COMPUTATIONAL ECONOMICS COMPUTATIONAL ECONOMICS DAVID A KENDRICK P RUBEN MERCADO HANS M AMMAN PRINCETON UNIVERSITY PRESS Princeton and Oxford Copyright © 2006 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Control Number: 2005934625 ISBN-13: 978-0-691-12549-7 ISBN-10: 0-691-12549-X British Library Cataloging-in-Publication Data is available This book has been composed in ITC Stone Sans and ITC Stone Serif by Princeton Editorial Associates, Inc., Scottsdale, Arizona Printed on acid-free paper ∞ pup.princeton.edu Printed in the United States of America 10987654321 FOR Alejandro Ann Colin Eveline Liselotte CONTENTS Preface Introduction PART I Once Over Lightly … Growth Growth Model in Excel Finance Neural Nets in Excel Microeconomics Partial Equilibrium in Mathematica Transportation in GAMS Database Databases in Access Finance Thrift in GAMS (with Genevieve Solomon) Portfolio Model in MATLAB PART II Once More … Microeconomics General Equilibrium Models in GAMS Game Theory Cournot Duopoly in Mathematica (with Daniel Gaynor) 10 Stackelberg Duopoly in Mathematica (with Daniel Gaynor) 11 Genetic Algorithms and Evolutionary Games in MATLAB Finance 12 Genetic Algorithms and Portfolio Models in MATLAB Macroeconomics 13 Macroeconomics in GAMS Agent-Based Computational Economics 14 Agent-Based Model in MATLAB Environmental Economics 15 Global Warming in GAMS Dynamic Optimization 16 Dynamic Optimization in MATLAB PART III Special Topic: Stochastic Control Stochastic Control 17 Stochastic Control in Duali 18 Rational Expectations Macro in Duali APPENDIXES A Running GAMS B Running Mathematica C Running the Solver in Excel D Ordered Sets in GAMS E Linearization and State-Space Representation of Hall and Taylor’s Model F Introduction to Nonlinear Optimization Solvers G Linear Programming Solvers H The Stacking Method in GAMS I Running MATLAB J Obtaining the Steady State of the Growth Model References Index PREFACE One of the best ways to learn computational economics is to computational economics One of the best ways to computational economics is to begin with existing models that you modify as you experiment with them This is the approach we take in this book Each chapter presents an economic model First we discuss the economics and mathematics of the model and then analyze its computational form This process enables one to learn the economics and the mathematics of the problem area as well as the computational methods that are used in that area For example, in studying economic growth we make use of a Ramsey-type model The economics of growth theory are first discussed along with the equations that model this process Then the software representation of the model is presented so that the reader can see how the model can be solved on a computer The student can then modify the model in order to analyze its sensitivity to various parameters and functional specifications In the process of experimenting with the model one can gain an improved understanding of both the software and the economic modeling This book grew out of undergraduate and graduate level courses in computational economics taught by us at the University of Texas, ISEG (Argentina), and the University of Amsterdam Several teaching assistants and students also participated in the development of various chapters, notably Daniel Gaynor and Genevieve Solomon The book is intended for use by advanced undergraduates and professional economists and even, as