Sequential Model of Economic System and Constrained Pareto Optimality

Received October 11, 2016 Received in revised from December 02, 2016 Accepted February 14, 2017 Available online 15 March 2017 The concept of general economic equilibrium of sequential structures of economic system involves development of the general economic equilibrium model that includes the structure of asset markets. The assets are instruments of income redistribution through the different states of nature of economic system. General economic equilibrium model with the asset markets should provide, on the one hand, an explanation of the relative prices of commodities and, on the other hand, the possibility of analyzing income transfers through the different states of nature and the process of asset prices formation. After defining the appropriate general economic equilibrium model which involves asset markets in the absence of arbitrage process, further analysis is focused on the examination of Pareto optimality of economic system. The existence of imperfect market conditions in the form of incomplete asset markets reduces possibility of income transfers through the different states of nature of economic system, which results that equilibrium allocation is not Pareto optimal allocation. Therefore, in this paper focus is on a constrained form of Pareto optimality in a sequential model of economic system. JEL classification: D50, D52, D61 DOI: 10.14254/1800-5845/2017.13-1.10

of market exchange are based on the concept of general equilibrium of sequential structure of economic system 1 .
The sequential models should in a more realistic way describe economic reality, since in these models economic system is viewed through both time and space dimensions as the possible states of the economic system. Model defined in this paper is a model of sequential market structure consisting of spot commodity markets and asset markets, where assets provide consumers (agents) ability to redistribute income through different periods of time. Unlike a perfectly competitive equilibrium determined at the initial point in time that remains the same for all future periods, in the model developed in this work it will be considered a series of equilibrium states of economic system. General economic equilibrium models with the asset markets should provide the possibility of analyzing income transfers between time periods, and the process of asset prices formation. After defining the appropriate general economic equilibrium model which involves asset markets in the absence of arbitrage process, further analysis is focused on the examination of Pareto optimality of economic system.
With the aim of making our model more realistic, in our analysis of economic system we assume that certain level of uncertainty in the economic processes exists.
In our analysis, we look at two points in time, where at the initial point in time ( 0  t ), consumers (agents) are not sure which of the states will be realized in the following period of time ( 1  t ). The uncertainty that exists for consumers at time Depending on how dividends are paid, assets are divided into two classes -nominal assets, for which dividends are paid in units of account, and real assets, for which dividends are paid in the form of commodity bundle. Nominal assets are defined as a contract that promises delivery of a nominal value, which we present in the form of dividend vectors in each state is a set of assets. Based on defined set of assets and particular dividend vector, dividend value matrix could be defined in the following way: where dividend vectors are represented in the columns of the matrix which describes the asset structure. If we extend dividend value matrix, given by the expression (1), with the price vector at time 0 t  we get extended dividend value matrix W : Matrix   V s generates subspace of income transfers   V s 1 In the context of general equilibrium, sequential structure of the economic system was analyzed by Hicks (1939) and later by Stigum (1969), Grandmont (1974Grandmont ( , 1977 and Drèze (2009).
Consumer portfolio  is determined by net income vector r for all expected states of nature 0 S s  as a linear combination of columns of extended dividend value matrix W : The linear subspace that crosses over columns of extended dividend value matrix represents subspace of income transfers Since the rank of matrix   With this definition we set up the basics of the asset market structure. If market structure is complete, consumer can achieve any net income vector for each state of nature of the economic system. Conversely, if asset market structure is incomplete, then at least one net income vector will be always unavailable to consumer. Since we introduced the assets in economic system in order to give agents (consumers) ability to transfer income through time using dividends earned by a specific portfolio, we also assume that there is an asset market at the initial point in time 0  t . However, we have not analyzed the problem of equilibrium prices formation on asset markets. The necessary condition for the existence of equilibrium of asset markets, and indirectly of commodity markets, is absence of arbitrage process. In other words, we have to show that absence of arbitrage implies the existence of the discount factors at time 0  t such that the asset prices at time 0  t equal to their discounted values, both for complete and incomplete asset market structure.
Suppose that the asset structure is given by dividend value matrix   V s (expression (1) Suppose that there is a dividend value matrix   V s and a vector of asset prices q so that a pair of     , V s q is no-arbitrage process. Furthermore, if we choose any discount factor vec- , then for each asset S j  we have that the price of the asset j is given with the following expression: With next theorem the above -mentioned statements are presented.
Theorem 3. (Fischer (1972), Ross (1976) If you continue to analyze the first row of extended dividend value matrix W at time 0  t , we see that for each asset S j  asset price q j represents the discounted value of dividend at time , V s q represents no -arbitrage process. If there is a complete asset market structure such that rankW S  , then: (i) there is a unique discount factor vector

