EXERCISES IN GENERAL EQUILIBRIUM MODELING USING GAMS

Transcription

1 EXERCISES IN GENERAL EQUILIBRIUM MODELING USING GAMS by Hans Löfgren International Food Poliy Researh Institute Washington, D.C. Draft Revised August 24, 1999 Adapted at TIPS Marh 8, Modified by Dirk van Seventer and Rob Davies. The main modifiation has been to break up the original exerise 3 into 3A and 3B. We would like to thank Hans Lofgren for allowing us to work from his original text.

2 TABLE OF CONTENTS INTRODUCTION AND OVERVIEW 3 EXERCISE 1: A SIMPLE WALRASIAN MODEL 6 INTRODUCTION 6 VERBAL MODEL PRESENTATION 6 MATHEMATICAL MODEL STATEMENT 7 DATA BASE 9 TASK 10 HINTS AND SUGGESTIONS 10 APPENDIX: EXERCISE A1 14 INTRODUCTION 14 TASK 14 HINTS 14 EXERCISE 2: INTRODUCING INTERMEDIATE DEMAND 16 INTRODUCTION 16 DATA BASE 16 TASKS 16 HINTS AND SUGGESTIONS 17 EXERCISE 3: INTRODUCING THE SAVINGS-INVESTMENT BALANCE AND ACTIVITY SPECIFIC WAGES 18 INTRODUCTION 18 EXERCISE 3A: INTRODUCING THE SAVINGS-INVESTMENT BALANCE 18 INTRODUCTION 18 DATA BASE 18 TASKS 19 HINTS AND SUGGESTIONS 20 EXERCISE 3B: INTRODUCING ACTIVITY SPECIFIC WAGES AND UNEMPLOYMENT 21 INTRODUCTION 21 DATA BASE 21 TASKS 21 HINTS AND SUGGESTIONS 22 EXERCISE 4: INTRODUCING GOVERNMENT 24 INTRODUCTION 24 DATA BASE 24 TASKS 25 HINTS AND SUGGESTIONS 26 EXERCISE 5: INTRODUCING THE REST OF THE WORLD 29 INTRODUCTION 29 DATABASE 29 TASKS 42 HINTS AND SUGGESTIONS 42 REFERENCES 44

3 INTRODUCTION AND OVERVIEW 2 Computable General Equilibrium (CGE) models are a lass of eonomy-wide models that are widely used for poliy analysis. The term "omputable" refers to the fat that the model solution an be omputed, a prerequisite when a model is used for applied purposes. A general equilibrium model expliitly reognizes that an exogenous hange (in poliy or from some other soure, for example world markets) that has an impat on any one part of the eonomy an give rise to reperussions throughout the system. The use of a CGE model is preferable when these reperussions are essential in assessing the impat of these shoks. Mathematially, a standard CGE model onsists of a set of simultaneous nonlinear equations. Eonomially, its starting point is Walras' neolassial world. However, CGE models used for applied poliy analysis, inluding food poliy, tend to deviate onsiderably from this starting point, inorporating a relatively large amount of detailed real-world struture. CGE models have been used to analyze the impat of poliy shifts and external shoks in a wide range of ontexts where an understanding of the eonomy-wide reperussions of the hange is required. This paper presents a revised version of a set of exerises initially developed for use in a Master's level ourse in CGE modeling taught by the author while at the Amerian University in Cairo. Earlier versions have also been used in the graduate program in Eonomis at George Washington University, Washington, D.C. and in a training program for researhers at l'institut d'enomie Quantitative, Ministère du Développement Eonomique, Tunis. The materials are also appropriate for advaned undergraduate ourses. The purpose of the exerises is to develop the ability of the reader to onstrut, modify and ondut simulations with CGE models in the GAMS language, the format of whih is losely linked to standard mathematial notation. 3 This manual omes with a CD ROM with a limited- 2 Support for this researh from the Ford Foundation is gratefully aknowledged with speial thanks to its former representative for the Middle East and North Afria, David Nygaard. I would also like to thank Maureen Kilkenny, Sherman Robinson, Moataz El-Said, Anne-Sophie Robilliard, Rebea Harris, and Peter Wobst for helpful omments. Moataz El-Said also provided valuable researh assistane. 3 GAMS (the General Algebrai Modeling System) onsists of a language ompiler and integrated solvers. GAMS has strong data proessing failities and an be used to solve a wide variety of optimization and simultaneous equation models. It is one of the most popular softwares for solving CGE models. The web site of the GAMS Development Corporation ( provides information on how to aquire the full-apaity versions of the software as well as aess to a wide range of GAMS-related resoures, inluding the manual (Brooke, Kendrik, Meeraus, and Raman, 1998). 3

