Walrasian Equilibrium Computation, Network Formation, and the Wen Theorem

Transcription

1 Review of Development Economics, 6(3), 45 47, 00 Walrasian Equilibrium Computation, Network Formation, and the Wen Theorem Shuntian Yao* Abstract The paper studies the Walrasian equilibrium theory of division of labor for large economies with weakly convex production techniques and with the presence of transaction costs. Earlier results published by Wen are revised and generalized so that the new versions can be applied to a much larger category of economic models. A simple example demonstrates how the theoretical results can be used for Walrasian equilibrium computation and equilibrium network structure identification.. Introduction Economic growth and development are two of the most challenging issues facing all developed and developing countries. All countries require a good international economic environment, and each country itself needs to improve its domestic economic infrastructure. Both these issues can be related to so-called economic networking theory, which studies how economic agents should choose their activities, how they should choose their trading partners, and how they could improve transaction efficiency; and in general, how they can maximize their utility with their given resources. As is well known, given some economic resource, different economic infrastructures or different economic networking decisions can lead to totally different economic outcomes. China can serve as a convincing example. Following the introduction of economic reforms, China has developed a more efficient economic network for its domestic market and for its international affairs. The per capita GNP of the Chinese economy has more than tripled over the past 0 years or so. Most recently, because of the rapid development of the computer industry and Internet communication, all economic agents are potentially linked to a worldwide e-commerce network. This, on the one hand, gives more freedom to any economic agent (including any developing country) to make decisions. However, the decision process becomes more complicated because the agent has to deal with more variables than before. In this sense, the study of economic networking theory has become more important for development economists. The topics addressed by this paper are some of the basic analytical issues in the general-equilibrium theory of economic networks. As will be seen later, the results presented here provide a systematic approach for computation of general equilibria and for identification of equilibrium network structures. As pioneers in applying inframarginal analysis to increasing-returns-to-scale economic models, Xiaokai Yang and Yew Kwang Ng, together with many of their colleagues, have made a series of helpful contributions to the general-equilibrium theory of division of labor and economic networking theory (e.g., Yang and Ng, 998; Wen, 998a,b; Ng, 00). They have investigated many typical examples, with interesting results. For example, they have shown that, as the transaction costs in an economy * Yao: Division of Applied Economics, Nanyang Business School, Nanyang Technological University, Singapore. Tel: ; astyao@ntu.edu.sg. I wish to thank Mei Wen for helpful discussion., 08 Cowley Road, Oxford OX4 JF, UK and 350 Main Street, Malden, MA 048, USA

2 46 Shuntian Yao became sufficiently low, a higher level of division of labor should lead to a higherefficiency general equilibrium. To concentrate on the relationship between economic efficiency and specialization, in most of their examples Yang and his colleagues consider simple models with a population of identical individuals and with a limited number of goods. The most interesting feature of the general equilibria of these examples is that, when the transaction costs are low, the ex-ante identical individuals, aiming at utility maximization, divide themselves into several groups, each group choosing a different configuration and trading with the other groups of individuals. Such a configuration is characterized by the subset of goods to be produced, the subset of goods to be sold, and the subset of goods to be bought. In the Yang Ng framework, determining the different basic configurations is the most important step in both the equilibrium existence proof and the equilibrium computation (Sun et al., 000). In this respect, the following result referred to as the Wen theorem by Yang and his colleagues (Yang, 00), plays a very important role: In any Walrasian market with any price vector given, any rational agent with convex production technique will never sell more than one of her produces, and will never simultaneously produce and buy the same good. With the help of this, they can divide the population of ex-ante identical agents into several groups, each consisting of those selling one single good. They can then compute an equilibrium at which all agents in the same group choose the same optimal decision, and a trading network is formed. However, as I will show in the next section, what Mei Wen proved (998a, p ; and 998b) does not have the general implications