You just highlighted why doubling the rates might work for many people. If you charge $100/hr, the client may simply assume you can't handle the higher level/harder work. If you potentially can.. then doubling the rate will make the client feel comfortable giving it to you.
The same pattern happens in salaried work, A manager needs to show that the expensive new senior hire was worth the money. So they give the new hire all the good projects and leave the (cheaper) current employees high and dry. Further, management is incentivized to show the new hire's work in a positive light during performance review as they would be admitting that the manager made a mistake otherwise.
Understanding this dynamic will make you happier at work. Sometimes you need to leave so that you can be the great new hire, sometimes you need to demand more money/promotion so that management sees your worth. People inherently assume that you are worth something approximating your salary.
The same pattern happens in salaried work, A manager needs to show that the expensive new senior hire was worth the money. So they give the new hire all the good projects and leave the (cheaper) current employees high and dry. Further, management is incentivized to show the new hire's work in a positive light during performance review as they would be admitting that the manager made a mistake otherwise.
Understanding this dynamic will make you happier at work. Sometimes you need to leave so that you can be the great new hire, sometimes you need to demand more money/promotion so that management sees your worth. People inherently assume that you are worth something approximating your salary.