Question

Consider a firm making choices over two inputs, L and K with prices w and r, respectively, and that produces output Q:

(a) What is the MRTS(K,L) for the following production function: Q = 2K²L?

(b) What is the MRTS(K,L) for the following production function: Q = K + 4L^1/2?

Answer 1

So MRTS is all about the various combination of inputs that can be used to produce the same level of output in a production unit, keeping all other economic variables constant, and we know the formula for MRTS it is defined as the negative ratio of change in capital divided by the change in labor.

we can write the formula for MRTS in two different ways that are described below

= - (change in capital/change in labor) = marginal productivity of labor / marginal productivity of capital   

$MRTS\,\,\,\,\,=\,\,\,\,\,-\frac{\Delta K}{\Delta L}\,\,\,\,\,=\,\,\,\,\,\frac{MP_L}{MP_K}$ ............................(1)

Marginal productivity of labor = MPL

Marginal productivity of capital = MPK

(A)

now to find the MRTS we will use the formula that we have written in equation no (1), and to calculate the value of MRTS we will differentiate the given equation, ' Q = 2K2L ' by both with respect to labor and capital to calculate the value of MPL and  MPK   and then we will put the value in equation no. (1)

so to find out the MPL we have to differentiate the given production function equation with respect to 'L'

$\frac{dQ}{dL}\,\,\,\,=\,\,\,\,\frac{d(2K2L)}{dL}\,\,\,\,=\,\,\,\,2K*2\,\,\,\,=\,\,\,\,4K$ .............................(2)

Similarly to find out the MPK we have to differentiate the given production function equation with respect to 'K'

$\frac{dQ}{dK}\,\,\,\,=\,\,\,\,\frac{d(2K2L)}{dK}\,\,\,\,=\,\,\,\,2*2L\,\,\,\,=\,\,\,\,4L$ ................................(3)

on putting the value of equation no (2) and (3) in equation no (1) to get the required MRTS

$MRTS\,\,\,\,\,=\,\,\,\,\,\frac{4K}{4L}\,\,\,\,\,=\,\,\,\,\frac{k}{l}$ ................................(4)

(B)

now again to find the MRTS we will use the formula that we have written in equation no (1), and to calculate the value of MRTS we will differentiate the given equation, ' Q = K + 4L1/2 ' by both with respect to labor and capital to calculate the value of MPL and  MPK   and then we will put the value in equation no. (1)

so to find out the MPL we have to differentiate the given production function equation with respect to 'L'

$\frac{dQ}{dL}\,\,\,\,=\,\,\,\,\frac{d( K + 4L1/2)}{dL}\,\,\,\,=\,\,\,\,0\,\,+\,\,2\,\,\,\,=\,\,\,\,2$ .............................(5)

Similarly to find out the MPK we have to differentiate the given production function equation with respect to 'K'

$\frac{dQ}{dK}\,\,\,\,=\,\,\,\,\frac{d( K + 4L1/2)}{dK}\,\,\,\,=\,\,\,\,1\,\,+\,\,0\,\,\,\,=\,\,\,\,1$ ..........................(6)

on putting the value of equation no (5) and (6) in equation no (1) to get the required MRTS

$MRTS\,\,\,\,\,=\,\,\,\,\,\frac{2}{1}\,\,\,\,\,=\,\,\,\,\frac{2}{1} \,\,\,\,\,=\,\,\,\,\, 2$