# Essay on Arrow-Debreu Private Ownership Economy

Published: 2021-07-01
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Arrow-Debreu model claims that in certain economic conditions such as perfect competitive market or demand independence, there is a set price at a point where the entire supply and all demand of a commodity are equal (Basile, Graziano, & Pesce, 2013). The model assumes that differentiation of products depends on the time it is brought to the market for trading. In order to apply the Arrow-Debreu model, it is assumed that the number commodities in the market are known. In a privately owned economy, all products traded on the market are the property of the consumers and the stable price of a commodity at a particular time and place is at the point where the total demand is equal to the aggregate supply. The corresponding point of demand and supply is modeled as per the Walrasian equilibrium. The model assumes satisfaction of the Walrasian equilibrium because the market operation is assumed to be a perfect market with free competition and consumers choose what to a consumer as per their preference.

Walrasian Equilibrium

Walrasian equilibrium it is also known as the competitive equilibrium because it assumed to be the equilibrium that arises in a market with many buyers and sellers. The prices of the commodities keep on changing with time depending on the change in supply and demand (Bryant, 2010). The equilibrium is based on assumptions such as retailers targeting to maximize profits, consumers maximizing utility at a particular budget and that all commodities in the market are bought at a given time. The assumptions if satisfied means that the price set on a product is at equilibrium point.

Fixed Point

A fixed point is an element of a function of a domain that is mapped to it by the function (Khamsi, & Kirk, 2011). It is also recognized as the invariant point. It can change from time to time depending on the point mapped by the function. For examples, if t is the fixed point in function f(x) then the fixed point remain f (t) = t.

Fixed Point Theorem

A fixed point theorem is a result that shows that a function has at least one invariant point (Khamsi, & Kirk, 2011). Fixed point theorem takes place in some conditions that ensure that their fixed point in a function.

Brouwers Fixed Point Theorem

Brouwers fixed point theorem is a fixed point on a topology. It states that in a continuous function has one fixed point and is named after Luitzen Brouwer (Khamsi, & Kirk, 2011).

Kakutanis Fixed Point Theorem

Kakutanis fixed point theorem is a fixed theorem that acts as a generalization of Brouwers fixed point theorem (Khamsi, & Kirk, 2011). The generation of the Brouwers theorem is done in two ways, one the point to point mapping and the continuous mapping in the upper continuity. First, it associates each input with more than one output.

Market Excess Demand

Market excess demand is the amount of a commodity that is higher than what is available on the market. It happens when the consumers require more of a product than what the market is offering at a particular time (Trockel, 2012). It leads to an increase in prices because it makes the aggregate supply less than the total demand at a given time. It can occur when there are higher numbers of consumers than what the seller could predict. For example, when the in a particular area there are many visitors than it was anticipated. Another reason why excess demand can arise may be when there is a speculation that an individual product may vanish out of the market shortly, making consumers to purchase more of the commodity to cover future deficits.

Homogeneity of Degree Zero

Homogeneity of degree zero means when there are two independent variables, when one change, the other remain constant. That is the change in one independent variable with a constant figure leads to a change of the other variable with a value raised to power zero (Trockel, 2012). Since mathematically a change in a number to power zero means that it remains the same, then, in homogeneity of degree zero in a market occurs there are two independent variable one commodity changes, and the other remains constant. For example, when the demand for a product in market increases, and supply remain the same, homogeneity of degree zero is experienced because the two factors are interdependent.

Walras Law

Walras law suggests that the when there is excess supply in a market, it should be matched with an excess demand in another market to achieve an equilibrium point. The matching of the excess supply and excess demand in different markets bring stability at the prices of the commodity (Trockel, 2012). For example, suppose there is an excess supply in a certain market, the suppliers look for another market that has where that can sell the excess commodities to maximize profit. The excess demand in a secondary market allows them to look for a new market for their commodities and equilibrium is achieved when the new excess supply is equal to the excess demand in the other market. The Walras law assumes that there is the homogeneity of degree zero, that the price of the commodity does not change, and there is quasi-convex of the commodity. The assumptions allow that excess demand and extra supply in two different markets to cancel out.

