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I wish to understand the following paragraph (from section $2.7$ of Debreu's Theory of Value):

Imagine that a certain good circulates as money at location $s$, at date $t$, and let $k$ be the index of the commodity thus defined. To obtain the price at $s$, at $t$ of the $h$th commodity $p^{s,t}_h$ i.e., the number of units of that money which must be paid at $s$, at $t$ in order to have one unit of the $h$th commodity available, one would divide $p_h$ by $p_k$.

The paragraph seems to be pointing to the equation $$p_h^{s,t}=\frac{p_h}{p_k}$$

The problem is that up until this point Debreu has defined a commodity in terms not only of the product itself (say, an apple), but also on the time and place at which it is available (so that an "apple in New York in September" and "an apple in Chicago in June" are different commodities, and thus have different indexes). Thus the price $p_h$ associated to the $h$th commodity already corresponds to a location and date.


My concrete doubts are:

  1. What is the meaning then of $p_h^{s,t}$?

  2. Why is $p_h^{s,t}=p_h/p_k$?

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1 Answer 1

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To address both your doubts:

  1. $p_h^{s,t}$ is read as the price of good $h$ in state of the world $s$ (which Debreu defines as location) at time $t$.

  2. the paragraph notes that every price is normalized by the index commodity price $p_k$ (which is just a price index). We can think of this as a way to remove the average from each price thus when we define $p_h^{s,t}=\frac{p_h}{p_k}$ we obtain the relative price based on state of the world $s$ at time $t$.

Reiterating on 2. $p_k$ takes on the interpretation of "average price" here and this normalization allows for expression of state and time varying components.

I hope this helps.

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  • $\begingroup$ Sorry, I still do not understand. Perhaps an example may help. Say that $p_h$ is the price of apples in New York in September. What could $p_h^{s,t}$ be? $\endgroup$
    – Sam
    Mar 16, 2022 at 12:33
  • $\begingroup$ @Leo using your example: $p_h$ is the price of apples. when we write $p_h^{s,t}$ we say this is the price of apples in new york in september $\endgroup$
    – EconJohn
    Mar 16, 2022 at 15:44
  • $\begingroup$ Would this not contradict what was said previously in the chapter? "Thus a good at a certain date and the same good at a later date are different economic obiects [...] a good at a certain location and the same good at another location are different economic objects [...] In the case now discussed a commodity is therefore defined by a specification of all its physical characteristics, of its availability date, and of its availability location. As soon as one of these three factors changes, a ditferent commodity results." $\endgroup$
    – Sam
    Mar 16, 2022 at 15:52
  • $\begingroup$ Debreu then follows to explain that $p_h$ refers to the price of a certain commodity, yet as previously explained, a commodity must specify a location and time, thus I do not see how $p_h$ may refer simply to the price of an apple (as opposed to something as the price of an apple in New York in September) $\endgroup$
    – Sam
    Mar 16, 2022 at 15:55

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