In this post I want to say something about the methodology behind the CBO’s potential output forecasts. Here’s the tl;dr:

Officially, the CBO forecasts are based on a production function, which requires use of a number of unobservable parameters and questionable assumptions. But with one important exception, use of the the production function has no effect on the final estimate of potential output. The results are always very close to what you would get by simply extrapolating the trend of labor productivity.

The post is based on various CBO documents on their forecasting methodology, mainly this one, this one and this one, and on the relevant sections of the most recent Budget and Economic Outlook. It’s also much too long, mainly negative critique, and basically unnecessary to the larger argument I’m developing. Much of the post is devoted to the neoclassical production function (a serious demerit); since I’m far from an expert on it, there’s a nontrivial chance of embarrassing mistakes. You can keep reading or not.

The CBO produces annual of estimates potential output for the next ten years, and separate estimates of actual output for next five years; after that actual output is assumed to be at potential. These estimates are in two parts: first, the nonfarm business sector, and second everything else — government, farms, the foreign sector, households, owner-occupied housing, and nonprofits. The potential laborforce is based on population projections from the Social Security Administration and historical laborforce participation rates for different demographic groups. Productivity in the nonfarm business sector, which accounts for about 75 percent of GDP, is estimated using a production function. Productivity in the remaining sectors is based on a simple trend.

Personally I prefer the simple trend approach. Output and employment are observables and productivity is a simple ratio of them. Linear extrapolation of past trends certainly has limitations, and we may be able to do better; but it is straightforward to interpret an gives us a reasonable baseline.

Of course, you have to decide where to start the trend from. For its trend-based variables — including most of the inputs to the NFB production function as well as productivity in the other sectors —the CBO calculates trends from the start of the last full business cycle:

CBO projects potential output by projecting its components: potential labor, capital services, and potential total factor productivity. For those variables, CBO’s projections over the next decade are based mainly on its estimates of trends during the most recent full business cycle and the as-yet-incomplete current cycle.

So last year’s trend variables were based on the period 2001-2014, this year they’re based on 2001-2015, and next year they will be based on the period 2001-2016. But if it turns out that this is the peak of the business cycle, then the following year’s trends will be based on 2008-2017. The fact that the periods for the trends are adjusted this way, rather than smoothly, will tend to produce larger than usual revisions in the first forecast after the NBER announces a new business cycle peak. I’m not sure how much these discontinuities matter in practice.

Other than potential laborforce, almost all the inputs to the model are extrapolations from trend. For the non-business sectors, this mostly just means extrapolating labor productivity. For the nonfarm business sector, the estimate of potential labor is then combined with the production function to get an estimate of potential output. Walking through how this is done is interesting illustration of the poor fit between economic theory and economic measurement.

The production function is the standard *Y = A F(L, K)* where *Y *is potential GDP, *L* is the flow of labor services, *K* is the flow of capital services and *A* is total factor productivity. *F* is a Cobb-Douglas function, meaning *Y = A L ^{(1-a)} K^{a}*. For percent growth in output this is approximated by a linear equation, which is what CBO actually uses:

*%Δ Y = %Δ A + (1-a) %Δ L + a %Δ K*

where %Δ* *refers to the percentage change in the variable. In other words, the growth in potential GDP is equal to the growth of total factor productivity plus some fraction of the growth in the labor input and some fraction of the growth in the capital input, with the two fractions adding up to one.

So to get from the laborforce to potential output, we need the following:

1. The Cobb-Douglas parameter *a;*

2. The utilization rate of the capital stock;

3. The flow of capital services per unit of capital stock in use;

The growth of the capital stock, which requires:

4. The rate of investment;

5. The rate of depreciation;

6. Any other change in the capital stock (including changes in non-produced capital); and

7. Total factor productivity *A*.

Of these, only investment is directly observable. And if we are disaggregating then we need not just one value for each, but many, since their behavior is presumably different across industries. [1]

CBO’s approach is, first, to set *a *at 0.3 by assumption. This is justified by the assumption that factor incomes are equal to marginal product, plus the “fact” that payments to owners of capital account for 30 percent of nonfarm business output and payments to labor for 70 percent. [2]

Second, capital is assumed to always be fully utilized. This assumption seems reasonable at first glance, since we are supposed to be interested in potential; but it’s not consistent with having a NAIRU that varies over time. More importantly, it raises question about what happens when physical means of production cease to be used because demand shifts, firms go bankrupt, etc. Either we say that capital has disappeared, in which case the evolution of the capital stock no longer can be derived from investment and steady depreciation; or we have a large stock of unused capital, so we can no longer assume utilization returns to normal on any reasonable timescale.

