Chp3 (varian) - cost min Flashcards
What is the Euler equation of the cost min problem?
MP1/w1 = MP2/w2
Show that the cost function is increasing in w.
Note that w in this context is the input costs.
Let x and x’ be the cost-minimizing bundles associated with the costs w and w’ where w ≤ w’.
By minimizing
wx ≤ w’x
And
wx’ ≤ w’x
Putting this together yields,
wx≤w’x’
That is
c(w,y) ≤ c(w’y)
as required.
Show that the cost function is H.M 1.
We like to show that if $x$ is the cost-minimizing bundle at $w$, then $x$ also minimizes costs at $tw$, where $t$ is a positive constant.
Suppose that was not the case, so $x’$ is instead the cost-minimizing bundle of $tw.$ This would yield:
twx ≥ twx’
However, this would also imply that
wx ≥ wx’
This would however contradict the fact that $x’$ is the cost-minimizing bundle at $w$.
Hence, multiplying with a positive scaler does not change the composition of the cost-minimizing bundle, thus costs rise by exactly a factor of $t$. That is
c(tw,y) = twx = tc(w,y)
Show that the cost function is concave
Show a diagram with costs $c(w,y)$ on the $y$-axis and input price $w_i$ on the $x$-axis. Then if input costs rise, the costs will rise 1:1 linearly if the firms choose to do nothing at all, i.e., they do not cost-minimize by substituting for other inputs. This would be represented by a positive 45-degree line. If the firm however would cost-minimize when the price $w_i$ increases, then they can not do worse than the 45-degree line. This is represented by a concave function under the 45-degree line. wich at most tangents the 45-line.
Formal approach: See notion
Let $w’’ = \alpha w + (1-\alpha)w’$. Then,
$$
\begin{align}
c(w’‘,y) &= w’‘x(w’‘,y)
&=(\alpha w+(1-\alpha)w’)x(w’‘,y)
&=\big[\alpha wx(w’‘,y)+(1-\alpha)w’x(w’‘,y)
&\geq \alpha w x (w,y) + (1-\alpha)w’x(w’,y)\big]
& = \alpha c(w,y) + (1-\alpha)c(w’,y)
\end{align}
$$
Which states that:
$$
c(\alpha w + (1-\alpha)w’,\bar y) \geq \alpha c(w,\bar y) + (1-\alpha)c(w’‘\bar y)
$$
