The production function for the personal computers of DISK, Inc., is given by

q = 10K0.5L0.5

where q is the number of computers produced per day, K is hours of machine time, and L is hours of labor input.

DISK’s competitor, FLOPPY, Inc., is using the production function

q = 10K0.6L0.4

a. If both companies use the same amounts of capital and labor, which will generate more output?

b. Assume that capital is limited to 9 machine-hours, but labor is unlimited in supply. In which company is the marginal product of labor greater? Explain.

Disk Inc’s production function is:

q = 10K^0.5L^0.5

FLOPPY Inc’s production function is:

q = 10K^0.6L^0.4

Here, both the companies use the same amounts of capital and labor, i.e., (say) K = 50 units and L = 50 units.

The output produced by Disk Inc’s if K = 50 units and L = 50 units is:

q = 10K^0.5L^0.5

= 10(50)^0.5(50)^0.5

= 500 units

The output produced by FLOPPY Inc’s if K = 50 units and L = 50 units is:

q = 10K^0.5L^0.5

= 10(50)^0.5(50)^0.5

= 500 units

So, if both the companies use the same amount of inputs, then the output produced by both is 500 units.

The capital level is fixed at 9 machine hours.

Disk Inc’s production function is

q = 10(9)^0.5L^0.5

= 30 x L^0.5

The marginal product of labor for Disk Inc is $\frac{\partial q}{\partial L}=\frac{30\times 0.5}{L^{0.5}}=\frac{15}{L^{0.5}}$

Now, if the value of L = 50 units, then MPL for Disk Inc is: $\frac{\partial q}{\partial L}=\frac{30\times 0.5}{L^{0.5}}=\frac{15}{{50}^{0.5}}=2.1213$

FLOPPY Inc’s production function if K = 9 machine hours is:

q = 10K^0.5L^0.4

= 10(9)^0.5L^0.4

= 37.37192 x L^0.4

The marginal product of labor for Floppy Inc is:

$$\begin{gathered} \frac{\partial q}{\partial L}=\frac{37.37192\times 0.4}{L^{0.6}} \\ =\frac{14.948768}{L^{0.6}} \\ \mathrm{Now},\mathrm{if}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{L}=50\mathrm{units},\mathrm{then}\mathrm{MPL}\mathrm{for}\mathrm{Floppy}\mathrm{Inc}\mathrm{is}: \\ \frac{\partial q}{\partial L}=\frac{37.37192\times 0.4}{{50}^{0.6}} \\ =\frac{14.948768}{{50}^{0.6}}=3.12604 \end{gathered}$$

Hence, it can be concluded from the above numerical example that MPL for Floppy Inc. is greater than the MPL for Disk Inc.