Wednesday, October 5, 2016

Romer’s Identification Problem in a Nutshell.


Unidentfied Economic Object. [A]
In his recent paper, Paul Romer mentioned several problems in macroeconomics. Some are clear enough and anyone can understand them: unmeasurability and unobservability of variables, for instance. I've written about that.

Others are more technical; the "identification problem", among them.

So, what is the "identification problem"?

David Ruccio explains using a simple supply and demand model. It's very didactic, so readers can follow easily (hell, even I did!), to get a general idea. It has a drawback, though: Prof. Ruccio explains the identification problem in a supply/demand setting, not in the specific situation Prof. Romer wrote about.

Based on Prof. Ruccio's explanation, I'll provide more concrete details using a "toy" model.

Second caveat: my intention is to give a fair, good faith exposition, to the extent of my capability (if I miss the point it's my own responsibility). Given the justified resentment against mainstream economics, this approach may test the patience of some readers (particularly those used to crackpot, ill-tempered stuff). I can only beg them to bear with me.

With those caveats in place, read first Prof. Ruccio's post (never fear: it's short) and come back when you are ready.

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Figure 1 illustrates the kind of data (hypothetical here) economists and statisticians really observe and measure: prices and quantities of, say, ACME Blue Widgets v2.0 traded over time. (Examples readers might have seen are stock market data: share price/volume).

Note: how about the dates? Answer..

The scatter plot in Figure 2 re-arranges our "data" (the "x"s), following the standard convention: quantity in the horizontal axis, price in the vertical. Note that the data lost their time dimension (not that in this particular case it added much structure).


A data point is singled out (the "+") to present the usual explanation:  the intersection of supply (Su(q)) and demand (De(q)) schedules.

Given a set of assumptions (see Appendix), one can "see" both schedules in Fig. 2. Their equations:

(1)           De(q) = AD + BD*q

And

(2)           Su(q) = AS + BS*q

It may be useful to think of this in terms of two black boxes (mechanisms transforming inputs into outputs). The input -- common to the two boxes -- is q: a count of Blue Widgets. The output of each box is a price: De(q) and Su(q), measured in units/$. It works like this: one inputs a quantity q, and each black box outputs a price (De(q) and Su(q)). In fact, one could find the equilibrium points (like "+" in Fig. 2) by trying different inputs and stopping when supply and demand matched.

Easy, yes?

Let's peek inside each black box, however. Whatever changes inputs and outputs go through, the black boxes themselves must remain unchanged. The exercise would be pointless otherwise, yes? The formulae on the RHS of equations (1) and (2) -- AD+BD*q and AS+BS*q -- are the fixed mechanisms inside the black boxes. The values AD, BD, AS, and BS are called "parameters" and are constant for each black box (again, see Appendix).

Given this new information (i.e. equations (1) and (2)), one doesn't need to use a trial-and-error procedure to find "+"; its coordinates can be found with a little algebra:

                        AD - AS
(3)                q' = -------
                        BS - BD

And:

                      AD*BS - BD*AS
(4)              p' = -------------
                         BS - BD

Readers with some basic maths training are invited to prove (or disprove, for that matter) those two equations: think of it as a fun -- but optional -- challenge. To solve your first economic model may boost your self-esteem (just don't let it go to your head, though: this isn't a really difficult one, that's why even a subhuman worker like me managed).

Whether readers take the challenge or not, they are urged to at least test those equations with numbers: pick your trusty calculator or open your spreadsheet software and calculate (3) and (4) for AS = 2, BS = 1, AD = 10, and BD = -1 (check that those values satisfy the assumptions in the Appendix). The results should be q' = 4 and p' = 6, respectively: the coordinates of "+".

Evidently, entering different values for AS, BS, AD, and BD may yield different results. So, if you don't get q' = 4 and p' = 6, it's your own fault. :-)

Strictly speaking, you don't need any calculation to see that. Just have a look at equations (3) and (4): different values (parameters!) could mean different results. Those parameters also appear in the curves De(q) and Su(q): see equations (1) and (2). Different parameters there mean different curves with certainty. The numerical values in this little calculation exercise were the ones I used to produce Fig. 2.

To put things differently: the black boxes must change in order to determine the data points represented in Fig. 2. Is it a change in AS? Or maybe it was AD? How about a combination? Can't BS and BD change? Why do these things change, anyway?

Typically, mainstream macroeconomists would wave their hands and say: it's an exogenous shock. Sounds impressive, yes? A neoclassical would likely point to supply (AS and BS) with absolute conviction: technology shocks, the Confidence Fairy/uncertainty, say. An equally self-confident Keynesian would probably say is more a demand thing (AD and BD): Animal Spirits.

