Question

If output is produced according to Q = min{4L,6K}, the price of K is $24, and the price of L is $20, then the cost-minimizing combination of K and L capable of producing 72 units of output is?

Answer 1

At equilibrium the slope of the isoquant is equal to the price ratio of the inputs. That is

Assuming w=20 and r=24,

$\frac{w}{r}=\frac{20}{24}=\frac{5}{6}$

The equation of the isoquant is

$Q=4L+6K$

$\therefore MP_L=4 \\MP_K=6$

$\therefore \frac{MP_L}{MP_K}= \frac{4}{6}$

$\therefore \frac{MP_L}{MP_K}< \frac{w}{r}$

As the isoquant is straight line we should have corner solution. The slope of the isocost represented by the price ratio is greater than the slope of the isoquant. At equilibrium the isocost must lie below the isoquant. The producer will only employ capital. Then the production technology is given as

$72=6K \ \therefore K=12$

Capital Q-72 C1 C2 Labor

To have a corner solution, the cost must be either C1 or C2. As C2>C1, the producer will choose C1 or produce only with K.

Another reason for choosinh capital is as the inputs are substitute, the producer will choose that input which have higher marginal produce. In this case MP_L< MP_K, Thus, the producer will choose only K.

Answer 2

[Q = 72] Q = mind 4L, 6k} => 72 = min f 42,6k? At Equilibrium 4 L=6K = 72 42=72 L - 18 6K=72 k = 12 = Henne cost minimising combination of inputs is, CL, k) = (18,12)