Are the marginal conditions for Pareto optimality sufficient?

Question

In introductory microeconomics textbooks, it is argued that the following three conditions are necessary for the Pareto optimality;

1. Exchange Efficiency (MRS must be equal among all individuals)
2. Input Efficiency (MRTS must be equal among all firms)
3. Output Efficiency (MRS must be equal to MRT for all individuals)

Are they sufficient for Pareto optimality? Where can I find a more rigorous treatment of this discussion?

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Following Giskard's suggestion, I consulted MWG and found 16.F helpful. It suggests that under certain conditions, an allocation is Pareto efficient if and only if

1. $\frac{\partial u_i/\partial x_{li}}{\partial u_i/\partial x_{l'i}}=\frac{\partial u_{i'}/\partial x_{li'}}{\partial u_{i'}/\partial x_{l'i'}}$ for all $i,i',l,l'$ $\cdots(1)$
2. $\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}=\frac{\partial F_{j'}/\partial y_{lj'}}{\partial F_{j'}/\partial y_{l'j'}}$ for all $j,j',l,l'$$\cdots(2)$
3. $\frac{\partial u_i/\partial x_{li}}{\partial u_i/\partial x_{l'i}}=\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}$ for for all $i,j,l,l'$$\cdots(3)$

where $F_j$ is such that the production set of firm $j$ is $Y_j=\{y:F_j(y) \le 0\}$.

I still need to work on filling the gap between these expressions and the abovementioned three conditions. I will update once I solve it, but I appreciate any help.

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1. It is immediate that (1) is equivalent to Exchange Efficiency.

2. I assume that when MRTS is discussed it is implicitly assumed that each firm $j$ produces one good ($l_j$). Hence, $F_j(y)=y_{l_j}-f_j(-y_{-l_j})$ for some production function $f_j$. Then MRTS of $l$ for $l'$ ($l,l'\neq l_j$) is $\frac{\partial f_j/\partial y_{lj}}{\partial f_j/\partial y_{l'j}}=\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}$. Thus, (2) implies Input Efficiency. What about the opposite direction?

We probably need to define $MRTS^j_{ll'}$ when either $l$ or $l'$ is $l_j$, the product of firm $j$. If we define $MRTS^j_{l_jl'}=\frac{1}{\partial f_j/\partial y_{l'}}$ and $MRTS^j_{ll_j}=\partial f_j/\partial y_{l}$, then the equivalence holds. Does this extension of the definition make sense? Does the equalization of MRTS under perfect competition still hold with this extended definition?

3. What is the relationship between MRT and $\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}$?

Given (2), it is intuitive that the market MRT is the same as each firm's MRT, $\frac{\partial F_j/\partial y_{lj}}{\partial F_j/\partial y_{l'j}}$, which implies the equivalence. How can I describe it rigorously?

Answer 1

These are all conditions that cover local deviations. Without further assumptions, usually ones involving convexity & closedness of sets and/or concavity & continouity of functions they are not sufficient.
(For a counterexample, see below.)

The usual recommendation for graduate level treatment of this topic is
Microeconomic Theory by Andreu Mas-Colell, Michael Dennis Whinston, Jerry R. Green.

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Counterexample

Consider an economy with just one consumer (or many identical consumers) with $U(x,y) = x + y$. Let us assume that inputs are always used efficiently, and the resulting frontier of the set of production possibilities is $$ \begin{eqnarray*} y = 2 - x & \text{ when } & x>1 \\ \\ \ln y = 1 - x & \text{ when } & x \leq 1. \end{eqnarray*} $$ When production & consumption are $(x,y) = (3/2,1/2)$, we have MRT$=$MRS, but producing $(x,y) = (0,e)$ is also feasible and yields a higher utility since $e > 2$. The trick is that the set of feasible outputs was not concave.

Additional details:
For simplicity's sake we did not deal with inputs here, but it is quite possible to define production functions that would result in such a set of feasible outputs:
Assume that the only input is labor ($L$), there are 2 units available, and
$f_x(L) = L$,
$f_y(L) = L$ when $L<1$, $f_y(L) = e^{L-1}$ when $L\geq1$.