A Two-Sector Model of an Open Economy. B. Essama-Nssah Poverty Reduction Group (PRMPR) The World Bank October 16, 2005 Module 2

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1 A Two-Sector Moel of an Open Economy B. Essama-Nssah Poverty Reuction Group (PRMPR) The Worl Bank October 6, 2005 Moule 2

2 Outline Structure Assumptions Analytical Expression Numerical Implementation Closure Calibration Valiation Exercise 2

3 Assumptions A small open economy Cannot influence its terms of trae with rest of the worl. Two sectors of prouction: Sector prouces an export goo not sol omestically. Sector 2 prouces a omestic goo use for both intermeiate an final consumption. 3

4 Assumptions The eman for primary factors of prouction, labor an capital, is a consequence of profit maximization (or cost minimization) by firms in each sector subect to technological constraints escribe by Cobb-Douglas prouction function. The eman for intermeiate inputs is proportional to the level of output. The intermeiate goo is a composite goo mae up of omestic goo an imports. 4

5 Assumptions Two representative househols Rural househol represents 60 percent of the population, an owns a fraction θ RL of labor an a fraction θ RK of capital Urban househol represents 40 percent of the population, an owns a fraction (- θ RK ) of capital an a fraction (- θ RL ) of labor. 5

6 Assumptions Two representative househols Each househol spens its income on a composite goo (mae of omestic an imports) so as to maximize utility (or minimize expeniture). This income inclues factor income, transfer from the government an from the rest of the worl. Imperfect substitutability between omestic goo an imports. 6

7 Assumptions Government revenue inclues an inirect tax on omestic sales of the omestic goo, an tariffs on importe final an intermeiate goos. It balances the buget by re istributing its income to consumers. Factors an goos market are perfectly competitive. Full employment prevails. 7

8 Assumptions The above moel is a moifie version of a stylize general equilibrium moel evelope by Devaraan, Lewis an Robinson (990). It can be interprete as an extene version the Salter-Swan moel which provies a founation for the stuy of the impact of macroeconomic imbalances an austment policies on the real sector of a small open economy. 8

9 Analytical Expression Cobb-Douglas technology X i α i α li AK ki L ; α + α ; i e,. i i ki li Deman for labor li ( PVAi X i ) Li α ; i w Deman for capital e,. ki( PVAX i i) Ki α ; i r e,. 9

10 Analytical Expression Net proucer price PVAi PXi a2 ipq2, i e, a 2i is the amount of aggregate intermeiate goo (Q 2 ) per unit of output in sector i. Proucer price of exports PX R( + e t e ) π e 0

11 Analytical Expression Armington commoity aggregation ( for final an 2 for intermeiate) Import eman functions Deman for omestic component of the aggregation [ ] 2, ; ) ( + D M B Q s ρ ρ ρ β β [ ] 2, ; ) ( ) ( ) ( ) ( + Q PM PD PM B M s β β β [ ] 2, ; ) ( ) ( ) ( ) ( ) ( + Q PD PD PM B D s β β β

12 Analytical Expression Elasticity of substitution, + ρ, 2 Domestic price of imports m PM R( + tm ) π ;, 2 Price of omestic sales PD PX ( + tx );, 2 Proucer price of omestic goo PX 2 X PD ( + D tx ) ;, 2 2

13 Analytical Expression Price of composite goos PQ s ( PDD + PMM )/ Q ;, 2 Income of each househol inclues a fraction of all labor an capital income, a fraction of government revenue, an a share of the balance of payment in local currency. Yh θ ( wls) + θ ( rks) + θ Y + θ RS ; h r, u hl hk hg G hf f 3

14 Analytical Expression Government revenue exclues export subsiies Y G 2 tm m ( Rπ M ) + tx πe Househol eman for final goo Total eman for intermeiate goo (by proucers) X Yh Q h ; h r, u; Q Q PQ Q2 a2i X i; i e,. i h te( R h X e ) 4

15 5 Analytical Expression Home goo market equilibrium Material balance for composite goos Factor market equilibrium Balance of payments 2 ;, Q Q s 2 2, ; D X i i i i e i K KS L LS, ; ; m f e e M S X + 2 π π

16 Analytical Expression Note factor eman uner cost minimization L i αli αki r w α ki ( X / A); i e, i i K i αki αli w r α li ( X / A); i e, i i 6

17 Numerical Implementation Closure Use of EViews requires a one-to-one mapping between equations an enogenous variables. The process of etermining which variables to make enogenous an which exogenous is known generally as closure. 7

18 Numerical Implementation In the Walrasian framework, only relative prices matter. If we multiply all prices (the price of omestic goo, the exchange rate an returns to factors of prouction) by a constant, all nominal values will be multiplie by the same constant an real variables will remain unchange. Hence the system is sai to be homogeneous of egree zero. 8

19 Numerical Implementation We can therefore choose a numéraire goo an fix its price exogenously. We select the exchange rate an make it exogenous. The small country assumption implies that worl prices are exogenous. Finally, we make the balance of trae exogenous as well. 9