mentioned above. Moreover, some of her claims appear to be false.. Wen s Result and a Counter Example What is referred to as the Wen theorem is her following result in Wen (998a): Consider an economy E with a population of individuals and a finite number of goods and with the presence of transaction costs. Assume that the following assumptions hold. () Every individual in E has a strictly quasi-concave and twice continuously differentiable utility function. () The production function f i of every individual i satisfies either (i) f i ( ) > 0, f i ( ) > 0, and f i is continuous for l i > 0; or (ii) f i (l i ) = a i l i - b i, where a i,b i > 0. (3) Every transaction function has positive first derivative and nonnegative second derivative in the whole domain. Then given any price vector p for the goods, the optimal decision of any individual does not involve buying and selling the same good, does not involve selling more than one good, and does not involve self-providing and buying the same good. In Wen s proof, her arguments under assumption (i) are correct, but the requirements in (i) are too strong. Actually, in order to derive the consequences we need only assume the strict convexity of the production functions (see later). It is well known that production functions with positive second derivative in the whole domain just form an extremely tiny subset of the set of strictly convex production functions. In most of the examples studied by Yang, Ng, and their colleagues, the conditions in (i) were not satisfied, and Wen s result was inappropriately applied. With regard to Wen s assumption (ii), what Wen should have used is the following functional form:

3 f ( l )= max { 0, al -b}; i i i i i otherwise the meaning of negative quantities of the produce needs to be explained. The main reason for Wen to choose the functional form in (ii) seems to be that the inverse of the production function is required in her arguments for not involving buying and selling the same good (Wen, 998a, p. 74). But under her assumption that the transaction efficiency k is below, claim in her analysis could be easily established without using the existence of the inverse production function. Wen s claim (selling one good at most) is not correct under assumption (ii). In her arguments for claim, she did pay attention to (ii). However, in discussing the sign of p j a j - p k a k she ignored the case of p j a j - p k a k = 0, and it is then that claim is false! Let us turn to Wen s claim 3 (not self-providing and buying the same good). On the one hand her argument relies on the conclusion of claim, which is false under assumption (ii). On the other hand, to establish the consequence, she requires the strict inequality: Ê pi0 ˆ X ( l)= f ()+ l Á ( - ) + ( - )> Ë p f l l k pi0 j j i j p kf j i l l j WALRASIAN EQUILIBRIUM COMPUTATION 47 The inequality could be false for a weakly convex transaction function k j under assumption (ii) (where all the f i =0), because k j =0 will lead to X j (l) = 0. Thus Wen s arguments in claim 3 are only good for the case under assumption (i). j A Counter Example To see that Wen s claim is false under her assumption (ii), let us consider an economy E with four goods {,, 3, 4} and a set of agents I. Assume that one of the agents has an initial endowment of unit of labor, and has the same utility function given by u(w, x, y, z) = wxyz, where w, x, y, and z are, respectively, the quantities of good, good, good 3, and good 4 she consumes. Assume that this agent has a set of production functions q j = max{0, L j - 0.}, j =, and q j = max{0, L k - 0.6}, k = 3, 4. Here L j is the amount of labor allocated for good j s production, and L k is the amount of labor for good k s production. Finally assume that, because of the presence of the transaction costs, when this agent buys y j units of any good j, the amount she actually receives is 0.6y j. Let us imagine that a price vector p = (,,, ) is announced by the referee. We look at the optimal decision of this agent. If she chooses autarky, because of the fixed learning costs, she can never produce these four goods simultaneously all in positive amounts. As a result her maximal utility in an autarky is 0. On the other hand, if she wants to trade, in view of her production technology and the market prices she should never produce good 3 or good 4; instead she should produce good or good or both, selling part of her produces in exchange for the other two goods. We first consider the case in which she produces good only. We may assume that she allocates all her labor towards good production, producing 0.9 units of good, then selling x units of good for buying good, selling y units of good again for buying good 3, and selling y units of good again for buying good 4. Because of the transaction costs, she will consume x - y units of good, 0.6x units of good, and 0.6(y) units of each of the other two goods. As a result she gets a utility of u = (0.9 - x - y)(0.6x)(.y). By elementary calculus it is easy to verify that her utility is maximized at x = 9/40 and y = 9/40. The maximal utility in this case is 4(0.6) 3 (9/40) 4 = 0.00.