Collection of discount factor vectors
is orthogonal to each column vector of the matrix W and represents orthogonal subspace of space W . It follows that The orthogonal subspace if asset market structure is incomplete, in which no -arbitrage portfolio  is given by Definition 2. Following theorem provides a description of subspace of income transfers in absence of arbitrage when asset market structure is incomplete.
Theorem 5. (Magill and Quinzii (2001) This theorem asserts the existence of the income transfers subspace in absence of arbitrage when asset market structure is incomplete. If the rank of the extended dividend value matrix rankW J  , then exist H linearly independent vectors that form a basis of vector subspace which is orthogonal to the subspace M . Based on specified orthogonal vector subspace We can now introduce the basic assumptions concerning the consumption set, the vector of initial consumer wealth and consumer utility function.
Basic assumptions 3 : is a set of functions with domain C taking values from the set R and have twice differentiable property; represents the Hessian function of u in point c .
Since we introduced the basic assumptions, we can now formulate a mathematical concept of consumer that satisfies the established assumptions and of economic system with the asset structure in case that uncertainty exists. Definition 6. Consumer is defined as ordered triplet is utility function of consumer.
Definition 7. Economic system with the asset structure under uncertainty is determined with ordered triplet ( , , ) is consumer, and with the dividend value matrix specified by expression (1). It could be represented in the following form: In further analysis, it is necessary to determine the portfolio selection that ensures consumer expenditure plan. Next theorem gives conditions that must be fulfilled by economic system  if portfolio selection could be able to ensure consumption plan.
Theorem 8 (Magill and Quinzio (2002)). Let   be economic system with the asset structure and let a pair represents the spot market equilibrium of economic system  , where the subspace of income transfers is equal to the extended dividend value matrix, (Sum of consumer portfolio selections in economic system is equal to zero), and for the consumer is a solution of consumer portfolio selection problem, which has the following form: represents solution of consumer portfolio selection problem on spot -markets, which is given in the following form: The conditions of Theorem 8 imply that when the economic system is in equilibrium, where the set of net income vectors is generated by the assets, the ordered pair represents an equilibrium solution of consumer problem with the following form: and, , Theorem 8 claims that a collection of net income vectors , as a subspace of income transfers, is generated by the assets in economic system.

THE EXISTENCE OF SOLUTION IN THE SEQUENTIAL MODEL OF ECONOMIC SYSTEM
Having defined the economic system with asset structure under uncertainty, our further analysis will be focused on the study of the existence of general economic equilibrium of incomplete market structure with the dividend payment in the nominal value. With Theorem 8 we decompose consumer problem in the real and financial flows of economic system. We will now explore the existence of equilibrium of the financial system and the existence of no -arbitrage equilibrium of economic system.
Consumer as the owner of the portfolio Consumer i chooses the pair of vectors -the consumption vector i c and portfolio vector i Therefore, portfolio vector i  shows a method of financing the consumption vector i c . Equilibrium of financial system is given by the following definition.
Definition 9. The equilibrium of the financial system and vector of asset prices  q , such that the following relations are satisfied: Thus, market equilibrium equations   Having defined the equilibrium of the financial system and the equilibrium of no -arbitrage economic system, the next theorem gives the relationship between these two equilibriums.
Theorem 11 (Magill and Quinzio (2002) Theorem 11 claims that there is interdependence between equilibrium model of the financial system and the equilibrium model of no-arbitrage economic system. Our further analysis is focused on the application of direct access to the proof of existence of equilibrium solution in the model of the financial system. Furthermore, in the proof of existence of equilibrium in the model of the financial system we limit our analysis to the case of no -arbitrage market exchange processes. The importance of introducing restrictive assumptions about the absence of arbitrage in the process of asset prices formation is clearly evident in the following well -known theorem.
Theorem 12 (Magill and Quinzio (2002) Theorem 12, as a fundamental theorem of asset pricing, claims that for consumer (agent) with a monotone utility function there is solution of the problem of maximum if and only if no -arbitrage condition is satisfied on the financial markets. Specifically, for each financial instrument j , discount factor vector   defines profit from purchasing one unit of asset j by price j q at time ) occurs. Consequently, the previous theorem states that the existence of arbitrage process is equivalent to the existence of a positive discount factor vector.
In the rest of the paper, we will limit our analysis on a set of asset prices formed in the absence of arbitrage. The set of no -arbitrage prices is defined in the following way: In order to prove the existence of an equilibrium of the financial system, we define a demand function for financial instruments: Now, on the basis of established basic assumptions (A3) -(A5) of the utility function of consumers i and of Berge theorem of maximum (Border (1985)), and since the budgetary corre- is half -continuous from above and below, we have that the demand function   Based on established basic assumptions (A1) -(A5) for utility function, and on the assumption that rank of the dividend value matrix is equal to the number of assets, i.e.

