Yes, being politically free is the same thing as being economically prosperous, because when a people are free to do as they please, they all at in their own interests. That results in all people getting rich. When a person can act the way they please with their life, they are prosperous, not just economically.
Increasingly, it is really those with the most money and the economically prosperous that are given the most freedom politically. Their speech is deemed to matter more in reality, and their freedom to speech out is greater in that they have less to fear in terms of being punished for speaking a controversial opinion. If they are targeted for political speech, they also have the means to rectify the situation, whereas poorer people may not be able to afford lawyers or court fees.
In many cases, yes, I do believe that being politically free is the same as being economically prosperous. Countries that have the most political freedoms are often the richest. This is not always the case (for example, China) but it is the case in the vast majority of countries. It is good for the economy to have happy people in your country.
In this great land of blood, sweat and liberty we have been freely given, recogonition, by our forefathers (whomever they may be ) the great gift of having recognized divinely given rights, enshrineded in all documents of American constittution .and amoung them the right to vote, independant of any outside coercion.
The two parts of this questions have traditionally gone hand in hand, at least in recent times, but, in reality, there is no reason either should lead to another. Many democracies today are struggling economically today, while dictatorships can prosper (often, at least, if they have massive access to natural resources).