Fundamental theorems of welfare economics

In principle, there are two commonly found versions of the fundamental theorems, one relating to an exchange economy in which endowments are exogenously given, and one relating to an economy in which production occurs. The production economy is more general and entails additional assumptions. The assumptions are all based on the standard graduate microeconomics textbook.[36]

The fundamental theorems do not generally ensure existence, nor uniqueness of the equilibria.

The first fundamental theorem holds under general conditions.[38] A formal statement is as follows: If preferences are locally nonsatiated, and if ${\displaystyle (\mathbf {X^{*}} ,\mathbf {Y^{*}} ,\mathbf {p} )}$ is a price equilibrium with transfers, then the allocation ${\displaystyle (\mathbf {X^{*}} ,\mathbf {Y^{*}} )}$is Pareto optimal. An equilibrium in this sense either relates to an exchange economy only or presupposes that firms are allocatively and productively efficient, which can be shown to follow from perfectly competitive factor and production markets.[38]

Given a set ${\displaystyle G}$ of types of goods we work in the real vector space over ${\displaystyle G}$, ${\displaystyle \mathbb {R} ^{G}}$ and use boldface for vector valued variables. For instance, if ${\displaystyle G=\lbrace {\text{butter}},{\text{cookies}},{\text{milk}}\rbrace }$ then ${\displaystyle \mathbb {R} ^{G}}$ would be a three dimensional vector space and the vector ${\displaystyle \langle 1,2,3\rangle }$ would represent the bundle of goods containing 1 unit of butter, 2 units of cookies and 3 units of milk.

Suppose that consumer i has wealth ${\displaystyle w_{i}}$ such that ${\displaystyle \Sigma _{i}w_{i}=\mathbf {p} \cdot \mathbf {e} +\Sigma _{j}\mathbf {p} \cdot \mathbf {y_{j}^{*}} }$ where ${\displaystyle \mathbf {e} }$ is the aggregate endowment of goods (i.e. the sum of all consumer and producer endowments) and ${\displaystyle \mathbf {y_{j}^{*}} }$ is the production of firm j.

Preference maximization (from the definition of price equilibrium with transfers) implies (using ${\displaystyle >_{i}}$ to denote the preference relation for consumer i):

if ${\displaystyle \mathbf {x_{i}} >_{i}\mathbf {x_{i}^{*}} }$ then ${\displaystyle \mathbf {p} \cdot \mathbf {x_{i}} >\mathbf {w_{i}} }$

In other words, if a bundle of goods is strictly preferred to ${\displaystyle \mathbf {x_{i}^{*}} }$ it must be unaffordable at price ${\displaystyle \mathbf {p} }$. Local nonsatiation additionally implies:

if ${\displaystyle \mathbf {x_{i}} \geq _{i}\mathbf {x_{i}^{*}} }$ then ${\displaystyle \mathbf {p} \cdot \mathbf {x_{i}} \geq \mathbf {w_{i}} }$

To see why, imagine that ${\displaystyle \mathbf {x_{i}} \geq _{i}\mathbf {x_{i}^{*}} }$ but ${\displaystyle \mathbf {p} \cdot \mathbf {x_{i}} <w_{i}}$. Then by local nonsatiation we could find ${\displaystyle \mathbf {x'_{i}} }$ arbitrarily close to ${\displaystyle \mathbf {x_{i}} }$ (and so still affordable) but which is strictly preferred to ${\displaystyle \mathbf {x_{i}^{*}} }$. But ${\displaystyle \mathbf {x_{i}^{*}} }$ is the result of preference maximization, so this is a contradiction.

An allocation is a pair ${\displaystyle (\mathbf {X} ,\mathbf {Y} )}$ where ${\displaystyle \mathbf {X} \in \Pi _{i\in I}\mathbb {R} ^{G}}$ and ${\displaystyle \mathbf {Y} \in \Pi _{j\in J}\mathbb {R} ^{G}}$, i.e. ${\displaystyle \mathbf {X} }$ is the 'matrix' (allowing potentially infinite rows/columns) whose ith column is the bundle of goods allocated to consumer i and ${\displaystyle \mathbf {Y} }$ is the 'matrix' whose jth column is the production of firm j. We restrict our attention to feasible allocations which are those allocations in which no consumer sells or producer consumes goods which they lack, i.e.,for every good and every consumer that consumers initial endowment plus their net demand must be positive similarly for producers.

Now consider an allocation ${\displaystyle (\mathbf {X} ,\mathbf {Y} )}$ that Pareto dominates ${\displaystyle (\mathbf {X^{*}} ,Y^{*})}$. This means that ${\displaystyle \mathbf {x_{i}} \geq _{i}\mathbf {x_{i}^{*}} }$ for all i and ${\displaystyle \mathbf {x_{i}} >_{i}\mathbf {x_{i}^{*}} }$ for some i. By the above, we know ${\displaystyle \mathbf {p} \cdot \mathbf {x_{i}} \geq w_{i}}$ for all i and ${\displaystyle \mathbf {p} \cdot \mathbf {x_{i}} >w_{i}}$ for some i. Summing, we find:

${\displaystyle \Sigma _{i}\mathbf {p} \cdot \mathbf {x_{i}} >\Sigma _{i}w_{i}=\Sigma _{j}\mathbf {p} \cdot \mathbf {y_{j}^{*}} }$.

