Recently, Scott Sumner wrote a blog post with this chart in it:

I thought it would be interesting to see how well this relationship held over the period that Sumner didn't include in his chart. Here it is:

It's interesting to note that the relationship doesn't look so good when you look at the entire sample in which all of the data is available. This is aside from that fact that the idea that the NGDP/Wage ratio would track unemployment is part of basic neoclassical theory and has nothing to do with wage stickiness.

Start with a simple Cobb-Douglas production function with employment and capital:

$$(1)\: Y_t = F(K_{t-1},L_t) = K_{t-1}^\alpha L_t^{1-\alpha} $$

Assume that the firm maximizes profits, $Y_t - w_t L_t - r_{t-1} K_{t-1}$ and you get the following first order condition for labor:

$$(2)\: w_t = (1-\alpha)\left(\frac{Y_t}{L_t}\right) $$

Dividing by $Y_t$ will give the nominal wage to NGDP ratio (since the nominal wage to NGDP ratio is the same as the real wage to RGDP ratio), which is

$$(3)\: \frac{w_t}{Y_t} = \frac{1-\alpha}{L_t} $$

It's clear from this that, in a simple neoclassical model, the nominal wage to NGDP ratio is expected to be negatively correlated with employment and, therefore, positively correlated with unemployment -- which is coincidentally the exact thing that Scott's chart shows. Variations in the nominal wage to NGDP ratio are not, in fact, vindications of the musical chairs model.

I thought it would be interesting to see how well this relationship held over the period that Sumner didn't include in his chart. Here it is:

It's interesting to note that the relationship doesn't look so good when you look at the entire sample in which all of the data is available. This is aside from that fact that the idea that the NGDP/Wage ratio would track unemployment is part of basic neoclassical theory and has nothing to do with wage stickiness.

Start with a simple Cobb-Douglas production function with employment and capital:

$$(1)\: Y_t = F(K_{t-1},L_t) = K_{t-1}^\alpha L_t^{1-\alpha} $$

Assume that the firm maximizes profits, $Y_t - w_t L_t - r_{t-1} K_{t-1}$ and you get the following first order condition for labor:

$$(2)\: w_t = (1-\alpha)\left(\frac{Y_t}{L_t}\right) $$

Dividing by $Y_t$ will give the nominal wage to NGDP ratio (since the nominal wage to NGDP ratio is the same as the real wage to RGDP ratio), which is

$$(3)\: \frac{w_t}{Y_t} = \frac{1-\alpha}{L_t} $$

It's clear from this that, in a simple neoclassical model, the nominal wage to NGDP ratio is expected to be negatively correlated with employment and, therefore, positively correlated with unemployment -- which is coincidentally the exact thing that Scott's chart shows. Variations in the nominal wage to NGDP ratio are not, in fact, vindications of the musical chairs model.

Nice catch. Simply posting the rest of the available data in response to Sumner's graphs could be a cottage industry ...

ReplyDeleteJohn, I followed you here to some extent (the math, but not all the background info), but it seems like your back and forth with Sumner never had a satisfying resolution, specifically the one located here:

ReplyDeletehttp://www.themoneyillusion.com/?p=31383

(that had the original plot you reproduce above).

He left off suggesting you read his post on the topic of what an RBC criticism of musical chairs would be.

At the point I wrote this, I had so many running arguments with Sumner on his blog that I lost track and abandoned them all. I'll probably jump back to commenting on his blog next times he writes something I find particularly vexing. Since I started commenting on his blog, about 3 months ago, our arguments have rarely come to anything that could be considered closure. He's prety set in his ways when it comes to not believing in liquidity traps, suggesting that the Federal Reserve is at least partially, if not mostly or wholly, complicit in the Great Recession, etc. I challenge him in the hope that he'd at least acknowledge that liquidity traps are theoretically valid, but that seems to be a bit of a pipe dream.

DeleteWhat seems "vexing" to you often seems like a Zen koan to me. When I see you two interact, I get my popcorn and enjoy the show.

DeleteSpeaking of which, are you moving to Japan as a foreign exchange student?

No, not an exchange program.

Delete