4 48 Shuntian Yao The second case is that she produces good only. By symmetry it is easy to see that her maximal utility in this case is also 4(0.6) 3 (9/40) 4. Now consider the third case in which this agent produces both good and good. In view of her production technique and the transaction costs, when these two goods are produced she will no longer buy one of them by selling the other; i.e., her consumption of these two goods will be all self-provided. On the other hand, she will sell part of her produce for buying good 3 and good 4. By symmetry, she does not care which good she sells as long as she has sufficient of it left for self-consumption. Let us assume that she self-consumes x units of each of good and good. Since she can produce 0.8 units of these two goods as a whole, she will sell x units of them in the market, which is sufficient for buying x of each of the other two goods. (Note that the price of good or good is twice as high as that of good 3 or good 4.) Because of the transaction costs, her utility will be u = x [0.6(0.8 - x)], which is maximized at x = 0..The maximal utility is (0.) (0.4) = Compared with the first two cases, this case gives the highest utility she can achieve. To sum up, with the price vector p = (,,, ) given, this agent has a continuum of different optimal decisions: allocating l units of labor for good s production and - l units of labor for good s production (0.3 l 0.7), keeping 0. units of good and 0. units of good for self-consumption, selling all the remainder of them for buying 0.4 units of each of good 3 and good 4. In particular, when l = 0.5, she sells 0. units of each of good and good. Thus Wen theorem does not hold in this example. 3. Revised Theorems Although most readers may be familiar with the definitions and properties of weakly convex functions and strictly convex functions, for convenience I present a brief review here. For more details refer to Stromberg (98), for example. Definition. A function f:[0, ] Æ R is said to be weakly convex if, for any x, x Œ[0, ],x< x and any l Œ (0, ): f( lx+ ( -l) x ) lf( x)+ ( -l) f( x ). () Moreover, f is said to be strictly convex if in the above inequality the sign is replaced with <. Proposition. Assume that f:[0, ] Æ R is weakly convex. Then for any x Œ [0, ], Dx > 0, and a > 0 such that x + a +Dx, it holds that f( x+ a+ Dx)- f( x+ a) f( x+ a)- f( x). Dx a () Moreover, if f is strictly convex, then the above inequality holds with replaced by >. Proof. Choose l =Dx/(a +Dx) and x =x + a +Dx in (). It is easy to verify that - l = a/(a +Dx) and that lx + ( - l)x =x + a. As a result we have f ( x x + a ) a x f x a D ( )+ + a x f ( x + a + D x ). D + D (3)

5 WALRASIAN EQUILIBRIUM COMPUTATION 49 On multiplying both sides of (3) by (a +Dx) and re-arranging terms, we obtain [ ] [ + ] a f( x+ a+ Dx)- f( x+ a) Dx f( x a)- f( x). On dividing both sides of (4) by adx, () thus follows. (4) We now consider a Walrasian market with m goods. Assume that a price vector p >> 0 for all these goods is given. Assume that an agent is endowed with unit of labor. A decision by her consists of 3m nonnegative variables: L j, Dx j, Dy j, j =,...,m, where L j is the amount of labor she allocates for good j s production, Dx j is the amount of good j she sells, and Dy j is the amount of good j she buys. We will denote by f j her production function for good j, and by g j her transaction function for good j. Assume that this agent has a continuous utility function given by u = u(z,...,z m ), where z j is the amount of good j she consumes. Statement of Theorem Consider a Walrasian market with m goods with or without transaction costs; i.e., g j (y j ) < y j or g j (y j ) = y j, respectively. Assume that the utility function u is continuous and is nondecreasing in each of the variables. Assume that the production functions f j are all defined and weakly convex on [0, ]. Then for any given price vector p >> 0, the maximal utility U(p) for this agent can always be achieved by a decision to sell no more than one