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Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature Introduction to computable general equilibrium modeling 1 th Educational Workshop of the Simulation Lab October,1 2015 Dr. Renger van Nieuwkoop Center for Energy Policy and Economics Department of Management, Technology and Economics ETH Zürich

Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

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Page 1: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.-1

Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Introduction to computable generalequilibrium modeling1th Educational Workshop of the Simulation Lab

October,1 2015

Dr. Renger van NieuwkoopCenter for Energy Policy and Economics

Department of Management, Technology and EconomicsETH Zürich

Page 2: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.0

Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Content

1 Introduction

2 A simple general equilibrium model with two sectors and twoprimary factors of production factors

3 Steps in CGE, the 2 x 2 model

4 Literature

Introduction to CGE modelling October,1 2015 0 / 61

Page 3: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.1

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Computable General Equilibrium (CGE) Models

• Numerical simulation model based on general equilibrium theory.

• Elements of general equilibrium theory:• Multiple interacting agents (firms, consumers, government).• Individual behavior based on optimization• Most agent interactions are mediated by markets and prices• Equilibrium occurs when endogenous variables (e.g., prices) adjust

such that:1 agents, subject to the constraints they face, cannot do better by

altering their behavior (taking prices as given)2 markets (generally, not always) clear, for example, supply equals

demand in each market.

• Computable: empirically calibrated to socioeconomic data,elasticities, and policy.

Introduction Introduction to CGE modelling October,1 2015 1 / 61

Page 4: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.1

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Computable General Equilibrium (CGE) Models

• Numerical simulation model based on general equilibrium theory.• Elements of general equilibrium theory:

• Multiple interacting agents (firms, consumers, government).• Individual behavior based on optimization• Most agent interactions are mediated by markets and prices• Equilibrium occurs when endogenous variables (e.g., prices) adjust

such that:1 agents, subject to the constraints they face, cannot do better by

altering their behavior (taking prices as given)2 markets (generally, not always) clear, for example, supply equals

demand in each market.

• Computable: empirically calibrated to socioeconomic data,elasticities, and policy.

Introduction Introduction to CGE modelling October,1 2015 1 / 61

Page 5: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.1

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Computable General Equilibrium (CGE) Models

• Numerical simulation model based on general equilibrium theory.• Elements of general equilibrium theory:

• Multiple interacting agents (firms, consumers, government).• Individual behavior based on optimization• Most agent interactions are mediated by markets and prices• Equilibrium occurs when endogenous variables (e.g., prices) adjust

such that:1 agents, subject to the constraints they face, cannot do better by

altering their behavior (taking prices as given)2 markets (generally, not always) clear, for example, supply equals

demand in each market.

• Computable: empirically calibrated to socioeconomic data,elasticities, and policy.

Introduction Introduction to CGE modelling October,1 2015 1 / 61

Page 6: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.2

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

CGE models are great, but...

unearthing the features of CGE models that drive [theirresults] is often a time consuming exercise. This is becausetheir sheer size, facilitated by advances in computertechnology, makes it difficult to pinpoint the precise sourceof a particular results. They often remain a black box.Indeed, frequently, authors are themselves unable to explaintheir results intuitively and, when pressed, resort touninformative answers..

— Panagariya and Duttagupta (2001), “The "gains" from preferential tradeliberalisation in the CGEs: where do they come from?”

Introduction Introduction to CGE modelling October,1 2015 2 / 61

Page 7: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.3

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

However,

• Although model might be “huge”, the algebraic foundation issimple.

• Often we use a model, because we can’t find the answer in ananalytical way.

• Models are to be used, not to be believed... (Henri Theil)

Introduction Introduction to CGE modelling October,1 2015 3 / 61

Page 8: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.3

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

However,

• Although model might be “huge”, the algebraic foundation issimple.

• Often we use a model, because we can’t find the answer in ananalytical way.

• Models are to be used, not to be believed... (Henri Theil)

Introduction Introduction to CGE modelling October,1 2015 3 / 61

Page 9: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.3

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

However,

• Although model might be “huge”, the algebraic foundation issimple.

• Often we use a model, because we can’t find the answer in ananalytical way.

• Models are to be used, not to be believed... (Henri Theil)

Introduction Introduction to CGE modelling October,1 2015 3 / 61

Page 10: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.4

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Why are CGE models popular?

• Combination of economic theory with data• CGE models provide a consistent framework to accommodate a

large number of structural model features & dimensions requiredfor a quantitative analysis of economic policy

• Rigorous "micro-foundation" constitutes key improvement overtraditional Keynesian macroeconomic models and Input-Outputmodels (price-dependent economic behavior, "deep" behavioralparameters)

• General equilibrium perspective: interacting behavior of differenteconomic agents in different markets and respecting overallresource constraints of the economy, scope of ceteris paribusassumptions

• CGE models are a useful tool to conduct cost-benefit analysis,investigate distributional implications, and quantify trade-offsbetween economic policies.

Introduction Introduction to CGE modelling October,1 2015 4 / 61

Page 11: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.4

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Why are CGE models popular?

• Combination of economic theory with data• CGE models provide a consistent framework to accommodate a

large number of structural model features & dimensions requiredfor a quantitative analysis of economic policy

• Rigorous "micro-foundation" constitutes key improvement overtraditional Keynesian macroeconomic models and Input-Outputmodels (price-dependent economic behavior, "deep" behavioralparameters)

• General equilibrium perspective: interacting behavior of differenteconomic agents in different markets and respecting overallresource constraints of the economy, scope of ceteris paribusassumptions

• CGE models are a useful tool to conduct cost-benefit analysis,investigate distributional implications, and quantify trade-offsbetween economic policies.

Introduction Introduction to CGE modelling October,1 2015 4 / 61

Page 12: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.4

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Why are CGE models popular?