Next, CBO estimates the flow of capital services per unit of capital. This isn’t a parameter we usually think much about. The fact that the “*K*” in the production function is a flow while we usually think of “capital” as a stock, isn’t something you have to worry about when you’re just working with formal models. [3] The easiest way to deal with this is to imagine a “corn economy” where some of this year’s crop is consumed and some is planted but there are no lasting assets. Next simplest is to make the flow of capital services just equal to depreciation, so that the contribution to current production is equal to the amount of capital used up; now we are imagining a silo of corn. This second approach is basically what CBO does, though the formula is more complex and takes into account taxes as well as depreciation. The not-crazy justification of this is that total returns should be equal across different kinds of assets — no one will invest in an asset that depreciates rapidly unless they get a higher return to compensate. But of course this assumes away anything that might produce variation in *ex ante* returns — market power, differences in access to finance, etc. — not only across firms but also over time. In particular, variation in interest rates should, in theory, lead to variation in the minimum return for investment, so the expected return will be different for different vintages of capital. It also ignores the rather important fact that *ex post* income and depreciation may depart from expectations. And of course there is the problem that unlike investment, depreciation does not involve an observable flow of money payments. As the CBO dryly notes, “settling on an appropriate definition for economic depreciation … and then measuring it as defined is extremely difficult.”

Finally, total factor productivity is simply a residual, set at whatever level is needed to make output derived from the production function equal actual output when the economy is deemed to be at potential — which means, whenever unemployment is equal to the NAIRU.

There is a great deal you could criticize here. Go through step by step and you can make a long list of things left out, which can be safely ignored when constructing a simple model but not when, as here, you are trying to describe an actual economy.

The more fundamental problem is that, except for investment, none of the terms are directly observable. So the model has too many degrees of freedom. Let’s say investment is steady but we see a decline in productivity growth. Do we interpret that as: a change in the parameter *a*; or a fall in the utilization rate of the capital stock; or a reduction in the flow of capital services per unit of stock (perhaps due to a shift toward longer-lived capital or some other change in composition); or an increase in the depreciation rate (either faster “normal” wearing out or some specific development that leads to existing capital being scrapped); or a decline in total factor productivity? Any of them could produce the same slowdown in output per worker. How do you decide which it is?

The truth is, it doesn’t matter.

No one sensible believes that there is a “real” production function out there, Cobb-Douglas or Lincoln-Douglas or anything else. *K* — as anyone sensible will agree — is not actually a physical quantity. At best these are metaphors, ways of summarizing all the various activities people carry out in order to transform the material an social world in lasting ways and, under capitalism, to generate future flows of money payments. We don’t need to worry too much that “a smaller stock of capital” and “a lower flow of capital services per unit of capital” are observationally indistinguishable; they were always just two equivalent ways of describing the same change in activity. The production function — its defenders will say — is a shorthand, a stripped-down representation of a complex social reality whose value is precisely in its “unrealistic” simplicity. OK, fair enough. But some shorthands are clearer and more useful than others. The objection to the production function is not that it isn’t true — my imagined interlocutor is right — no map gives a “true” picture of the territory. The objection to it is that it’s a pretty poor simplification that replaces one variable with half a dozen.

One way to see the problem here is to look at the role of total factor productivity (TFP). TFP growth is set at whatever level is needed to make potential productivity grow at the same rate as actual productivity over each full business cycle, regardless of what happens to the capital stock. So as far as the CBO’s historical estimates of potential output go, the production function is doing no work at all.