4 apaity version of GAMS, inluding solvers for linear, non-linear, and mixed-omplementarity problems. The approximate bakground requirements are omputer literay, basi familiarity with GAMS, knowledge of maro and miro theory at the intermediate level, and basi mathematis for eonomists (inluding the ability to derive first-order onditions for onstrained optimization problems). To arry out these exerises, you need a personal omputer of the IBM-type and the GAMS software. All the software now omes with GAMS Integrated Development Environment (GAMSIDE), a windows interfae that most GAMS modellers now use. 4 These exerises should be studied in onjuntion with other materials on general equilibrium modeling and relevant eonomi theory. 5 The ontent of the Exerises is outlined in Table 1. The approah is to start with a very simple model and subsequently modify it step by step. With very few exeptions, the models of this volume are based on data presented in the form of Soial Aounting Matries (SAMs). In Exerise 1, the mathematial statement of the simple model and its SAM are provided; the reader's task is to implement the model in GAMS. In the model statements for these Exerises (as well as most models in the CGE literature), the model elements (equations, parameters, and variables) are defined over sets. The Appendix to Exerise 1 inludes Exerise A1 in whih part of the model of Exerise 1 is rewritten in "longhand," i.e., without any referenes to sets; the purpose is to remind the reader of what is hidden behind referenes to sets and elements in model statements. Starting from Exerise 2, umulative modifiations are introdued into the model of Exerise 1. The model of the final Exerise 5, whih inludes a ritial minimum of real-world features, provides a starting point for more detailed ountry-speifi models that an be used for poliy analysis. 4 Some older modellers prefer to run GAMS in a DOS environment, whih an be done using any text editor that an generate ASCII files. The Mirosoft MS-DOS Editor, inluded with the MS-DOS and Mirosoft Windows operating systems is fully adequate. Alternatively, modelers may use the Windows based Integrated Development Environment (IDE) as a substitute.?? 5 Informative surveys of CGE models inlude Dealuwe and Martens(1988), Robinson (1989), and Bandara (1991). Referenes to and examples of CGE-based analyses of food poliy in developing ountries done at IFPRI s Trade and Maroeonomis Division are found in the relevant setion of IFPRI s website ( Extensive treatments of CGE methods are found in Dervis, de Melo and Robinson (1982), Shoven and Whalley (1992), Dixon, Parmenter and Wiloxen (1992), and Ginsburgh and Keyzer (1997). Condon, Dahl and Devarajan (1987), and Devarajan, Lewis and Robinson (1994) fous on the implementation of CGE models in GAMS. 4

5 Table 1. Outline of Exerise ontent Exerise # Content/New feature 1 Simple CGE model A1 Optional: Simple CGE model in "longhand" 2 Intermediate demands 3a 3b Savings and investments; Ativity-speifi wages; labor unemployed 4 Government 5 Rest of World (open eonomy) In Exerises 2-4, the student is provided a verbal desription of the model hange (with various hints) and a new SAM, and any supplementary data (if needed), on the basis of whih he or she is asked to first present a new mathematial statement and, subsequently, implement the modified model in GAMS. For the more omplex Exerise 5, the task is limited to implementing the new model in GAMS; the new mathematial statement is provided. The model of the final Exerise 5 inludes a ritial minimum of real-world features. Hene, it provides a starting point for more detailed ountry-speifi models that an be used for poliy analysis. The numbers in the SAMs are fititious. CGE modeling is not a spetator sport. It is the hope of the author that, by working through these Exerises, the reader an take an important step toward using CGE models as a tool for poliy analysis. 5

6 EXERCISE 1: A SIMPLE WALRASIAN MODEL INTRODUCTION This exerise involves implementing a simple CGE model in GAMS. The model is presented below, in both verbal terms and the form of a mathematial statement. The model presentation is followed by a SAM whih inludes the data needed to solve the model using "alibration": on the basis of a data set for a base period, the parameters of the model are estimated in a manner whih enables the model (general equilibrium) solution to preisely repliate the base-year data set. Behavioral parameters are alibrated as if the base-year eonomy was indeed in equilibrium. The funtional forms for the various relationships embodied in this Exerise have been seleted so as to assure that all parameters an be derived from the aompanying SAM. (With a few exeptions, this is also true for the rest of the exerises in this manual.) VERBAL MODEL PRESENTATION We assume that produers maximize profits subjet to prodution funtions with primary fators as arguments while households maximize utility subjet to budget onstraints. Cobb-Douglas funtions are used for both produer tehnology and the utility funtions from whih household onsumption demands are derived. Fators are mobile aross ativities, available in fixed supplies, and demanded by the produers at market-learing pries (rents). On the basis of fixed shares (derived from base-year data), fator inomes are passed on in their entirety to the households, providing them with their only inome. The outputs are demanded by the households at market-learing pries. The model satisfies Walras law in that the set of ommodity market equilibrium onditions is funtionally dependent. Any one of these onditions an be dropped; in the proposed model, we drop the equilibrium ondition for the non-agriultural ommodity. The model is homogeneous of degree zero in pries; to assure that only one solution exists, a prie normalization equation, in this ase fixing the onsumer prie index (CPI), has been added. After these adjustments, the model has an equal number of endogenous variables and independent equations. Given this definition of the prie normalization equation, all simulated prie hanges an be diretly interpreted as hanges vis-a-vis the CPI. The model is disaggregated into two 6

7 households (urban and rural), two fators (labor and apital), and two ativities and assoiated ommodities (agriulture and non-agriulture). The expliit distintion between ativities and ommodities failitates model alibration but is not neessary for the CGE models of this volume. It is, however, needed for models that deviate from a one-to-one mapping between ativities and ommodities, i.e., for models where at least one ativity produes more than one ommodity and/or at least one ommodity is produed by more than one ativity. The label "simple" is well deserved sine the model does not inlude a government, intermediate demands, savings, investment, or an outside world. MATHEMATICAL MODEL STATEMENT The mathematial statements and the GAMS input files that aompany this volume follow the urrent standard notation used in CGE models developed at IFPRI s Trade and Maroeonomi Division. All endogenous variables are written in upperase Latin letters whereas parameters (inluding variables with fixed or exogenous values) have lower-ase Latin or Greek letters. Subsripts refer to set indies, with one letter per index. Supersripts are part of the parameter name (i.e., not an index). In terms of letter hoies, variables and parameters for ommodity and fator quantities start with the letter q; for ommodity and fator pries, the first letters are p and w, respetively. 6 Sets f F fators {LAB labor CAP apital} h H households {U-HHD urban household R-HHD rural household} a A ativities {AGR-A agriultural ativity NAGR-A non-agriultural ativity} C ommodities {AGR-C agriultural ommodity NAGR-C non-agriultural ommodity} 6 For a disussion of style in eonomi modeling, see Kendrik (1984). 7

