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

Answer:

Step-by-step explanation:

Given

Economic system Q is given by

$Q=cL^aK^b$

also $a+b=1$

if $Q=12,200$

$a=\frac{1}{6}$

$b=\frac{5}{6}$

$c=42$

substitute these values

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

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

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

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

differentiate w.r.t to L to get $\frac{dK}{dL}$

$\frac{dK}{dL}=(\frac{12,200}{42})^{\frac{6}{5}}\times (-5)\times L^{-6}$

$\frac{dK}{dL}=-5(\frac{12,200}{42})^{\frac{6}{5}}\times \frac{1}{L^6}$

$\frac{dK}{dL}=-\frac{4515.466}{L^6}$