<p>To address both your doubts:</p>
<ol>
<li><span class="math-container">$p_h^{s,t}$</span> is read as the price of good <span class="math-container">$h$</span> in state of the world <span class="math-container">$s$</span> (which
Debreu defines as location) at time <span class="math-container">$t$</span>.
<br><br></li>
<li>the paragraph notes that every price is normalized by the index commodity price <span class="math-container">$p_k$</span> (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 <span class="math-container">$p_h^{s,t}=\frac{p_h}{p_k}$</span> we obtain the relative price based on state of the world <span class="math-container">$s$</span> at time <span class="math-container">$t$</span>.</li>
</ol>
<p>Reiterating on 2. <span class="math-container">$p_k$</span> takes on the interpretation of &quot;average price&quot; here and this normalization allows for expression of state and time varying components.</p>
<p>I hope this helps.</p>

CC BY-SA 4.0
2022-03-16T12:33:53.937Z
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?
Sam
CC BY-SA 4.0
2022-03-16T15:44:02.227Z
@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_
EconJohn
CC BY-SA 4.0
2022-03-16T15:52:46.247Z
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._"
Sam
CC BY-SA 4.0
2022-03-16T15:55:12.200Z
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_)
Sam
CC BY-SA 4.0
2022-03-16T05:11:54.633Z
2022-03-16T05:11:54.633Z
EconJohn
<p>I wish to understand the following paragraph (from section <span class="math-container">$2.7$</span> of <a href="https://cowles.yale.edu/sites/default/files/files/pub/mon/m17-all.pdf" rel="nofollow noreferrer">Debreu's <em>Theory of Value</em></a>):</p>
<blockquote>
<p>Imagine that a certain good circulates as money at location <span class="math-container">$s$</span>, at date <span class="math-container">$t$</span>,
and let <span class="math-container">$k$</span> be the index of the commodity thus defined. To obtain the
price at <span class="math-container">$s$</span>, at <span class="math-container">$t$</span> of the <span class="math-container">$h$</span>th commodity <span class="math-container">$p^{s,t}_h$</span> i.e., the number of units of that
money which must be paid at <span class="math-container">$s$</span>, at <span class="math-container">$t$</span> in order to have one unit of the <span class="math-container">$h$</span>th
commodity available, one would divide <span class="math-container">$p_h$</span> by <span class="math-container">$p_k$</span>.</p>
</blockquote>
<p>The paragraph seems to be pointing to the equation <span class="math-container">$$p_h^{s,t}=\frac{p_h}{p_k}$$</span></p>
<p>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 &quot;apple in New York in September&quot; and &quot;an apple in Chicago in June&quot; are different commodities, and thus have different indexes). Thus the price <span class="math-container">$p_h$</span> associated to the <span class="math-container">$h$</span>th commodity already corresponds to a location and date.</p>
<hr />
<p>My concrete doubts are:</p>
<ol>
<li><p>What is the meaning then of <span class="math-container">$p_h^{s,t}$</span>?</p>
</li>
<li><p>Why is <span class="math-container">$p_h^{s,t}=p_h/p_k$</span>?</p>
</li>
</ol>

<mathematical-economics><general-equilibrium><price><commodities>
Understanding a paragraph in Debreu's Theory of Value
Sam
# Understanding a paragraph in Debreu's Theory of Value

(Asked by: username_0 on 2022-03-15)

I wish to understand the following paragraph (from section $2.7$ of [Debreu's *Theory of Value*](https://cowles.yale.edu/sites/default/files/files/pub/mon/m17-all.pdf)):

> 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 $p\_h^{s,t}=p\_h/p\_k$?

**Tags:** mathematical-economics, general-equilibrium, price, commodities


---

## 1 answers

### Answer by username_1 on 2022-03-16. Score: 1

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: @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_  
> **username_0**: 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._"  
> **username_0**: 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_)  

---

