貨幣聯盟中最適的財政政策合作 - 政大學術集成

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(1)國立政治大學社會科學院經濟學系碩士論文 Department of Economics College of Social Sciences National Chengchi University Master Thesis. 政治大. 立貨幣聯盟中最適的財政政策合作. ‧ 國. 學. The optimal coordination of fiscal policy. ‧. in a monetary union. n. Ch. 朱詩閔. engchi. er. io. sit. y. Nat. al. i n U. v. Chu, Shih-Min. 指導教授：黃俞寧博士 Advisor: Hwang, Yu-Ning, Ph.D.. 中華民國 100 年 7 月 July, 2011.

(2) 謝辭在研究所學習的這兩年成長了很多，謝謝政治大學給我這個機會和環境。首先，很感謝黃俞寧老師在撰寫論文的期間，在我產生疑問時給予我耐心的指導，讓我對模型架構有更深入的了解，也在我遇到困難時給我一個溫暖的笑容鼓勵我繼續前進，真的很謝謝老師的指導。感謝口試評審委員台灣大學陳南光老師及政治大學蕭明福老師，您們給予論文許多寶貴的建議，提供我重新思考並且讓文章能更趨完整。感謝蕭翰屏、林銘峰、賴柏勳同學在我撰寫論文時給予我的幫助及支持。感. 政治大學生涯能充滿快樂的回憶。也很感謝大學、高中及國中的好友陪著我一起經歷研立謝慈恬、家瑋、淳雅、芳倩、月雲等許多碩班同學的陪伴與照顧，讓我碩士的求. ‧ 國. 學. 究所這兩年的苦與樂。你們的陪伴給我很大的支持，讓我在疲憊的時候能夠有力量繼續往前邁進，謝謝你們。. ‧. 最後我要感謝我的家人，謝謝媽媽和阿公、阿嬤的鼓勵，謝謝爸爸一直以來. sit. y. Nat. 對我的支持，謝謝你們幫助我不斷的成長，讓我人生的每個階段都很美麗。. al. er. io. 能夠這樣一步步的完成一篇論文，真的很感謝大家對我的鼓勵與支持，僅將. n. 此論文獻給關心我的家人及朋友。. Ch. engchi. i n U. v. 朱詩閔謹誌政治大學經濟學系碩士班中華民國一百年七月. i.

(3) 中文摘要本研究目的是在動態隨機一般均衡模型中，討論在貨幣聯盟中，一個中央的財政政府面對衝擊時如何反應。我們根據 Gali and Monacelli (2008)的架構並加入一個基金機制來模擬會員國間的財政合作。此基金機制設定為有一中央財政政府向各會員國收取固定的基金費用並將此基金費用全部重新分配給各會員國，故基金在每一期都會結清。在這樣的設定下，聯盟的財政合作和個別國家政府面對波動時的反應相同。. 政治大. 關鍵詞：DSGE、貨幣聯盟、財政合作. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i n U. v.

(4) Abstract The objective of this paper is to investigate how the central fiscal authority copes with shocks in a monetary union with a dynamic stochastic general equilibrium (DSGE) model. We follow the framework of Gali and Monacelli (2008) and set a fund mechanism to simulate one cast of fiscal coordination among member countries. The central fiscal authority raises the constant fund payment from all member countries and redistributes it to member states, so the budget of the transfer is balanced in each period. Under our design of fund mechanism, we find that this cast of fiscal. 政治大. coordination plays the same rule as the government sector.. 立. Keywords: DSGE, monetary union, fiscal coordination. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. iii. i n U. v.

(5) Contents 1.. Introduction………………………………………………………………………1 1.1. Motivation…………………………………………………………………..1 1.2. Literature review……………………………………………………………2. 2.. The model………………………………………………………………………...4 2.1. Goods market……………………………………………………………….4 2.2. Household…………………………………………………………………...6. 政治大 Technology……………………………………………………….9 立. 2.3. Firm…………………………………………………………………………9 2.3.1.. Price setting……………………………………………………10. 學. ‧ 國. 2.3.2.. 2.4. Government and fiscal coordination………………………………………10 Equilibrium dynamics…………………………………………………………..12. ‧. 3.. sit. y. Nat. 3.1. Market clearing condition…………………………………………………12. The social planner’s problem…………………………………………………...15. al. v i n The policy tradeoffs in theC currency union underU h e n g c h i sticky prices………………...17 n. 5.. io. 4.. er. 3.2. Inflation dynamics………………………………………………………....14. 5.1. Union member’s tradeoffs…………………………………………………18. 5.2. Union-wide tradeoffs………………………………………………………19 6.. The optimal monetary and fiscal policies in the currency union……………….20. 7.. Conclusion………………………………………………………………………23. Reference……………………………………………………………………………..24 Appendix……………………………………………………………………………..25. iv.

(6) 1. Introduction. 1.1. Motivation. The US subprime mortgage crisis that was caused by high-risk mortgage loans in 2007 brought about the global depression. This serious crisis was spread quickly around the world followed by the insolvency of Iceland in the European monetary. 政治大 economic fundamentals and raised the uncertainty of the EMU. The bankruptcy of 立. union (EMU). The credit problems in Iceland stripped off people’s confidence in its. Iceland makes us deliberate on the shortcomings of the EMU. In lack of the individual. ‧ 國. 學. monetary policy, the member countries can only rely on the fiscal policy to fight for. ‧. the recession. Whether the currency union requires further coordination of the fiscal. sit. y. Nat. policy becomes the core of the concerns.. io. er. At present, the EMU has a common monetary authority that targets on the union-wide inflation, but each member decides their own fiscal policy without. al. n. v i n C h countries whichUlose the monetary policy as a coordination. Therefore, the member engchi tool of economy stabilization can only use fiscal policy to against the impact of a. worldwide depression. This situation causes members to raise a large amount of debt and then lead their solvency to exacerbate rapidly. Because they have a common currency, the insolvency problems will influence all the EMU countries like domino. Hence, the certain fiscal deterioration in a depression circumstance is the upmost topic in the European monetary union. To this result, we consider the fiscal coordination of a central fiscal authority of the currency union to examine the effects of fiscal coordination in a currency union. Therefore, the objective of this paper is to investigate how the central fiscal 1.

