Exchange between two consumers

EXCHANGE

in this chapter we analyse a scenario where there is a market with two consumers, who exchange among themselves two commodities. In order to represent that situation graphically, we use a tool: the Edgeworth box. We are going to indicate the two consumers as A and B and the goods that are exchanged 1 and 2. The width of the box represents the total quantity of good 1 available for consumers on the market while the height of the box represents the quantity of good 2. The quantities of goods consumed by consumer A are measured from the lower left corner while the choices of consumer B are measured from the upper right corner. This means that the quantity consumer A holds of good 1 is measured as the vertical distance starting from the lower left corner while the quantity of good two possessed by the same consumer is measured as the horizontal distance from the same corner. If we look from consumer B perspective, we have to look at the graph upside-down: the quantity held by him of good 2 is the vertical distance from the upper right corner and his consumption of good 1 is measured as the horizontal distance from the same corner. Both the consumers could initially have some quantities of the goods available on the market. What the consumer has before trade takes place is called the initial endowment allocation. For consumer A it is defined as (ω1A , ω2A) while, for consumer B, it is defined as (ω1B , ω2B ). What the consumers end up consuming is called the final allocation of the goods. The quantities of a good that a consumer wants to have represents his gross demand and, in order to represent it for consumer A, we write: XA= (x1A, x2A). While his net demand (called also excess demand),which is the change in the quantities of the goods the consumer wants to obtain thanks to trade calculated as the difference of the consumer’s gross demand and what he already has in his endowment, is represented as: eA=(x1A- ω1A , x2A - ω2A). In that notation the superscript represents the good considered while the subscript represents the consumer we are referring to. This means that x1A represents consumer A’s consumption of good 1 and x2A represents A’s consumption of good 2. Similarly, consumer B’s consumption is denoted as XB = (x1B , x2B) and his net demand is indicated with: eB=(x1B- ω1B , x2B - ω2B). We can define an allocation as the consumption of both the consumers XA and XB.. When the consumers consume together all the goods available on the market we have a feasible allocation. Mathematically an allocation is feasible if:

x1A + x1B = ω1A + ω1B

x2A + x2B = ω2A + ω2B

This means that, if in the market there are 10 oranges and 14 apples and consumer A chooses to have 4 oranges and 10 apples (4,10), consumer B must have 6 oranges and 4 apples (6,4) in order to make the allocation feasible. Both the consumers could initially have some quantities of the goods available on the market. We can represent the preferences of consumer A in the way we are used to (did you miss our lecture on consumer preferences?). The preferences of consumer B are represented upside-down because we have to draw them keeping in mind that his point of view is opposite to the one of consumer A. This means that if we move in the box toward the upper right corner A will be better off while, if we do the opposite and therefore move nearer to A’s origin consumer B will consume more and therefore be better off.

WHAT HAPPENS WHEN THE CONSUMERS TRADE

As said, before trade happens, consumers possess some quantity of the goods: the endowment. Now, if we depict the indifference curves for both the consumers that pass through the endowment, we can notice that all the combinations of the goods, which are above or to the right of the endowment, make the consumer better off because they correspond to indifference curves higher than the one passing through the endowment. The same happens for consumer B: all the combination of goods that are lower and to the left (by looking at the graph from our point of view and not from consumer B’s) of the endowment will make him better off. The area which is between the indifference curves of both consumers

passing through the endowment of both consumers makes both A and B better off. Since every time that goods are distributed among consumers such that we draw indifference curves that intersect each other there is room for improvement for both the consumers, we need to find a point on the graph where the consumers reach the maximum utility possible and therefore have no incentive to continue trading. This condition occurs when the two indifference curves are tangent. The allocation that satisfies that condition is called a Pareto efficient allocation. If from that point consumer A wants to increase his utility he will harm consumer B because, in order to move his indifference curve up he would need to move the allocation closer to B’s origin and therefore decreasing his utility. Obviously the Pareto efficient allocation depends on the original endowment because each consumer will in the end possess goods that are valued the same as the endowment. This means that, if we change the original endowment into one that is worse for consumer A (so to an allocation closer to his origin), the Pareto efficient allocation will change. Therefore, in a market there are many possible Pareto efficient allocations depending on how resources are distributed at the beginning. The set of all the possible Pareto efficient allocations creates the Pareto set or contract curve which can be represented graphically as a curve going from A’s origin to B’s one. Notice that the origin themselves are Pareto efficient allocations: if we give everything to consumer A (meaning we are at B’s origin), he is on the highest indifference curve possible while B is on his lowest one. Because of that distribution of resources A can not gain anything by trading with B and B can not increase his own utility without decreasing the one of consumer A.

