The ABCs of RBCs: An Introduction to Dynamic Macroeconomic Models

by George McCandless

Harvard University Press, 2008 Cloth: 978-0-674-02814-2 | eISBN: 978-0-674-03378-8 Library of Congress Classification HB141.M395 2008 Dewey Decimal Classification 339.015195

ABOUT THIS BOOK | REVIEWS | TOC

ABOUT THIS BOOK

The ABCs of RBCs is the first book to provide a basic introduction to Real Business Cycle (RBC) and New-Keynesian models. These models argue that random shocks—new inventions, droughts, and wars, in the case of pure RBC models, and monetary and fiscal policy and international investor risk aversion, in more open interpretations—can trigger booms and recessions and can account for much of observed output volatility.

George McCandless works through a sequence of these Real Business Cycle and New-Keynesian dynamic stochastic general equilibrium models in fine detail, showing how to solve them, and how to add important extensions to the basic model, such as money, price and wage rigidities, financial markets, and an open economy. The impulse response functions of each new model show how the added feature changes the dynamics.

The ABCs of RBCs is designed to teach the economic practitioner or student how to build simple RBC models. Matlab code for solving many of the models is provided, and careful readers should be able to construct, solve, and use their own models.

In the tradition of the “freshwater” economic schools of Chicago and Minnesota, McCandless enhances the methods and sophistication of current macroeconomic modeling.

REVIEWS

The ABCs of RBCs fills an important gap in the textbook literature in modern macroeconomics. It presents, in a self-contained, agile fashion, a variety of dynamic macroeconomic models that are currently used to understand business cycles in developed as well as developing countries.
-- Martin Uribe, Duke University

The ABCs of RBCs is a focused text that does exactly what it sets out to do, which is to explain the current techniques for the analysis of real business cycle models. A highly useful primer.
-- Kim Border, California Institute of Technology

For every economist who ever felt stymied by the process of numerically approximating equilibria of [RBC] models, and for all current and future graduate students who are or who will be aiming to learn this technology, this book is a truly wonderful treatment--a user's guide--to moving from the set of first-order conditions and budget and resource constraints that characterize the solutions to these models to putting these models on a computer and making them operational...This is a textbook in the truest sense--every word, diagram, and table have been assembled to make learning this technology as transparent as possible. For the most part, the exposition is truly outstanding. As you read the text, you feel as if they were detailed notes written by someone in the process of learning and distilling a lot of information about a difficult and nuanced topic. And this is exactly how it was done, as noted by McCandless in the text...While there is some overlap with other treatments in the literature, there are aspects of the book that are done better here than elsewhere...The book excels exactly where many texts do not by being remarkably transparent and explicit. If you occasionally feel that advanced economics textbooks are a bit like those impossible "how-to-assemble furniture instructions" we have all sweated over, this book is for you...This book would be a terrific fit in a first year graduate course in macroeconomics...or used in a second year course that focuses on quantitative analysis of macroeconomic models.
-- Lee E. Ohanian Journal of Economic Literature