(7) authority copes with shocks in a monetary union with a dynamic stochastic general equilibrium (DSGE) model whether the fund mechanism is successful to reduce the impacts of shock. In accordance with Robio and Roldan (2003) which built an explicit form of fiscal coordination to simulate its effect in a monetary union by using the data of the EMU, we set a cast of fund mechanism into a monetary union to simulate a fiscal coordination among member states. Gali and Monacelli (2008) modeled the currency union as a continuum of small open economies in a micro-founded model in the presence of nominal rigidity. They. 政治大 respectively. We follow their framework and embed a fund mechanism in the union . 立. evaluate the role of monetary and fiscal policies in the country and union level, 1. Under the fund mechanism, there is a central fiscal authority that raises the. ‧ 國. 學. constant fund payment from all member countries and redistributes it to member. ‧. states. Hence, the local governments can focus on their provision of public goods and. sit. y. Nat. the fluctuation of output could be absorbed by the net transfer. We find that the net. io. coordination is not necessary for a monetary union.. n. al. 1.2. Literature review. Ch. engchi. er. transfer only substitutes some parts of local government spending. This kind of fiscal. i n U. v. Several papers have discussed the influence of monetary and fiscal policies together in a currency union. Beesta and Jensen (2005) addressed the interaction between monetary and fiscal policies in a two-country, micro-founded model of a monetary union with sticky prices. They discussed the mechanism of the fiscal stabilization policy and commitment and identify the gains of it. Their results 1. We refer to Chatterjee and Turnovsky (2005) developed the public capital fashion that accumulates from two sources: constant government expenditure and transfer from the rest of the world, for the fund transfer between the fiscal authority and individual countries. 2.

(8) suggested that there were not trivial gains from the fiscal stabilization and commitment. Sánchez (2010) also discussed the interplay of monetary and fiscal policies in a multinational currency union and focused on the free-rider problems which lead to the failure of the fiscal coordination. Therefore, he suggested that the fiscal budget considerations as the Stability and Growth Pact could against the fluctuations caused by shocks. In recent years, several literatures focused on the fiscal coordination among countries in a monetary union. For instance, Kirsanova, Satchi and Vines (2004). 政治大 provide macroeconomic stabilization in the face of asymmetric shocks in a monetary 立 pointed out that an active fiscal policy could significantly improve welfare and. union. Oros (2008) proved that no matter the demand or supply shocks, the fiscal. ‧ 國. 學. coordination was an optimal shock absorber in a closed monetary union with a. ‧. two-country, static Keynesian model. Valeria and Pompeo (2010) found that if. sit. y. Nat. member countries were hit by asymmetric shocks, the fiscal cooperation and. io. er. state-contingency were necessary to obtain an efficient outcome.. Robio and Roldan (2003) used the change in the unemployment rate as an. al. n. v i n C hthe monthly data U indicator to redistribute funds with of 11 countries in EMU showed engchi. that the insurance mechanism would lead a higher stabilizing effect the more asymmetric was the shock. Gali and Monacelli (2008) assumed a common policymaker to determine the monetary and fiscal policies together in the union-wide. Therefore, the common policymaker was not only a central bank but also a central fiscal authority. Under their investigation, the fiscal policy had a country-specific stabilization role in the equilibrium. Some empirical studies have formed the specific fashion of fiscal coordination and discuss the effects in the European Union. But there is no discussion of the form 3.

(9) of fiscal coordination in DSGE model. Therefore, we want to capture the stabilized function of fiscal coordination in a monetary union. We include a fund mechanism in a union with a DSGE model to discuss how the central fiscal authority works in a monetary union. The rest of this paper is structured as follows. In Section 2, we depict the model specifically. In Section 3, we derive the market clearing condition and have the dynamic equilibrium for output and inflation in country level and union-wide, respectively. Then, in Section 4, we discuss the social planner’s problems under. 政治大 5. In Section 6, we obtain the optimal monetary and fiscal policies and the path of the 立. flexible prices. We show the tradeoffs between policies under sticky prices in Section. fund mechanism. Finally, Section 7 concludes.. ‧. ‧ 國. 學. 2. Model. sit. y. Nat. io. er. 2.1. Goods market. al. n. v i n C hthe EMU, the majority In the currency union like of the countries are small engchi U. relative to the whole union. The fiscal policy which is determined independently by each country itself has little influence on the other countries in the currency union. So we follow the framework of Gali and Monacelli (2008) to consider the currency union as a continuum of small open economies represented by the unit interval, which is indexed by i ∈ [0,1] . Later in Section 2.4, we set a fund mechanism into a monetary union so that each country gets the net transfer from the central fiscal authority. For simplicity, the member countries in the currency union are symmetric. The household consumes the domestic goods, Cii,t and imported goods, CFi ,t from 4.

(10) the rest of the member countries in the union.. (Cii,t )1−α ( CFi ,t ). α. C ≡ i t. (1). (1 − α )(1−α ) α α. where α ∈ [0,1] is the weight of imported goods in the composite consumption and also represents the openness of country i . Cii,t and CFi ,t are the country i ’s consumption of domestic goods and imported goods given by CES function as follow: 1. Cii,t ≡ ( ∫ Cii,t ( j ). ∈−1 ∈. 0. ∈. dj )∈−1. (2). where j ∈ [0,1] denotes the type of goods. ∈> 1 is the elasticity of substitution between. 政治大. the different type of goods. 1. 立 C ≡(. 0. i f ,t. 1. ∫C 0. i f ,t. ( j). ∈−1 ∈. dj ). ∈ ∈−1. (3). 學. ≡ exp ∫ c df ; i f ,t. ‧ 國. C. i F ,t. where c if ,t = log C if ,t and f ∈ [0,1] . CFi ,t is the country i ’s consumption of goods that. ‧. produced in the rest of countries in the union. Minimizing expenditure, we can obtain. where Pt ≡ ( ∫ Pt ( j ) 0. i. 1−∈. al. i f ,t. n. 1. i. Pt f ( j ) −∈ i C ( j ) = ( f ) C f ,t Pt. dj ). 1 1−∈. sit. i i ,t. Ch. and Pt. f. (∫ n ( jc ) h Pg dji) ≡e 1. 0. f. 1−∈. t. er. io. Pt i ( j ) −∈ i C ( j ) = ( i ) Ci ,t ; Pt. y. Nat. the demand function for domestic goods and imported goods of good j as:. i n U. 1 1−∈. v. (4). denote the price indexes of the. good j which is produced in country i and country f , respectively. Then, the Eq. (4) can be rewritten as follow:. ∫. 1. 0. Pt i ( j )Cii,t ( j )dj =Pt i Cii,t ;. ∫. 1. 0. Pt f ( j )C ff ,t ( j )dj = Pt f C ff ,t. (5) 1. Then, the optimal allocation of imported goods is ∫ Pt f C if ,t df = Pt *CFi ,t , where 0. 1. Pt * ≡ exp ∫ ptf df . Pt * is the union-wide price index. 0. We define the consumer price index (CPI) of country i is Pci,t ≡ ( Pt i )1−α ( Pt * )α , then we can obtain the optimal allocation of expenditures of domestic and imported goods 5.

