Conditions under Which General Equilibrium Exists
Competitive allocation and competitive/walrashian/general Equilibrium
In every economy, there are theories that try to explain the equilibrium of the markets systems. General equilibrium gives a picture of the behavior of supply demand and prices in an economy that comprise many markets. It helps in explaining the equilibrium model pricing and situations in which the general equilibrium will hold (Balasko 2011).
Competitive equilibrium also referred to as the walrashian equilibrium is involved in the study of the good markets whereby there are elastic prices and numerous traders. It is mainly based on competitive environment whereby traders decides on an amount which is often lesser compared to the over-all amount in the market. Thus, the individual transactions do not affect the prices of the commodity. This equilibrium is based on two elements, which are the price function and the allocation matrix. The price function is always a vector that represents the bundle of commodities, and it gives a positive real number that gives a representation of the price (Ellickson 1994).
The allocation matrix describes the vector commodities that are allotted to each agent in the market. To ensure there is an equilibrium in the market the allocation matrix should have certain characteristics. These include satisfaction whereby all the agents accept the certain bundle they are given with also out being reluctant even if they are offered another bundle at even a lower price. This means they are willing to get their bundle any time no matter what. The allocation matrix should ensure market clearance whereby the demand equals the supply. These ensure that there is no surplus, and neither is there a shortage. In addition, it should ensure individual rationality whereby the position of the agents becomes better than before the trading took place. In the situation, it ensures that financially, all agents benefit, and no one go to a loss. In this competitive allocation, the budget should balance in that the agents should afford their allocation given the endowment.
Generally, a competitive allocation should be Pareto efficient. This is where the allocation of resources to an individual will end up affecting another in the same market. If an economy state is Pareto efficient, it means that all individuals are maximizing their utility (Franklin n.d). An efficient allocation can be sustainable in any competitive equilibrium. Allocation efficiency is ensured if in the competitive equilibrium the marginal benefits are equal to the marginal cost. The cost of producing a certain commodity should not exceed the benefits as this will affect the allocation efficiency (Bewley 2009).
Excess Demand Function
The function that expresses the excess demand is referred to as the excess demand function. The excess demand function determines if the product has equilibrium. This function presents itself as the excess quantity that is demanded over the quantity that is supplied in terms of the product price and also other determinants that affect the price directly or indirectly (Levine 1996).
The price of a product is said to be at equilibrium if the variance between the demand and the supply function totals to zero. This implies that equilibrium exists whereby the excess demand function is zero. If the demand equals the supply, the market is said to clear whereby there is no surplus leading to wastage and no shortage of the commodity. An example is whereby a farmer produces two tons of maize, and a milling company demands two tones to produce maize flour. In this case, the demand of the milling has been met fully by the farmer and the farmer has no surplus since all the maize he produces is bought by the milling company, and also, it ensures that the milling does not have a shortage since the farmer is able to produce the amount the company requires (Library of Economics and Liberty 1999).
The equilibrium price of a product in a market determines if there will be a surplus or a shortage. If the price of a product is higher than the equilibrium price, there will be a surplus in the market. Moreover, when the price is lower, the excess demand function will be positive thus there will be a shortage of the commodity in the market. If the farmer who produces increases the price of the maize, then the milling company will not be able to purchase the maize thus there will be surplus, and the farmer has to bring the price back to the equilibrium price to ensure that there is no surplus and no shortage of the milling company. The price change in a market is always proportional to the excess demand function. Thus, the price changes always have an effect on any commodities equilibrium.
The Walra’s law states that the sum of all values of excess demand in all markets should encapsulate to zero whether or not the economy is in equilibrium (Thomas 1997). This law implies that if there is a negative excess demand in one market then in another market a positive excess demand must be existing (Thomas 1997). If all markets in an economy are at equilibrium except one then the law of Walra is not fulfilled thus even the other market should at equilibrium to ensure all the markets are at equilibrium.
This principle in a general equilibrium shows that asserting all budgets constraints all the values of excess demand should be at zero. An economy is at a general equilibrium if all markets are in a partial equilibrium. In this case, the markets have no surplus or shortages because at the current price of the commodity the demand equals the supply. For example, if the current price of a shoe is 10$ and all the manufacturers of that specific product are willing to sell 5000 shoes at that price, and the buyers are willing to buy the 5000 shoes at 10$ then a partial equilibrium exists since there is no shortage neither is there a surplus. This kind of market meets the Walra’s law as the excess demand function sums up to zero putting into consideration all determinants in the market.
Walra’s Law ensures that the allocation matrix characteristic of budget balance is met. In an economy, it is met if every agent’s budget constraints hold with equality. The agents budget constraints state that the expenditure should be less than or equal the revenue. Summing up all the markets value of the agents costs should be less or equal all the benefits the agents will get after trading that commodity. This ensures that the agents do not suffer losses but acquire profit from trading that commodity. This then indicates that the agents will not acquire the commodities and neither will they sell them for free thus in this case the excess demand function will be zero and thus equilibrium will be attained.
