Question

The output of an economic system​ Q, subject to two​ inputs, such as labor L and capital​ K, is often modeled by the​ Cobb-Douglas production function Q=cL^a K^b. When a+b=​1, the case is called constant returns to scale. Suppose Q=12,200​, a = 1/6​, b= 5/6​, and c=42. Find the rate of change of capital with respect to labor, dK/dL.

Answer 1

Explanation:

Given

Economic system Q is given by

`Q=cL^aK^b`

also
`a+b=1`

if
`Q=12,200`

`a=(1)/(6)`

`b=(5)/(6)`

`c=42`

substitute these values

`12,200=42* (L)^{(1)/(6)}K^{(5)/(6)}`

`(L)^{(1)/(6)}K^{(5)/(6)}=(12,200)/(42)`

`K^{(5)/(6)}=\frac{12,200}{42(L)^{(1)/(6)}}`

`K=((12,200)/(42))^{(6)/(5)}* (1)/(L^(5))`

differentiate w.r.t to L to get
`(dK)/(dL)`

`(dK)/(dL)=((12,200)/(42))^{(6)/(5)}* (-5)* L^(-6)`

`(dK)/(dL)=-5((12,200)/(42))^{(6)/(5)}* (1)/(L^6)`

`(dK)/(dL)=-(4515.466)/(L^6)`