(11) as follows:. Pt i C if ,t= (1 − α) Pci,t Cti. Pt *CFi ,t = α Pci,t Cti. ;. (6). Combining Eq. (5) and Eq. (6), we have the optimal allocation of expenditures in country i as:. ∫. 1. 0. Pt i ( j )Cii,t ( j )dj + ∫. 1 1. ∫. 0 0. Pt f ( j )Ct f ( j )djdf =Pt i Cii,t + Pt *CFi ,t =Pci,t Cti. (7). 2.2. Household. 政治大. The infinitely-lived representative household in country i maximizes the. 立. discounted sum of utilities as follow:. ‧ 國. 學. ∞. E0 ∑ β tU (Cti , N ti , Gti ) t =0. (8). ‧. where β ∈ (0,1) is the subjective discount factor of the representative household. We. sit. y. Nat. assume the government consumption has impact on the household’s utility. Especially,. io. n. al. er. Gti in our study includes the constant provision of public goods and the net transfer. i n U. v. from the central fiscal authority. Namely, the household’s utility is also influenced by the fund mechanism.. Ch. engchi. The form of period utility based on consumption, government spending and labor is given by. U (Cti , N ti , Gti ) = (1 − χ ) log Cti + χ log Gti −. ( N ti )1+ϕ 1+ ϕ. (9). where Cit is the individual consumption and parameter χ ∈ [0,1) is the share of government spending relative to individual consumption in the utility. Notice that, the government spending, Gti , contains the provision of public and the net transfer. So, the net transfer also impacts on the household’s utility. N ti is the labor supply of the 6.

(12) representative household. ϕ > 0 is the elasticity of the marginal disutility of labor supply. The budget constraint of the representative household is formed as follow:. ∫. 1. 0. Pt i ( j )Cii,t ( j )dj + ∫. 1 1. ∫. 0 0. Pt f ( j )C if ,t djdf + Et {Qt ,t +1 Dti+1} ≤ Dti + Wt i N ti − Tt i. (10). where Dti denotes the payoff of the portfolio that the representative household held at the end of period t . Qt ,t +1 is the stochastic discounted factor. By using an optimal allocation of expenditure as Eq. (7), the household’s budget constraint becomes: (11) 政治大 Maximizing the household’s 立 discounted sum of utility, Eq. (9) which is subjected. Pci,t Cti + Et {Qt ,t +1 Dti+1} ≤ Dti + Wt i N ti − Tt i. ‧ 國. 學. to Eq. (11), we obtain the first-order conditions as below: Wt i Pci,t. ‧. Cti ( N ti )ϕ= (1 − χ ). Nat. n. al. (13). er. io. sit. y. Pci,t Cti β ( i )( i ) = Qt ,t +1 Ct +1 Pc ,t +1. (12). i n U. v. The household would purchase the riskless bond which the nominal return is. Rt* =. Ch. engchi. 1 . Thus, we can derive the Euler equation from Eq. (13) to: Et {Qt ,t +1}. β Rt* Et {(. Pci,t Cti )( )} = 1 Cti+1 Pci,t +1. (14). Taking log-linearization on Eq. (12) and Eq. (14), we can rewrite them as below:. wti − pci ,t =cti + ϕ nti − log(1 − χ ). (15). = cti Et {cti+1} − (rt* − Et {π ti+1} − ρ ). (16). The bilateral terms of trade, S. i f ,t. Pt f = i , is defined as a ratio of the price of a Pt 7.

(13) bundle of goods produced in country f in terms of the goods in country i . Accordingly, the effective terms of trade of country i is given by 1 1 Pt * exp ∫ ( ptf − pti )df = exp ∫ s if ,t df S =i = 0 0 Pt i t. (17) 1. where s if ,t = log S if ,t and we also have sti = ∫ s if ,t df . 0. We can see the relationship between the CPI and the domestic price level by the definition of the terms of trade as Pci,t = Pt i ( Sti )α , or in logs as follow:. pci= pti + α sti ,t. (18). 政治大 By defining the inflation is π= p − p , we can derive the domestic inflation 立 i t. i t. i t −1. ‧ 國. π ci= π ti + α sti ,t. 學. from Eq. (18) as below:. (19). ‧. The CPI inflation is influenced by the domestic inflation and the change of the. Nat. sit. y. terms of trade. According to Eq. (19), we find that the parameter α which denotes the. n. al. er. io. openness of country i has the positive effect on the CPI inflation. That is, the opener the country i is, the more CPI inflations changed.. Ch. engchi. i n U. v. Under the symmetric assumption, we obtain the Euler equation of country f from Eq. (13) as β (. Pc ,ft Ct f )( ) = Qt ,t +1 . Combining these two equations, we derive the Ct f+1 Pc ,ft +1. international risk sharing condition:. Cti = ϑi Ct f ( S if ,t )1−α. (20). where ϑi is the constant coefficient that depends on initial conditions. We assume initial conditions are symmetric so that we have ϑ= ϑ= 1 . Integrating Eq. (20) over i f. f in logs on this condition, we have: 8.

(14) cti = ct* + (1 − α ) sti. (21). 1. where ct* = ∫ ctf df denotes the aggregate union-wide consumption index. We find that 0. the terms of trade has less influence when country i is opener.. 2.3. Firm. 2.3.1. Technology We assume that each firm only hires labors N ti ( j ) to produce good j in country i. 政治大 so that the production function is given by 立. ‧ 國. 學. Yt i ( j ) = Ati N ti ( j ). (22). where i, j ∈ [0,1] and Ati is a country-specific technology shock which we assume it. ‧. follows the AR(1) process in logs = as ati ρ a ati−1 + ε ti , where ati = log Ati and ε ti is a white. er. io. sit. y. Nat. noise.. Under the assumption of technology, the real marginal cost of goods produced by. n. al. firms in country i is:. MCti =. (1 − τ )Wt Ati Pt i i. Ch. engchi. i n U. v. i. (23). where τ i is the subsidy of employment to the firms. 1. We define Yt ≡ [ ∫ Yt ( j ) i. 0. i. ∈−1 ∈. ∈ ∈−1. dj ]. is the aggregate output index of country i . The. total amount of labor hired to produce domestic goods in country i is given by. Yt i Z ti = N ∫= N ( j )dj 0 Ati i t. 1. i t. (24). 9.

