Foreign Exchange Reserves Demand Model Based on Chinese Government Utility Maximization and Analysis of Chinese Foreign Exchange Reserves *

At the end of 2010 China’s foreign exchange reserve reached $2847.34 billion, the author designs the maximum government utility function when consider the China government buys a part foreign exchange if company earns, it means that the China government will increase Ren-Min-Bi Yuan. And it will cause inflation. The inflation will cause disutility to government. Finally it gets the optimal fuction. VAR Regress finds the fitted value and actual value of foreign exchange reserves is nearly equal within 99.8%. The thesis gets the long term equilibrium relation of the nature logarithm of variables by VEC model, which are foreign exchange reserves, standard error of export, marginal propensity to import, the opportunity cost for foreign exchange reserves, marginal output to export. Using the sample datas in China 1980-2006 and VEC, we can find that 1) the government-holding foreign exchange reserves has positive correlation with the export standard error, 2) the government-holding foreign exchange reserves has positive correlation with the marginal propensity to import. The data and regression method are all different, but all have the positive correlation between the foreign exchange reserves and export standard error.


Introduction
This template, In order to study demand model of optimal foreign exchange reserves based on Chinese government utility maximization and analysis of Chinese foreign exchange reserves, the thesis arranges as the followings: the first section designed the demand model of optimal foreign exchange reserves based on Chinese government utility maximization, the second section analyzed the unit root test and cointegration of optimal foreign exchange reserves based on government utility maximization, the third section analyze VAR and VEC model of optimal foreign exchange reserves based on government utility maximization, the fourth section is the summary of this thesis.

Demand Model of Optimal Foreign Exchange Reserves Based on Chinese Government Utility Maximization
In order to study the demand model of optimal foreign exchange reserves based on Chinese government utility maximization, the thesis arrange as the followings: First, the authors analyzes the significance and objectives of the research, then review the literature simply.Based on Kelly, Michael G. (1970), consider the China government buy part foreign exchange as oreign exchange reserves if company earn oreign exchange, it means that the China government will increase supply of Ren-Min-Bi Yuan, and it will cause inflation.The inflation will cause disutility to government that is a negative government utility, combining with the outputting fluctuation, the foreign exchange reserves make the resource to be unused and lead to a negative social welfare, then design the following government utility function.

Literature Review
The Heinz Robert Heller (1966) constructed the demand of reserves.Heller's model is the beginning of cost revenue.After that time, many scholars have developed Heller's pattern and obtained their results.He pointed out that holding the reserves exist opportunity cost.The expression is the loss of investment benefit owing to holding reserves.When marginal cost equals to marginal proceeds，international reserve achieved the proper scope [1].Clark (1970) has extended Heller's method and developed a random pattern, considered the opportunity cost for holding foreign exchange reserves, then constructed the government utility function.He reached: 1) the random error item of standard error has positive correlated between foreign exchange reserves and optimal foreign exchange reserves; 2) It has negative correlated between optimal foreign exchange reserves and marginal propensity to import; 3) The opportunity cost for holding foreign exchange reserves more, the optimal foreign exchange reserves less [2].
Kelly.Michael G. (1970) thought that holding foreign exchange reserves has an opportunity cost.According to this, he constructed the utility function.At last, he achieved the optimal foreign exchange reserves function be means of the utility function.He found that there has positive correlated between the foreign exchange reserves holding by government and the standard error of export, in the same time has positive correlated with the average propensity to import [3].
Guobo Huang (1995) has collected the economic data about China during 1980 to 1990.He researched the international reserve scale of China by using of ECM and discovered that: 1) The Chinese foreign exchange reserve has the negative correlation with import, that is when the import increased the foreign exchange reserves will reduce; 2) It has negative correlation with average propensity to import, that is when the average propensity to import increased the foreign exchange reserves will reduce; 3) The Chinese government has an ability to adjust the foreign exchange reserves, but the ability will reduce without considering the net foreign exchange reserves of China Bank [4].
Yu Yongding (1997) found Chinese foreign exchange reserves are higher than reasonable and optimal foreign exchange reserves [5].Wu (1998) combined the ratio analysis method and the factor analysis method to study the determinant of reasonable Chinese foreign exchange reserves.He assumed that the Chinese demand foreign exchange reserve consists of four aspects: foreign exchange demanded for imports, foreign exchange demanded for repaying the total foreign debt balance, exchange demanded for profits return from foreign direct investment and foreign exchange reserves demand for the country's intervention in the foreign exchange currency market.He also established a linear equation model.The purpose of his thesis was to determine reasonable foreign exchange reserves for China, so he did not determine the equation's parameters with time-serial data or test the equation [6].
According to Xu (2001), the amount of currency in circulation imposes a more notable impact on the foreign exchange reserve in the short-term than does money and quasi-money.In addition, he also found that a long-term equilibrium relationship between the average propensity to import and the demand for foreign exchange reserves does not exist, yet the variation of a short-term average propensity to import exerts comparatively notable negative impacts on foreign exchange reserves.The reason is that under the current foreign exchange supervision system, an increase in imports means that a country would have to sell more foreign exchange, which would result in a decrease in the volume of foreign exchange reserves that are held [7].
Victoria miller (2006) support China to use foreign exchange reserves to save Chinese banks and Asian bank [8].M. Ramachandran (2006) finds the asymmetric control over capital flows and asymmetric intervention in favour of strengthening export competitiveness in an era of persistent capital inflows seem to be responsible for large stockpile of reserves in India [9].
Adnan Kasman and Duygu Ayhan (2008) find that exchange rate Granger causes foreign exchange reserves in the long-run nominal [10].
Victor Pontines and Ramkishen S. Rajan (2011) find that Asian central banks react more strongly to currency appreciations than depreciations and more to nominal effective exchange rates (NEERs) than to bilateral US dollar rates.This rationalizes the relative exchange rate stability and the sustained reserve accumulation in the region [11].

