Question

In Example \(6.4,\) wheat is produced according to the production function $$q=100\left(K^{0.8} L^{0.2}\right)$$ a. Beginning with a capital input of 4 and a labor input of \(49,\) show that the marginal product of labor and the marginal product of capital are both decreasing. b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

Short answer

Both the Marginal Product of Labor and the Marginal Product of Capital are decreasing when evaluated at K=4, L=49. The production function exhibits constant returns to scale.

Step-by-step solution

Calculate the Marginal Products of Labor and Capital

The marginal product of an input is the additional amount of output that can be gained by increasing that input by one unit, while keeping all other inputs constant. It is found by taking the partial derivative of the production function with respect to that input. The marginal product of labor (MPL) is found by taking the partial derivative of the production function with respect to labor:\MPL = \frac{\partial q}{\partial L} = 20 K^{0.8} L^{-0.8} = \frac{20 K^{0.8}}{L^{0.8}} \Similarly, the marginal product of capital (MPK) is found by taking the partial derivative of q with respect to K:\MPK = \frac{\partial q}{\partial K} = 80 K^{-0.2} L^{0.2} = \frac{80 L^{0.2}}{K^{0.2}}.

Show that MPL and MPK are Decreasing

We can show that the marginal products of labor and capital are decreasing by substituting the given values of K and L into the equations for MPL and MPK and demonstrating that an increase in either input results in a decrease in the corresponding marginal product. Given K=4 and L=49, MPL at L=49: \MPL(49) = \frac{20 \cdot 4^{0.8}}{49^{0.8}} Similarly, the MPL at L=50 would be: \MPL(50) = \frac{20 \cdot 4^{0.8}}{50^{0.8}} Since 50 > 49, if \MPL(50) < \MPL(49), then MPL is decreasing.A similar process can be repeated for MPK.

Determine the Returns to Scale

Returns to scale refers to how output changes if all inputs are increased proportionally. If the sum of the exponents of K and L in the production function is greater than 1, the function exhibits increasing returns to scale. If the sum is less than 1, it exhibits decreasing returns to scale. If the sum is equal to 1, it exhibits constant returns to scale. In this production function, the sum of the exponents of K and L is 0.8+0.2=1, which implies constant returns to scale.

Key concepts

Marginal Product of Labor

When diving into microeconomics, the concept of the marginal product of labor (MPL) is fundamental. Imagine you're running a business and you want to understand how hiring one more employee will affect your production. MPL helps here: it measures the change in output resulting from adding one more unit of labor, like a worker, while keeping all other inputs constant.

The technical way to calculate MPL is by taking the partial derivative of our production function with respect to labor. This might sound intimidating, but it’s simply the rate at which output changes as labor increases, assuming we hold everything else steady. For our wheat production, the function looks like this: \[MPL = \frac{20 \times K^{0.8}}{L^{0.8}}\].

In simpler terms, if you keep the amount of capital (machinery, equipment, etc.) the same but add one more worker, MPL tells you how much more wheat you'd produce. The tricky part to watch out for, and a critical insight for economics students, is that MPL often decreases as you keep adding workers. This is because of the Law of Diminishing Returns: at some point, each new worker contributes less to output than the previous one did, usually because they start getting in each other’s way or there's just not enough machinery to keep all workers equally productive.

Marginal Product of Capital

In parallel to labor, the marginal product of capital (MPK) is another staple of the production function in microeconomics. It deals with the additional output you gain from increasing your capital by one more unit - think adding an extra tractor to your fleet on the farm - while keeping your labor constant.

MPK is computed similarly to MPL, through partial differentiation of the production function, but this time with respect to capital: \[MPK = \frac{80 \times L^{0.2}}{K^{0.2}}\].

So, if you buy one more piece of equipment, while the number of workers stays fixed, MPK tells you how much additional wheat you can expect to get. Again, it's important to note that MPK can also decrease as you add more capital. This might happen, for example, when there is not enough labor available to use the additional capital effectively, which can lead to underutilization and a less than proportionate increase in production.

Returns to Scale

The concept of returns to scale takes a broader view: it looks at what happens to output when we increase all production inputs simultaneously. If doubling labor and capital more than doubles output, you have increasing returns to scale. If doubling inputs less than doubles output, you're looking at decreasing returns to scale. Finally, if output doubles when inputs are doubled, that is constant returns to scale.

In our wheat production scenario, we've got a sum of the exponents in our production function (0.8 for capital and 0.2 for labor) that adds up to 1. This gives us constant returns to scale. Essentially, in this case, scaling up all resources proportionally will scale up output in an exact 1-to-1 ratio. If we doubled the size of our farm and our workforce, we’d expect to double our production of wheat.

Understanding returns to scale is crucial not just for microeconomics students but also for business owners and managers; it helps in making decisions about expansion and understanding the scalability of different production processes.