good. Moreover, if u is strictly increasing in each of the variables and all the f j are strictly convex on [0, ], then any utility-maximization decision made by her must involve no more than one good being sold. Proof of Theorem First consider the case that u is nondecreasing and the f j are weakly convex. Assume that u is maximized by a decision in which there are two or more goods being sold. It suffices to show that this agent can reduce the number of goods sold and still achieve the same maximal utility. Assume that originally good and good are sold by her. Let L and L, respectively, be the amounts of labor she allocates for the production of these two goods. Let x and x, respectively, be the quantities of these two goods produced. Let Dx and Dx, respectively, be the amounts of them being sold. Let l j be the amount of labor satisfying f j (l j ) = x j -Dx j, j =,. Let Dl j = L j - l j. We then have Dxj = fj( lj + Dlj)- fj( lj), j =,. (5) Let p j be the price of good j. Then the contribution to the budget by selling these two goods is where  pj[ fj( lj + Dlj)- fj( lj) ]= k Dl + k Dl j= k j [ ] pj fj( lj + Dlj)- fj( lj) =. Dl j, (6) (7) Proof of the first part of the theorem It suffices to show that this agent can achieve a looser (or at least not tighter) budget for buying through reallocating labor on the

6 40 Shuntian Yao production of these two goods, selling just one of them but not changing the selfconsumed amounts of each, and not changing the other part of her decision on production and selling of the other goods (if any).we need to consider two different cases. Case (i): k k In this case we will consider her decision to sell good only. Imagine that she now allocates just l units of labor for good s production, producing x -Dx units of good all for self-consumption; and she allocates l +Dl +Dl units of labor on good s production, consuming the same amount x -Dx = f (l ) as before. By selling the other part of good, she receives an amount of money [ ] p f( l + Dl + Dl)- f( l) = p[ f( l + Dl + Dl)- f( l + Dl) ]+ p[ f( l + Dl)- f( l) ] p[ f( l + Dl + Dl)- f( l + Dl) ] = Dl + Dl p f ( l + D l )- f ( l ) Dl Dl p[ f( l + Dl)- f( l) ] ( Dl + Dl) = ( Dl + Dl ) k Dl k Dl + k Dl. Thus we can see by selling good only, she can really guarantee a budget at least as good as before, and, as a result, she can achieve a utility either greater than or at least equal to the original one. Note that, in deriving the above inequality, Proposition was applied to obtain [ ] [ + ] = p f( l + Dl + Dl)- f( l + Dl) Dl p f( l Dl)- f( l) k. Dl [ ] (8) Case (ii): k > k In this case we consider her decision to sell good only. Assume that she allocates l +Dl +Dl units of labor for good s production, and l units of labor for good s production. Using a very similar argument to that for case (i), we can conclude that by selling good only, she can guarantee a looser budget and can achieve a higher utility. On combining cases (i) and (ii), the first part of the theorem is proved. Proof of the second part of the theorem Simply note that when strict convexity is assumed, the expression (8) becomes a strict inequality. As a result, this agent can always achieve a higher budget by reducing the number of goods sold. She can thus buy more and consumes more. In view of the strictly increasing property of her utility function, she can thus achieve a higher utility. Statement of Theorem Consider a Walrasian market with m goods as described in Theorem. Assume that all the transaction functions g j are weakly convex (including the linear case). If all the f j of the agent are weakly convex and her utility function u is nondecreasing in each of the variables, then for any given price vector p >> 0 her maximal utility U(p) can be always achieved by a decision to not buy and produce the same good simultaneously. Moreover, if the f j are strictly convex and u is strictly increasing in each variable, then any utility-maximization decision by her must not allow any good being simultaneously produced and purchased.