(15) Yt i ( j ) dj . According to Gali and Monacelli (2008) Appendix A, we can 0 Yi t. where Z ti ≡ ∫. 1. derive the relationship between the output level and the aggregate labor in logs as follows: i y= ati + nti t. (25). 2.3.2. Price setting. 政治大 possibility, (1 − θ ) , that firm 立j resets its price. Accordingly, we can also image that a. Following the staggered price-setting structure of Calvo (1983) model, there is a. ‧ 國. 學. proportion of (1 − θ ) firms reset their price and θ firms keep their price the same as i. before in each period. Let P t presents the price which firm j resets in period t. Firm j. ‧. i. adjusts its price P t in period t by maximizing its discounted profit that is given by. y. Nat. ∞. max ∑θ k Et {Qt +k [Yt i+k ( Pt − MCti+k )]} i. sit. n. al. which is subjected to the demand constraints of good j :. Yt i+ k ( j ) ≤ (. i t. Ch. P −∈ i ) [Ci ,t + k ( j ) + CiF,t + k ( j )] i Pt + k. engchi. (26). er. k =0. io. Pt. i. i n U. v. (27). We can derive the optimal price setting (in log) of country i in period t on the first-order condition as follow: ∞. p t = µ + (1 − βθ )∑ ( βθ ) k Et {mcti+ k + pti+ k } i. (28). k =0. where µ = log. ∈ is the log of optimal markup. ∈ −1. 10.

(16) 2.4. Government and the fiscal coordination. In our study, we try to investigate how the fund mechanism works in the currency union. To this end, we assume that there is a central fiscal authority that raise the constant fund payment, Ft from all member countries and redistributes all of the receipts to member states, Oti . We assume that all the members pay the same amount to the fund in period t so that the difference of the net transfer among member countries only depends on the gains from fund, Oti . Thus, the net transfer from the. 政治大. central fiscal authority to each member:. 立. i TR= Oti − Ft t. (29). ‧ 國. 學. where TRti denotes the country i ‘s net transfer. To the central fiscal authority, the. ‧. 1. budget should be balanced in each period, Ft = ∫ Sti di . 0. y. Nat. io. sit. Following Chatterjee and Turnovsky (2005), we assumed that the public. n. al. er. expenditure for each country is accumulated from two sources: the constant. i n U. v. government spending which is financed domestically by the lump sum tax and the net. Ch. engchi. transfer received from the central fiscal authority. We let the government spending stable to prevent the country from bankruptcy like Iceland, so we have the government spending G constant. Thus, the government consumption fluctuation caused by shocks entirely responds to the net transfer. We can see how the fund mechanism operates in the country i as follow,. Gti = G + Oti − Ft = G + TRti. (30). The budget constraint of the government sector of country i is:. G + Ft = Tt i + Oti. (31) 11.

(17) The government consumption index of country i is given by CES function as follow: 1. Gti = ( ∫ Gti ( j ). ∈−1 ∈. 0. ∈. dj )∈−1. (32). where Gti ( j ) is the quantity of good j that country i ’s government domestic purchases. Minimizing the government expenditure, we can obtain the optimal government spending of good j in country i :. Gti ( j ) = (. Pt i ( j ) −∈ i ) Gt Pt i. (33). 3. Equilibrium dynamics. 政治大. 立. ‧ 國. 學. 3.1. Market clearing condition. ‧ y. Nat. sit. The market clearing condition for good j in country i is given by 1. al. n. 0. er. io. Yt i ( j ) = Cii,t ( j ) + ∫ Ci ,ft ( j )d +fGti ( j ). i n U. f 1 Pc ,t Pci,t i Pt ( j ) −∈ = ( i ) [(1 − α )( i )Ct + α ∫ ( i )Ct f df + Gti ] 0 P Pt Pt t i. i. Ch. engchi. v. 1 P ( j) = ( t i ) −∈[(1 − α )( Sti )α Cti + α ( Sti )α ∫ ( Sti )1−α Ct f df + G + TRti ] 0 Pt. =(. (34). Pt i ( j ) i i α )[Ct ( St ) + G + TRti ] Pt i. and based on Eq. (20), this condition holds for all i, j ∈ [0,1] in each period. Thus, we obtain the function of aggregate goods in terms of the consumption, the terms of trade and the net transfer in country i as follow:. = Yt i Cti ( Sti )α + G + TRti. (35). We make use of a first-order Taylor expansion around steady state to obtain the 12.

(18) market clearing condition in log-linear. i  yti = (1 − γ − φ )(cti − α sti ) + φ tr t. (36). where "  " denotes the deviation of log variables from their steady state values which. G TR Yt i  i are denoted without time subscript, e.g. yt ≡ log i = yt − y i and γ ≡ and φ ≡ Y Y Y denote the government spending share and the net transfer share in steady state value, respectively. We can rewrite the domestic output equation Eq. (36) by using Eq. (21) and the definition of the terms of trade as follow:. 政治大.  i + (1 − γ − φ )c* − (1 − γ − φ )( p i − p* ) i yt= φ tr t t t t. 立. (37). The domestic output is positively related to the net transfers and the union-wide. ‧ 國. 學. consumption. If the union-wide consumption is more than its steady state value, the. ‧. output of the country i will increase. We also find that the output is negatively related. y. Nat. to the domestic prices and positively to the union-wide prices. We integrate the. * t. al. n. y = φ tr  + (1 − γ − φ )c t *. er. io. union level as below:. sit. domestic output, Eq. (37), over i ∈ [0,1] to have the market clearing equation in the. * t. Ch. engchi. i n U. v. (38). i * 1 i  *t = 1 tr where y t ≡ ∫ y t di and tr ∫  t di . In this case, the budget of the transfer is 0. 0. balanced in each period. Taking the difference equation of Eq. (38) and combining it with the integration of Eq. (16), the dynamic IS equation is given by y* E { y* } − (1 − γ − φ )(r * − E {π * } − ρ ) − φ E {tr  *t +1} = t t t t +1 t t t +1. (39). 1. where π t*+1 = ∫ π ti+1di .We assume the deviation of output and net transfer in the union 0.  *t +T } lim{ y* } 0 so that we can solve the Eq. would close in the infinite, i.e. lim{ = tr = t +T T →∞. T →∞. 13.