8 Parameters ad a effiieny parameter in the prodution funtion for ativity a pi onsumer prie index (CPI) wts weight of ommodity in the CPI shry h f share for household h in the inome of fator f qfs f supply of fator f α f a share of value-added for fator f in ativity a β h share in household h onsumption spending on ommodity yield of output per unit of ativity a θ a Variables P PAa Q QA a QF fa Equations market prie of ommodity prie of ativity a output level in ommodity level of ativity a demand for fator f from ativity a QH h onsumption of ommodity by household h WF f prie of fator f YF h f inome of household h from fator f YH h inome of household h Prodution and Commodity Blok Ativity prodution funtion QA a = ad α f a a QF f a a A f F (1) Fator demand α PA QA WF f = QF fa a a fa f F, a A (2) Ativity prie PAa = θa P a A (3) C Commodity output Q Institution Blok = θ QA C (4) a a A Fator inome 8

9 YF shry WF QF Household inome YH =, hf hf f fa a A h f F Household demand QH hf h H f F (5) = YF h H (6) β YH System onstraint blok Fator market equilibrium h h h =, P QFfa = qfs f f F a A Output market equilibrium Q h H h C h H (7) (8) = QH C (9) Prie normalization equation wts P = pi C C (10) DATA BASE The data base of the model is presented in Table

10 Table 1.1: Soial Aounting Matrix for Exerise 1 AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD Total AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD Total TASK Using GAMSIDE, input the above model using GAMS syntax. Solve the model in GAMS and verify that the solution an repliate the above SAM. HINTS AND SUGGESTIONS 1. Before attempting to do this Exerise, the reader should be familiar with GAMS, at least at the level of the tutorial Chapter 2 in the User's Guide (Brooke, Kendrik, Meeraus, and Raman, 1998, pp. 5-28). The rest of this guide is an indispensable referene (f. Footnote 3). 2. You may run the input file in GAMS at any point in the proess of onstruting the model. If the model is inomplete and hene not solved, GAMS will nevertheless hek that the input onforms with its syntax, report any errors, and, in the absene of errors, arry out other instrutions, inluding displays. To ath errors at an early stage, it is often helpful to inspet the results of displays of elements (sets, variables and parameters), that have been defined via operations. 3. A timesaving devie when developing the model is to use mehanial searhes for text segments, inluding ****, used to indiate errors in the output file of GAMS (whih by default is alled myfile.lst if the input file is alled myfile or myfile.gms). 4. The above mathematial statement is provided in a format that an be easily implemented in GAMS. It is advisable to use the same notation (subjet to various minor transformations) sine this will save effort (no additional notation is needed) and make it easier to move between the mathematis and GAMS. 7 For example, the first equation, with PRODFN as its delared name, may be input as follows: PRODFN(A).. QA(A) =E= ad(a)*prod(f, QF(F,A)**alpha(F,A)); 5. In order to failitate SAM-related omputations, it is helpful to generate a global set, 7 Note that GAMS is ase-insensitive. Nevertheless, it is probably easier to read a GAMS statement where the distintion between variables and parameters is evident, for example with parameters in lower ase and variables in upper ase (the proposed onvention). 10

11 here named AC, inluding all elements in the sets for fators, ativities, ommodities, and households. Sets for the latter items are subsequently delared and defined as subsets of the global set. 6. When building CGE models, it is often useful to have idential sets with different names. In GAMS, the ALIAS ommand may be used to reate a set that is idential to a set that already has been defined. (In the suggested answer, ALIAS is used to define sets idential to AC, C and F.) 7. The models in this volume are formulated as a set of simultaneous equations and solved using a solver for nonlinear mixed-omplementarity problems (PATH or MILES). 8 Alternatively, simultaneous-equation models may be solved as nonlinear optimization problems. To follow this approah, you may define an additional equation, OBJFN, as: OBJFN.. OBJ =E= DUMMY**2; where OBJ is an unonstrained variable, and DUMMY is a nonnegative variable. The solution statement should be hanged to SOLVE CGE1 USING NLP MINIMIZING OBJ. In this setting, the other equations would define the onstraint set with only one feasible solution (idential to the solution to the orresponding simultaneous-equation model). OBJ is minimized when DUMMY has a value of zero. 8. Do not input the row and olumn totals of the SAM. Instead, to remove one soure of errors and model infeasibility, ompute row and olumn totals and hek that they are idential. 9. After having input the SAM, start using it to define the parameter values and initial variable levels. Initial variable levels are helpful for two reasons. First, it gives the GAMS solver a good starting point, failitating its searh for the solution. Seond, it makes it easier to pinpoint reasons for why a model fails to generate the benhmark equilibrium. The latter aspet is disussed further under point Note that parameters (in this model for example ad a ) may be defined using preeding definitions; when this approah is followed, it matters in whih order the parameters and initial variable values are defined. 11. Assume that base-year fator and output pries are at unity and assign parameter and variable quantities on this basis. (This amounts to hoosing the unit for eah real flow so as to assure that its orresponding prie is one.) 12. If you have implemented the model orretly, there should be no "signifiant" disrepanies between left-hand and right-hand sides of the equations when GAMS plugs in your parameter values and initial variable levels. You hek this by looking for three asterisks (***) in the "equation listing. If the left-hand side value (indiated 8 In the general ase, a mixed-omplementarity problem onsists of a set of simultaneous equations that are a mixture of strit equalities and inequalitites, with the latter linked to bounded variables. The urrent model is a speial ase sine all equations are strit equalities. For details and a mathematial definition, see Rutherford (1995). 11

