Keep in mind that we're looking at the three sector Keynesian model. And like all models, some assumptions are made. I'll mention these assumptions as we go along. Also, I'm using underscores ("_") to denote subscripts. (Haven't gotten around to using LaTeX on BoS, lol.)

C_0appears because of the following formula, which is assumed to be true:

C = C_0 + cY, whereC_0is a fixed amount of consumption andcis the marginal propensity to consume.

This plots as a straight line if you plotCagainstY, with the slope being the marginal propensity to consumec. And this makes sense, because you expect that as the national income increases, consumers in the economy would generally spend more.

C_0is a fixed amount of consumption that occurs in the economy anyway. This makes sense, as there are other things outside ofcandYwhich influences consumption. More technically, this term is calledexogenous consumption(don't memorise this), and is assumed in the model to take into account factors influencing consumption in the economy thatisn'tbased on the MPCcor the national incomeY. (That's why it's called the exogenous; it's based on factors outside of our model.)

Now our second assumption is that the amount of investment in the economy is fixed. That is to say:

I = I_0, where I_0 is a fixed number.

In this three-sector model, we decide to include the business sector by introducing the investment termI, but since we're not looking at the relationship betweenIand anything else, we decide to keep it constant in this model. (In other words, the entire business sector is captured in this model exogeneously; we assumed it's only influenced by factors outside of our model, and cannot be explained by any of the other terms in our model.) In the notes you have provided, the1_0is a typo; it's supposed to beI_0.

I hope that answers your question.

But in general, the reason why terms likeC_0andI_0appear in a model is not based on pure logic, but rather a subjective judgment on the part of the person who is mathematically modelling. We can choose to, say, instead of modelling investment as fixed, to model investment as a function ofc,Y, and other variables, but this would make the model a lot more complicated and messy. We could also choose to ignore investment completely and get rid of theI, but this would make the model weaker (in the sense that it completely neglects an important sector). So Keynes, who developed this model, decided to strike a balance between simplicity and strength of the model by including investment, but leaving it only as a fixed constant, to suit his purpose for the model (which was to demonstrate ideas like the multiplier effect).

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