I don't think so. If I tell my friends "hey this app is cool, we should all buy it" I think that could work. It's only $2, you spend way more than that doing most anything with friends anyway.
If all have to buy a copy, the decision to buy would become a collective one which the chance of buying this app would be even lower.
Further, as this is the first app by the author, there is no track record as a reference.
Hope you understand my reasoning for why social apps should be free at least at the beginning.