# 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??

• I would have thought the diagram was clear enough (at least on the right hand side) where both curves are increasing and the $S$ curve is the higher curve. So the $S$ curve must be steeper upwards when moving right from the tangent (otherwise it would be lower than the $C$ curve) and the $C$ curve must be steeper downwards when moving left (otherwise it would be higher than the $S$ curve) – Henry Jan 7 at 22:57

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) &< 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) &< 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) &< 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} &< \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} &< \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} &< -\frac{\partial S}{\partial p_i} \end{align}

or in other words,

$$\frac{\partial C}{\partial p_i} > \frac{\partial S}{\partial p_i} ~~\mbox{for}~~ p_i < p_i^0 \tag{2}$$

or in plain English: $$S$$ is flatter than $$C$$ for $$p_i < p_i^0$$. You can repeat the same argument for $$p_i > p_i^0$$

• Thanks. Do you have any idea about the economical interpretation of this inequality? – a kd Jan 8 at 9:10
• @akd It is kind of difficult to say without more context, but probably the author is going after some form of elasticity discussion – caverac Jan 8 at 9:20