Greg's Algebra
How much attain workers gain from a majuscule taxation cut? This inquiry has reverberated inwards oped pages as well as blogosphere, amongst the commons vitriol at anyone who mightiness fifty-fifty speculate that a dollar inwards taxation cuts could heighten reward past times to a greater extent than than a dollar. (I vaguely recall to a greater extent than blogosphere give-and-take which I at nowadays can't find, I welcome links from commenters. Greg was every bit good polite to link to it.)
Greg Mankiw posted a actually lovely niggling event of how this is, inwards fact, a rather natural result.
However, Greg posted it every bit a niggling puzzle, as well as the average reader may non direct maintain taken pen as well as newspaper out to solve the puzzle. (I volition acknowledge I had to accept out pen as well as newspaper too.) So, hither is the respond to Greg's puzzle, amongst a niggling of the background fleshed out.
The production applied scientific discipline is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the instant equality defines \(f(k)\). For event \(K^{\alpha}L^{1-\alpha}=(K/L)^{\alpha}L\) 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 get down social club weather condition 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 \[ X=\tau\left[ F(K,L)-wL\right] \] hence taxes per worker are \[ x=\tau\left[ f(k)-w\right] =\tau f^{\prime}(k)k. \] Now, permit us alter the taxation rate. The static -- neglecting the alter inwards majuscule -- terms of the taxation change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To honour the alter inwards wages, differentiate that get down social club 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 honour the alter inwards capital, differentiate that get down social club condition, as well as recall the supposition that the provide to majuscule is fixed at \(r\), hence \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now utilisation this on the correct manus 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 alter inwards the taxation rate.) Each dollar (per worker) of static taxation losses raises reward past times \(1/(1-\tau)\). It's ever greater than one. For \(\tau=1/3\), each dollar of taxation cutting raises reward past times $1.50. Influenza A virus subtype H5N1 number greater than i does non hateful you're a moron, incapable of addition, a stooge of the corporate class, etc.
The event is gorgeous, because all the production utilisation parameters drib out. Usually you lot direct maintain to calibrate things similar the parameter \(\alpha\) as well as hence scrap most that.
This is non the same every bit the Laffer curve, which I mean value causes to a greater extent than or less of the confusion. The inquiry is non whether i dollar of static taxation cutting produces to a greater extent than than a dollar of revenue. The inquiry is whether it raises majuscule plenty to create to a greater extent than than a dollar of wages.
This is also a lovely niggling event for people who decry math inwards economics. At a verbal level, who knows? It seems plausible that a $1 taxation cutting could never heighten reward past times to a greater extent than than $1. Your caput swims. Influenza A virus subtype H5N1 few lines of algebra later, as well as the declaration is clear. You could never attain this verbally.
You mightiness object though that nosotros utilisation the dynamic wage ascent over the static taxation loss. However, that (at to the lowest degree inwards my hands) does non Pb to hence beautiful a result. Also, the political as well as blogosphere declaration is over how much reward volition ascent relative to the static taxation losses. Moreover, the dynamic taxation loss is lower. So Greg's calculation is a lower outflow on the ascent inwards reward relative to the truthful loss inwards taxation revenue.
Update: Thanks to a Jason Furman tweet, I was inspired to continue 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 \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] hence \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, as well as using \(\alpha=1/3\), at nowadays $1 inwards majuscule taxation loss gives ascent to $2.00 inwards extra wages, non simply $1.50. Thanks Jason!
Jason goes on to say this "misses much of what matters inwards taxation policy," a signal amongst which I heartily agree. The signal of Greg's, as well as my post, though, was a answer to the commentary that anyone that persuasion that lowering majuscule taxes could mayhap heighten reward at all, permit solitary i for one, permit solitary to a greater extent than than i for one, was a "liar", evil, stupid, as well as hence forth. Among other things, lowering majuscule taxes tin lav heighten wages, as well as to a greater extent than than i for i inwards real unproblematic models. It has lots of other effects which nosotros tin lav discuss. I withal similar zero, move the code, move all the rotten cronyist exemptions, inwards a revenue neutral reform. But that's for to a greater extent than or less other day.
Update 2: vitriol at anyone who mightiness fifty-fifty speculate that a dollar inwards taxation cuts could heighten reward past times to a greater extent than than a dollar. (I vaguely recall to a greater extent than blogosphere give-and-take which I at nowadays can't find, I welcome links from commenters. Greg was every bit good polite to link to it.)
Casey Mulligan's blog is a must read on this issue, both for to a greater extent than intellectual history, as well as a graphical analysis. Be certain to click Casey's "algebra here" link, or directly here to encounter how he does this algebra past times machine.
Update 3: inwards answer to a correspondent's asking for the persuasion inwards words: Influenza A virus subtype H5N1 corporation invests upwards to the signal that the after-tax provide on its investment equals the provide investors need to give the corporation capital. So, permit us suppose the taxation charge per unit of measurement is i half. To give investors a 5% return, the corporation must pursue projects that earn a 10% earlier taxation return. Suppose nosotros eliminate this tax. Now, novel projects, that offering a provide betwixt 5% as well as 10% perish profitable. The fellowship borrows or issues stock, as well as buys novel machines, factories, etc. These novel machines as well as factories brand workers to a greater extent than productive. The theatre wants to hire to a greater extent than workers to run the novel machines. But at that spot are solely hence many workers available inwards the economy, as well as everyone is doing the same thing. Firms bid against each other for the workers, raising wages. Eventually reward rise, hence the theatre has the same number of workers, but each i is to a greater extent than productive because they direct maintain to a greater extent than machines at their disposal. Lowering corporate taxes raises wages.