Think of it this way. Suppose you are making a forecast of the US presidential election. You construct a very complicated model, based on polling data, unemployment, frequencies of keyword scraped from google news, the heights of the candidates, combined with your own proprietary algorithm. Let’s call the resulting estimate of the Democrat’s chances *x*. But your forecast is not *x*. It’s *x + y*, where *y* is defined as 50% – *x*. Well, *x + y = x* + 50% – *x* = 50%. It doesn’t matter how smart or stupid your use of data was in constructing *x*, your forecast will always be it’s a tossup.

That’s the role TFP plays in CBO’s estimates of potential output. It makes sure potential comes out the same no matter what values you pick for *a* and the various *K*s.

But even if the production function plays no role in the historical estimates, it could still change the forecasts. In growth form the production function is linear – it’s just a weighted sum of the various inputs. If the inputs are all forecast by extrapolating the trend, then we will get the same result as if we just extrapolated the sum itself, and the production function will be entirely irrelevant. But if they’re not, we will not. So are the inputs in fact set by extrapolating the trend? As far as I can tell, the answer is yes, mostly; but two are not.

As I noted above, CBO’s forecast of the potential labor force is not a simple extrapolation of trend, but is based on the Social Security Administration’s demographic forecasts. This is why CBO was already forecasting below-trend growth of the laborforce before 2009. And the way these forecasts are incorporated into the final estimate of potential GDP does depend on the production function. For the non-business part of the economy, where the CBO estimates productivity as a trend, a one point slowdown in laborforce growth reduces the output forecast by the same one percentage point. But for the nonfarm business sector, where the production function is used, a one point slowdown in laborforce growth reduces output by only 0.7 percentage points. (This is because each unit of labor is now combined with more capital. Robert Waldmann, who perhaps really does believe in production functions, makes the same argument here. [4]) So we see that where the production function does depart from simple extrapolation, it does so in the wrong direction. It suggests that slower laborforce growth should be associated with faster productivity growth, when in fact we’ve seen the two fall together. [5]

The second input that is not simply extrapolated is the federal budget position. Projecting federal spending and revenue is CBO’s core function and incorporating these projections into its economic forecasts is the only substantive argument for using a production function. The reason the federal budget position matters is that the CBO assumes that business investment is determined by national saving, including government saving. Over the long run, each dollar of federal deficit is supposed to reduce business investment by about 75 cents, with effects on potential output in future periods that depend on the specifics of the production function. Other than that, the evolution of the capital stock is governed by linear trends.

As far as I can tell, the size of this negative effect of federal deficits on productivity growth is the only thing that depends on the specifics of the production function. *K* in the function includes a number of distinct kinds of capital goods, each providing a flow of capital services at a different rate. But since the equation for growth is linear — it’s just a weighted sum of the inputs — and since the forecast for all the other inputs is just extrapolated from trend, the result is just the same as if you’d simply used trend growth of labor productivity. That means all the questions you might raise about the production function — does the capital share really reflect its marginal product; are there really constant returns to scale; are returns to capital really equal across industries and over time — are irrelevant. Operationally, what the CBO does is equivalent to estimating labor productivity growth based on trend, and then adding an assumption that productivity falls when the federal deficit rises.

Now, it’s true that CBO’s public materials aren’t sufficiently detailed to fully replicate their estimate. It’s certainly possible I’ve misunderrstood what they’re doing. But the view that the whole business of *A F(K, L)* is just a weirdly roundabout way of saying that government deficits are bad for productivity growth is supported by CBO’s own explanations for why they use it.

At the end of the 2001 paper, which is still the methodological reference given in the most recent estimates, they have a sensible discussion of the pros and cons of using a production function. The big argument against — which they acknowledge quite frankly — is the great difficulty of measuring most of the inputs into the function. “Analysts who thought that measurement error was a severe problem,” they conclude, “would model potential output as a function of labor input and labor productivity.” That is right. On the pro side they offer three arguments. Two of these are essentially circular — if you don’t model output as a function of *L* and *K*, then you can’t talk about the contributions of labor and capital. The one substantive argument for the production function approach is that

it supplies a projection for potential output that is consistent with CBO’s projection for the federal budget. That consistency allows CBO to incorporate the effects of changes in fiscal policy …. Fiscal policy has obvious effects on aggregate demand in the short run… However, fiscal policy will also influence the growth in potential output over the medium term through its effect on national saving and capital accumulation. Because the growth model explicitly includes capital as a factor of production, it captures that effect.