• Combination of economic theory with data• CGE models provide a consistent framework to accommodate a

large number of structural model features & dimensions requiredfor a quantitative analysis of economic policy

• Rigorous "micro-foundation" constitutes key improvement overtraditional Keynesian macroeconomic models and Input-Outputmodels (price-dependent economic behavior, "deep" behavioralparameters)

• General equilibrium perspective: interacting behavior of differenteconomic agents in different markets and respecting overallresource constraints of the economy, scope of ceteris paribusassumptions

• CGE models are a useful tool to conduct cost-benefit analysis,investigate distributional implications, and quantify trade-offsbetween economic policies.

Introduction Introduction to CGE modelling October,1 2015 4 / 61

Page 13: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.4

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Why are CGE models popular?

• Combination of economic theory with data• CGE models provide a consistent framework to accommodate a

large number of structural model features & dimensions requiredfor a quantitative analysis of economic policy

• Rigorous "micro-foundation" constitutes key improvement overtraditional Keynesian macroeconomic models and Input-Outputmodels (price-dependent economic behavior, "deep" behavioralparameters)

• General equilibrium perspective: interacting behavior of differenteconomic agents in different markets and respecting overallresource constraints of the economy, scope of ceteris paribusassumptions

• CGE models are a useful tool to conduct cost-benefit analysis,investigate distributional implications, and quantify trade-offsbetween economic policies.

Introduction Introduction to CGE modelling October,1 2015 4 / 61

Page 14: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.5

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

CGE models for energy and environmental policy do what?

For example...• Evaluate the impacts on: macroeconomic variables,trade flows,

industrial activity, labor markets, factor prices, commodity prices,and regional or household welfare

• of changes in: carbon emission quota schemes, energy taxes,technology subsidies, international terms of trade, energy pricingpolicy, and terms of trade.

Introduction Introduction to CGE modelling October,1 2015 5 / 61

Page 15: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.5

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

CGE models for energy and environmental policy do what?

For example...• Evaluate the impacts on: macroeconomic variables,trade flows,

industrial activity, labor markets, factor prices, commodity prices,and regional or household welfare

• of changes in: carbon emission quota schemes, energy taxes,technology subsidies, international terms of trade, energy pricingpolicy, and terms of trade.

Introduction Introduction to CGE modelling October,1 2015 5 / 61

Page 16: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.6

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Types of CGE models

Treatment of time• Static• Fake dynamics• Recursive dynamcis• Ramsey model:Steady state• Overlapping generations

Regionalization• Single country with closure rules• Multi-regional model

Introduction Introduction to CGE modelling October,1 2015 6 / 61

Page 17: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.7

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Building blocks of a CGE model

A CGE model is an economic model that typically combines thefollowing:• Firms that attempt to maximize profits and minimize costs• Households who maximize "welfare" (e.g., consumption) by

demanding commodities according to price

• Markets that mediate behavior of economic agents (e.g., pricesadjust until supply and demand are equal)

• Government that collects taxes and spends revenue onconsumption and transfer to households (government is usuallyrepresented as a "passive" agent)

Introduction Introduction to CGE modelling October,1 2015 7 / 61

Page 18: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.7

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Building blocks of a CGE model

A CGE model is an economic model that typically combines thefollowing:• Firms that attempt to maximize profits and minimize costs• Households who maximize "welfare" (e.g., consumption) by

demanding commodities according to price• Markets that mediate behavior of economic agents (e.g., prices

adjust until supply and demand are equal)

• Government that collects taxes and spends revenue onconsumption and transfer to households (government is usuallyrepresented as a "passive" agent)

Introduction Introduction to CGE modelling October,1 2015 7 / 61

Page 19: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.7

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

Building blocks of a CGE model

A CGE model is an economic model that typically combines thefollowing:• Firms that attempt to maximize profits and minimize costs• Households who maximize "welfare" (e.g., consumption) by

demanding commodities according to price• Markets that mediate behavior of economic agents (e.g., prices

adjust until supply and demand are equal)• Government that collects taxes and spends revenue on

consumption and transfer to households (government is usuallyrepresented as a "passive" agent)

Introduction Introduction to CGE modelling October,1 2015 7 / 61

Page 20: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.8

1. Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature

The circular flow

Introduction Introduction to CGE modelling October,1 2015 8 / 61

Page 21: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.9

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Outline

1 Introduction

2 A simple general equilibrium model with two sectors and twoprimary factors of production factors

3 Steps in CGE, the 2 x 2 model

4 Literature

2 x 2 GE model Introduction to CGE modelling October,1 2015 9 / 61

Page 22: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.10

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

A simple general equilibrium model with two sectors andtwo primary factors of production factors (2 x 2 GE model)

2 x 2 GE model Introduction to CGE modelling October,1 2015 10 / 61

Page 23: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)

• 2 commodities which can be produced using productiontechnologies X = F (K ,L) and Y = G(K ,L)

• F and G are homogenous of degree one (=exhibit constantreturns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 24: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)• 2 commodities which can be produced using production

technologies X = F (K ,L) and Y = G(K ,L)

• F and G are homogenous of degree one (=exhibit constantreturns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 25: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)• 2 commodities which can be produced using production

technologies X = F (K ,L) and Y = G(K ,L)• F and G are homogenous of degree one (=exhibit constant

returns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 26: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)• 2 commodities which can be produced using production

technologies X = F (K ,L) and Y = G(K ,L)• F and G are homogenous of degree one (=exhibit constant

returns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 27: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)• 2 commodities which can be produced using production

technologies X = F (K ,L) and Y = G(K ,L)• F and G are homogenous of degree one (=exhibit constant

returns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 28: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)• 2 commodities which can be produced using production

technologies X = F (K ,L) and Y = G(K ,L)• F and G are homogenous of degree one (=exhibit constant

returns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 29: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.11

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Model description

• 2 factors of production: capital (= K ) and labor (= L)• 2 commodities which can be produced using production

technologies X = F (K ,L) and Y = G(K ,L)• F and G are homogenous of degree one (=exhibit constant

returns to scale) consistent with the assumption of perfectcompetition

• Production in each industry X and Y is represented by aprofit-maximizing firm (or a continuum of homogeneous firms)

• A representative consumer/household earns income I fromsupplying (inelastically) fixed endowments of K and L to firmsand demands commodities X and Y so as to maximize welfareW = U(X ,Y )

• Let pX , pY , pK , pL and pW denote the price of X , Y , K , L and W ,respectively

• Static model, closed economy, no government.

