Greg's Algebra
How much arrive at workers gain from a majuscule revenue enhancement cut? This inquiry has reverberated inwards oped pages as well as blogosphere, alongside the commons vitriol at anyone who powerfulness fifty-fifty speculate that a dollar inwards revenue enhancement 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 besides polite to link to it.)
Greg Mankiw posted a actually lovely piffling illustration of how this is, inwards fact, a rather natural result.
However, Greg posted it equally a piffling puzzle, as well as the average reader may non direct keep taken pen as well as newspaper out to solve the puzzle. (I volition acknowledge I had to direct keep out pen as well as newspaper too.) So, hither is the answer to Greg's puzzle, alongside a piffling of the background fleshed out.
The production technology scientific discipline is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the instant equality defines \(f(k)\). For illustration \(K^{\alpha}L^{1-\alpha}=(K/L)^{\alpha}L\) 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 starting fourth dimension social club weather 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 \[ X=\tau\left[ F(K,L)-wL\right] \] thence taxes per worker are \[ x=\tau\left[ f(k)-w\right] =\tau f^{\prime}(k)k. \] Now, allow us alter the revenue enhancement rate. The static -- neglecting the alter inwards majuscule -- toll of the revenue enhancement change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To detect the alter inwards wages, differentiate that starting fourth dimension 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 detect the alter inwards capital, differentiate that starting fourth dimension social club condition, as well as recollect the supposition that the supply to majuscule is fixed at \(r\), thence \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now purpose 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 revenue enhancement rate.) Each dollar (per worker) of static revenue enhancement losses raises reward past times \(1/(1-\tau)\). It's ever greater than one. For \(\tau=1/3\), each dollar of revenue enhancement cutting raises reward past times $1.50. Influenza A virus subtype H5N1 number greater than ane does non hateful you're a moron, incapable of addition, a stooge of the corporate class, etc.
The illustration is gorgeous, because all the production business office parameters driblet out. Usually you lot direct keep to calibrate things similar the parameter \(\alpha\) as well as and thence combat most that.
This is non the same equally the Laffer curve, which I intend causes about of the confusion. The inquiry is non whether ane dollar of static revenue enhancement cutting produces to a greater extent than than a dollar of revenue. The inquiry is whether it raises majuscule plenty to arrive at to a greater extent than than a dollar of wages.
This is also a lovely piffling illustration for people who decry math inwards economics. At a verbal level, who knows? It seems plausible that a $1 revenue enhancement 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 arrive at this verbally.
You powerfulness object though that nosotros purpose the dynamic wage rising over the static revenue enhancement loss. However, that (at to the lowest degree inwards my hands) does non atomic number 82 to thence beautiful a result. Also, the political as well as blogosphere declaration is over how much reward volition rising relative to the static revenue enhancement losses. Moreover, the dynamic revenue enhancement loss is lower. So Greg's calculation is a lower saltation on the rising inwards reward relative to the truthful loss inwards revenue enhancement revenue.
Update: Thanks to a Jason Furman tweet, I was inspired to decease on 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 \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] thence \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, as well as using \(\alpha=1/3\), at nowadays $1 inwards majuscule revenue enhancement loss gives rising to $2.00 inwards extra wages, non but $1.50. Thanks Jason!
Jason goes on to say this "misses much of what matters inwards revenue enhancement policy," a indicate alongside which I heartily agree. The indicate of Greg's, as well as my post, though, was a response to the commentary that anyone that watch that lowering majuscule taxes could mayhap heighten reward at all, allow lonely ane for one, allow lonely to a greater extent than than ane for one, was a "liar", evil, stupid, as well as thence forth. Among other things, lowering majuscule taxes tin john heighten wages, as well as to a greater extent than than ane for ane inwards real elementary models. It has lots of other effects which nosotros tin john discuss. I all the same similar zero, shipping away the code, shipping away all the rotten cronyist exemptions, inwards a revenue neutral reform. But that's for about other day.
Update 2: vitriol at anyone who powerfulness fifty-fifty speculate that a dollar inwards revenue enhancement 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 besides 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 meet how he does this algebra past times machine.
Update 3: inwards response to a correspondent's asking for the watch inwards words: Influenza A virus subtype H5N1 corporation invests upward to the indicate that the after-tax supply on its investment equals the supply investors need to give the corporation capital. So, allow us suppose the revenue enhancement charge per unit of measurement is ane half. To give investors a 5% return, the corporation must pursue projects that earn a 10% earlier revenue enhancement return. Suppose nosotros eliminate this tax. Now, novel projects, that offering a supply betwixt 5% as well as 10% larn 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 in that place are exclusively thence 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, thence the theatre has the same number of workers, but each ane is to a greater extent than productive because they direct keep to a greater extent than machines at their disposal. Lowering corporate taxes raises wages.
Greg Mankiw posted a actually lovely piffling illustration of how this is, inwards fact, a rather natural result.
However, Greg posted it equally a piffling puzzle, as well as the average reader may non direct keep taken pen as well as newspaper out to solve the puzzle. (I volition acknowledge I had to direct keep out pen as well as newspaper too.) So, hither is the answer to Greg's puzzle, alongside a piffling of the background fleshed out.
The production technology scientific discipline is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the instant equality defines \(f(k)\). For illustration \(K^{\alpha}L^{1-\alpha}=(K/L)^{\alpha}L\) 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 starting fourth dimension social club weather 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 \[ X=\tau\left[ F(K,L)-wL\right] \] thence taxes per worker are \[ x=\tau\left[ f(k)-w\right] =\tau f^{\prime}(k)k. \] Now, allow us alter the revenue enhancement rate. The static -- neglecting the alter inwards majuscule -- toll of the revenue enhancement change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To detect the alter inwards wages, differentiate that starting fourth dimension 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 detect the alter inwards capital, differentiate that starting fourth dimension social club condition, as well as recollect the supposition that the supply to majuscule is fixed at \(r\), thence \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now purpose 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 revenue enhancement rate.) Each dollar (per worker) of static revenue enhancement losses raises reward past times \(1/(1-\tau)\). It's ever greater than one. For \(\tau=1/3\), each dollar of revenue enhancement cutting raises reward past times $1.50. Influenza A virus subtype H5N1 number greater than ane does non hateful you're a moron, incapable of addition, a stooge of the corporate class, etc.
The illustration is gorgeous, because all the production business office parameters driblet out. Usually you lot direct keep to calibrate things similar the parameter \(\alpha\) as well as and thence combat most that.
This is non the same equally the Laffer curve, which I intend causes about of the confusion. The inquiry is non whether ane dollar of static revenue enhancement cutting produces to a greater extent than than a dollar of revenue. The inquiry is whether it raises majuscule plenty to arrive at to a greater extent than than a dollar of wages.
This is also a lovely piffling illustration for people who decry math inwards economics. At a verbal level, who knows? It seems plausible that a $1 revenue enhancement 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 arrive at this verbally.
You powerfulness object though that nosotros purpose the dynamic wage rising over the static revenue enhancement loss. However, that (at to the lowest degree inwards my hands) does non atomic number 82 to thence beautiful a result. Also, the political as well as blogosphere declaration is over how much reward volition rising relative to the static revenue enhancement losses. Moreover, the dynamic revenue enhancement loss is lower. So Greg's calculation is a lower saltation on the rising inwards reward relative to the truthful loss inwards revenue enhancement revenue.
Update: Thanks to a Jason Furman tweet, I was inspired to decease on 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 \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] thence \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, as well as using \(\alpha=1/3\), at nowadays $1 inwards majuscule revenue enhancement loss gives rising to $2.00 inwards extra wages, non but $1.50. Thanks Jason!
Jason goes on to say this "misses much of what matters inwards revenue enhancement policy," a indicate alongside which I heartily agree. The indicate of Greg's, as well as my post, though, was a response to the commentary that anyone that watch that lowering majuscule taxes could mayhap heighten reward at all, allow lonely ane for one, allow lonely to a greater extent than than ane for one, was a "liar", evil, stupid, as well as thence forth. Among other things, lowering majuscule taxes tin john heighten wages, as well as to a greater extent than than ane for ane inwards real elementary models. It has lots of other effects which nosotros tin john discuss. I all the same similar zero, shipping away the code, shipping away all the rotten cronyist exemptions, inwards a revenue neutral reform. But that's for about other day.
Update 2: vitriol at anyone who powerfulness fifty-fifty speculate that a dollar inwards revenue enhancement 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 besides 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 meet how he does this algebra past times machine.
Update 3: inwards response to a correspondent's asking for the watch inwards words: Influenza A virus subtype H5N1 corporation invests upward to the indicate that the after-tax supply on its investment equals the supply investors need to give the corporation capital. So, allow us suppose the revenue enhancement charge per unit of measurement is ane half. To give investors a 5% return, the corporation must pursue projects that earn a 10% earlier revenue enhancement return. Suppose nosotros eliminate this tax. Now, novel projects, that offering a supply betwixt 5% as well as 10% larn 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 in that place are exclusively thence 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, thence the theatre has the same number of workers, but each ane is to a greater extent than productive because they direct keep to a greater extent than machines at their disposal. Lowering corporate taxes raises wages.
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