There has been a lot of discussion lately about how much a cutting inwards the taxation on uppercase volition growth wages. So I idea I would pose a relevant practise for my readers.
An opened upwards economic scheme has the production constituent y = f(k), where y is output per worker together with k is uppercase per worker. The uppercase stock adjusts thus that the after-tax marginal production of uppercase equals the exogenously given globe involvement charge per unit of measurement r.
r = (1-t)f '(k).
Wages are develop yesteryear the marginal production of labor, which (by Euler's theorem) equals
w = f(k) -f '(k)*k.
We cutting the taxation charge per unit of measurement t. Because f '(k)*k is the taxation base, the static toll of the taxation cutting (per worker) is
dx = -f '(k)*k*dt.
How much volition the taxation cutting growth wages? In particular, what is dw/dx? The commencement individual to e-mail me the right respond volition larn a shout-out on my blog.
By the way, the same calculation would apply to the steady-state of a Ramsey model of a shut economy, where r would live on interpreted equally the charge per unit of measurement of fourth dimension preference.
Bonus question: If in that place are positive externalities to uppercase accumulation, equally suggested by who has been thinking along like lines, was the commencement to e-mail me the right answer:
dw/dx = 1/(1 - t).
So if the taxation charge per unit of measurement is ane third, together with thus every dollar of taxation cutting to uppercase (on a static basis) raises reward yesteryear $1.50.
And if DeLong together with Summers are right that in that place are positive externalities to capital, the lawsuit volition live on larger than $1.50.
Update 2: Influenza A virus subtype H5N1 friend asks to encounter the proof. Here goes. Start amongst my instant equation
w = f(k) -f '(k)*k.
Take the full differential of this equation to get
dw = -k*f "(k)*dk.
This equation relates the alter inwards reward to the alter inwards capital. To uncovering dk, usage my commencement equation
r = (1-t)f '(k).
Take the full differential together with solve for dk to obtain
dk = {f '(k)/[(1-t)*f "(k)]}*dt
This equation relates the alter inwards uppercase to the alter inwards the taxation rate. Substitute this human face into the dw equation to obtain
dw = -[k*f '(k)/(1-t)]*dt.
This equation relates the alter inwards reward to the alter inwards the taxation rate. The 3rd equation inwards the model tin live on rewritten as
dt = dx/[-f '(k)*k].
This equation relates the alter inwards the taxation charge per unit of measurement to the static revenue loss. Substitute this human face into the preceding equation to yield the result
dw/dx = 1/(1 - t).
I must confess that I am amazed at how precisely this turns out. In particular, I produce non accept much intuition for why, for example, the respond does non depend on the production function.
By the way, this derivative (dw/dx) is slightly unlike from what Casey calls the Furman ratio inwards his post. Casey looks at the ratio of the wage alter to the dynamic revenue loss, whereas dw/dx is the ratio of the wage alter to the static revenue loss. We mightiness telephone yell upwards dw/dx the static Furman ratio. The dynamic Furman ratio is typically larger.
Update 3: Alan Auerbach emails me the next comment:
Just to house this final result inwards context, it's a combination of (1) the criterion final result that inwards a modest opened upwards economic scheme project bears 100% of a modest uppercase income tax; together with (2) the fact that starting at a positive taxation rate, the burden of a taxation growth exceeds revenue collection due to the first-order deadweight loss.
An opened upwards economic scheme has the production constituent y = f(k), where y is output per worker together with k is uppercase per worker. The uppercase stock adjusts thus that the after-tax marginal production of uppercase equals the exogenously given globe involvement charge per unit of measurement r.
r = (1-t)f '(k).
Wages are develop yesteryear the marginal production of labor, which (by Euler's theorem) equals
w = f(k) -f '(k)*k.
We cutting the taxation charge per unit of measurement t. Because f '(k)*k is the taxation base, the static toll of the taxation cutting (per worker) is
dx = -f '(k)*k*dt.
How much volition the taxation cutting growth wages? In particular, what is dw/dx? The commencement individual to e-mail me the right respond volition larn a shout-out on my blog.
By the way, the same calculation would apply to the steady-state of a Ramsey model of a shut economy, where r would live on interpreted equally the charge per unit of measurement of fourth dimension preference.
Bonus question: If in that place are positive externalities to uppercase accumulation, equally suggested by who has been thinking along like lines, was the commencement to e-mail me the right answer:
dw/dx = 1/(1 - t).
So if the taxation charge per unit of measurement is ane third, together with thus every dollar of taxation cutting to uppercase (on a static basis) raises reward yesteryear $1.50.
And if DeLong together with Summers are right that in that place are positive externalities to capital, the lawsuit volition live on larger than $1.50.
Update 2: Influenza A virus subtype H5N1 friend asks to encounter the proof. Here goes. Start amongst my instant equation
w = f(k) -f '(k)*k.
Take the full differential of this equation to get
dw = -k*f "(k)*dk.
This equation relates the alter inwards reward to the alter inwards capital. To uncovering dk, usage my commencement equation
r = (1-t)f '(k).
Take the full differential together with solve for dk to obtain
dk = {f '(k)/[(1-t)*f "(k)]}*dt
This equation relates the alter inwards uppercase to the alter inwards the taxation rate. Substitute this human face into the dw equation to obtain
dw = -[k*f '(k)/(1-t)]*dt.
This equation relates the alter inwards reward to the alter inwards the taxation rate. The 3rd equation inwards the model tin live on rewritten as
dt = dx/[-f '(k)*k].
This equation relates the alter inwards the taxation charge per unit of measurement to the static revenue loss. Substitute this human face into the preceding equation to yield the result
dw/dx = 1/(1 - t).
I must confess that I am amazed at how precisely this turns out. In particular, I produce non accept much intuition for why, for example, the respond does non depend on the production function.
By the way, this derivative (dw/dx) is slightly unlike from what Casey calls the Furman ratio inwards his post. Casey looks at the ratio of the wage alter to the dynamic revenue loss, whereas dw/dx is the ratio of the wage alter to the static revenue loss. We mightiness telephone yell upwards dw/dx the static Furman ratio. The dynamic Furman ratio is typically larger.
Update 3: Alan Auerbach emails me the next comment:
Just to house this final result inwards context, it's a combination of (1) the criterion final result that inwards a modest opened upwards economic scheme project bears 100% of a modest uppercase income tax; together with (2) the fact that starting at a positive taxation rate, the burden of a taxation growth exceeds revenue collection due to the first-order deadweight loss.
Most people forget nearly the instant indicate when contention nearly where betwixt 0 together with 100% of a taxation cutting goes to project vs. capital, together with this is exacerbated yesteryear the fact that distribution tables assume revenue alter = burden change, except inwards special cases (such equally where a cutting inwards uppercase gains taxes is presumed to lose petty or no revenue).
Update 4: John Cochrane weighs in.
Update 5: Steven Landsburg weighs in.
Sumber https://gregmankiw.blogspot.com/
Update 4: John Cochrane weighs in.
Update 5: Steven Landsburg weighs in.