2 x 2 GE model Introduction to CGE modelling October,1 2015 11 / 61

Page 30: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.12

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Definition: competitive equilibrium

A competitive equilibrium consists of a non-negative consumptionplan {CX ≥ 0,CY ≥ 0} for the representative consumer, anon-negative production plan {KX ,LX ,X ,KY ,LY ,Y} for each firm andnon-negative prices {pK ,pL,pX ,pY} such that:

1 for every firm the set of inputs used and outputs producedmaximize profit at those prices given the firms productiontechnology (profit maximization),

2 for each consumer the consumption bundle maximizes utility forthose prices given the budget constraint (utility maximization),

3 for each market (factors and goods), demand does not exceedsupply (market clearance).

2 x 2 GE model Introduction to CGE modelling October,1 2015 12 / 61

Page 31: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.13

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Two ways of formulating economic models:

1 as an constrained optimization problem

2 as a complementarity problem: square system ofequations/inequalities and unknowns

2 x 2 GE model Introduction to CGE modelling October,1 2015 13 / 61

Page 32: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.13

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Two ways of formulating economic models:

1 as an constrained optimization problem2 as a complementarity problem: square system of

equations/inequalities and unknowns

2 x 2 GE model Introduction to CGE modelling October,1 2015 13 / 61

Page 33: Introduction to computable general equilibrium …Introduction to computable general equilibrium modeling Dr. Renger van Nieuwkoop.0 Introduction2x2 GE modelSteps in CGE, the 2x2 modelLiterature

Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.14

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium as a constrained optimization problem

Equilibrium of this economy can be described as solution to thefollowing constrained optimization problem:

maxKX ,KY ,LX ,LY ,pK ,pL,pX ,pY

W = U(X ,Y )

s.t.

X = F (KX ,LX )

Y = G(KY ,LY )

KX + KY ≤ KLX + LY ≤ L

pX X + pY Y ≤ pK (KX + KY ) + pL (LX + LY )

where all prices and quantities are constrained to be non-negative.

2 x 2 GE model Introduction to CGE modelling October,1 2015 14 / 61

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Introduction tocomputable general

equilibrium modeling

Dr. Renger vanNieuwkoop

.15

Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Two ways of formulating economic models:

1 as an constrained optimization problem2 as a complementarity problem: square system of

equations/inequalities and unknowns

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Advantages of complementarity-based approach

While solving economic equilibrium problems as constrainedoptimization problems works for simple problems, the usefulness ofthis approach breaks down quickly as the model becomes morecomplicated:• It is not always clear what to maximize/what objective function is

(e.g., multiple households, multiple countries).

• Use of competitive market models can be based on optimizationapproach but problematic if economy is not characterized byperfect competition.

• Incorporation of “second-best” features (e.g., distortions due togovernment interventions) is not straightforward with optimizationapproach.

• Optimization cannot handle mixing of primal (e.g., physicalquantities such as power generation, gas flows) and dualvariables (e.g., prices).

• Can not easily incorporate corner solutions, i.e. price orquantity=0.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Advantages of complementarity-based approach

While solving economic equilibrium problems as constrainedoptimization problems works for simple problems, the usefulness ofthis approach breaks down quickly as the model becomes morecomplicated:• It is not always clear what to maximize/what objective function is

(e.g., multiple households, multiple countries).• Use of competitive market models can be based on optimization

approach but problematic if economy is not characterized byperfect competition.

• Incorporation of “second-best” features (e.g., distortions due togovernment interventions) is not straightforward with optimizationapproach.

• Optimization cannot handle mixing of primal (e.g., physicalquantities such as power generation, gas flows) and dualvariables (e.g., prices).

• Can not easily incorporate corner solutions, i.e. price orquantity=0.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Advantages of complementarity-based approach

While solving economic equilibrium problems as constrainedoptimization problems works for simple problems, the usefulness ofthis approach breaks down quickly as the model becomes morecomplicated:• It is not always clear what to maximize/what objective function is

(e.g., multiple households, multiple countries).• Use of competitive market models can be based on optimization

approach but problematic if economy is not characterized byperfect competition.

• Incorporation of “second-best” features (e.g., distortions due togovernment interventions) is not straightforward with optimizationapproach.

• Optimization cannot handle mixing of primal (e.g., physicalquantities such as power generation, gas flows) and dualvariables (e.g., prices).

• Can not easily incorporate corner solutions, i.e. price orquantity=0.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Advantages of complementarity-based approach

While solving economic equilibrium problems as constrainedoptimization problems works for simple problems, the usefulness ofthis approach breaks down quickly as the model becomes morecomplicated:• It is not always clear what to maximize/what objective function is

(e.g., multiple households, multiple countries).• Use of competitive market models can be based on optimization

approach but problematic if economy is not characterized byperfect competition.

• Incorporation of “second-best” features (e.g., distortions due togovernment interventions) is not straightforward with optimizationapproach.

• Optimization cannot handle mixing of primal (e.g., physicalquantities such as power generation, gas flows) and dualvariables (e.g., prices).

• Can not easily incorporate corner solutions, i.e. price orquantity=0.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Advantages of complementarity-based approach

While solving economic equilibrium problems as constrainedoptimization problems works for simple problems, the usefulness ofthis approach breaks down quickly as the model becomes morecomplicated:• It is not always clear what to maximize/what objective function is

(e.g., multiple households, multiple countries).• Use of competitive market models can be based on optimization

approach but problematic if economy is not characterized byperfect competition.