Existence of Competitive Equilibrium
A competitive equilibrium exists in an exchange economy with divisible goods according to Arrow-Debreu model. These divisible goods must satisfy a certain condition that is convex preferences from all agents and the desirability of the goods. The convex preferences correspond to the idea that averages are better than extremes. In a situation where all agents have strictly convex preferences, the price vector is continuous and so is t demand function (Arrow & Debreu 1954).
The desirability of the goods means that if the goods were available free, all agents would try all their best to acquire as much as possible thus surplus in this situation will be limited. For divisible goods, the prices should be strictly positive to meet the assumption of desirability. Competitive equilibrium for divisible goods only exists if the goods are substitutes and not complement. This means that if the price of one commodity (A) increases then the demand of the other (B) will either remain constant or increase but it will not decrease. This is because B is a substitute for A and if agents cannot buy B, they can use commodity A in place of it.
For indivisible goods, a competitive equilibrium exists if the utility functions of all agents are Gross substitutes. This is whereby if an agent has all his prices set and a price of one increase he or she may decide he does not want that product and decide to get another product, which is a substitute but cannot decide that he does not want the third item in the group whose price remained constant. The largest set of a gross substitute that contains the unit demand guarantees the existence of a competitive equilibrium.
It is possible to find a competitive equilibrium for indivisible items whereby all function is gross substitutes by use of ascending auction. This is whereby the auctioneer establishes their price vector and buyers choose their bundles based on the prices of the auctioneer. In this situation every item has a binder and if one item has an over demand the price is increased and the buyers have to bind all over again thus in a competitive equilibrium is found, and there is neither a surplus neither a shortage.
Uniqueness of a competitive Equilibrium
For uniqueness of equilibrium to exist, it has to meet certain conditions. These include the presence of wealth effects, which generates the possibility of having multiple equilibriums in the economy. This wealth effect helps in differentiating between a general equilibrium and partial equilibrium. In a unique competitive equilibrium if the price of a product changes then it has several effect that include the attractiveness of the commodity. If the price increases, the attractiveness might decrease and vice versa. Price changes also affect the wealth distribution among the individual agents. This is because the price changes affect both the demand and the supply. The price changes ensure that there is more than one set of prices for a commodity that constitutes equilibrium.
The uniqueness of equilibrium is also determined by either the number of equilibrium is finite or odd. A regular economy is finite, and it is unique. According to Debreu, most of the economies are regular which means most of them competitive equilibriums are unique. Another characteristic of a unique competitive equilibrium is that the excess demand function should be expressed as preference property which is much stronger than an individual is or as a gross substitute property (Pearce & Wise 1973). To ensure uniqueness, all competitive equilibrium should have the same positive local index. The stability of the equilibrium also affects the uniqueness of the competitive equilibrium.
Generally, for equilibrium to exist in any economy it has to meet a certain condition. In addition, many other variables affect the stability and uniqueness of equilibrium. Ensuring that the Walra’s law has been met, and the excess demand function is zero then the commodity will achieve its equilibrium. Most economies have many traders and prices of the commodities in the market are flexible thereby it implies that in many economies competitive equilibrium exists. Although many researchers have questioned the Walra’s law, it makes perfect sense in ensuring that in a market there will not exist any surplus no matter what, and neither will there be a shortage of a certain commodity in the market. Largely, the general equilibrium tries to explain how different pricing models affect the different equilibriums of a commodity in a market. For maximal benefits in an economy then the best solution is the use of competitive equilibrium, which involves competitive allocation. With this ensures trading in markets does not stagnate, and all the involved parties are gaining and money is circulating in the economy. This theorem of general equilibrium can only be used in microeconomics as it uses a bottom-up approach that majorly deals with the individual markets and the agents involved in handling the commodities. For one to understand the competitive equilibrium using excess demand, then they ought to understand its key properties, which are homogenous of the degree zero, and the fact that it satisfies the Walra’s law on budget constraints.
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Ellickson, B 1994, Competitive equilibrium: theory and applications, Cambridge University Press, Cambridge.
Franklin, MF n.d, The stability of general equilibrium –What do we know and why is it important? Routledge, London.
Library of Economics and Liberty, 1999, Supply and demand, markets and prices. Available from: <http://www.econlib.org/library/topics/college/supplyabddemand.html>. [8 January 2016].
Palley, T 1997, Walra’s Law and Keynesian macroeconomics, public policy department. Available from: <http://www.thomaspalley.com/docs/articles/macro_theory/walras_law.pdf>. [8 January 2016].
Pearce, IS & Wise, J 1973, On the uniqueness of competitive equilibrium: Part 1, unbounded demand, vol 41, no. 5 pp. 817-828.
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