a first exposure to computational economics, for graduate students We expect the coming years to see the development of undergraduate courses with a focus on economic modeling along the lines we have outlined in these pages Moreover, we envisage the development of a two-course sequence in computational economics in graduate programs The introductory course would have a broad economicmodeling focus with an approach similar to that used in some of the chapters in this book The second course would focus on algorithms and numerical methods Part of our motivation for writing this book is spelled out in a couple of paragraphs that are taken from a paper the three of us wrote entitled “Computational Economics: Help for the Underestimated Undergraduate.”1 These comments— although written for that paper—apply equally well to the present work: The ubiquitous personal computer has filtered deeply through the lives of college undergraduates; however undergraduate education in economics has so far failed to take full advantage of this sweeping change We are underestimating the learning ability and insufficiently challenging a whole generation of undergraduate students in economics Our thesis is that computational economics offers a way to improve this situation and to bring new life into the teaching of economics in colleges and universities With its early focus on algorithms, computational economics seemed well-suited for a relatively small group of graduate students and unlikely to have much impact on undergraduates However, that is changing as we are discovering that computational economics provides an opportunity for some students to move away from too much use of the lecture-exam paradigm and more use of a laboratory-paper paradigm in teaching undergraduate economics This opens the door for more creative activity on the part of the students by giving them models developed by previous generations and challenging them to modify those models The modifications can be altering the models to make them applicable to the student’s interest or finding weaknesses in the model that can be strengthened by changes in the structure of the model In the process the students become much more involved in their own education The organization of the chapters in the book reflects primarily the outlines of the courses at the University of Texas, which are designed to allow students to find an area of computational economics of particular interest and to pursue that area Since some of them are interested in microeconomics, others in macroeconomics, and others in finance, an effort is made to give a quick and broad exposure to models across a range of fields early in the semester Then the range is covered again later in the semester in greater depth The book is structured to follow this pattern In Part I there is a “once over lightly” treatment of computational economics examples from a number of fields This is then repeated in greater depth and complexity in Part II Part III covers an Solver dialog box, 17f, 26, 33–36 Solver options dialog box, 18f, 35, 35f target capital stock, 20–21, 23 terminal capital stock target, 17, 18, 22 time horizons in, calculating, 22–23 turnpike property, 22 Use Automatic Scaling, 36 expectations formation, 248 rational model See rational expectations model Fackler, Paul L., 4, 422 Fair, Ray C., 363–66, 419 Fair model, 247 Fair–Taylor method, 4, 363–66, 364f, 369–73 Fanchon, P., 360, 423 feedback gain