This exercise explores in some more detail the relationship between production technologies and marginal product of labor.
A: We often work with production technologies that give rise to initially increasing marginal product of labor that eventually decreases.
(a) True or False: For such production technologies, the marginal product of labor is increasing so long as the slope of the production frontier becomes steeper as we move toward more labor input.
(b) True or False: The marginal product of labor becomes negative when the slope of the production frontier begins to get shallower as we move toward more labor input.
(c) True or False: The marginal product of labor is positive so long as the slope of the production frontier is positive.
(d) True or False: If the marginal product of labor ever becomes zero, we know that the production frontier becomes perfectly flat at that point.
(e) True or False: A negative marginal product of labor necessarily implies a downward sloping production frontier at that level of labor input.
B: We have thus far introduced two general forms for production functions that give rise to initially increasing and eventually decreasing marginal product.
(a) The first of these was given as an example in the text and took the general form f (ℓ) = α (1− cos (βℓ)) for all ℓ ≤ π/β ≈ 3.1416/β and f (ℓ) = 2α for all ℓ > π/β ≈ 3.1416/β), with α and β assumed to be greater than 0. Determine the labor input level at which the marginal product of labor begins to decline. (Hint: Recall that the cosine of π/2 ≈ 1.5708 is equal to zero.)
(b) Does the marginal product of labor ever become negative? If so, at what labor input level?
(c) In light of what you just learned, can you sketch the production function given in (a)? What does the marginal product of labor for this function look like?
(d) The second general form for such a production function was given in exercise 11.5 and took the general form f (ℓ) = βℓ2 −γℓ3. Determine the labor input level at which the marginal product of labor begins to decline.
(e) Does the marginal product of labor ever become negative? If so, at what labor input level?
(f) Given what you have learned about the function f (ℓ) = βℓ2 −γℓ3, illustrate the production function when β = 150 and γ = 1. What does the marginal product of labor look like?
(g) In each of the two previous cases, you should have concluded that the marginal product of labor eventually becomes zero and/or negative. Now consider the following new production technology: f (ℓ) = α/ (1+ e− (ℓ−β)) where e ≈ 2.7183 is the base of the natural logarithm. Determine the labor input level at which the marginal product of labor begins to decline.
(h) Does the marginal product of labor ever become negative? If so, at what labor input level?
(i) Given what you have discovered about the production function f (ℓ) = α/ (1+e− (ℓ−β)), can you sketch the shape of this function when α = 100 and β = 5? What does the marginal product of labor function look like?
(i) Given what you have discovered about the production function f (ℓ) = α/ (1+e− (ℓ−β)), can you sketch the shape of this function when α = 100 and β = 5? What does the marginal product of labor function look like?