The compromise includes a one-year cut in the payroll tax by 2 percentage points. The tax cut will be entirely in the employees' share. Why do you think they designed the policy in this way? Was it the right choice
From Greg Mankiw's blog describing the agreement between Obama and republicans, which trades a cut in payroll taxes for a cut of taxes for income over $250,000.
Mankiw notes that for most goods making the buyer or seller pay the tax shouldn't matter. However, wages tend not to change fast particularly they tend not to drop (economists call this sticky wages).
In short to summarize Mankiw by cutting the employees' share that means more money in the pocket of anyone who gets a pay check (hey that's me!). If you cut the employers share it means they could hire workers for 2% less than they are paying now or use the savings for investment in new capital. So as I understand it giving the tax break to workers should increase spending faster, while giving it to employers should decrease unemployment faster.
As an economist #2 seems better, as a person keeping 2% more of my income sounds better. So perhaps the best political decision is to give the money to the employee.