Circular Flow with Inventories I´m trying to fully understand the basic circular flow and the equivalence among production, income and expenditure, but considering the accumulation of inventories (still the most basic version, without financial sector or government).

When there´s the hypothesis that the firms sell everything they produce, the equivalence among product, expenditure and income is obvious, but what if the firms don´t sell everything and inventory (either planned or unplanned) is formed?

Product Y is the value of everything that is produced even if it isn't sold. All of the output is consumed either by the household sector or there is accumulation of inventories which is interpreted as an investment done by the firms, so that Product = Expenditure. So far, so good, but this value must turn into income (wages, since the model considers just work as a production factor) and now there is less income than product (because less output was sold, which leads to lower wages) and the equality between product and income doesn't hold anymore. Would inventory somehow generate this missing piece of income? Is it impossible to maintain the model as simple as this and still keep the equivalence? If not, what´s the minimum amount of information necessary to add?

Also, if it´s true that a flow of real goods in one direction must accompany a flow of money in the other direction in this simple model, is there an interpretation for the "acquisition" of inventories that fit in this idea or it should be abandoned?

EDIT. Some additional thoughts:

As I understand, the idea of circular flow contains 3 direct causalities (which are equalities):
1. Production -> Income
2. Income -> Expenditure
3. Production -> Expenditure
All of these happen in the same period of time. The next causality connects 2 different periods and it is not necessarily an equality. The production here is the level of production of the next period:
4. Expenditure -> Production (a behavioral relation and not an identity like the previous ones)

"3" analyzes the destiny of the values of all produced goods. Everything that is produced ends up with someone. If there´s inventory, the "trick" that considers unsold goods also as an expenditure (by the firms) leads to the equality Y = C + Inventory. There isn't a monetary counterpart in the "acquisition" of inventory by the firms, so it really seems just an exception to balance the identity.

"2" analyzes what people do with their incomes. If inventory is formed, it means the workers didn't spend everything and there's some savings S inside their houses. They're the only ones who receive income, so that wages are the only form of income. Income = C + S. Comparing with the previous equation, it makes sense that the value of inventory is the same as the amount of money under the mattresses of the household sector.

Now I´m struggling only with "1", which seems like a philosophical relation and not mathematical like the others. The factors of production "turn into" a finished good. The firm begins the period with a plan to produce N items and spends a certain amount to achieve that. The resultant payment of the factors acts as a definition of "value of production" Y? Is payment of factors in some sense the same thing as "costs of production"? Is the equality production=income a tautology?

About "4", without creation of money, it seems that in this simple example the production has an upper limit because just a certain amount of income can be paid. So, the production of the next period can only be lower. In this next period, if workers decide to spend their savings, the production can raise again later.
The statement 


  and now there is less income than product (because less output was sold, which leads to lower wages)


misses that the workers are paid to work a certain number of hours in production, and that production happened, so current wages are unaffected.

Meanwhile, the firm’s income is unaffected by the inventory build, since it is a form of investment, and thus the wages used to produce the inventory are effectively capitalised on the balance sheet.

Incomes might only fall in subsequent periods, as production is lowered in response to the unplanned growth in inventories.

As such, there is no real mystery to be explained.