Because ${\displaystyle \mathbf {Y^{*}} }$ is profit maximizing, we know ${\displaystyle \Sigma _{j}\mathbf {p} \cdot y_{j}^{*}\geq \Sigma _{j}p\cdot y_{j}}$, so ${\displaystyle \Sigma _{i}\mathbf {p} \cdot \mathbf {x_{i}} >\Sigma _{j}\mathbf {p} \cdot \mathbf {y_{j}} }$. But goods must be conserved so ${\displaystyle \Sigma _{i}\mathbf {x_{i}} >\Sigma _{j}\mathbf {y_{j}} }$. Hence, ${\displaystyle (\mathbf {X} ,\mathbf {Y} )}$ is not feasible. Since all Pareto-dominating allocations are not feasible, ${\displaystyle (\mathbf {X^{*}} ,\mathbf {Y^{*}} )}$ must itself be Pareto optimal.[38]

Note that while the fact that ${\displaystyle \mathbf {Y^{*}} }$ is profit maximizing is simply assumed in the statement of the theorem the result is only useful/interesting to the extent such a profit maximizing allocation of production is possible. Fortunately, for any restriction of the production allocation ${\displaystyle \mathbf {Y^{*}} }$ and price to a closed subset on which the marginal price is bounded away from 0, e.g., any reasonable choice of continuous functions to parameterize possible productions, such a maximum exists. This follows from the fact that the minimal marginal price and finite wealth limits the maximum feasible production (0 limits the minimum) and Tychonoff's theorem ensures the product of these compacts spaces is compact ensuring us a maximum of whatever continuous function we desire exists.

The second theorem formally states that, under the assumptions that every production set ${\displaystyle Y_{j}}$ is convex and every preference relation ${\displaystyle \geq _{i}}$ is convex and locally nonsatiated, any desired Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers.[38] Further assumptions are needed to prove this statement for price equilibria with transfers.

The proof proceeds in two steps: first, we prove that any Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers; then, we give conditions under which a price quasi-equilibrium is also a price equilibrium.

Let us define a price quasi-equilibrium with transfers as an allocation ${\displaystyle (x^{*},y^{*})}$, a price vector p, and a vector of wealth levels w (achieved by lump-sum transfers) with ${\displaystyle \Sigma _{i}w_{i}=p\cdot \omega +\Sigma _{j}p\cdot y_{j}^{*}}$ (where ${\displaystyle \omega }$ is the aggregate endowment of goods and ${\displaystyle y_{j}^{*}}$ is the production of firm j) such that:

i. ${\displaystyle p\cdot y_{j}\leq p\cdot y_{j}^{*}}$ for all ${\displaystyle y_{j}\in Y_{j}}$ (firms maximize profit by producing ${\displaystyle y_{j}^{*}}$)

ii. For all i, if ${\displaystyle x_{i}>_{i}x_{i}^{*}}$ then ${\displaystyle p\cdot x_{i}\geq w_{i}}$ (if ${\displaystyle x_{i}}$ is strictly preferred to ${\displaystyle x_{i}^{*}}$ then it cannot cost less than ${\displaystyle x_{i}^{*}}$)

iii. ${\displaystyle \Sigma _{i}x_{i}^{*}=\omega +\Sigma _{j}y_{j}^{*}}$ (budget constraint satisfied)

The only difference between this definition and the standard definition of a price equilibrium with transfers is in statement (ii). The inequality is weak here (${\displaystyle p\cdot x_{i}\geq w_{i}}$) making it a price quasi-equilibrium. Later we will strengthen this to make a price equilibrium.[38] Define ${\displaystyle V_{i}}$ to be the set of all consumption bundles strictly preferred to ${\displaystyle x_{i}^{*}}$ by consumer i, and let V be the sum of all ${\displaystyle V_{i}}$. ${\displaystyle V_{i}}$ is convex due to the convexity of the preference relation ${\displaystyle \geq _{i}}$. V is convex because every ${\displaystyle V_{i}}$ is convex. Similarly ${\displaystyle Y+\{\omega \}}$, the union of all production sets ${\displaystyle Y_{i}}$ plus the aggregate endowment, is convex because every ${\displaystyle Y_{i}}$ is convex. We also know that the intersection of V and ${\displaystyle Y+\{\omega \}}$ must be empty, because if it were not it would imply there existed a bundle that is strictly preferred to ${\displaystyle (x^{*},y^{*})}$ by everyone and is also affordable. This is ruled out by the Pareto-optimality of ${\displaystyle (x^{*},y^{*})}$.

