<p>Assuming you are referring to the definition from the original <a href="https://web.stanford.edu/class/msande311/arrow-debreu.pdf" rel="nofollow noreferrer">Arrow &amp; Debreu</a> paper <span class="math-container">$x_i$</span> represents the units of good <span class="math-container">$x_i$</span> consumed.</p>

<p>I would not call <span class="math-container">$Y$</span> technology, according to Arrow &amp; Debreu <span class="math-container">$Y$</span> is set of production plans - so by that they mean production (which may make some assumptions on technological nature of the process).</p>

<p>Also, from your second paragraph I am a bit confused what do you mean by value. In mathematics value is some definite mathematical object like integer 5. In that sense yes, in the general model they are variables but once you would try to apply it to concrete problem <span class="math-container">$x$</span> and <span class="math-container">$y$</span> would have some values, if you are referring to some human values then no. In economics human values or preferences are taken as given and expressed through utility function, in Arrow-Debreu paper <span class="math-container">$u(x)$</span>.</p>

<p>A bit confused what you mean by value equilibrium on macroeconomic scale, assuming you mean general equilibrium, that depends on exact model you are following. Arrow-Debreu model is an example of basic general equilibrium model so you can use the variables you see there. If you would want to estimate it then currently one of the most popular ways is to estimate Dynamic Stochastic General Equilibrium (DSGE) models- these include a lot of variables. In such model you try to explicitly model demand, supply and monetary policy. The number of variables there can be so large that it’s not feasible for me to list it rather you may want to see an example of <a href="https://scholar.google.nl/scholar?hl=en&amp;as_sdt=0%2C5&amp;as_vis=1&amp;q=smets%20wouters%202003&amp;btnG=#d=gs_qabs&amp;u=%23p%3DY7hCUupcC4MJ" rel="nofollow noreferrer">Smets-Wouters</a> DSGE model for Euro-area economy.</p>

CC BY-SA 4.0
2019-11-20T10:17:35.980Z
2019-11-20T13:01:46.943Z
2019-11-20T13:01:46.943Z
1muflon1
<p>I'm very inexperienced in mathematical economics, so when I came across the idea of the consumption set in Arrow-Debreu, I was a little confused. So for each element x in consumption set X, what does x represent? This also applies to element y in technology set Y. Are x and y values? </p>

<p>And as a side note, how can one analyze something like a Value Equilibrium on a macroeconomic scale? What kind of variables and analysis goes into looking at equilibrium/efficiency of allocations?</p>

<mathematical-economics>
Consumption Set in Arrow-Debreu
kimsanity7
# Consumption Set in Arrow-Debreu

(Asked by: username_0 on 2019-11-20)

I'm very inexperienced in mathematical economics, so when I came across the idea of the consumption set in Arrow-Debreu, I was a little confused. So for each element x in consumption set X, what does x represent? This also applies to element y in technology set Y. Are x and y values?

And as a side note, how can one analyze something like a Value Equilibrium on a macroeconomic scale? What kind of variables and analysis goes into looking at equilibrium/efficiency of allocations?

**Tags:** mathematical-economics

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## 1 answers

### Answer by username_1 on 2019-11-20. Score: 0 | Accepted answer

Assuming you are referring to the definition from the original [Arrow & Debreu](https://web.stanford.edu/class/msande311/arrow-debreu.pdf) paper $x\_i$ represents the units of good $x\_i$ consumed.

I would not call $Y$ technology, according to Arrow & Debreu $Y$ is set of production plans - so by that they mean production (which may make some assumptions on technological nature of the process).

Also, from your second paragraph I am a bit confused what do you mean by value. In mathematics value is some definite mathematical object like integer 5. In that sense yes, in the general model they are variables but once you would try to apply it to concrete problem $x$ and $y$ would have some values, if you are referring to some human values then no. In economics human values or preferences are taken as given and expressed through utility function, in Arrow-Debreu paper $u(x)$.

A bit confused what you mean by value equilibrium on macroeconomic scale, assuming you mean general equilibrium, that depends on exact model you are following. Arrow-Debreu model is an example of basic general equilibrium model so you can use the variables you see there. If you would want to estimate it then currently one of the most popular ways is to estimate Dynamic Stochastic General Equilibrium (DSGE) models- these include a lot of variables. In such model you try to explicitly model demand, supply and monetary policy. The number of variables there can be so large that it’s not feasible for me to list it rather you may want to see an example of [Smets-Wouters](https://scholar.google.nl/scholar?hl=en&as_sdt=0%2C5&as_vis=1&q=smets%20wouters%202003&btnG=#d=gs_qabs&u=%23p%3DY7hCUupcC4MJ) DSGE model for Euro-area economy.


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