   
To prove the existence of equilibrium of financial system we use statements that are contained in the proof of existence of equilibrium solutions given by Talman and Thijssen (2004).
Theorem 13 (Talman and Thijssen (2004)). Suppose that we have a linear problem, whereby F q  represents a solution to the linear programming problem, whose primary form is . We can conclude that the aggregate excess demand function, for the equilibrium price vector as a certain fixed point, is equal to zero, i.e.   0 , which is supposed to be proved.
Thus, we have shown that there is equilibrium of the financial system with a normalized vector of no -arbitrage prices. Furthermore, since we perform decomposition of consumer problem into the real and financial flows (Theorem 8), and since we found that there is interdependence between equilibrium model of the financial system and no-arbitrage equilibrium model of the economic system (Theorem 11), and that there is an equilibrium solution of no -arbitrage equilibrium model, we deduce that there is an equilibrium in general economic equilibrium model with incomplete market structure.

Extended sequential model of economic system
Now we consider a more complex model of the economic system, where we introduce the assumption that there is central authority in the economic system through which money is put into the economic system. In the previous part, we observe market processes of economic system through two periods of time. Now, each of these time periods is divided into three periods in which we observe market processes of economic system. In the first period at time . In the second period at time 0  t consumers buy and sell assets and in the third period consumers buy commodities using money they received in previous period. At time 1  t , for each state of nature of economic system, the same processes occur, expect in the second period, when consumers receive dividends from their assets. Now we can define consumer problem in which money is involved: If we compare the consumer problem defined by the expression (11) with the consumer problem given by the expression (18), we can notice that latter one contained additional system of constraints related to the amount of money generated by the sale of consumer initial wealth. The total amount of money in economic system for each state of nature is   represents an equilibrium solution of consumer problem given by the expression (18); (ii) commodity market equilibrium for all states of nature of economic system Further, based on theorem of existence of the discount factor (Theorem 3), there is a discount factor . Since for no -arbitrage process In Theorem 8 we performed a decomposition of consumer problem into the real and financial flows of the economic system. Based on this decomposition, we formed no -arbitrage equilibrium model of the economic system, and the equilibrium model of the financial system. Furthermore, consumer problem which includes money, using theorem of the existence of the discount factor, has been transformed into a no -arbitrage equilibrium model. Now, using statements about the relationship between no -arbitrage equilibrium model of economic system and model of the financial system (Theorem 11), as well as the theorem that there is an equilibrium solution of financial system model (Theorem 13), we can conclude that there is monetary equilibrium in the model of economic system with the asset structure and money under uncertainty, which has the form of ordered n -tuple of vectors