TABLE OF CONTENTS

PART ONE BASIC MODELS AND SOLUTION METHODS
1 The Basic Solow Model 7
1.1 The Basic Model 7
1.2 Technological Growth 10
1.3 The Golden Rule 11
1.4 A Stochastic Solow Model 12
1.5 Log-Linear Version of the Solow Model 14
1.5.1 Capital 15
1.5.2 Output 16
1.6 Reprise 18
2 Savings in an OLG Model 19
2.1 The Basic OLG Model 20
2.1.1 An Example Economy 26
2.2 Dynamics 27
2.3 A Stochastic Version 28
2.4 Reprise 32
2.5 Matlab Code Used to Produce Figure 2.2 32
3 Infinitely Lived Agents 33
3.1 A Robinson Crusoe Economy with Fixed Labor 34
3.1.1 Variational Methods 34
3.2 A Robinson Crusoe Economy with Variable Labor 38
3.2.1 The General Model 38
3.2.2 Solution for a Sample Economy 40
3.3 A Competitive Economy 41
3.4 The Second Welfare Theorem 44
3.4.1 An Example Where the Representative Agent Economy and the
Decentralized Economy Are Not Equal 45
3.5 Reprise 49
4 Recursive Deterministic Models 50
4.1 States and Controls 51
4.2 The Value Function 52
4.3 A General Version 55
4.4 Returning to Our Example Economy 58
4.4.1 Another Version of the Same Economy 59
4.5 An Approximation of the Value Function 60
4.6 An Example with Variable Labor 63
4.7 Reprise 66
4.8 Matlab Code for Figures 4.2 and 4.3 67
5 Recursive Stochastic Models 69
5.1 Probability 70
5.2 A Simple Stochastic Growth Model 71
5.3 A General Version 74
5.3.1 The Problem of Dimensionality 76
5.4 The Value Function for the Simple Economy 77
5.4.1 Calculating the Value Functions 78
5.5 Markov Chains 80
5.6 Reprise 86
5.7 Matlab Code 87
6 Hansen's RBC Model 89
6.1 Hansen's Basic Model 90
6.2 Log Linearization Techniques 94
6.2.1 The Basics of Log Linearization 95
4 6.2.2 Uhlig's Method of Log Linearization 98
6.3 Log-Linear Version of Hansen's Model 100
6.3.1 Solution Using Jump Variables 104
6.3.2 Calibration of the Log-Linear Model 106
6.3.3 Variances of the Variables in the Model 109
6.4 Hansen's Model with Indivisible Labor 112
6.4.1 Stationary State 115
6.4.2 Log-Linear Version of the Indivisible Labor Model 118
6.5 Impulse Response Functions 120
6.6 Reprise 124
6.7 Appendix 1: Solving the Log-Linear Model 124
6.8 Appendix 2: Blanchard and Kahn's Solution Method 128
6.8.1 General Version 129
6.8.2 Stochastic Shocks 131
6.8.3 Hansen's Model and Blanchard-Kahn 132
6.8.4 The Generalized Schur Method 134
6.9 Matlab Code 142
6.9.1 Solution to Basic Hansen Model 142
6.9.2 Approximating the Variances 143
6.9.3 Code for Appendix 2 144
7 Linear Quadratic Dynamic Programming 146
7.1 Taylor Approximations of the Objective Function 147
7.2 The Method of Kydland and Prescott 148
7.2.1 AnExample 151
7.2.2 Solving the Bellman Equation 154
7.2.3 Calibrating the Example Economy 155
7.3 Adding Stochastic Shocks 157
7.3.1 The Example Economy 160
7.3.2 Calibrating the Example Economy 163
7.4 Hansen with Indivisible Labor 166
7.5 Impulse Response Functions 172
7.5.1 Vector Autoregressions 174
7.6 An Alternative Process for Technology 176
7.7 Reprise 178
7.8 Matlab Code 178
PART TWO EXTENSIONS OF THE BASIC RBC MODEL
8 Voney: Cash in Advance 183
8.1 Cooley and Hansen's Model 184
8.2 Finding the Stationary State 190
8.3 Solving the Model Using Linear Quadratic Methods 195
8.3.1 Finding a Quadratic Objective Function 196
8.3.2 Finding the Economy Wide Variables 199
8.4 Solving the Model Using Log Linearization 202
8.4.1 The Log Linearization 202
8.4.2 Solving the Log-Linear System 205
8.4.3 Impulse Response Functions 209
8.5 Seigniorage 210
8.5.1 The Model 212
8.5.2 The Stationary State 215
8.5.3 Log-Linear Version of the Model 218
8.6 Reprise 222
8.7 Appendix 1: CES Utility Functions 223
8.8 Appendix 2: Matrix Quadratic Equations 230
8.9 Matlab Code for Solving the CES Model with Seigniorage 233
9 Money in the Utility Function 236
9.1 The Model 237
9.2 Stationary States 240
9.3 Log-Linear Version of the Model 242
9.4 Seigniorage 246
9.4.1 The Full Model 248
9.4.2 Stationary States 248
9.4.3 Log Linearization 251
9.5 Reprise 256
10 Staggered Pricing Model 258
10.1 The Basic Model 259
10.1.1 The Final Goods Firms 259
10.1.2 The Intermediate Goods Firms 261
10.1.3 TheFamily 266
10.1.4 Equilibrium Conditions 267
10.1.5 The Full Model 269
10.2 The Stationary State 270
10.3 Log Linearization 273
10.3.1 fog Linearization of the Firm's Problem 273
10.3.2 The Final Goods Pricing Rule 273
10.3.3 The Intermediate Goods Pricing Rule 273
10.3.4 Inflation Equation (Phillips Curve) 275
10.3.5 Log Linear Version of the Model 277
10.4 Solving the Log Linear Model 279
10.4.1 Impulse Response Functions 285
10.5 Inflation Adjustment for Nonoptimizing Firms 290
10.5.1 The Stationary State 291
10.5.2 Log Linearization 293
10.5.3 Solving the Model 295
10.5.4 Impulse Response Functions 300
10.6 Reprise 304
11 Staggered Wage Setting 306
11.1 The Labor Bundler 307
11.1.1 First-Order Conditions for Families 310
11.1.2 TheRest of the Model 312
11.1.3 Equilibrium Conditions 313
11.1.4 The Full Model 314
11.2 The Stationary State 315
11.3 Log Linearization 317
11.4 Solving the Model 321
11.4.1 Impulse Response Functions 325
11.5 Reprise 327
12 Financial Markets and Monetary Policy 329
12.1 Working Capital 331
12.1.1 Households 331
12.1.2 Firms 333
12.1.3 Financial Intermediaries 334
12.1.4 The Full Model 334
12.1.5 The Stationary State 336
12.1.6 Log Linear Version of the Model 340
12.1.7 Impulse Response Functions 345
12.1.8 Economy with Annual Inflation of 100 Percent 347
12.1.9 Comparative Impulse Response Functions 349
12.2 Central Banking and Monetary Policy Rules 352
12.2.1 The Model with a Taylor Rule 353
12.2.2 Stationary States 356
12.2.3 Log-Linear Version and Its Solution 357
12.2.4 Comparing a Taylor Rule to a Friedman Rule 362
12.3 Reprise 368
13 Small Open Economy Models 370
13.1 The Preliminary Model 371
13. 1. I1 The Household 371
13.1.2 The Firm 374
i 13.1.3 Equilibrium Conditions 374
13.1.4 Stationary State 374
13.1.5 The Dynamic (Log-Linear) Model 376
13.2 Model with Capital Adjustment Costs 379
13.3 Closing the Open Economy 386
13.3.1 Interest Rates and Country Risk 386
13.3.2 The Dynamic Version 388
13.4 The "Closed" Open Economy with Money 393
13.4.1 The Open Economy Conditions 394
13.4.2 The Household 395
13.4.3 Firms 396
13.4.4 Equilibrium Conditions 396
13.4.5 The Full Model 397
13.4.6 The Stationary State 398
13.4.7 Log-Linear Version ofFull Model 400
13.5 Reprise 410
References 411
Index 417

