Consider a firm that takes heterogeneous labor and capital inputs L1, L2… and K1, K2…, using these to produce some output Y. Define a firm production function Y=F(K1, K2…, L1, L2…) as the maximal output that can be produced using the given vector of outputs – and note the implicit optimization condition in that definition, which means that production functions are not simply *technical* relationships. What conditions are required to construct an aggregated production function Y=F(K,L), or more broadly to aggregate across firms an economy-wide production function Y=F(K,L)? Note that the question is not about the definition of capital per se, since defining “labor” is equally problematic when man-hours are clearly heterogeneous, and this question is also not about the more general capital controversy worries, like reswitching (see Samuelson’s champagne example) or the dependence of the return to capital on the distribution of income which, itself, depends on the return to capital.

(A brief aside: on that last worry, why the Cambridge UK types and their modern day followers are so worried about the circularity of the definition of the interest rate, yet so unconcerned about the exact same property of the object we call “wage”, is quite strange to me, since surely if wages equal marginal product, and marginal product in dollars is a function of aggregate demand, and aggregate demand is a function of the budget constraint determined by wages, we are in an identical philosophical situation. I think it’s pretty clear that the focus on “r” rather than “w” is because of the moral implications of capitalists “earning their marginal product” which are less than desirable for people of a certain political persuasion. But I digress; let’s return to more technical concerns.)

It turns out, and this should be fairly well-known, that the conditions under which factors can be aggregated are ridiculously stringent. If we literally want to add up K or L when firms use different production functions, the condition (due to Leontief) is that the marginal rate of substitution between different types of factors in one aggregation, e.g. capital, does not depend on the level of factors not in that aggregation, e.g. labor. Surely this is a condition that rarely holds: how much I want to use, in an example due to Solow, different types of trucks will depend on how much labor I have at hand. A follow-up by Nataf in the 1940s is even more discouraging. Assume every firm uses homogenous labor, every firm uses capital which though homogenous within each firms differs across firms, and every firm has identical constant returns to scale production technology. When can I now write an aggregate production function Y=F(K,L) summing up the capital in each firm K1, K2…? That aggregate function exists if and only if every firm’s production function is additively separable in capital and labor (in which case, the aggregation function is pretty obvious)! Pretty stringent, indeed.

Fisher helps things just a bit in a pair of papers from the 1960s. Essentially, he points out that we don’t want to aggregate for *all* vectors K and L, but rather we need to remember that production functions measure the maximum output possible when all inputs are used most efficiently. Competitive factor markets guarantee that this assumption will hold in equilibrium. That said, even assuming only one type of labor, efficient factor markets, and a constant returns to scale production function, aggregation is possible if and only if every firm has the same production function Y=F(b(v)K(v),L), where v denotes a given firm and b(v) is a measure of how efficiently capital is employed in that firm. That is, aside from capital efficiency, every firm’s production function must be identical if we want to construct an aggregate production function. This is somewhat better than Nataf’s result, but still seems highly unlikely across a sector (to say nothing of an economy!).

Why, then, do empirical exercises using, say, aggregate Cobb-Douglas seem to give such reasonable parameters, even though the above theoretical results suggest that parameters like “aggregate elasticity of substitution between labor and capital” don’t even exist? That is, when we estimate elasticities or total factor productivities from Y=AK^a*L^b, using some measure of aggregated capital, what are we even estimating? Two things. First, Nelson and Winter in their seminal book generate aggregate date which can almost perfectly be fitted using Cobb-Douglas even though their model is completely evolutionary and does not even involve maximizing behavior by firms, so the existence of a “good fit” alone is, and this should go without saying, not great evidence in support of a model. Second, since ex-post production Y must equal the wage bill plus the capital payments plus profits, Felipe notes that this identity can be algebraically manipulated to Y=AF(K,L) where the form of F depends on the nature of the factor shares. That is, the good fit of Cobb-Douglas or CES can simply reflect an accounting identity even when nothing is known about micro-level elasticities or similar.

So what to do? I am not totally convinced we should throw out aggregate production functions – it surely isn’t a coincidence that Solow residuals for TFP match are estimated to be high in places where our intuition says technological change has been rapid. Because of results like this, it doesn’t strike me that aggregate production functions are measuring *arbitrary* things. However, if we are using parameters from these functions to do counterfactual analysis, we really ought know better exactly what approximations or assumptions are being baked into the cake, and it doesn’t seem that we are quite there yet. Until we are, a great deal of care should be taken in assigning interpretations to estimates based on aggregate production models. I’d be grateful for any pointers in the comments to recent work on this problem.

