Economics Questions

Re: Help a sympathetic Keynesian out on ABCT

Sam Silverspring — Sat, 09 Jul 2011 00:23:15 GMT

First off, the "Hayekian Triangle" in Prices and Production is linear due to two critical assumptions.

1) "The amount of original factors applied during successive stages of the process is constant" (Footnote 44 of Lecture 2; Page 231 of my Mises Institute edition of P&P, - if you have another edition it's the long footnote with the integral in it)

2) Hayek "intentionally neglected interest" in that lecture (p. 231 of my Mises Institute edition as well).

So, if we imagine that the only input is labour (which Hayek often does), which is continually applied over time in a production process AT A CONSTANT RATE, then the slope of the Hayekian triangle should, in this simplified model, be equal to the wage rate. So, yeah, I can see why one could be confused by the relationship between the interest rate and the Hayekian triangle. In it's original presentation, THERE WAS NO INTEREST RATE (You get to that in lecture 3, I believe, and by then he stops using triangles).

To double check why the Hayekian triangle should grow linearly with a slope equal to the wage rate, let's think of time discretely (in days) as an approximation of how the Hayekian triangle works in continuous time. Suppose stage one of some process takes one day, uses one unit of labour, and that the wage rate per day per unit of labour is a dollar; if the interest rate is zero (I use interest synonymously with accounting profit here), then the value of the intermediate good produced by that unit of labour better be a dollar (If it's less, the interest rate is negative; if it's more, then there IS interest). As we jump to the next stage of production, one would need to spend another dollar on another unit of labour, and then spend another dollar purchasing the intermediate products produced at stage one. As long as interest is still assumed to be zero, then the value of the intermediate goods produced at this stage (2) is 2 dollars. Obviously, stage 3 goods will be worth $3, and so on. I hope you get the idea.

When Hayek DOES account for interest (See either Pure Theory of Capital or the much underrated "Relationship Between Investment and Output"), then the "triangle" is not necessarily a straight line. If labour input application is constant, but the rate of interest is positive, then the triangle should have an exponential slope (Why? To account for the "profit" or "interest" earned at each stage of production).

Now, Daniel also asked whether the "triangle" need be a triangle. I assume that are wondering whether the value of the capital stock could decrease at later stages of production, so you end up with, well... a sort of curvy line moving up and down. Well, I can answer this question somewhat simply if we assume all capital goods are "goods in process"; i.e. unfinished goods on their way to becoming final output (for example, not yet fully matured wine). Suppose later stages of production require very little labour, and early stages require lots of labour (as you were considering). Even then, you will get a sort of “triangular” shape, since the later stages of production need to purchase the “capital” (or goods in process) produced by the earlier stages to continue the production chain. Therefore, to have a constant rate of profit or interest prevailing in the system, the value of the goods produced at the later stages of production need to be greater than the value of those produced at earlier stages. So, in this case, you will always have a sort of triangular shape. Or put differently, if time is the x-axis, and value is the y-axis, the slope will always be positive.

However, if durable capital goods are used in production, is the Haykian triangle still a “triangle” then? Well; here’s where things get tricky, for now we must think about how we are to “time date” an input’s “investment period” to accurately draw our Hayekian triangle.

Now this is an abstract point, but bear with me. Hayek defines investment period as “the interval between the application of a unit of input and the maturing of the quantity of output due to that input” (Pure Theory of Capital, p. 69). In a goods in process model, this is easy to specify; it’s the amount of time that passes between, say, unit of labour crushing or picking some grapes, and when the wine from those grapes matures and is sold. But if one imagines a production process involving, say, a durable machine, then the time relationship is not so simple as it was in the goods in process model; a unit of labour is responsible for output at many FUTURE dates, and therefore may have many different investment periods.

