Marginal rate of Substitution. For a production function, the Marginal Rate of Substitution $($MRS$)$ of its

inputs is the ratio of its marginal products. To be specific, for a production function $F(L,K),$ the MRS between the

labor $L$ and the capital $K$ is given by

$MR{S}_{LK}=\frac{MPL}{MPK},$ where $MPL=\frac{\text{d}\text{e}\text{l}\text{F}}{\text{d}\text{e}\text{l}\text{L}}$ and $MPK=\frac{\text{d}\text{e}\text{l}\text{F}}{\text{d}\text{e}\text{l}\text{K}}.$

The marginal rate of substitution $,MR{S}_{LK},$ represents how many units of capital should be added to maintain the

same production level when the labor input is reduced by $1$ unit.

Exercise $3.($a$)$ Consider the level curve $($an isoquant curve$)F(L,K)=Q_0.$ Compute the slope of its tangent line, $\frac{dK}{dL},$

in terms of $MR{S}_{LK}.$

$\frac{3}{4}$

$($b$)$ Consider the production function

$F(L,K)=(\frac{1}{4}{L}^{-\frac{1}{4}}+\frac{3}{4}{K}^{-\frac{1}{4}}{)}^{-4}.$

Compute $MR{S}_{LK}$ at $(L,K)=(625,10000).$ Suppose that $(624,$tilde$(K))$ lies on the level curve $F(L,K)=2^{12}.$

Estimate tilde$(K).$ Find the tangent line of the level curve $F(L,K)=2^{12}$ at $(L,K)=(625,10000).$

Exercise $4.($Exercises about the chain rules.$)$ Consider the production function

$F(L,K)=(\frac{1}{4}{L}^{-\frac{1}{4}}+\frac{3}{4}{K}^{-\frac{1}{4}}{)}^{-4}.$

Suppose that due to worker recruiting pace and investment conditions, the inputs $L$ and $K$ vary with time

$t$ and interest rate $r,$ via the following expressions:

$L(t,r)=6t^2+250r,$ and $,K(t,r)=\frac{10t^2}{r}.$

$($a$)$ Calculate the rate of change of output $q$ with respect to $t$ when $t=10$ and $r=0\mathrm{.}1.$ What is the meaning

of this rate?

$($b$)$ Calculate the rate of change of output $q$ with respect to $r$ when $t=10$ and $r=0\mathrm{.}1.$ What is the meaning

of this rate?