The following theorem from Mas-Collel, Whinston and Green's Microeconomic Theory (3rd Edition) assumes $\frac {\partial {w}}{\partial {p}}$ to be zero. I would to like to know the reason behind such assumption.


$p$ = price

$w$ = wealth

$x$ = Walrasian demand function

$D$ = derivative matrix

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    $\begingroup$ Hi and welcome to Economics SE. Because images are non-searchable to other users we ask that you type in all relevant information from the image. Until such an edit is made I am voting to close this question. $\endgroup$ – Giskard Oct 30 '15 at 15:39
  • $\begingroup$ So, why might you think that price would have an effect on your current wealth? Is that intuitive? A price change doesn't take away your current wealth, though it will might change your optimal bundle. $\endgroup$ – Kitsune Cavalry Oct 30 '15 at 17:56
  • $\begingroup$ @denesp, I appreciate your feedback and willingness to keep SE neat, but I ask you to reconsider your views. All relevant words from the image have been typed in. These are: Walrasian demand, derivative matrix, Mas-Collel and Whiston. Little outside this set of words would be ever used in a search. The use of images is important to estabilish the context behind the author's words, as some terms differ in meaning across textbooks. I post book prints in almost every question in Math SE and I've never gotten flagged for that. Is there a rule compendium somewhere in this site? $\endgroup$ – Incognito Oct 30 '15 at 20:59
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    $\begingroup$ @BrunoSchiavo I think you make some good points. I retract my vote but I also ask you to post about this suggestion on meta.economics.stackexchange.com, that is where the guideline on closing image questions originated. meta.economics.stackexchange.com/questions/1365/… $\endgroup$ – Giskard Oct 30 '15 at 23:08
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    $\begingroup$ @BrunoSchiavo Yes, as I said price changes your optimal bundle, but not by changing your wealth. So why should $\frac{\partial w}{\partial p}$ not be zero? $\endgroup$ – Kitsune Cavalry Oct 31 '15 at 17:02

The answer is that $w$, like $p$, is treated as a parameter in this model. It can only be changed by exogenous external forces that are not part of the mechanism included in the model. In contrast, the demand $a$ is a variable that is entirely determined (endogenously) within the model given the prevailing values of the parameters.

Suppose that $p\cdot a=w$. You are correct to think that if we increase the price, $p$, then we break the budget constraint ($p\cdot a> w$). Within the scope of this particular model, only $a$ is variable and therefore the budget constraint must be restored to equality by a decrease in (some elements of) $a$.

Now, the question why is it so is somewhat philosophical because one could, indeed, build an alternative model that includes some mechanism for determining $w$. For example, we could allow consumers not only to buy goods, but also to supply labour and to negotiate wage rates with their employer. Such models exist within the domain of general equilibrium (a basic treatment of which can also be found in your copy of Mas Collel et al.).

However, there are at least a few reasons to think that the model in question is interesting and relevant despite providing no explanation for the determination of $w$:

  • Incomes might change in response to the overall price level (i.e. in an inflationary environment salaries are likely to increase). But they are much less likely to change in response to isolated or temporary changes in relative price. For example, the price of fruits changes on an regular cycle depending on when and where they are in season. It would be odd to think that people's income changes in response to this cyclic behaviour. But it would also be odd to think that people's consumption plans did not respond to these price changes. this model allows us to study how people vary demand in the face of such isolated price fluctuations.
  • At least in the short-run, most consumers are much more able to change their consumption decisions (i.e. to vary $a$) than to change their income. Affecting the latter will often require negotiations, change of job, or even retraining—all of which take time. Thus, the model is likely to be informative about the immediate and short-run effects of a price change. In the data, consumption usually reacts relatively quickly to the economic cycle, whereas wages tend to be "sticky".
  • In modelling we face a constant trade-off between building a model that is realistic and building one that is tractable/parsimonious. The simple consumer demand model has the virtue of yielding a striking number of sensible predictions from a model that is comparatively straightforward. This has various payoffs (easier to communicate to policy makers, easier to use for empirical work, easier to teach to students, easier to use as the foundation for more applied modelling, etc.)
  • There are many mechanisms by which $w$ could be determined. For example, it could be union-firm bargaining. Or perhaps firms have all the bargaining power and workers are paid only their marginal product. Or perhaps the consumers are entrepreneurs whose income comes from profits/investments rather than labour. Each alternative would require a different model, whereas this model is agnostic on these points and allows us to focus on another dimension of the problem.
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  • $\begingroup$ Is this somehow related to the concept of Coeteris Paribus? $\endgroup$ – Incognito Oct 31 '15 at 12:14
  • $\begingroup$ Is the assumption of exogeneity of $p$ and $w$ disclosed somewhere in the excerpt? How did you know $p$ and $w$ are both exogenous? $\endgroup$ – Incognito Oct 31 '15 at 12:17
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    $\begingroup$ @BrunoSchiavo The exogeneity of $p$ and $w$ are introduced in Section 2.D. The authors are quite explicit that "...prices are beyond the influence of the consumer." The assumption that $w$ is exogenous is not made so explicitly, but is implied by the definition of a budget set in Definition 2.D.1 and the subsequent statement that the consumer's problem is to choose an $x$ taking as given the constraint that $x$ has to lie within the budget set (this assumption is made a bit more explicit in Section 3.D where the consumer's maximisation problem is explicitly set-up). $\endgroup$ – Ubiquitous Oct 31 '15 at 13:20

The demand function $x(p,w)$ is defined for all parameters $p,w$. For example by nonnegativity of demand and the budget constraint $x(p,0) = 0$. The price and wealth parameters can be changed independently of each other as well as together. The answer to your question ends here.

You seem to be describing notion of Slutsky compensation where the consumer receives monetary compensation after the price change so that she can purchase the bundle she would have purchased without the price change. The existence of such a compensation however is not a necessity. If prices rose my salary would still not change (at least for a while), and I would still have a demand function with the new prices and my original salary as inputs.

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