The production function for the personal computers of DISK, Inc., is given by $$q=10 K^{0.5} L^{0.5}$$ 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 $$q=10 K^{0.6} L^{0.4}$$ 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.

We are given that both companies use the same amounts of capital and labor. Let's denote the common amount of capital and labor used by both companies as \(K\) and \(L\) respectively. Hence, we should substitute these values into the production functions for both companies and compare the output. DISK, Inc's production function is \(q=10 K^{0.5} L^{0.5}\) and FLOPPY, Inc's production function is \(q=10 K^{0.6} L^{0.4}\). Since \(K\) and \(L\) are the same for both companies, the company with the higher exponents on its capital and labor in its production function would produce more. In this case, it is FLOPPY, Inc. because 0.6 + 0.4 > 0.5 + 0.5.

The second part of the question requires us to find the marginal product of labor, when capital is limited to 9 machine hours and labor is unlimited. The marginal product of labor (MPL) is the additional amount of output produced when one more unit of labor is increased, while holding other inputs fixed. It can be found by taking the derivative of the production function with respect to labor \(L\). For DISK, Inc., the MPL can be obtained from its production function as follows: \(MPL_{DISK} = 0.5 * 10 * K^{0.5} * L^{-0.5}\). Substituting K = 9 into this equation, we get \(MPL_{DISK} = 0.5 * 10 * 9^{0.5} * L^{-0.5}\). Similarly, for FLOPPY, Inc., the MPL can be obtained from its production function as follows: \(MPL_{FLOPPY} = 0.4 * 10 * K^{0.6} * L^{-0.6}\). Substituting K = 9 into this equation, we get \(MPL_{FLOPPY} = 0.4 * 10 * 9^{0.6} * L^{-0.6}\). As L is unlimited, we can disregard \(L^{-0.5}\) and \(L^{-0.6}\) for comparison. Hence, DISK, Inc. will have the greater MPL, if the exponents of \(L\) were positive, for this to change FLOPPY, Inc.'s exponent of \(L\) should be larger.

From these calculations, we can infer that although FLOPPY, Inc. will produce more output if both companies used the same amount of capital and labor, DISK, Inc. would have a greater increase in output if an additional hour of labor was used, all else equal. This is due to the higher coefficient (0.5 compared to 0.4) on labor in the production function of DISK, Inc. Thus, DISK's production function is more sensitive to changes in labor.