Question

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.

Short answer

a. If both the companies use the same amount of inputs, both will produce the same quantity of output.

b. The marginal product of labor is greater for Floppy Inc.

Step-by-step solution

The output generated by each company 

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 value of MPL for each company 

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.