(19) (39) as below: ∞. y*= φ tr  *t − (1 − γ − φ ) E {r * − π * − ρ } ∑ t t +k t +k +1 t. (40). k =0. Thereby, we find that both of the union-wide transfers and the expected long-term rate have effects on output in the union level. The intensity of those effects depends on the steady state share of government spending and net transfers.. 3.2. Inflation dynamics. 政治大 domestically dynamic inflation 立 equation in terms of the marginal cost by making use Under the staggered price-setting of Calvo (1983) model, we can obtain the. ‧ 國. 學. of the difference to Eq. (28) as below:  it = π ti β Et {π ti+1} + λ mc. ‧. (1 − βθ )(1 − θ )  i . mct is the deviation of the real marginal cost from its steady. y. Nat. θ. sit. where λ ≡. (41). io. er. state value that is defined below.. We can derive the real marginal cost equation by combining Eq. (16) with the Eq.. n. al. (23) in logs as follow:. Ch. engchi. i n U. v. mc = w − p − a + log(1 − τ ) = ( w − pci ,t ) + ( pci ,t − pti ) − ati + log(1 − τ i ) i t. i t. i t. i t. i. i t. (42). Combining Eq. (25) and Eq. (36) with Eq. (42) in Taylor expansion, we can derive the deviation of the real marginal cost from its steady state is in terms of output and net transfer. i φ  it ( 1  it − (1 + ϕ )a i = mc + ϕ ) y t − tr t 1− γ −φ 1− γ −φ. (43). Substituting Eq. (43) into Eq. (41), we obtain the new Keynesian Phillips curve (NKPC) as a function of the net transfers in the country level:. 14.

(20) i λφ  i 1 π ti β Et {π ti+1} + λ ( = + ϕ ) y t − tr t − λ (1 + ϕ )ati 1− γ −φ 1− γ −φ. (44). By integrating the country’s NKPC over i ∈ [0,1] , we obtain the new Keynesian Phillips curve of the union: * λφ  * 1 π t* β Et {π t*+1} + λ ( = + ϕ ) y t − tr t − λ (1 + ϕ )at* 1− γ −φ 1− γ −φ. (45). We can find that no matter in the country level or in the union level, if the inflation is stable under the common monetary policy, the shocks were offset by the net transfer in order to stabilize the output level. In this section, we obtain the dynamic equilibrium equations of output and. 政治大. inflation in the country level, Eq. (37) and Eq. (44) and in the union level, Eq. (40). 立. and Eq. (45) in this section. The difference between Gali and Monacelli (2008) and. ‧ 國. 學. our study is the setting of government sector. Therefore, the equilibrium dynamics of the output and the inflation in our study are the functions of the net transfer instead of. ‧. the government spending in Gali and Monacelli (2008).. io. sit. y. Nat. n. al. er. 4. The social planner’s problem. Ch. engchi. i n U. v. In this section, we want to discuss what the efficient allocation is to the social planner as a whole union under flexible prices (i.e., there is no market power distortion). The social planner seeks to maximize the utility of the whole union as follow: 1. max ∫ U (Cti , N ti , Gti )di. (46). 0. which is subjected to the technology and resource constraints: Yt i = Ati N ti. (47). 1. Yt i =+ Cii,t ∫ Ci ,ft df + Gti 0. Under the preference formed as Eq. (9), we obtain the optimal first-order 15.

(21) condition of labor, consumption and government spending as below: ( N ti )ϕ (1 − χ )(1 − α ) = = Ati Cii,t. 1 (1 − χ )α χ = ∫0 Ci,ft df Gti. (48). We assume N ti = 1 to simplify. Then, the optimal conditions, Eq. (48) can be rewritten as:. Yt i = Ati. (49). Cii,t =− (1 χ )(1 − α ) Ati. (50). 政治大. Ci ,ft= (1 − χ )α Ati. 立. Gti = χ Ati. (51) (52). ‧ 國. 學. Substituting Eq. (50) and Eq. (51) into Eq. (1), we can have the optimal consumption for the social planner as follow:. ‧. Cti= (1 − χ )( Ati )1−α ( At* )α. y. Nat. 1. sit er. al. n. We let the”. io. where At* = ∫ Ati di is the union-wide supply shocks. 0. (53). i n U. v. ”present the efficient equilibrium under flexible prices. The. Ch. engchi. following conditions must be satisfied to obtain the efficient allocation with no distortion. i ∈ −1 = mct ∈. (1 − τ i ) i i ϕ i α = i C t ( N t ) (S t ) At (1 − χ ) i. i. (1 − τ i ) i i ϕ Y t − G − TR t = i ) Ct (N t ) ( i At (1 − χ ) Ct. (54). i. i 1−τ i G + TR t = (1 − ( ))( N t )1+ϕ i 1− χ Yt. Substitute Eq. (49), Eq. (50) and Eq. (52) into Eq. (54), we find that the subsidy 16.

(22) must set a level as follow to eliminate the market power distortion: i. τ =. 1 ∈. (55). Based on the previous conditions, Eq. (54) and Eq. (55), we obtain the country-specific optimal allocation for all i ∈ [0,1] as follow: i. Nt =1. (56). i. Y t = Ati. (57). i. G t = χ Ati. (58). i. C t= (1 − χ )( Ati )1−α ( At* )α. 立. 政治大. (59). By using Eq. (58), we can find that the government spending equilibrium in the. ‧ 國. 學. country level is affected by country-specific shocks. Under our setting of the fund mechanism, the expenditure of public goods is constant so that the shocks influence. ‧. on the net transfer instead.. y. Nat. n. al. er. io. government spending is:. sit. According to Eq. (57) and Eq. (58), we can derive the constant weight of. i. i. Gt. G TR t == χ + i =+ γ φ i i Yt Yt Yt. Ch. engchi. i n U. v. (60). In the union level, we obtain the optimal allocation by integrating Eq. (49), Eq. (52) and Eq. (53) over i ∈ [0,1] as below: *. Y t = At*. (61). *. C t= (1 − χ ) At*. (62). *. G t = χ At*. (63). We find that the efficient variables in the union level are affected by the union-wide shocks. In the next section, we discuss the variables under sticky prices 17.

