Understanding Short-Run Production Functions in Microeconomics

Let's unpack the concept of short-run production functions in microeconomics using a clear example that you'll likely encounter in your studies at the University of Central Florida (UCF). So, what does that even mean? Well, in microeconomics, a production function describes the relationship between the quantity of inputs used in production and the resulting quantity of output. It’s fundamental stuff, but understanding how to manipulate these functions, especially in the short run, is key to doing well on your ECO2023 exam.

Here's the scenario: you have a production function represented by ( Q = 0.4K^{0.5}L^{0.5} ). Sounds complex, but hang tight! In this setup, ( Q ) is the quantity of output produced, ( K ) is the amount of capital, and ( L ) is the amount of labor. You might notice that we often keep one input constant when talking about short-run production. In this case, we have fixed capital at 100 units.

So, how do we derive the firm's short-run production function? To simplify, first, let’s plug the fixed number for capital into the original production function. Isn’t that logical? When capital ( K ) is held constant at 100, we can calculate ( K^{0.5} ):

[ K^{0.5} = 100^{0.5} = 10 ]

After this calculation, we substitute that back into the production equation. It may sound tedious, but bear with me! You will essentially scale the function down to something more manageable. Our new function looks like this:

[ Q = 0.4 \cdot 10 \cdot L^{0.5} ]

When we simplify it, we arrive at:

[ Q = 4L^{0.5} ]

Now here’s where it gets even more interesting. The value ( 4 ) is a scaling factor derived directly from the fixed capital, reflecting how labor influences output. This equation shows you how production changes with varying levels of labor while capital remains constant.

When it comes to exam time, you'll want to recognize that this result aligns with Option B from the multiple-choice answers, confirming that the short-run production function when capital is fixed at 100 is ( Q = 4L^{0.5} ).

Have you noticed how understanding functions like this deepens your insights into the microeconomics field? Mastering these relationships not only prepares you for exams but also provides you real-world skills! Whether you’re working for a startup or a multimillion-dollar corporation, knowing how production works can make a massive difference.

To wrap it up, getting a grip on short-run production is not just about racking up points on an exam—it’s about equipping yourself with essential tools for the future. Consider this a stepping stone on your journey through the vibrant world of economics. So, ensure you practice this concept and others in your study groups, keeping that bright bulb of curiosity shining. Good luck!