Examples of how economists come up with a production function for a firm?

I am a young man studying economics on his own. In every microeconomics text I find, they teach you what a production function is, but they never show examples (or practice problems) on how one comes up with a production function of a firm on his/her own. Even in practice problems, a production function is already given. I can never find practice problems that help you practice finding an actual production function (ex: A firm has 3 ovens, 2 computers, 10 special extractors, and 15 workers in an olive oil plant. Find the production function for this firm).

I personally want to see how an economist makes up his/her own production function of a firm he/she is observing. I know this takes some sort of mathematical ability. I personally am not that good at modeling real life situations into functions yet (I am practicing that as well). This is why I want to practice this particular skill.

Can anyone please show me examples of this? Does anyone have any good websites or books that show great examples of this, along with practice problems? I have found nothing in almost every book/website I've found. Please help. Thank you.

• I don't know if anyone is actually doing this, but if they do they would do it with econometrics, not microeconomics. The production functions in microeconomic theory are used to illustrate economic concepts such as the shutdown point and the input demand curve, not to come up with exact predictions. – Giskard Oct 12 '15 at 10:15
• Please specify whether you're after individual (micro) production functions, or aggregate production functions that are used in macroeconomics. – FooBar Oct 14 '15 at 9:07
• I am after microeconomic production functions. In particular, I'd like to better my mathematical modelling for microeconomics. – Copyright Dec 18 '15 at 5:52

The "production function" in economics and econometrics is not an engineering concept. It is a highly abstract construction where the myriads of inputs used are lumped together in very few categories: "capital" and "labor" being the most usual ones, while sometimes inputs critical to an industry like say "Energy" may be singled out and appear in the production-function formulation. At such an abstract level, it is easily implementable without the need to go into particulars of each firm.

Moreover, in many if not most of the cases, what is modeled is the production of value added, not the total output (either in quantities or in values).

Applied work commissioned by specific (usually very large) firms, and also, applied research into "adjustment costs" or what is called "insider econometrics", may go some steps towards creating much more complex production functions, that essentially try to take into account the actual physical relations and interactions between the various inputs. But these are very specialized areas -remember that the Economics discipline ultimately is about markets and economic systems -the "individual" (firm or person) level is researched mainly in order to have sound foundations for the aggregation that follows.

Actually there is a large literature on the estimation of production functions. Usually, one starts out with assuming a particular functional form (CES, Cobb-Douglas, etc.) and then estimating the parameters of the model. As indicated, the mathematics of this estimation are often nontrivial. However, here are some early references which can provide you with an entry point to the literature:

Aigner, D., Lovell, C. K., & Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1), 21-37.

Førsund, F. R., Lovell, C. K., & Schmidt, P. (1980). A survey of frontier production functions and of their relationship to efficiency measurement. Journal of Econometrics, 13(1), 5-25.

It's about mathematical modelling. We model real-world phenomenon by applying mathematics. We approximate real-world parameters using these models.

For modelling the production function of a commodity you need to do, chronologically,

1. For the commodity, observe the factory, its laborers, the owner, the machines, the plant-size, the productions, the production process, etc..

2. Now, since you want the production function of the commodity, you look for the factors that influence the amount of production of units the most. Discard those factors that have negligible influence.

3. Now, make a function(f) of these selected factors treating them as independent variables, observing the relationships among them and the f.

4. Test your function for a huge number of instances by comparing the real measurement and the functional value on each instance. Look at the error part in the comparison. The error has to be low. Too much error discards the function.

5. Negligible error establishes the function, otherwise you go to step 1 and restart.

{When you show your established function, mention all the assumptions that have been made.}

• Thank you very much for your response. Is there anywhere I can practice this? Are their any textbooks available where it gives one problems in modeling? My interest is to better my mathematical modelling ability for uses in economic analysis (in this case, microeconomic analysis). Most textbooks that I have found do not do this. – Copyright Dec 18 '15 at 5:51
• @Copyright - There are lots of books on modelling. You will also need statistics, probability along with modelling. But, first of all, you must know economics well. And, you will need some teachers too. They will guide you, otherwise you will be lost in your studies. – ReekMaths Jan 2 '16 at 18:11