Barter Economy and the Core ∗†
Ram Singh
‡September 22, 2014
This Write-up is not for circulation
1 Basics
Consider a pure exchange economy. That is, there is no production - either because it has already taken place or the initial endowments themselves are consumable goods.
We want to find out the nature of equilibrium outcome under such an economy. In a 2×2 pure exchange economy the initial endowments can be written as e1 = (e11, e12) and e2 = (e21, e22)). Suppose (e11, e12) = (8,2) and (e21, e22) = (0,7). That is, to start with, the first individual has 8 units of first good and only 2 units of the second good. As to individual 2, she has 7 units of the second good only.
In principle, it is possible to distribute, i.e., allocate the two goods between these two individual differently from the initial allocation. Formally, an allocation is a distri- bution of available goods among the individuals consistent with the initial endowments.
For example, allocation x1 = (3,5) and x2 = (5,3) is one of the many possible allo- cations. This allocation gives to the first individual 3 units of first good and 5 units of the second good; it is the other way around for the second individual. Let us de- note an allocation by x = (x1,x2), where x1 = (x11, x12), and x2 = (x21, x22). We say that an allocation x = (x1,x2) is feasible if it distributes the available goods among the individuals in a way that is consistent with the initial endowments: Formally, if x11+x21 ≤e11+e21 and x12+x22 ≤e12+e22.
Note that out of 9 nine units of good two, the allocationx1 = (3,5), andx2 = (5,3)
‘wastes’ one unit of the good, as this unit is consumed by neither of the individuals.
As intuition would suggest, an allocation x= (x1,x2) is non-wasteful if x11+x21 = e11+e21
x12+x22 = e12+e22
∗To learn more about the issues covered here, you can read Debreu and Scarf (1963), Aumann (1964), Scarf (1967) and Feldman (1974).
†Please bring typos and errors, in any, to my attention.
‡Delhi School of Economics, University of Delhi. Email:
We can easily extend the above concepts toN×M economy, i.e., economy consisting of N individuals and M goods/commodities. Let an individual be indexed by i, i = 1,2, ..., N; and a commodity be indexed by j, j = 1,2, ..., M. Each individual i is endowed with a bundle of M commodities denoted by ei = (ei1, ei2, ..., eiM).
So, for theN×M economy, an endowment is a vector of vectors: e = (e1,e2, ...,eN), where e1 = (e11, e12, ..., e1M); ei = (ei1, ei2, ..., eiM), etc. Using these notations, the total endowment of goods can be re-written as:
good 1 : e11+e21+...+eN1 =
N
X
i=1
ei1
good j : e1j +e2j +...+eNj =
N
X
i=1
eij For theN ×M economy, an allocation is can be denoted by:
x= (x1, ...,xN),
where x1 = (x11, x12, ..., x1M), xi = (xi1, xi2, ..., xiM), etc. Moreover, we will call an alloca- tion, x = (x1, ...,xN), to be non-wasteful w.r.t. good 1 if PN
i=1xi1 = PN
i=1ei1. More generally,
Definition 1 Allocation x= (x1, ...,xN) is non-wasteful if
N
X
i=1
xij =
N
X
i=1
eij
for all j = 1, ..., M
We can meaningfully restrict our attention to non-wasteful allocations. Therefore, we define the set of feasible allocation as follows.
Definition 2 Set of feasible (non-wasteful) allocations is the set F(e) =
(
x= (x1, ...,xN)| xij ≥0 and
N
X
i=1
xij =
N
X
i=1
eij )
, (1)
for all i= 1, ..., N and j = 1, ..., M.
We will make the following assumptions about the choice sets and the individual preferences:
• The set of alternatives is the set of feasible (non-wasteful) allocations, i.e., the set defined in (1).
• Individuals have (self-interested) preferences defined over the set of allocations defined in (1).
