Assessing the Impact of Government Policy on Widget Consumption and Widget Sector Capital Usage

Let supply and demand for widgets (y) be given by the following two equations, respectively:


(1) yt = αt + β x t + ε t


(2) yt = γ + δ x t + Γ z t + u t


Where x is the relative price of widgets, z is a government procurement policy for widgets, and ε and u are serially uncorrelated mean zero errors, E(ε u) = 0. Note that there is a time varying constant in the supply equation, α t.


How would one analyze the impact of a public policy, such as an increase in government procurement of widgets to place in public places, on the total number of widgets consumed?


First, solve the system for the endogenous variables. Suppose I want to know the reduced form expression for the quantity of widgets purchased. The invert the second equation, solving for x, and substituting into the first (supply) equation. This leads, after solving:


(3) yt = [α t(δ-γ)/(δ-β)] + [(βΓ)/(δ-β)] zt + (δ/(δ-β))(εt – (β/δ) u t)


Suppose I wanted to forecast widget consumption next year, taking into a government policy involving z. How would one best undertake this exercise? For myself, I would take the time differential of the above expression (3). Let the Δ(.) operator indicate the time difference, hence Δ yyt+1-yt.

(4) Δ y = [Δ α (δ-γ)/(δ-β)] + [(βΓ)/(δ-β)] Δ z + (δ/(δ-β))(Δ ε – (β/δ) Δ u )


Suppose additionally capital services demand (i.e., the derived factor demand) is given by:


(5) kt = Θ y t + et


Where e is another random error term. Since the error terms are random, and serially uncorrelated, then my best guess of the change in widget consumption (and widget industry capital usage) in the absence of a change in government procurement is:


(6) Δ y = Δ α [(δ-γ)/(δ-β)]


(6a) Δ k = Θ × Δ α [(δ-γ)/(δ-β)]


And my best guess of widget consumption with the government policy change is:


(7) Δ y = Δ α[ (δ-γ)/(δ-β) ] + (βΓ)/(δ-β) Δ z


(7a) Δ k = Θ × {Δ α[ (δ-γ)/(δ-β) ] + (βΓ)/(δ-β) Δ z}


Where I would use estimates of δ , β , γ , Γ , and Θ from the literature on widget supply and demand, presumably obtained by way of econometric studies. Δ α would be a variable based upon forecasts, presumably based upon observations on real time data.


Of course, what we observe in reality is:


(8) Δ y = Δ α [(δ-γ)/(δ-β)] + [(βΓ)/(δ-β)] Δ z + (δ/(δ-β))(Δ ε – (β/δ) Δ u)


(8a) Δ k = Θ × {Δ α [(δ-γ)/(δ-β)] + (βΓ)/(δ-β) Δ z + (δ/(δ-β))(Δ ε – (β/δ) Δ u )} + Δ e


Wherein (8) differs from (7) by virtue of the unpredictable errors, (δ/(δ-β)(Δ ε – (β/δ) Δ u )). Employment differs by Θ [(δ/(δ-β)(Δ ε - (β/δ) Δ u)] + Δ e


Thus far, I don’t think many economists would have trouble with this methodological approach, although they could clearly argue against this particular set of exogenous variables in, say, the demand equation; or they could reasonably argue that expectations of future widget demand (as well as future government procurement policies with respect to widgets) should matter. But the key is thinking structurally, and about shocks to the three structural equations. In this analysis, I don’t think one would want to compare (8) against (6), nor (8a) against (6a), since one is conflating shocks with policy effects.


Now, inexplicably, in current discourse, the troubles begin…


Now, let’s think about what happens if (1) is short run aggregate supply of real GDP, and (2) is aggregate demand, where x is the price level, and z is government spending. Equation (3) now defines the output-labor relationship, of the nature examined in this post. The critics of the Administration’s approach to estimating the number of jobs created are doing one of two things: (i) they are comparing (8a) against (6a) when the errors have been large, so the composite error (δ/(δ-β)) × (Δ ε – (β/δ) Δ u) is large and negative; or (ii) they are arguing for different estimates of the relevant coefficients. By far, (i) is more popular discourse (such as discussed in this post). Regarding (i), I’ll note that there was a substantial deterioration in everybody’s expectations regarding the course of the economy in January and February [1]. The more relevant comparison would be (8) against (6) (and (8a) against (6a)) where the shocks (that have now been realized) are added in. That is, compare (8) against (6′), and (8a) against (6a’):


(6′) Δ y = Δ α [(δ-γ)/(δ-β)] + (δ/(δ-β))(Δ ε – (β/δ) Δ u)


(6a’) Δ k = Θ × {Δ α [(δ-γ)/(δ-β)] + (δ/(δ-β))(Δ ε – (β/δ) Δ u )} + Δ e


More reasonable critiques rely upon (ii), although I have not seen detailed quantitative analyses which break down the sources of mis-prediction. (There is a third route, which involves arguing as an article of faith that there is going to be no, or negative, effect of the government widget procurement policy on widget consumption).

