Efficiency | ECON 510

Pareto Efficiency

Definition

Definition: In an economy with $n$ agents, a utility profile $u^\ast=\left(u_{1}^\ast,\ldots,u_{n}^\ast\right)$ in the set of feasible utilities $\mathcal U$ is said to be efficient, if and only if, when $u_{i}^\ast\lt \tilde u_{i}$ for some agent $i$ and some alternative utility profile $\tilde u$ in $\mathcal U$ then there exists another agent $j$ such that $u_{j}^\ast\gt \tilde u_{j}$.

Examples of inefficient outcomes: the Nash equilibria of our Equilibrium examples.

Public good. In the public good example with equal wealth, $w_1=w_2=w$, the Nash equilibrium is $g_1^e=g_2^e=\frac w3$, so $G^e=\frac{2w}3$ and $u_i^e=\frac{2w}3\cdot\frac{2w}3=\frac{4w^2}9$. If instead each contributes half her wealth, $g_1=g_2=\frac w2$, then $G=w$ and $u_i=\frac w2\cdot w=\frac{w^2}2\gt\frac{4w^2}9$: both are better off.
Peer effects. In the peer-effects example with $a_1=0$, $a_2=1$ and $\lambda=1$, the Nash equilibrium is $x^e=\frac13$ and $y^e=\frac23$, so $u_1^e=u_2^e=1-\frac19-\frac19=\frac79$. If instead both move closer to each other, $x=\frac25$ and $y=\frac35$, then $u_1=u_2=1-\frac4{25}-\frac1{25}=\frac45\gt\frac79$: both are better off.
Contest. In the contest example, the Nash equilibrium is $x^e=y^e=\sqrt{\pi/8}$, so each player wins with probability $\frac12$ and $u_i^e=\frac\pi2-\frac\pi8=\frac{3\pi}8$. If instead both halve their effort, $x=y=\frac12\sqrt{\pi/8}$, each still wins with probability $\frac12$ but pays only $\frac\pi{32}$, so $u_i=\frac\pi2-\frac\pi{32}=\frac{15\pi}{32}\gt\frac{3\pi}8$: both are better off.

Welfare Functions

Definition: A welfare function is a strictly increasing map from the player's payoffs into the real line

$$W:\mathbb{R}^n \longrightarrow \mathbb{R}$$ $$(u_1,u_2,\ldots, u_n) \longmapsto W(u_1,u_2,\ldots, u_n)$$

$$\diff{}{u_i}W \gt 0 \text{ for every player }i=1,2,\ldots, n.$$

An iso-welfare curve is a region in the utility space such that the welfare function is constant, $W(u_1,u_2,\ldots, u_n)=\text{cte.}$

Welfare Functions and Pareto Efficiency

Proposition Efficiency 1

If $\mathbf u^\ast$ in $\mathcal U$ maximizes $W(\mathbf u)$ then $\mathbf u^\ast$ is Pareto efficient.

Proof (by contradiction)

Suppose that $\mathbf u^\ast$ maximizes $W$ over $\mathcal U$ but is not Pareto efficient. Then there are an alternative profile $\tilde{\mathbf u}$ in $\mathcal U$ and an agent $i$ with $u_i^\ast\lt\tilde u_i$ such that no agent $j$ has $u_j^\ast\gt\tilde u_j$, that is,

$$\tilde u_i\gt u_i^\ast\quad\text{and}\quad\tilde u_j\ge u_j^\ast\ \text{ for every agent } j.$$

Since $W$ is strictly increasing in each player's payoff, raising $u_i$ without lowering any other payoff raises welfare: $W(\tilde{\mathbf u})\gt W(\mathbf u^\ast)$. But $\tilde{\mathbf u}$ is in $\mathcal U$, so $\mathbf u^\ast$ does not maximize $W$ over $\mathcal U$, a contradiction. Hence $\mathbf u^\ast$ is Pareto efficient. $\blacksquare$

We also have the partial converse,

Proposition Efficiency 2

If the space of utility profiles that are feasible, $\mathcal U$, is a convex set and $u^\ast$ is Pareto efficient then there are $n$ positive weights $w_i\gt 0$ such that $u^\ast$ maximizes the social welfare function that is a weighted average of the utilities, $W(u)=\sum w_i \cdot u_i$.

Definition: A set $A$ is convex, if for any two points $x$ and $y$ in $A$, we have that the line connecting these points is also in $A$, $\alpha x +(1-\alpha)y$ is in $A$ for any $0\lt\alpha\lt 1$.

Welfare Functions and Pareto Efficiency

Click on an efficient point.

$a$ changes the welfare function $W=a\,u_1+(1-a)\,u_2$, and $c$ changes the iso-welfare curve $W=c$.

$a$

$c$

Efficient Trade

An Example

Recall our pure trade economy with Adam and Eve, where both have utility $u^i(c_0,c_1)=\sqrt{c_0}+\delta\sqrt{c_1}$ and the total endowments are $S_0=109$ and $S_1=29$.

By Proposition Efficiency 2, a Pareto efficient allocation maximizes a weighted welfare function:

$$\max_{\substack{c_0,\,c_1\\ \mathrm{subject\, to} \\[1mm] -c_0\le 0\\[1mm] -c_1\le 0\\[1mm] c_0\le 109\\[1mm] c_1\le 29 }} \alpha\left(\sqrt{c_0}+\delta\sqrt{c_1}\right)+(1-\alpha)\left(\sqrt{109-c_0}+\delta\sqrt{29-c_1}\right)$$

The Lagrangian is:

$$\begin{align*}\mathcal{L}=&\;\alpha\left(\sqrt{c_0}+\delta\sqrt{c_1}\right)+(1-\alpha)\left(\sqrt{109-c_0}+\delta\sqrt{29-c_1}\right)\\ &-\lambda_1(-c_0)-\lambda_2(-c_1)-\lambda_3(c_0-109)-\lambda_4(c_1-29)\end{align*}$$

The first-order conditions are:

$$\begin{align*} \diff{\mathcal{L}}{c_0}&=\frac{\alpha}{2\sqrt{c_0}}-\frac{1-\alpha}{2\sqrt{109-c_0}}+\lambda_1-\lambda_3=0\\[2mm] \diff{\mathcal{L}}{c_1}&=\frac{\alpha\,\delta}{2\sqrt{c_1}}-\frac{(1-\alpha)\delta}{2\sqrt{29-c_1}}+\lambda_2-\lambda_4=0 \end{align*}$$

Efficient Trade

An Example (continuation)

Notice that for interior solutions (constraints not binding, Recipe #0), the FOC simplify to:

$$\begin{align*} \frac{\alpha}{2\sqrt{c_0}}&=\frac{1-\alpha}{2\sqrt{109-c_0}}\\[2mm] \frac{\alpha\,\delta}{2\sqrt{c_1}}&=\frac{(1-\alpha)\delta}{2\sqrt{29-c_1}} \end{align*}$$

We can simplify this expression to:

$$\frac{\dfrac{\alpha\,\delta}{2\sqrt{c_1}}}{\dfrac{\alpha}{2\sqrt{c_0}}}=\frac{\dfrac{(1-\alpha)\delta}{2\sqrt{29-c_1}}}{\dfrac{1-\alpha}{2\sqrt{109-c_0}}}$$

Or

$$\frac{c_0}{c_1}=\frac{109-c_0}{29-c_1}\Leftrightarrow MRS^A=MRS^E$$