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General equilibrium by Leon Walras

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    In this essay, Edgeworth Box is used to analyze how general equilibrium is achieve in a pure exchange economy. This essay will be divided into five parts 1) definition of general equilibrium, 2) definition of a pure exchange economy, 3) definition and explaination of how Edgeworth box is used in a simple model with a pure exchange economy. 4) explaination of Pareto improvement, Pareto efficiency and Pareto optima in the Edgeworth Box. 5) Analyzing how general equilibrium is achieved in a pure exchange economy.

    General equilibrium

    General equilibrium theory as proposed by (Leon Walras, 1874), studies how the supply and demand in an economy with multiple markets fit together while working as a whole to come into equilibrium simultaneously.

    This theory analyzes the mechanism by which the choices of economic agents are coordinated across all markets. With many markets in the economy, it is essential for general equilibrium analysis to take into account for the change in price caused by the change in supply and demand in one market. As the resulting change in that one market will also affect the supply and demand of the other markets, which in turn result in changes of prices in the other markets as well.

    Because interactions among markets creates feedback effects between markets, it can result in a significant non-linearities in the market, which in turn causes complicated equilibrium price behavior. Therefore, it is important to further understand how the general equilibrium model works in a pure exchange economy. Pure exchange economy

    A pure exchange economy, as written by (Jehle and Reny, 2011) is an economy model with no production. The model assumes that there is already supply of goods meaning supply is already fixed. The model also assumes that agents in a pure exchange economy are consumers who already possessed a certain amount of goods initially. The only economic activities in this model is trading and consumption between consumers. In a pure exchange economy, there will be a few consumers and each consumer is described by their utility function on baskets of goods and by the initial basket of goods they own. With their initial possession of goods, they will trade the goods among themselves to make themselves better off. In short , this model shows the efficiency and inefficiency of how goods are allocated among consumers.

    Edgeworth Box
    A useful tool used in general equilibrium analysis is the Edgeworth Box. An Edgeworth Box, named after Francis Ysidro Edgeworth[3], is used to depict the interaction of two people trading two different goods. One of the special feature of the Edgeworth box diagram which makes it different is that it has two orgins. It merges the indifference map between two people in the trade by inverting one person diagram so as to analyze their trading behaviour. An illustration can be done by studying the general equilibrium from an extended example of the most simple possible general equilibrium model with a pure exchange economy that has no production using the Edgeworth Box. Figure 1.1 is called an Edgeworth Box.

    The economy consists of two people, John (J) and Mary(M), and there are two types of goods, coke and pepsi. The total amount of coke in the economy is measured on the horizontal axis of the Edgeworth box, and similarly, the total amount of pepsi is measured as the height of the box. Any point in the Edgeworth Box represents an allocation of the two goods to the two individuals. Since the Edgeworth Box model uses the concept of initial endowment. So let’s take point “z” in Fig 1.1 as the initial endowment. At the endowment point “z”, it represents the amount of coke and pepsi that John and Mary have before the trade. At “z”, John gets Jx litres of coke and Jy litres of pepsi, while the remaining litres of coke (Mu) and pepsi (Mw) initially go to Mary. We can also add both John and Mary individual’s preferences, in the form of indifference curves to Fig 1.1. The blue colour indifference curve belongs to John and the red colour indifference curve belongs to Mary. The arrows indicates the direction of their increasing utility. Notice that John utility increases as his indifference curve move towards the north-east direction while Mary utility increases as her indifference curve move towards the south-west direction. This observation shows that John like the allocations to the north-east better and Mary like the allocations to the south-west better.

    Fig 1.1

    Consider another example in Fig 1.2, There are two indifference curves, the blue curve for John and the red curve for Mary that pass thru the initial endowment point “z” in Fig 1.2. The shaded area in between John and Mary indifference curves are allocation that would make both of them better off than their initial endowment. If both John and Mary exchange goods, and to be specific if John(J) gives some pepsi to Mary(M) in exchange for coke such that they move to point (h) in the shaded area, then both of them would be better off than before(initially). This type of exchange that makes both of them better off or at least one of them better off without causing the other person to be worse off is called Pareto improvement. In other words, we can say that the allocation point (h) is a Pareto improvement than allocation point (z). However, not all allocations in the Edgeworth Box can be Pareto compared, in a way that either one of them is Pareto improvement better than another. Consider another case, at allocation (k) in Fig 1.2. John has a higher utility as compared to in allocation (z) or (h), also at (k), Mary has a lower utility as compared to at (z) or (h). Hence, ((z) and (k)) and ((h) and (k)) are not Pareto comparable.

