Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by


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1 Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes in prices (resulting from changes in supply or demand) in one market typically a ect the demand or supply in other markets, resulting in changes in prices in these markets as well. While this fact is implicitly assumed in partial equilibrium analysis, the adjustment process in other markets is ignored in this mode of market analysis. In general equilibrium analysis, we track all adjustments simultaneously. The interactions of markets lead to feedback e ects between markets which can result in signi cant nonlinearities in the market. These nonlinearities, in turn, can give rise to complicated equilibrium price behavior, even when underlying preferences and production processes are nicely behaved. To illustrate this, we will begin our study of general economic equilibrium by looking at an extended example of the simplest possible general equilibrium model that of a pure exchange economy having no production. An Extended Example Our extended example involves a pure exchange economy without production. There are two goods in the economy, denoted x and y; and two types of agents who are identical in terms of their basic economic characteristics (but who may, if we wish, have di erent hair color, eye color, etc.). Economic characteristics in the model are the agents preferences and endowments. Agent Characteristics Indexing agents by i = ;, we denote the endowment of an agent of type i by! i = [! x i ;!y i ] R +: The endowment is simply the goods that the agent owns at the outset, prior to trading with other agents. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type agents utility is given by u (x ; y ) = Type agents utility is given by u (x ; y ) = " x + " 7 7 y # : # x + y : You may recognize these utility functions as being of the CES (constant elasticity of substitution) type. Note that our assumption that agents of each
2 type have the same economic characteristics is re ected in the assumption that each type of agent has the same preferences and endowments. Allocations An agent of type i 0 s consumption bundle is denoted (x i ; y i ) for i = ; : An equaltreatment allocation for the economy is a pair of consumption bundles [(x ; y ) ; (x ; y )] which speci es the percapita consumption of each type of agent, together with the assumption that each agent type gets the same consumption bundle. (The reason for imposing this assumption will become clear shortly.) An allocation is said to be feasible if it uses no more than the total resources available. Thus, for the equal treatment allocation to be feasible requires that Nx + Nx N! x + N! x and Ny + Ny N! y + N!y or (since there is a factor of N on both sides) and x + x! x +! x y + y! y +!y : It should be apparent from this calculation that as long as we work with equal treatment allocations, it is su cient to work in per capita terms, which reduces the model to one in which we need only deal with two agents (one representative agent of each type). Our reason for assuming that there are N agents (and that N is large) is to justify the assumption that agents act as price takers, so that our economy s markets are competitive. We will impose one additional (and innocuous) assumption on the model by normalizing the total amount of each good to one unit, so that! x +! x =! y +!y = : This assumption is innocuous since the allocations then simply become shares of the total amount of each good allocated to each agent. Markets Since we are interested in marketmediated exchange, we assume that there are two competitive spot markets for the two goods. Agents observe the posted exchange prices on the market and make decisions as to whether they wish to sell some amount of a good they have in excess. If they choose to do so, they may use the proceeds from the sale to purchase goods they are de cient in. To