20 Numerical Implementation The moel also respects Walras law: if all economic agents satisfy their buget constraints an all, but one, markets are in equilibrium, then the last market must also be in equilibrium. Alternatively, if all markets are in equilibrium an all, but one, buget constraints are bining, then the last buget constraint is bining as well (Dinwiy an Teal 988). 20

21 Numerical Implementation Hence, we can rop one equilibrium conition from the system an it will still solve. We rop the balance of trae equation by assigning to it a ummy enogenous calle WALRAS. This ummy must equal zero when the law is satisfie by a solution of the moel. 2

22 Numerical Implementation Calibration To make the moel fully computable we resort to calibration. This process solves the relevant equations for values of the parameters that are consistent with base year ata. 22

23 Numerical Implementation Base Year SAM Export Domestic Final Intermeiate Labor Capital Rural Urban Worl Total Export Domestic Final Intermeiate Labor Capital Rural Househol Urban Househol Worl Total Source: Aapte from Devaraan, Lewis an Robinson (990) 23

24 Numerical Implementation Calibration For the parameters of the Cobb-Douglas function: wli αli ; αki ( αli); i e,. PVAX an A X i i αki αli Ki Li i i ; i e,. 24

25 Numerical Implementation Calibration Parameters of the Armington aggregation ρ β + PD PM Q D B [ ] ρ ρ β M + ( β ) D Q ρ Calibrate Parameters for the Two-Sector Moel α L α K A β M β D B Export Domestic Final Intermeiate

26 Numerical Implementation Calibration Distribution of factor income in base year SAM: θ RL 0.70, θ UL 0.30, θ RK 0.0, an θ UK 0.90 Distribution of government transfers: θ RG 0.60 an θ UG 0.40 Distribution of foreign transfers: θ RF 0.20 an θ UF

27 Numerical Implementation Source Text for Calibration alphal_xpt (WR * LD_xpt) / VA_xpt alphak_xpt - alphal_xpt a_xpt XS_xpt / (LD_xpt^alphal_xpt * KD_xpt^alphak_xpt ) alphal_mg (WR * LD_mg) / VA_mg alphak_mg - alphal_mg a_mg XS_mg / (LD_mg^alphal_mg * KD_mg^alphak_mg ) betam_fnl / ( + ( PD_fnl / PM_fnl) * (XM_fnl / XD_fnl)^(- / sigma_fnl) ) beta_fnl ( - betam_fnl) ces_fnl XQ_fnl / ( betam_fnl * XM_fnl^(-rho_fnl) + beta_fnl * XD_fnl^(-rho_fnl) )^(- / rho_fnl) betam_ntr / ( + ( PD_ntr / PM_ntr) * (XM_ntr / XD_ntr)^(- / sigma_ntr) ) beta_ntr ( - betam_ntr) ces_ntr XQ_ntr / ( betam_ntr * XM_ntr^(-rho_ntr) + beta_ntr * XD_ntr^(-rho_ntr) )^(- / rho_ntr) 27

28 Numerical Implementation Source Coe for the Main Moel XS_xpt a_xpt * ( (LD_xpt^alphal_xpt) * (KD_xpt^alphak_xpt) ) XS_mg a_mg * ( (LD_mg^alphal_mg) * (KD_mg^alphak_mg) ) LD_xpt * WR (alphal_xpt * PVA_xpt * XS_xpt) KD_xpt * RK (alphak_xpt * PVA_xpt * XS_xpt) LD_mg * WR (alphal_mg * PVA_mg * XS_mg) KD_mg * RK (alphak_mg * PVA_mg * XS_mg) XQ_ntr io_ntr_xpt * XS_xpt + io_ntr_mg * XS_mg PVA_xpt PX_xpt - io_ntr_xpt * PQ_ntr PVA_mg PX_mg - io_ntr_mg * PQ_ntr PX_xpt EXR * ( + te_xpt) * PWE_xpt PX_mg * ( + tx) * XS_mg PD_fnl * XD_fnl + PD_ntr * XD_ntr XM_fnl (XQ_fnl / ces_fnl) * (betam_fnl / PM_fnl )^sigma_fnl * ( betam_fnl^sigma_fnl * ( PM_fnl)^( - sigma_fnl) + beta_fnl^sigma_fnl * (PD_fnl)^( - sigma_fnl) )^(sigma_fnl / ( - sigma_fnl)) XD_fnl (XQ_fnl / ces_fnl) * (beta_fnl / PD_fnl )^sigma_fnl * ( betam_fnl^sigma_fnl * ( PM_fnl)^( - sigma_fnl) + beta_fnl^sigma_fnl * (PD_fnl)^( - sigma_fnl) )^(sigma_fnl / ( - sigma_fnl)) PM_fnl EXR * ( + tm_fnl) * PWM_fnl XM_ntr (XQ_ntr / ces_ntr) * (betam_ntr / PM_ntr )^sigma_ntr * ( betam_ntr^sigma_ntr * ( PM_ntr)^( - sigma_ntr) + beta_ntr^sigma_ntr * (PD_ntr)^( - sigma_ntr) )^(sigma_ntr / ( - sigma_ntr)) XD_ntr (XQ_ntr / ces_ntr) * (beta_ntr / PD_ntr )^sigma_ntr * ( betam_ntr^sigma_ntr * ( PM_ntr)^( - sigma_ntr) + beta_ntr^sigma_ntr * (PD_ntr)^( - sigma_ntr) )^(sigma_ntr / ( - sigma_ntr)) 28

