# Income distribution and Euler’s theorem

Forgive me, a quick "technical" integration which could be useful for appreciating some nuances of today's debate and will certainly help you with my *slides* on Sunday.

We have often talked about the neoclassical growth model, based on the neoclassical production function. In its standard version, this function exhibits constant returns to scale. It means that, as I explained to you when talking about Neoclassical Growth for Dummies , the increases in output are proportional to those in inputs. Imagining that the level of production Y depends on the quantity of capital K and that of labor L, that is, that it is:

Y = *f* (K, L)

we will have that if we multiply the inputs by a certain number *t* , the output is also multiplied by *t* :

*f* ( *t* K, *t* L) = *t* Y

and therefore, for example, setting *t* = 2, if we double the inputs the output doubles:

*f* (2K, 2L) = 2Y

(and I told you this at the time and the good ones remember it).

Functions of this type are called "homogeneous of the first degree" by mathematicians and Euler's theorem applies to them, according to which the value of Y is given by the sum of the values of the arguments, each multiplied by the respective derivative:

And here I lost almost all of you along the way, but that's okay, I continue undaunted on the path of truth (which is the new honesty), and whoever loves me follows me!

What does this arcane formula mean?

Meanwhile, for the more curious,I refer you to its proof in the general case .

And that's math.

The economy, and the distribution of income, enter into the reasoning when we remember that, as we explained at the time to Lampredotto, who is once again going crazy on social media with his delirious intention of abandoning the floating ship for the sinking one , in the model of competitive equilibrium the factors of production are remunerated at their respective marginal productivity. The table was this:

and the explanation was in the post on BDSM (dear, it's not what you think!).

From what we have said so far (and in the linked posts), two properties of the standard neoclassical model follow:

- in equilibrium, work will be remunerated at the value of its marginal productivity;
- in equilibrium, all the product will be distributed.

The second thing derives from the fact that, in fact, the sum of the products of the quantities of factors used, multiplied by the respective remunerations, coincides, look, with the total production. So in equilibrium all production is distributed, and everyone receives in proportion to how much they contributed to production.

There is another little piece of technique that might be useful to someone. In functions of this type, marginal productivity is proportional to average productivity. If you consider for example the most used of the production functions, the Cobb-Douglas :

you can easily verify that the marginal productivity of labor (the derivative of *Y* with respect to *L* ) is given by:

(the passages are on Wikimmmm, which obviously does them for capital – it's a clear political message – but if you've survived this far you also know how to do them for work).

What does this beautiful story mean (which we could have dwelt on, for example by developing all the steps directly here, for which I instead refer you to the sources cited)?

It means that in theory we should expect real wages to evolve proportionally to average labor *productivity* (APL). If this does not happen, there are two cases:

- or the production function does not have constant returns (for example because it has increasing ones, i.e. because as the inputs increase the economies of scale cause the output to increase more than proportionally);
- or the world does not work as in the neoclassical model (for example because instead of being remunerated on the basis of their marginal productivity the factors of production are remunerated on the basis of social power relations).

You will say: yes, everything is beautiful, perhaps everything is understandable, with difficulty, but what does this have to do with the things we talk about so much today?

It has a lot to do with it (or, as those who write "I can't do it", say) a lot! Haven't you ever heard that the wage crisis is linked to the productivity crisis, that the problem of wage stagnation is a problem of productivity stagnation?

Well!

Whoever tells you this is telling you that the world is neoclassical, that every factor of production (including you who are reading this) is remunerated based on its marginal productivity, and therefore that if the remuneration of the factors does not increase it depends on the fact that the their productivity has not increased (i.e. that you, dear reader, sucked and got what you deserved, which is not much).

But if we instead observed that while labor productivity has increased, labor remuneration (real wages) has not, what should we conclude?

The conclusion is not difficult, but we will draw it together on Sunday…

*This is a machine translation of a post (in Italian) written by Alberto Bagnai and published on Goofynomics at the URL https://goofynomics.blogspot.com/2024/10/distribuzione-del-reddito-e-teorema-di.html on Thu, 24 Oct 2024 15:07:00 +0000. Some rights reserved under CC BY-NC-ND 3.0 license.*