How can I prove that the equilibrum point $D(p)=S(p)$ is pareto efficient?

The definition of pareto efficient: there is no way to make any person better off without hurting anybody else

$D(p)$ is the market demand curve and $S(p)$ is the market supply curve

I´m working with Hal Varian intermediate microeconomics and he proves that any amount less than the equilibrium amount cannot be Pareto efficient :

At any amount of output less than the competitive amount $q^*$ there is someone who is willing to supply an extra unit of the good at a price that is less than the price that someone is willing to pay for an extra unit of the good. If the good were produced and exchanged between these two people at any price between the demand price and the supply price, they would both be made better off. Thus any amount less than the equilibrum amount cannot be Pareto efficient, since there will be at least two people who could be better off

But I´m having trouble proving that the equilibrium point is pareto efficient

I would really appreciate if you can help me with this problem. Any comments, or suggestions would be highly appreciated

  • 1
    $\begingroup$ A general approach to such proofs is by contradiction. Have you tried this? Also, if you know anything about the utility function (does it satisfy local non satiation?) that could be another method. $\endgroup$ – 123 Jun 8 '16 at 3:39

A rough guideline:

You already know the first case. At $q < q_c$ (competitive quantity), we have

$$S(p) > S(p_c) = D(p_c) > D(p)$$

This is assuming you have "normal" (monotonic) assumptions on the supply and demand curves, which are pretty important to this proof. Notice that $S(p) = q_s; D(p) = q_d$ and in this case $q = q_s < q_d$, that is, there is a shortage. At any higher price $p_0 \in (p, p_c]$, there is a Pareto improvement. A new seller will sell at this higher price and there will exist some buyer willing to still buy it.

The other case is analogous. At $q > q_c$, now

$$D(p) > D(p_c) = S(p_c) > S(p)$$

You can use a similar argument as Varian's. At any lower price, there will be a buyer willing to buy the good and a seller still willing to supply the good.

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