Introduction to Economic Analysis Chapter 14 General Equilibrium

Chapter 14 General Equilibrium General equilibrium puts together consumer choice and producer theory to sets of prices that clear many markets . It was pioneered by Kenneth Arrow , Gerard , and Lionel Mackenzie in the late . Many economists consider general equilibrium to be the pinnacle of economic analysis . General equilibrium has many practical applications . For example , a study of the impact of carbon taxes uses general equilibrium to assess the effects on various sectors of the economy . Box LEARNING OBJECTIVES . How are several prices simultaneously determined ?

What are the efficient allocations ?

Does a price system equilibrium yield efficient prices ?

The box considers a , exchange That is , two people have utility functions of two goods and endowments ( initial allocations ) of the two goods . The box is a graphical representation of the exchange problem facing these people and also permits a straightforward solution to their exchange problem . Figure box URL books 338

The box is represented in Figure The box . Person is located in the lower left ( southwest ) corner , and Person in the upper right ( northeast ) corner . The good is given on the horizontal axis , the good on the vertical . The distance between them is the total amount of the good that they have between them . A point in the box gives the allocation of the the distance to the lower left to Person , the remainder to Person . Thus , for the point illustrated , Person obtains ( and Person obtains ( The total amount of each good available to the two people will be fixed . What points are efficient ?

The economic notion of efficiency is that an allocation is efficient if it is impossible to make one person better off without harming the other person that is , the only way to improve utility is to harm , and vice versa . Otherwise , if the consumption is inefficient , there is a rearrangement that makes both parties better off , and the parties should prefer such a point . Now , there is no sense of fairness embedded in the notion , and there is an point in which one person gets everything and the other gets nothing . That might be very unfair , but it could still be the case that improving must necessarily harm . The allocation is efficient if there is no URL , org books ( 909 339

waste or slack in the system , even if it is wildly unfair . To distinguish this economic notion , it is sometimes called . We can the points by fixing Person utility and then asking what point , on the indifference of Person , maximizes Person utility . At that point , any increase in Person utility must come at the expense of Person , and vice versa that is , the point is . An example is illustrated in Figure An efficient point . Figure An ( point In Figure An efficient point , the of Person is drawn with a dark , thick line . This utility level is . It acts like the budget constraint for Person . Note that Person face the opposite way because a movement southwest is good for , since it gives him more of both goods . Four are for Person , and the highest feasible , which leaves Person getting the fixed utility , has the point illustrated with a large dot . Such points occur at of the . This process of identifying the points that are can be carried out for every possible utility level for Person . What results is the set of URL books 340

points , and this set is also known as the contract curve . This is illustrated with the thick line in Figure The contract curve . Every point on this curve maximizes one utility given the other utility , and they are characterized by the in the . The contract curve need not have a simple shape , as Figure The contract curve illustrates . The main properties are that it is increasing and ranges from Person consuming zero of both goods to Person consuming zero of both goods . Figure The Example Suppose that both people have utility . Let the total endowment of each good be one , so that . Then Person utility can be written as , and utility is ( Then a point is efficient if ( xy ( Thus , solving for , a point is on the contract curve ( URL books 341

Thus , the contract curve for the case depends on a single parameter ( oi ( It is for a Variety of examples ( a and ) in Figure Contract curves with utility . Figure Contract curves with utility KEY TAKEAWAYS The box considers a , exchange The box is a graphical representation of the exchange problem facing these people and also permits a straightforward solution to their exchange problem . A point in the box is the consumption of one individual , with the balance of the endowment going to the other . efficiency is an allocation in which making one person better off requires making someone else worse are no gains from trade or reallocation . In the box , the points arise as tangents between of the individuals . The set of such points is called the contract curve . The contract curve is always increasing . URL books 342

EXERCISES . If two individuals have the same utility function concerning goods , is the contract curve the diagonal line ?

Why or why not ?

For two individuals with preferences , when is the contract curve the diagonal line ?

Equilibrium With Price System LEARNING OBJECTIVES . How are prices in the economy determined ?