The ABCs of RBCs: An Introduction to Dynamic Macroeconomic Models

by George McCandless

Harvard University Press, 2008 Cloth: 978-0-674-02814-2 eISBN: 978-0-674-03378-8

The ABCs of RBCs is the first book to provide a basic introduction to Real Business Cycle (RBC) and New-Keynesian models. These models argue that random shocks—new inventions, droughts, and wars, in the case of pure RBC models, and monetary and fiscal policy and international investor risk aversion, in more open interpretations—can trigger booms and recessions and can account for much of observed output volatility.

George McCandless works through a sequence of these Real Business Cycle and New-Keynesian dynamic stochastic general equilibrium models in fine detail, showing how to solve them, and how to add important extensions to the basic model, such as money, price and wage rigidities, financial markets, and an open economy. The impulse response functions of each new model show how the added feature changes the dynamics.

The ABCs of RBCs is designed to teach the economic practitioner or student how to build simple RBC models. Matlab code for solving many of the models is provided, and careful readers should be able to construct, solve, and use their own models.

In the tradition of the “freshwater” economic schools of Chicago and Minnesota, McCandless enhances the methods and sophistication of current macroeconomic modeling.

REVIEWS

The ABCs of RBCs fills an important gap in the textbook literature in modern macroeconomics. It presents, in a self-contained, agile fashion, a variety of dynamic macroeconomic models that are currently used to understand business cycles in developed as well as developing countries.
-- Martin Uribe, Duke University

The ABCs of RBCs is a focused text that does exactly what it sets out to do, which is to explain the current techniques for the analysis of real business cycle models. A highly useful primer.
-- Kim Border, California Institute of Technology