• Incorporation of “second-best” features (e.g., distortions due togovernment interventions) is not straightforward with optimizationapproach.

• Optimization cannot handle mixing of primal (e.g., physicalquantities such as power generation, gas flows) and dualvariables (e.g., prices).

• Can not easily incorporate corner solutions, i.e. price orquantity=0.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium as a mixed complementarity problem

• Rutherford (1995) and Mathiesen (1985) showed that manyeconomic models, including general equilibrium models, can becast as a mixed complementary problem.

• Mixed Complementarity Problem (MCP):

Given a function F: Rn −→ Rn, find z ∈ Rn such that F (z) ≥ 0,z ≥ 0, and zT F (z) = 0.

• Example: x f (x) = 0, x (5− x) = 0. Solution: x = 0 (f (x) 6= 0) orx = 5 (f (x) = 0).

• “Mixed”: solution is a mix of equalities f (z) = 0 and inequalitiesf (z) > 0.

• “Complementarity”: z and f (z) are a complementary pair. z is anassociated variable to a certain condition.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium as a mixed complementarity problem

• Rutherford (1995) and Mathiesen (1985) showed that manyeconomic models, including general equilibrium models, can becast as a mixed complementary problem.

• Mixed Complementarity Problem (MCP):

Given a function F: Rn −→ Rn, find z ∈ Rn such that F (z) ≥ 0,z ≥ 0, and zT F (z) = 0.

• Example: x f (x) = 0, x (5− x) = 0. Solution: x = 0 (f (x) 6= 0) orx = 5 (f (x) = 0).

• “Mixed”: solution is a mix of equalities f (z) = 0 and inequalitiesf (z) > 0.

• “Complementarity”: z and f (z) are a complementary pair. z is anassociated variable to a certain condition.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium as a mixed complementarity problem

• Rutherford (1995) and Mathiesen (1985) showed that manyeconomic models, including general equilibrium models, can becast as a mixed complementary problem.

• Mixed Complementarity Problem (MCP):

Given a function F: Rn −→ Rn, find z ∈ Rn such that F (z) ≥ 0,z ≥ 0, and zT F (z) = 0.

• Example: x f (x) = 0, x (5− x) = 0. Solution: x = 0 (f (x) 6= 0) orx = 5 (f (x) = 0).

• “Mixed”: solution is a mix of equalities f (z) = 0 and inequalitiesf (z) > 0.

• “Complementarity”: z and f (z) are a complementary pair. z is anassociated variable to a certain condition.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium as a mixed complementarity problem

• Rutherford (1995) and Mathiesen (1985) showed that manyeconomic models, including general equilibrium models, can becast as a mixed complementary problem.

• Mixed Complementarity Problem (MCP):

Given a function F: Rn −→ Rn, find z ∈ Rn such that F (z) ≥ 0,z ≥ 0, and zT F (z) = 0.

• Example: x f (x) = 0, x (5− x) = 0. Solution: x = 0 (f (x) 6= 0) orx = 5 (f (x) = 0).

• “Mixed”: solution is a mix of equalities f (z) = 0 and inequalitiesf (z) > 0.

• “Complementarity”: z and f (z) are a complementary pair. z is anassociated variable to a certain condition.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium as a mixed complementarity problem

• Rutherford (1995) and Mathiesen (1985) showed that manyeconomic models, including general equilibrium models, can becast as a mixed complementary problem.

• Mixed Complementarity Problem (MCP):

Given a function F: Rn −→ Rn, find z ∈ Rn such that F (z) ≥ 0,z ≥ 0, and zT F (z) = 0.

• Example: x f (x) = 0, x (5− x) = 0. Solution: x = 0 (f (x) 6= 0) orx = 5 (f (x) = 0).

• “Mixed”: solution is a mix of equalities f (z) = 0 and inequalitiesf (z) > 0.

• “Complementarity”: z and f (z) are a complementary pair. z is anassociated variable to a certain condition.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium conditions in a MCP format

An equilibrium in complementarity format is represented by anon-negative vector of activity levels, a non-negative vector of prices,and a non-negative vector of incomes such that:

1 Zero profit conditions (no production activity makes a positiveprofit, output is an associated variable):

−profit ≥ 0, output ≥ 0, outputT(-profit) = 02 Market clearance conditions (excess supply (supply minus

demand) is non-negative for all goods and factors, price is anassociated variable):

supply− demand ≥ 0, price ≥ 0,priceT(supply− demand) = 0

3 Income definition (expenditure equals income):

income = value of endowments

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

• Complementarity-based solution approach: Defines theequilibrium as the solution to a (squared) system of equations.

• Two types of equations: zero profit conditions and marketclearance conditions.

• These can be formulated using optimal choice functions for1 profits in production (= revenue/price minus costs)2 excess demand (= demand minus supply)

that embody the underlying optimizing behavior of economicagents.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (I)

• Economic equilibrium problems are thus represented as asystem of n equations/inequalities in n unknowns.

• Complementarity involves

a. associating each equation with a particular variable, called thecomplementary variable.

b. if the variables are restricted to be non-negative (prices andquantities), then the equations are written as weak inequalities.

• if the equation holds as an equality in equilibrium, then thecomplementary variable is generally strictly positive.

• if the equation holds as a strict inequality in equilibrium, thecomplementary variable is zero.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (I)

• Economic equilibrium problems are thus represented as asystem of n equations/inequalities in n unknowns.

• Complementarity involves

a. associating each equation with a particular variable, called thecomplementary variable.

b. if the variables are restricted to be non-negative (prices andquantities), then the equations are written as weak inequalities.

• if the equation holds as an equality in equilibrium, then thecomplementary variable is generally strictly positive.

• if the equation holds as a strict inequality in equilibrium, thecomplementary variable is zero.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (I)

• Economic equilibrium problems are thus represented as asystem of n equations/inequalities in n unknowns.

• Complementarity involvesa. associating each equation with a particular variable, called the

complementary variable.

b. if the variables are restricted to be non-negative (prices andquantities), then the equations are written as weak inequalities.