12 as LHS = <value> ) is signifiantly different (let s say by more than 1.0E-5) from the right-hand side value in the preeding equation (after =E=), there is some problem assoiated with the definitions of the parameters and variables whih appear in the equation in question. Errors may be aught by displaying the values of all parameters and variables in any problemati equation and heking whether the values are ompatible with the SAM. 13. The number of (single) equations and (single) variables, reported as part of the "model statistis, should be idential (at 24 for the proposed solution) A value of unity for all fator and ommodity pries (that were initialized at this level) is a reliable indiator that the initial model solution repliates the initial equilibrium as aptured by the initial SAM. To test that the model is robust, it is a good idea to solve it a ouple of times with different values for seleted parameters. To hek that the model indeed is homogeneous, the initial value of pi may be multiplied by some positive fator; ompared to the base, the pries of the model solution should all be hanged by the value of the fator while all quantities should stay unhanged. 15. In addition to model statement and base solution, the GAMS input file with the suggested answer inludes a LOOP where two simulations are arried out, the BASE and a simulation for whih the apital stok is inreased by 10%. Report parameters are reated to show values for variables and the fator supply parameter (whih was hanged in the experiment). Variable and parameter values may also be used to generate report parameters that provide additional information not defined by any single model variable, for example national aount statistis This first Exerise may be the toughest one; it is ertainly the one requiring the largest amount of new GAMS ode. It is important that you try to do it before heking the suggested answer sine "learning by doing" is the name of the game. However, keep in mind that, although the suggested answer tries to embody good modeling praties, 9 By default, the GAMS variable ount also inludes variables that are fixed (have exogenous values) unless the model attribute holdfixed has been speified with a value of one, in whih ase only endogenous variables are inluded in the ount (see Brooke et al, 1998, pp ). If the model inludes fixed variables, this attribute makes it straightforward to verify that the number of endogenous variables and equations is equal. In this manual, fixed variables and the holdfixed attribute appear in the proposed GAMS solutions to Exerises For a relatively omprehensive set of report parameters for a CGE model, see the GAMS listing of the Cameroon CGE model in Chapter 3 of Devarajan et al. (1994). 12

13 it is merely a suggested answer. Different formulations may seem preferable to other users. In many ways, "style" in modeling and language involve similar onsiderations, inluding room for taste differenes. 13

14 APPENDIX: EXERCISE A1 INTRODUCTION The notation of the model of Exerise 1 is set-driven, i.e., referene is frequently made to sets and set indies. 11 The purpose of this Exerise is to provide pratie in interpreting what is hidden behind the veil of set notation and to demonstrate the gains from set notation in the form of more onise and more easily modified models. For example, the disaggregation of models based on set notation an be hanged simply by varying the set definitions, the SAM and other data, without any other hanges in the input file. 12 The starting point for this Exerise is the GAMS input file with your orret answer to Exerise 1. TASK In the GAMS statement, rewrite two of the equations in Exerise 1, PRODFN and FACDEM, in "longhand,, i.e., instead of making referene to sets and indies, refer to speifi set elements by name. Leave the rest of the model statement unhanged. Run the model and hek that the solution is idential to Exerise 1. HINTS 1. You have to delare and define a total of six equations. (In the suggested answer, they are named PRODFN1, PRODFN2, FACDEM1, FACDEM2, FACDEM3, and FACDEM4.) Put asterisks in the first harater position of the lines where referene is made to the equations PRODFN and FACDEM (their definitions and delarations) Write the prodution funtion for the agriultural ativity as follows: 11 Reall that subsripts (but not supersripts) are set indies. 12 If one more prodution ativity and ommodity is added, the omplete urrent model in long-hand would require six new equations and modifiations for five old equations. To add one more ativity and ommodity in the set-driven statement would merely require the addition of a ouple of additional lines in the set definitions. Irrespetive of approah, it would be neessary to further disaggregate the SAM. 13 When an asterisk ('*') is put in the first harater position, GAMS ignores the rest of the line but reprodues it in the output file. This is a useful devie for inluding short omments or exluding unused parts of a program without deleting them. (An alternative approah, preferable for longer omments, is to blok off a setion with $ONTEXT and $OFFTEXT before and after the setion, respetively. The two "dollar" statements must start in the first harater positions of their respetively lines.) 14

15 PRODFN1.. QA('AGR-A') =E= ad('agr-a')*qf('lab','agr-a')**alpha('lab','agr-a') *QF('CAP','AGR-A')**alpha('CAP','AGR-A'); 15

