Liberals tend to believe that they are fairer, however, the conservative and more moral way of fairness in valuing equity, or proportionality, and understanding that everyone is an individual, and that is is important to consider an individual's situation, and meeting an individual's needs as is truly unfair when people are asked to contribute more than they should expect to receive in return (not to say conservatives disapprove of them willing to contribute more), or when people receive more than they contribute. .
For liberals, since they do not see everyone as an individual, their version of fairness reflects that thus they advocate for social justice. To make matters worse as the world is becoming more liberal, people (including adults) are using fairness as an assertion of wounded entitlement. Thus liberals demand that “The rich to pay their fair share" as an entitled to what the rich have earned.  Another myth liberals have is that they believe that Free enterprise is unfair, but true fairness is not redistributing wealth; fairness is rewarding merit. So Free enterprise, like freedom of speech, freedom of assembly, and freedom of religion makes our country fairer, not less. 
Usually, complaints about unfairness have nothing to do with justice, but are simply a reaction to finding out that you have to take responsibility for your life; that you are accountable for your actions; that your choices have consequences; that you have to work for money; that you have to fix something you broke; that you do not get rewards that others earned while you played video games. None of this is unfair. - Charles J. Sykes, 50 Rules Kids Won't Learn in School, page 12