Applied general equilibrium

# Applied general equilibrium

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In mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...

, applied general equilibrium (AGE) models were pioneered by Herbert Scarf
Herbert Scarf
Herbert Eli "Herb" Scarf is an American mathematical economist and Sterling Professor of Economics at Yale University. He is a member of the American Academy of Arts and Sciences...

at Yale
YALE
RapidMiner, formerly YALE , is an environment for machine learning, data mining, text mining, predictive analytics, and business analytics. It is used for research, education, training, rapid prototyping, application development, and industrial applications...

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
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...

model with empirical data, to provide "“a general method for the explicit numerical solution of the neoclassical model”
(Scarf with Hansen 1973: 1)

Scarf's method iterated a sequence of simplical 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.

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 a sense 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), and then was developed through the 1970s by Scarf’s students and others.

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 described by functions with continuous second derivatives or convexity or both. Of course, "global Newton method
Newton's method
In numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...

s" for essentially convex and smooth functions and path-following methods for diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

s converged faster than did robust algorithms for continuous functions, when the smooth methods are applicable.

## AGE and CGE models

AGE models, being based on Arrow–Debreu general equilibrium theory, work 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. 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.