matrix G input window, in Duali, 380, 382f rules, 385 feed-forward model, 26 finance, in Excel, 25–36 in GAMS, 91–117 in MATLAB, 119–46, 223–45 firm behavior, theory of, 37, 38, 41, 46–49 firm.nb file, 47 fiscal policy, 348 Fischer, S., 342n, 376n, 377n, 418 Fisher, P G., 363, 419 fitness_gagame function, in MATLAB, 211–14 fitness_gaportfol function, in MATLAB, 230–31 Flaschel, Peter, 360, 418 flow variables, 299 fmincon function, in MATLAB, 130, 131, 132, 134–35, 404 for loops, in MATLAB, 123, 127, 243 forcing term, 294–95, 301 form specifications dialog box, in Duali, 344, 345f, 369, 370f, 371, 374, 375f Fortran, 323 Froeb, Luke M., 188, 419 functions, in MATLAB code, 207 See also individual functions by name evolutionary games, 210–18 portfolio models, 227–31 G and g dialog box, in Duali, 379–80, 381f GA See genetic algorithms game representation, 205t game theory, in Mathematica, 174–76 Cournot quantity competition, 177–88 Stackelberg leadership model, 189–99 in MATLAB, 201–21 GAMS, alias statement, 100, 103, 142, 165 all keyword, 61 card keyword, 100 Cobb-Douglas function, 157, 162 criterion function, 93–96, 103, 105, 106–10 DICE model See DICE model display statement, 61, 101 dual variables, 64 environmental economics, 291–308 Equations keyword, 60, 107, 143–45, 146 Evanchik model, 98–110 finance, 91–117, 149–71 general equilibrium models, 149–71 global warming model See global warming model growth models, 23, 291 Hall and Taylor model, 253–65, 360 input-output model, 149–53, 154–55, 170, 171 lagged variables, 255 loss function, 259, 260 macroeconomics, 247–65 Markowitz model using, 142–46 microeconomics, 55–66, 149–71 minimizing keyword, 61 Model keyword, 60–61, 146 multiple solutions, 60, 65 nonlinear programming, 309 ORANI model, 170 ord keyword, 100 ordered sets, 394–95 parameter keyword, 58–59, 65–66, 145 PARAMETER section in SAM-based model, 165 parameter statement, 58–59, 65–66 portfolio models, 119, 142–46 Positive Variable statement, 106 production price model, 149, 150–71 running, 389–90 scalar keyword, 59, 142–143, 145 SET specification, 253, 254–56 shadow prices, 64 simultaneous equations, 151–52 Solve keyword, 61, 109–10, 146 Solve statement, 257 Solve Summary, 62–64, 110 stacking method, 61, 257, 310, 411–12 table keyword, 58, 59–60, 65–66, 102, 105, 143 table statement, 59–60, 65–66 thrift model, 91–117 transportation model in, 55–66 using keyword, 61 Variables keyword, 60, 106–7, 143, 146 Garson, G D., 36, 419 GAUSS, 310, 332 Gaynor, Daniel, 173–88, 189–99 General Algebraic Modeling System See GAMS general equilibrium models, 267, 302 computable, 160–70 experiments, 170–71 in GAMS, 149, 156–70 Johansen-style, 149, 166–71, 340 SAM-based, 149, 161–66 genetic algorithms, 4, 122n and evolutionary games in MATLAB, 201–21 and portfolio models in MATLAB, 223–45 Gibbons, Robert, 173n, 175n, 185n, 188, 420 Gilli, Manfred, 289, 420 global dynamics approach, 360 global optimization algorithms, 223 global warming model, 291, 292f in GAMS, 298–302, 301t, 304–8 mathematical model, 291–98 Goffe, Bill, 67, 241n, 420 Goldberg, D., 34, 221, 245, 420 Gomis, Pedro, 360, 420 gradient method approach, in MATLAB, 129 gradient optimization function, in MATLAB, 119, 223 Greek letter symbols, in Mathematica, 42, 47 growth models, 9–23 computational, 9, 12–23, 13f, 15f–18f in Excel, 9–23, 291, 297 in GAMS, 23, 291 mathematical, 9, 10–12, 12f, 297 numerical, 9, 22 one-sector, 9, 291, 296 steady state, 414–15 