Computing Economic Equilibria with Smooth Homotopy Methods
計算經濟均衡的光滑同倫算法
Student thesis: Doctoral Thesis
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Award date  28 Aug 2019 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(f206f1a871cb4c2c97c085e0d847622f).html 

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Abstract
Since the existence of economic equilibrium was proved in the 1950s, the general equilibrium theory has become one of the cornerstones of modern economic studies. It is widely used to analyze problems in areas as diverse as international trade, public finance, asset pricing, transportation, etc. The computation of economic equilibrium in general equilibrium models plays a very important role in practice, and attracts a lot of attention in economics, operations research and computer science. The equilibrium is defined as prices at which all agents in the economy maximize their utilities and the market of all commodities clears. With the market clearing condition and the optimality condition for each agent, the problem of finding equilibria can usually be reformulated as finding zeros of the socalled aggregate excess demand function.
Homotopy pathfollowing methods are considered to be the only theoretically sound algorithms for computing equilibria for economies of general cases. In topology, a homotopy means two continuous functions can be continuously deformed into the other. The idea of homotopy method is to solve an easy problem that gradually deforms to the original problem. To ensure the convergence of homotopy methods for computing economic equilibria, the regularity of homotopy system is required, which follows immediately if the economy has continuously differentiable excess demand functions. However, excess demand functions are not differentiable in many cases. In this thesis, we investigate three different cases of the general equilibrium model, pure exchange economies under some nonsmooth utilities, economies with constant returns production, and economies with incomplete assets markets. Based on a standard homotopy argument, smooth pathfollowing algorithms are developed for all these economies.
Many common used utilities are nondifferentiable, including Leontief utilities and separable piecewise linear utilities. Although several efficient algorithms have been developed in the literature for some nondifferentiable utilities, these algorithms are typically not applicable to economies where consumers have different types of utilities. We find that the nondifferentiability can be smoothed with a new regularization technique, which is naturally compatible with homotopy methods. In this way, standard homotopy method can then be applied in computing an equilibrium under some nondifferentiable utilities.
For economies with production technologies, firms make production plans to maximize their profits according to prices over commodities. The behavior of firms makes the economy more complicated and precludes the direct application of homotopy methods for computing equilibria. We look into economies that have constant returns to scale production, and convert equilibrium condition into that of a corresponding pure exchange economy, together with some complementarity conditions. Then, a smooth homotopy method can be applied to these two problems simultaneously.
For economies with incomplete assets market, things are much more complicated, since the excess demand functions can be discontinuous. The discontinuity happens at prices where some asset becomes redundant in terms of nominal returns. In a mathematical view, the discontinuity occurs when the asset return matrix drops rank and households' budget sets drop dimensions. We show that this problem can be resolved by introducing a new return matrix which has constant rank and restricting the prices in a smooth submanifold of the Euclidean space. Then, a continuous differentiable excess demand function is derived, and a smooth pathfollowing algorithm can be developed for computing an equilibrium.
Homotopy pathfollowing methods are considered to be the only theoretically sound algorithms for computing equilibria for economies of general cases. In topology, a homotopy means two continuous functions can be continuously deformed into the other. The idea of homotopy method is to solve an easy problem that gradually deforms to the original problem. To ensure the convergence of homotopy methods for computing economic equilibria, the regularity of homotopy system is required, which follows immediately if the economy has continuously differentiable excess demand functions. However, excess demand functions are not differentiable in many cases. In this thesis, we investigate three different cases of the general equilibrium model, pure exchange economies under some nonsmooth utilities, economies with constant returns production, and economies with incomplete assets markets. Based on a standard homotopy argument, smooth pathfollowing algorithms are developed for all these economies.
Many common used utilities are nondifferentiable, including Leontief utilities and separable piecewise linear utilities. Although several efficient algorithms have been developed in the literature for some nondifferentiable utilities, these algorithms are typically not applicable to economies where consumers have different types of utilities. We find that the nondifferentiability can be smoothed with a new regularization technique, which is naturally compatible with homotopy methods. In this way, standard homotopy method can then be applied in computing an equilibrium under some nondifferentiable utilities.
For economies with production technologies, firms make production plans to maximize their profits according to prices over commodities. The behavior of firms makes the economy more complicated and precludes the direct application of homotopy methods for computing equilibria. We look into economies that have constant returns to scale production, and convert equilibrium condition into that of a corresponding pure exchange economy, together with some complementarity conditions. Then, a smooth homotopy method can be applied to these two problems simultaneously.
For economies with incomplete assets market, things are much more complicated, since the excess demand functions can be discontinuous. The discontinuity happens at prices where some asset becomes redundant in terms of nominal returns. In a mathematical view, the discontinuity occurs when the asset return matrix drops rank and households' budget sets drop dimensions. We show that this problem can be resolved by introducing a new return matrix which has constant rank and restricting the prices in a smooth submanifold of the Euclidean space. Then, a continuous differentiable excess demand function is derived, and a smooth pathfollowing algorithm can be developed for computing an equilibrium.