• if the equation holds as an equality in equilibrium, then thecomplementary variable is generally strictly positive.

• if the equation holds as a strict inequality in equilibrium, thecomplementary variable is zero.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (I)

• Economic equilibrium problems are thus represented as asystem of n equations/inequalities in n unknowns.

• Complementarity involvesa. associating each equation with a particular variable, called the

complementary variable.b. if the variables are restricted to be non-negative (prices and

quantities), then the equations are written as weak inequalities.

• if the equation holds as an equality in equilibrium, then thecomplementary variable is generally strictly positive.

• if the equation holds as a strict inequality in equilibrium, thecomplementary variable is zero.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (I)

• Economic equilibrium problems are thus represented as asystem of n equations/inequalities in n unknowns.

• Complementarity involvesa. associating each equation with a particular variable, called the

complementary variable.b. if the variables are restricted to be non-negative (prices and

quantities), then the equations are written as weak inequalities.• if the equation holds as an equality in equilibrium, then the

complementary variable is generally strictly positive.

• if the equation holds as a strict inequality in equilibrium, thecomplementary variable is zero.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (I)

• Economic equilibrium problems are thus represented as asystem of n equations/inequalities in n unknowns.

• Complementarity involvesa. associating each equation with a particular variable, called the

complementary variable.b. if the variables are restricted to be non-negative (prices and

quantities), then the equations are written as weak inequalities.• if the equation holds as an equality in equilibrium, then the

complementary variable is generally strictly positive.• if the equation holds as a strict inequality in equilibrium, the

complementary variable is zero.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Exercise Complementarity

We have a partial model with a linear supply and demand function.1 Knowing that the marginal costs of the producer should be

greater/smaller (?) than the selling price and supply isgreater/smaller (?) than demand. Draw how an equilibrium mightlook like (including corner solutions)

2 Write down the complementarity conditions.

Some help:How do we know which inequality is associate with which variableand the direction of the inequality? Economic theory tells you whichvariable must be associated with which inequality and which way theinequality goes.

a. ask the question whether or not a particular direction of theinequality is consistent with economic equilibrium.

b. ask the question, “what must be true if the inequality is strict inequilibrium”?

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

SolutionThree outcomes of partial equilibrium example

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (II)

• Supply of good X with price P. The supply curve exploits thefirm’s optimization decision, equating price with marginal cost:P = MC.

MC ≥ P with the complementarity condition that X ≥ 0 .

• Note that the price equation is complementary with a quantityvariable.

• Suppose that Cost = aX + (b/2)X 2. Marginal cost is then givenby MC = a + bX .

a + bX ≥ P complementarity with X ≥ 0 .

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (III)

• Optimizing consumer utility for a given income and prices willyield a demand function of the form X = D(P,M) where M isincome.

X ≥ D(P,M) with the complementarity condition that P ≥ 0 .

• Note that the quantity equation is complementary with a pricevariable.

• We will suppress income and assume a simple function:X = c + dP where c > 0, d < 0.

X ≥ c + dP complementarity with P ≥ 0 .

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Supply-demand problem illustrating complementarity (IV)

• MC < P cannot be an equilibrium, there is a profit opportunityand more of the good will be produced.

• MC > P can be an equilibrium: the good is unprofitable and isnot produced. Thus X is complementary with the supplyequation.

• X < D(P,M) cannot be an equilibrium, excess demand willcause the price to rise and more will be supplied.

• X > D(P,M) can be an equilibrium, this must mean that thegood is free, and P = 0. Thus P is complementary with thedemand equation.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Consumer behavior

• Standard microeconomic model of consumer choice describes aworld inhabited by many rational individuals (consumers).

• These individuals are capable of making consistent decisionsabout all kinds of issues, and act independently of all others.

• Preferences of a given individual are assumed to be describedby a utility function.

• Consumers are price-takers, no strategic interaction.• They maximize utility subject to an individual budget constraint.• The result of their optimization can be expressed as a set of

demand functions, which completely characterize thepreferences.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Optimal behavior of consumers• Utility maximization problem:

maxX ,Y

U(X ,Y ) s.t. pX X + pY Y ≤ I

• At the optimum (ignoring corner solutions), the marginal rate ofsubstitution, i.e., the rate that the consumer is willing to trade good Y forgood X, holding utility constant, must equal the ratio of the prices of thetwo goods:

∂U(X ,Y )/∂X∂U(X ,Y )/∂Y︸ ︷︷ ︸

MRS

=pX

pYfor X ,Y > 0 .

• Dual expenditure minimization problem:

minX ,Y

pX X + pY Y s.t. U(X ,Y ) = 1

• Solution to this problem is known as the Hicksian compensated demandfunction: HX (pX , pY ) , HY (pX , pY )

• Value function for this optimization problem is the unit expenditurefunction (minimum expenditure necessary to achieve one unit of utility,given goods prices):

E(pX , pY ) ≡ pX HX (pX , pY ) + pY HY (pX , pY )

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Producer behavior

• Firms are characterized by technology: way of converting inputsinto outputs.

• Technology of a firm is represented by a production function.• Firms have to choose both the amount of inputs and the amount

of outputs, in order to maximize profits.• Firms use factor inputs.• Factors are owned by the households and are in fixed supply.• Firms are price-takers, no strategic interaction.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Complex CES function

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Optimal behavior of firms• Objective of the firm is to maximize profits:

maxK≥0,L≥0

pX X︸ ︷︷ ︸revenue

− (pL LX + pK KX )︸ ︷︷ ︸costs

s.t. G (K , L) ≥ X .

• At the optimum (ignoring corner solutions), marginal rate of technicalsubstitution, i.e., the amount by which input K should be decreased inorder to keep output constant following an increase in input L, mustequal the ratio of the two input prices:

∂G(K , L)/∂K∂G(K , L)/∂L︸ ︷︷ ︸

MRTS

=pK

pLfor K , L > 0 .