3 formalize this idea, let p x and p y denote the prices of each good. agent s budget constraint is then given by A typical p x x i + p y y i p x! x i + p y! y i : This inequality states that the value of whatever the agent purchases must be less than or equal to her income, which in this model is simply the value of her total endowment of each of the goods. One way to think of this market is that every agent brings their endowment to market and sells all of it. They receive a credit on the bank account for the value of this endowment at the going prices. They can then spend this credit on the market for each good as they wish. Note that one particular purchase is always feasible for our agent: they can always simply buy back their endowment. Whether the agent wishes to trade (i.e. buy a bundle (x i ; y i ) di erent from their endowment bundle) will depend on whether the agent s demand at the going prices is di erent from the endowment bundle (! x i ;!y i ). The demand bundle is de ned as the solution to the agent s constrained utility maximization problem max u i (x i ; y i ) (x i;y i) subject to p x x i + p y y i p x! x i + p y! y i : h i A competitive equilibrium for the economy is then an allocation ^x = (bx i ; by i ) N i= together with prices bp = [bp x ; bp y ] such that the allocation is feasible and solves each agent s budgetconstrained utility maximization problem at prices bp: E ciency We say that an allocation x is e cient or Pareto optimal if it is feasible and there is no other feasible allocation y such that u i (y) u i (x) for all i, with at least one strict inequality. Hence, an e cient allocation is one in which it is impossible to make some subset of agents better o (in utility terms) without making some other agent worse o. In our simple model with two goods and two types of agents, it is easy to characterize the e cient allocations, using the concept of the Edgeworth box diagram. A typical Edgeworth box diagram is illustrated below. In the diagram, the size of the box is determined by the total amounts of each good available, which we denote as r x for the rst good, and r y for the second. By de nition, r i =! i +! i, for i = x; y. Any point in the box represents a feasible allocation, since if agent gets (x ; y ) ; then agent gets [(r x x ) ; (r y y )] : We can also graph each agent s indi erence map in the box, as shown above, taking the lower left corner as the origin for agent, and the upper right as the origin for agent. The e cient allocations occur at the tangency points of the two agents indi erence curves. To see this, start at an allocation such as x 0 in the diagram above. This allocation is not
4 e cient, since moving in a northwesterly direction will make both agents better o. To see that the e cient allocations require that indi erence curves be tangent, consider reallocations that keep agent indi erent between x 0 and the reallocation. This amounts to moving along the indi erence curve through x 0 ; to the northwest. Clearly, this makes agent no worse o, while it makes agent strictly better o. This remains true until we reach an allocation at which agent two s indi erence curve is just tangent to agent s. Equivalently, this requires that the two agents utility gradients be colinear. The locus of tangencies in the Edgeworth box is called the contract curve. The contract curve contains all of the e cient allocations. We can calculate the contract curve to our example by requiring that allocations be feasible (which we do by stipulating that agent gets whatever we don t give agent ), and that the two agents marginal rates of substitutions at the allocation be the same. 4
5 The MRS s for each agent are give by MRS = MRS = 7 y x ry y : 7 r x x Taking r x = r y = and equating, we get 7 y x = y 7 x or 7 ( x ) = y x y which when solved for y gives us y = h 7 + x 7 i x : As an exercise, plot the contract curve for this model. Marketmediated Exchange We are interested in examining the competitive equilibrium for this model. To do this, we rst calculate the demand functions for each agent, add them up to get the aggregate demand function, and then equate aggregate demand to aggregate supply and solve for the competitive equilibrium price. Note that we really only need to nd the aggregate demands for one of the two goods, since Walras Law implies that once one market clears, so does the other. This is a consequence of the budget constraints, and can be demonstrated as follows. Consider the two agents budget constraints: p x x i + p y y i = p x! x i + p y! y i : Adding over the two agents (or over all the agents of each type), we have p x x + p y y + p x x + p y y = p x! x + p y! y + p x! x + p y! y ) p x (x + x ) + p y (y + y ) = p x (! x +! x ) + p y (! y +!y ) ) p x (x + x! x! x ) + p y (y + y! y! y ) = 0: Now, suppose prices are such that the market for the rst good clear, x + x! x! x = 0: Then, necessarily, p y (y + y! y! y ) = 0; which requires either that p y = 0 or y + y! y! y = 0. We can rule out the rst case by noting that with strictly monotonic preferences, if one good becomes free, the demand for it becomes in nite. Hence, we must have the second case, which tells us that the market for good clears. 5
