Short-run and Long-run cost against the price of the variable input This is an image of page 133 of Microeconomics; Gravell and Rees.In the fourth line there is a statement about the slope of those curves. Can anyone please explain me the reason??
Everything comes from the constraint

$$
C(p, \gamma^0) \leq S(p, \gamma^0, z^0_k) \tag{C.17}
$$

By construction at $p_0$ these two things are equal:

$$
C(p^0, \gamma^0) = S(p^0, \gamma^0, z^0_k) \tag{1}
$$

Consider a very small positive number $\delta p_i$, and note that (I dropped the other components of $p$ since they remain constant)

\begin{align}
C(p_i^0 - \delta p_i^0, \gamma^0) &amp;&lt;  S(p^0_i - \delta p_i^0, \gamma^0, z^0_k) \\ 
C(p_i^0 - \delta p_i^0, \gamma^0) - C(p_i^0, \gamma^0) &amp;&lt;  S(p^0_i - \delta p_i^0, \gamma^0, z^0_k) - C(p_i^0, \gamma^0)\\
C(p_i^0 - \delta p_i^0, \gamma^0) - C(p_i^0, \gamma^0) &amp;&lt;  S(p^0_i - \delta p_i^0, \gamma^0, z^0_k) - S(p_i^0, \gamma^0, z_k^0) \\
\frac{C(p_i^0 - \delta p_i^0, \gamma^0) - C(p_i^0, \gamma^0)}{\delta p_i^0} &amp;&lt;  \frac{S(p^0_i - \delta p_i^0, \gamma^0, z^0_k) - S(p_i^0, \gamma^0, z_k^0)}{\delta p_i^0}\\
\lim_{\delta p_i^0\to 0^+}\frac{C(p_i^0 - \delta p_i^0, \gamma^0) - C(p_i^0, \gamma^0)}{\delta p_i^0} &amp;&lt;  \lim_{\delta p_i^0\to 0^+} \frac{S(p^0_i - \delta p_i^0, \gamma^0, z^0_k) - S(p_i^0, \gamma^0, z_k^0)}{\delta p_i^0} \\
-\frac{\partial C}{\partial p_i} &amp;&lt; -\frac{\partial S}{\partial p_i}
\end{align}

or in other words,

$$
\frac{\partial C}{\partial p_i} &gt; \frac{\partial S}{\partial p_i} ~~\mbox{for}~~ p_i &lt; p_i^0 \tag{2}
$$

or in plain English: $S$ is flatter than $C$ for $p_i &lt; p_i^0$. You can repeat the same argument for $p_i &gt; p_i^0$