The concept of “investment period” turns out to be crucial to Hayek’s triangular construction. Basically, “time” in the Hayekian triangle is not really “time” per se, but rather the “investment period” of various units of input. So, to even draw a Hayekian triangle you need to solve a really funny sort of imputation problem, where, given the value of each capital good and input, you determine how long a moment in time’s input SHOULD be invested to allow for compound interest to be earned on all units of input invested for x units of time. More briefly, this is simply another way of dealing with the commonly quoted problem with Austrian Capital Theory; which production processes are “early” stages, and which production processes are “late” stages. For more detail on this, you’ll probably want to just consult Pure Theory of Capital, but for what my opinion is worth, I think Hayek does, come up with a technically consistent way to deal with the problem (I could be wrong though). Not sure if it would help much in statistical applications though, since Hayek is dealing with an idealized stationary state.

On that note, for all intensive purposes, I think the Hayekian triangle should be thought of as a goods in process model only. You can bring durable goods in if you like, but if you want it to be a consistent model, it is not going to be mathematically tractable, or at the very least, it confuses more than it illuminates.

One final note; Garisson has straight line Hayekian triangles, not exponential Hayekian triangles, even though he does account for interest. This is because he assumes simple interest, not compound interest, rules in the system.

Re: Help a sympathetic Keynesian out on ABCT

Clayton — Fri, 08 Jul 2011 23:04:23 GMT

As an aside, I've often thought to myself that Austrians would do well to invest in studying ordinal arithmetic. A great deal of Austrian assumptions are of the form "A must be greater than B" - ordinal quantities. Even though Austrian arguments are not arithmetic, they are still formal and, therefore, amenable to abbreviation through the tools of formal methods. Ordinal arithmetic would probably help make some of these arguments more compact.

Clayton -

Re: Help a sympathetic Keynesian out on ABCT

Clayton — Fri, 08 Jul 2011 22:59:47 GMT

You seem to be thinking there is some kind of arithmetic or mathematical relationship between consumption and stages, i.e. a geometric slope. The actual shape, straightness and slope of the hypotenuse are irrelevant except that spending on each higher stage of production must necessarily be less than spending on lower stages of production (else, businesses operating in that stage of production are doing so systematically unprofitably which would be a very strange state of affairs). The "slope" of the hypotenuse will become steeper as interest rates lower and the cost of new investment, therefore, decreases, thus leading to a "lengthening" of the time structure of production. Since the amount of available resources in the economy are roughly the same after a lengthening of the time structure as existed before, a narrowing of consumption must follow. As total production expands (due to the increased length of the time-structure, that is, increased division of labor), the base of the triangle can again lengthen.

I don't see what there is to be confused about except that you seem to be interpreting the triangle as a geometric figure, which it is not.

Clayton -

Re: Help a sympathetic Keynesian out on ABCT

Jonathan M. F. Catalán — Fri, 08 Jul 2011 19:59:34 GMT

The "step diagram" is also from Prices and Production.

Re: Help a sympathetic Keynesian out on ABCT

Smiling Dave — Fri, 08 Jul 2011 19:57:51 GMT

Now I think about it, I believe Rothbard had a diagram of steps, not of a triangle, to explain ABCT. So that he at least did not consider that hypotenuse very important.

But it's up to the more proficient to correct or confirm me on this.

Re: Help a sympathetic Keynesian out on ABCT

Smiling Dave — Fri, 08 Jul 2011 19:49:56 GMT

Dan,

The interest-rate sensitivity of the capital structure means, in simple English, that if interest rates are lowered, some parts of the structure are going to be invested in that otherwise would not, specifically those further away from final consumer goods, because those need the most money to build. And indeed that is one way of describing the major insight of the ABCT. But that is not related strongly to the linearity of the hypotenuse, as we are about to explain.

