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