Information for questions $9-12$

The following are some equations that may be useful for this group of questions. They assume that we graph isoquants and isocost lines with labor on the horizontal axis, and capital on the vertical axis. In all cases, "slope" refers to the absolute value of slopes that may actually be negative.

Marginal Product of Labor: $M{P}_L=\frac{\text{d}\text{e}\text{l}\text{f}(L,K)}{\text{d}\text{e}\text{l}\text{L}}$

Marginal Product of Capital: $M{P}_K=\frac{\text{d}\text{e}\text{l}\text{f}(L,K)}{\text{d}\text{e}\text{l}\text{K}}$

Marginal Rate of Technical Substitution $(=$ slope of the isoquant$)$:

$MRTS=-\frac{K}{L}=\frac{M{P}_L}{M{P}_K}$

Slope of the isocost line: $\frac{w}{r}$

Implication of profit maximization: $\frac{M{P}_K}{r}=\frac{M{P}_L}{w}.$

A firm has production function: $Q=f(L,K)=2L^{\frac{1}{3}}{K}^{\frac{2}{3}},$ where $L$ are units of labor, $K$ are units of capital, and $Q$ is output. Throughout this problem, the firm is using $L=8$ units of labor and $K=1$ unit of capital. The price of labor is $w=1,$ and the price of capital is $r=20.$