theoretical, 22–23 Hahn, Frank, 157, 417 Hall, Robert E., 339, 347, 347n, 348n, 420 Hall and Taylor model, 248–53 in GAMS, 253–60, 360 program code, 261–65 linearization and state-space representation, 396–402 nonlinear, 342 in state-space form, 340–42, 343n handcrafted feedback rules, 385 Hansen, Lars Peter, 385, 420 Haro, A., 360, 420 Harris, R Lee, 171, 421 Hattori, Miwa, 132, 139–41, 310, 333 Havenner, Arthur, 350, 424 Hemphill, Carter, 122 Herbert, Ric D., 360, 420 Hercules system, 161n Hicks, John R., 247n, 420 Holly, S., 360, 363, 377n, 385, 419, 420 Howe, M., 385, 423 Howitt, P., 23, 417 Hughes Hallett, Andrew J., 5, 360, 363, 377n, 385, 419, 420 I-A form model, 367–70 income distribution, 154 incomplete information games, 174 INFEASIBLE, in GAMS Solve Summary, 62, 110 initagents function, in MATLAB, 279–80 initial capital stock, 16f, 18, 21, 22, 23 initial conditions, 10, 11, 12, 322 initpopdet function, in MATLAB, 227–28 initpoprand_gagame function, in MATLAB, 210–11 initpoprand_gaportfol function, in MATLAB, 232–34, 235f initsugarscape function, in MATLAB, 274, 275–79, 277f, 279f input statements, in Leontief function, 38–40 input-output model, 149–53, 154–55, 171 input-output relationship, in Access database example, 84–86, 84t intertemporal optimization models, 247–48 inv function, in MATLAB, 327 IS-LM models, 247–52 isoprofit curves, 197, 199f Johansen, Leif, 166, 169–70, 420 Johansen-style computable general equilibrium, 149, 166–71, 340 joins, in databases, 68–69, 69t, 77–78, 78f, 80 experiment, 83 Jones, C., 23, 420 Judd, Kenneth L., 4, 23, 289, 420 Juillard, Michel, 363, 420 Kahn, C M., 363, 418 Kalman filter, 321, 355–56 Kalvelagen, Erwin, 145 Kaplan, Todd, 188, 419 Kendrick, David A., 2n, 3, 4, 5, 23, 56, 57, 66, 67, 70, 79, 84, 85, 137–38, 161, 169–70, 247n, 316, 319n, 320, 321, 323, 332, 333, 337, 339n, 341, 348n, 350, 350n, 359, 360, 365, 385, 417, 418, 420, 421, 422 Keynes, John Maynard, 247n, 421 Kim, Seung-Rae, 142–46, 171, 421 Kline, Kevin, 122 Koopmans, Tjalling, 55, 85, 421 Kozicki, Sharon, 385, 421 labor demand curve, 49, 49f lagged variables, 255, 361, 371 Lagrangian functions, in Mathematica, 43–44 structure, 44n Lahiri, Supriya, 302, 418 learning mechanisms, 260, 350, 355, 360 LeBaron, Blake, 289, 421 Lee, Myong Hwal, 359, 421 Leontief, Wassily, 149, 421 Leontief function, 37–40, 40f, 41, 53 contour lines, 40f Leontief-type models, 154–55 Leontief.nb file, 38 Letson, David, 303, 421 LEVEL column, in GAMS output file, 62, 111 Lin, Kuan Pin, 332, 421 Lin, Lihui, 23, 422 linear models, 242, 363, 365 and nonlinear models compared, 169, 170t linear programming, 1, 3, 61 solvers, 407–10, 408f–409f linearization Hall and Taylor model, 396–402 and nonlinear model compared, 169, 170t technique, 161 Lofgren, H., 171, 421 loss function, 259, 260 Lucas, R., 332, 423 macroeconomic models, 1, 242, 321–23, 349, 361 control theory methods and, 360 in GAMS, 247–65 in MATLAB, 267–89, 310, 323–31, 333–36 stabilization and, 337 trend-deviation, 385 macroeconomics models standard, 247–52 Macrosolve, 259n main window, in Duali, 344f, 369, 370f Maksymonko, Paul, 122 Malinvaud, Edmund, 171, 421 Manne, Alan S., 302, 421 Margarita, S., 36, 418 MARGINAL column, in GAMS output file, 63, 64 Marimon, Ramon, 5, 421 MarketEquil.nb file, 49–50 Markowitz, Harry, 119, 421 Markowitz model, 121, 122, 223 GAMS code for, 139–41 MATLAB code for, 139–41 using optimization function, 130–36 Mathematica, Clear