Greg's algebra

How much do workers gain from a capital tax cut? This question has reverberated in oped pages and blogosphere, with the usual vitriolat anyone who might even speculate that a dollar in tax cuts could raise wages by more than a dollar. (I vaguely recall more blogosphere discussion which I now can't find, I welcome links from commenters. Greg was too polite to link to it.)

Greg Mankiw posted a really lovely little example of how this is, in fact, a rather natural result.

However, Greg posted it as a little puzzle, and the average reader may not have taken pen and paper out to solve the puzzle. (I will admit I had to take out pen and paper too.) So, here is the answer to Greg's puzzle, with a little of the background fleshed out.

The production technology is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the second equality defines \(f(k)\). For example \(K^{\alpha}L^{1-\alpha}=(K/L)^{\alpha}L\) is of this form. Firms maximize \[ \max\ (1-\tau)\left[ F(K,L)-wL \right] -rK \] \[ \max\ (1-\tau)\left[ f\left( \frac{K}{L}\right) L-wL \right] -rK \]

The firm's first order conditions are \[ \partial/\partial K:(1-\tau)f^{\prime}\left( \frac{K}{L}\right) \frac{1}{L}L=r \] \[ (1-\tau)f^{\prime}\left( k\right) =r \] \[ \partial/\partial L:f\left( \frac{K}{L}\right) -f^{\prime}\left( \frac {K}{L}\right) \frac{K}{L^{2}}L=w \] \[ f(k)-f^{\prime}(k)k=w. \] Total taxes are \[ X=\tau\left[ F(K,L)-wL\right] \] so taxes per worker are \[ x=\tau\left[ f(k)-w\right] =\tau f^{\prime}(k)k. \] Now, let us change the tax rate. The static -- neglecting the change in capital -- cost of the tax change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To find the change in wages, differentiate that first order condition, \[ \frac{dw}{d\tau}=\left[ f^{\prime}(k)-f^{\prime\prime}(k)k-f^{\prime }(k)\right] \frac{dk}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau}. \] To find the change in capital, differentiate that first order condition, and remember the assumption that the return to capital is fixed at \(r\), so \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now use this on the right hand side of the \(dw/d\tau\) equation, \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{f^{\prime}(k)}{(1-\tau )f^{\prime\prime}(k)}=-\frac{kf^{\prime}(k)}{1-\tau}=-\frac{1}{1-\tau}\frac {dx}{d\tau}. \] Dividing, \[ \frac{dw}{dx}=-\frac{1}{1-\tau} \] (Greg has a +, since he defined a negative change in the tax rate.) Each dollar (per worker) of static tax losses raises wages by \(1/(1-\tau)\). It's always greater than one. For \(\tau=1/3\), each dollar of tax cut raises wages by $1.50. A number greater than one does not mean you're a moron, incapable of addition, a stooge of the corporate class, etc.

The example is gorgeous, because all the production function parameters drop out. Usually you have to calibrate things like the parameter \(\alpha\) and then argue about that.

This is not the same as the Laffer curve, which I think causes some of the confusion. The question is not whether one dollar of static tax cut produces more than a dollar of revenue. The question is whether it raises capital enough to produce more than a dollar of wages.

This is also a lovely little example for people who decry math in economics. At a verbal level, who knows? It seems plausible that a $1 tax cut could never raise wages by more than $1. Your head swims. A few lines of algebra later, and the argument is clear. You could never do this verbally.

You might object though that we use the dynamic wage rise over the static tax loss. However, that (at least in my hands) does not lead to so beautiful a result. Also, the political and blogosphere argument is over how much wages will rise relative to the static tax losses. Moreover, the dynamic tax loss is lower. So Greg's calculation is a lower bound on the rise in wages relative to the true loss in tax revenue.

Update: Thanks to a Jason Furman tweet, I was inspired to keep going. Here is the dynamic result: \[ \frac{dx}{d\tau}=kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] \frac{dk}{d\tau} \] We had \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] so \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, and using \(\alpha=1/3\), now $1 in capital tax loss gives rise to $2.00 in extra wages, not just $1.50. Thanks Jason!

Jason goes on to say this "misses much of what matters in tax policy," a point with which I heartily agree. The point of Greg's, and my post, though, was a response to the commentary that anyone that thought that lowering capital taxes could possibly raise wages at all, let alone one for one, let alone more than one for one, was a "liar", evil, stupid, and so forth. Among other things, lowering capital taxes can raise wages, and more than one for one in very simple models. It has lots of other effects which we can discuss. I still like zero, burn the code, burn all the rotten cronyist exemptions, in a revenue neutral reform. But that's for another day.

Update 2: Casey Mulligan's blog is a must read on this issue, both for more intellectual history, and a graphical analysis. Be sure to click Casey's "algebra here" link, or directly hereto see how he does this algebra by machine.

Update 3: in response to a correspondent's request for the idea in words: A corporation invests up to the point that the after-tax return on its investment equals the return investors demand to give the corporation capital. So, let us suppose the tax rate is one half. To give investors a 5% return, the corporation must pursue projects that earn a 10% before tax return. Suppose we eliminate this tax. Now, new projects, that offer a return between 5% and 10% become profitable. The company borrows or issues stock, and buys new machines, factories, etc. These new machines and factories make workers more productive. The firm wants to hire more workers to run the new machines. But there are only so many workers available in the economy, and everyone is doing the same thing. Firms bid against each other for the workers, raising wages. Eventually wages rise, so the firm has the same number of workers, but each one is more productive because they have more machines at their disposal. Lowering corporate taxes raises wages.

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