• Dual cost minimization problem:

minK ,L

pK K + pL L s.t. G(K , L) = 1

• Solution to this problem is known as the conditional factor demandfunction: ZX (pK , pL), ZY (pK , pL)

• Value function for this optimization problem is the unit cost function(minimum costs to produce one unit of output X , given input prices):

CX (pK , pL) ≡ pK ZX (pK , pL) + pL ZY (pK , pL)

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Optimal choice functions for firms and households

• Applying the envelope theorem the partial derivatives of thevalue functions for firms and households, i.e., the unit cost andunit expenditure functions, deliver the optimal reaction of therespective choice variable to a change in prices.

• Unit capital demand by sector i = X ,Y :

∂Ci(pK ,pL)

∂pK=: ai,K (pK ,pL)

• Unit labor demand by sector i = X ,Y :

∂Ci(pK ,pL)

∂pL=: ai,L(pK ,pL)

• Consumer demand for good i = X ,Y per unit of utility:

∂E(pX ,pY )

∂pi=: di(pX ,pY )

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Links between consumers and firms

• Decisions by consumers and firms are linked through:• factor markets: firms buy production factors from consumers• product markets: consumers buy goods & services from firms

• GE model tracks origination and spending of income• Firms receive income by providing goods & services.• Consumers get income by providing factor service to the firms.

• arising from optimizing, price-dependent, and mutuallyconsistent decisions of economic agents on markets.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium conditions for the 2 x 2 model (I)Following the three types of equilibrium conditions, the solution of the simple2 x 2 GE model is given by the following square system of equations:

Non-positive profits for X:

−(pX − CX (pK , pL)) ≥ 0, X ≥ 0, X (−(pX − CX (pK , pL))) = 0

write shorthand for this:

−(pX − CX (pK , pL)) ≥ 0 ⊥ X (1)

Non-positive profits for Y:

−(pY − CY (pK , pL)) ≥ 0 ⊥ Y (2)

Non-positive profits for W:

−(pW − E(pX , pY )) ≥ 0 ⊥ W (3)

Market clearing for X:

X − dX (pX , pY )W ≥ 0 ⊥ pX (4)

Market clearing for Y:

Y − dY (pX , pY )W ≥ 0 ⊥ pY (5)

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium conditions for the 2 x 2 model (II)

Market clearing for W:

W − I/pW ≥ 0 ⊥ pW (6)

Market clearing for K:

K − (aX ,K (pK ,pL)X + aY ,K (pK ,pL)Y ) ≥ 0 ⊥ pK (7)

Market clearing for L:

L− (aX ,L(pK ,pL)X + aY ,L(pK ,pL)Y ) ≥ 0 ⊥ pL (8)

Income balance:I = pK K + pL L . (9)

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Equilibrium conditions for the 2 x 2 model (III)

• Solving the system of weak inequalities (1)–(9) for the unknownsX , Y , W , pX , pY , pK , pL, pW , I yields the general equilibrium ofthe model

• There are “off-the-shelf” algorithms for solving complementarityproblems (e.g. GAMS/MILES and GAMS/PATH solver)

• If a zero profit condition holds as a strict inequality in equilibrium,i.e. profits for that activity are negative, that activity will not used.If a zero profit conditions holds with equality, its associatedactivity level is positive.

• If a market-clearing condition holds as a strict inequality, supplyexceeds demand for that good or factor in equilibrium so its pricemust be zero. If a market-clearing condition holds with equality,the price of the associated good is strictly positive.

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Introduction 2. 2 x 2 GE model Steps in CGE, the 2 x 2 model Literature A simple general equilibrium model with two sectors and two primary factors of production factors

Price normalization

• Equilibrium determines only relative prices but not absolute price level.

• If all prices are multiplied by the same scalar, consumption choices donot change: demand function is homogeneous of degree zero. (thisfollows from continuity and monotonicity assumptions on consumers’preferences).

• Problem: Prices are only defined up to scalar, i.e. there are infinitelymany prices that solve the system of equations (system isover-identified).

• We need one more equation to fix the absolute price level. Thisequation defines the units of account, or the numeraire price.

• But then we have more equations than unknowns. Thus, one of theequations can be dropped and we end up with an exactly identifiedsystem.

• Solution: Walras’ law.

• Consumer fully expends his wealth.• Implication: If all markets but one are cleared, then the remaining

market must also be cleared.

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Outline

1 Introduction

2 A simple general equilibrium model with two sectors and twoprimary factors of production factors

3 Steps in CGE, the 2 x 2 model

4 Literature

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Moving from GE to CGE: Implementing the 2 x 2 Model inGAMS

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Steps in developing and applying a CGE model

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Calibration of a CGE model and SAM

• Calibration of a CGE model is the process of selecting parametersvalues such that the model endogenously reproduces a benchmarkdataset, typically in the form of a Social Accounting Matrix (SAM)starting from the Input Output Table (IOT)

• A SAM represents data on flows of all economic transactions that takeplace within an economy (including external sector transactions)

• Numbers in a SAM represent the values (price times quantity) ofeconomic transactions at a point in time (within a given period of time,i.e. a year). Normalization of prices and quantities.

• Theoretically, a SAM always balances, i.e it is micro-consistentrepresenting an equilibrium (zero profit, market clearance and incomebalance conditions imply that row sums and column sums are equal tozero)

• Empirically, matrix balancing methods have to be applied to obtain amicro-consistent SAM which can be used for model calibration

• Specification of “free” parameters, i.e., parameters which are notcalibrated to reproduce the benchmark dataset, typically draws on other“appropriate” sources

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Schematic overview of a National Accounting Matrix

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A stylized SAM

• For the example of the simple 2 x 2 model, we represent theinitial data by a SAM which consists of:

• Columns corresponding to the production sectors in the model (X ,Y , W ) and the representative consumer CONS

• Rows corresponding to markets with prices pX , pY , pK , pL and pW

as complementary variables .