Greg Mankiw posted a actually lovely niggling event of how this is, inwards fact, a rather natural result.
However, Greg posted it every bit a niggling puzzle, as well as the average reader may non direct maintain taken pen as well as newspaper out to solve the puzzle. (I volition acknowledge I had to accept out pen as well as newspaper too.) So, hither is the respond to Greg's puzzle, amongst a niggling of the background fleshed out.
The production applied scientific discipline is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the instant equality defines \(f(k)\). For event \(K^{\alpha}L^{1-\alpha}=(K/L)^{\alpha}L\) 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 get down social club weather condition 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 \[ X=\tau\left[ F(K,L)-wL\right] \] hence taxes per worker are \[ x=\tau\left[ f(k)-w\right] =\tau f^{\prime}(k)k. \] Now, permit us alter the taxation rate. The static -- neglecting the alter inwards majuscule -- terms of the taxation change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To honour the alter inwards wages, differentiate that get down social club 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 honour the alter inwards capital, differentiate that get down social club condition, as well as recall the supposition that the provide to majuscule is fixed at \(r\), hence \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now utilisation this on the correct manus 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 alter inwards the taxation rate.) Each dollar (per worker) of static taxation losses raises reward past times \(1/(1-\tau)\). It's ever greater than one. For \(\tau=1/3\), each dollar of taxation cutting raises reward past times $1.50. Influenza A virus subtype H5N1 number greater than i does non hateful you're a moron, incapable of addition, a stooge of the corporate class, etc.
The event is gorgeous, because all the production utilisation parameters drib out. Usually you lot direct maintain to calibrate things similar the parameter \(\alpha\) as well as hence scrap most that.
This is non the same every bit the Laffer curve, which I mean value causes to a greater extent than or less of the confusion. The inquiry is non whether i dollar of static taxation cutting produces to a greater extent than than a dollar of revenue. The inquiry is whether it raises majuscule plenty to create to a greater extent than than a dollar of wages.
This is also a lovely niggling event for people who decry math inwards economics. At a verbal level, who knows? It seems plausible that a $1 taxation cutting could never heighten reward past times to a greater extent than than $1. Your caput swims. Influenza A virus subtype H5N1 few lines of algebra later, as well as the declaration is clear. You could never attain this verbally.
You mightiness object though that nosotros utilisation the dynamic wage ascent over the static taxation loss. However, that (at to the lowest degree inwards my hands) does non Pb to hence beautiful a result. Also, the political as well as blogosphere declaration is over how much reward volition ascent relative to the static taxation losses. Moreover, the dynamic taxation loss is lower. So Greg's calculation is a lower outflow on the ascent inwards reward relative to the truthful loss inwards taxation revenue.
Update: Thanks to a Jason Furman tweet, I was inspired to continue 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 \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] hence \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, as well as using \(\alpha=1/3\), at nowadays $1 inwards majuscule taxation loss gives ascent to $2.00 inwards extra wages, non simply $1.50. Thanks Jason!
Jason goes on to say this "misses much of what matters inwards taxation policy," a signal amongst which I heartily agree. The signal of Greg's, as well as my post, though, was a answer to the commentary that anyone that persuasion that lowering majuscule taxes could mayhap heighten reward at all, permit solitary i for one, permit solitary to a greater extent than than i for one, was a "liar", evil, stupid, as well as hence forth. Among other things, lowering majuscule taxes tin lav heighten wages, as well as to a greater extent than than i for i inwards real unproblematic models. It has lots of other effects which nosotros tin lav discuss. I withal similar zero, move the code, move all the rotten cronyist exemptions, inwards a revenue neutral reform. But that's for to a greater extent than or less other day.
Update 2: vitriol at anyone who mightiness fifty-fifty speculate that a dollar inwards taxation cuts could heighten reward past times to a greater extent than than a dollar. (I vaguely recall to a greater extent than blogosphere give-and-take which I at nowadays can't find, I welcome links from commenters. Greg was every bit good polite to link to it.)
Casey Mulligan's blog is a must read on this issue, both for to a greater extent than intellectual history, as well as a graphical analysis. Be certain to click Casey's "algebra here" link, or directly here to encounter how he does this algebra past times machine.
Update 3: inwards answer to a correspondent's asking for the persuasion inwards words: Influenza A virus subtype H5N1 corporation invests upwards to the signal that the after-tax provide on its investment equals the provide investors need to give the corporation capital. So, permit us suppose the taxation charge per unit of measurement is i half. To give investors a 5% return, the corporation must pursue projects that earn a 10% earlier taxation return. Suppose nosotros eliminate this tax. Now, novel projects, that offering a provide betwixt 5% as well as 10% perish profitable. The fellowship borrows or issues stock, as well as buys novel machines, factories, etc. These novel machines as well as factories brand workers to a greater extent than productive. The theatre wants to hire to a greater extent than workers to run the novel machines. But at that spot are solely hence many workers available inwards the economy, as well as everyone is doing the same thing. Firms bid against each other for the workers, raising wages. Eventually reward rise, hence the theatre has the same number of workers, but each i is to a greater extent than productive because they direct maintain to a greater extent than machines at their disposal. Lowering corporate taxes raises wages.
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