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15 thoughts on “Assessing the Impact of Government Policy on Widget Consumption and Widget Sector Capital Usage

  1. biofuel

    Does this kind of modeling correctly predict effects of government policy on widget consumption in past instances of government intervention, where the inputs and outcomes are known? I have no clue whether this modeling is sound, but I would be receptive to the argument that it has been successful in predicting past outcomes post festum. Although even then, I would take any predictions as a hypothesis worth testing, a guideline.

  2. Tom

    Your analogy of ARRA job creation/retention to a widget procurement policy would make more sense if the ARRA was a workfare program. To reflect the indirect nature of ARRA’s influence on the job market I guess you would need to add a few more Greek letters to your equations. Also, you don’t seem to account at all for the effects on the economy of raising funds from it, which might be unimportant for widget procurement but is obviously very important for the ARRA.
    I think it’s obvious that the ARRA has cushioned employment to some extent, but it’s just as obvious that there’s no way to accurately measure how much. Which, anyway, is only one of many things we need to consider when judging whether the ARRA was a good idea. There is also the question of whether the work that people are doing under the ARRA is economically beneficial and growth-positive or economically parasitic and growth-negative. There are also questions about the effects of funding the ARRA, which go beyond job numbers. For me the biggest concern is the way that Treasury borrowing has become dependent on Fed money creation, not only in the direct sense of the Fed’s recent Treasury purchase program but also and more importantly in the indirect sense that without steady money creation, the Treasury’s borrowing demands would be overbearing on capital markets and would quickly drive up interest rates. The Fed seems to be trapped into continuing radical monetary stimulus indefinitely.
    But you seem to prefer to limit discussion of the ARRA to the narrow question of the quantity of jobs created or saved.

  3. Buzzcut

    You know what’s confusing?
    Menzie is on a “created or saved” jihad based on economic theory, but it seems to me that the government reports are trying to show the actual numbers “created or saved” quantitatively, not based on a model.
    Certainly, the critics are making quantitative judgements, not ones based on modeling. They’re going in and looking at the numbers and saying things like, “hey, there is no 00 congressional district in New Hampshire!”. ;)
    Is that impression incorrect? Are these reports simply the result of economic models?

  4. Menzie Chinn

    Buzzcut: You are confused because you have not been reading carefully my posts (or are unable to comprehend them); I have been discussing one of the two ways the Administration, in particular, the CEA, has been tracking the impact of the ARRA. I have not made reference to tracking via Recovery.gov in any of my posts.

    To edify yourself, please consult: [1], [2]. I am hopeful that this will clear up your confusion.

  5. Buzzcut

    Menzie, maybe I’m wrong here, but I’d be willing to bet REAL MONEY that a poll of your commenters would find that most of us are hearing criticism of the actual “created or saved” numbers at recovery.gov.
    So your little jihad is really not addressing the substantive criticism that is out there.
    Now, maybe an actual accounting of the jobs “created or saved” is too difficult to do, or whatever. But I think few are persuaded by these simplistic math models, one way or the other.
    As an aside, I’d also be willing to bet that an accounting of the number of comments to your posts would find that any post with math has very few comments, relative to, say, the ones where you do accounting of the number of dead soldiers in Iraq.
    Graphs good. Algebra bad. ;) That’s my gut feel from reading this blog for darn near three years now.
    Haven’t seen one of those Iraq posts in awhile. Wonder why.
    When are we going to see a similar accounting from you on Afghanistan?

  6. Menzie Chinn

    Buzzcut: Thank you for your insights. Actually, if my objective function were to elicit comments from people such as yourself, your comment would be of some consequence. But I’m trying to affect informed opinion, and so algebra, I think, is hitting exactly the right audience.

    By the way, what’s the point of referring to a jihad?

  7. 2slugbaits

    Buzzcut,
    Maybe the algebra posts don’t get a lot of comments because the math makes the arguments relatively clear to those who understand the algebra and incomprehensible to those who don’t understand. You either get comparative analysis or you don’t.
    If we’re voting on topics, personally, I’d like to see more of Menzie’s econometric stuff…JDH’s too!