7 WALRASIAN EQUILIBRIUM COMPUTATION 4 Proof of Theorem Proof of the first part of the theorem Assume that p is given, and that in her utilitymaximization decision the agent produces and purchases good simultaneously. According to Theorem, we may assume she sells just one good, say good. (Obviously in any case there is no need for her to sell and buy the same good.) Let p j be the market price of good j. Let x = f (l ) be the amount of good she produces for herself. Let y be the amount of good she buys. Let x = f (l +Dl ) be the amount of good she produces, and x -Dx = f(l ) be the amount of good she would just require if she had no need to buy y units of good. Thus Dl is the amount of labor she can save if she does not buy good. We then have y = ( p p) [ f( l + D l)- f( l) ]. (9) Because of the presence of the transaction costs, the actual amount of good she receives from the market is Ê p Dx g D (0) p f l l f l ˆ = [ ( + )- ( ) ] Ë. Let us fix l and define h(x) = g ((p /p )[f (l + x) - f (l )]). Note that h is a weakly convex function; this is because f (l + x) is weakly convex and increasing in x, and (p /p )[f (l + x) - f (l )] is linear in f (l + x) and hence weakly convex and increasing in x. As a composite function of two weakly convex increasing functions, h is also weakly convex. With our new notation we can write Dx = h( Dl), () and consider the following two new decision options of the agent. In option (i), she saves Dl units of labor from the production of good, not selling Dx units of good to exchange for good but adding the saved amount of labor Dl to l for good s production. As a result she will consume f (l +Dl ) units of good and the same amounts of good and all the other goods as before. In option (ii), she does not produce good at all, adding l units of labor saved to l +Dl for good production, selling f (l +Dl + l ) - f (l ) units of good to exchange for good.as a result she will consume h(dl + l ) units of good and the same amounts of good and all the other goods as before. The first part of the theorem will be proved if we can show that at least one of f (l +Dl ) and h(dl + l ) is not less than f (l ) + h(dl ), which is the original amount of good she consumes. Define k and k by f( l) h l h l h 0 k = k = ( D ) ( D )- (), =, () l Dl Dl where we have used the fact that h(0) = 0. We first consider the case k k. With the help of Proposition we have f( L+ Dl)- f( l) f l f( l+ Dl)= Dl + ( ) l Dl l k ( Dl + l ) k l + k Dl = f ( l )+ h( Dl ), which implies that the new option (i) is better than (or at least not worse than) her old decision.

8 4 Shuntian Yao We now consider the case k > k. We have h( Dl + l)- h( Dl) h( Dl + l)= l + h( Dl) l h( Dl)- h() 0 l + h( Dl) Dl = k l + h( Dl )> k l + h( Dl )= f ( l )+ h( Dl ), which implies that option (ii) is better than her old decision. As a result, the first part of Theorem is proved. Proof of the second part of the theorem Simply note that the strict convexity of the production functions implies the strict convexity of h. As a result either option (i) or (ii) will lead to a higher amount of good being consumed than in her original decision. 4. Equilibrium Computation and Network Formation In most published computed examples of general-equilibrium models of a pure exchange economy, the convexity of the consumption sets is assumed and the agents preferences are assumed to be strictly quasiconcave. With such assumptions, given any price vector, each individual has but a unique optimal decision. To compute the optimal decision of every individual in terms of the prices, an equilibrium price vector can be then determined by the market-clearing conditions. In all these examples, net trade occurs only between ex-ante different agents, either because they have different utility functions, or because they have different initial endowments. In Walrasian models where the convexity of the consumption sets or the convexity of the production sets is lacking, although equilibrium existence can still be established for large economies with the convexification idea, no systematic computation method has been proposed. The difficulty is that, given any price vector, an agent usually has quite a few different optimal decisions, all leading to the same maximal utility, but any convex combination of these decisions is no longer feasible to her. On the other hand, if all ex-ante identical agents in the population choose the same optimal decision, the market would not be cleared. As a result, a general equilibrium can be achieved only when the mass of ex-ante identical agents divide into several groups, each group with