16 EXERCISE 2: INTRODUCING INTERMEDIATE DEMAND INTRODUCTION In this Exerise, intermediate demands are added to the CGE model presented in Exerise 1. DATA BASE The new SAM is shown in Table 2.1. New payment flows, representing payments for intermediate goods, have been added in the ells at ommodity rows and ativity olumns Table 2.1: Soial Aounting Matrix for Exerise 2 AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD TOTAL AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD TOTAL intersetions. The aounts in the SAM are unhanged. TASKS 1. Mathematial statement. Modify the mathematial statement so that the model inorporates intermediate demands. For both setors, assume Leontief tehnology, i.e., that a fixed input quantity is needed per unit of output. 2. GAMS. After having onvined yourself that the answer to (1) is without errors (ompare it to the suggested answer), implement the model in GAMS. This involves all or parts of the following: modifying the SAM; adding/modifying delarations and definitions for sets, parameters, variables (for the latter, the definition involves defining the initial levels), and equations; displaying (and heking) the results of new omputations; solving the model without errors for the base ase and for a simple experiment (the latter to hek that the model is robust); and onfirming that it repliates the base data. Make sure that for eah new element (set, variable, parameter or equation), you go through the same steps as for the orresponding 16

17 elements already present in the initial model. One the model is alibrated to the SAM, run the same experiment as for Exerise 1. HINTS AND SUGGESTIONS 1. Mathematial statement a. When modifying the statement, hek that the hanges in the number of variables and equations are equal (so that the number of variables and independent equations remains equal). Compared to the Exerise 1 model, the suggested model has six additional equations and variables, the total number being 30. b. The suggested answer inludes the following new elements: Parameters Variables ia a quantity of as intermediate input per unit of output in ativity a PVA a value-added (or net) prie of ativity a QINTa quantity of ommodity as intermediate input in ativity a There are no new sets but two new equations, for value-added pries and intermediate demands. Some hanges in other equations are also needed. 2. GAMS a. Be systemati when you modify the model: for every parameter/variable you delare, make sure you don't forget to inlude it in the equations, or to define and display its value/initial level. Chek that the displayed values oinide with the values you expet. b. Note that both PVAa and P annot have initial (or equilibrium) levels of unity. In the suggested answer, we follow the onvention of keeping the initial value of P at unity and defining PVA at a level orresponding to the share of total output value aruing to the primary fators. a. As for Exerise 1, define the parameter ada so that the prodution (or, more preisely, value-added) funtion for ativity a on its own will generate the base level for QA a. d. Among the hints for Exerise 1, pay speial attention to point 12 regarding how to trak down reasons for why the model fails to repliate the base-year equilibrium. 17

18 EXERCISE 3: INTRODUCING THE SAVINGS-INVESTMENT BALANCE AND ACTIVITY SPECIFIC WAGES INTRODUCTION Few applied CGE models fall short of expliitly overing savings and investment. In our gradual proess of onstruting an applied model, we start by adding this aspet, using the CGE model of Exerise 2 as our starting point. Moreover, in the previous models, the wage (prie) of eah fator was assumed to be uniform aross all ativities that use it; in other words, every ativity paid the average wage. In the real world, wages tend to be "distorted" in the very broad sense that they differ aross ativities. A treatment that permits this feature (with no distortions as a speial ase) is also introdued in this exerise. We will assume that wages are distorted for labor but uniform aross ativities for apital in a setting with full (or fixed) employment for both fators. When doing the Exerise, follow one general aspet of good modeling pratie: introdue hanges in different areas one at a time. Consequently, we split this exerise in two and will introdue savings and investment first, followed by ativity speifi wages. EXERCISE 3A: INTRODUCING THE SAVINGS-INVESTMENT BALANCE INTRODUCTION Few applied CGE models fall short of expliitly overing savings and investment. In our gradual proess of onstruting an applied model, we start by adding this aspet, using the CGE model of Exerise 2 as our starting point. DATA BASE The SAM, displayed in Table 3.1, inludes one new aount alled savings-investment (S-I). 14 Its row reeives payments from the household (the only saver in this simple eonomy); its olumn shows spending on ommodities used for investment. Note that investment is defined 14 The hange in the assumption about the apital market is not linked to any hange in the SAM sine the SAM merely reports payment flows, i.e., the SAM does not say anything about the behavioral rules of the 18

19 in terms of the ommodities used in the prodution of the apital stok, not the ativity of destination (the ativity that reeives the investment goods as an addition to its apital stok). This means that the model only applies to a period so short that there is not enough time for new investments to provide additional prodution apaity. For a model relevant to a longer time period (for example a multi-period model), it would also be neessary to expliitly onsider the resulting hanges in apital stok. Table 3.1: Soial aounting matrix for Exerise 3 AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD S-I TOTAL AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD S-I TOTAL TASKS 1. Mathematial statement. Expand the mathematial statement so that savings-investment is inluded. For savings-investment, assume the following: household inome is alloated in fixed shares to savings and onsumption; investment is savings-driven, i.e., the value of total investment spending is determined by the value of savings investment spending is alloated to the two ommodities in a manner suh that the ratio between the quantities is fixed. Together, assumptions (b) and () mean that when savings values and/or the pries of investment ommodities hange, there is a proportional adjustment in the quantities of eonomy, inluding the workings of the apital market. 19