Kelly, Michael G. (1970) Design of Government Utility Function
Kelly, Michael G. (1970) designed the government util-ity function without considering the China government buys part foreign exchange if company earn, it means that the China government will increase Ren-Min-Bi Yuan, and it will cause inflation.The inflation will cause disutility to government.
Because the change of foreign exchange reserves equal to export increment subtract import increment, so where R is the foreign exchange reserves, X is the export, M is the import.Suppose that the reaction coefficient of import to export is ，where Y is income Because the marginal propensity to import is V is the variance in Equation (3); Equation (3) suggests that: the variance of foreign exchange reserves equal to the expectation of the squared of change amount of foreign exchange reserves Use the same arguments that: in Equation ( 4), where Y is income The government won't hope to run out the foreign exchange reserves, or we can say that the government hope keep lowest foreign exchange reserves R¹, so its target is to keep enough foreign exchange reserves as more as they can, to make it almost impossible that the foreign exchange reserves they kept lower than their target amount R¹, and satisfied the following Equation, e in Equation ( 5) is a very minor number, for any arbitrarily given and ruled probability density function, arbitrarily given e, educes which suggests that if the variance of foreign exchange reserves is greater, in order to keep a arbitrarily given probability e of the ruled foreign exchange reserves R¹, the average of foreign exchange reserves required is greater.
Suppose that probability e is positive with V(R), and is negative with where S(X) is the standard error of the export X From Equation (4) we can get Because keeping amount of foreign exchange reserve means the resource being left unused, so it may make output reduced, so the output that the relationship of the government whether keep foreign exchange reserve is, where Y¹ is the output of government does not hold foreign exchange reserve, Y is the output of government hold foreign exchange reserve, where R is foreign exchange reserve, i is the opportunity cost for holding foreign exchange reserves.Kelly, Michael G. (1970) designed the following government utility fuction.

Considered the Inflation Made by the Chinese Government Buy Part Foreign Exchange If Company Earn and Design the Government
Utility Function Based on Kelly, Michael G. (1970), consider the Chinese government buy part foreign exchange if company earn, it means that the Chinese government will increase supply of Ren-Min-Bi Yuan, and it will cause inflation.The inflation will cause disutility to government that is a negative government utility.
Combining with the outputting fluctuation, the foreign exchange reserves make the resource to be unused and lead to a negative social welfare, then design the following government utility function.
where β > 0 is a coefficient,  is the inflation,  is change amount of inflation or increasing amount.is the average fluctuation outputs when government does not hold foreign exchange reserve and hold foreign exchange reserve, is the fluctuation of outputs and average outputs.

