Q.Let the production function of a firm be $Q = 5\,L^{\frac{1}{2}}K^{\frac{1}{2}}$. Find out the maximum possible output that the firm can produce with 100 units of $L$ and 100 units of $K$.

Let’s start with something you already know. Imagine you are baking a cake. The two main things you need are flour (let’s call it capital — the mixer, the oven, the tools) and labour (your own time and effort). If you double the flour and double your effort, you expect roughly double the cake, right? But what if you only double the flour but keep your effort the same? You’ll get more cake, but not twice as much — because the extra flour has less labour to work with. That diminishing “oomph” from adding just one input is the core intuition behind the Cobb-Douglas production function.

Now, in economics, a production function is a mathematical way to say: given a certain amount of capital (machines, buildings, tools) and labour (workers, hours), what is the maximum output a firm or an economy can produce? The Cobb-Douglas form is the most famous one, and it looks like this:

Here:

• $Q$ = total output (the number of cakes, or GDP for a whole country)
• $K$ = capital (value of machines, factories, etc.)
• $L$ = labour (number of workers, or total hours worked)
• $A$ = total factor productivity — a catch-all for technology, efficiency, and everything else that isn’t capital or labour. If $A$ rises, you get more output from the same $K$ and $L$.
• $\alpha$ (alpha) and $\beta$ (beta) are numbers between 0 and 1. They tell you how responsive output is to each input.

The special property that makes Cobb-Douglas so useful is that $\alpha + \beta$ tells you about returns to scale. If $\alpha + \beta = 1$, doubling both $K$ and $L$ exactly doubles $Q$ — this is called constant returns to scale. If the sum is less than 1, doubling inputs gives less than double output (decreasing returns); if greater than 1, you get more than double (increasing returns). In most NCERT-style problems, you’ll see $\alpha + \beta = 1$, often written as $\beta = 1 - \alpha$.

Why does this matter? Because it lets you separate the contribution of each input. For example, if you know $\alpha = 0.3$, then a 10% increase in capital raises output by only $0.3 \times 10\% = 3\%$, holding labour constant. That’s the diminishing marginal product in action — each extra unit of capital adds less than the previous one, because labour is fixed.