In the OLG framework, assume that the incomes in the (one-commodity) economy are paid to workers in the form of the commodity by the representative private firm, with full employment in the economy. Assume that each worker produces one unit of the commodity (without requiring physical capital). Specifying any other assumptions that you need to make, derive the implications for prices, the return on money, the efficiency of monetary expansion and the optimal rate of monetary expansion for the following cases:

(i) a constant labor force;

(ii) the labor force grows at the rate n;

(iii) the labor force is constant, but its average productivity rises at a constant rate τ over time.