Depreciation and Income Shares

I would like now to wade briefly into a debate over the gap between growth in productivity and wages by introducing a bit of modeling fun. It seems clear that— in recent decades— although wage income has grown more slowly than GDP there has been little difference between the growth rate of NDP (GDP net of capital depreciation) and the growth rate of total labor compensation.

On the other hand, inequality of compensation has increased quite a bit— driving a large wedge between pay at the top and pay of the ordinary worker. None of this is news. Somewhat less clear is whether net product is more or less appropriate as a comparison. At first blush, workers still have to produce the whole of output no matter how much investment goes to replacing depreciating capital. It might make sense for labor compensation to rise in step with gross production. But...

Differential confusion and a note on “Standard Neoclassical pedagogy”

In my Comment to Standish and Keen, I asserted that it must be that if $P$ is defined as a function $P\!\left(Q\right)$ then it must be that ${\partial P}/{\partial q_i}$ must be zero because $P$ is not a function of $q_i$, but one of $Q$ alone. Standish and Keen wish to argue that if we hold $Q=\sum_i{q_i}$, then ${\partial P}/{\partial q_i}={dP}/{dQ}$, which is assumed to be negative. It is my contention that this may appear correct at first blush, but this compact notation masks hidden assumptions about the underlying economic model.

I tried to explain this in Section 4.2, but it appears my message was lost. Here, I am going to clarify the notation a bit. When specifying the inputs to a function, I am going to use square brackets. Evaluation of a function will employ parentheses. That is, $f\!\left[x\right]$ should be read “$f$— a function of $x$” while $f\!\left(y\right)$ should be read “$f$— a function of one variable evaluated at $y$.”

For example, though Wilfred Kaplan states in Advanced Calculus (3^rd ed.) that

If $z=f\!\left(x,y\right)$ and $x=g\!\left(u,v\right)$, $y=g\!\left(u,v\right)$, then $$ \frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u} $$

Kaplan also clarifies on the following page that $z\!\left[u,v\right]=f\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$ “is the function whose derivative with respect to $u$ is denoted by ${\partial z}/{\partial u}$.”

That is, when $z$ is evaluated at $\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$ the result is an entirely new function. Kaplan’s chain rule could be written more clearly (if pedantically)

If $z\!\left[x,y\right]=f\!\left(x,y\right)$ and $x\!\left[u,v\right]=g\!\left(u,v\right)$, $y\!\left[u,v\right]=h\!\left(u,v\right)$, then $$ \frac{\partial z^*\!\left[u,v\right]}{\partial u}=\frac{\partial z\!\left[x,y\right]}{\partial x}\frac{\partial x\!\left[u,v\right]}{\partial u}+\frac{\partial z\!\left[x,y\right]}{\partial y}\frac{\partial y\!\left[u,v\right]}{\partial u} $$ where $z^*\!\left[u,v\right]=z\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)=f\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$

Farmer’s Folly: The Sequel

The post below is part of an exchange with Roger Farmer with origins which predate the start of this blog. The ultimate question is should (or even can) the government control asset markets for purposes of managing the rate of inflation. I believe Farmer’s call for such interventions is misguided.

More specifically, Farmer declares that the fall in the stock market in 2008 “caused” the Great Recession. What he seems to mean is that current movements in stock prices can be shown to help predict future movements in unemployment. Unfortunately, there is evidence that the relationship has broken down in recent years. Indeed, Farmer dismisses my concern that his initial model produces poor forecasts by making this very point. Furthermore, it is difficult to distinguish between stock prices as forward-looking, as opposed to forward-causing. Thus, even if the actual association today may be discerned, it is not clear that if, say, the Federal Reserve bought up stocks to keep prices high that such action would actually lead to much reduction in unemployment.

---

In a new working paper (PDF) UCLA’s Roger Farmer responds to last year’s investigation into his claim that declines in the stock market caused the Great Recession.(PDF) Farmer apparently failed to grasp the nature of the critique.

In his original paper, Farmer claimed to have found a stable relationship between the movements in S&P 500 and unemployment rates, and that the data “leads me to stress asset market intervention as a potential policy resolution to the problem of high and persistent unemployment.” In other words, the government should deliberately prop up the stock market as a way of boosting the economy. Farmer appealed to the apparent forecasting power of his model to support his policy preference.

In response we countered that his visual evidence of forecasting power was deceptive– playing off the serial correlation in the data to trick the naïve observer. Rather, his model was not in fact powerful, as was demonstrated by the fact that a simpler model that ignored stock prices produced superior forecasts. Our analysis showed that even if Farmer’s model was correct, movements in the stock market fail to explain– let alone cause– the Great Recession. Finally, we pointed out that the intervention necessary to prevent the recession was implausibly large to be considered serious.

Farmer now:

• Asserts as fact certain properties of his data shown to be consistent with, but unsupported by his analysis.
• Argues that the asserted properties require the use of a particular type of model.
• Suggests that despite using the proper kind of model, his model was “seriously mispecified” by failing to account for a structural break.
• Reasserts that despite this structural break the observed relationship is somehow “structurally stable.”
• Abandons the “correct way to model” and employs pre-break data in an effort to support the uncontroversial position that stock market data may help forecast unemployment.

We agree that his model may have failed due to structural breaks. In fact, post-2008 data may be completely different in structure than data prior, and therefore any model based on previous data is liable to produce forecasts only spuriously related to the post-2008 economy. In any case, we believe this undermines both his assertion that stock prices caused the Great Recession and his proposed policy solution.

-1/12 is a large friggin’ number

Suppose $$S_n\equiv\sum^n_{i=1}{i}=1+2+\cdots+\left(n-1\right)+n$$ You may have figured out that you can rearrange the sum as $$S_n=n+\left(n-1\right)+\cdots+2+1$$ This is interesting because the first terms in each arrangement sum to $n+1$, the second terms also sum to $n-1+2=n+1$. In fact, this is true of all $n$ terms so we find that $$ S_n+S_n=n\left(n+1\right) $$ Fine. But what about $S_\infty$? The sum of all natural numbers, it is obviously bigger than any natural number. Can we more precisely describe $S_\infty$? Perhaps.