a suitable measure, and when agents in different groups choose different optimal decisions. An important feature of these economic models is that net trades may occur between ex-ante identical agents. The importance of the Wen theorem is that it provides a systematic approach for the equilibrium computation where convexity is lacking and convexification is required. To explain the application of the two new theorems established in the last section, I next consider a large economy E with m goods and a continuum set I = [0, ] of ex-ante identical consumer producers, each being endowed with unit of labor. I will use the same notations as before, f j for the production functions, g j for the transaction functions, and u for the utility function. Denote by Lj i the labor units that i allocates for good j s production, by xj i the amount of good j that i sells, and by yj i the amount of good j that i buys. We will assume weak convexity of the production functions, linearity of the transaction functions, and a continuous and nondecreasing utility function. In addition we also assume the following property of u: for any given j and any given z > 0,...,z j- > 0, z j+ > 0,...,z m > 0, it holds that

9 WALRASIAN EQUILIBRIUM COMPUTATION 43 lim zj Æ uz (,..., zj-, zj, zj+,..., zm)=. (3) It is easy to see that, as a special example, the Cobb Douglas utility functions satisfy all the requirements mentioned here. Definition. Given a large economy E as described above, a price vector p >> 0 together with a decision d i = (L i j, x i j, y i j: j =,...,m) for each and every agent i Œ I is said to be a Walrasian equilibrium of E if (i) the decision of every consumer producer is a utility-maximizing decision under p, and (ii) the market for every good is cleared under p and the abovementioned individuals decisions. While the existence of a Walrasian equilibrium is an easy consequence following the idea of the classical equilibrium theory for large economies with the convexification trick (Zhou et al., 999; Hildenbrand, 974), the computation problem, as mentioned above, has never been systematically addressed before. The existence proof itself does not provide any computational approach. On the other hand, the revised Wen theorems do provide a systematic approach for equilibrium computation. The general algorithm consists of three steps:. Divide the population of agents into m groups with the measure of the mass w j in group j to be determined later. Assume that every agent in group j is allowed to sell good j only if she does want to sell anything. (She is allowed, however, to produce any good and buy any good.) Assume that all the agents in the same group choose the same optimal decision, and compute a constrained optimal decision d j (p) and the constrained maximal utility U j (p) for every member in each group j in terms of the prices (p,...,p m ). Here we use the word constrained because each agent in group j is allowed to sell good j only.. Compute a price vector p* which equates the maximal utilities of the agents across all different groups: U (p*) =...= U m (p*) = U(p*). 3. Determine the measure of the mass w* j 0 of each group by the market-clearing conditions under the price vector p*. (Note that w* j = 0 means that no agents are in group j, and good j is not sold in the market although it may still be produced.) Proposition. Assume that the price vector p* and the weights w* j have been computed in the above algorithm. Then p* together with the d* j d j (p*) (chosen by every agent in group j) is a Walrasian equilibrium. 3 Proof. The maximal utility U( p*) that every agent receives in the above algorithm is under the constraint that she is allowed to sell only one specific good. We need to show that this is actually the maximal utility she can achieve under p* even if such a constraint is removed. In fact, according to Theorem, given p*, it is always possible for her to achieve the (unconstrained) maximal utility U by choosing a decision to sell no more than one good. Let this be good j; then U = U j (p*) = U(p*). The result of Proposition 3 thus follows. The trade network of this Walrasian equilibrium can be constructed in the following procedure: (a) Vertices. Assume that at the abovementioned equilibrium, I is divided into m groups: I = G... G m, but only the first n groups are of positive measure, which means that at the equilibrium only the first n goods are traded. To construct