commands, 193 Evaluate function, 51 Plot[ ] function, 45–46, 51–52, 181–82, 184–85 Plot3D function, 39 SetAttributes commands, 193 Simplify command, 188, 190 Solve function, 45, 180, 190 MATLAB, 1, 337 axis square statement, 279 bitand function, 206 bitcmp function, 207 bitor function, 206 bitshift function, 206 Clear commands, 128 crossover function, 215–16, 231 fmincon function, 130, 131, 132, 134–35 max function, 126 ones(m, n) function, 132–33 Optimization Toolbox, 119, 130 randperm function, 274–75 scripts, 207 see function, 281–85 Simulink system and, 360 subplot function, 281 max function, in MATLAB, 126 May, Robert, 221, 422 McAdam, Peter, 5, 420 McCarl, Bruce A., 303, 422 McKibbin, Warwick J., 303, 422 measurement errors, 337, 350 Meeraus, Alexander, 56, 57, 66, 302, 418, 421 memory management, 324 Mercado, P Ruben, 23, 137–38, 247n, 337, 339n, 350, 421, 422 method count window, in Duali, 382, 383f method dialog box, in Duali, 357, 358f, 381, 382f microfoundations, 247 Min[ ] function, in Mathematica, 39 minimizing keyword, in GAMS, 61 Miranda, Mario J., 4, 422 mixed strategy equilibrium, 175n Model keyword, in GAMS, 60–61, 146 model size dialog box, in Duali, 345, 346f, 352f, 371, 371f, 379, 380f MODEL STATUS, in GAMS, 62 monetary policy, 348, 385 monopoly markets, 173 monopoly outcome, 193–95, 196–97 Monte Carlo optimization, 3–4 in Duali, 351, 355–59, 378–79, 379f, 382–84 in MATLAB, 119, 122–24, 136, 137–38, 223 Moore, George, 363, 417 moveagent function, in MATLAB, 275, 285–86 multiple solutions, in GAMS, 60, 65 multiple-equation linear models, 385 multiplicative noise terms, 337 multiplicative uncertainty, 319–21, 350 mutation function, in MATLAB, 216–17, 231 Nash, John, 177 Nash equilibrium, 175–76, 218 Cournot quantity competition and, 177–78, 183, 185–88 evolutionary game and, 205 neighbor function, in MATLAB, 281–85 neural nets, 2, activation and combination functions, 27f estimation problems, 34 in Excel, 25–36 hidden layer, 26–27, 28 input layer, 26–27, 28 interconnection topology, 26 layers, 26f learning scheme, 26 natural, 26 output layer, 26–28 processing elements, 26 share prices, 30f, 31f spreadsheet, 32f neurons, 26–27 new query dialog box, in Access, 76, 76f Newton method, 18, 27, 33, 241 NLP See nonlinear programming nonconvex problem, 244f nonlinear optimization solvers, 3, 403–6, 404f nonlinear programming, in GAMS, 309 solver, 257 nonnegativity constraints, 134 Nordhaus, William D., 291, 301–2, 422 See also DICE model NORMAL COMPLETION, in GAMS Solve Summary, 62, 110 normal form game, 177 normport function, in MATLAB, 228–30 Nowak, Martin, 221, 422 numerical growth models, 9, 22 object oriented programming, 267 OLF See open-loop feedback oligopolistic markets, 173, 189–99 Cournot quantity competition and, 177–88 static models, 176 one-sector growth model, 9, 291, 296 one-shot quantity games, 176, 197 one-shot simultaneous move game, 177 ones(m, n) function, in MATLAB, 132–33 open-loop feedback, 332, 350–51, 353–59 optimal control, 4, 301, 309, 331–32, 363, 374–75, 375–76 optimal control techniques, 373–84 Optimization Toolbox, in MATLAB, 119, 130 options variable, in MATLAB, 134 ORANI model, in GAMS, 170 ord keyword, in GAMS, 100 ordered sets, in GAMS, 394–95 Oudiz, G., 363, 422 Paez, Pedro, 171, 422 parameter keyword, in GAMS, 58–59, 145 parameter statement, in GAMS, 58–59, 65–66, 105 parentsdet function, in MATLAB, 214, 231 parentsrand function, in MATLAB, 234–37, 