For every economist who ever felt stymied by the process of numerically approximating equilibria of [RBC] models, and for all current and future graduate students who are or who will be aiming to learn this technology, this book is a truly wonderful treatment--a user's guide--to moving from the set of first-order conditions and budget and resource constraints that characterize the solutions to these models to putting these models on a computer and making them operational...This is a textbook in the truest sense--every word, diagram, and table have been assembled to make learning this technology as transparent as possible. For the most part, the exposition is truly outstanding. As you read the text, you feel as if they were detailed notes written by someone in the process of learning and distilling a lot of information about a difficult and nuanced topic. And this is exactly how it was done, as noted by McCandless in the text...While there is some overlap with other treatments in the literature, there are aspects of the book that are done better here than elsewhere...The book excels exactly where many texts do not by being remarkably transparent and explicit. If you occasionally feel that advanced economics textbooks are a bit like those impossible "how-to-assemble furniture instructions" we have all sweated over, this book is for you...This book would be a terrific fit in a first year graduate course in macroeconomics...or used in a second year course that focuses on quantitative analysis of macroeconomic models.
-- Lee E. Ohanian Journal of Economic Literature

TABLE OF CONTENTS

PART ONE BASIC MODELS AND SOLUTION METHODS
1 The Basic Solow Model 7
1.1 The Basic Model 7
1.2 Technological Growth 10
1.3 The Golden Rule 11
1.4 A Stochastic Solow Model 12
1.5 Log-Linear Version of the Solow Model 14
1.5.1 Capital 15
1.5.2 Output 16
1.6 Reprise 18
2 Savings in an OLG Model 19
2.1 The Basic OLG Model 20
2.1.1 An Example Economy 26
2.2 Dynamics 27
2.3 A Stochastic Version 28
2.4 Reprise 32
2.5 Matlab Code Used to Produce Figure 2.2 32
3 Infinitely Lived Agents 33
3.1 A Robinson Crusoe Economy with Fixed Labor 34
3.1.1 Variational Methods 34
3.2 A Robinson Crusoe Economy with Variable Labor 38
3.2.1 The General Model 38
3.2.2 Solution for a Sample Economy 40
3.3 A Competitive Economy 41
3.4 The Second Welfare Theorem 44
3.4.1 An Example Where the Representative Agent Economy and the
Decentralized Economy Are Not Equal 45
3.5 Reprise 49
4 Recursive Deterministic Models 50
4.1 States and Controls 51
4.2 The Value Function 52
4.3 A General Version 55
4.4 Returning to Our Example Economy 58
4.4.1 Another Version of the Same Economy 59
4.5 An Approximation of the Value Function 60
4.6 An Example with Variable Labor 63
4.7 Reprise 66
4.8 Matlab Code for Figures 4.2 and 4.3 67
5 Recursive Stochastic Models 69
5.1 Probability 70
5.2 A Simple Stochastic Growth Model 71
5.3 A General Version 74
5.3.1 The Problem of Dimensionality 76
5.4 The Value Function for the Simple Economy 77
5.4.1 Calculating the Value Functions 78
5.5 Markov Chains 80
5.6 Reprise 86
5.7 Matlab Code 87
6 Hansen's RBC Model 89
6.1 Hansen's Basic Model 90
6.2 Log Linearization Techniques 94
6.2.1 The Basics of Log Linearization 95
4 6.2.2 Uhlig's Method of Log Linearization 98
6.3 Log-Linear Version of Hansen's Model 100
6.3.1 Solution Using Jump Variables 104
6.3.2 Calibration of the Log-Linear Model 106
6.3.3 Variances of the Variables in the Model 109
6.4 Hansen's Model with Indivisible Labor 112
6.4.1 Stationary State 115
6.4.2 Log-Linear Version of the Indivisible Labor Model 118
6.5 Impulse Response Functions 120
6.6 Reprise 124
6.7 Appendix 1: Solving the Log-Linear Model 124
6.8 Appendix 2: Blanchard and Kahn's Solution Method 128
6.8.1 General Version 129
6.8.2 Stochastic Shocks 131
6.8.3 Hansen's Model and Blanchard-Kahn 132
6.8.4 The Generalized Schur Method 134
6.9 Matlab Code 142
6.9.1 Solution to Basic Hansen Model 142
