I thought some examples might help.
Imagine your total compensation for your work consists of wage and health care benefits and noting else. Ignore all taxes. Suppose your wage is $1,000 per week and the total premium for your health insurance is $100 per week. Imagine your employer pays $50 toward your insurance and the other $50 is removed from your paycheck.
Question: What’s your total compensation?
Answer: $1,000 (in wage) minus $50 (your contribution toward health insurance) plus a health insurance policy valued at $100. The grand total of compensation is $1,050, though $100 of it is in the form of an insurance policy. (That your employer pays $1,000 in wage and $50 toward insurance is another way to the same answer.)
Now suppose your employer shifts its $50 contribution toward insurance to you and makes no other change in compensation.
Question: Now, what’s your total compensation?
Answer: $1,000 (wage) less $100 (premium), plus $100 in the value of an insurance policy. The total is $1,000, exactly what your employer pays. It went down by $50, exactly the amount shifted to you.
Question: Does that mean if your employer had not shifted the $50 to you, your compensation would still be $1,050?
Answer: No. Total compensation level is set by supply and demand in the labor market. To stay competitive, your employer has to offer the market price (total compensation). That price was once $1,050. Now it is $1,000. How that happens to be split between wage and health insurance depends on other factors. But the total is set by the market (obviously not precisely, week by week, person by person, but on average, over time, and in aggregate). That’s why wage and premium offset each other dollar for dollar even if total compensation changes, as shown above. So,
Final Question: If the labor market equilibrium total compensation is $1,000 but, for some reason, your employer insists on paying the full premium, not requiring any employee contribution, what’s your take home pay?
Answer: $900. The other $100 in total compensation comes to you from the insurance policy your employer buys for you.