as the inflation rate,
fiscal and monetary policy. t is time. When the policy of , say, prices falls below the
equilibrium price, u<u**, the convergence probability also falls below the
Pareto-improving probability, p  p * =1 and q*=1. Investors expect the prices to
revert to the equilibrium and buy the stocks and foreign currency; this is the
contrarian strategy for the long run, if the central bank will increase the money supply
and decreases the interest rate towards the optimal rate. During recovery, by use of the
futures contract and momentum strategy, the investors buy the prior winners, sell the
losers, and expect the prices to rise and revert to the equilibrium price. Thus, the
maxmin or minmax solution (x*, u**) is
stabilized by investors and central banks.
Example 2: The foreign exchange rate q*=1 equals the purchasing power parity.
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This minmax solution is equal to the maxmin solution, when the first-order coefficient
is
1 >0 and the second one is  2  0. The instability is indicated by the
insignificant second-order derivative. Without intervention, the floating real exchange
rate is the competitive equilibrium, and is robust to time-varying parameters.
Suppose q(t)=F(q(t-1), t).
 2F/  t  q(t-1)= 1 >0. Beyond the equilibrium,
 3F/  t  q2(t-1)=  2 <0 if q(t-1)>q*=1. Below the equilibrium,  3F/  t  q2(t-1)=
 2 >0 if q(t-1)<q*=1. q is the real exchange rate of U.K. pound per US dollar, which is
adjusted by the value-added price deflators. The quarterly data of the real exchange
rate over 1978.1 to 1996.4 are used, as reported by International Financial Statistics.
Suppose t denotes time or quarters.
Our unique equilibrium of the real
exchange rate, q*=1, is estimated as follows:
 q=q(t)-q(t-1)=  0 +(  1 - 1)q(t-1) +  2 q2(t-1)
=-0.17+0.46q(t-1)-0.23q2(t-1)+v3(t)
(0.8) (3.38)
(1.2)
= -0.23(q(t-1) - (0.46)/2(0.23))2+ v3(t)
(4.8)
for q*=1.0

R 2=0.8; D.W.=1.71
where Student t statistics are in the underlying parentheses. It is noteworthy that the
Dickey-Fuller test is applicable for the linear model.
By treating (q(t-1) - 1)2 as
one explanatory variable, we need to reestimate the nonlinear model. The coefficient

of determination is R 2=0.88.
The Dickey Fuller(DF) t test is t=1.92. The
constant intercept is statistically insignificant. Our dynamic quadratic regression,
however, provides the unique stable prediction, q*=1, for the nonstationary difference.
Our solution converges to the equilibrium E(q)=1, which is robust to time-varying
parameters  (t), while the chaos model provides multiple solutions, depending on
the initial values and noises.
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Remark: Andrews(1993, p.153) shows that the half life of an unit shock is
log(1/2)/log(  ) and denotes the speed of adjustment of perturbations, where x(t)=
 x(t-1)+v(t). Our maximum x*=F(x2) corresponds to Andrews’ (1993) median of the
variable x(t) due to his lack of the optimal-control context.
In the equation of foreign exchange rate, the convergence probability is

R 2=0.8. The convergence rate is  =0.93  1+ 1 . t=1/ 1 =1/0.46=2 years is the
half life speed of adjustment of an unit shock.
lim 1 =  /2=1/2  0.46,
t t *
where
dq/dt=  0 + 1 q(t-1)+  2 q2(t-1+v(t).
Suppose q*(0)=q(t)exp(-xt)+O(1/xt2)=1;
where x is the convergence rate. log p=log (1/2)=tlog x. t=log (1/2)/x* is the
half-life time of adjustment by 50 percent towards the purchasing power parity.
p(x*)=exp  x(t)=  2 (t- t*)2+  4 (x(t-1) – x*(t-1))2+v(t)
for 0  t  t*=- 1 /2  2 ; x*=-  3 /2  4 .
where t* is the convergence time. 0  t  t*<n. n is the sample size.
p(x*)=1=  2 tt*=-  2 t(- 1 /2  2 )= t( 1 /2)
t= (2/ 1 )=2/0.46=4 years, denoting the one life ( or time) of the maximum distance of
q(t) from the purchasing power parity q*(0)=1. In contrast, Murray(2002) shows
three to five years, and others find that about two years is the half-life distance of the
real exchange rate converges to the purchasing power parity.
Example 3: Exchange Rates and Monetary and Fiscal Policy
The foreign exchange rate is determined by the share of real money balances and
that of the government spending in output.
Such monetary and fiscal policies
dominate the effects of output fluctuations, net exports, and capital flows(Hsieh,
2003b). Suppose S is the market nominal exchange rate of pounds per U.S. dollar.
Dynamic quadratic regression is estimated as:
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S(t)=-1.33+0.64S(t-1)+10.17M(t-1)/P(t-1)Y(t-1) –15.51(M(t-1)/P(t-1)Y(t-1))2
(-1.33) (4.66)
R2=0.89
(1.89)
(-1.57)