6 We will, therefore, limit our attention to the market for the rst good (i.e.good x). To nd the individual demand functions, we start from the rstorder conditions for the budget constrained utility maximization problem, in the form that requires the marginal rate of substitution to be equal to the price, with the budget constraint satis ed. For the rst agent, these conditions require 7 y = p x x p y p x x + p y y = p x! x + p y! y : Note from these conditions that the price level is indeterminate, in the sense that if we scale the price of each good by the same amount, it leaves the price ratio unchanged, and the scaling factor appears on both sides of the budget constraint. This is just the usual result from microeconomics which states that only relative prices matter, once we take account of the fact that people earn their incomes by selling goods and services they are endowed with at going market prices. In this situation, we are free to normalize prices by taking one of the goods as the numeraire, measuring the value of the other good in terms of how many units of the numeraire good it exchanges for. We will let good y be the numeraire, and normalize prices by setting p x =p y = r; the relative price of good in terms of good. With this modi cation, the rstorder conditions become 7 y x = r rx + y = r! x +! y : Solving these two equations simultaneously, we obtain the demand functions x (r;! ) = r!x +! y r + 7 r y (r;! ) = r!x +! y + 7 r : Carrying out the parallel calculation for agent, we nd x (r;!) = r!x +! y r + 7 r y (r;! ) = r!x +! y + 7 r : As an exercise, carry out the calculations involved in solving the rstorder conditions for the demand functions. By Walras Law, we need only consider the market for good x. market, the aggregate demand is For this x (r;! ) + x (r;!) = r!x +! y r + 7 r 6 + r!x +! y r + 7 r :
7 To nd the competitive equilibrium for this market, we de ne the aggregate excess demand function z x (r;!) = x (r;! ) + x (r;! )! x! x = r!x +! y r + 7 r + r!x +! y r + 7 r! x! x : At the competitive equilibrium price ^r; we will have z x (^r;!) = 0: Hence, we need to nd the zeroes of the excess demand function. To do this, let us simplify our notation. Let = =7; and = r : Then z x (r;!) =! x +! y + +! x +! y +! x! x! x +! y h i + +! x +! y + = i! [ + ] h x +! x : In this expression, we can factor an out of each term in the denominator and cancel this against a factor of in the numerator to get z x (r;!) =! x +! y h i + [ + ] +! x +! y + h + i! x! x : Now, for the purposes of nding equilibrium, the monotonicity of preferences implies that any equilibrium price must be positive, which means that > 0: This in turn implies that the denominator of the rst term of z x (r;!) is always positive, so it is su cient to nd the zeroes of the polynomial equation! x +! y + +! x +! y = 0 (taking account of the fact that! x +! x = ): When we multiply the terms out and collect like powers of ; we get! 5 [! x +! x ] + x +!x + [! y +!y ] +!y +!y = 0: Again, using the facts that! x +! x = ; and! y +!y = this reduces to! x +!x + +!y +!y = 0: We can simplify this expression further by noting that since the initial endowment is itself an allocation, we have! x =! x ; and! y =!y : Making these substitutions, we have "! x # + + +! y = 0 7
8 or, putting the expression in canonical form, " # " # +! x! x + +! y! x = 0: As an exercise, carry through the details of this set of calculations to go from the initial expression for z x (r;!) to the equation above. Cases: Consider the case rst where! x = and! y = 0 (so that agent owns all of the rst good, and none of the second). Then the excess demand polynomial becomes + = 0: By inspection, one root of this equation is = : Dividing the polynomial by ( ) ; we get + = ( ) + ( ) + : With = =7; this is ( ) = ( ) = ( ) : Thus, the excess demand function has three real roots, = ; = =4; and = 4=: These correspond to competitive equilibrium prices of, (=4) ; and (4=) respectively. This situation is shown graphically in Figure. Now consider what happens when! x = 7=49 and! y = =49: Substituting these values in the excess demand polynomial, we get the equilibrium equation + = 0 which factors into ( ) + = 0: This equation has only one real root, so for this distribution of endowments, the competitive equilibrium is unique. To understand why we get these di erent results, consider the graph of the excess demand function for good under the rst distribution of endowments, shown in gure. 8
