# Intro to Econ: Fourth Lecture – Market Allocations and Market Values (or Prices)

In the last class (summary) we have discussed trade and that, under certain conditions, trade leads to Pareto improvements (which means that at least one person is better off and no one is worse off). I now want to discuss what economists call a market, market prices or, better, market values, and a market allocation. The difference between the idea of a market and bilateral trade is that bilateral trade is, well, bilateral (i.e. always between two people), whereas a market is, at least in some form, a central meeting place in which all participants interact at the same time in this one place by making offers and counteroffers to possibly many other participants. We have two options of how to deal with such a market. One is to try to capture the dynamic protocol of interaction that underlies the market place. This is difficult and probably depends on the exact market we are interested in. So we will not do this here. I also do not know of any very convincing general model of this kind, but there are some for special cases. The other option is to state what we think will be the likely outcome of any such market interaction. Note that what we write down next is an assumption or definition and not derived from any more basic set of assumptions.

In my discussion of this topic, I again follow fairly closely chapter 3 of Ariel Rubinstein’s “Economic Fables”. The key new concept is that of a market “value” of objects. Note that the “value” for a thing in our world (the world of human beings) always derives from people wanting to have it. A souvenir in the shape of one of the famous Egyptian pyramids may have value in the world of humans, but would probably have little, if any, value in the world of rabbits. Also it is hard to define a single objective value of a thing. People tend to have different subjective values for things. Consider a veal cutlet. To some people, those who can be seen to occasionally eat one, a veal cutlet seems to have a fairly high value. To a vegetarian, however, a veal cutlet has little value (if any). If a vegetarian doesn’t like someone else eating a veal cutlet, then we have yet another problem, one of externalities (which I will address in a later lecture).

It turns out that in our conception of a market one can nevertheless meaningfully identify a single value for each object. I would not call it an objective value and it is also not really the value that the object adds to society. It is, however, the value as it will enter the gross domestic product (GDP) accounting calculation.

To explain all this I come back to the example of the three kids and their three presents:

$\begin{tabular}{c|ccc} & Eva & Franz & Maria \\ \hline present: & pirate & nurse & ghost \\ \hline 1 & nurse & pirate & pirate \\ 2 & ghost & ghost & nurse \\ 3 & pirate & nurse & ghost \\ \end{tabular}$

The present is the initial allocation given to the three kids and below each child in the table is that child’s preference ranking over the three possible presents.

A market outcome is then defined as follows: There is a market value (or price) for each possible figure. Each child sells the figure initially allocated to him or her for the price of this figure and buys her favorite figure among those that he or she can afford given all the prices such that at the end each child has exactly one figure. The final allocation is then called the market allocation at the given market values (or prices).

In the example, is the outcome of Eva and Franz trading their figures a market allocation? If so, what are the market values underlying this market outcome? You may want to think about it for a while.

This is how it works. For this trade to result in a market allocation (according to the above definition) the market values or prices for the three figures have to satisfy certain conditions. Let us start with Eva. She initially owned the pirate and now has the nurse. Prices must be such that she can afford the nurse by selling the pirate. So we have $p_N \le p_P$, if we let $p_N$ denote the price of the nurse and $p_P$ the price of the pirate. As Eva is not interested in the ghost (given she gets the nurse) her choices reveal nothing about the price of the ghost, which we denote by $p_G$. Let us now look at Franz’s choices. He initially owned the nurse and now has the pirate. Analogously to what we inferred from Eva’s choices we now get that $p_N \ge p_P$ and we also do not learn anything about the price of the ghost from Franz’s choices as he does not value the ghost above the pirate. Let us now look at Maria. She still has the ghost. How can we “explain” this with prices? Well, it must be that she can afford neither the nurse nor the pirate. Altogether, we must have that $p_N = p_P > p_G$. This works! We have found market values (or prices) as well as a market allocation starting from the initial allocation. In this situation, the market value of the nurse and the pirate is equal and higher than that of the ghost (because no one really likes the ghost all that much). The fact that Maria likes the pirate more than she does the nurse does not enter the market values. A single set of market values allows us to rationalize the eventual allocation as a market allocation.

Can there be another market allocation starting from this initial allocation? The answer is no. This is, in fact, a very general finding but I will not go into the exact boundaries of when this result is true and when not. Let us see how this works in the present context. We have found two other Pareto efficient allocations that derived from a series of bilateral trades. Why are they not market outcomes according to the above definition? Consider the case that first Franz and Maria trade and then Eva and Franz. This leads to a final allocation of Eva having the ghost, Franz the pirate, and Maria the nurse. What conditions would the supposed market values or prices of the three figures have to satisfy for this final outcome to be called a market outcome according to the above definition? Let us start with Eva. She must be able to afford the ghost with her pirate and yet not be able to afford the nurse. This means that $p_N > p_P \ge p_G$. Let us now look at Maria. She must be able to afford the nurse with her ghost. We must have that $p_G \ge p_N$. Putting this together we get that $p_G \ge p_N > p_P \ge p_G$. But this is impossible! So, no single set of market values can explain this final allocation starting from the initial allocation. What is the problem here? Well, Eva could interject when she sees Franz and Maria trading. When Franz is about to offer Maria his nurse for her ghost, Eva could wave her pirate in front of Franz’s face and state that she would also be willing to accept the nurse in exchange. It seems possible that Franz then would rather trade with Eva as she has the more valuable figure for him (and he still has to give up the same figure in both cases). Maria could do little to prevent that. Whether this necessarily has to happen when three children trade their toys, I do not know, but this in any case is the going definition of a market outcome and market prices. It strikes me as not entirely silly.

To finish this I want to come back to the free trade zone discussion. Suppose we only have Franz and Maria in the free trade zone. They would trade nurse and ghost and this would also be the market allocation with market prices such that the price for the nurse and the ghost are equal: $p_G = p_N$. The two figures are equally valuable in this market. Now if Eva were to enter this free trade zone before Franz and Maria trade, the market allocation would be the one we discussed above in which Eva and Franz trade nurse and pirate and Maria is stuck with her ghost. In this case $p_P = p_N > p_G$. In this market, the ghost is no longer as valuable as the nurse and, thus, not as valuable as it was before. The reason for this is that Franz now found, in Eva’s pirate, a better and for him affordable substitute for the ghost.

The video (in German) is here: