How much hit workers gain from a upper-case alphabetic character revenue enhancement cut? This interrogation has reverberated inward oped pages in addition to blogosphere, amongst the commons vitriol at anyone who mightiness fifty-fifty speculate that a dollar inward revenue enhancement cuts could enhance reward yesteryear 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 picayune illustration of how this is, inward fact, a rather natural result.
However, Greg posted it every bit a picayune puzzle, in addition to the average reader may non own got taken pen in addition to newspaper out to solve the puzzle. (I volition acknowledge I had to own got out pen in addition to newspaper too.) So, hither is the response to Greg's puzzle, amongst a picayune of the background fleshed out.
The production technology scientific discipline is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the minute 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 gild 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 \[ X=\tau\left[ F(K,L)-wL\right] \] thus 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 inward upper-case alphabetic character -- toll of the revenue enhancement change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To uncovering the alter inward wages, differentiate that starting fourth dimension gild 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 uncovering the alter inward capital, differentiate that starting fourth dimension gild condition, in addition to call back the supposition that the supply to upper-case alphabetic character is fixed at \(r\), thus \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now role this on the correct mitt 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 inward the revenue enhancement rate.) Each dollar (per worker) of static revenue enhancement losses raises reward yesteryear \(1/(1-\tau)\). It's ever greater than one. For \(\tau=1/3\), each dollar of revenue enhancement cutting raises reward yesteryear $1.50. H5N1 number greater than i 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 drib out. Usually you lot own got to calibrate things similar the parameter \(\alpha\) in addition to and thus fighting well-nigh that.
This is non the same every bit the Laffer curve, which I intend causes approximately of the confusion. The interrogation is non whether i dollar of static revenue enhancement cutting produces to a greater extent than than a dollar of revenue. The interrogation is whether it raises upper-case alphabetic character plenty to hit to a greater extent than than a dollar of wages.
This is also a lovely picayune illustration for people who decry math inward economics. At a verbal level, who knows? It seems plausible that a $1 revenue enhancement cutting could never enhance reward yesteryear to a greater extent than than $1. Your caput swims. H5N1 few lines of algebra later, in addition to the declaration is clear. You could never hit this verbally.
You mightiness object though that nosotros role the dynamic wage rising over the static revenue enhancement loss. However, that (at to the lowest degree inward my hands) does non Pb to thus beautiful a result. Also, the political in addition to 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 jump on the rising inward reward relative to the truthful loss inward revenue enhancement revenue.
Update: Thanks to a Jason Furman tweet, I was inspired to proceed 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 \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] thus \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, in addition to using \(\alpha=1/3\), at nowadays $1 inward upper-case alphabetic character revenue enhancement loss gives rising to $2.00 inward extra wages, non simply $1.50. Thanks Jason!
Jason goes on to say this "misses much of what matters inward revenue enhancement policy," a indicate amongst which I heartily agree. The indicate of Greg's, in addition to my post, though, was a response to the commentary that anyone that sentiment that lowering upper-case alphabetic character taxes could maybe enhance reward at all, allow lone i for one, allow lone to a greater extent than than i for one, was a "liar", evil, stupid, in addition to thus forth. Among other things, lowering upper-case alphabetic character taxes tin lavatory enhance wages, in addition to to a greater extent than than i for i inward rattling elementary models. It has lots of other effects which nosotros tin lavatory discuss. I yet similar zero, sack the code, sack all the rotten cronyist exemptions, inward a revenue neutral reform. But that's for approximately other day.
Update 2: vitriol at anyone who mightiness fifty-fifty speculate that a dollar inward revenue enhancement cuts could enhance reward yesteryear 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, in addition to a graphical analysis. Be certain to click Casey's "algebra here" link, or directly here to run across how he does this algebra yesteryear machine.
Update 3: inward response to a correspondent's asking for the sentiment inward words: 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 i 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% in addition to 10% kicking the bucket profitable. The fellowship borrows or issues stock, in addition to buys novel machines, factories, etc. These novel machines in addition to factories brand workers to a greater extent than productive. The theater wants to hire to a greater extent than workers to run the novel machines. But in that place are alone thus many workers available inward the economy, in addition to everyone is doing the same thing. Firms bid against each other for the workers, raising wages. Eventually reward rise, thus the theater has the same number of workers, but each i is to a greater extent than productive because they own got to a greater extent than machines at their disposal. Lowering corporate taxes raises wages.
Greg Mankiw posted a actually lovely picayune illustration of how this is, inward fact, a rather natural result.
However, Greg posted it every bit a picayune puzzle, in addition to the average reader may non own got taken pen in addition to newspaper out to solve the puzzle. (I volition acknowledge I had to own got out pen in addition to newspaper too.) So, hither is the response to Greg's puzzle, amongst a picayune of the background fleshed out.