20 investment demand for eah ommodity, generating an investment value equal to the savings value. The set of equilibrium onditions that is funtionally dependent now inludes, the savingsinvestment balane. It would be possible to drop one of these equations. In the suggested answer, another approah is seleted. Instead of dropping one of these equations, a variable alled WALRAS is introdued in the savings-investment balane. This approah is ommonly followed in this lass of models. The model still has an equal number of variables and equations. If the model works, the savings-investment balane should hold, i.e.,the value of WALRAS should be zero. 2. GAMS After having produed an error-free mathematial statement, implement the model in GAMS, using the same systemati approah that was developed for Exerise 2. Proeed with the introdution of savings-investment. When the base solution works well also with this hange, solve for the experiment with a 10% inrease in the apital stok. Verify that the value for WALRAS remains (very lose to) zero also for this solution. HINTS AND SUGGESTIONS 1. Mathematial statement. a. Introdue the hanges in a step-wise manner, in eah step keeping trak of hanges in the number of variables and equations. b. For the savings-investment modifiation, the following hanges may aomplish the task: Parameters: Introdue new parameters for household savings shares (alled mps ) and base-year setoral investment quantities (alled qinv ); h Variables: Add new variables for quantities of investment demand and a fator introduing proportional hanges in investment quantities (referred to as QINV and IADJ, respetively). Equations: Inlude new equations to define QINV and to impose balane between savings and investment values. The investment equation may be written as QINV = qinv IADJ. 20

21 2. GAMS. Before introduing savings and investment, add S-I to elements in the set AC and use the new SAM. The suggested GAMS model has 34 equations and variables. EXERCISE 3B: INTRODUCING ACTIVITY SPECIFIC WAGES AND UNEMPLOYMENT INTRODUCTION In the previous models, the wage (prie) of eah fator was assumed to be uniform aross all ativities that use it; in other words, every ativity paid the average wage. In the real world, wages tend to be "distorted" in the very broad sense that they differ aross ativities. A treatment that permits this feature (with no distortions as a speial ase) is introdued in this exerise. We will assume that wages are distorted for labor but uniform aross ativities for apital in a setting with full (or fixed) employment for both fators. In addition, we onsider a model set-up with a flexible market-learing wage by assuming that labor is unemployed with fixed, ativity-speifi real wages and the quantity of labor supply as the market-learing variable. DATA BASE For labor, the number of workers employed is 100 for agriulture and 50 for nonagriulture. For apital, we ontinue to assign quantities on the basis of the assumption that the wage is unity for both ativities. There are no hanges in the SAM assoiated with the hange in the fator treatment. TASKS 1. Mathematial statement. Expand the mathematial statement so that eah ativity pays fixed shares of average base wages for labor and apital. For the fator markets (both labor and apital), assume that eah ativity pays an endogenous wage expressed as the produt of an endogenous (eonomywide) wage variable (for the base equal to the average wage) and an exogenous distortion fator. For 21

22 the speial ase of no distortions, the distortion fator is equal to one for all ativities. In eah fator market, variations in the average wage lear the market. 2. GAMS After having produed an error-free mathematial statement, implement the model in GAMS, using the same systemati approah that was developed for Exerise 3A. Proeed with the introdution of ativity speifi wages. When the base solution works well also with this hange, solve for the experiment with a 10% inrease in the apital stok. Continue verifying that the value for WALRAS remains (very lose to) zero also for this solution. HINTS AND SUGGESTIONS 1. Mathematial statement. 1. Introdue the hanges in a step-wise manner, in eah step keeping trak of hanges in the number of variables and equations. 2. The hange in the treatment of fator markets may involve the following: Parameters: Define and delare a new distortion fator ( wfdist fa ) that represents the ratio between the wage for fator f in ativity a and the average wage for fator f ; Equations: To assure that payments for fators are made at distorted wages, multiply the average wage variable ( WF ) by the distortion fator in every equation where the f wage variable appears. (The definition of the distortion fator implies that WF f wfdist indeed defines the wage for fator f in ativity a.) fa Introdue fixed, ativity-speifi real wages and the quantity of labor supply as the market-learing variable. A relatively flexible approah is suggested. The hanges are: i. The parameters wfdist fa and qfs f are turned into variables, written as WFDIST fa and QFS f, respetively. ii. Among the fator wage and quantity variables, the following are fixed: 22

23 WFDIST lab, a, WF lab, QF ap, a and WF ap. This approah is relatively flexible sine, by seletively fixing fator wage and quantity variables, it an handle a variety of losure rules (inluding the one used as default in this exerise). 2. GAMS For fators quantities and wages, you may go through the following steps: On the basis of the information in the data base and stated assumptions, define initial levels for the ativity-speifi fator demand variable ( QF ) and the fator supply parameter ( qfs f ). fa Define initial levels for the average wage variable ( WF f ) and ativity-speifi wages (an auxiliary parameter that only is used to failitate alibration, in the proposed solution alled wfa fa ). Define the wage distortion parameter ( wfdist fa ) as the ratio between wfa fa and WF f. For eah ativity-fator ombination, verify that the produt WFf wfdist fa QFfaequals the SAM payment from the ativity to the fator. 15 The suggested GAMS model has 34 equations and variables. 15 The market-learing eonomy-wide wage variable was initialized at the level of the average base wage. Generally speaking, it will not oinide with the eonomy-wide average wage for any experiment unless (1) wfdist fa equals one for all ativities and/or (2) there is no hange in the employment shares for the different ativities. (This is onfirmed by the results for Exerise 3a.) 23