This is the only example they give of the production function giving a different result than a simple extrapolation of trend productivity growth. I think that’s because it’s the only one there is.

The point: It’s a mistake to think that CBO’s use of a production function grants the model any special authority, or makes it a “better” forecast in the sense of being more likely to capture actual developments. It’s also a mistake to think that the production function offers any meaningful protection from forecasters’ tendency to attribute any prolonged slowdown to structural factors. But it’s equally a mistake to spend a lot of time criticizing the specific assumptions embodied in the production function, or to try to reconstruct it with better ones. You might think the production function is an important link between the CBO’s forecasts and rigorous economic theory. Or you might think it’s an unfortunate concession to obscurantist economic ideology. But as far as the actual forecasts go, it doesn’t have any noticeable consequences either way.

For me personally, the main reason to think about this is to be reassured that we don’t need to think about it. It’s perfectly reasonable to discuss the long-term evolution of output with employment and labor productivity as primitives.

[1] As it happens CBO uses seven categories of capital. It’s sort of a weird list: computers, software, communications equipment, other equipment, nonresidential structures, inventories, and land. I think breaking out the new-economy stuff was a response to the model’s serious underprediction of GDP growth in the late 1990s.

[2] Here’s how they explain their decision to just assume a value for *a:*

The model uses some parameters—most notably, the coefficients on labor and capital in the production function—that are imposed rather than econometrically estimated … An alternative method for determining the appropriate values of the coefficients is to use an econometric approach … CBO chose not to follow that approach, in part because of several estimation problems. For instance, in econometric estimates of production functions, a correlation is likely between the explanatory variables (particularly capital) and the error term in the regression…. Another problem is that in order to estimate the coefficients statistically, the capital input must be adjusted (over time) to reflect how intensively the capital stock is used. That adjustment introduces a source of measurement error because of the difficulty in accurately measuring the rate at which the capital stock is used. Since most analysts who employ the econometric approach check to see whether their estimates are reasonable by comparing them with income shares, the payoff to using the econometric approach is small.

The CBO economists are quite clear on the disconnect between the production function and observable economic reality. To estimate the unobservable parameter *a* you’d first need to know several other parameters, which aren’t observable either. Note also the circularity of the whole enterprise. To test whether a factor’s income share really reflect its marginal product, you’d first need to estimate marginal products. And how do you know if your estimate is a good one? You see if it matches with income shares. If, as CBO concludes here, “the payoff is small” to trying to get the right parameters for the production function, it’s hard to see how there can be much payoff to using it at all.

[3] Here’s CBO’s discussion of this issue:

Production theory dictates that the capital input in the production function should measure the flow of capital services available for production. If every capital asset were leased in a rental market every year, estimating the capital input would be relatively simple: rental payments would provide a basis for gauging the value of the capital services provided (analogous to the wages paid to workers per period for their labor). However, most assets are owned, not leased, so the transfer of capital services from owner to user cannot be observed by data-collection agencies.

This is a good statement of the problem, but the “analogy” between rental payments and wages doesn’t make sense. The labor input is not in fact measured by wages, even though they are perfectly observable. It’s measured in hours. This is points up a fundamental incoherence in the production-function story. As Joan Robinson always pointed out, theory needs price and quantity to be two distinct magnitudes, but since capital is only measured as value there is no way even in principle to observe price and quantity independently. One corollary, important in the current context, is that changes in payments to capital are treated axiomatically as reflecting changes in the real flow of capital services, while changes in the flow of payments to labor are at least sometimes regarded as departures from the real value of labor services as a result of a an economy operating above or below capacity. The question of why overheating shows up in the payments to labor and not to capital is very seldom acknowledged, let alone answered.

[4] To be fair, his series of posts criticizing European estimates of potential output are really good — much better than the post you’re currently reading.

[5] This will be the subject of the next post in this series