R 2=0.88; D.W.=1.60 1st order autocor.=0.17
where the values in the parentheses are t statistic. Below the optimum,
(M/PY)**=-10.17/(2)(-15.51)  30%, an increase in money supply denotes the credit
capital in advance,
tends to increase output, and leads to the appreciation of
domestic currency.
dS(t)/dt=133.23+0.48S(t-1)-19.40(M(t-1)/P(t-1)Y(t-1) –0.30)2
(1.74)
(3.49)
(2.00)
(1.38)
+0.08(G(t-1)/Y(t-1) – 27%)2
(2.72)
R2=0.33

R 2=0.24; D.W.=1.85 1st order autocor.=0.057
where dS(t)/dt=133.23+0.48S(t-1)=0.48(S(t)-2.5), implying that the non-homogenous
solution is the equilibrium exchange rate, S=2.5 pounds per U.S. dollar.  6 =0.08>0
implies that the flexibility and volatility of government spending has a positive impact
upon the appreciation of the U.S. dollar.
Remark: In simulations we test the alternative money supply, either M/PY=40% or
24%. These alternative performances become statistically insignificant,
have a
lower convergence probability, and cannot shift the trend of the U.S. dollar
depreciation. In 1965-2001, the Japanese Yen currency has a long-term and large
swing. The U.S. dollar has a long-term trend of depreciation. The market exchange
rate of British pounds per dollar continues to appreciate from 2.80 pounds in 1965 to
1.15 pounds per U.S. dollar in 2001.
Table 1: The crowding-out effect and turning points in the United States
____________________________________________________________________
The dependent variable
Investment rate I/Y:
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 (I/Y)/  (G/Y)<0 if G/Y>10%;
 (I/Y)/  (T/Y)<0 if T/Y>24%;
Consumption ratio C/Y:
 (C/Y)/  (T/Y)<0 if T/Y>37%;
Government spending share G/Y:
 (G/Y)/  (M/PY)<0 if G/Y>37%;
_________________________________________________________________
Notations: I is investment; Y is output; G is government spending; T is tax revenues;
M is money supply; P is the GDP price deflator.
Note:In the steady state, if the government spending share exceeds 24%, the
consumption share is maximized but the investment share is crowded out. The
consumption share is C/Y=1-(I/Y)-(G/Y)=100%-1%-23%=76%, if G/Y=23% and
I/Y=1%. The optimum income tax rate is around T/Y=17% i.e., ((37%+10%)/2),
while government spending switches between 10%<G/Y<37%.
Example 4: Output, Inflation Rate, and Monetary Policy
Contractionary monetary policy leads to declines in output growth and
depreciation within two years in 2001-2002 in Taiwan and in 1929-1932 in the United
States.
In the equation of monetary growth upon output growth in Taiwan,
 x(t)=19.5+0.63  x(t-1)-0.413  x(t-2)-0.089(  M/M(t))- 8%)2
(2.98) (5.73)
(-4.03)
(-3.02)
+1.07(x(t)-9%)(  P/P(t))-3%)2+0.013(x(t)-9%)(  M/M(t))-8%)2
(5.20)
(3.10)
 2 =0.71;

