Рrofit function (стр. 2 из 2)

h(p) = n_L(p) - n_s(p, 2*) = n_s(p, z(p)) - 7T_S(p, z*) > 0

for all prices p. At the price p* the difference between the short-run and long-run profits is zero, so that h(p) reaches a minimum at p = p*. Hence, the first derivative must vanish at p*. By Hotelling's lemma, we see that the short-run and the long-run net supplies for each good must be equal at p*.

But we can say more. Since p* is in fact a minimum of h(p), the second derivative of h(p) is nonnegative. This means that

d²n_L(p*) d²n_s(p*,z*) _>Q

dp² dp² ~

Using Hotelling's lemma once more, it follows that

dVLJP*) dy_s(p*,z*) ₌d²ir_L{p*) d²n_s(p*,z*) _>Q

dp dp dp² dp² ~

This expression implies that the long-run supply response to a change in price is at least as large as the short-run supply response at z* = z(p*).

Notes

The properties of the profit function were developed by Hotelling (1932), Hicks (1946), and Samuelson (1947).