(23) based on the flexible price equilibrium.. 5. The policy tradeoffs in the currency union under sticky prices. In this section, we discuss the tradeoffs under sticky prices. The tradeoffs arise for two reasons here: (1) the sticky prices and (2) the common currency. In the presence of the staggered price setting following Calvo (1983), the sluggish price adjustment leads the variables to deviate from their efficient level under flexible price. 政治大 the monetary union use the common currency. This situation implies that the nominal 立. equilibrium we derive in the last section. On the other hand, the member countries in. exchange rate is fixed and the terms of trade is unable to adjust for achieving the. ‧ 國. 學. efficient level. Because of these two reasons, the policymakers have to make choices. ‧. depending on the country-specific and the union-wide tradeoffs, respectively. Then,. er. io. sit. y. Nat. we derive the tradeoffs in terms of the fiscal gap and the output gap.. 5.1. Union members’ tradeoffs. n. al. Ch. engchi. i n U. v. Letting the”  ”denotes the variables’ log deviation from their flexible prices i i equilibrium, i.e., x t ≡ xti − x t , for all variables, x . We define the fiscal gap as follow:. f i ≡ tr  it − y i t t. (64).  it and y i are the transfer gap and the output gap of the country i . where the tr t i. i. i. Under the fact that y t − y i = g t − g i = tr t − tr i = ati and Eq. (43), we can have the real marginal cost in terms of output gap and fiscal gap:  it = ( 1 + ϕ ) y i − φ tr  it = ( 1 + φ + ϕ ) y i − φ f i mc t t t 1− χ 1− χ 1− χ 1− χ 18. (65).

(24) We can rewrite the inflation dynamic equation, Eq. (41), by substituting the Eq. (65) with the real marginal cost term as follow: i 1+ φ λφ  i = π ti β Et {π ti+1} + λ ( + ϕ ) y t − ft 1− χ 1− χ. (66). By this new Keynesian Phillips curve for each union member, we can find that if the inflation is stabilized by the union-wide monetary policy, then the fluctuation of the output gap must reflect on the fiscal gap (i.e., the net transfer would cope with the fluctuation of output ). Notice that, the local government's public spending is constant so that only the net transfer from the fund mechanism responds to the output. 政治大 Combining the differential 立 market clearing conditions, Eq. (37) and Eq. (38) and. fluctuation 2.. ‧ 國. 學. substituting Eq. (6) and Eq. (17) with consumption, we can derive that the change in output gap is in terms of the change in fiscal gap, inflation and shocks.. ‧. io. sit. Nat. * i 1− χ φ )( f t − f t ) − ( )(π ti − π t* ) − (ati −at* ) = ( 1−φ 1−φ. y. i *  it −tr  *t ) − (1 − γ − φ )(π i − π * )  y t − y t =−(1 − φ )(ati −at* ) + φ (tr t t. (67). al. er. where the “ ∆ ” denotes xt − xt −1 , for all variables x . Depending on Eq. (67), if there are. n. v i n C hthe inflation or theUfiscal gap will respond to the asymmetric shocks in the union, engchi. shocks for keeping the output stable.. 5.2. Union-wide tradeoffs. We can obtain the new Keynesian Phillips curve in the union level by integrating Eq. (65) over i ∈ [0,1] as * 1+ φ λφ  * = π t* β Et {π t*+1} + λ ( + ϕ ) y t − ft 1− χ 1− χ. 2. (68). In contrast with Gali and Monacelli (2008), the output gap responds to the the government spending. 19.

(25) We derive the union-wide output gap function by the dynamic equation, Eq. (39):  *t +1} y* E { y* } − (1 − χ )(r * − E {π * } − rr *t ) − φ E {tr = t t t t +1 t t t +1. (69). *. where rr t is the union’s natural rate of interest as follow, * * * * * 1 ( Et { y t +1} − φ Et {tr t +1}) = ρ+ ρ + Et { y t +1} = ρ + Et {a t +1} rr t = (1 − χ ) * * According to the outcome that we obtain in the next section, y= π= 0 , we can t t. *. find the rule of central bank is as= rt rr t +ν π π t* . By combining Eq. (68) and Eq. (69), we can have the interaction of fiscal gap and inflation as:. 立. 政治大. ‧ 國. 學. ∞ * * 1+ φ λφ 1 + φ [ = π t* β Et {π t*+1} − λ ( + ϕ )∑ Et (rt*+ k − π t*+ k +1 − rr t + k ) + − (1 − ϕ )] f t 1− χ 1− χ 1− χ k =0 ∞ ∞ * * λφ 1 + φ λ 1+ φ [ ( − (1 − ϕ )]∑ Et β k { f t + k } − + ϕ )∑ (1 − β k +1 ) Et {rt*+ k − π t*+ k +1 − rr t + k } 1− χ 1− χ 1− β 1− χ k 0= k 0 =. =. ‧. According to the interaction among the fiscal gap, the inflation and the interest. sit. y. Nat. rate determined by the central bank, we can find that the union-wide inflation rises. n. al. er. io. when the fiscal gap is positive in order to against the depression. Then, the central. i n U. v. bank of union needs to dampen the inflation by means of raising the interest rate in. Ch. engchi. the future. The contractionary monetary policy leads to a decline in output so that the common fiscal authority would run a positive fiscal policy again. Thereby, we discuss the optimal monetary and fiscal policy of a currency union simultaneously in next section.. 6. The optimal monetary and fiscal policies in the currency union. In this section, we have a common policymaker who determines the optimal monetary and fiscal policy together in a currency union. The objective of this 20.