 =0.69;
n=76; D.W.=2.20;
(3.2)
1st order autocorrelation =-0.10.
where, below the optimum inflation  P/P(t)=3%, as the growth rate of money
supply  M/M(t) decreases, output growth x(t) declines. Money growth, inflation,
and output growth are positively related.
The convergence probability is  2 =0.71. The speed of adjustment to
perturbations is  =0.63 and   1+ 1 does not always hold for output growth
 x(t), while 1 =10.30, where it is assumed that  x(t)/  x(t-1)=  and 
x(t)= 1 x(t).
Suppose  x(t)/  x(t-1)=  =1/2 denotes a half-life convergence probability
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of fifty percent after perturbations and suppose that  x(t)/  x(t-1)=  .
t
life probability implies
The half
0< lim 1 =  /2 <1
t t *
The cumulative convergence probability is p=  (x(t)/  x(t-1))=   1, as
t
t   . We estimate the convergence probability, p=exp(dx/dt), and the convergence
rate, x. The convergence time is
t={log(Y(t)/Y(0))}/x
for x*=- 1 /2  2 and Y(t)=Y(0)exp(xt).
p=exp(dx/dt)=  2 (t-t*)2=  2 t(- 1 /2  2 )= t(- 1 /2)
for 0  t  t*  x(t*)
where t=p(-2/ 1 )=(1/2)( -2/ 1 )=1/ 1 is an approximate half life, p=1/2, of a unit
shock when the state variable x(t) converges to the equilibrium x*, time t also
converges to t* but has no independent explanatory power.
For the half-life speed of adjustment, t =log(1/2)/   log(1/2)/log(1+ 1 )
= -0.69 /0.63  1 year. The monetary impact is lim 1 =  /2=2.00/2  1; and t=
t t *
1/ 1 =1 year is half life of an unit shock after monetary perturbations in the sense of
x  dlogY/dt  dlogM/dt, denoting the incentive-compatible equivalence. Our
convergence acceleration implies that
 x(t)/x(t-1)= 1 =10.30 and that delay time
is t=1/ 1 =1/10.30=0.1 a year and denotes a half life cycle of adjustment speed. Thus,
when the inflation falls below the optimal inflation, the monetary contraction is a
significant cause for depression with various delay times between one month and one
year.
5. Concluding Remarks
The contribution of this paper is an efficiency test of real exchange rate regimes
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through the second-order Lagrangian multiplier, nonlinear stability of sign, and
convergence probability. We test the stability of the general competitive equilibrium
in a continuous sense, when the equilibrium is
optimal policy
incentive-compatible with the
in according with the conditional convergence probability. The
first-order solution, positive, or negative, may be misleading, when inequality
constraints imply that resources are underutilized or potential and can be
Pareto-improving. The control variable also indicates the location. The minmax
solution is the equilibrium, where the (individual) firms minimize the welfare loss,
and when the (aggregate) central bank maximizes the social welfare. The positive ( or
negative) sign of the first-order coefficient of the welfare function is not instable but
denotes the maxmin or minmax solution.
When Lagrange uses the first-order negative coefficient to obtain the negative
solution, the Lagrangian multiplier is a show cost. We transform the Lagrange
multiplier into an equilibrium price, which comprise the first-order and the
second-order coefficients. We compare the equilibrium levels, which is attained by
various strategies under evaluation, when there do not coexist the positive and
negative impacts and the supply and demand curves. Beyond (or below) the
equilibrium, the central banks may contract (or expand) the money supply, when
output growth declined due to high or low inflation. The macro policy is
incentive-compatible with micro investment strategies, which otherwise may fail to
revert to the positive equilibrium or the fair value. In the concave domain, beyond (or
below) the equilibrium, the uncertainty of expansionary policy has a negative (or
positive) coefficient impact upon output. The performance is measured through
convergence probability, is indicated by the coefficient of determination and estimated
by
dynamic quadratic regression. An example illustrates the coexistence of the
(Ricardian) identity, which becomes the (non-Ricardian) inequality at disequilibrium
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of budget constraints.
Note:Consider the position vector of spiral waves which is a nonlinear equation:
x=F(x,u)={cos  (i,t)x+sin  (i,t)u}{-sin  (i,t)x+cos  (i,t)u}
where 0<  <90o denotes the tangent angle of the position vector F(x,u). x is
a
tangent vector; u is a normal vector. In linear models, the Hessian matrix H(x) implies
that when the second-order derivative
maximum, x*= min max F(x,u),
du
x
cofficient is negative, the solution x* is
if  2  2F(x(t),u(t))/  x2<0. In our nonlinear
models, the solution is a minmax one when there is the same sign for the first-order
coefficient of state variable and that third order coefficient of the policy. The solution
may be maximal or minimal; depending upon the second-order coefficient,
 2F(x(t),u(t))/  u2<0 or  2F(x(t),u(t))/  x  u>0. The same maximum is attained
according to the envelope theorem, when the time first-order derivative is positive,
 F(x(t),u(t))/  t >0, and  2F(x(t),u(t))/  t  logY=  F(x(t),u(t))/  x>0
if
dlogY/dt  x(t). Thus a common optimal policy holds for the output growth and for
output levels.
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