Реферат: Рrofit function

Рrofit function

it has more factors to adjust in the long run than in the short run. This intuitive proposition can be proved rigorously.

For simplicity, we suppose that there is only one output and that the input prices are all fixed. Hence the profit function only depends on the (scalar) price of output. Denote the short-run profit function by ns(p,z) where z is some factor that is fixed in the short run. Let the long-run profit-maximizing demand for this factor be given by z(p) so that the long-run profit function is given by itl(p) = ^s{Pi z{p))- Finally, let p* be some given output price, and let z* = z(p*) be the optimal long-run demand for the 2-factor at p*.

The long-run profits are always at least as large as the short-run profits since the set of factors that can be adjusted in the long run includes the subset of factors that can be adjusted in the short run. It follows that

h(p) = nL(p) - ns(p, 2*) = ns(p, z(p)) - 7TS(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

d2nL(p*) d2ns(p*,z*) >Q

dp2 dp2 ~

Using Hotelling's lemma once more, it follows that

dVLJP*) dys(p*,z*) = d2irL{p*) d2ns(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).