Final published version (RePEc IDEAS. The “F. Fisher” on this paper is the former Clark Medal winner and well-known IO economist Franklin Fisher; rare is it to find a nice discussion of capital issues written by someone who is firmly part of the economics mainstream and completely aware of the major theoretical results from “both Cambridges”. Tip of the cap to Cosma Shalizi for pointing out this paper.

Hi,

see this sequence of journal articles by Fisher’s co-author Felipe and the growth economist Jonathan Temple:

http://www.tandfonline.com/doi/abs/10.1080/.VCEnM_ldWPY

http://www.tandfonline.com/doi/abs/10.1080/02692171.2010.512146?src=recsys#.VCEoZvldWPY

http://www.tandfonline.com/doi/abs/10.1080/02692171.2010.511454#.VCEnRPldWPY

from memory Temple argues that empirical applications of production functions may be more informative than Fisher and Felipe allow, and that the “accounting identities imply data fits Cobb Douglas” argument requires an auxiliary assumptions that may not hold (I hope I have got this right, it’s been a long time since I read them)

there are also some interesting papers coming out with ‘microfoundatons’ for agg prod functions. here’s one

it’s not actually the one I had in mind, but I have mentally mislaid it.

I remember learning about the problems of aggregating over heterogeneous agents from Grandmont, e.g. http://ideas.repec.org/p/cwl/cwldpp/987.html in the context of demand

From your linked paper:

«These facts have strong consequences for the unicity and stability of equilibrium as well as for the prevalence of the weak axiom of revealed preference in the aggregate in a private ownership Walrasian exchange model.»

That’s very funny, in a sort of nerdy way. I can imagine the author at some point bannging on his desk and then shouting “bullshit! JUST BULLSHIT!” for a while and when he calmed down he found the strength to restrain himself and write the euphemism above. 🙂

Thanks – I am vaguely familiar with that paper, but should look more closely.

Hmm, I returned to read some more on this in a spare moment, and find my links above don’t seem to be working. here is a non-gated version of Temple’s aggregate production functions and growth economics.

Yes, this is a very good summary.

«why the Cambridge UK types and their modern day followers are so worried about the circularity of the definition of the interest rate, yet so unconcerned about the exact same property of the object we call “wage”»

Oh no! This is a terrible misunderstanding.

You may have gotten the impression above by following the Cambridges Capital Controversies, but that was a very special case.

It was a special case because the stake was whether the neoclassical approach embedded in Arrow-Debreu-Lucas makes sense, and the first thing one says about it is that it does not because it lacks a theory of capital, and wiwthout a theory of capital, it cannot be called a theory of the political economy.

That’s the first thing that comes to mind because what has been carefully excised from neoclassical theories is *time*, and without time of course one cannot have a theory of capital (beyond initial endowments). And then one cannot have a theory of money or liquidity or interest rates either, they become just a “veil”.

Time has been carefully excised from the neoclassical approach because including its effects in the theory it is well nigh impossible to prove the central truthiness of Economics, that the distribution of income solely depends on marginal productivity, absent government distortionary intervention.

So the discussion became a discussion about capital, because that is both a very big deal in itself, not something that neoclassical can claim is irrelevant in the big scheme, and at the same time entirely mysterious in the neoclassical model, because it has to be, else the central truthiness of Economics cannot be asserted.

«all that matters is the central truthiness of Economics, upholding which is also central to a well-rewarded and richly sponsored career in academic Economics.x

This link is a report from the field:

http://rwer.wordpress.com/2013/06/30/doctor-x-pure-shit-and-the-royal-societys-motto/

«every firm’s production function must be identical if we want to construct an aggregate production function. This is somewhat better than Nataf’s result, but still seems highly unlikely across a sector (to say nothing of an economy!).»

This “seems highly unlikely” seems to me like a huge understatement indeed, because we must consider not just the empirical likelihood of that condition being satisfied (something that neoclassicals care very little about), but also its effect on the mechanics of the neoclassical model itself.

Because in the neoclassical model it is assumed that there are infinite markets across all of eternity with infinite suppliers in each of them in order to give perfect competition without which the central truthiness of Economics cannot be asserted.

But how can there be infinite suppliers in infinite markets if they all must have the same production function? That means that in effect there is only supplier which get arbitrarily sliced into infinite subsets. But then how can there be perfect competition?

It just does not make sense.

IIRC the above is pointed out in Steve Keene’s book.

But “aligned” Economists never point out the above “technical assumptions” because all that matters is the central truthiness of Economics, upholding which is also central to a well-rewarded and richly sponsored career in academic Economics.