WHEN TRADE CAN HAPPEN

Until now we supposed in our analysis that consumers are willing to exchange the two goods they can trade in order to satisfy one the necessities of the other and vice versa. But that needs not to be true: each consumer in fact is not trying to help the other one but he is trying to maximize his own utility. Furthermore, it is not always the case that what a consumer wants to sell (or to buy) is bought (or sold) by the other consumer meaning that both consumers could desire to buy the same commodity (and therefore no one would sell it) or both could want to sell it (and therefore no one would buy it). The decision of a consumer to buy or sell a commodity and how much of it are dictated by the prices of the goods on the market. Imagine that we have someone, an auctioneer, who can change the prices of the goods in the market. If, for a certain set of prices, the consumers can not reach a situation where what a consumer wants the other wants to sell, trade does not happen and therefore the market is said to be in disequilibrium. The auctioneer, in order to let trade happen will have to change the prices of the two commodities. If together both the consumers demand too much of a certain good exceeding the quantity available on the market, the auctioneer will raise the price of that good so that consumers will buy less of it. If instead, together both the consumers demand a too small quantity of a certain good so that the quantity demanded of that good is less than the total amount of it available on the market, the auctioneer will decrease the price of that good so that both consumers have incentives to buy more of it. This process continues until a competitive equilibrium (called also a Walrasian equilibrium) is reached. The problem of maximizing each consumer’s utility is analogous to the one we considered when we were dealing only with only one consumer (did you miss our lecture on consumer choice?). Remember that in the end consumers chose a point where the budget line (imposed by the market) was tangent to the indifference curve. Mathematically the consumer’s Marginal rate of substitution (MRS) had to be equal to the relative price (p1/p2). The same principle is applied here but now we need to consider two consumers instead of one. In this case, when looking at consumer’s A perspective, the slope of the budget line is computed in the same way as -p1/p2. Because each consumer tries to maximize his own utility given prices, it means that both of them will choose a point where his own indifference curve is tangent to the budget line but, because the budget line is one for both the consumers and because, by being a straight line, has constant slope, the two indifference curves in the equilibrium must be tangent to each other.

WHAT HAPPENS IN EQUILIBRIUM

The demand for a good depends on the prices in the market for that good and, therefore, it is a function of them that we are going to indicate as for consumer A who is choosing how much to consume of good 1 as: x1A (p1, p2). The prices that lead to an equilibrium in the market are called equilibrium prices and are indicated with (p*1, p*2). As said earlier, in order to have a feasible allocation the quantity demanded by both consumer of a good must be equal to the quantity available in the market. This is depicted by the following relationship:

x1A (p*1, p*2) + x1B (p*1, p*2) = ω1A + ω1B

x2A (p*1, p*2) + x2B (p*1, p*2) = ω 2A + ω 2B

if we put the endowment on the left side of the equation we can rewrite the equation as:

[x1A (p*1, p*2) - ω 1A] + [x1B (p*1, p*2) - ω 1B] = 0

[x2A (p*1, p*2) – ω 2A] + [x2B (p*1, p*2) – ω 2B] = 0

Written like that, the relationship indicates that the sum of the excess demands of both the consumers must be 0 so that what a consumer wants to buy (which corresponds to a positive excess demand) is exactly equal to what the other consumer is willing to sell (which corresponds to a negative excess demand) so that their sum is equal to 0. By writing everything using the excess demand notation we have:

e1A(p1 , p2) + e1B(p1 , p2) = 0

e2A(p1 , p2) + e2A(p1 , p2) = 0

the sum of the excess demands for both consumers for each good can be indicated with the aggregate excess demand function that is indicated for good 1 as z1= e1A(p1 , p2) + e1B(p1 , p2). Therefore we can rewrite the previous relationship as:

z1(p*1 , p*2) = 0

z2(p*1 , p*2) = 0

if both the aggregate excess demands are 0, their sum will also be 0. This conclusion is not only true in the equilibrium condition. According to Walra’s law in fact for every combination of prices the sum of the aggregate excess demands must be 0. This relationship is expressed mathematically as:

p1z1(p1,p2) + p2z2(p1,p2)≡0