Demand Model of Optimal Foreign Exchange Reserves Based on Chinese Government Utility Maximization
Choose variable E(R) and S(Y), and combine with the constraint condition Equation ( 8) and design the followinig F fuction.
From Equation (16) gets: Put Equation (18) in Equation (17 As a matter of convenience, arranges h as the followings From Equation ( 19) and Equation (20) gets Put Equation (21) in Equation ( 8) From Equation ( 22) gets Because of Conditions as above satisfied binary function approach maximum, that is the government expectation efficacy function E(U) exist a maximum, the optimal foreign exchange reserves based on government utility maximization could fixed by Equation (24).
Be convenient order, let

Analyze the Optimal Expectation Foreign Exchange Reserve to the Sensitivity of Variable and Parameter Based on Government Utility Maximization
The foreign exchange reserve which government expectation utility maximization E(U) exist a maximum is the optimal expectation foreign exchange reserve E(R), to the sensitivity that income to export, approach the partial derivative.
So the sensitivity ) that optimal foreign exchange reserve to reaction coefficient that import to export is depend on the size of f mg  , that is depend on the size of the reac- that import to export.The optimal foreign exchange reserve susceptibility , approach the partial derivative.
The optimal expectation foreign exchange reserve susceptibility to the export standard deviation S (X), approach the partial derivative, X1 is the export amount per year (Dimension of R, X2, S (X), M, dM is 100 million RMB Dollars) because , so: X2 is the export amount per year dX2 is the export increasing amount per year is the average value of X2 S(X) is the absolute value of the deviation between the average value of X2 and X2 The optimal expectation foreign exchange reserve susceptibility  to the opportunity cost for government holding foreign exchange reserves i, approach the partial derivative, dM is the import increasing amount per year , g is the reaction factor from income to export, where Y is income 2 i yt yr   is the opportunity cost for holding foreign exchange reserves, using the difference between profit rate and the reserve earnings yield before capital tax in China to compotator.

Brief Summaries
The thesis finishes the following researches: 1) The author designs the government utility function when consider the China government buy part foreign exchange if company earn, it means that the China government will increase Ren-Min-Bi yuan, and it will cause inflation.The inflation will cause disutility to government; 2) The thesis gets the optimal foreign exchange reserves by Maximum the government utility.
2 yt is the profit rate before capital tax in China.yr is the reserve earnings yield, using one year treasury bill rate-inflation rate in USA yr br ir   br is one-year Treasury bill yield rate in USA.ir is the inflation in USA; take the yearly average inflation in USA as the standard.
The value of m, i ,g be changed to the positive value when they are negative value by inserting reasonable value, so we use datas as Appendix (The principle is, if the column datas m, g are -0.001,-0.01, -1, -10, -100, then change to 0.1, 0.01, 0.001, 0.001, 0.0001.0.0001, 0.000001; if the column data i is -0.001, then change to 0.00001) This thesis takes the unit root test and cointegration analysis for demand variable optimal foreign exchange reserves based on government utility maximization.

About the Unit Root Test for Variable 3.1. About Data and Express and Process for Data
LNR, LNSX, LNM, LNI, LNG, these variables' defining in several is LNR = LOG (R); LNSX = LOG (SX) In order to discuss conveniently, the authors use as the following marks to express the different economic variable in this thesis.Dimension of R, X2, S (X), M, dM is 100 million US Dollar SX, m, i, g, from Figures 1(a) and  (b).We can see that LNR, LNSX, LNM, LNI, LNG these variables are very unstable.

R is the foreign exchange reserve in China
ER is equal to exchange rate The upper results express that there are exist 2 cointegration equations with a 5% significant level.

Brief Summarizes
Unit root test the nature logarithm of variables which the optimal foreign exchange reserves function include, the thesis find they are I(1),they exist co integration.3.
See from Table 3 the forecast error is very minor, from 0.02% to 0.1%, and the forecasting is highly accurate, from 99.98% to 99.99%.

