An economic system where the government or state takes the economic decisions is called a centrally planned economy.
Productive efficiency can be defined as optimal method of producing goods i.e producing goods at the lowest cost. In short run productive efficiency is achieved by producing goods at the lowest point on the short run average cost curve and in the long run it refers to producing goods at the lowest point on the long run average cost curve. Productive efficiency means taking advantage of economies of scale.
Allocative efficiency can be defined as optimal distribution or allocation of goods and services in society. This occurs at an where Price (Marginal Benefit) equals Marginal Cost
A centrally planned economy should be better at allocative efficiency rather than productive efficiency because the main focus of centrally planned economy is fair distribution of limited resources available in the society. So, a centrally planned economy would optimally distribute goods/ services (Allocative Efficiency)
Consider a scenario where all resources of the society are diverted towards producing a socially unwanted goods (example cigarettes) which is being produced efficiently at lowest cost (productive efficiency), it will not benefit the society as society needs more of food and health care products.
That is why a centrally planned economy will always be better at allocative and not productive efficiency.
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