10 44 Shuntian Yao the network, we first draw n vertices v,...,v n, each denoted by a small circle, with v j representing group j. Inside the small circle of each vertex we put w* j, which is the measure of the mass of agents in group j(j =,...,n). (b) Reflexive arcs. If each of the members in group j produces amount x j h of good h for self-consumption, then an arc jj labeled with h is constructed, starting from and ending at the same vertex v j. (c) Nonreflexive arcs. If each of the members in group j buys an amount y j k of good k, then an arc kh labeled with k is constructed, starting from vertex v k and ending at vertex v j. Thus the equilibrium network structure constructed above is a weighted digraph. I explain all the details of equilibrium computation and network construction with a simple example in the next section. 5. A Simple Example Consider a large economy E with three goods {,, 3} and a continuum of ex-ante identical agents I = [0, ]. Each agent is endowed with unit of labor. The production functions of each agent are given by f j (L j ) = L j, j =,, and f 3 (L 3 ) = max{0, L 3-0.5}. The transaction functions are given by g j (y j ) = ky j, j =,, 3, k Π[0, ]. The utility function for every agent is given by u(z, z, z 3 ) = z z z 3. With the symmetry in good and good, one can expect that the equilibrium price vector has the form of (,, p). According to the revised Wen theorems, we can divide the population into three groups: G, G, and G 3, with agents in each G j selling only good j (if they sell anything at all) and not simultaneously producing and buying the same good. Assume that the measures of masses in group j is w j. By symmetry we may assume w = w = w, and w 3 = - w. The value of w will be determined in the final stage. Under the price vector (,, p), consider the decision of any agent in group G. She may choose autarky, allocating l units of labor for the production of each of good and good, and - l units of labor for the production of good 3, achieving a utility of u = l (0.5 - l). By elementary calculus it is easy to verify that the best she can do in an autarky structure is to choose l = /6, so - l = 4/6, yielding a utility of /6. The second option for the agent is to produce and sell part of good, to self-provide good, and to buy good 3 from the market. Assume that she allocates l units of labor for good s production and - l units of labor for good s production, selling x units of good in order to buy x/p units of good 3. Her utility is then u = (l - x)( - l)(kx/p), which is maximized at l = /3 and x = /3. The maximal utility is k/7p. The third option for the agent is to produce and sell part of good, to self-provide good 3, and to buy good. Assume that she allocates l units of labor for good s production and - l units of labor for good 3 s production, selling x units of good in order to buy x units of good. Her utility is then u = (l - x)(kx)(0.5 - l), which is maximized at l = /3 and x = /6. The maximal utility is k/6, which is not greater than the maximal utility in autarky no matter what value k assumes. The last option is to produce and sell part of good, and to buy both good and good 3. Assuming she sells x units of good, buying y units of good and y 3 units of good 3. The budget constraint is x = y + py 3. If we write y = y, then y 3 = (x - y)/p.her utility is u = ( - x)(ky)[k(x - y)/p], which is maximized at x = /3 and y = /3. The maximal utility is k /7p, which is not greater than the maximal utility in the second option no matter what value k assumes. To sum up, we have:

11 WALRASIAN EQUILIBRIUM COMPUTATION 45 (a) Any agent in group will choose autarky if k/p < /8, and choose the second option if k/p > /8. (By symmetry) (b) Any agent in group will choose autarky if k/p < /8, and choose self-providing good and buying good 3 if k/p > /8. Now consider the decision of any agent in group 3. If she chooses autarky, her maximal utility is /6. If she produces and sells part of good 3, self-provides good, and buys good, she has a utility of u = ( - l)(kpx)(l x), where l is the amount of labor allocated for good 3 s production, - l is the amount of labor allocated for good s production, and x is the amount of good 3 she sells in order to buy px units of good. The maximal utility is kp/6, achieved at l = 5/6 and x = /6. If she produces and sells part of good 3, self-provides good, and buys good, she ends up with the same maximal utility. Finally she may produce good 3 only and buy both good and good. Let x be the amount of good 3 she sells, y the amount of good she buys; then (px - y) is the amount of good she buys. Her utility is then u = ky[k(px - y)](0.5 - x), which is maximized at x = /3 and y = p/6. The maximal utility is (kp) /6. To sum up, we have: (c) Any agent in group 3 will choose the autarky if kp <, and choose the last option (buying both good and good ) if kp >. If trade does occur among the three groups, at any equilibrium, individuals across all groups must have the same maximal utility, otherwise there is an incentive for individuals of some group to switch to another group. Thus, comparing (a) and (b) with (c), any equilibrium price vector for an active trade market must equate k/7p with (kp) /6, from which one can solve p = /k /3, which is the equilibrium price for good 3. Now it is not difficult to verify that, with the equilibrium price vector (,, /k /3 ):. If k < / 8, we have both k/p < /8 and kp <, and the only Walrasian equilibrium network is the autarky structure.. If k > / 8, we have both k/p > /8 and kp >, and one Walrasian equilibrium network is with three vertices v, v, and v 3. In v, every agent produces /3 units of good and /3 units of good, selling /3 units of good in order to buy k /3 /6 units of good 3. In v, each agent produces /3 units of good and /3 units of good, selling /3 units of good in order to buy k /3 /6 units of good 3. In v 3 each agent allocates all her labor for good 3 s production, producing 0.5 units of good 3, selling /3 units of it in order to buy p/6 units of each of good and good. Finally to determine the measure of the mass of each group, we use the market-clearing condition, say for good 3: ( - w)(/3) = w(k /3 /6). We thus obtain w = /( + k /3 ). 3. In particular, when the transaction costs for good 3 is zero (k = ), the equilibrium price vector is (,, ), the measures of the masses in the three groups are all equal to /3, and the equilibrium utility for every agent is /54 four times as large as the maximal utility from autarky! The equilibrium structures are depicted in Figure. It is interesting to note that, in the final two equilibrium structures, the costs of high production technology (or high fixed learning costs) paid by the good 3 producers are compensated by the high equilibrium price they receive for their produce.

12 46 Shuntian Yao k</8 / 3 /8 / <k< 3 3 k= Figure. The Equilibrium Network Structures Remarks. (i) In this example we do not have an equilibrium structure with complete division of labor for any k <, which results from the linearity of the production functions for good and good. In fact it is not rational to sell one of them to buy another, unless the transaction cost is 0. (ii) In most of situations there is no unique equilibrium structure. 6. Conclusion These results can be applied to economic models with weakly convex production functions. They can be applied to the computation of a Walrasian equilibrium of a large economy with a continuum of ex-ante identical consumer producers. In fact the results can also be applied to economic models with finitely many types of ex-ante different agents, although the argument and the computation will be much more complicated. References Hildenbrand, Werner, Core and Equilibria of a Large Economy, Princeton, NJ: Princeton University Press (974).

13 WALRASIAN EQUILIBRIUM COMPUTATION 47 Ng, Yew-Kwang, Infra-marginal versus Marginal Analysis of Networking Decisions and E- Commerce, a speech at the International Symposium of E-Commerce and Networking Decisions, Monash University (00). Stromberg, Karl R., An Introduction to Classical Real Analysis, Dordrecht: Kluwer (98). Sun, Yang, and Yao, Theoretical Foundation of Economic Development Based on Networking Decisions in the Competitive Market, Harvard Center for International Development working paper 7 (999). Wen, Mei, An Analytical Framework of Consumer Producers, Economies of Specialization and Transaction Costs, in Arrow, Ng, and Yang (eds), Increasing Returns and Economic Analysis, London: Macmillan (998a):70 85., The Dichotomy between Production and Consumption Decisions and Economic Efficiency, University of Melbourne Economics Department working paper (998b). Yang, Xiaokai and Siang Ng, Specialization and Division of Labor: A Survey, in Arrow, Ng, and Yang (eds), Increasing Returns and Economic Analysis, London: Macmillan (998):3 63. Yang, Xiaokai and Yimin Zhao, Endogenous Transaction Costs and Evolution of Division of Labor, Monash University Economics Department working paper (998). Zhou, Sun, and Yang, General Equilibrium in Large Economies with Endogenous Structure of Division of Labor, Monash University Economics Department working paper (998). Notes. Note that linear functions are included in the class of weakly convex functions.. Note that, unlike in case (i), here we have a strict inequality k > k. 3. For the general existence proof of the p* and the w* j, refer to Sun et al. (999).