6.9.2 Approximating the Variances 143
6.9.3 Code for Appendix 2 144
7 Linear Quadratic Dynamic Programming 146
7.1 Taylor Approximations of the Objective Function 147
7.2 The Method of Kydland and Prescott 148
7.2.1 AnExample 151
7.2.2 Solving the Bellman Equation 154
7.2.3 Calibrating the Example Economy 155
7.3 Adding Stochastic Shocks 157
7.3.1 The Example Economy 160
7.3.2 Calibrating the Example Economy 163
7.4 Hansen with Indivisible Labor 166
7.5 Impulse Response Functions 172
7.5.1 Vector Autoregressions 174
7.6 An Alternative Process for Technology 176
7.7 Reprise 178
7.8 Matlab Code 178
PART TWO EXTENSIONS OF THE BASIC RBC MODEL
8 Voney: Cash in Advance 183
8.1 Cooley and Hansen's Model 184
8.2 Finding the Stationary State 190
8.3 Solving the Model Using Linear Quadratic Methods 195
8.3.1 Finding a Quadratic Objective Function 196
8.3.2 Finding the Economy Wide Variables 199
8.4 Solving the Model Using Log Linearization 202
8.4.1 The Log Linearization 202
8.4.2 Solving the Log-Linear System 205
8.4.3 Impulse Response Functions 209
8.5 Seigniorage 210
8.5.1 The Model 212
8.5.2 The Stationary State 215
8.5.3 Log-Linear Version of the Model 218
8.6 Reprise 222
8.7 Appendix 1: CES Utility Functions 223
8.8 Appendix 2: Matrix Quadratic Equations 230
8.9 Matlab Code for Solving the CES Model with Seigniorage 233
9 Money in the Utility Function 236
9.1 The Model 237
9.2 Stationary States 240
9.3 Log-Linear Version of the Model 242
9.4 Seigniorage 246
9.4.1 The Full Model 248
9.4.2 Stationary States 248
9.4.3 Log Linearization 251
9.5 Reprise 256
10 Staggered Pricing Model 258
10.1 The Basic Model 259
10.1.1 The Final Goods Firms 259
10.1.2 The Intermediate Goods Firms 261
10.1.3 TheFamily 266
10.1.4 Equilibrium Conditions 267
10.1.5 The Full Model 269
10.2 The Stationary State 270
10.3 Log Linearization 273
10.3.1 fog Linearization of the Firm's Problem 273
10.3.2 The Final Goods Pricing Rule 273
10.3.3 The Intermediate Goods Pricing Rule 273
10.3.4 Inflation Equation (Phillips Curve) 275
10.3.5 Log Linear Version of the Model 277
10.4 Solving the Log Linear Model 279
10.4.1 Impulse Response Functions 285
10.5 Inflation Adjustment for Nonoptimizing Firms 290
10.5.1 The Stationary State 291
10.5.2 Log Linearization 293
10.5.3 Solving the Model 295
10.5.4 Impulse Response Functions 300
10.6 Reprise 304
11 Staggered Wage Setting 306
11.1 The Labor Bundler 307
11.1.1 First-Order Conditions for Families 310
11.1.2 TheRest of the Model 312
11.1.3 Equilibrium Conditions 313
11.1.4 The Full Model 314
11.2 The Stationary State 315
11.3 Log Linearization 317
11.4 Solving the Model 321
11.4.1 Impulse Response Functions 325
11.5 Reprise 327
12 Financial Markets and Monetary Policy 329
12.1 Working Capital 331
12.1.1 Households 331
12.1.2 Firms 333
12.1.3 Financial Intermediaries 334
12.1.4 The Full Model 334
12.1.5 The Stationary State 336
12.1.6 Log Linear Version of the Model 340
12.1.7 Impulse Response Functions 345
12.1.8 Economy with Annual Inflation of 100 Percent 347
12.1.9 Comparative Impulse Response Functions 349
12.2 Central Banking and Monetary Policy Rules 352
12.2.1 The Model with a Taylor Rule 353
12.2.2 Stationary States 356
12.2.3 Log-Linear Version and Its Solution 357
12.2.4 Comparing a Taylor Rule to a Friedman Rule 362
12.3 Reprise 368
13 Small Open Economy Models 370
13.1 The Preliminary Model 371
13. 1. I1 The Household 371
13.1.2 The Firm 374
i 13.1.3 Equilibrium Conditions 374
13.1.4 Stationary State 374
13.1.5 The Dynamic (Log-Linear) Model 376
13.2 Model with Capital Adjustment Costs 379
13.3 Closing the Open Economy 386
13.3.1 Interest Rates and Country Risk 386
13.3.2 The Dynamic Version 388
13.4 The "Closed" Open Economy with Money 393
13.4.1 The Open Economy Conditions 394
13.4.2 The Household 395
13.4.3 Firms 396
13.4.4 Equilibrium Conditions 396
13.4.5 The Full Model 397
13.4.6 The Stationary State 398
13.4.7 Log-Linear Version ofFull Model 400
13.5 Reprise 410
References 411
Index 417