9 For this case, consider the standard tatonnement procedure for nding the competitive equilibrium prices. Pick an initial price r 0 : If the excess demand at that price is positive, increase the price. If it is negative, decrease the price. Continue adjusting prices in this way until you arrive at an equilibrium. If we apply this procedure to any initial price with r 0 less than, we converge to the equilibrium at (=4) : If we start with r 0 > ; we go to the equilibrium at (4=) : For this tatonnement procedure, the equilibrium with r = is unstable; starting in any neighborhood of r = ; the tatonnement diverge from that neighborhood. Note the curious fact here: if r is close to but not equal to, then increasing the price increases excess demand, while decreasing the price decreases excess demand. To see why this happens, we need to look at the relationship between income and substitution e ects in the model. Recall that for this case, agent makes all of her income from the sale of the rst good, while agent makes her income from the sale of the second. To analyze this, we will consider the e ects of changes in the price of good in various cases, which are illustrated in Figure 4. Consider situation A, where the price of good one is low relative to that of good two, and we increase the price of good one slightly. Increasing the price 9
10 of good one has two e ects. First, it makes good one relatively more expensive than good. Second, it increases agent s income since the value of what she sells is now greater. The increase in relative prices will generate a standard negative substitution e ect, while the increase in value of agent s endowment will lead to a positive income e ect (since these are normal goods). Whether agent s overall demand for good increases or decreases depends on which of these two e ects dominates. In case A, it should be clear from the diagram that the substitution e ect dominates, and increasing the price of good decreases agent s demand for good. Since agent experiences no income e ects due to the increase in the value of good, but does experience the substitution e ect, her demand for good one also falls. Hence, aggregate demand for good one falls when the price rises. In situation B, the income e ect dominates and an increase in the price of good one generates an increase in agent s demand for good one. While agent s demand for the good continues to decrease when the relative price increases (i.e. substitution e ect dominates), agent s increase in demand is larger, so aggregate demand increases. This is a simple example of how the distribution of wealth can a ect the equilibrium of the economy. You should contrast this to the case where! x = 7=49 and! y = =49: If you check, you will nd that this endowment distribution (with agent having! x = =49 and! y = 7=49) lies on the contract curve and is Pareto optimal to start with. For this distribution of endowments, agents will not nd trade pro table. This situation is frequently referred to as a notrade 0
11 equilibrium. We saw above that in this case, the equilibrium is unique; in fact, it is easy to show that any notrade equilibrium must be unique. One of things that economists are frequently interested in is how equilibrium changes when we change the underlying fundamentals of the economy. For the simple model we are examining here, the fundamentals include the distribution of endowments, and the preferences of the agents. We will focus for the remainder of this section on analyzing how changes in the distribution of endowments a ect the equilibrium of the economy. Looking at the e ects of changing preferences can be done using similar techniques. To begin our analysis, let us introduce the set E (r;!) = (r;! x ;! y ) R ++ j Z x (r;!) = 0 : The set E (r;!) is called the equilibrium set of the economy when resources are xed but the distribution of endowments can be varied. By studying the properties of this set, we can provide information about the feasibility of comparative
12 static analysis of equilibrium when we change the distribution of wealth. We ve seen from our analysis above that changing the wealth distribution (even continuously) can lead to changes in the number of (discrete) equilibria, which implies that any relationship between equilibrium prices and endowments must include some discontinuities. Can we determine how prevalent these discontinuities are? If they occur only rarely, then we can be con dent that for the most part, small changes in the distribution of endowments lead to small changes in the equilibrium price (or prices). On the other hand, if there are many discontinuities, then small changes in the distribution of wealth can lead to large, radical changes in the nature of the economic equilibrium. Our jumping o point for this analysis is the equilibrium polynomial " # ^Z (;! x ;! y ) = +! x " #! x + +! y! x = 0: If we let P = " #! x we can write the polynomial as Q = +! y! x This equation can be put into the form by making the substitution + P P + Q = 0: w + Aw + B = 0 w = + P where and A = P [ + P ] B = P + 9P + 7Q : 7 Note that we have implicitly de ned a mapping from our original parameter space to the space of parameters (w; A; B) : To see that this mapping is locally