• Each preference is/can be represented by a (self-interested) utility function. If a preference is complete, transitive and continuous, then it will be represented by a utility function.1
• Each preference relation and thereby each utility function is monotonic.
2 Outcome under Barter
What are the achievable outcomes under Barter? We will try to answer this question as- suming that individuals can freely decide whether to trade/exchange with one-another or not. Moreover, they have all the information relevant for the trade - who has what, who is willing to trade and who is not, etc.
The familiar Pareto optimal criterion will help us answer the above question. Going back to our 2×2 economy, an allocation (x1,x2) is Pareto superior to the endowment, (e1,e2), if ui(xi) ≥ui(ei) holds for i= 1,2, and ui(xi) > ui(ei) holds for at least one i= 1,2.
Remark 1 If (x1,x2) is Pareto superior to (e1,e2), then: (e1,e2) will be Pareto infe- rior to (x1,x2); and (e1,e2) CANNOT be Pareto Optimum. However, (x1,x2) may or may not be Pareto Optimum.
Definition 3 Allocation (x1,x2) is Pareto Optimum, if there is no (feasible) allocation (y1,y2) such that (y1,y2) is Pareto superior to (x1,x2).
Note, in general, there can be several Pareto Optimum allocations.
2.1 Barter in an Ideal World
Presumably, people would want trade with one another so to move from Pareto inferior allocations to Pareto optimum ones. Therefore, as a first cut, it seems plausible that the outcome under barter should be a subset of the Pareto optimal outcomes. Indeed this should happen when there are no restrictions on exchanges. To see this, let’s use the following example.
Example 1 Suppose the two-person two-goods economy consists of you (as first in- dividual) and me (the second person). Let endowments be: e1 = (1,9), and e2 = (9,1). Our preferences are ‘similar’ and are represented by utility functions That is, u1(x11.x12) = x11.x12 and u2(x21.x22) =x21.x22, respectively.
Consider the allocation: x1 = (3,3), and x2 = (7,7). Clearly, it is feasible. And, u1(x1) ≥ u1(e1), and u2(x2) ≥ u2(e2). In fact u2(x2) > u2(e2). That is, allocation
1Note: When the set of alternatives is finite then a preference relation can be represented by a utility function as long as it is complete and transitive.
x= (x1,x2) is Pareto superior to e= (e1,e2)
In the above example, allocation x= (x1,x2) = ((3,3),(7,7)) makes both of us at least as much as well off as is the case at e= (e1,e2), one of us is strictly better off at x. Therefore nobody would mind moving frome toxand one of us strictly prefers this shift. (The intuition suggests that we will not stay put at e). Formally, put we can say that the allocation e = (e1,e2) is blocked by the allocation x= (x1,x2). Clearly, x = ((3,3),(7,7)) is not the only allocation that blocks e = (e1,e2). Try to find out the other allocations that can block e= (e1,e2).
For a two-person two-goods economy: Allocation y= (y1,y2) will be blocked, if any of the following is true:
u1(e1) > u1(y1) or;
u2(e2) > u2(y2) or;
there exists a feasible allocation (x1,x2) such that
ui(xi)≥ui(yi) for every individual i=1,2. And;
uj(xj)> uj(yj), for aleast one j=1,2.
The last condition is a formal way of saying that (x1,x2) is Pareto superior to (y1,y2).
Here it will be useful to use an Edgeworth box (let’s borrow one from Jehle and Reny, page 183). In this diagram, your origin isO1, and my origin isO2. You can check that if allocationy= (y1,y2) lies below your (first person’s) indifference curve passing through e, you will findyunacceptable - you are better off holding on to whatever you have got. That is, the condition (1) stated above will be true. On the other hand, if allocation y = (y1,y2) lies above my indifference curve passing through e, I will find y unacceptable. Now, condition (2) will hold and again y will be rejected. Finally, suppose ylies within the area/lense formed by our ICs through e. If ydoes not lie on the segment c−c(suppose it is a point like B), in that case, two of us could gain by moving to a point like D. That is, in that case, condition (3) will hold. You can check that for any allocation x on the segment c−c, none of the above conditions can hold.