CONSTRAINED PARETO OPTIMALITY OF SEQUENTIAL MODEL OF ECONOMIC SYSTEM
The assets offer consumers the ability to transfer income through the different states of nature of economic system. However, in the case of incomplete asset market structure, i.e. when there is a reduced ability to transfer income through the different states of nature of the economic system, the question is whether the equilibrium allocation are Pareto optimal allocation. For example, Borch (1962) in his work "Equilibrium and reinsurance market," claimed that in case of incomplete asset market structure the existence of Pareto optimal allocation can not be guaranteed. Diamond (1967), in his paper "The role of a stock market and a general equilibrium model with technological uncertainty", showed that in the case of incomplete market structure with one product in each state of economic space competitive equilibrium is constrained Pareto optimal allocation. Introduced assumption is restrictive and does not represent real processes of economic system. However, in order to facilitate evidence and understanding of a particular form of Pareto optimality, we are forced to introduce some restrictions that differ from the actual processes in the economic system.
Constrained Pareto optimality consists in seeking ways of redistribution (reallocation) of assets by adjustments of prices and consumption allocation on spot markets that would lead to Pareto improvement 6 . Geanakoplos and Polemarchakis (1986), in their work "Existence, Regularity, and Constrained Suboptimality of Competitive Asset Allocations when the Market is Incomplete", showed that the equilibrium of economic system with incomplete market structure is constrained suboptimal. In doing so, they presented that certain Pareto improvements can be achieved by an appropriate redistribution of initial consumers wealth, and then introduced some restrictions on asset markets. So, instead of asking, "Is some equilibrium of economic system optimal?" it could be questioned "whether a particular consumption allocation is optimal when we imposed restrictions due to market imperfections?". Geanakoplos and Polemarchakis (1986), and later, Geankoplos, Magill, Quinzio and Dreze (1990), see economic policy measures of government (as centralized body) as a way for achieving Pareto improvements. However, the question is whether the government in the role of central planning authority with the measures of their economic policy can anticipate all the resulting adjustments on the spot markets as well as all individual utility effects.
Our further analysis is focused on simplified version of the general economic equilibrium with incomplete market structure that is presented by Magill and Quinzio in their first version of the work in 1996 (Magill and Quinzio (2002)). The aim is to investigate the possibility of existence of constrained Pareto optimality in imperfect market conditions. We start from the assumption that there are two period of time - (future), finitely large number of possible states of nature of economic system, finitely number of assets, where the structure of the dividend value matrix   V s is fixed, i.e. is exogenously given, and that there is only one consumption good in each state of nature in the spot markets of economic system. Also, we assume that the distribution of goods to consumers at time 1  t is realized on the basis of the selected portfolio. Financial instruments represent a fundamental mechanism for redistribution of risk at time 1  t . The set of feasible consumption allocations with constraints is given by the next definition.
Definition 16. The consumption allocation Namely, after discounting all spot values using the discount factor vector solution to the general economic equilibrium model of real flows if and only if there is an equilibrium solution to the asset markets equilibrium model. Evidence of the existence of equilibrium solutions have been carried out under the assumption of the absence of arbitrage in asset markets. The importance of introducing restrictive assumptions about the absence of arbitrage in asset markets is evident from the Fundamental Theorem of Asset Pricing.
Given the assumption that the structure of assets is exogenously determined, the research in this paper was primarily concentrated on inquiry into Pareto optimality of the defined general economic equilibrium model with incomplete asset markets. The space of income transfers is defined by the number of linearly independent vectors of dividend value matrix. In addition, the vector of income transfers belongs to the area of dividend value matrix, while the gradient of indirect utility function of consumers, which is the vector of marginal utility of income, belongs to the orthogonal subspace of space dividend value matrix. In the general economic equilibrium model with incomplete asset markets gradients of indirect consumers utility function are not proportional, i.e. there is no compatibility between present value vectors (there is no singular normalized discount factor vector). Thus, the equilibrium consumption allocation is not Pareto optimal allocation. The existence of imperfect market conditions in the form of incomplete asset market reduces possibility of income transfers throughout different states of nature of observed economic system, which results that equilibrium allocations are not Pareto optimal allocations.
The absence of Pareto optimal allocations in defined models required the examination of the allocation of consumption, which is Pareto optimal with the additional restrictions imposed, i.e. research has been focused on finding a limited form of Pareto's optimum in conditions of incomplete asset market. The set of feasible allocations of consumption has been redefined into the set of feasible allocations of consumption in the space of dividend value matrix. Thus defined set of achievable allocations of consumption, relative to space of dividend value matrix, we have defined limited Pareto optimal allocation of consumption and further proven that equilibrium solution to defined model of equilibrium on asset markets represent a limited Pareto optimal allocation relative to space of dividend value matrix. Furthermore, we have proven that equilibrium allocation of consumption is limited Pareto optimal allocation with equilibrium assets price vector if and only if the gradients of indirect utility function of all consumers are proportional as a normalized portfolio, i.e. normalized portfolio gradients represent marginal rate of substitution between investments in assets and consumption.
Based on these results, it can be concluded that development of asset market and adequate redistribution of consumers' wealth is necessary for every economic system that has a goal to obtain Pareto optimality. Additional research in this field should focus on more detailed analysis of influence of various economic policy measures on redistribution of consumer wealth, as well as development of asset markets and development and implementation of new forms of assets into economic system. Adequate implementation of economic policy measures would reduce influence of imperfect market conditions, with obtaining the goal of Pareto optimality of the observed economic system.