# Wealth transfer by Social Planner in GE

I am reading the definition of a price equilibrium with transfer in MWG on page 548, 524-5.

Consider two-consumer exchange economy. If social planner wants to transfer wealth, how does she achieve this?

Two consumers are born with some apples and oranges. So, the social planner lets these consumers sell at the going market place of what they are endowed with. Then, does she collect taxes and transfer wealth in this way?

It seems like this wealth transfer can be achieved also by directly reallocating the apples and oranges distributed among consumers.

I am not exactly getting the terms like "there be some wealth distribution" or "equilibrium with transfers", "wealth transfers". What does it mean to say, "there is an assignment of wealth levels with ...."?

We have an allocation $(x^*,y^*)$ and a price system $p$ forming a price equilibrium with transfers.The assignment of wealth levels $(w_1,\ldots,w_I)$ has to satisfy $$\sum_i w_i=p\cdot\bar{\omega}+\sum_j p\cdot y_j^*.$$ One way to interpret the wealth levels is that the planner hands out coupons that can be traded at existing prices for commodities and chooses the production plans of the economy. But one can interpret the wealth distribution also as a distribution of the initial endowments and of firm shares. Let $w=\sum_i w_i$ and for each consumer $i$, let $m_i=w_i/w$, which is just the share consumer $i$ has in the aggregate wealth. Now give each consumer an endowment $\omega_i$ of $m_i \bar{\omega}$ and for each firm $j$ a share $\theta_{ij}$ equal to $m_i$. For the resulting private ownership economy, $(x^*,y^*)$ and $p$ constitute a Walrasian equilibrium, so one has reduced redistributing wealth to redistributing endowments and shares in firms. To see that everything works out, note that $$w_i=m_i w=m_i\bigg(p\cdot\bar{\omega}+\sum_j p\cdot y^*_j\bigg)=p\cdot m_i\bar{w}+\sum_j m_i p\cdot y_j^*=p\cdot \omega_i+\sum_j\theta_{ij} p\cdot y_j^*.$$