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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 ...."?

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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 $$w_i=m_i w=m_i\bigg(p\cdot\bar{\omega}+\sum_j p\cdot y^*_j\bigg)=p\cdot m_i\bar{w}+\sum_j m_i p\cdot y_j^*=p\cdot \omega_i+\sum_j\theta_{ij} p\cdot y_j^*.$$

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  • $\begingroup$ okay thanks. Just few clarifying points. When we say, for an allocation and price vector to constitute a PE with transfer, we require there be some wealth distribution such equilibrium occurs. I want to understand the timing of these "assignments". Does the following example correspond to what MWG is defining? Consider an Edgeworth box economy. 2 consumers are born with some apples and oranges. Social planner observes this. Somehow there is an allocation, which social planner deems desirable (MWG Figure 15.B.13). $\endgroup$ – Frank Swanton Feb 8 '18 at 15:45
  • $\begingroup$ To achieve this socially desirable allocation, SP let consumers sell all of their endowments at some price vector p so that their is wealth in their pockets instead of apples and oranges. This is where SP jumps in and construct some tax scheme to transfer wealth from one consumer to another. Once this happens, we attain a PE with transfer. $\endgroup$ – Frank Swanton Feb 8 '18 at 15:47
  • $\begingroup$ But then, couldn't SP always do this? Transfer wealth via tax scheme or just move around apples and oranges such that she enforces any socially desirable outcome? $\endgroup$ – Frank Swanton Feb 8 '18 at 15:48
  • $\begingroup$ I think the part of confusion I am having is, in a typical GE problem, like problem sets, you don't really mess around with the endowed amount of oranges and apples. You just take them as given along with the preferences. So in that case, that is exactly what MWG says at the bottom of p548 : a Walrasian eq'm is a special case of PE with transfers since consumer's wealth level is determined by the initial endowment vector and by profit shares without any FURTHER wealth transfers. $\endgroup$ – Frank Swanton Feb 8 '18 at 15:50
  • $\begingroup$ @FrankSwanton People are not born with apples and oranges. That is an assignment of property rights, a specific institutional setting. This is data about the economy you do not need to figure out whether something is a price equilibrium with transfers. Problems of timing are hard to answer since general equilibrium theory doesn't come with an entirely satisfactory account of how prices come to be. $\endgroup$ – Michael Greinecker Feb 8 '18 at 21:28

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