QUESTION
Q1- explain and illustrate the following terms:
1- An Arrow-Debreu private ownership economy
2- Walrasian equilibrium
3- A fixed point
4- A fixed point theorem
5- Brouwer’s fixed point theorem
6- Market excess demand
7- Homogeneity of degree zero
8- Walras Law
9- Continuity of a function
10- The unit simplex
Q2-Tutorial Reading 1: Read, summarise and think about: J. Geanakoplos (2008), “Arrow-Debreu model of general equilibrium”, (in) The New Palgrave Dictionary of Economics, 2nd Edition, pg. 3 ● (A.6) and associated discussion then from ● (A.12) to bottom of pg. 5, then pg. 6 ‘Existence of equilibrium’.
SOLUTION
1. An Arrow-Debreu private ownership economy
The Arrow-Debreu model characterises the players in the economy as the producers and consumers. With the objective of the producer to maximise profits at a given price level of choice and on the other hand consumer maximization his interests by purchasing goods at the price level of choice. In a private ownership economy however the wealth is distributed among the consumers according to the shares held by the different consumers in the economy. The producers are the recipient of two things firstly he is entitled to the salary for working in the corporation .These are the principal agents in the economy and there is no interference of any other agent in the economy. According to Debreu the equilibrium in such an economy can be achieved at
I. Firstly when the producer is able to maximise profits at determined price level of choice
II. Secondly , the consumer is able to maximise his interests at the given price level that constraint consumer choice
III. Thirdly there is no existence of excess demand implying that the excess demand is 0.
It should however be noted that the equilibrium in this case is referred to a set of prices and plans.
(Currie et al ,1990)
2. Walrasian Equilibrium
The Walras equilibrium shows all points of the contract curve that are pareto efficient. The Walras equilibrium is a measure of the utility maximization of particular households. Let us explain the Walras equilibrium graphically, the figure below shows the pareto optimality condition. The point A represents the pareto optimality condition, there is a set of goods corresponding represents Walras equilibrium. There is a budget line P drawn between the two households, when the budget line shifts from P to P0 then the pareto optimality condition is achieved. The income is distributed, such that household 2 should be taxed be obtain pareto optimality. Point A represents pareto optimality condition and the Walrus equilibrium.
Figure 1 : Representation of Walras Equilibrium
(Søren, 2002)
3. The Fixed Point, Fixed Point Theorem and Brouwer’s fixed point theorem
The solutions proposed for particular multimarket systems may not be applicable when the market structure is abstract and the demand curve that they are faced can be kinked, these are also difficult to be numerically implemented as well. The Brouwer’s fixed point theorem point theorem states that there should be a point to point mapping of the contonous , close bounded convex set into itself that has a fixed point. For example there is a F(x) and there exists a fixed point x1 then the mapping can be defined as x1 = f(x1) . According to the Bruwer’s fixed pint theorem in the short run the number of firms in the industry are pre-determined but however the profits of individual firms are not 0
Figure 2 : Graphical Presentation of the Brouwer’s fixed point theorem
In the above figure the A close bounded convex set has been mapped The function F(x) is a continuous function , therefore the point where the 45 degree intersects the function F(x) is the point x1 , and at this point the multi market equilibrium is said to be achieved.
(Henderson,2003)
4. Market Excess Demand
A comparative statistics explains the shift in the demand curve , let us consider if the price p is the equilibrium price , then any price above p (p1 )for a good is unstable in the market in the good is said to have positive excess demand. In other words it can be said that if the supply of the goods is reduced in such a way that it leads to excess demand, then it would result in an increase in the equilibrium price to p1 and the price will continuously move away from p1 because of unstable market conditions.
Figure 3: Walrus and Marshall Stability and Instability
3(a) Marshall stable Walrus unstable
3(b) Marshall unstable Walrus stable
Thus in the figures the excess demand condition holds, figure 3(a) shows the convergence to price p1 in the case faced with the excess demand situation, the shortage will lead to an increase in the price. Therefore is faced with an upward sloping demand curve. On the other hand if it is faced with the excess demand leads to a reduction in the price of the good, hence causes the firm to face a downward sloping demand curve. Thus this has raised certain points of controversy between the Walrus and Marshall Ian theory.
(Blaug,2010)
5. Homogeneity of Zero
The homogeneity of Zero implies the situation such that when at all price levels the value of the excess demand is 0.
6. Walrus Law
The Walrus law states that the aggregate excess demand is 0. This means at all prices the value of the excess demand is 0. This can mathematically be expressed as
p1z1 (p1, p2) +p2z2 (p1, p2) =0
This implies that the equation applies to the principal agents in the economy i.e Agent A and Agent B
Applying the equation with the Budget constraint we have
p1z1A (p1, p2) +p2z2A (p1, p2) =p1w1A+p2w2A =0
Therefore we have p1eA(p1,p2)+p2eA(p1,p2)= 0 The same also applies to agent B. thus it can be concluded as long p1>0 the excess demand for individuals in the economy is also 0 at any equilibrium price of 1 or 2.
(people.stfx.ca – accessed on 3/5/2012)
7. Continuity of a Function
A function f is X continuous at a point where,
Lim f(x)=f(x0)
A function x is said to be continuous if F is continuous at every point where x ϵ X. A continuous curve represents a curve such that there is no disconnected point between them. A continuous function is such that if a small change in x produces a small change in F(x).
(Lecture notes – Micro Economic Theory – accessed on 3/5/2012)
8. The unit simplex
The unit simplex is a part of the general equilibrium theory and is partly associated with the Arrow-Debreu private ownership model. The unit indicates a 3 goods economy and the relative price level has no effect on the real price level. The relative price matters because the of the assumption of the homogeneity of the 0.
(Mark Blaug, 2010)
Part 2 An Arrow-Debreu private ownership economy
The Arrow-Debreu model characterises the players in the economy as the producers and consumers. With the objective of the producer to maximise profits at a given price level of choice and on the other hand consumer maximization his interests by purchasing goods at the price level of choice. In a private ownership economy however the wealth is distributed among the consumers according to the shares held by the different consumers in the economy. The producers are the recipient of two things firstly he is entitled to the salary for working in the corporation .These are the principal agents in the economy and there is no interference of any other agent in the economy. According to Debreu the equilibrium in such an economy can be achieved at
I. Firstly when the producer is able to maximise profits at determined price level of choice
II. Secondly , the consumer is able to maximise his interests at the given price level that constraint consumer choice
III. Thirdly there is no existence of excess demand implying that the excess demand is 0.
It should however be noted that the equilibrium in this case is referred to a set of prices and plans.
(Currie et al ,1990)
References
- Currie, Martin, and Ian Steedman. Wrestling with time: problems in economic theory. Manchester: Manchester University Press, 1990. Print.
- Henderson, H. Microeconomic Thoery 3/E. Delhi: Tata Mcgraw Hill, 2002. Print.
- Blaug, Mark. Famous figures and diagrams in economics. Cheltenham: Edward Elgar, 2010. Print.
- “Intermediate Microeconomics II, ECON 301.” /people.stfx.ca/. N.p., n.d. Web. 4 May 2012. <http://people.stfx.ca/tleo/MicroIILectu
- “Lecture Notes1 Microeconomic Theory.” econweb.tamu.edu. N.p., n.d. Web. 3 May 2012. <http://econweb.tamu.edu/tian/micro1.pdf>.
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