Under Conditions Of Non-zero Transaction Costs, General Equilibrium Analysis

After Arrow and Debreu (1954) first strictly proved the existence of Walrasian equilibrium under some relatively strong assumptions, many works to weaken those strong assumptions have already been done.Transaction cost is unavoidable in reality. However, transaction cost is neglected in Arrow and Debreu (1954), or say, Arrow and Debreu (1954) considers positive transaction cost as an implicit premise. Therefore, it is a natural desire of many economists to accommodate transaction cost in general equilibrium model.Foley (1970) has first discussed the existence of general equilibrium with buying and selling prices and transaction costs, which means the transaction cost is linear and the fixed term of it is zero; Hahn (1971) and Starrett (1973) have constructed sequence economy models with budgets' balancing at each date; Kurz (1974) has analyzed transaction cost in a barter exchange equilibrium model; etc. After Kurz (1974), there isn't any other crucially significant contribution in this field.This dissertation shares some similarity with Kurz's (1974) to some extent. In Kurz's (1974) model, he focuses only on a barter exchange economy with zero fixed term transaction cost. However, in contrast to it, the models in this dissertation will on one hand consider all cases, on the other hand, will accommodate the positive fixed term of transaction cost.Why is a difference between zero fixed term and positive fixed term of transaction cost very crucial? It is because that if the fixed term is zero the continuity and convexity of nominal preference will remain but if the fixed term is positive the continuity and convexity will be broken.Therefore, introducing positive fixed term of transaction cost means introducing many singularities into the models, which means a big challenge from both the perspectives of economics and mathematics.To put it briefly, this dissertation reaches three main conclusions: 1) The market supply and demand functions will be discontinuous at many price points, but the move of supply and demand curves will be continuous with respect to price, and at those price points incurring discontinuity, the corresponding traders' utility or profit will be indifferent between making a trade and doing nothing. 2) There must exist a market equilibrium price, at which, however, one of the necessary conditions for the market absolutely and soundly clearing must be either that the cost of negotiation and arrangement among related traders is zero or to adopt some planning way to deal with the special situation. 3) Transaction cost spoils the satisfaction of the first fundamental theorem and second fundamental theorem of welfare economics as well as it ruins the social aggregate welfare through three approaches.In detail, chapter 1 will discuss the simplest case where the economy has only two kinds of commodities and there is no production and money. In this barter economy, transaction cost with positive fixed term is gradually incorporated in the model.In this chapter, some concepts and related notations are first defined. Evidently, because of transaction cost the real consumption of each consumer will usually differ from the nominal consumption (namely the actual purchase from the market). Facing this difference, two approaches may be selected to deal with the problem: one is to establish a model in the nominal terms while the other is to establish a model in real terms. This dissertation chooses the former.I was deeply inspired by Edgeworth Box when I invent the tool of analytic geometry, and the process from Edgeworth Box to a general pattern is briefly summarized. Then, the case with zero fixed term transaction cost is discussed to give a first basic idea about the technique.And then, the positive fixed term is taken into account The positive fixed term of transaction cost won't change the continuity of die nominal budget constraint set, but the continuity of nominal preference is broken. This will give rise to the discontinuity of the offer curve at two critical points, one of which is called upper critical point while the other is called lower critical point. However, the utility is continuous at the upper or lower critical point on the offer curve.Different consumer has different upper and lower critical points. Therefore, the market supply and demand functions (or say, market excess demand function) have to discontinue at many relevant points. 'Finally, this chapter discusses die properties of market equilibrium. It is concluded mat, after considering die positive fixed term of transaction cost, the equilibrium price stilt exists, but a perfect and sound existence of market clearing needs some more condition additional to those conditions in Arrow-Debreu's model.Chapter 2 expands the horizon from two commodities into multi-commodities as welt as lets money be a base for each transaction.For the purpose of making sense gradually, it first introduces the method to tackle a static nonlinear programming problem in question with dynamic programming as a basic pattern under the case without transaction cost. The medium decision variable is very important When good h is under discussion, this dissertation selects the payment for all other goods as the medium decision variable.In the first step, a model is established to choose the maximized solution of die consumptions of other goods under given consumption of good h and value of medium decision variable, and the outcome utility is called partial indirect utility function.In the second step, an indifference map of the partial indirect utility function is first drawn in die 2-dimension space measuring the consumption of good A and the medium decision variable. It is significantly proven the partial indirect utility function is quasiconcave with respect to the consumption of good A and the medium decision variable. Then, like that in the barter economy, in the 2-dimension space measuring the consumption of good h and the medium decision variable, a offer carve reflecting the relationship between die price and die best consumption of good A is drawn naturally.Then, die zero fixed term transaction cost is incorporated in die model via transform from real term to nominal term. Since die continuity remains, no surprise arises.And then, in turn the positive fixed term transaction cost is taken into account. Of course, die nominal preference is discontinuous at relevant points, so dial the partial indirect utility function is discontinuous as well. So, the continuity of the partial indirect utility function is analyzed at once. After die analysts, die indifference map of die partial indirect utility function is shown in the plane, whose discontinuity is revealed.Based on this discontinuous indifference map, die offer curve reflecting die relationship between the price and die best nominal consumption of good h is drawn in die same way, which is discontinuous at many points. By contrasting with die case of barter economy, each consumer's upper optical points probably become not only oae. And die rest is similar to the case of barter economy.Chapter 3 deals vefth the transaction cost in production sector. Of course, the problem with production is also static nonlinear programming, which can be similarly tackled with dynamic programming technique.The benchmark pattern is first discussed by introducing two concepts, partial profit indifference line (curve) and partial feasible border. The offer curve reflecting the relationship between the prices and the actual sale is drawn as well.There are two ways to entirely accommodate transaction cost: one is to incorporate transaction cost in partial feasible border while the other is to incorporate transaction cost in partial profit indifference map. Both of these two ways are discussed.In addition to these two ways, there is another way to tackle transaction cost concisely, i.e. to treat transaction cost as part of productive cost. In this way, the long-run cost function will have its positive fixed term.Anyway, after consider positive fixed term transaction cost, the sale function will likely discontinue at some points. This discontinuity property in production sector is similar to that in consumption sector, which make it reasonable to combine these two sectors together to get the market supply and demand function (or say market excess demand function).It is proven that the market supply and demand functions are discontinuous with respect to the price vector, but their curves will continuously change their shape as the price vector continuously changes. That is, the upper continuity of the mapping from a price vector to its corresponding partial equilibrium price (see definition 2.4) vector is guaranteed.Chapter 4 will first prove the existence of market equilibrium price vector. Then, it will discuss the condition for a perfect and sound existence of market clearing.After these, two important properties of welfare economics will be discussed, and it is proven that the two fundamental theorems of welfare economics are violated if the transaction cost function isn't constant.Finally, it will be moreover concluded that transaction cost will ruin the social aggregate welfare through three approaches: the first is that transaction cost will depreciate consumers' resources so as to reduce consumers' budget payout ability; the second is that transaction cost will deplete scarce resources directly in agents' exchanging activities; the third is that transaction cost will reduce producers' profit so as to indirectly reduce consumers' income.