• The SAM:

X Y W K L CONS rowsum

X (pX ) 100 100Y (pY ) 100 100W (pW ) 200 200L(pL) 25 75 100K (pK ) 75 25 100

CONS (Inc) 100 100 200

colsum 100 100 200 100 100 200

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

CES and calibrated share form• Production and utility activities are typically described by

constant-elasticity-of-substitution functions (CES) which have thefollowing form:

y = f (x) =

(n∑

i=1

αi xρi

)1/ρ

(10)

where x1, . . . , xn denote inputs and the elasticity of substitution is givenby σ = 1/(1− ρ).

• The equivalent calibrated share form that is based on the observedquantities, prices and budget shares from the data:

f (x) = y

(n∑

i=1

θi

(xi

x i

)ρ)1/ρ

where θi =pi x i∑n

j=1 pj x j, i 6= j , denotes the benchmark value share of input i

in total production value given input prices pi . Benchmark quantities andprices are denoted with a bar.

• The associated cost / expenditure function is given by:

c(p) = c

(n∑

i=1

θi

(pi

pi

)1−σ)1/(1−σ)

. (11)

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

How can we calibrate the model to observed data?

Suppose we want to calibrate a technology that combines inputs X and Y toproduce W = f (X ,Y ).

• Observed benchmark data available as an input for calibration:

1 quantities of inputs and output2 prices of inputs and output

• Information on curvature of isoquant, i.e., the elasticity of substitution.

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Calibration

A firm produces output y with factor inputs x i at factor prices pi . Whatvalues of αi are consistent with this information, taking ρ (σ) as given?

αi =pi(x i/y)1−ρ

c

where c is benchmark unit cost:

c =∑

i

pix i .

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Price normalization

• Equilibrium determines only relative prices but not absolute pricelevel.

• If all prices are multiplied by the same scalar, consumptionchoices do not change: demand function is homogeneous ofdegree zero. (this follows from continuity and monotonicityassumptions on consumers’ preferences).

• Problem: Prices are only defined up to scalar, i.e. there areinfinitely many prices that solve the system of equations (systemis over-identified).

• We need one more equation to fix the absolute price level. Thisequation defines the units of account, or the numeraire price.

• But then we have more equations than unknowns. Thus, one ofthe equations can be dropped and we end up with an exactlyidentified system.

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

General equilibrium exchange model (I)1 $title General equilibrium exchange model with alternative choice of2 numeraire

5 SET h Households /h1*h3/6 g Goods /g1*g3/;

8 PARAMETER9 theta(g,h) ’Share parameters’,

10 sigma(h) ’Substitution parameters’,11 omega(g,h) ’Endowment parameters’;

13 * Generate arbitrary data:

15 theta(g,h) = uniform(0,1);16 alias (g,gg);17 theta(g,h) = theta(g,h)/sum(gg,theta(gg,h));18 sigma(h) = uniform(0,3);19 omega(g,h) = uniform(0,1);

22 VARIABLE23 P(g) ’Market price for good g’,24 Y(h) ’Household income’,25 C(h) ’Unit cost of consumption’,26 D(g,h) ’Uncompensated demand’27 XI(g) ’Market excess demand’;

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General equilibrium exchange model (II)1 EQUATION2 income(h) ’Income definition’3 cost(h) ’Unit expenditure function’4 demand(g,h) ’Uncompensated demand function’5 market(g), ’Market excess demand’6 equil(g) ’Equilibrium condition (excess demand=0)’;

8 income(h).. Y(h) =e= sum(g, P(g)*omega(g,h))9 ;

11 cost(h).. C(h) =e= sum(g, theta(g,h) * P(g)**(1-sigma(h)))**(1/(1-sigma(h)));

13 demand(g,h).. D(g,h) =e= Y(h)/C(h) * theta(g,h) * (C(h)/P(g))**sigma(h);

15 market(g).. XI(g) =e= sum(h, D(g,h)) - sum(h, omega(g,h));

17 equil(g).. XI(g) =e= 0;

19 MODEL edgeworth /income.Y, cost.C, demand.D, market.XI, equil.P /;

21 * Fix one price index as numeraire. For example, good "g1":

23 P.FX("g1") = 1;

25 SOLVE edgeworth using mcp;

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

Price normalization

• Absolute equilibrium prices for alternative numeraires (based on modelwalras.gms):

• As illustrated in the left side of the table, when the exchange model is properlyspecified, the same relative equilibrium prices are returned irrespective of thenumeraire specification.

• Likewise the “Walras check”, the imbalance in the omitted numeraire marketclearance condition is zero.

• Conversely, when an imbalance is introduced in the model, both absolute andrelative prices depend on the numeraire choice and the “Walras check” isnonzero, as indicted on the right side of the table.

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Introduction 2 x 2 GE model 3. Steps in CGE, the 2 x 2 model Literature Steps in Developing and Applying a CGE Model

The 2 x 2 GE model in GAMS

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The 2 x 2 GE model in GAMS (I)1 * Declare elasticity parameters:

3 PARAMETERS4 sigma_x "Elasticity of substitution between capital and labor (X

sector)" /.5/5 sigma_y "Elasticity of substitution between capital and labor (Y

sector)" /.5/6 sigma_w "Elasticity of substitution between X and Y (W sector)" /.5/7 lendow "Labor endowment multiplier" /1/;

9 * Declare variables and equations for the 2x2 model:

11 POSITIVE VARIABLES12 X ’X sector output index’13 Y ’Y sector output index’14 W ’Welfare index’15 PX ’Price index for commodity X’16 PY ’Price index for commodity Y’17 PL ’Price index for primary factor L’18 PK ’Price index for primary factor K’19 PW ’Price index for welfare’20 HH ’Household income and expenditure’;

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The 2 x 2 GE model in GAMS (II)Model code