24 EXERCISE 4: INTRODUCING GOVERNMENT INTRODUCTION Up to now, the modeled eonomy has not inluded a government, an essential ator in applied poliy analysis. In this Exerise, this defet is remedied. The government of the model earns its revenues from inome and sales taxes and spends it on onsumption and transfers to the households. Government savings is the differene between its revenues and spending. DATA BASE The model is built around the SAM shown in Table 4.1. Labor employment quantities are the same as for Exerise 3B (100 for agriulture and 50 for non-agriulture). The introdution of the government is behind the hanges in the SAM struture. There are new aounts for the government (GOV) and the two tax types, taxes on inomes (YTAX) and sales (STAX). In the tax rows, inome taxes are olleted from the household and sales taxes from the ommodity aounts (AGR-C and NAGR-C). In the tax olumns, this inome is passed on to the government. The government olumn shows that the government uses this revenue to over the ost of government ommodity onsumption (payments to AGR-C and NAGR-C), transfers to the households (payments to U-HHD and R-HHD), and (negative) government savings (payment to S-I). Note that, in the rows of the ommodity aounts, the demanders buy ommodities at market pries; in the olumns of the ommodity aounts, these payments are split between the sales tax aount and the ativities (paid for output valued at produer pries). One important part of government onsumption, government payment for the labor servies of its administrators and other employees, does not appear expliitly in the SAM. They may be viewed as working for a government servie ativity that produes a ommodity whih is purhased by the aount for the government (institution). In this SAM, this ativity-ommodity pair is part of the non-agriultural ativity and its ommodity. In more disaggregated, real-world SAMs, the government servie ativity and ommodity typially have their own aounts. 24

25 Table 4.1: Soial Aounting Matrix for Exerise 4 AGR-A NAGR-A AGR-C NAGR-C LAB CAP AGR-A 255 NAGR-A 350 AGR-C NAGR-C LAB CAP U-HHD R-HHD GOV S-I YTAX STAX TOTAL U-HHD R-HHD GOV S-I YTAX STAX TOTAL AGR-A 255 NAGR-A 350 AGR-C NAGR-C LAB 177 CAP 208 U-HHD R-HHD GOV S-I YTAX STAX 58 TOTAL TASKS 1. Mathematial statement. Present a statement for a model, based on the above SAM, that inludes a government. Assume the following: The inome tax is a fixed share of the gross inome of eah household. A fixed share of post-tax inome is saved and the rest is spent on onsumption. 25

26 Sales taxes are fixed shares of (mark-ups on) produer ommodity pries. The government onsumes fixed ommodity quantities, paying market pries (inluding the sales tax). Government transfers to the households are CPI-indexed, i.e., they an simply be fixed in nominal terms indexation to the CPI is automati sine the CPI level is fixed via the prie normalization equation. 16 Government savings is a residual, assuring balane between government outlays (inluding savings) and revenues. It is omputed as the differene between expenditures (exluding savings) and revenues. 2. GAMS One you have produed a orret mathematial statement, implement the model in GAMS. Make sure that the model an repliate the database and solves for an experiment where the quantities of government onsumption of eah ommodity are inreased by 20%. HINTS AND SUGGESTIONS 1. Mathematial statement. To model the government, go through the following steps: Sets: A new set, I (with an idential set named I ) defines institutions (urrently the two households and the government; the rest of the world will be added in Exerise 5). It is referred to in the modeling of transfers between institutions. Parameters: The new parameters, with suggested notation parenthesized, define government ommodity onsumption ( qg ), sales and inome tax rates ( tq and ty h, respetively), and transfers from institution i to institution i ( tr ii ). The transfer parameter aptures transfers from the government to the households; in the equations where it appears, referene is made expliitly to the relevant subset (H) and elements (GOV) If so, the model is, stritly speaking, no longer homogeneous of degree zero in pries. (Why?) If you would like to maintain homogeneity, multiply the government transfer parameters by pi. 17 Alternatively, it would have been possible to delare this as a government transfer parameter with only the 26

27 Variables: The new variables, with the symbols parenthesized, denote produer pries exlusive of the sales tax ( PX ), government revenue (YG ), and government expenditures ( EG ). The sales tax introdues a wedge between the prie reeived by the produers ( PX ) and that paid by the demanders whih inludes the sales tax (the old symbol P is used to define the latter prie). Thus, one important task is to hange the variable P in the Exerise 3 model to PX in the urrent model whenever referene is made to what the produer reeives (and not to what the demander pays). Equations: New equations are needed to define government revenue and expenditures. Modifiations are introdued in the equations for household inome (government transfers are a new inome soure), household onsumption demand (owing to the presene of inome taxes), ommodity market equilibrium (to aount for government onsumption), and the savings-investment balane (sine the government represents a new soure of savings). 2. GAMS To model the government, the following hints may failitate your task: 1. Augment the set AC with aounts for the government, and the two tax types. 2. Delare and define the new set for institutions. 3. Let the initial values be unity for all pries exept PVAa and P (i.e., pries with an initial value of unity). PX is among the 4. Calibrate the rate of the sales tax ( tq ) as the ratio between the tax payment and output value exluding the sales tax. reeiving set of institutions, h, in its domain. However, in a more omplex model with many paying institutions, this approah would be tedious, requiring the definition of a separate parameter for eah paying institution. The advantage of defining it over broadly defined sets of paying and reeiving institutions is inreased flexibility one single parameter an handle a wider variety of ontexts and fewer hanges are needed elsewhere in the model. This will be evident in Exerise 5 where the rest of the world is added to the set of paying institutions, transferring money to both households and the government. 27

28 5. Given the values of tq and values and the values ofia, you an find PVA. a PX, you an ompute the initial value of P. Given these a 6. Given that ommodity market pries ( P ) paid by the demanders are no longer unity, it is now neessary to onsider this prie expliitly when omputing values for parameters and variables linked to ommodity quantities. 7. Note that the household savings rate ( mps h ) should now be omputed as the ratio between household savings and household disposable (post-tax) inome. In addition to the 34 variables and equations of the previous exerise (CGE3B), we now have variables and equations for two demander pries, government expenditure and government revenue. The suggested model thus has 38 variables and equations. 28