Are these prices efficient ?

The contract curve provides the set of efficient points . What point will actually be chosen ?

Let start with an endowment of the goods . An endowment is just a point in the box that represents the initial ownership of both goods for both people . The endowment is marked with a triangle in Figure Individually rational efficient points . Note this point indicates the endowment of both Person and Person because it shows the shares of each . Figure ( URL books 343

Figure Individually rational efficient points also shows for persons and going through the endowment . Note the for is concave toward the point labeled , and the for is concave toward the point labeled . These utility define a reservation utility level for each utility they could get alone , without exchange . This no exchange state is known as autarky . There are a variety of efficient points that give these people at least as much as they get under autarky , and those points are along the contract curve but have a darker line . In Figure Individually rational efficient points , starting at the endowment , the utility of both players is increased by moving in a southeast is , down and to the the contract curve is reached . This involves Person getting more ( movement to the right ) in exchange for giving up some ( movement down ) Thus , we can view the increase in utility as a trades some of his some of Person . In principle , any of the darker points on the contract curve , which give both people at least as much as they achieve under autarky , might result from trade . The two people get together and agree on exchange that puts them at any point along this segment of the curve , depending upon the bargaining skills of the players . But there is a particular point , or possibly a set of points , URL books 344

that results from exchange using a price system . A price system involves a price for trading , and vice versa , that is available to both parties . In this , prices define a straight line whose slope is the negative of the price ( the for is the reciprocal ) Figure with ( Figure Equilibrium with a price system illustrates trade with a price system . The in the center is the point on the contract curve connected to the endowment ( triangle ) by a straight line ( the price line ) in such a way that the straight line is tangent to both and at the contract curve . This construction means that if each person took the price line as a budget constraint , they would maximize their utility function by choosing the point labeled . The fact that a price line exists , that ( i ) goes through the endowment and ( ii ) goes through the contract curve at a point tangent to both people utility , is relatively easy to show . Consider lines that satisfy property ( ii ) and let see if we can one that goes through the endowment . Start on the contract curve URL books 345

at the point that maximizes utility given reservation utility , and you can easily see that the price line through that point passes above and to the right of the endowment . The similar price line maximizing utility given reservation utility passes below and to the left of the endowment . These price lines are illustrated with dotted lines . Thus , by continuity , somewhere in between is a price line that passes through the endowment . The point labeled represents an equilibrium of the price system , in so far as supply and demand are equated for both goods . Note , given the endowment and the price through the endowment , both parties maximize utility by going to the . To see this , it may help to consider a version of the that only shows Person and the price line . Figure ' I ) Figure Illustration of price system equilibrium removes player , leaving only player and the price line through the endowment . The price line through the endowment is the budget facing each player at that price . Note that , given this budget line , player , who gets more as player gets less , maximizes utility at the middle . That is , taking the price as given , player would choose the given player endowment . URL books 345

The logic for player is analogous . This shows that if both players believe that they can buy or sell as much as they like at the of the price through the , both will trade to reach the . This means that if the players accept the price , a balance of supply and demand emerges . In this sense , we have found an equilibrium price . In the box , we see that , given an endowment , it is possible to reach some point using a price system . Moreover , any point on the contract curve arises as an equilibrium of the price system for some endowment . The proof of this proposition is startlingly easy . To show that a particular point on the contract curve is an equilibrium for some endowment , just start with an endowment equal to the point on the contract curve . No trade can occur because the starting point is gain by one party entails a loss by the other . Furthermore , if a point in the box represents an equilibrium using a price system ( that is , if the quantity supplied equals the quantity demanded for both goods ) it must be efficient . At an equilibrium to the price system , each player is tangent to the price line and , hence , tangent to each other . This implies that the equilibrium is . Two of the three propositions are known as the first and second welfare theorems of general equilibrium . The first welfare theorem of general equilibrium states that any equilibrium of the price system is . welfare theorem of general equilibrium states that any efficient point is an equilibrium of the price system for some endowment . They have been demonstrated by Nobel laureates Kenneth Arrow and Gerard , for an arbitrary number of people and goods . They also demonstrated the third , for any endowment , there exists an equilibrium of the price system with the same high level of generality . URL books 347