13 onetoone, calculate the derivative of the mapping (; P; Q) 7! (w; A; B) : Since 4 w + p A 5 = 4 P ( + P ) B 7 P + 9P + 7Q 5 the derivative matrix is J = P P + P This matrix will be nonsingular as long as P 6= : You can check directly that when P = ; this corresponds to (and any value of!y ), so that along the vertical line in the Edgeworth box over! x : = 0:88 we will lose the invertibility of the mapping. But this set has measure zero in the twodimensional Edgeworth box, so that the two invertible mappings on either side of the line of singularity can be "glued" together by continuity. Let h (r;! x ;! x ) = (w; A; B) be the map de ned above, and consider (subject to the caveat on inverting the invertibility of the mapping) the set n ^E = (w; A; B) R j ^Z o h (; P; Q) = 0 : In canonical coordinates, then, the set ^E is de ned by 5 : ^E = (w; A; B) R j w + Aw + B = 0 : Figure 5 shows a Matlab plot of this set looking down at the surface ^E from above in the direction of the w axis. Note how for values of (A; B) toward the right side of the diagram, the value of w for which w + Aw + B = 0 is unique, while toward the left side of the diagram, there are multiple roots to this equation. We can characterize the parameter values of (A; B) for which the number of equilibria change by de ning the socalled natural projection : ^E! (A; B) R which associates the zeros of the equilibrium polynomial with parameters A and B: The mapping is given explicitly by (w; A; B) = w + Aw + B; A; B : It should be obvious that for any (w; A; B) ^E; this expression is precisely [0; A; B] which is just the projection of ^E on the parameter space. Now, from the diagram, it should be clear that the number of zeroes of the equilibrium polynomial change precisely when we cross one of the folds of the surface. This is equivalent to the statement that the natural projection fails to map onto the
14 parameter space at these points. This can perhaps be best seen from looking at a crosssection of the surface, as in Figure 6. At the two fold points, maps the tangent line at the fold point into a single point. Anywhere else along the curve, it maps the tangent line onto the B axis. From advanced calculus, we know that for a mapping or transformation to exhibit this kind of singular behavior, it must be that the Jacobian matrix D (w;a;b) has less than maximal rank. For the mapping above, this matrix takes the form w + A w D (w;a;b) = : 0 0 This fails the have rank precisely when w + A = 0: Putting this together with the requirement that (w; A; B) ^E; the locus of points at which singularity occurs will be the solutions to q Since w = w + Aw + B = 0 w + A = 0: A ; substituting this into the cubic and solving for A yields A = B : 4
15 This equation is plotted in Figure 7. In the diagram, values of B are plotted along the horizontal axis, with values of A plotted along the vertical axis. Positive values of A correspond to the case where the cubic has only one real root. Negative values of A lying inside the cusp gure above correspond to the case where there are real roots. Negative A 0 s outside the cusp correspond to unique roots, while combinations of A and B lying on the cusp correspond to repeated roots. We call the cusp gure itself the critical set, and denote it by : You can also view the cusp as the projection on the A B plane of the fold region of the surface ^E; as indicated in Figure 8, where the fold region is highlighted. One interesting thing to note here is that the fold region is thin, that is, it has measure zero relative to the natural measure or area on the plane or on a dimensional surface. This tells us that if we pick an endowment distribution at random, then the probability that we will pick one in is zero. Hence, for regular economies (i.e. those not lying in ), we can be sure that su ciently small variations in the underlying distribution of endowments will generate continuous variations in the equilibrium prices, so that standard comparative static analysis will be valid, with probability one. 5
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