Question

given the following production function, calculate the technical rate of substitution and the optimal choice of capital and labor if the production goal is 140 units of output:

q = 15K + 7E

MP_E = 7

MP_K = 15

price of K = $20

price of E = $8

Answer 1

Formula:

Technical rate of substitution(RTS) = MPE/MPK (Assuming K is on y axis and E(Labor) is on axis)

=> Technical rate of substitution = MPE/MPK = 7/15 = 0.47

Note that If K is in x axis and E is on Y axis then RTS = MPK/MPE = 15/7

q = 15K + 7E

Hence K and E are perfect substitutes and firm values K(Capital) 15/7 times over E(Worker). hence as they are perfect substitutes hence If cost of hiring capital(r) is more than 15/7 times cost of hiring worker(w) then he will use only worker and If cost of hiring capital(r) is less than 15/7 times cost of hiring worker(w) then he will use only Capital.

Here cost of hiring capital(r) = 20 and cost of hiring worker(w) = 8 => r = 2.5w

=> r = 2.5w > (15/7)w. Hence using above discussion Firm will use only worker

=> q = 15K + 7E => 140 = 15*0 + 7E => E = 20

Hence Optimal Labor = 20 and Optimal Capital = 0