What would happen if total welfare were evenly distributed? Suppose that there is a market with a large number of consumers and producers. Assume that the government knows the reservation prices of  all people in the market. Suppose that the government uses this information to tax all the consumer and producer surplus created in the market (by making everyone pay or receive the reservation price). The government distributes the total welfare equally to everyone. 

What effect would this have on the economy compared to the situation without government intervention?
Firstly, let us fix the mode of taxation as lump-sum taxes. This means that after the redistribution we will still have an efficient allocation assuming there are no externalities.

Next, think about what happens to each individuals' income. Individuals with a high demand elasticity for the good will in general be taxed less than individuals with a low demand elasticity. If the demand elasticity is unrelated to the original income level, this means that this mode of taxation does not necessarily increase equality in income across individuals.

This may not be bad, because the redistribution may still be welfare enhancing if the marginal utility of income of high elasticity individuals is higher than the marginal utility of income of low elasticity individuals. However, we know that marginal utilities are unrelated to demand, since one can always apply any monotone transformation to the utility function and obtain the same demand functions. Thus, both equality of income and welfare effects are unclear.

Finally, we may consider what happens to demand in that particular market. Again, the effects are unclear. This time, it is the presence of income effects which obfuscate the picture. An individual with a positive income transfer may consume more or less of the good, depending on whether it is an inferior good or not.

In this answer, we have thus seen that without making stronger assumptions on preferences, we cannot say much about the effects of such a redistribution even if we assume lump-sum redistribution. If we go into second-best analysis, things may get even more complicated.