These two convex, non-intersecting sets allow us to apply the separating hyperplane theorem. This theorem states that there exists a price vector ${\displaystyle p\neq 0}$ and a number r such that ${\displaystyle p\cdot z\geq r}$ for every ${\displaystyle z\in V}$ and ${\displaystyle p\cdot z\leq r}$ for every ${\displaystyle z\in Y+\{\omega \}}$. In other words, there exists a price vector that defines a hyperplane that perfectly separates the two convex sets.

Next we argue that if ${\displaystyle x_{i}\geq _{i}x_{i}^{*}}$ for all i then ${\displaystyle p\cdot (\Sigma _{i}x_{i})\geq r}$. This is due to local nonsatiation: there must be a bundle ${\displaystyle x'_{i}}$ arbitrarily close to ${\displaystyle x_{i}}$ that is strictly preferred to ${\displaystyle x_{i}^{*}}$ and hence part of ${\displaystyle V_{i}}$, so ${\displaystyle p\cdot (\Sigma _{i}x'_{i})\geq r}$. Taking the limit as ${\displaystyle x'_{i}\rightarrow x_{i}}$ does not change the weak inequality, so ${\displaystyle p\cdot (\Sigma _{i}x_{i})\geq r}$ as well. In other words, ${\displaystyle x_{i}}$ is in the closure of V.

Using this relation we see that for ${\displaystyle x_{i}^{*}}$ itself ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})\geq r}$. We also know that ${\displaystyle \Sigma _{i}x_{i}^{*}\in Y+\{\omega \}}$, so ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})\leq r}$ as well. Combining these we find that ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})=r}$. We can use this equation to show that ${\displaystyle (x^{*},y^{*},p)}$ fits the definition of a price quasi-equilibrium with transfers.

Because ${\displaystyle p\cdot (\Sigma _{i}x_{i}^{*})=r}$ and ${\displaystyle \Sigma _{i}x_{i}^{*}=\omega +\Sigma _{j}y_{j}^{*}}$ we know that for any firm j:

${\displaystyle p\cdot (\omega +y_{j}+\Sigma _{h}y_{h}^{*})\leq r=p\cdot (\omega +y_{j}^{*}+\Sigma _{h}y_{h}^{*})}$ for ${\displaystyle h\neq j}$

which implies ${\displaystyle p\cdot y_{j}\leq p\cdot y_{j}^{*}}$. Similarly we know:

${\displaystyle p\cdot (x_{i}+\Sigma _{k}x_{k}^{*})\geq r=p\cdot (x_{i}^{*}+\Sigma _{k}x_{k}^{*})}$ for ${\displaystyle k\neq i}$

which implies ${\displaystyle p\cdot x_{i}\geq p\cdot x_{i}^{*}}$. These two statements, along with the feasibility of the allocation at the Pareto optimum, satisfy the three conditions for a price quasi-equilibrium with transfers supported by wealth levels ${\displaystyle w_{i}=p\cdot x_{i}^{*}}$ for all i.

We now turn to conditions under which a price quasi-equilibrium is also a price equilibrium, in other words, conditions under which the statement "if ${\displaystyle x_{i}>_{i}x_{i}^{*}}$ then ${\displaystyle p\cdot x_{i}\geq w_{i}}$" imples "if ${\displaystyle x_{i}>_{i}x_{i}^{*}}$ then ${\displaystyle p\cdot x_{i}>w_{i}}$". For this to be true we need now to assume that the consumption set ${\displaystyle X_{i}}$ is convex and the preference relation ${\displaystyle \geq _{i}}$ is continuous. Then, if there exists a consumption vector ${\displaystyle x'_{i}}$ such that ${\displaystyle x'_{i}\in X_{i}}$ and ${\displaystyle p\cdot x'_{i}<w_{i}}$, a price quasi-equilibrium is a price equilibrium.

To see why, assume to the contrary ${\displaystyle x_{i}>_{i}x_{i}^{*}}$ and ${\displaystyle p\cdot x_{i}=w_{i}}$, and ${\displaystyle x_{i}}$ exists. Then by the convexity of ${\displaystyle X_{i}}$ we have a bundle ${\displaystyle x''_{i}=\alpha x_{i}+(1-\alpha )x'_{i}\in X_{i}}$ with ${\displaystyle p\cdot x''_{i}<w_{i}}$. By the continuity of ${\displaystyle \geq _{i}}$ for ${\displaystyle \alpha }$ close to 1 we have ${\displaystyle \alpha x_{i}+(1-\alpha )x'_{i}>_{i}x_{i}^{*}}$. This is a contradiction, because this bundle is preferred to ${\displaystyle x_{i}^{*}}$ and costs less than ${\displaystyle w_{i}}$.

Hence, for price quasi-equilibria to be price equilibria it is sufficient that the consumption set be convex, the preference relation to be continuous, and for there always to exist a "cheaper" consumption bundle ${\displaystyle x'_{i}}$. One way to ensure the existence of such a bundle is to require wealth levels ${\displaystyle w_{i}}$ to be strictly positive for all consumers i.[38]

• Convex preferences
• Varian's theorems – a competitive equilibrium is both Pareto-efficient and envy-free.
• General equilibrium theory