# Applied general equilibrium

Applied general equilibrium

Applied General Equilibrium (AGE) models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow-Debreu General equilibrium model with empirical data, to provide "“a general method for the explicit numerical solution of the neoclassical model”(Scarf with Hansen 1973: 1)

Scarfs method was an algorithm (based on simplical subdivisions) that would narrow a ‘net’ around the possible solution to the quantified general equilibrium problem. With enough iteration, the net was tightened sufficiently to choose a cut off point, giving a price vector that could clear the market [ [http://www.newschool.edu/cepa/publications/workingpapers/SCEPA%20Working%20Paper%202008-1%20Kahn.pdf See Mitra-Kahn (2008)] , pages 20-26 for a quick review of the history of AGE models and their relation to Computable general equilibrium models] .

Brouwer's Fixed Point theorem 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 asense to be explained below, a fixed point of such a mapping (Scarf 1967a: 1326).

Scarf never built an AGE model, but hinted that “these novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environment”(Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price vector could be solved for any changes in policies (or exogenous shocks), giving the equilibrium ‘adjustments’ needed for the prices. This method was first used by Shoven and Whalley (1972 and 1973), but was used up through the 1970s by Scarf’s students and students’ students [A list is provided in Kehoe et al. (2005: 5): Ph.D. Students: Terje Hansen, Timothy Kehoe, Rolf Mantel, Michael Todd, Ludo van der Heyden and John Whalley, and the ‘Yale core’: Andrew Feltstein, Ana Matirena-Mantel, Marcus Miller, Donald Richter, Jaime Serra-Puche, John Shoven and John Spencer.] .

Scarf’s simplex method was quite appealing to the economic theorists at the time, who wanted constructive proofs for the existence of equilibrium. Lance Taylor [ [http://www.newschool.edu/cepa/publications/workingpapers/SCEPA%20Working%20Paper%202008-1%20Kahn.pdf See Mitra-Kahn (2008)] , page 22] noted that as a solution algorithm, it ignored 2nd derivatives and curvature information, thus being much less effective than Newton methods for the highly convex model specifications which AGE modelers at the time were creating.

Furthermore, in a recent article, Velupillai (2006) proves how the AGE models, can not be precisely solved numerically and points out that "From a very elementary (classical) recursion theoretic standpoint it is easy to show the absence of a computable (and constructive) content" (Velupillai 2006: 366).

AGE and CGE models

AGE models developed separately and independently from Computable general equilibrium (CGE) models, as illustrated in Mitra-Kahn (2008). However, today the two terms are used inter-changeably, and the Arrow-Debreu theoretical background of AGE models, is then attributed to CGE models, which are in widespread use across the world. This is a confusion as the AGE literature has no connections with the CGE models until the mid 1980's when AGE models are going out of fashion due to the high cost of implementing them, and the AGE modellers begin to adopt the CGE methods of using Social Accounting Matrices for their data set-ups, and macro balancing equations as opposed to Arrow-Debreu and General Equilibrium Theory.

AGE models, being based on Arrow-Debreu general equilibrium theory works in a different manner than CGE models. The model first establishes the existence of equilibrium through the standard Arrow-Debreu exposition, and then inputs data into all the various sectors, and then apply Scarf’s algorithm (Scarf 1967a, 1967b and Scarf with Hansen 1973) to solve for a price vector that would clear all markets instantly. This algorithm would narrow down the possible relative prices through a simplex method, which kept reducing the size of the ‘net’ 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.

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. This process is uncontroversial, but also completely separate from AGE modelling and formal general equilibrium theory.

References

Bibliography

*Scarf, H.E., 1967a, “The approximation of Fixed Points of a continuous mapping”, SIAM Journal of Applied Mathematics 15: 1328-43
*Scarf, H.E., 1967b, “On the computation of equilibrium prices” in Fellner, W.J. (ed.), "Ten Economic Studies in the tradition of Irving Fischer", New York, NY: Wiley
*Scarf, H.E. with Hansen, T, 1973, "The Computation of Economic Equilibria", Cowles Foundation for Research in economics at Yale University, Monograph No. 24, New Haven, CT and London, UK: Yale University Press
*Mitra-Kahn, Benjamin H., 2008, " [http://www.newschool.edu/cepa/publications/workingpapers/SCEPA%20Working%20Paper%202008-1%20Kahn.pdf Debunking the Myths of Computable General Equilibrium Models] ", "SCEPA Working Paper" 01-2008
*Kehoe, T.J., Srinivasan, T.N. and Whalley, J., 2005, Frontiers in Applied General Equilibrium Modeling, In hounour of Herbert Scarf, Cambridge, UK: Cambridge University Press
*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.", "Journal of Public Economics" 1 (3-4), November, pp. 281-321
*Shoven, J.B. and Whalley, J., 1973, “General Equilibrium with Taxes: A Computational Procedure and an Existence Proof”, "The Review of Economic Studies" 40 (4), October, pp. 475-89
*Velupillai, K.V., 2006, “Algorithmic foundations of computable general equilibrium theory”, "Applied Mathematics and Computation" 179, pp. 360-69

ee also

* Computable general equilibrium models or CGE models
* General equilibrium theory
* Arrow-Debreu

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