We will say that an allocation belongs to the Core, if it cannot be blocked in the sense described above. In the above example, recall the allocation (x1,x2) with x1 = (3,3) and x2 = (7,7) cannot be blocked. Therefore, it belongs to the Core. In other words, it is Core allocation.
You can verify that for a two-person two-goods economy, an allocationx= (x1,x2) belongs to the Core, if and only if the following two conditions are satisfied: First, each individual, i, i= 1,2, must (weakly) prefer xi at least as much asei. That is,
ui(x1)≥ui(e1) holds for each i= 1,2 (2)
Second, the allocation x = (x1,x2) must be Pareto Optimum. That is, there should NOT be any feasible allocation, z= (z1,z2), such that
ui(zi)≥ui(xi) for every i=1,2. And;
uj(zj)> uj(xj), for some at least one j=1,2 (3) The (2) and (3) are necessary and sufficient conditions for an allocation to be in the Core of 2×2 economy. So, we seem to have made some progress on finding the Core of a 2×2 economy. Really? For the above example, how many unblocked allocations are there?
Next question is: How can we find the Core of aN ×M economy? Let us expand our economy 2×2 economy to make itN×M. You can check that the above conditions (2) and (3) are relevant for N ×M economy as well. It is easy to see that the above two conditions remain necessary condition, for N ×M economy as well. That is, an allocation x= (x1,x2, ...,xN) will belong to the Core, only if
• Every i prefers xi at least as much as ei, i = 1,2, ..., N - otherwise i will reject the proposed allocation x, which is to say that individual i will blockx.2
• Allocation x= (x1,x2, ...,xN) is Pareto optimum - If x= (x1,x2, ...,xN) is not Pareto optimum, then there will be some allocation z = (z1,z2, ...,zN) that is Pareto superior to x. So, everybody would want to move to z. That is, all individuals together will block x.
However, forN×M economy the above conditions are not sufficient to give us the set of Core allocations. To see why, let us modify our earlier example as follows:
Example 2 The 2×2 economy of Example 1, is expanded to make it the following three-person, two-goods economy: Endowments are e1 = (1,9), e2 = (9,1), and e3 = (5,5), respectively. Preferences are represented by u1(x11.x12) = x11.x12, u2(x21.x22) = x21.x22, and u3(x31.x32) = x31.x32, respectively.
Now, consider the allocation x= (x1,x2,x3), where
x1 = (3,3), x2 = (7,7), andx3 = (5,5).
Let us ask the following questions about allocation x= (x1,x2,x3): Is it feasible?
Does every individual i prefer xi over ei? Is allocation x = (x1,x2,x3) Pareto Opti- mum? You should be able to show that answers to these questions are: ‘Yes’, ‘Yes’, and ‘Yes’. Therefore, allocation x = (x1,x2,x3) satisfies the two conditions stated above. In fact, you can see that x= (x1,x2,x3) is Pareto-superior to the endowment vector, e= (e1,e2,e3).
2This can be called ‘individual rationality’ constraint.
Still, allocationx= (x1,x2,x3) does NOT belong to the Core. To see why, consider a Coalition of 1 and 3, i.e., consider the set S = {1,3}. Recall, e1 = (e11, e12) = (1,9) and e3 = (e31, e32) = (5,5). That is, individuals 1 and 3 together have 6 units of the first good and 14 units of the second good. Therefore, by working together and using whatever is available between two of them, they can afford the following allocation:
y1 = (2,5) = (y11, y21) for individual 1 andy3 = (4,9) = (y31, y32) for person 3. Note that for the setS ={1,3}, the following is true about allocationsxandy: y11+y31 =e11+e31 and y21+y32 =e12+e32. That is,
X
i∈S
yji =X
i∈S
eij for all j = 1,2
Moreover, u1(y11, y21) > u1(x11, x12), i.e, u1(y1) > u1(x1) and u3(y31, y23) > u3(x31, x12), i.e., u3(y3)> u3(x3). Using mathematical notations, we can say that in this example the following claims hold:
ui(yi) = ui(y1i, y2i) ≥ ui(xi1, xi2) =ui(xi) for all i∈S ={1,3}
uj(yj) =uj(y1j, yj2) > ui(xj1, xj2) =uj(xj) for at least some j ∈S ={1,3}.