The length of the hypotenuse at any segment is determined by two factors, how much money comes in and goes out at each stage, and how much time is spent at each stage. So, yes, it measures return on investment per amount of time. However, that is not the only factor influencing a decision to invest at a particular stage. After all, if that hypotenuse is linear, there is no reason to chooose one stage over another, for they all give the same return on investment per unit of time, right? And AE assumes that there is a huge difference in the various stages, and the interest rates charged to borrow money will determine which stage gets chosen. One huge factor is that the higher stages are more expensive to build usually. It costs more to build an auto factory than an auto. So no matter the shape of the hypotenuse at any stage [within reason], looking at the big picture, it may well behoove an entrepeneur to invest in the higher stages now when he gets money cheap to do it. In addition, low interest rates, absent a central bank meddling, means consumption is going down. So that even if return on investment for existing final consumption is high, there will be no point in putting more money into that stage, for that market is saturated right now.

the problem is that the wrong factories where built"

It's not clear to me at all that that is the primary problem.

It's what the ABCT is saying bottom line, right? Investment in higher orders of production which was not justified by consumer desires.

There is also the question of certain particular industries being the "wrong choice", like too many house built in our new millenium. But that also follows from ABCT. Because this view of lower interest rates meaning [absent bank meddling] that consumers want to underconsume is only an aggregate picture. A particular industry may have increased consumer interest. In other words, everyone may want smart phones badly, but not much of anything else. So that the smart phones companies should keep churning them out right now, as opposed to the big picture of everyone else churning out less lower order goods. And so too, the bank's distorted picture will be giving an incorrect view of many industries, but not all of them.

Re: Help a sympathetic Keynesian out on ABCT

Jonathan M. F. Catalán — Fri, 08 Jul 2011 18:39:17 GMT

Who uses the Hayekian triangle as an "analytical assumption"? That is, who purports to explain the existing structure of production by using the Hayekian triangle, aside from using the triangle as a means of illustrating roughly what a structure of production looks like?

Obviously. But he's still making what seems to be a specific assumption about the interest rate elasticity of supply here, does he not?

I feel you're missing the point. He's making that "specific assumption" so he can provide a model as a pedadogical device. Hayek never intended for anybody to assume that his Hayekian triangle was an accurate portrayal of what a real life structure of production looks like. Hayek, again, stresses the fact that the Hayekian triangle is meant for simplicity and not accuracy.

Re: Help a sympathetic Keynesian out on ABCT

Daniel Kuehn — Fri, 08 Jul 2011 18:34:37 GMT

Jonathan -

re: "I don't know what this means."

It means that people publishing today, in 2011, use this as an analytic assumption - NOT just a pedagogical device.

re: "Hayek can't comment on the "interest sensititivity of each stage of production", because it's not the stage that is sensitive to the rate of interest, but the individuals who operate within each stage."

Obviously. But he's still making what seems to be a specific assumption about the interest rate elasticity of supply here, does he not?

Re: Help a sympathetic Keynesian out on ABCT

Jonathan M. F. Catalán — Fri, 08 Jul 2011 18:26:52 GMT

OK, my concern is you don't have to go back to Prices and Production to get this assumption.

I don't know what this means.

And unlike the question of a curved or straight LM curve, it seems to me the straightness of the Hayekian triangle says something qualitative about the interest rate sensitivity of each stage of production (it's akin to Hayek assuming a fairly elastic interest rate elasticity of supply).

Hayek can't comment on the "interest sensititivity of each stage of production", because it's not the stage that is sensitive to the rate of interest, but the individuals who operate within each stage. The best Hayek can do is model some theoretical circumstance and he just assumes that the spread is "perfect" (in the sense that the economy is at equilibrium); i.e. that profit at each stage has been maximized and that it doesn't pay for new entrepreneurs to enter any of the stages.

Hayek's more complicated models are curvilinear, though, not straight; some models are used to depict different aspects of the same theory, though. For example,

Re: Help a sympathetic Keynesian out on ABCT

Daniel Kuehn — Fri, 08 Jul 2011 18:18:28 GMT

Jonathan -

re: "Hayek's main purpose in Prices and Production is only to bring to the table the theory of a capital structure based on the pricing process"

OK, my concern is you don't have to go back to Prices and Production to get this assumption.