29 Numerical Implementation Source Coe for the Main Moel PM_ntr EXR * ( + tm_ntr) * PWM_ntr PD_fnl PX_mg * ( + tx) PQ_fnl * XQ_fnl PD_fnl * XD_fnl + PM_fnl * XM_fnl PD_ntr PX_mg * ( + tx) PQ_ntr * XQ_ntr PD_ntr * XD_ntr + PM_ntr * XM_ntr Y_rhh shlab_rhh * (WR * LS) + shcap_rhh * (RK * KS) + tr_rhh_gov * Y_gov + tr_rhh_row * SAV_row XQ_fnl_rhh * PQ_fnl Y_rhh Y_uhh shlab_uhh * (WR * LS) + shcap_uhh * (RK * KS) + tr_uhh_gov * Y_gov + tr_uhh_row * SAV_row XQ_fnl_uhh * PQ_fnl Y_uhh Y_gov tx * PX_mg * XS_mg + tm_fnl * EXR * PWM_fnl * XM_fnl + tm_ntr * EXR * PWM_ntr * XM_ntr WALRAS EXR * PWM_fnl * XM_fnl + EXR * PWM_ntr * XM_ntr - EXR * PWE_xpt * XS_xpt - EXR * SAV_row XQ_fnl XQ_fnl_rhh + XQ_fnl_uhh RK * KS RK * KD_xpt + RK * KD_mg WR * LS WR * LD_xpt + WR * LD_mg 29

30 Numerical Implementation Valiation Once a general equilibrium moel has been mae computable through the process of calibration, an be fore it is use for any policy analysis, it is important to conuct a series of valiation tests esigne to show that the moel is behaving the way it is suppose to behave. 30

31 Numerical Implementation Valiation Moel behavior epens on the unerlying theory. Given the principles of optimization an equilibrium unerlying this simple Walrasian moel, we can perform two basic tests: homogeneity an Walras law. 3

32 Numerical Implementation Valiation Multiplying the numéraire price, here the exchange rate by 2, will ouble all nominal values while leaving all real values unchange. Finally, the equation that is roppe from the system because of Walras law must hol a posteriori (i.e. after a solution). Alternatively, the ummy variable WARLAS must equal zero. This guaranties that there are no leakages. 32

33 Exercise De Melo an Robinson (989) Present the ata in table 2 as a SAM. Use the ata to calibrate the following analytical expression which is an interpretation of a simple one-sector moel with symmetric prouct ifferentiation for both imports an exports. 33

34 34 Exercise CET prouction possibility frontier Where the constant elasticity of transformation is: Export supply [ ] φ φ φ γ γ + e e x s X X A X + φ ω ( ) ) ( ) / ( ω ω ω ω ω ω ω γ γ γ + e e e e x s e P P P A X X

35 35 Exercise Domestic sales Domestic price of export GDP eflator ( ) ) ( ) / ( ω ω ω ω ω ω ω γ γ γ + e e x s P P P A X X e e R P π ( ) e e s x X P P X X P +

36 36 Exercise Armington aggregation Deman for imports Deman for omestic goo [ ] ρ ρ ρ β β + x m m q s D Q B Q ( ) ) ( ) / ( β β β + m m m m q s m P P P B Q Q ( ) ) ( ) / ( β β β + m m q s x P P P B Q D

37 Exercise Domestic price of imports P m Rπ m Total expeniture on the composite consumption goo P Q + q s ( Pm Qm P Dx Househol income Y P X + RS h x s f ) 37

38 Exercise Deman for composite consumption goo Q Y P Goos market equilibrium D x Balance of trae h q Q Q X s π Q π X S m m e e f 38

39 Exercise Assume full employment an use the consumer price inex as a numéraire price. Calibration Results for the Core Moel of an Open Economy obs OMEGA SIGMA BETAD BETAM CES CET GAMMAD GAMMAE PHI RHO

40 Exercise Simulate the welfare an structural impact of a 0 percent increase in foreign transfers Welfare an Structural Implications of a 0% Increase in Foreign Transfers Omega Sigma EXR PD PX RPD RPX XD XE XM XQ

41 References e Melo Jaime an Robinson Sherman Prouct Differentiation an the Treatment of Foreign Trae in Computable General Equilibrium Moels of Small Economies. Journal of International Economics, Vol. 27: Devaraan, Shantayanan, Jeffrey D. Lewis, an Sherman Robinson Policy Lessons from Two-sector Moels. Journal of Policy Moeling 2 (4): Dinwiy, C.L. an F.J. Teal The Two- Sector General Equilibrium Moel: A new Approach. Oxfor: Philip Allan. 4

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