Now suppose someone asks individuals 1, 2 and 3 to give up their endowments and accept the above bundles (x1, x2 and x3), respectively. Individuals 1 and 3, i.e., those in the set S ={1,3}would not agree - they can serve their interests better by forming a ‘coalition’, pooling their endowment so as to consume (y1 and y3), respectively. In such a scenario, we will say that the set S = {1,3} is a blocking coalition against allocation x= (x1,x2,x3).
Therefore, even if no individual can deviation/objection against a Pareto Optimum allocation, there can be subgroups who can profitably deviate from it. Therefore, a Pareto Optimum allocation. may not belong to the Core - it will not belong to the Core if there exists a blocking coalition against it.
Now, we are ready to provide a general definition of a Blocking Coalition and the Core.
Definition 4 Blocking Coalition: A set S ⊆ {1, ..., N} is called a blocking coalitions for x= (x1,x2, ...,xN) if there is some vector ysuch that
X
i∈S
yji = X
i∈S
eij for all j = 1, ..., M
ui(yi) =ui(yi1, ..., yMi ) ≥ ui(xi1, ..., xiM) = ui(xi) for all i∈S
uj(yj) = uj(yj1, ..., yMj ) > uj(xj1, ..., xjM) =uj(xj) for at least some j ∈S Definition 5 Core: Consider a pure exchange economy (ui(.),ei)i∈N. The Core, C(ui(.)i∈N,e), is a subset of allocations that cannot be blocked. That is,x∈C(ui(.)i∈N,e) if and only if there is no blocking coalition against x.
3 The Core and its Size
How can we go about finding the Core of an economy? What can we say about its size? Let us address these issues.
In view of the above, we know that to find the Core we just need to consider the set of Pareto optimal allocations. Further, we can safely restrict attention to only those Pareto optimal allocations that satisfy conditions described in (2) and (3) above. How- ever, the set of Pareto optimal allocations satisfying these conditions can be large - in principle, there can be infinitely many allocations satisfying these conditions. There- fore, satisfying these conditions, we need to check possibility of a blocking coalition for each allocation. Discuss on Example 3 suggests that the set of possible blocking coalition increases with the number of individuals in the economy. You can check that if there are just 25 individuals, there can be as many as 225 −1 coalitions that can potentially block an allocation.
Moreover, the above discuss suggests that the size of the Core changes with the number of individuals in the economy. Recall, the allocation that gives (3,3) to in- dividual 1 and (7,7) to individual 2 belongs to the Core when N = 2, but does not belong to Core when N = 3. This inference is further corroborated by the following example, in which we replicate the economy described in Example 2.
Example 3 There are four individuals and two goods. Utility functions are: ui(xi1.xi2) = xi1.xi2, for i = 1, ...,4. Endowments are: e1 = (1,9), e2 = (9,1), e3 = (1,9), and e4 = (9,1), respectively.
That is, we have introduced two twins - individual 3 is twin of 1, and 4 is identical to 2! Now, consider the allocation x= (x1,x2,x3,x4), where
x1 = (3,3) =x3 and x2 = (7,7) =x4.
Note that the allocation x = (x1,x2,x3,x4) is Pareto optimum. Moreover, we have already seen that allocation withx1 = (3,3) and x2 = (7,7) belongs to the Core when economy consists of individuals and 1 and 2 only. In view this, we may be tempted to think that the allocation x = (x1,x2,x3,x4) belong to the Core of 4×2 economy of Example 4. But, this is not true.