How do they know? Nothing in the data presented in Fig. 1 suggests how or why parameters change.

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But, wait a moment … On second thoughts, I think I may have made a mistake. I'm getting old and I don't feel so confident any more about the numbers behind Fig. 2: try evaluating (3) and (4) for AS = 0, BS = 1.5, AD = 8, and BD = -0.5.

Notice something?

It's not just that the De(q) and Su(q) curves and their parameters (to say nothing of their functional forms!) are volatile. They are unobservable, too, except for a single point: "+". In my example, two different pairs of schedules/black boxes determine that point (see Fig. 3). This is not the entire truth: in fact, infinitely many black boxes do and the only thing they all share and one observes and measures is that point itself.


With the data available (Fig. 1), one cannot identify those curves: estimate their parameters. One cannot statistically fit a curve to the data. That's what the identification problem is. Those are unidentified economic objects.

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It's for that reason that classical economists attempted to base economics on measurable variables.

To be fair with mainstreamers/orthodox (whether Keynesians or not), that fondness for unmeasurable, unobservable, ghostly things is not limited to them: many an illustrious and opinionated heterodox shares it, without given it a moments' thought, in fact, without having the slightest idea what they are talking about. But they are experts: they don't need that.

Talk about economics moving backwards.



Appendix.

The supply and demand diagram included in Fig. 2 follows common conventions:
  1. The quantity (q) of a good (units, for instance) is shown in the horizontal axis; the vertical axis shows its price (p) in $/unit. This practice is idiosyncratic and a bit irritating, but otherwise harmless, I think;
  2. The demand and supply curves (or schedules) are continuous, represented for simplicity as straight lines. This is not necessary, but it's common and convenient. It doesn't affect the argument presented here. Given point 1, both are prices, strange is it may sound!
  3. The parameters (i.e. AD, BD, AS, and BS) fulfil the following constrains: AD > AS >= 0, BS > 0 > BD. This guarantees that the solution of the equation De(q) = Su(q) (i) exists, (ii) is unique, and (iii) takes on positive values for quantity and prices: equations (3) and (4).

Image Credit.
[A] "Grainy B&W image of supposed UFO, Passaic, New Jersey". July 1952. Author: George Stock. Source: Wikipedia. File in the public domain.

7 comments:

  1. there are two 7s and two 10s in figure 1, but no 1 or 9. why?
    -the oo

    ReplyDelete
    Replies
    1. Good binoculars/eyeballs, Birdwatcher!

      The date "data" (!) was generated using my spreadsheet's RANDBETWEEN(Bottom, Top) function, for Bottom = 0 and Top = 12. As a matter of fact, all the "data" are random numbers.

      Twice the same "date" means two sales that day (i.e. day 7 and day 10): say, on day 7 Joe bought 3 widgets, at the price of $1 ea in Sydney; that day Mary, in Brisbane, bought 8, paying $5 apiece. No "date", means no sales that day (days 1 and 9).

      Should I add a brief note to make that clear to other readers, do you think?

      Delete
  2. makes sense. put a link to question/answer.
    -the oo
    ps. jeeezuz. a catholic answer!

    ReplyDelete
  3. its me again, magpie. sorry to be a pain. i tried to get q' & p'. never managed. please explain.
    - the oo

    ReplyDelete
  4. No worries, Pauline H. :-)

    This is what you do. You need to make supply [Su(q)] equal demand [De(q)]:

    Su(q) = De(q)

    Makes sense? Replace the RHS from equations (1) and (2):

    AS + BS*q = AD + BD*q

    Now "isolate" q in the LHS:

    AS + BS*q - BD*q = AD <--- I just "moved" BD*q to the LHS

    I'll leave the rest for you to do. Begin by "moving" the other things to the RHS and then factoring q in the LHS. (Just a warning: it may need some simplification.)

    When you get q "alone" in the LHS you've got the solution: q'. It must be (3).

    The next step is to find p'. You can use either (1) or (2) (why?). Say, using (2):

    Su(q') = AS + BS*q'.

    Just write the solution you found for q' into the RHS of that last equation (look one line above)

    AS + BS*(solution_you_found) < --- this is what I mean.

    Likewise, it may need some simplification at the end. The final result must look like (4).

    If you have any more difficulties, never fear, the Magpie is here!!!

    ReplyDelete
  5. got it! thanx.
    - the oo
    ps. didnt appreciate the 'pauline'. hmph :)

    ReplyDelete