And unlike the question of a curved or straight LM curve, it seems to me the straightness of the Hayekian triangle says something qualitative about the interest rate sensitivity of each stage of production (it's akin to Hayek assuming a fairly elastic interest rate elasticity of supply). The answer qualitatively changes - it's not just semantics over the pedagogy, otherwise I wouldn't be as concerned about it.

Re: Help a sympathetic Keynesian out on ABCT

Daniel Kuehn — Fri, 08 Jul 2011 18:14:23 GMT

Smiling Dave -

My primary concern is this - the interest-rate sensitivity of the capital structure and the linearity of the hypotenuse are quite closely related concepts.

If the interest-rate sensitivity of the capital structure is abandoned, what is left of ABCT? This is coming from someone who has thus far been fairly convinced it's at least one of a couple processes going on over the business cycle.

re: " If you have time to answer personal q's, how does it feel learning this stuff? Do you approach it as a newborn babe, or place some burden of proof than that on the Austrians, since they are trying to topple a previous picture of how things work?"

There's a lot to complain about in the Austrian school, but the business cycle theory is one element that makes a good deal of sense. Of course, though, once a sensible theory has been articulated it ought to be held to a burden of proof. I wish there was more of that, but more gets published every day.

re: "What is your opionion of the key insight, in my opinion, that animal spirits or confidence or tightening of the money supply or whatever are not a satisfactory explanation for why a recession begins, and that an answer has been found by Austrians?"

As I read the evidence, the standard Keynesian and monetarist story stands up well to the data. I'm also reasonably well convinced that the elongation of the capital structure occurs over the business cycle (or, more accurately, with interest rate fluctuations). What I still feel in the dark about is the extent to which this is significant for the business cycle.

re: "Also to the corrolary, that given that the problem is that the wrong factories where built"

It's not clear to me at all that that is the primary problem.

Re: Help a sympathetic Keynesian out on ABCT

Jonathan M. F. Catalán — Fri, 08 Jul 2011 18:12:30 GMT

How individual stages respond to changes in prices has been up for debate, and ultimately depends on how individual market agents respond to these changes in prices. No curve can model a real economy, because any curve is necessarily a posteriori (it can only model a past structure of production). Hayek's main purpose in Prices and Production is only to bring to the table the theory of a capital structure based on the pricing process, and then illustrate what happens to that structure with monetary changes (and thus distortions in how the pricing process reflects preferences).

Re: Help a sympathetic Keynesian out on ABCT

Daniel Kuehn — Fri, 08 Jul 2011 18:08:04 GMT

This may be the case, but it seems hard to do anything with the pedagogical device in that case.

When the interest rate changes, with a linear capital structure, the entire capital structure shifts. If we abandon the linearity and assume, perhaps, a steep curvature of the capital structure we allow for the possibility that any particular stage is highly insensitive to the interest rate if the interest rate costs at that particular stage are negligible relative to other costs (indeed - this is why the capital structure wouldn't be linear - if the interest costs were negligible. If they were substantial we would expect a highly triangular capital structure).

I guess I just feel like it seems like it chalks a lot up to pedagogy. In other words, the principle is still sound but its relevance seems to require a lot more clarification about the nature of the actual capital structure.

Re: Help a sympathetic Keynesian out on ABCT

Daniel Kuehn — Fri, 08 Jul 2011 18:03:26 GMT

nirgrahamUK -

A uniform rate of profit is one thing, but even if we put the fact that a uniform rate of profit never emerges in practice that still doesn't guarantee a linear relationship between time and the factors of production added at each stage. What could the rate of profit possibly have to do with that?

Re: Help a sympathetic Keynesian out on ABCT

Jonathan M. F. Catalán — Fri, 08 Jul 2011 17:11:43 GMT

The Hayekian triangle is a pedagogical device Hayek developes in Prices and Production. More complicated models are presented in The Pure Theory of Capital.