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Does the 2nd welfare theorem hold with Leontief preferences? If not, which of the assumptions does not hold?

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Hint:

Second Welfare Theorem: Suppose that for every individual $i = 1,...,N$ the utility function $U^i(c_i)$ is continuous, locally non-satiated, and quasi-concave. If $\left\{c_1^P,...,c_N^P\right\}$ represents a Pareto optimal allocation in which $c_i^P >> 0$ for every $i = 1,...,N$, then there exists a set of prices, $p^E \geq 0$ and $p^E \neq 0$ and a set of transfer payments, $\left\{\tau_1,...,\tau_N\right\}$ such that:

  • $\left\{c_1^P,...,c_N^P\right\}$ is feasible

$$\sum^N_{i=1}c_i^P \leq \sum^N_{i=1}\omega_i^P$$

  • For every individual $i = 1,...,N$, the consumption bundle, $c_i^P$, solves:

$$\max_{c_i} U^i(c_i) \ s.t. \ p^E \cdot c_i = p^E \cdot \omega^i - \tau_i$$

  • and $\sum^n_{i=1}\tau_i = 0$

Or in other words, for any efficient (Pareto) allocation, there exists a price vector such that the price vector and optimal bundle represent a quasi-equilibrium with transfers.

Also note that you need preferences to be convex and locally non-satiated and the production set must be convex. So now what happens when $U_i(c_i) = \min \left\{\frac{c_1^P}{\alpha_1},...,\frac{c_N^P}{\alpha_N}\right\}$? It is indeed continuous, locally non-satiated and quasi-concave.

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