# 個經HW 5
## Q1
星號題

## Q2

$-\frac{aL^{\rho-1}}{bK^{\rho-1}}$

## Q3
oirginal: $\frac{df}{dL}$ and $\frac{df}{dK}$
After it goes to $xL$ and $xK\ \longrightarrow$ $\frac{df(xL,xK)}{dxL}=\frac{x^gdf(L,K)}{xdL}=x^{g-1}\frac{df(L,K)}{dL}$
Similarly, $\frac{df(xL,xK)}{dxK}=\frac{x^gf(L,K)}{xdK}=x^{g-1}\frac{df(L,K)}{dK}$

## Q4
$\frac{dK}{dL}=-\frac{MP_L}{MP_K}$
又由5.7可知$MP_L和MP_K$都是homogeneous of degree g-1
因此可以知道當scale翻了x倍時$-\frac{MP_L}{MP_K}=\frac{dK}{dL}$不變，因此MRTS只會和$\frac{K}{L}$有關

## Q5
**記得改成 partial**

$x^gf(L,K)=f(xL,xK)\rightarrow gx^{g-1}f(L,K)=x^{g-1}\frac{df(L,K)}{dL}L+x^{g-1}\frac{df(L,K)}{dK}K$
可以得到 $gf(L,K)=\frac{df}{dL}L+\frac{df}{dK}K$
Constant return to scale: $f(L,K)=\frac{df}{dL}L+\frac{df}{dK}K$
$MP_L$是勞工工資，$MP_K$是資本租金

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CH 7
## Q1
We know
$AC(q) = AVC(q) + AFC(q)$
$AFC(q_2) > AFC(q_1)\ \ if \ \ q_2 < q_1$ (Obviously)

and
$\forall q\neq q_1, AVC(q_1) < AVC(q)$
$\forall q \neq q_2, AC(q_2) < AC(q)$
since AVC and AC are both U-shaped

We want to prove $q_1 \leq q_2$

Suppose not => $q_1 > q_2$

by assumption
$AVC(q_2)>AVC(q_1)$
$AC(q_1)>AC(q_2)$

$-AVC(q_1)>-AVC(q_2)$

thus
$AC(q_1)-AVC(q_1) > AC(q_2)-AVC(q_2)$
=> $AFC(q_1) > AFC(q_2)$

Which leads to contradiction.

## Q2, Q3

大於等於跟大於 (resp. 小於) 可以注意一下

## Q4
$F=400$、$AVC=200-6q+0.3q^2$、$AC=200-6q+0.3q^2+\frac{400}{q}$、$MC=200-12q+0.9q^2$、$AFC=\frac{400}{q}$

## Q5