Hi, sorry you got a late response you may already know this but for thoose who will resd the thread
Productive efficiency involves firms producing at the lowest possible long run costs
And allocative effieciency represents the most effecient aloocation of scarce rescources for an economy in the sense that for any combination of scarce rescources the production of goods and services that occurs is mot valued by society
Explained simply:
Productive efficeny is just ways a business tries to increase its productivity and lower it production costs (which is how much money it takes to produce a good or service. For eg, if i wanna sell cakes i need to buy materials which cost me money and thats part of my production cost. ) in the long term meaning the future, a good example of a firm acheving this is via the implementation of technoloy, by implenting it, because it costs less than labour force it makes production costs cheaper only in the long term. As the costs of buying it right now is expensive and over time it will get cheaper
Allocative effeciency is pretty much the ways in which rescources that are not unlimited are best used to satisfy the things that are most in demand.
Hope that helps