Continuity of a Function

Continuity of a function means that a function, for example, y = 2x + b does not have spaces of jumps in between it. It does not leave a space at any point in areas where it covers it does not leave any space (Basile, Graziano, & Pesce, 2013). The act that it does not leave any space means that the small change in the variable factors in and it has an impact on the change in the other variable. For example, in the production function, a small bit of change in the inputs leads to a change in the output.

Continuity of Correspondence

Continuity of correspondence is not immediate for various correspondents. The continuity is either upper or lower hemi continuity. For the upper hemi continuity to exist a convergent set of points are combine with another set that is convergent and the image at the domain has limited sequences of range. It means the upper hemi continuity has an open game while the lower one has a shrinking set of sequences. In a case of the single correspondences, the continuity is almost the same as the one for a function, and it combines both the lower and upper hemi continuity (Basile, Graziano, & Pesce, 2013). It does not have explosion observed in the upper hemi continuity and shrinking seen in lower hemi continuity.

Unit Simplex

Unit simplex is the smallest set that is convex in shape that and has vertices. The fundamental properties of a unit simplex are the vertices and the having a convex set. They unit simplexes can combine to for simplex that is regular or standard.

Unit Sphere

The unit sphere is a set of points from one central point to another, and there has to be a distance between them. For example, a unit sphere can exist in a ball from one central point to the other points especially to the circumference.

ArrowDebreu Model of General Equilibrium

Debreu sees the possibility of measuring prices and commodity as key to the possibility of applying mathematics in economics. Prices are the only measurable factor that makes it possible for one compare two products. The relative unit price of two commodities gives the ArrowDebreu model its definition. The general equilibrium is seen as practices to the real work because of its consideration of customers preferences and the free will of the producers to supply commodities to the market. The ArrowDebreu models determination of prices of a commodity is also seen as practical in the real competitive market because the price other factors such as consumer income and the desire of traders to make a profit are primarily driven the two forces. The demand and supply forces work well because of assumptions that are practical such as the knowledge of the prices. The possibility of predicting makes the ArrowDebreu equilibrium work because firms do not make decisions based on the prices of commodities, but of the focused demand.

Walrasian Equilibrium

(1, 2 ) = ln(1 ) + (1 ) ln(2 ) and = (1 , 2 ) 0

(1 , 2 ) = ln(1 ) + (1 ) ln(2 ) and = (1 , 2 ) 0

(1, 2 ) = ln(1 ) + (1 ) ln(2 )

For consumer 1 (1, 2 ) = (1) + (1 ) (2)

Consumer 2 ln(1 ) + (1 ) ln(2 )

P2 = ln(1) + (1 ) ln(2)

At equilibrium demand and supply are equal

ln(1) + (1 ) ln(2) = (1) + (1 ) (2)

Therefore 2*/ 1* = (1 )1 + (1 )1/{ 2 + 2}

The resulting expression is central to Arrow Debreu Microeconomics because it comes into equilibrium of demand and supply for the consumption of the two goods.

No Such Thing as a Condition Free Theorem

A theorem is the something that has been proven to be true several methods that have been studied. All theorems that have been set have several conditions that make it be seen as true. This makes the statement that there is no such a thing as condition free theorem true. For examples, Sonnenschein-Mantel-Debreu theorem is based on conditions such as the boundary, continuity, Walras model, condition and homogeneity of degree zero. The two conditions are necessary for making the theorem true. For examples assumes highlights that there is interdependence of markets. The conditions make the theory eliminate any possible ambiguity that may arise, for example, the boundary condition ensures that demand does not become negative and the Walras model gives the diversity of the markets. The homogeneity of degree zero provides it gives the sellers or consumers the desire to explore other markets and remain restricted to the same market.

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References

Basile, A., Graziano, M., & Pesce, M. (2013). On fairness of equilibria in economies with differential information. Theory And Decision, 76(4), 573-599. http://dx.doi.org/10.1007/s11238-013-9386-z

Bryant, W. (2010). General Equilibrium: Theory And Evidence (1st ed.). World Scientific.

Khamsi, M., & Kirk, W. (2011). An Introduction to Metric Spaces and Fixed Point Theory (1st ed.). New York, NY: John Wiley & Sons.

Trockel, W. (2012). Market Demand: An Analysis of Large Economies with Non-Convex Preferences (4th ed.). Berlin, Heidelberg: Springer Berlin Heidelberg.

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