(26) policymaker is to minimize the average utility loss function of the whole union. We use the second order approximation of the utility formed in Eq. (9) to obtain the loss function as follow:. = W. i i 1 ∞ t 1 λ i 2 φ β ∫ ( (π t ) + (1 + ϕ )( y t ) 2 + ( f t ) 2 )di ∑ 0 ∈ 2 t =0 (1 − χ ). (70). Then, we choose the optimal fiscal policy { f t } and the optimal monetary policy i. i {rt*} which can be determined by { y t , π ti } by minimizing Eq. (70) which is subjected to. Eq. (66), Eq. (67) and the aggregate constraints given by. 政治大. 1 1 i 1 i * * π t* = ∫ π ti di ; y t = ∫ y t di ; f t = ∫ f t di 0. 0. 0. 立. (71). where we let the Lagrange multipliers {ψ πi ,t ,ψ yi ,t ,ψ π* ,t ,ψ *y ,t ,ψ *f ,t } associate with the. ‧ 國. 學. constraints in Eq. (66), Eq. (67) and Eq. (71) for all i ∈ [0,1] and t = 0,1, 2...... . We obtain the first-order conditions as follow: 1− χ i )ψ y ,t −ψ π* ,t = 0 1−φ. π ti +ψ πi ,t + (. sit. φ i λφ φ ( )f t +( )ψ πi ,t − ( )(1 − β L−1 )ψ yi ,t +ψ *f ,t = 0 1− χ 1− χ 1−φ. n.  1− χ  i *  ψ y ,t = ψ π ,t  1−φ . Ch. engchi. er. io. i 1+ φ + ϕ )ψ πi ,t + (1 − β L−1 )ψ yi ,t −ψ π* ,t = (1 + ϕ ) y t − λ ( 0 1− χ. al. (72). y. Nat. λ. ‧. ∈. i n U. v. (73) (74). (75). (1 − β L−1 )ψ yi ,t = ψ *y ,t. (76). φ. ψ *f ,t ( )(1 − β L−1 )ψ yi ,t = 1−φ. (77). Integrating Eq. (72), and then substituting it with Eq. (75), we obtain ∈. λ. 1. π t* + ∫ ψ πi ,t di = 0. (78). 0. Integrating Eq. (73), and then substituting it with Eq. (76), we obtain. 21.

(27) *  1+ φ  * (1 + ϕ ) y t = λ ψ π ,t  1− χ . (79). Integrating Eq. (74), and then substituting it with Eq. (77), we obtain f * + λψ * = 0 π ,t t. (80). We can derive the relationship between the fiscal gap and the output gap of the union by integrating Eq. (74) and combining it with Eq. (77) as * 1− χ f * =−(1 + ϕ )[ ] y t t 1 + φ + ϕ (1 − χ ). (81). Using the dynamic IS-type equation, Eq. (69) and the optimal condition, Eq. (78),. 政治大. Eq. (79), Eq. (80) and Eq. (81), the equilibrium under the optimal policy will satisfy:. 立. * * * f= y= π= 0 t t t. (82). ‧ 國. 學. for all t . As this result, we find that the inflation, output gap and fiscal gap in the. ‧. union level are zero which means the variables will be in their efficient flexible-price. sit. y. Nat. equilibrium value in the union level. By the outcome of Eq. (82), we have the rule of *. n. al. er. io. monetary policy, = rt rr t +ν π π t* as we discussed in section 5.2.. i n U. v. We can derive how the fund mechanism works in a country by combining Eqs. (73), (74), (76) and (77):. Ch. f i λ[ 1 + φ + (1 + ϕ )]ψ i − (1 + ϕ ) y i = π ,t t t 1− χ. engchi. (83). By using the definition of the fiscal gap, Eq. (64), we derive the transfer's rule as:  it λ[ 1 + φ + (1 + ϕ )]ψ i − ϕ y i = tr π ,t t 1− χ. (84). If there is an undesirable shock which causes the output gap to be negative, the net transfer will be positive to reduce the fluctuation of output. Thus, the fund mechanism is countercyclical. Further, we can find that in the country level, i i y= f= 0 cannot be equilibrium when the prices are less fully flexible, ψ πi ,t > 0 . t t. 22.

(28) Gali and Monacelli (2008) consider the monetary and fiscal policies in the union level simultaneously by a singular policymaker. Under this cast of fiscal coordination, the output’s fluctuation entirely affects on the government spending so that the local government still faces the shocks by themselves. In our model, the local government's public expenditure is constant so that the fluctuation which is caused by productive shocks is absorbed by the central fund mechanism. If we consider the multiplier term, 1+ φ + (1 + ϕ )]ψ πi ,t as given, the path of the fiscal gap and the output gap is the same 1− χ. λ[. as the result of Gali and Monacelli (2008). Consequently, we can find that the net. 政治大 cast of fiscal coordination plays 立 the same rule as the government sector.. transfer only substitutes some parts of government spending in country level. This. ‧. ‧ 國. 學. 7. Conclusion. sit. y. Nat. In this paper, we investigate the fiscal coordination in a monetary union with. n. al. er. io. micro-founded dynamic stochastic general equilibrium (DSGE) model. We follow the. i n U. v. framework of Gali and Monacelli (2008) and set a fund mechanism to simulate a. Ch. engchi. fiscal coordination under staggered price-setting. Every local government can get a net transfer from the fund and the central fiscal authority redistributes all receipts to every member state in each period. We find that if we have a common policymaker to determined monetary and fiscal policies together, the net transfer only substitutes parts of the government spending. Hence, under our design of fund mechanism which the budget of transfer is balanced in each period, this type of fiscal coordination plays the same rule as the government sector. If we set the receipts do not entirely transfer to the member countries, the fund mechanism will be inefficient. We conclude this paper by providing some other interesting issues for future 23.

(29) researches. The fund mechanism can also be considered in infinite period rather than balanced in each period. Or we can design the other form of fund, such as the net transfer is based on a proportion of output.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 24. i n U. v.

(30) Reference. Bajo-Rubio, O. and C. Díaz-Roldán (2003). Insurance mechanisms against asymmetric shocks in a monetary union a proposal with an application to EMU. Recherches économiques de Louvain, 69, 73-96.. Beetsma, R.M.W.J. and H. Jensen (2005). Monetary and fiscal policy interactions in a micro-founded model of a monetary union. Journal of International Economics, 320-352. 立. 政治大. Calvo, G. (1983). Staggered prices in a utility-maximizing framework. Journal of. ‧ 國. 學. Monetary Economics, 12(3), 383-398. ‧. sit. y. Nat. Chatterjee, S. and S.J. Turnovsky (2005). Financing public investment through. io. International Economics, 13(1), 20-44. n. al. Ch. engchi. er. foreign aid: Consequences for economic growth and welfare. Review of. i n U. v. Gali, J. and T. Monacelli (2008). Optimal monetary and fiscal policy in a currency union. Journal of International Economics, 76(1), 116-32. Gali, J. and T. Monacelli (2005). Monetary policy and exchange rate volatility in a small open economy. Review of Economic Studies 72 (3), 707-734. Kirsanova, T., M. Satchi and D. Vines (2004). Monetary union: Fiscal stabilization in the face of asymmetric shocks. CEPR Discussion Paper (4433) 25.

