How are workers harmed, from firms' payroll tax revenue aimed at reducing workers' tax?

Question

Source: p 130, Question 6.5, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw
= (page unknown), Question 6.5, Principles of Microeconomics, 4 Ed, 2008, by N Gregory Mankiw

p 125, Case Study: a payroll tax [...] is a tax on the wages that firms pay their workers. ... When a payroll tax is enacted, the wage received by workers falls, and the wage paid by firms rises.

$5.$ A senator wants to raise tax revenue and make workers better off. A staff member proposes raising the payroll tax paid by firms and using part of the extra revenue to reduce the payroll tax paid by workers. Would this accomplish the senator’s goal? Explain.

Given Answer: Reducing the payroll tax paid by firms and using part of the extra revenue to reduce the payroll tax paid by workers would not make workers better off, because the division of the burden of a tax depends on the elasticity of supply and demand and not on who must pay the tax. Because the tax wedge would be larger, it is likely that both firms and workers, who share the burden of any tax, would be worse off.

I reddened the transfer of tax revenue from firms, and yellowed workers' lost wages. Why are workers worsened, by the transfer of tax revenue from firms (the red as above) intended to repay workers' lost wages (yellow)? The workers receive extra money, right? Please advise if I erred, but I think Question 5 can be graphed as above (I modified Figure 8, p 126).

Answer 1

I attached three figures. Figure 1 shows the labor market without the tax. You can see the surplus that the firms and workers receive. They enjoy a surplus because some labor would be hired at even higher wages and some labor would be willing to work at lower wages. The sum of these surpluses gives the 'welfare' of the actors in the labor market.

In figure 2 the government introduces a tax on labor. Whether this is collected from firms or workers does not matter as long as there is full information and all actors are rational. As you can see the surpluses are reduced. Even if you add together the surpluses and the tax collected by the government you do not get the welfare you had in the situation without tax. This is because the equilibrium amount of labor has decreased due to the tax. The welfare loss this creates is called the deadweight loss. This shows that these kind of taxes introduce some inefficiency. However that does not mean that all are worse off. If the government gives all the money to the secretary of Treasury he will be very happy. Similarly, if the government transfers all the tax to the workers (long live the proletariat) their total surplus may be bigger than in the situation without taxes. To see this, compare the area of the green polygons in figure 1 and figure 3. If the deadweight loss is too large this may not be possible.

What you could say is that you cannot compensate both firms and workers at the same time, as welfare is lost.

Answer 2

As the answer says, it depends on the elasticity. The following is a short counter example.

A Short Example

Suppose that the firm is perfectly elastically demanding labor based on

$$L^D (w) = 5 - w$$

And workers supplies labor:

$$L^S (w) = w$$

In equilibrium, we have $L = 2.5, w = 2.5$. Total income = $2.5^2$ Now install a tax:

$$L^D (w) = 5 - (1+t)*w$$

Then, we have that $w = \frac{5}{1+1+t}$ and $L = \frac{5}{1+1+t}$

Hence total labor income is: $wL = \left(\frac{5}{2+t}\right)^2$. We give the payroll taxes to the workers, who then receive $(1+t)*wL$. Let $t=1$.

Total worker income is then $2*\left(\frac{5}{3}\right)^2$ = 50/9. Previously, they got 50/8. They new total income, including payroll taxes, is smaller than before.

So what is the math indicating?

The reason behind this is ignored externalities.

• The payroll tax increases the labor cost, making the firms want to hire less
• The workers do not take into account that they are going to get the payroll taxes out later on, so they do not compensate for the higher labor costs by accepting smaller wages (see last paragraph)
• Given higher costs, we produce less. There is a deadweight loss (DWL) caused by disturbing taxation (production changed), and the only question is: how this DWL is shared. This depends on the elasticities of supply and demand w.r.t. wages. In this scenario, some of the DWL was taken over by the workers. They got partly compensated through taxes, but not by enough

Negative externality

Why doesn't the worker take into account that the more they work, the higher the taxes they receive are going to be?

Basically, because each worker is very very small compared to the total labor force. Think about it this way: Compensating for the increase in labor cost means that the worker has to work for a smaller wage. This will hurt him a lot. The benefits of this $t\cdot wL$ are going to be shared with all other fellow workers. That means that his own share of his "sacrifice" is going to be small. His private gain is too small from doing this. On the other hand, he would receive the positive externality from all the other workers working more. To wit, he would receive some of their tax share - but since the other workers' incentives aren't set correctly either, they don't work "as much" either.

Answer 3

I will set up a naive static/short run model to examine the case (so this post may be a bit long - I will try to dispense with some algebraic steps). I will use convenient functional forms, which are nevertheless consistent with usual assumptions.

