# Why Did Consumption TFP Stagnate?

I've been trying to think more about why consumption-sector TFP flatlined from about 1980 forward. What I mentioned in the last post about this was that the fact that TFP was constant does not imply that technology was constant.

I then speculated that technology in the service sector may not have changed much over the last 30 years, partly explaining the lack of consumption productivity growth. By a lack of change, I mean that the service sector has not found a way to produce more services for a given supply of inputs, and/or produced the same amount of service with a decreasing supply of inputs. Take something that is close to a pure service - a back massage. A one-hour back massage in 1980 is almost identical to a one-hour back massage in 2014. You don't get twice (or any other multiple) of the massage in 2014 that you got in 1980. And even if the therapist was capable of reducing back tension in 30 minutes rather than 60, you bought a 60-minute massage.

We often buy time when we buy services, not things. And it isn't so much time as it is attention. And it is very hard to innovate such that you can provide the same amount of attention with fewer inputs (i.e. workers). Because for many services you very specifically want the attention of a specific person for a specific amount of time (the massage). You'd complain to the manager if the therapist tried to massage someone else at the same appointment.

So we don't have to be surprised that even technology in services may not rise much over 30 years. But there were obviously technological changes in the service sector. As several people brought up to me, inventory management and logistics were dramatically changed by IT. This allows a service firm to operate "leaner", with a smaller stock of inventory.

But this kind of technological progress need not show up as "technological change" in doing productivity accounting. That is, what we call "technology" when we do productivity accounting is not the only kind of technology there is. The "technology" in productivity accounting is only the ability to produce more goods using the same inputs, and/or produce the same goods using fewer inputs. It doesn't capture things like a change in the shape of the production function itself, say a shift to using fewer intermediate goods as part of production.

Let's say a firm has a production function of ${Y = AK^{\alpha}L^{\beta}M^{\gamma}}$ where ${A}$ is technology in the productivity sense, ${K}$ is capital, ${L}$ is labor, and ${M}$ is intermediate goods. Productivity accounting could reveal to us a change in ${A}$. But what if an innovation in inventory management/logistics means that ${\gamma}$ changes?

If innovation changes the shape of the production function, rather than the level, then our TFP calculations could go anywhere. Here's an example. Let's say that in 1980 production is ${Y_80 = A_{1980}K_{80}^{.3}L_{80}^{.3}M_{80}^{.4}}$. Innovation in logistics and inventory management makes the production function in 2014 ${Y_14 = A_{2014}K_{14}^{.4}L_{14}^{.4}M_{14}^{.2}}$.

Total factor productivity in 1980 is calculated as

$$TFP_{80} = \frac{Y_{80}}{K_{80}^{.3}L_{80}^{.3}M_{80}^{.4}}$$

and total factor productivity in 2014 is calculated as

$$TFP_{14} = \frac{Y_{14}}{K_{14}^{.4}L_{14}^{.4}M_{14}^{.2}}.$$

TFP in 2014 relative to 1980 (the growth in TFP) is

$$\frac{TFP_{14}}{TFP_{80}} = \frac{Y_{14}}{K_{14}^{.3}L_{14}^{.3}M_{14}^{.4}} \times \frac{K_{80}^{.3}L_{80}^{.3}M_{80}^{.4}}{Y_{80}} \times \frac{M_{14}^{.2}}{K_{14}^{.1}L_{14}^{.1}}$$

which is an unholy mess. The first fraction is TFP in 2014 calculated using the 1980 function. The second fraction is the reciprocal of TFP in 1980, calculated normally. So the first two fractions capture the relative TFP in 2014 to 1980, holding constant the 1980 production function. The last fraction represents the adjustment we have to make because the production function changed.

That last term could literally be anything. Less than one, more than one, more than 100, less than 0.0001. If ${K}$ and ${L}$ rose by a lot while ${M}$ didn't go up much, this will lower TFP in 2014 relative to 1980. It all depends on the actual units used. If I decide to measure ${M}$ in thousands of units rather than hundreds of units, I just made TFP in 2014 go down by a factor of 4 relative to 1980.

Once the production function changes shape, then comparing TFP levels across time becomes nearly impossible. So in that sense TFP could definitely be "getting it wrong" when measuring service-sector productivity. You've got an apples to oranges problem. So if we think that IT innovation really changed the nature of the service-sector production function - meaning that ${\alpha}$, ${\beta}$, and/or ${\gamma}$ changed, then TFP isn't necessarily going to be able to pick that up. It could well be that this looks like flat or even shrinking TFP in the data.

If you'd like, this supports David Beckworth's notion that consumption TFP "doesn't pass the smell test". We've got this intuition that the service sector has changed appreciably over the last 30 years, but it doesn't show up in the TFP measurements. That could be due to this apples to oranges issue, and in fact consumption TFP doesn't reflect accurately the innovations that occurred.

To an ambitious graduate student: document changes in the revenue shares of intermediates in consumption and/or services over time. Correct TFP calculations for these changes, or at least provide some notion of the size of that fudge factor in the above equation.