Let S = {1,2,3}, y1 = (3,4) = y3 and y2 = (5,11). I leave it to you to ver- ify that individuals in set S can re-arrange their allocations to arrive at allocation y= (y1,y2,y3). Moreover, set S satisfies all the conditions of a blocking coalition. So, set S can use allocation y to block allocationx. The result? xis no more part of the Core.
As the above examples suggest, the size of the Core shrinks as N increases. Edge- worth (1881) himself had predicted this. However, the formal proofs were provided by was subsequently works of Debreu and Scarf (1963); Aumann (1964), among others.
To sum up, a particular allocation or a particular distribution of resources, say x = (x1,x2, ...,xN) belongs to Core if it has the following property: No group or subgroup of individuals can do better by deviating from or refusing to accept the al- location x = (x1,x2, ...,xN). In other words, if individuals find themselves at the allocation x = (x1,x2, ...,xN), there will be no deviations from it. In that sense, the Core is set of possible equilibrium allocations.
We have seen that the location and the size of the Core depends on the nature of individual preferences, the initial distribution of endowment/wealth, the number of individuals in the economy. The above discussion also suggests that description of the Core is complex. Since, it is hard to fully describe the individual elements of the Core of Barter economy - beyond saying that they satisfy the conditions mentioned earlier.
Some of the complexity of the Core is due to the fact that in the above discussion utilities are ‘non-transferable’ across individuals. That is, once the endowments are used (consumed) and converted into utility, it is not possible to transfer utility across individuals. In the following, we will show that for the contexts involving ‘transferable’
utilities the Core may be described more neatly.
3.1 The Core when Utilities are Transferable
Example 4 Production: Consider the following three firms/farmers producing one good: The endowments of FOP are: e1 = (e11, e12) = (1,9), e2 = (e21, e22) = (9,1), and e3 = (e13, e32) = (5,5) Production Functions are: q1(x11.x12) = x11.x12; q2(x21.x22) = x21.x22; and q3(x31.x32) = x31.x32. Assume that the factors of production (FOPs) can be shifted/employed across farms/plants as desired. Suppose, the farmers’ (owners’) consume the output produced, and utility ofqunits of output iskq. That is, the utility functions are: ui(qi) = qi, where i= 1,2,3.
Note that in the above example, after endowments (FOPs) are converted into out- put level(s), it is still possible to transfer the resulting output and therefore utilities across individuals. It is in this sense, the utilities are transferable. Let,
¯
eSj denote the total endowment of FOP, j, jointly owned by a group or Coalition, S. Therefore, we can write
¯
eS1 =X
i∈S
ei1 and e¯S2 =X
i∈S
ei2
Note that the maximum possible production level for Coalition, S, is QS = ¯eS1.¯eS2. (Find out this quantity for each subgroup of farmers.) The maximum production oc- curs when the three farmers/firms go for joint production by pooling their endowments.
The maximum output level is 225 units. Since output levels can be transferred across individuals, you can plausibly assume that each group or coalition, S, will produce
QS = ¯eS1.¯eS2 units. The maximum production level (225 units) is when they work to- gether.
In this context, the Core is the set of those allocations (divisions of 225 units) that cannot be blocked/objected to by an individual farmer. You can see that allocation (0,0,225) does not belong to the Core - since individual 1 and 2 both can enjoy higher utility by going alone, as well as by working together. Does allocation (45,45,135) belong to the Core? You can check that no farmer can do better by individually refusing to accept this - the maximum farmers 1 and 2 can do is produce 9 units each, farmer three can produce maximum 25 units. However, farmers 1 and 2 together can produce 100 units and if they go for equal division they can get 50 units each.
Therefore, S = {1,2} is a blocking coalition against allocation (45,45,135), which in turn means that this allocation cannot belong to the Core.
