92:. The model first establishes the existence of equilibrium through the standard ArrowâDebreu exposition, then inputs data into all the various sectors, and then applies Scarfâs algorithm (Scarf 1967a, 1967b and Scarf with Hansen 1973) to solve for a price vector that would clear all markets. This algorithm would narrow down the possible relative prices through a simplex method, which kept reducing the size of the ânetâ within which possible solutions were found. AGE modelers then consciously choose a cutoff, and set an approximate solution as the net never closed on a unique point through the iteration process.
60:
Scarf never built an AGE model, but hinted that âthese novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environmentâ (Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price
68:
Scarf's fixed-point method was a break-through in the mathematics of computation generally, and specifically in optimization and computational economics. Later researchers continued to develop iterative methods for computing fixed-points, both for topological models like Scarf's and for models
64:
Most contemporary applied general equilibrium models are numerical analogs of traditional two-sector general equilibrium models popularized by James Meade, Harry
Johnson, Arnold Harberger, and others in the 1950s and 1960s. Earlier analytic work with these models has examined the distortionary
95:
CGE models are based on macro balancing equations, and use an equal number of equations (based on the standard macro balancing equations) and unknowns solvable as simultaneous equations, where exogenous variables are changed outside the model, to give the endogenous results.
61:
vector could be solved for any changes in policies (or exogenous shocks), giving the equilibrium âadjustmentsâ needed for the prices. This method was first used by Shoven and
Whalley (1972 and 1973), and then was developed through the 1970s by Scarfâs students and others.
47:
Scarf's method iterated a sequence of simplicial subdivisions which would generate a decreasing sequence of simplices around any solution of the general equilibrium problem. With sufficiently many steps, the sequence would produce a price vector that clears the market.
65:
effects of taxes, tariffs, and other policies, along with functional incidence questions. More recent applied models, including those discussed here, provide numerical estimates of efficiency and distributional effects within the same framework.
55:
states that a continuous mapping of a simplex into itself has at least one fixed point. This paper describes a numerical algorithm for approximating, in a sense to be explained below, a fixed point of such a mapping (Scarf 1967a:
112:
201:
Kehoe, T.J., Srinivasan, T.N. and
Whalley, J., 2005, Frontiers in Applied General Equilibrium Modeling, In honour of Herbert Scarf, Cambridge, UK: Cambridge University Press
115:, Ludo van der Heyden and John Whalley, and Andrew Feltstein, Ana Matirena-Mantel, Marcus Miller, Donald Richter, Jaime Serra-Puche, John Shoven and John Spencer.
148:
Reprint of the 1990 edition . Classics in
Applied Mathematics, 45. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. xxvi+388 pp.
204:
Shoven, J. B. and
Whalley, J., 1972, "A General Equilibrium Calculation of the Effects of Differential Taxation of Income from Capital in the U.S.",
44:
with empirical data, to provide "âa general method for the explicit numerical solution of the neoclassical modelâ (Scarf with Hansen 1973: 1)
198:, Cowles Foundation for Research in economics at Yale University, Monograph No. 24, New Haven, CT and London, UK: Yale University Press
249:
244:
111:
A list of Scarf's students appears in Kehoe et alia (2005: 5): Ph.D. Students: Terje Hansen, Timothy Kehoe, Rolf Mantel,
214:
Shoven, J.B. and
Whalley, J., 1973, âGeneral Equilibrium with Taxes: A Computational Procedure and an Existence Proofâ,
180:
Scarf, H.E., 1967a, âThe approximation of Fixed Points of a continuous mappingâ, SIAM Journal on
Applied Mathematics
153:
36:
in 1967, in two papers, and a follow-up book with Terje Hansen in 1973, with the aim of empirically estimating the
80:
converged faster than did robust algorithms for continuous functions, when the smooth methods are applicable.
174:
52:
41:
88:
AGE models, being based on ArrowâDebreu general equilibrium theory, work in a different manner than
37:
17:
224:
Velupillai, K.V., 2006, âAlgorithmic foundations of computable general equilibrium theoryâ,
69:
described by functions with continuous second derivatives or convexity or both. Of course, "
159:
8:
187:
Scarf, H.E., 1967b, âOn the computation of equilibrium pricesâ in
Fellner, W.J. (ed.),
73:
149:
156:
33:
77:
238:
124:
76:" for essentially convex and smooth functions and path-following methods for
29:
173:
Cardenete, M. Alejandro, Guerra, Ana-Isabel and Sancho, Ferran (2012).
89:
70:
131:, K. J. Arrow and M. D. Intrilligator, North-Holland, Amsterdam,
189:
Ten
Economic Studies in the tradition of Irving Fischer
236:
146:Introduction to numerical continuation methods.
175:Applied General Equilibrium: An Introduction.
237:
196:The Computation of Economic Equilibria
83:
226:Applied Mathematics and Computation
13:
194:Scarf, H.E. with Hansen, T, 1973,
129:Handbook of Mathematical Economics
14:
261:
211:(3â4), November, pp. 281â321
144:Allgower, Eugene L.; Georg, Kurt
127:, Global analysis and economics,
167:
216:The Review of Economic Studies
138:
118:
105:
1:
221:(4), October, pp. 475â89
99:
53:Brouwer's Fixed Point theorem
7:
206:Journal of Public Economics
28:) models were pioneered by
22:applied general equilibrium
10:
266:
250:Fixed points (mathematics)
245:General equilibrium theory
42:general equilibrium theory
58:
18:mathematical economics
191:, New York, NY: Wiley
135:(1981), pp. 331--370.
50:
84:AGE and CGE models
38:ArrowâDebreu model
231:, pp. 360â69
257:
162:
142:
136:
122:
116:
109:
265:
264:
260:
259:
258:
256:
255:
254:
235:
234:
170:
165:
143:
139:
123:
119:
113:Michael J. Todd
110:
106:
102:
86:
78:diffeomorphisms
34:Yale University
12:
11:
5:
263:
253:
252:
247:
233:
232:
222:
212:
202:
199:
192:
185:
178:
169:
166:
164:
163:
137:
117:
103:
101:
98:
85:
82:
74:Newton methods
9:
6:
4:
3:
2:
262:
251:
248:
246:
243:
242:
240:
230:
227:
223:
220:
217:
213:
210:
207:
203:
200:
197:
193:
190:
186:
183:
179:
176:
172:
171:
161:
158:
155:
154:0-89871-544-X
151:
147:
141:
134:
130:
126:
125:Stephen Smale
121:
114:
108:
104:
97:
93:
91:
81:
79:
75:
72:
66:
62:
57:
54:
49:
45:
43:
39:
35:
31:
30:Herbert Scarf
27:
23:
19:
228:
225:
218:
215:
208:
205:
195:
188:
181:
168:Bibliography
145:
140:
132:
128:
120:
107:
94:
87:
67:
63:
59:
51:
46:
25:
21:
15:
239:Categories
100:References
90:CGE models
184:: 1328â43
177:Springer.
160:2001018
152:
71:global
56:1326).
150:ISBN
229:179
40:of
32:at
26:AGE
16:In
241::
219:40
182:15
157:MR
20:,
209:1
133:1
24:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.