The production technology scientific discipline is \[Y=F(K,L)=f(k)L;k\equiv K/L\] where the minute 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 gild 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 \[ X=\tau\left[ F(K,L)-wL\right] \] thus 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 inward upper-case alphabetic character -- toll of the revenue enhancement change, per worker, is \[ \frac{dx}{d\tau}=f^{\prime}(k)k. \] To uncovering the alter inward wages, differentiate that starting fourth dimension gild 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 uncovering the alter inward capital, differentiate that starting fourth dimension gild condition, in addition to call back the supposition that the supply to upper-case alphabetic character is fixed at \(r\), thus \(dr/d\tau=0\) \[ -f^{\prime}(k)d\tau+(1-\tau)f^{\prime\prime}(k)dk=0 \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)}. \] Now role this on the correct mitt 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 inward the revenue enhancement rate.) Each dollar (per worker) of static revenue enhancement losses raises reward yesteryear \(1/(1-\tau)\). It's ever greater than one. For \(\tau=1/3\), each dollar of revenue enhancement cutting raises reward yesteryear $1.50. H5N1 number greater than i 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 drib out. Usually you lot own got to calibrate things similar the parameter \(\alpha\) in addition to and thus fighting well-nigh that.
This is non the same every bit the Laffer curve, which I intend causes approximately of the confusion. The interrogation is non whether i dollar of static revenue enhancement cutting produces to a greater extent than than a dollar of revenue. The interrogation is whether it raises upper-case alphabetic character plenty to hit to a greater extent than than a dollar of wages.
This is also a lovely picayune illustration for people who decry math inward economics. At a verbal level, who knows? It seems plausible that a $1 revenue enhancement cutting could never enhance reward yesteryear to a greater extent than than $1. Your caput swims. H5N1 few lines of algebra later, in addition to the declaration is clear. You could never hit this verbally.
You mightiness object though that nosotros role the dynamic wage rising over the static revenue enhancement loss. However, that (at to the lowest degree inward my hands) does non Pb to thus beautiful a result. Also, the political in addition to 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 jump on the rising inward reward relative to the truthful loss inward revenue enhancement revenue.
Update: Thanks to a Jason Furman tweet, I was inspired to proceed 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 \[ \frac{dw}{d\tau}=-kf^{\prime\prime}(k)\frac{dk}{d\tau} \] \[ \frac{dk}{d\tau}=\frac{f^{\prime}(k)}{(1-\tau)f^{\prime\prime}(k)} \] thus \[ \frac{dx}{dw}=\frac{kf^{\prime}(k)+\tau\left[ f^{\prime}(k)+kf^{^{\prime \prime}}(k)\right] \frac{dk}{d\tau}}{-kf^{\prime\prime}(k)\frac{dk}{d\tau}} \] \[ \frac{dx}{dw}=-\frac{f^{\prime}(k)(1-\tau)f^{\prime\prime}(k)}{f^{\prime \prime}(k)f^{\prime}(k)}-\frac{\tau\left[ f^{\prime}(k)+kf^{^{\prime\prime} }(k)\right] }{kf^{\prime\prime}(k)} \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{f^{\prime}(k)}{kf^{\prime\prime }(k)}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{\alpha k^{\alpha-1}}{\alpha (\alpha-1)k^{\alpha-1}}\right] \] \[ \frac{dx}{dw}=-(1-\tau)-\tau\left[ 1+\frac{1}{\alpha-1}\right] \] \[ \frac{dx}{dw}=\frac{\left( \alpha-1\right) \left( \tau-1\right) -\tau\alpha}{\alpha-1} \] \[ \frac{dx}{dw}=-\frac{1-\tau-\alpha}{1-\alpha} \] Inverting, in addition to using \(\alpha=1/3\), at nowadays $1 inward upper-case alphabetic character revenue enhancement loss gives rising to $2.00 inward extra wages, non simply $1.50. Thanks Jason!
Jason goes on to say this "misses much of what matters inward revenue enhancement policy," a indicate amongst which I heartily agree. The indicate of Greg's, in addition to my post, though, was a response to the commentary that anyone that sentiment that lowering upper-case alphabetic character taxes could maybe enhance reward at all, allow lone i for one, allow lone to a greater extent than than i for one, was a "liar", evil, stupid, in addition to thus forth. Among other things, lowering upper-case alphabetic character taxes tin lavatory enhance wages, in addition to to a greater extent than than i for i inward rattling elementary models. It has lots of other effects which nosotros tin lavatory discuss. I yet similar zero, sack the code, sack all the rotten cronyist exemptions, inward a revenue neutral reform. But that's for approximately other day.
Update 2: vitriol at anyone who mightiness fifty-fifty speculate that a dollar inward revenue enhancement cuts could enhance reward yesteryear 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, in addition to a graphical analysis. Be certain to click Casey's "algebra here" link, or directly here to run across how he does this algebra yesteryear machine.
Update 3: inward response to a correspondent's asking for the sentiment inward words: 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 i 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% in addition to 10% kicking the bucket profitable. The fellowship borrows or issues stock, in addition to buys novel machines, factories, etc. These novel machines in addition to factories brand workers to a greater extent than productive. The theater wants to hire to a greater extent than workers to run the novel machines. But in that place are alone thus many workers available inward the economy, in addition to everyone is doing the same thing. Firms bid against each other for the workers, raising wages. Eventually reward rise, thus the theater has the same number of workers, but each i is to a greater extent than productive because they own got to a greater extent than machines at their disposal. Lowering corporate taxes raises wages.