VEC Model Estimation Results of Equilibrium Foreign Exchange Reserves
The upper equation express that, the elasticity that the Chinese foreign exchange reserves to export standard deviation (SX) is 8.396461487, it means that the export standard deviation increased 1%, the foreign exchange reserves will increase 8.396461487%.The elasticity that the Chinese foreign exchange reserves to the marginal propensity to import (m) is 6.531723289, it means that the marginal propensity to import increased 1%, the foreign exchange reserves will increase 6.531723289%.It can be explained that, the marginal propensity to import means the unit output increased will caused more import, so more reserves are necessary for the need of import.
The elasticity of the Chinese foreign exchange reserves to the net opportunity cost (i)1 is -9.319611202, it means that the net opportunity cost of hold foreign exchange reserves increased 1%; the Chinese foreign exchange reserves will reduce 9.319611202%.
The elasticity between the foreign exchange reserves and the reaction coefficient of income to export is -5.921618915, it means that the reaction coefficient of income to export increased 1%, the foreign exchange reserves will reduce 5.921618915.
It can be explained that, the reaction coefficient of income to export is g increased, means that the increased unit export will lead to the unit output increase, and then increased the export lead to the foreign exchange re-serves increase, so the government should cut the holding foreign exchange reserves to make the government utility maximization.The results of forecasts are shown in Table 4 and Figure 4.I analyzed the Table 4. to find that: 1) The values of forecasting foreign exchange reserves in long term equilibrium relation with either delay section K = 1 or K = 2 are the same; 2) The ratio R/FLR is equal to 1 express that the values of actual foreign exchange reserve and long-term equilibrium are the same.If it less than 1, it express that the actual value is less than the long-term equilibrium.If it more than 1, it express that the actual value is bigger than the long-term equilibrium; 3) From Table 4 we can find that the actual foreign exchange reserves bigger than the long-term equilibrium in 1982, 1985, 1987, 1990, 1995, 2002, 2006, and

Summarizes for This thesis
The thesis finishes the following researches: 1) The author designs the government utility function when consider the China government buy part foreign exchange if company earn, it means that the China government will increase Ren-Min-Bi yuan, and it will cause inflation.The inflation will cause disutility to government.
2) The thesis gets the optimal foreign exchange reserves by Maximum the government utility.
3) Unit root test the nature logarithm of variables which the optimal foreign exchange reserves function include, the thesis find they are I(1),they exist co integration.
4) VAR Regress finds the fitted value and actual value of foreign exchange reserves is nearly equal within 99.8%.
5) The thesis gets the long term equilibrium relation of the nature logarithm of variables by VEC model, which are foreign exchange reserves, standard error of export, marginal propensity to import, the opportunity cost for foreign exchange reserves, marginal output to export.
6) The Chinese actual foreign exchange reserves are bigger than the long-term equilibrium foreign exchange reserves in 1982,1985,1987,1990,1995,2002,2006.The Chinese actual foreign exchange reserves are more bigger than the long-term equilibrium foreign exchange reserves in 1982, 1990, 1995.They are nearly equal in 1994.The Chinese actual foreign exchange reserves are smaller than the long-term equilibrium foreign exchange reserves in 1981, 1986, 1999.The Chinese actual foreign exchange reserves are smaller than the long-term equilib-rium foreign exchange reserves in 1999 for weak world economy.
Different from the Kelly, Michael G. (1970) [3], Kelly, Michael G. (1970) designed the government utility function without considering the China government buy part foreign exchange if company earn, it means that the China government will increase Ren-Min-Bi yuan, and it will cause inflation.The inflation will cause disutility to government.Finally it gets the optimal fuction, we can find through the sample data about 46 countries from 1953 to 1965, (1) the government-holding foreign exchange reserves has positive correlation with the export standard error, ( 2) it has positive correlation with the average propensity to import [3].
This thesis designed the government utility function when consider the China government buy part foreign exchange if company earn, it means that the China government will increase Ren-Min-Bi yuan, and it will cause inflation.The inflation will cause disutility to government.Finally it gets the optimal fuction, using the sample datas in China 1980-2006 and VEC we can find: 1) The government-holding foreign exchange reserves has positive correlation with the export standard error; 2) It has positive correlation with the marginal propensity to import.The optimal fuction, data and regression method are all different, but all have the positive correlation between the foreign exchange reserves and export standard error, also is consonant with the results about the marginal propensity to import basically.
Different from Guobo Huang (1995), Guobo Huang (1995) collected the correlative economic datas from 1980 to 1990 in China, using ECM based on the quarterly datas to research the international reserves scale and discovered that: 1) The Chinese foreign exchange reserve has the negative correlation with import, that is when the import increased the foreign exchange reserves will reduce; 2) it has negative correlation with average propensity to import, that is when the average propensity to import increased the foreign exchange reserves will reduce.The thesis find that the elasticity that the Chinese foreign exchange reserves to the marginal propensity to import (m) is 6.531723289, it means that the marginal propensity to import increased 1%, the foreign exchange reserves will increase 6.531723289%.It can be explained that, the marginal propensity to import means the unit output increased will caused more import, so more reserves are necessary for the need of import.
Finally get the *This work was achievements of the current stage of 2010 Statistical Science key research program of China (No. 2010LB33) and 2010 talent to deepen teaching plan in Beijing University of Technology (01100054R6002).

