Labor Income Tax

Macro

$$l^s=w=\frac{1}{1-{\tau }_w}\frac{\beta c}{L-l}⟹L-\frac{\beta c}{(1-{\tau }_w)w}=l^s(c,w,{\tau }_w)$$ $$l^d:w=\frac{(1-\alpha )Y}{l}$$ $$l^s=l^d:l=L-\frac{\beta }{(1-{\tau }_w)}\frac{c}{(1-\alpha )Y}l$$ $$⟹l^{\ast }=\frac{L}{1+\frac{\beta }{(1-\alpha )(1-{\tau }_w)}\frac{c}{Y}}$$

When $\delta =0$, we have $\frac{c}{Y}=1-\gamma$

$$l^{\ast }=\frac{L}{1+\frac{\beta (1-\gamma )}{(1-\alpha )(1-{\tau }_w)}}⟹l^{\ast }({\gamma }_{+},{\tau }_{w-})$$

Details: we have to derive the $\frac{c}{Y}$

$$\dot{K}=I=RK+wl-{\tau }_{w}wl-T-C$$

note that $G=\gamma Y$

$$\dot{K}=Y-\gamma Y-c$$

Steady State:

$$\dot{K}=0⟹C=(1-\gamma )Y\text{when}\delta =0$$

==What if $\delta >0?$ (left as an exercise)==

Different effects of ${\tau }_w$ and $\gamma$

It’s clear that ${\tau }_w$ acts as a contractionary effect, and $\gamma$ acts as an expansionary effect

But the question is, what happens when ${\tau }_w$ and $\gamma$ change together?

• Government Budget: $G={\tau }_{w}wl+T$

But we ignore the lump sum tax? That is, we set $T=0$

$$⟹\gamma Y={\tau }_{w}wl$$

Considering that $Y=wl+rK$, so that the effect of ${\tau }_w$ must dominate the whole effect.

Mathematically,

$$\gamma ={\tau }_w\frac{wl}{Y}=(1-\alpha ){\tau }_w$$

(recall that $w=(1-\alpha )\frac{Y}{l}$)

$${\tau }_w=\frac{\gamma }{1-\alpha }$$ $$⟹\frac{\delta {\tau }_w}{\delta \gamma }=\frac{1}{1-\alpha }>1$$

Eg: If $\alpha =0.5$, then $\frac{\delta {\tau }_w}{\delta \gamma }=2$

In summary, the negative effect (Substitution Effect) would dominate the positive effect (Income Effect).

If the government increases $\gamma$ by $1\%$, then it has to increase ${\tau }_w$ by $2\%$

Recall the equation:

$$l^{\ast }=\frac{L}{1+\frac{\beta (1-\gamma )}{(1-\alpha )(1-{\tau }_w)}}⟹l^{\ast }({\gamma }_{+},{\tau }_{w-})$$

We then try to substitute: ${\tau }_w=\frac{\gamma }{1-\alpha }$ into $l^{\ast }$

We get:

$$l^{\ast }=\frac{L}{1+\beta \frac{1}{1-\frac{\alpha }{1-\gamma }}}⟹l^{\ast }({\gamma }_{-})$$

Policy Implications

1. Recall that when $T>0$, an increase in the government spending ratio $\gamma$ (holding constant ${\tau }_w$) has an expansionary effect on the macroeconomy.
2. When $T=0$, a simultaneous increase in $\gamma$ and ${\tau }_w$ causes a contractionary effect on the macroeconomy.

Exercise:

Derive $l^{\ast }$ for the case of $\delta >0$, Show that $l^{\ast }({\tau }_{w-},{\gamma }_{+})$

Answer:

$$l^{\ast }=\frac{L}{1+\frac{\beta }{(1-\alpha )(1-{\tau }_w)}\left(1-\gamma -\frac{\alpha \delta }{\rho +\delta }\right)}$$

When ${\tau }_w$ and $\gamma$ increase together, $l^{\ast }$ decreases (unless $\alpha \to 0$ or $\rho \to 0$)