29 EXERCISE 5: INTRODUCING THE REST OF THE WORLD INTRODUCTION In this final Exerise, we omplete the model by adding the rest of the world (RoW). Interation with the RoW takes plae in the form of imports, exports and transfers. Cruially, for demanders, imports and domesti output sold domestially are assumed to be imperfet substitutes. Similarly, for produers, imperfet transformability is assumed between exports and domesti output sold domestially. 18 Compared to the alternative of perfet substitutability (whih, for any given ommodity, only permits one-way trade), this treatment tends to generate more realisti responses in domesti pries, prodution and onsumption to hanges in international pries. The treatment of fator markets is the same as for Exerise 4. In ombination with an appropriately disaggregated SAM, and data for labor employment and elastiities, the model that is the output of this Exerise may provide the starting point for real-world applied poliy analysis. However, it is highly likely that hanges are needed to reflet better the struture of the modeled eonomy. Suh hanges, may, for example, inlude the introdution of prie ontrols and other features that invalidate the assumption that flexible pries lear perfetly ompetitive markets. In addition, available prodution and onsumption elastiities would typially suggest that the Cobb-Douglas funtions should be replaed by more flexible (and omplex) funtional forms. The task of implementing the model from srath on the basis of stated assumptions is quite omplex. Hene, we will provide, not only the SAM, but also a omplete mathematial statement with brief omments on new features. DATABASE The data base for the model onsists of the SAM found in Table 5.1, unhanged data for labor employment, and trade elastiities: elastiities of substitution between imports and domesti sales (of domesti output), and elastiities of transformation between exports and domesti sales. In the model, a value of 2 is used for both elastiities aross the two 18 Imperfet substitutability and transformability may arise from differenes in physial quality, differenes in time and plae of availability, and from aggregation biases. 29

30 ommodities. Table 5.1: Soial Aounting Matrix for Exerise 5 AGR-A NAGR-A AGR-C NAGR-C LAB CAP U-HHD R-HHD AGR-A 279 NAGR-A 394 AGR-C NAGR-C LAB CAP U-HHD R-HHD GOV S-I YTAX 20 5 STAX TAR 39 ROW 105 TOTAL GOV S-I YTAX STAX TAR ROW TOTAL AGR-A 279 NAGR-A 394 AGR-C NAGR-C LAB 177 CAP 208 U-HHD R-HHD GOV S-I YTAX 25 STAX 30 TAR 39 ROW 105 TOTAL The SAM itself inludes two new aounts, for the rest of the world (ROW) and for import tariffs (TAR). The row of the ROW aount shows that our spending on imported ommodities is the only inome soure of the RoW in its dealings with our ountry; the olumn of the same aount shows that the reeipts of our ountry from ROW onsist of export revenues, and 30

31 transfers to the households and the government. 19 The payments from ROW to S-I is foreign savings or the urrent aount defiit, i.e., the differene between our ountry's urrent (nonapital) foreign exhange expenditures and earnings. Mathematial statement The bulk of this statement onsists of the model equations (a total of 31), divided into "bloks" for pries, prodution and ommodities, institutions, and system onstraints. Explanatory boxes are provided below eah equation. New equations and other hanges ompared to the model of Exerise 4 are explained. The statement starts with alphabetial lists of sets, parameters, and variables that should serve as a referene as the reader goes through the equations Sets a A ativities C ommodities CM ( C) imported ommodities CNM ( C) non-imported ommodities CE ( C) exported ommodities CNE ( C) non-exported ommodities f F fators h H ( I) households i I institutions (households, government, and rest of world) Parameters ad a prodution funtion effiieny parameter aq shift parameter for omposite supply (Armington) funtion at shift parameter for output transformation (CET) funtion 20 pi onsumer prie index wts ommodity weight in CPI iaa quantity of as intermediate input per unit of ativity a mps h share of disposable household inome to savings pwe export prie (foreign urreny) 19 Neither of the two ommodities are both exported and imported. The phenomenon of two-way trade ("ross-hauling") is nevertheless ommonly observed in the real world at the level of ommodity aggregation used in applied models. It an be handled by the proposed approah without any modifiations in the model struture. 20 The aronym CET stands for onstant elastiity of transformation. 31

32 pwm qg qinv shry hf te tm tq tri i ty h import prie (foreign urreny) government ommodity demand base-year investment demand share for household h in the inome of fator f export tax rate import tariff rate sales tax rate transfer from institution i to institution i rate of household inome tax α f a value-added share for fator f in ativity a β h share of onsumption of household h for ommodity share parameter for omposite supply (Armington) funtion δ q δ t θ a share parameter for output transformation (CET) funtion yield of ommodity per unit of ativity a ρ q exponent (-1< q < ) ρ for omposite supply (Armington) funtion ρ t exponent (1< t < ) ρ for output transformation (CET) funtion elastiity of substitution for omposite supply (Armington) funtion σ q σ t Variables elastiity of transformation for output transformation (CET) funtion EG government expenditure EXR foreign exhange rate (domesti urreny per unit of foreign urreny) FSAV foreign savings IADJ investment adjustment fator PAa ativity prie PD domesti prie of domesti output PE export prie (domesti urreny) PM import prie (domesti urreny) PQ omposite ommodity prie PVA value-added prie PX produer prie QA a ativity level QD quantity of domesti output sold domestially QE quantity of exports QF f a quantity demanded of fator f by ativity a QFS f supply of fator f QH h quantity of onsumption for household h of ommodity QINT a quantity of intermediate use of ommodity by ativity a QINV quantity of investment demand 32