KEY TAKEAWAYS Autarky means consuming one endowment without trade . If the endowment is not on the contract curve , there are points on the contract curve that make both people better off . A price system involves a specific price for trading , and vice versa , that is available to both parties . Prices define a straight line whose slope is the negative of the price ( the price is the reciprocal ) There is a price that ( i ) goes through the endowment and ( ii ) goes through the contract curve at a point tangent to both people utility . Such a price represents a supply and demand equilibrium Given the price , both parties would trade to the same point on the contract curve . In the box , it is possible to reach some point using a price system . Moreover , any point on the contract curve arises as an equilibrium of the price system for some endowment . If a point in the box represents an equilibrium using a price system , it must be efficient . The first and second welfare theorems of general equilibrium are that any equilibrium of the price system is efficient and any point is an equilibrium of the price system for some endowment . General Equilibrium LEARNING OBJECTIVES . What happens in a general equilibrium when there are more than two people buying more than two goods ?

Does the case provide insight ?

We will illustrate general equilibrium for the case when all consumers have utility in an exchange economy . An exchange economy is an URL books 348 economy where the supply of each good is just the total endowment of that good , and there is no production . Suppose that there are people , indexed by , There are , indexed by , Person has utility , which we can represent using exponents a ( so that the utility of person can be represented as ( a ( where ( is person consumption of good Assume that a ( for all and , which amounts to assuming that the products are , in fact , goods . Without any loss of generality , we can require ( for each ( To see this , note that maximizing the function is equivalent to maximizing the function for any positive . Let ( be person endowment of good The goal of general equilibrium is to prices , for the goods in such a way that demand for each good exactly equals supply of the good . The supply of good is just the sum of the endowments of that good . The prices yield a wealth for person equal to ( We will assume that Na ( i ) for every pair of goods and i . This assumption states that for any pair of goods , there is at least one agent that values good and has an endowment of good i . The assumption ensures that there is always someone who is willing and able to trade if the price is sufficiently attractive . The assumption is much stronger than necessary but useful for exposition . The assumption also ensures that the endowment of each good is positive . The utility simplifies the analysis because of a feature that we already encountered in the case of two goods , which holds , in general , that the share of wealth for a consumer on good equals the exponent a ( Thus , the total demand for good is ( The equilibrium conditions , then , can be expressed by saying that supply ( sum of the endowments ) equals demand or , for each good , URL books 349

We can rewrite this expression , provided that ( and it must be , for otherwise demand is infinite ) to be ( i ) Let be the matrix whose ( term is ( i ) a ( Let be the vector of prices . Then we can write the equilibrium conditions as ( I ) where is the zero vector . Thus , for an equilibrium ( other than ) to exist , have an eigenvalue equal to and a corresponding that is positive in each component . Moreover , if such an pair exists , it is an equilibrium , because demand is equal to supply for each good . The actual price vector is not completely because if is an equilibrium price vector , then so is any positive scalar times Scaling prices doesn change the equilibrium because both prices and wealth ( which is based on endowments ) rise by the scalar factor . Usually economists assign one good to be a , which means that all other goods are indexed in terms of that good and the price is set to be . We will treat any scaling of a price vector as the same vector . The relevant theorem is the theorem . It states that if is a positive matrix ( each component positive ) then there is an eigenvalue A and an associated positive and , moreover , A is the largest ( in absolute value ) of . This conclusion does most of the work of demonstrating the existence of an equilibrium . The only remaining condition to check is that the eigenvalue is in fact , so that ( I ) Suppose that the eigenvalue is A . Then Ap . Thus for each , i ) pi or ( i ) pi . Summing both sides over , URL , org books 350

) i ) pi ( i ) i ) pi . Thus , A as desired . The theorem actually provides two more useful conclusions . First , the equilibrium is unique . This is a feature of the utility and does not necessarily occur for other utility functions . Moreover , the equilibrium is readily approximated . Denote by the product of with itself times . Then for any positive vector , While are very useful for large systems ( large numbers of goods ) the system can readily be computed exactly with small numbers of goods , even with a large number of individuals . Moreover , the approximation can be interpreted in a potentially useful manner . Let be a candidate for an equilibrium price vector . Use to permit people to calculate their wealth , which for person is ( i ) Given the wealth levels , what prices clear the market ?