    Fig 1.2

    Moving on ,we now go on to talk about Pareto efficiency. The aim of this is to consider how goods are allocated among individuals. Whenever an initial endowment leaves the possibility of making all individuals better off by redistributing the goods among themselves, then we can say that the initial allocation is inefficient. To be more precise, all allocation inside the Edgeworth Box that have the two indifference curves intersecting with each other are inefficient, Meaning that all individuals can be better off simultaneously than in their initial allocation.

    However, there can also be initial allocation which is impossible to make both individuals better off. Such an allocation can be refer to as Pareto efficient.

    Consider an allocation I in Figure 1.3. I is Pareto efficient, because starting from I, it is not possible to reallocate the goods to make both individuals better off.

    To better understand this, we can see that the area of allocations towards the north-east direction that the indifference curve passes thru I which is better for John (J) and the area of allocations towards the south-west that (M) indifference curve passes thru I that is better for Mary (M) do not intersect each other. We can also see that in Fig 1.3, points inside the Edgeworth Box where John(J) and Mary(M) indifference curves are tangent to each other are Pareto optima.

    However, there are also cases where Pareto optima is achieved even thou both the individuals indifference curve are not tangent to each other. As explained by (Mathur, Vijay K, 1991), the main decisive feature of a Pareto optimum is that the intersection of the sets of allocations that are preferred by John (J) and Mary (M) is empty. Even though most allocations in an Edgeworth box are Pareto inefficient, there will also be many other Pareto optima in the Edgeworth box.

    For example, in Fig 1.3 , I” and I”” are Pareto optima. This means that Pareto comparison are not possible among Pareto optima as no Pareto optimum is Pereto better than another Pareto optimum. as we can see that in Fig 1.3, I is better than I” and worse than I”” for John (J), and I is better than I”” and worse than I” for Mary (M) case.

    Fig 1.3

    Moving on to Fig 1.4, it explained that all Pareto optima can be connected and lie on a curve that connects the south-west with the north-east corner of the Edgeworth box. Referring back to Fig 1.4, This curve is called the contract curve. (Wyn Morgan, Micheal Katz and Harvey Rosen. 2009) explained that the locus of all the consumption efficient points is called the contract curve. When individuals trade with each other, they will often likely end up at some point on the contract curve and they will only stop their trading until they reach the contract curve, because before reaching the contract curve, individuals will still stand to have potential gains from trading among themselves that would be left unexploited. For example ,both John (J) and Mary (M) must agree to any exchange, and they will only agree to exchange if the resulting allocation will make both of them better off. Logically, they will use up all the possible gains from trading. They will not stop at a Pareto inefficient allocation. Therefore, John (J) and Mary (M) will eventually come to a part on the contract curve which is Pareto better than their initial endowment “n”. This part is known as the core and is denoted by the untidy scribble line of the contract curve in Fig 1.4. Fig 1.4

    Lastly, we now come to the final part of the essay which is to analysis the market exchange in a simple Edgeworth economy. Answering to the main question of this essay. Suppose that there is a market where two people John(J) and Mary(M) can exchange two goods coke (c) and pepsi(p). Using the Edgeworth Box, we will then explain how general equilibrium is achieved in a pure exchange economy. To begin this example, we start with an initial endowment “z” in the Edgeworth Box. Given the initial endowment “z”, we will start by finding the price for coke and pepsi such that at the given set of prices and their endowment, both John and Mary are mazimizing their utility. To find the price for coke and pepsi in Fig 2, we begin by giving an initial set of price for both coke(pc) and pepsi (pp). Suppose that the price of (pc) is $1 and the price of (pp) is $2. Given these market prices, each individual will generate a budget line and a set of feasible consumptions plan. At the given prices, John is willing to trade away one litre of pepsi for two litres of coke because he knows that the price of pepsi(pp) is twice the price of coke(pc).

