# Why are production functions linear in technology?

Economists often assume a production function of the form $$Y = A f(K, L)$$, where $$Y$$ is output, $$K$$ is capital, $$L$$ is labour and $$A$$ is technology. This form of production function can describe both nation-wide production or firm-wide production.

Now, my question is, why is the production function assumed to be linear in the technology $$A$$? For example, can't we have a production function of the form

$$Y = \sum_{i = 1}^n A^{\alpha_i} K^{\beta_i} L^{1-\beta_i} \quad$$

where $$\alpha_1, \beta_1, \alpha_2, \beta_2, \dots \alpha_n, \beta_n$$ are exogenous parameters?

It can, production functions do not need to be linear in technology. Production functions of the form:

$$F=AK^{\alpha}L^{1-\alpha}$$

are used because they are simple to work with, have some nice properties, and as the authors after which this function has its name (this is so called Cobb-Douglas production function), Cobb & Douglas (1928) this function reasonably well (given its simplicity) describes, and exhibits properties of, production functions in real life (at least to a point). Over years Cobb-Douglas production function became sort of a default function to be used in examples (even outside production e.g. Cobb-Douglas utility).

Moreover, when estimating productivity with parametric models it is often easier to work with linear(ized) production functions.

However, the above being said production function can have various shape or forms. Ultimately it is for every firm and country an empirical question how the production function looks like. However, remember scientists deal with models. Having 100% realistic production function with all non-linearities would likely offer no extra benefit comparing to having simplified 80-90% realistic production function. Unless you are working on some speical case where that matters you have to weight the pros and cons of realism vs clarity/workability/usability trade-off.

It is important to note that technology is an abstract concept and cannot be measured in concrete units, but simply with a positive scalar magnitude (the greater the technical development, the greater the quantity). With this in mind the proposed function:

$$Y = \sum_{i = 1}^n A^{\alpha_i} K^{\beta_i} L^{1-\beta_i}$$

It can always be rewritten as a "linear function" of the technology within each sector $$i$$, simply by defining $$\tilde{A}_i = A^{\alpha_i}$$, then you have a linear function on different technologies:

$$Y = \sum_{i = 1}^n \tilde{A}_i K^{\beta_i} L^{1-\beta_i}$$

where each term in the sum has been interpreted as the output in a specific sector with technology level $$\tilde{A}_i$$.