236f, 238f Park, H J., 260, 343n, 422 Parmenter, Brian, 170, 171, 419 partial equilibrium, in Mathematica consumer theory, 43–46 firm behavior theory, 46–49 market equilibrium, 49–53 utility and production functions, 37–43 Passinetti, L., 154n, 171, 422 passive learning mechanisms, 350, 355, 360 penalty weights, 94 Pesaran, M Hashem, 363, 418, 422 Phillips curve, 248, 250, 251 Pindyck, Robert S., 360, 422 Pindyck model, 367–70, 371 Plot[ ] function, in Mathematica, 46–47, 51–52, 181–82, 184– 85 Plot3D function, in Mathematica, 39 PlotLabel option, in Mathematica, 44n PlotRange option, 53 policy variables, 291, 292f in Duali, 340, 342–49, 350, 354, 355f, 360, 361, 363, 369, 372, 373f portfolio models, 93 in GAMS, 119, 142–46 in MATLAB, 119–46 advanced, 240–45 experiments, 245 genetic algorithms and, 223–45 refinements, 231–40 results, 231, 232f random generation, 127–29 selection, 126–27 Positive Variables keywords, in GAMS, 106 Powell, Alan, 171, 419 priority parameters, 94 Prisoners’ Dilemma game, 174, 175–76, 177 iterated, in MATLAB, 201, 204–5 production prices model, in GAMS, 149, 153–56 projection-updating mechanism, 332, 355 pure strategy Nash equilibrium, 186 Q matrix input window, in Duali, 379, 381f QLP See quadratic linear problem quadratic criterion functions, 121, 312, 315, 316, 322, 337 quadratic linear problem, 332, 342, 366, 374–75, 377 general form, 315–21 simple form, 311–15 quadratic programming, 4, 119 quadratic tracking, 93–96, 236, 246, 337, 369 quantity games, 173, 176 queries option, in Access, 72–79, 80f–82f Raman, Ramesh, 56, 57, 418 Ramsey model, 1, 9, 23, 296 RAND function, in Excel, 23 rand function, in MATLAB, 127 random number generator, in MATLAB, 127 random search, in MATLAB, 119, 129 random shock generation, 350, 355, 356 random terms dialog box, in Duali, 352f, 357, 357f, 377–78, 378f randperm function, in MATLAB, 274–75 Rardin, R., 410, 422 rational expectations model, 247, 248 closed economy model and, 362–63 in Duali, 366–73 dynamic simulation, 369–73 experiments, 385 optimal policy analysis, 373–84 solution methods, 363–66 reaction curve, 182f, 183f, 185f, 187f reaction function, 178, 181, 183–84, 186–87, 190, 199f recursive solution method, 309–10 Reinert, K A., 171, 423 relationships in databases, 68, 69t, 70–79, 82t, 83, 85–86, 89 See also Access, databases in Return command, in Mathematica, 38 Ricardo, David, 154, 155n, 422 Riccati equations, 325 Riccati matrices and vectors, 309n, 315, 316, 317, 318, 325–26, 328–29 RICE model, 302 Richels, Richard G., 302, 421 Robinson, Sherman, 171, 419, 421 Roland-Holst, D W., 171, 423 Ros, J., 23, 423 Rotemberg, J., 248, 423 Rust, John, Rustem, Berc, 385, 423 Sachs, J., 363, 422 Sala-i-Martin, X., 23, 418 Salas, Huber, 207, 310, 333 SAM-based model, 149, 161–66 Samuelson, Paul, 1, 55, 419 Sargent, Thomas J., 36, 332, 385, 420, 423 scalar keyword, in GAMS, 59, 142–43, 145 scaling, in Excel, 36 Schmidheiny, Kurt, 4–5 Schneider, Uwe A., 303, 422 Schwaitzberg, Scott, 137–38 Scott, Andrew, 5, 421 scripts, in MATLAB code, 207 see function, in MATLAB, 281–85 Sengupta, J., 360, 423 sequential move games, 176, 189 sequential moves, in game theory, 174 SET specification, in GAMS, 253, 254–56 Set Target Cell, in Excel, 17 SetAttributes commands, in Mathematica, 193 shadow prices in GAMS, 64 Shah, Vivek, 106n Shiells, C R., 171, 423 Shift-Enter command, 38 Shoven, J., 171, 423 Show Table dialog box, in Access, 76, 76f Shupp, F., 350, 