  8. ppcm

    The math model is a neutral vehicle for those whom understand and drive it.As a passenger I would be keen to know the point of destination,that is a model comprehensively addressing :
    The quality,the perenity, of the jobs created or maintained (alternative model in absence of monetary stimulus)
    The additional parameters moved along with monetary stimulus,supply and demand for money,the investments savings curve (corporate and private investments),the assets prices in relation to incomes and revenues.
    Ultimately they should converge towards equilibrium ?

  9. Buzzcut

    Menzie, the jihad description is just in reaction to recent trends in your commentary. Taking on Limbaugh, or Politico. Delving into the global warming controversy (but not commenting on the area that you might actually have experise on, the economics of the reaction to global warming).
    From a… how should I put it… anthropology of the overeducated… perspective (and not just you, but university intellectuals in general), it’s interesting to see what interests you. It’s like listening to “Talk of the Nation” and hearing what callers from Madison, Ann Arbor, or Berkley say.

  10. Menzie Chinn

    Buzzcut: Ah, I think your comment has illuminated for me fully the reasons for your discontent. I apologize for being educated. I’ve just scanned the past month’s worth of posts; it seems to me they are almost all related to economics. (And the Limbaugh and Politico posts you remark upon were about economics, weren’t they?)

    While it is true I’m not a climate scientist, I have posted on the economic impacts of climate change mitigation, here. Does that address your concerns.

    By the way, the correct spelling is “Berkeley”.

  11. Buzzcut

    (And the Limbaugh and Politico posts you remark upon were about economics, weren’t they?)
    That’s really not the point. Even if they are economics related, it’s interesting that a tenured academic at a major university feels the need to correct RUSH LIMBAUGH, for example.
    It’s kind of like when CNN fact checked a SNL skit about Obama.
    Some of the comments to that post where you regurgitate that flawed CBO analysis of cap and trade are very good.

  12. Menzie Chinn

    Buzzcut: I think of that post as correcting the WSJ editorial page. I am sure I’ll be doing that again in the future (it is a target rich environment, after all).

    I welcome your explanation of what specific parts of the CBO study are “flawed”.

  13. ohwilleke

    This kind of context free, constants absent, mathematical model manipulation is precisely the kind of theoretically rigorous but practically counterproductive kind of analysis that got macroeconomics into trouble in the first place. The notion that a general theory of widgets can be applied with any kind of useful generality is just usually wrong.
    The kind of widgets that the government buys, and the particular places that it puts them, turn out to matter a lot.
    Suppose that the kind of widget the government is buying is a sign that tells you how long it will take to get from point A to point B on the road that you are currently driving on given current traffic conditions. Places before the best possible turnoff, this sign could add a lot of value at a modest cost. Placed after that turnoff, it will only slightly appease hordes of angry and stuck motorists.
    Free bicycles placed in a dense urban environment with good bike lanes (e.g. Amsterdam) may dramatically increase bicycle useage. Free bicycles placed in a rural environment, in contrast, may do little to increase bicycle useage.
    Moreover, in the examples that matter, the demand curves will often be non-linear and kinked often in a way that is highly sensitive to some critical mass, and blur boundaries between related markets. For example, while municipal free wi-fi may damp private demand for private wireless routers, it may pump private demand for units made by the same company that allow privately owned laptops to access wi-fi signals.
    The value of the widgets purchased will also often be inextricably intertwined with other government policies implemented simultaneously. For example, reducing the supply of street parking to make bike lanes, could have a strong independent effect on the success of a free bike policy, and if the free bike policy reaches critical mass, factors that influence the demand for bikes, like conventions governing workplace dress codes in the impacted area, could greatly change the demand curve.
    When you talk about widgets in the abstract, the dry assumptions found in the math don’t seem unreasonable. But, when you start to try to apply those models to particular real life examples, they quickly lose their shine.

  14. Mattyoung

    I read through most of the background. Menzie did a great job of estimating the relation between GDP and employment. My minor objection is that he estimate his parameters from 2001 on, skipping the 91 recession.
    Every thing works except our estimate of GDP which is going to be subject to excessive revisions. Main point, if we were currently so good at estimating near term GDP we would not have crashed.
    So, getting a better estimate of GDP means getting an estimate of the economic constraints. Under the assumption that we are constrained by energy, then GDP will be highly correlated with oil imports by volume, which is what I look at:
    http://tonto.eia.doe.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=wttimus2&f=w
    Now that is noisy data, but my analog, optical regression analyzer tells me that something happened to boost oil imports slightly above the downward trend. Congress did manage to burn a little of the overshoot in oil inventories.

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