The Cabb$-$Douglas production function represents the relationship between two ar mare inputs and the culputs

that can be produced. In its most standard form for production of a single good with two factars, the function is

$q=F(K,L)=A{K}^{}{L}^{}$

where

$L$ is the quantity of labour used $($input variable$).$

$q$ is the quartity of the grod produced $($output$).$

$$ and $$ are positive constants called oulput elasticities of capital and labour, respectively. They describe the

change in the output resulted from a change in capital or labour.

$K$ is the quantity of capital used $($input variable$).$

A is a constant that represents the batal factor productivity, which accounts for the proportion of output not

explained by either capital or labour. It is a messure of economic efficiency.

a$)$ In microeconornics, I$$oquants are curves that contain al combinations of labour and capital which generate the

same level of dutput. In mathernatical language, they are level curves of the furction $q.$ Use Matlab is plot at least

three of these iscquants adding apprapriate labels $($remember$,$ and $$ are positive numbers, and $K$ and $L$ should

realistically be considered non$-$negafive values$).$

b$)$ The marginal productivity of labour $M{P}_L$ is defined as the rate of change of the output with respect to the input

labour. The marginal productivity of capital $M{P}_K$ is defined as the rate of change of the output with respect to the

input capital. Compule $M{P}_L$ and $M{P}_K$ for the function $q.($These functions tell us about the additional autput

oblained by increasing one of the inputs.$)$

c$)$ The marginal rate of technical aubstitution $($MRTS$)$ is defined to be the amourt by which the quantity of one

input has to be reduced when one extra unit of another input is used, so that the oulput remains constant. If we fix arn

culput $q=c,$ we can salve the equation $A{K}^{}{L}^{}=c$ for $K$ and define

$MRTS=\frac{dK}{dL}.$

I$)$ What is the geametric interpretation of this derivative for the graph of the isoquant $q=c?$

II$)$ It tums dut that

$MRTS=\frac{dK}{dL}=-(\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{L}})*(\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{K}}{)}^{-1}$

Find the expression for MRTS in terms of $K$ and $L.$

d$)$ We increase in scale $K$ and $L$ by a factor $r>1,$ that is$,(K,L)|(rK,rL).$

We say that there are

Increasing retums to scale if $F(rK,rL)>rF(K,L)$;

constant returne to scale if $F(rK,rL)=rF(K,L)$;

decreasing returns to scale if $F\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{K}}=0\mathrm{.}41=0\mathrm{.}59K,LA\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{K}}=\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{L}}=0\mathrm{.}2=0\mathrm{.}8K,LA\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{L}}==0\mathrm{.}47=0\mathrm{.}53==0\mathrm{.}72=0\mathrm{.}28,FF(rK,rL).$

Find $in$ what relation $$ and $$ should $beso$ that $F$ exhibits $an$ increasing, constant, and decreasing return $to$ scale.

You can confirm that you are $on$ the right track $by$ checking numerical arsswers $to$ some parts.

$b.$

Compule $\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{K}}$ for $=0\mathrm{.}41$ and $=0\mathrm{.}59$

$(This$ will $be$ a function $ofK,L,$ and the parameler $A)$

$\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{K}}=$

Compute $\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{L}}$ for $=0\mathrm{.}2$ and $=0\mathrm{.}8$

$(This$ will $be$ a function $ofK,L,$ and the parameter $A)$

$\frac{\text{d}\text{e}\text{l}\text{q}}{\text{d}\text{e}\text{l}\text{L}}=$

$B.$

$c.$ Find the $MRTS$ with $=0\mathrm{.}47$ and $=0\mathrm{.}53.$

$MRTS=$

$d.$ Far $=0\mathrm{.}72$ and $=0\mathrm{.}28,F$ exhibits

return $to$ scale.