Building on the work done so far, we now need to add some more elements of the model, such as government and firm behaviour. Firstly, regarding government behaviour, there is no need to define this in a different way to Woodford’s model. His flow of government debt is determined by the following equation:
Where bt is real government debt at time t; Gt is the government purchases of the composite good; Pt is the price of that composite good; τt is a proportional tax on sales of goods; and Yt is the quantity of the composite good produced by firms; and
is the gross rate of inflation;
Again, there is no need to move away from Woodford’s description of the behaviour of firms in his model. He states that firms take the sales tax as given, and following this, they distribute profits according to the following function:
Where Dt are the distributed profits at time t, which is shared by households; Wt(j) is the wage of labour type j in period t; and
Which is aggregate labour hired of type j.
The aggregate demand equation
Bringing these together, Woodford describes the market-clearing notion of the goods market as follows:
This equation states that aggregate demand is a function of consumption, which we have defined as coming from heterogeneous households (more on that later), government purchases of the composite good and Ξt is the cost involved in originating the real loans in this economy. This part of Woodford’s model is a crucial part, as it generates the financial and credit frictions that are central to his work. We will now turn to this intermediary sector to see if it needs to be adapted at all to account for De Grauwe’s version of heterogeneous households.