Suppose a firm’s technology is represented by the Cobb-Douglas production function F(L, K) = 5LK. The wage rate is $50 and the rental rate of capital is $10. What is the least-cost combination to produce 100 units of outpu Solution Minimize TC = wL + rK s.t Q = 5LK Plugging the value of Q in TC we get TC = wL + rQ/5L Differentiating TC w.r.t L and equating the equation equa to zero we get w= rQ/5L^2 L^2=rQ/5w Plugging the values, we get L = 2 units This is the cost minimizing labour Similarly, Plugging the value of L as Q/5K, we get TC = wQ/5K + rK Differentiating and equating it to zero we get K^2=wQ/5r K = 10 units. This is the cost minimizing capital..

1. Suppose a firm’s technology is represented by the Cobb-Douglas production function F(L, K) =
5LK. The wage rate is $50 and the rental rate of capital is $10. What is the least-cost
combination to produce 100 units of outpu
Solution
Minimize TC = wL + rK
s.t Q = 5LK
Plugging the value of Q in TC we get
TC = wL + rQ/5L
Differentiating TC w.r.t L and equating the equation equa to zero we get
w= rQ/5L^2
L^2=rQ/5w
Plugging the values, we get
L = 2 units This is the cost minimizing labour
Similarly,
Plugging the value of L as Q/5K, we get
TC = wQ/5K + rK
Differentiating and equating it to zero we get
K^2=wQ/5r
K = 10 units. This is the cost minimizing capital.