(31) Oros, C. (2008). Macroeconomic stabilization in a heterogeneous monetary union: some insights into the effects of fiscal policy coordination". Economics Bulletin, 5(34), 1-12. Traficante, G. (2010). Fiscal issues in a monetary union. Rivista Bancaria-Minerva Bancaria, 66(3), 5-27. Valeria, D.B. and D.P. Pompeo (2010). On the coordination of national fiscal policies. 政治大. in a monetary union. Economia Internazionale/International Economics, 63(3), 273-96. 立. ‧. ‧ 國. 學. Appendix. sit. y. Nat. We want to rewrite the utility function to loss function in terms of output gap,. io. 1. al. n. U t ≡ ∫ U (Cti , Gti , N ti )di. er. fiscal gap and inflation. We have the form of utility function as follow,. 0. 1. 1. 0. 0. C( Nh) en di g c h i. = (1 − χ ) ∫ Cti di + χ ∫ Gti di + ∫. i 1+ϕ t. 1. 1+ ϕ. 0. i n U. v. Next we derive the function of consumption, labor supply and government in second order approximation, respectively. 1. By using Eq. (34) and the fact, ∫ sti di = 0 , we can derive the union-wide Taylor 0. expansion of consumption as follow:. ∫. 1. 0. log= Cti di. =. ∫. 1. 0. i log(Yt i − G = t ) di. ∫. 1. 0. i log(Yt i − G − TR = t ) di. 1. ∫ log[(1 − γ − φ )Y ]di 0. Yt i − Y TRti − TR 1 φ log[(1 γ φ ) ] ( )( ) ( )( ) − − + − Y ∫0 1− γ −φ 1− γ −φ Y TR 1. 26. i. t.

(32) Yi −Y 2 TR i − TR 2 Yt i − Y TRti − TR φ 1 1 2φ )2 ( t ) +( )2 ( t ) − ( )( )]di − [( 2 1− γ −φ 1− γ −φ (1 − γ − φ ) 2 Y TR Y TR =. 1 φ y i − φ tr  ti ) − 1 [  ti − y i ) 2 }di {log[(1 γ φ ) ] ( )( ]( − − Y + tr t t ∫0 1− γ −φ 2 (1 − γ − φ ) 2 1 i 1 φ  ti ) − 1 [  ti − y i ) 2 + tips}di )( y t − φ tr ](tr  ∫ {( t 2 0 1− γ −φ 2 (1 − γ − φ ) 1. where γ ≡. G TR and φ ≡ denote the government spending share and the net transfer Y Y. share in steady state value, respectively. The union-wide Taylor expansion of government spending is given by 1. 1. G )di ∫ log(G + TR )di ∫ log(= i t. 0. i t. 0. 1. [log(G + TR) + (. φ. 政治大. ‧ 國. 立. )(. The union-wide Taylor expansion of labor is given by. 學. TRti − TR 1 φ 2 TRti − TR 2 )+ ( ) ( ) ]di ∫0 2 γ +φ γ +φ TR TR 1 1 φ i 1 φ 2 i 2 φ i )tr t + ( ) (tr t ) = ]di ∫ [( )tr t + tips]di = ∫ [lo G +gTR) + ( 0 0 γ +φ 2 γ +φ γ +φ. =. i i ( N ti )1+ϕ 1 i ϕ + N ti1+ϕ (n t + (n t ) 2 ) + N ti1+ϕ (n t ) 2 ]d 1+ ϕ 2 2. i. y. 0. [. sit. 1. Nat. ∫. =. ‧. 1+ϕ 1 (N ) ( N ti )1+ϕ ϕ N −N 2 1+ϕ N − N t ∫0 1 + ϕ di = ∫0 ( 1 + ϕ + ( Nt ) ( t N ) + 2 ( t N ) )di 1. io. [. n. al. er. i i N ti1+ϕ 1 i1+ϕ 2 i1+ϕ ∫0 1 + ϕ + Nt n t + 2 Nt (1 + ϕ )(n t ) ]di i i 1 1  ∫ (n t + (1 + ϕ )(n t ) 2 + tips )di 0 2 According to Eq. (24) and the lemma 1 in Gali and Monacelli (2008), 1. =. zt . Ch. engchi. i n U. v. ∈ var j { pt ( j )} , we can substitute the output into the labor as follow: 2. i i i n t = y t − ati + zti = y t + zti. Then, we obtain the union-wide Taylor expansion of labor in terms of output gap. 1 ( N ti )1+ϕ y i + z i + 1 (1 + ϕ )( y i ) 2 + tips )di (  di t ∫0 1 + ϕ ∫0 t t 2 1. According to the second order approximation of consumption, government spending and labor, we can rewrite the union-wide utility function as: 1. U t ≡ ∫ U (Cti , Gti , N ti )di 0. 27.

(33) ( N ti )1+ϕ = (1 − χ ) ∫ C di + χ ∫ G di + ∫ di 0 0 0 1+ ϕ 1 i  ti )di − 1 ( φ ) 1 (tr  ti − y i ) 2 di +φ 1 tr  ti di  ∫ ( y t − φ tr t ∫ ∫ 0 0 2 1− χ 0 1 i i 1 − ∫ [ y t + zti + (1 + ϕ )( y t ) 2 ]di + tips 0 2 1 i 1 1 φ  it − y i ) 2 ]di + tips = − ∫ [ zti + (1 + ϕ )( y t ) 2 + ( )(tr t 0 2 2 1− χ 1. 1. i t. 1. i t. Depending on lemma 2 shown in Gali and Monacelli (2008),. 1 ∈ ∞ t i 2 t i = β z β (π t ) , we can rewrite the discounted sum of utilities to the loss ∑ t λ∑ 2 t 0= = t 0 ∞. function as below: 1 ∞. ∞ i i i t t t 0 =t 0=t 0. 1. ∈. β ∫ [ (治 π ) + (1 + ϕ )( y ) ∑ β U (C , G , N )di =− 2 ∑ λ 政 t. t. 1. 0. i 2 t. 立. 大. i 2 t. +. φ 1− χ. 學 ‧. ‧ 國 io. sit. y. Nat. n. al. er. Wt = ∫. Ch. engchi. 28. i n U. v. ( ft i ) 2 ]di.

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