Method 2 ,
Using SC to choose K value residual error, T is the sample capacity; k is the maximum delay section.The principle of choosing K value is made AIC value minimum through k value increasing.Because of the limit of sample T, K is equal to 4 at most, seeing from the change of AIC and SC, when K = 4, the values of AIC and SC are minimum.So K = 4 is the best choice with the limit of sample T. This section finishes the following researches: VAR and VEC Model Analysis of the equilibrium Foreign Exchange Reserves Demand Based on Chinese data between 1985 and 2006.

Figure 3 .
Figure 3.The forecasting values of FR and the actual values of R.

Figure 4 .
Figure 4.The values comparing between the forecasting foreign exchange reserves and Actual foreign exchange reserves in long term equilibrium relation with delay section K = 1 and K = 2.
it is steady basically in 1994.The value of actual foreign exchange reserves is much bigger than the long-term equilibrium in these years1982, 1990, 1995.Because the actual value is much bigger than the long-term equilibrium in 1990, The situation supply basis for Ren-Min-Bi exchange rate reform of China in 1994, Ren-Min-Bi exchange rate select one kind of exchange rate and supervisory floating exchange rate institution.Because the actual value is much bigger than the long-term equilibrium in 1995, the situation supply basis for Ren-Min-Bi free exchange under current account in 1996; 4)From Figure4we can see that the Chinese actual foreign exchange reserves are bigger than the long-term equilibrium foreign exchange reserves in1982, 1985, 1987, 1990, 1995,  2002, 2006.The Chinese actual foreign exchange reserves are bigger than the long-term equilibrium foreign exchange reserves in 1982, 1990, 1995.They are nearly equal in 1994.The Chinese actual foreign exchange reserves are smaller than the long-term equilibrium foreign exchange reserves in 1981, 1986, 1999.The Chinese actual foreign exchange reserves are smaller than the long-term equilibrium foreign exchange reserves in 1999 for weak world economy.This section chooses the VAR Regress finds the fitted value and actual value of foreign exchange reserves is nearly equal within 99.8%.Finally it gets the long term equilibrium relation of the nature logarithm of variables by VEC model, which are foreign exchange reserves, standard error of export, marginal propensity to import, the opportunity cost for foreign exchange reserves, marginal output to export.
and for the arbitrarily

Table 1
to analyze unit root ( In order to convenient for readers, shown M, Y2,

Unit Root Test and Cointegration Analysis for Demand Variable of Optimal Foreign Exchange Reserves Based on Government Utility Maximization
r   in Appendix Table 2.

;
The meaning of Symbol Y1,dY2 and so on are the same from Appendix Table1; Dimension of Y1 is 100 million Renminbi (RMB), Dimension of Y2,DY2 is 100 million US Dollar.