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Microeconomics tutor LA explains production function.

A production function simply shows the relationship between inputs and output. Most economics students understand land, labor, capital, and entrepreneurship as factors of input. These factors of input are combined to produce output. The production function is simply a mathematical relationship that links our inputs to output in an exact way. A popularly used production function is presented by Cobb-Douglas equation $Y=K^{\alpha}L^{\beta}$ . This functional form is fairly flexible. As the values of $\alpha$ and $\beta$ change, we can get different marginal productivities (more on this later).

microeconomics tutor LA teaches production function

This is a simple production function which shows how output (Q) varies with units of labor.

Marginal products:

Producers (and economists) are also concerned with marginal products. That is, what happens to output as we add one more (or less) unit of a certain input, while keeping other inputs constant. For example, a farm based producer wants to know what happens to output of his farm when he hires one more labor unit, while keeping his land area, technology, equipment (tractors) constant. Mathematically, we can find this as the derivative of the output function with respect to that input.

$\frac{\partial Q}{\partial L} = \beta K^{\alpha}L^{\beta-1}=\frac{\beta K^{\alpha}L^{\beta}}{L^1}$
Given $\alpha$ and $\beta$ are both less than 1, we can see that MPL is diminishing in L. As we hire more units of Labor, each additional unit adds less and less to output. Hence, the marginal product of labor diminishes as we hire more labor.
diminishing mpl curve

Checking for returns to scale:
We also care about how scaling up (or down) of inputs change the level of output. For instance, if we increase all our inputs by 10 percent, will our output also increase by 10 percent? We classify returns to scale as increasing returns to scale (IRS), constant returns to scale (CRS) and diminishing returns to scale (DRS).
The question we are asking is mathematical. If using K and L produce output Y, then what happens if we use 2K and 2L input? Does it produce 2Y? More generally, we do not need to show this for 2K and 2L but for $\lambda K$ and $\lambda L$ producing $\lambda Y$ .

Constant returns to scale: Scaling up production by a certain proportion, say $\lambda$ increases output by the same proportion, so there is no gain or loss of productivity as the scale of production is ramped up.

We use production functions to sketch out isoquants – which shows all combinations of labor and capital that give a particular level of output. Work with our professional economics teachers to learn about production functions, and deriving isoquants.
isoquant with K and L