FIRMS
There are $i=1,...,n$ identical, price taking firms. In the short run they maximize the objective function

$$A\ln\ell_i - (1+s_f+\xi)w\ell_i\tag{1} $$

where $A$ includes any component of the production function that is fixed in the short run, $\ell_i$ is the amount of labor firm $i$ employs, $s_f$ is the Employer's Social Security Fees (SSF) as a percentage over the "mixed" wage $w$. $\xi$ is a possible change in this percentage, which I include from the start. The concept of "mixed wage" is central in actual labor markets: in most cases bilateral or union negotiations over the wage are carried in terms of the "mixed wage", not in terms of the "take home" wage.

Profit maximizing behavior will lead to market labor demand

$$L^d= n\cdot\frac {A}{(1+s_f+\xi)w} \tag{2}$$

WORKERS
There are $j=1,...,m$ workers, who posses one unit of labor and perform static maximization of the quasilinear utility function

$$U = c + \gamma \ln(1-\ell_j)\;\; s.t\;\; c= (1-s_w+\psi)w\ell_j \tag{3}$$

i.e. there is no consumption-saving decision here. $s_w$ is the "Employee's SSF" and $\psi$ is a possible change of this percentage (a positive $\psi$ implies lowering of the percentage) Utility maximization leads to

$$L^s = m\cdot \frac{(1-s_w+\psi)w - \gamma}{(1-s_w+\psi)w} \tag{4}$$

Assuming that the labor markets clears, we have

$$L^d = L^s \implies n\cdot\frac {A}{(1+s_f+\xi)w} = m\cdot \frac{(1-s_w+\psi)w - \gamma}{(1-s_w+\psi)w}$$

$$\implies (nA/m)\frac {(1-s_w+\psi)}{(1+s_f+\xi)} = (1-s_w+\psi)w - \gamma$$

$$\implies w^* = \frac {(nA/m)}{(1+s_f+\xi)} + \frac {\gamma}{(1-s_w+\psi)} \tag{5}$$

Equation $(5)$ provides the first major conclusion :

If we increase "Employer's SSF" ($\xi >0$), the equilibrium mixed wage will fall. But also, if we decrease "Employee's SSF" ($\psi >0$), the equilibrium mixed wage will also fall.

This is because the "take-home wage" will increase for any given level of mixed wage, and so the labor supply curve will shift outwards in the $(w, L)$ space. Of course this result depends critically on labor-market clearing.

What will happen to individual worker's income?

Dividing $(2)$ by $m$ and using the equilibrium wage, equilibrium labor employed per worker will be

$$\ell_j^* = \frac {nA/m}{(1+s_f+\xi)w^*}$$ and so equilibrium take-home (disposable) labor income per worker will be

$$DI^*=(1-s_w+\psi)w^*\ell_j^* = (1-s_w+\psi)w^*\frac {nA/m}{(1+s_f+\xi)w^*} \tag{6}$$

$$\implies DI^* = \frac {1-s_w+\psi}{1+s_f+\xi }(nA/m) \tag{7}$$

Let's now start to implement's the adviser's idea. We start by the situation where $\xi=\psi = 0$. We want to determine $\xi$ and $\psi$ so that disposable income increases. This requires

$$DI^* \uparrow \implies \frac {1-s_w+\psi}{1+s_f+\xi } > \frac {1-s_w}{1+s_f}$$

$$\rightarrow DI^* \uparrow \implies \psi > \xi \frac {1-s_w}{1+s_f} \tag {8}$$

Since $(1-sw)/(1+s_f) <1 $ we conclude that

We do not need to decrease the Employee's SSF percentage as much as we will increase the Employer's SSF percentage, in order to increase the worker's disposable income. But the decrease should satisfy $(8)$.

But we want also to increase total Social Security Fees collected. Total Social Security Fees are

$$SSF^* = m\cdot\ell_j^*\cdot w^* \cdot (s_f+\xi + s_w - \psi) $$

$$\implies SSF^* = m\cdot \frac {nA/m}{(1+s_f+\xi)w^*} \cdot w^*\cdot (s_f+\xi + s_w - \psi) \tag{9}$$

$$\implies SSF^* = nA\cdot \frac {s_f+\xi + s_w - \psi}{(1+s_f+\xi)} $$

The condition to increase social security fees is

$$SSF^* \uparrow \implies \frac {s_f+\xi + s_w - \psi}{(1+s_f+\xi)} > \frac {s_f+s_w}{(1+s_f)}$$

$$\implies (1+s_f)(\xi-\psi) > \xi(s_f+s_w)$$

$$\implies \xi + s_f\xi - (1+s_f)\psi > s_f\xi + s_w\xi$$

$$\rightarrow SSF^* \uparrow \implies \psi < \xi \frac {1-s_w}{1+s_f} \tag{10}$$

But $(10)$ is the exact opposite condition than $(8)$. So:

There exist no combination of $\xi, \psi$ that will increase worker's disposable income and increase total social security collections.

In other words, the adviser's proposal is infeasible. Of course, I do not claim that this result generalizes to all models -neither do I have at this point a clear view of what are the crucial assumptions on which this infeasibility results rests.