1 EQUATIONS2 PRF_X ’Zero profit for sector X’3 PRF_Y ’Zero profit for sector Y’4 PRF_W ’Zero profit for sector W’

6 MKT_X ’Supply-demand balance for commodity X’7 MKT_Y ’Supply-demand balance for commodity Y’8 MKT_L ’Supply-demand balance for primary factor L’9 MKT_K ’Supply-demand balance for primary factor L’

10 MKT_W ’Supply-demand balance for aggregate demand’

12 I_HH ’Income definition for HH’;

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The 2 x 2 GE model in GAMS (III)Model code

1 * Zero profit conditions:2 PRF_X.. (25/100)*PL**(1-sigma_x) + (75/100)*PK**(1-sigma_x)3 =G= PX**(1-sigma_x);4 PRF_Y.. (75/100)*PL**(1-sigma_y) + (25/100)*PK**(1-sigma_y)5 =G= PY**(1-sigma_y);6 PRF_W.. (100/200)*PX**(1-sigma_w) + (100/200)*PY**(1-sigma_w)7 =G= PW**(1-sigma_w);

9 * Market clearing conditions:10 MKT_X.. 100 * X =E= 0.5*(PW/PX)**sigma_w * W*200;11 MKT_Y.. 100 * Y =E= 0.5*(PW/PY)**sigma_w * W*200;12 MKT_W.. 200 * W =E= HH / PW;13 MKT_L.. 100 * lendow =G= 0.25*100*X*(PX/PL)**sigma_x14 + 0.75*100*Y*(PY/PL)**sigma_y;15 MKT_K.. 100 =E= 0.75*100*X*(PX/PK)**sigma_x16 + 0.25*100*Y*(PY/PK)**sigma_y;

18 * Income definition:19 I_HH.. HH =E= 100*lendow*PL + 100*PK;

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The 2 x 2 GE model in GAMS (IV)Model code

1 * Define model equations and assign variables to equations:2 MODEL SIMPLE_MCP /PRF_X.X, PRF_Y.Y, PRF_W.W, MKT_X.PX, MKT_Y.PY, MKT_L.PL,3 MKT_K.PK, MKT_W.PW, I_HH.HH /;

5 * Initialize variables:

7 X.L=1;Y.L=1;W.L=1;PW.l=1;PX.L=1;PY.L=1;PK.L=1;PL.L=1;PW.L=1;HH.L=200;

9 * Set a "numeraire":

11 PW.FX = 1;

13 * Solve statement:

15 SIMPLE_MCP.ITERLIM = 0;16 SOLVE SIMPLE_MCP USING MCP;

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The 2 x 2 GE model in GAMS (V)Model solution (output from 2_2GEmodel.lst)

2 LOWER LEVEL UPPERMARGINAL

4 ---- VAR X . 1.0000 +INF .5 ---- VAR Y . 1.0000 +INF .6 ---- VAR W . 1.0000 +INF .7 ---- VAR PX 1.000000E-10 1.0000 +INF .8 ---- VAR PY 1.000000E-10 1.0000 +INF .9 ---- VAR PL 1.000000E-10 1.0000 +INF .

10 ---- VAR PK 1.000000E-10 1.0000 +INF .11 ---- VAR PW 1.0000 1.0000 1.0000

EPS12 ---- VAR HH . 200.0000 +INF .

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Exercise

Suppose the labor productivity (or size of workforce) in the economyincreases by 20%. The following figure shows impacts for prices andquantities (comparing the “new” vs. initial equilibrium).

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Exercise

Suppose the labor productivity (or size of workforce) in the economyincreases by 20%. The following figure shows impacts for prices andquantities (comparing the “new” vs. initial equilibrium).

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20% increase in labor endowment: Output from GAMS model1 LOWER LEVEL UPPER

3 ---- VAR X . 1.0435 +INF .4 ---- VAR Y . 1.1429 +INF .5 ---- VAR W . 1.0909 +INF .6 ---- VAR PX . 1.0930 +INF .7 ---- VAR PY . 0.9112 +INF .8 ---- VAR PL . 0.8264 +INF .9 ---- VAR PK . 1.1901 +INF .

10 ---- VAR PW 1.0000 1.0000 1.0000 7.413234E-10

11 ---- VAR HH . 218.1818 +INF .

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20% increase in labor endowment: sensitivity with respect to elasticity ofsubstitution between capital and labor in sector X

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Introduction 2 x 2 GE model Steps in CGE, the 2 x 2 model 4. Literature

Outline

1 Introduction

2 A simple general equilibrium model with two sectors and twoprimary factors of production factors

3 Steps in CGE, the 2 x 2 model

4 Literature

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Books

Bewley, Truman F. (2007). General equilibrium, overlappinggenerations models and optimal growth theory. Harvard UniversityPress.

Ginsburgh, Victor A. and Michiel Keyzer (1997). The Structure ofApplied General Equilibrium Models. Cambridge MIT Press.

Hosoe, Nobuhiro, Kensi Gasawa, and Hideo Hashimoto (2010).Textbook of General Equilibrium Modeling. Palgrave Macmillan.

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Articles

Devarajan, Shantayanan and Delfin S Go (1998). “The SimplestDynamic General-Equilibrium Model of an Open Economy”. In:Journal of Policy Modeling 29.6, pp. 677–714.

Devarajan, S. et al. (1997). “Simple General Equilibrium Modeling”.In: Applied Methods for Trade Policy Analysis - A Handbook. Ed. byJ.P. Francois and K.A. Reiner. Cambridge University Press.Chap. Simple general equilibrium modeling.

Löfgren, Hans, Rebecca Lee Harris, and Sherman Robinson (2001).A standard computable general equilibrium (CGE) model in Gams.TMD Disccusion paper 75. Washington DC: Trade andMacroeconomics Division International Food Policy ResearchInstitute.

Wing, Ian Sue (2004). “Computable General Equilibrium Models andtheir Use in Ecomomy-WIde Policy Analysis: Everything You EverWanted to Know (But Were Afraid to Ask)”.

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