You can show that an allocation (y1, y2, y3) will be a Core allocation if and only if the following conditions are met with: y1 ≥9; y2 ≥9; y3 ≥25; y1+y2 ≥100; y1+y3 ≥ 84; y2+y3 ≥84. Formally, the set of Core allocations can be defined as:
C(e, qi) = {(y1, y2, y3)| yi ≥0 & X
i∈S
yi ≥QS = ¯eS1.¯eS2}, (4) for each S⊆ {1,2,3} (note, we are talking only about the non-empty subsets).
In this context, you can immediately tell whether an allocation is Core allocation or not. Is (55,55,115) a Core allocation?
3.2 Does Core always exist?
(This Subsection is strictly optional)
The above discussion suggests that the Core has many allocations - as a set it can be too big. However, depending on the rules of the game, in some context Core may not exits at all, i.e., the Core could be an empty set. As an illustration consider the following example.
Example 5 There are three farmers - 1, 2 and 3 - who own small but contiguous parcels of land. Individually they can use their parcels to produce a crop, say paddy.
Let this outcome be denoted by (t1, t2, t3) - ti denotes the output level produced by farmeriusing his land. Alternatively, they could plant any one of the three commercial crops; x, y, z. However, this will require them to pool their land. Further only one of the crops can be planted. If crop x is produced it will lead to the outcome allocation x = (x1, x2, x3); similarly for the other two crops. Therefore, pooling allocations are:
x = (x1, x2, x3), y = (y1, y2, y3) and z = (z1, z2, z3); Suppose, the production decision is to be taken unanimously. In case of any disagreement, farmers will use their land to produce t. The individual preference relations are as follows:
Table 1: (Preferences)
1 2 3
x1 y2 z3 y1 z2 y3 z1 x2 x3 t1 t2 t3
In this context, there are only four feasible allocations. Which of the above allo- cations are Pareto optimum? Which allocations are unblockable allocations? That is, which allocations belong to the Core?
Let’s change the rules of the game. Suppose, the production decision is to be taken unanimously. However, in case of partial disagreement the majority will prevail - in case of total disagreement farmers will use their land to produce t. Now, which of the above allocations are Pareto optimum? What is the the Core?
4 Barter Vs Market
4.1 Informational and logistical requirements
• Barter requires
– Search costs - you need to identify suitable trading partners.
– Successful negotiations - after identification of suitable trading partners, you need to negotiate a mutually beneficial and acceptable deal. In Example 1:
What is the number of mutually beneficial deals? What is the number of deals that will be acceptable to both parties?
• Market requires
– No search costs - buyers and sellers can go to market to carry out transac- tions.
– No cooperation or negotiations required - only decision making at individual level, given the prices of goods.
4.2 Relative Efficiency
• Under Barter, Pareto efficient outcome is unlikely, for large set of individuals. In that case, it will be very difficult to find suitable partners for mutually beneficial exchange of endowments. Even after partners have been identified, some of the negotiations are likely to fail. In real world, several factors can frustrate successful
bargaining among individuals. Therefore, bargaining among individuals will fail to lead to one of the Pareto efficient and therefore Core allocations.
• Under Market, Pareto efficient outcome more likely, especially for large set of individuals. Under market, there is no need to search and negotiate with other individuals. Also, when there is large number of buyers and sellers, they will be price-takers. So, individual decisions will be Pareto efficient.
Later in the course we will show that under ideal conditions the market can match Barter and do even better. These claims are valid with or without production. We will discuss the conditions under which these claims will be true.
4.3 Effect of Policy Interventions
Under Barter:
• Policy intervention only through reallocation of endowments Under Market:
• Policy intervention through reallocation of endowments as well as direct transfers of ’purchasing power’
Remark: Neither Barter nor Market can guarantee the intended outcome. Further, some endowments are not transferable - E.g. human capital.
Also,
• The superiority of markets over Barter holds only for private goods.
• In case of Common Property Resources, the opposite can hold.
