Total Factor Productivity as a Measure of Welfare

Posted by {"login"=>"dvollrath", "email"=>"[email protected]", "display_name"=>"dvollrath", "first_name"=>"", "last_name"=>""} on September 04, 2014 · 6 mins read

In response to my post regarding innovation and GDP, Areendam Chanda left a comment that hit at a key question:

Interesting – but if innovation frees up resources, what are we doing with the extra resources? To follow Mokyr’s own argument- driverless cars will reduce commuting times. What happens with that freed up time? I am presuming one could argue that it raises welfare by raising leisure time (unless we plan to spend that extra time working in which case it would show up in increased GDP). Furthermore, how would we actually figure out the effectiveness of R&D policy if we cannot come up with a proper measure of welfare (assuming we will no longer trust GDP)?

To recap, I said that innovation need not raise GDP because innovation may lower the input of resources used, rather than raise the amount of output produced. A different way of saying this is: innovation frees up resources. We can put those resources to use producing other goods/services, and then as Areendam says we may still see GDP go up. Or we can let those resources go idle - we can take more leisure time, as Areendam proposes. [By the way, this is also I think what Keynes and others had in mind when they speculated about 15 hour work weeks].

But if GDP is not necessarily correlated with innovation, then as Areendam asks, is there some way to measure the effectiveness of R&D/innovation on welfare? There is. And I'm borrowing this straight from a paper by Basu, Pascali, Shiantarelli, and Serven. The basic idea in their paper is that our residual productivity measure (the Solow residual) is directly related to welfare levels. If this is rising, then so is welfare. And this residual may rise even if GDP is falling because we are using fewer inputs.

I'm going to do severe injustice to their paper, but let me try to lay out the basic idea. Let utility be

$$ U = \ln{Y} - \beta \ln {X} $$

where ${Y}$ is GDP and ${X}$ is our provision of inputs to the market. The negative sign in front of ${\beta}$ means that we dislike providing inputs (working, saving, acquiring skills) to some extent.

But production depends on these inputs, as in

$$ Y = A X^{\alpha} $$

where ${A}$ is a measure of total factor productivity. As Basu et al point out, ${A}$ need not capture technology per se, but simply represents our efficiency at turning inputs ${X}$ into output ${Y}$. Putting this production function into the utility function, we get

$$ U = \ln{A} + (\alpha - \beta) \ln {X}. $$

First, note that utility depends on ${A}$ positively, no matter what. That is, if we get more productive, then welfare increases. Second, the effect of ${X}$ on utility is not clear. If the marginal gain from adding ${X}$ outweighs the utility cost (${\alpha>\beta}$), then we will want to maximize the amount of ${X}$ we provide to the market (save a lot, work a lot, acquire a lot of skills). But if the marginal gain of adding ${X}$ is small (${\alpha<\beta}$), then we'd prefer to provide as little ${X}$ as possible to the market. Note what this second case implies. We could well have ${A}$ rising, but ${X}$ falling, and this would be utility-maximizing. But GDP, ${Y = A X^{\alpha}}$ may go down, go up, or not change.

You can see this perhaps more easily if we assume that things balance completely, or ${\alpha = \beta}$. In that case, we are indifferent to how much input we supply, and utility is just ${\ln{A}}$. In this case, the level of GDP is completely indeterminate, as we can pick any level of ${X}$ and be happy with it. If we choose low ${X}$ (and so low GDP), great, we get to skip school, eat Cheetos, and watch TV. If we choose high ${X}$ (and so high GDP), great, we get lots of cool new stuff, but have to work for it. But welfare is identical no matter what we choose.

The key to welfare is the level of productivity. If that goes up, we are better off regardless of our decision regarding ${X}$. And therefore, in this model, GDP is not necessarily a measure of either welfare or innovation. Only in a very specific case, ${\alpha > \beta}$, is it the case that welfare is maximized by supplying inputs at the highest possible level. But who knows if we are in that case? It could well be that providing inputs is now more painful than it is worth, so as we innovate, and stop using inputs as much, we're getting higher welfare even though GDP is falling.

The upshot is that total factor productivity - A - is what we should be measuring and talking about when it comes to innovation and/or welfare. The level of inputs provided is a choice variable, and so therefore so is GDP. But as a choice variable, it isn't necessarily clear that more of it is better. We might find better things to do with our time than work/train/save if productivity continues to go up.