# More on utility functions for each household type

In this week’s post we are going to attempt to take the distinct utility functions from last week a step further. We finished last week with the description of separate utility functions for extrapolators and fundamentalists, based on the notion that they discounted the future in different ways.

As in the Woodford model, we will assume that households take the interest rates as given. As a reminder, there are two interest rates in the Woodford model, one for depositors and one for borrowers. For each period in our model, there will be households that are borrowers as well as a group of households that are savers.

To further our modelling process, we will combine Woodford’s equations 1.10-1.11 with our utility functions from last week. For extrapolators, as they are the borrowers in each period, they must satisfy the first order condition as follows: Where λt(i)=uc(ct), is the household’s marginal utility of real income in period t. A fundamentalist household is a saver in this model, where they must satisfy the following first order condition: These conditions are only subtly different from Woodford’s, as we have different betas and our borrowers and savers come from an extrapolator/fundamentalist distinction, rather than a random process.

Now we need to take into account the probabilities of each household changing type from one period to the next.  This is where we might see another subtle difference to the Woodford model. When he is describing the equilibrium evolution of marginal utility for each household, he takes into account the probabilities of household types in the next period. This makes sense in his framework, as household’s change type randomly, so their expectations of next period need to be balanced by these probabilities.

In the framework that I’m trying to build, however, it is the actual expectation of the future that defines the household. If a household believed it would change type in the next period, it would change type in the current period to account for this. Consequently, at this stage, it may not be necessary to include the probabilities of household type in the description of the equilibrium evolution.

On the other hand, these probabilities raise another issue that we need to address. In Woodford’s model, due to his inter-household contracting ability, he can make the claim that regardless of each household’s history, each household of a given type has the same marginal utility of income. In this respect, the model is tractable and this limits the concerns about keeping track of distributions as the model evolves across periods. In the present framework, assumptions or concepts need to be configured so that even as the probabilities change from one period to the next, we can still claim that all borrower households still have the same marginal utility of income.

Perhaps a concept from my Cubic IS curve may help here. In that model, I described households as being distributed along a spectrum according to ‘propensity to take on debt’. I outlined this characteristic as an inherent factor of each household – those with a high propensity to take on debt are more likely to be borrowers when responding to events around them, whereas those with a low propensity to take on debt are less likely to respond to those same events. In this regard, perhaps we can view the above equations more as representative households of each type, rather than trying to assume that each household of each type has exactly the same marginal utility of income.

Taking into account De Grauewe’s random ‘human psychology’ element of his probabilities, we could assume that the majority of borrower households in each period are distributed on the right hand side of that spectrum, while the majority of saver households are located on the left hand side. In each period, therefore, the utility functions could be the average of households of that type, resulting in functions for the representative household of that type.

This may be an intellectual cheat at this stage of the modelling process, so it may need to be returned to later. For now, I would like to move onto filling out the rest of the aggregate demand story with the government and aggregate debt dynamics.