423 sigmoid function, 28, 29f, 31 Silberberg, E., 410, 423 Simplify command, 188, 190 Sims, Christopher A., 363, 423 Simulink system, 360 simultaneous games, 176, 177 simultaneous moves, in game theory, 174 Social Accounting Matrix, model based on See SAM-based model Solomon, Genevieve, 91–117 Solow, Robert, 55, 419 Solve function, in Mathematica, 45, 47, 180, 190 Solve keyword, in GAMS, 61, 109–10, 146, 257 Solve Summary, in GAMS, 62–64, 110–13 Solver, in Excel dialog box, 2, 17f, 22, 26, 33–36, 33f running, 393 Solver options dialog box, in Excel, 18f, 35, 35f SOLVER STATUS, in GAMS, 61–62, 110 squasher function, 28 Sraffa, Piero, 154, 155n, 423 Stackelberg duopoly, in Mathematica, 174 and Cournot duopoly compared, 192–99 stacking method, in GAMS, 61, 257, 310, 411–12 stack.nb file, 189 state-space form, 337, 340–49 state-space representation, Hall and Taylor model, 396–402 stochastic control, 385 in Duali, 337–38, 349–59 with parameter updating, 355–59 stochastic terms dialog box, in Duali, 344, 344f, 351, 351f, 369f, 370f, 377, 378f Stokey, N., 332, 423 Stone, J R N., 161, 423 Stoutjesdijk, Ardy, 66 structure(s), data-type named, 267 subplot function, in MATLAB, 281 Suen, W., 410, 423 Sugarscape grow-back rule G1, 288 Sugarscape model, 267, 268–89 Suresh, Shyam Gouri, 201n Sutton, John, 170, 171, 419 table[ ] function, in Mathematica, 51–52 table keyword, in GAMS, 58, 59–60, 65–66, 102, 104–6, 143 Taylor, John B., 4, 9, 247, 339, 347, 347n, 348n, 372–73, 377, 385, 419, 420, 423 See also rational expectations model Taylor, Lance, 23, 421, 423 teaching methods, computational economics courses, 4–5 Terna, P., 36, 418 Tesfatsion, Leigh, 289, 418, 420, 423 Thompson, Gerald L., 4, 423 Thore, Sten, 4, 423 thrift model, in GAMS, 91, 98–110 Tinsley, Peter A., 350, 385, 421, 424 Tools menu, in Excel, 17 top statement, in MATLAB, 128 tracking function, 322, 323 transportation model in GAMS, 56–62, 56–64 trend-deviation macroeconomic models, 385 Tucci, Marco, 241n, 424 Turnovsky, Stephen, 350, 424 turnpike property, 22 two-point boundary value problems, 248 Uhlig, Harald, 9, 423 UNBOUNDED, in GAMS Solve Summary, 62, 110 uncertainty, 319, 337, 349–59, 351, 355, 359, 360 Use Automatic Scaling in Excel, 36 using keyword, in GAMS, 61 VAR section, in GAMS output file, 63, 110–11 VARIABLE section, in GAMS output file, 111–12 variables, declaring and defining in MATLAB, 125 Variables keyword, in GAMS, 60, 106–7, 143, 146 Varian, Hal R., 4, 424 variance costs, in MATLAB, 125–26 variance-covariance matrix, 321, 356 variations, calculus of, 309 Vincent, D P., 170, 171, 419 Wallis, K., 383, 424 Walras, Leon, 157 Walras’ law, 158 Werden, Gregory J., 188, 419 Whalley, J., 171, 423 Wilcoxen, Peter J., 171, 303, 419, 422 Winker, P., 289, 420 wizards, in Access, 76 Wolfram, Stephen, 53, 188, 424 Woodford, M., 248, 377, 385, 423, 424 Zadrozny, Peter, 363, 424 ZT elements input window, in Duali, 374, 375f ... References Index PREFACE One of the best ways to learn computational economics is to computational economics One of the best ways to computational economics is to begin with existing models that you... Algorithms and Portfolio Models in MATLAB Macroeconomics 13 Macroeconomics in GAMS Agent-Based Computational Economics 14 Agent-Based Model in MATLAB Environmental Economics 15 Global Warming in GAMS Dynamic... COMPUTATIONAL ECONOMICS COMPUTATIONAL ECONOMICS DAVID A KENDRICK P RUBEN MERCADO HANS M AMMAN PRINCETON UNIVERSITY