Demand for good gis ( i ) and the market clears , given the wealth levels , if ( i ) which is equivalent to . This an iterative process . Start with an arbitrary price vector , compute wealth levels , and then compute the price vector that clears the market for the given wealth levels . Use this price to recalculate the wealth levels , and then compute a new price vector for the new wealth levels . This process can be and , in fact , converges to the equilibrium price vector from any starting point . We this section by considering three special cases . If there are two goods , we can let an a ( and then conclude that a ( an . Then let Ny ( be the endowment of good Then the matrix is URL books 351

( A ( Ny ( an ) The of is ( The overall level of prices is not pinned scalar multiple of is also an equilibrium the relevant term is the price ratio , which is the price of Good in terms of Good , or ( We can readily see that an increase in the supply of Good , or a decrease in the supply of Good , decreases the price ratio . An increase in the preference for Good increases the price of Good . When people who value Good relatively highly are endowed with a lot of Good , the correlation between preference for Good , an , and endowment of Good is higher . The higher the correlation , the higher is the price ratio . Intuitively , if the people who have a lot of Good want a lot of Good , the price of Good is going to be higher . Similarly , if the people who have a lot of Good want a lot of Good , the price of Good is going to be lower . Thus , the correlation between endowments and preferences also matters to the price ratio . In our second special case , we consider people with the same preferences but who start with different endowments . Hypothesizing identical preferences sets aside the correlation between endowments and preferences found in the good case . Since people are the same , a ( Ag for all In this case , i ) whereas before Ny ( is the total endowment of good The matrix has a special structure , and in this case , is the equilibrium price vector . Prices are proportional to the preference for the good divided by the total endowment for that good . URL books 352

Now consider a third special case , where no common structure is imposed on preferences , but endowments are proportional to each other that is , the endowment of person is a fraction of the total endowment . This implies that we can write ( an equation assumed to hold for all people and goods Note that by construction , since the value represents share of the total endowment . In this case , we have ( i ) oi ( These matrices also have a special structure , and it is readily that the equilibrium price vector ( This formula receives a similar price of good is the strength of preference for good , where strength of preference is a weighted average of the individual preference , divided by the endowment of the good . Such an interpretation is guaranteed by the assumption of Douglas preferences , since these imply that individuals spend a constant proportion of their wealth on each good . It also the conclusion found in the case to more goods , but with the restriction that the correlation is now between wealth and preferences . The special case has the virtue that individual wealth , which is endogenous because it depends on prices , can be readily determined . KEY TAKEAWAYS General equilibrium puts together consumer choice and producer theory to find sets of prices that clear many markets . For the case of an arbitrary number of goods and an arbitrary number of with is a closed form for the demand curves , and the price vector can be found by locating an of a particular matrix . The equilibrium is unique ( true for but not true more generally ) The actual price vector is not completely identified because if is an equilibrium price vector , then so is any positive scalar times Scaling URL books 353

prices does change the equilibrium because both prices and wealth ( which is based on endowments ) rise by the scalar factor . The intuition arising from models may fail because of interactions with other preferences for a good ( shifting out demand ) changes the values of endowments in ways that then reverberate through the system . I . Consider a consumer with utility , where and facing the budget constraint . Show that the consumer maximizes utility by choosing for each good i . Hint the budget I I I , and us I I IA ) I I I I I . This function can now be maximized in an unconstrained fashion . Verify that the result of the maximization can be expressed , and , which . URL books 354