    Hence, John will have a budget line with a slope of -1/2 through the endowment point “z” denoted as B1 in Fig 2. Because both John(J) and Mary (M) face the same prices for the two goods. Both their budget line are B1. Again at B1 in Fig 2, we can see that John (J) is happier consuming bundle eJ rather than the initial bundle z because at bundle eJ, his utility is higher (we understand this because the indifference curve Je is further to the north east than Jz, the indifference curve that runs thru the initial endowment). However, to move John from z to eJ, he needs to sell some pepsi and buy some coke. To be more precise, John have to sell zi litres of coke and buy ieJ litres of coke. Moving on to Mary (M), we can see that in Fig 2, her initial endowment is also at “z” and her most preferred bundle is eM at the indifference curve Me. Notice that the indifference curve Me is more towards the south-west as compared to the indifference curve Mz (Mary utility increases as her indifference curve move further towards the south-west). However, to move her from z to eM, she would have to sell eMk litres of pepsi and purchase zk litres of coke.

    With all these being explained, our main aim is to achieved general equilibrium when pc = $1 and pp = $2. However, at these given prices, John and Mary wants to be at different points in the Edgeworth Box, but the economy does not allow John and Mary to be at two different allocation simultaneously. Also , because both of them wants to sell pepsi, therefore causing the market to have an excess supply of pepsi and also because both of them want to buy coke, it will cause the coke market to have excess demand. All these factors, will cause the quantity supplied to not be equals to the quantity demanded in both market. Therefore, after these analysis, we have come to a conclusion that general equilibrium is not achieved when (pc) = $1 and (pp) = $2.

    Fig 2

    We will then go on to talk about how the market will adjust to the above Fig 2 disequilibrium situation. The excess supply of pepsi and excess demand of coke will cause the price (pc) to rise in relative to (pp) and this change in prices will affects both John and Mary budget line causing it to change. The new change budget line will still pass thru the endowment point “z” and is steeper as compared to B1 resulting from the price increase in (pc)/(pp). Given the new price change, John(J) and Mary(M) will again find a new bundle to maximize their utility. They will continue to do so , until quantity supplied is equals to quantity demanded in both market. Because if the quantity supplied and quantity demanded in both market are not equals, than the price ratio will continue changing and both of them will continue to find new bundles given the new change price ratio until quantity demanded in the market equals to the quantity supplied. (Truman F. Bewley, 2007) mentioned that, “The economy is said to be in general equilibrium when the total demand for each commodity equals the total supply.” Therefore, general equilibrium is only achieve, when both individuals are maximizing their utility and their utility maximisation decisions are consistent with the equality of supply and demand in both markets.

    Consider a new set of price in Fig 3, given that the price of coke = (pc*) and price of pepsi =(pp*). This result to a new budget line denoted as B2 in Fig 3. Given these new prices, both John (J) and Mary(M) are maximizing their utility at bundle e*. Besides, at e* their choice are compatible because the quantity supplied and quantity demanded are equal in the market. Therefore , pc*/pp* will be the general equilibrium price ratio. Going back to bundle e*, notice that both John(J) and Mary (M) indifference curve are tangent to each other on the same slope B2 ,the reason being is because as utility maximizers, both John and Mary will each set their marginal rate of substitution (MRS) to be equals to the price ratio and because both of them faces the same price, their MRS must be equal. Their MRS will be just minus the slope of the indifference curve. With this example in Fig 3 , we have now achieved general equilibrium in a pure exchange economy using the Edgeworth Box.

    Fig 3

    Walras. 1954. Elements of Pure Economics (trans Jaffe). Homewood : Richard Irwin. GEOFFREY A. JEHLE and PHILIP J. RENY. 2011. Advanced Microeconomic Theory 3rd edition. England : Pearson Education Limited. Andrew Schotter. 2009. Mircroeconomics : A Modern Approach. Canada : Cengage Learning. Truman F. Bewley. 2007. General Equilibrium, Overlapping Generations Models, and Optimal Growth Theory. United States : Harvard University Press. Wyn Morgan, Micheal Katz and Harvey Rosen. 2009. Mircoeconomics 2nd European edition. United Kingdom : MacGraw-Hill Education. Mathur, Vijay K. 1991. “How Well Do We Know Pareto Optimality?”Journal of Economic Education,22 (2): 172–8.

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