It’s a very embarrassing question I have to ask, but I simply cannot get my head around this. In most of the DSGE models with housing and collateral constraints, housing services H_t enter the utility function of households.
I understand that households accumulate a certain amount of housing stock via H_t - (1-\delta)H_{t-1}, which provides them with a stream of housing services H_t and they can also use the stock as collateral for loans.
It seems in the literature, when it comes to the utility we call H_t a service, originating from the stock, but when we look at the budget and collateral constraint we call H_t the housing stock. Hence, I’m confused to what H_t actually refers to, stock or service? If we assume a one to one relationship, between housing stock and service, then this would be understandable. However, many papers assume that housing services are proportional to the actual housing stock, which should imply that we cannot easily say that housing services = housing stock.
Sorry for the trivial question and let me know if my question wasn’t clear enough.
Usually, only proportionality is assumed. You can easily normalize services to be equal to the stock, because the weight of the housing service in utility is a free parameter that then captures both the weight of housing services and the factor of proportionality between the housing stock and the resulting service flow.
where all A's are shocks, C_t is consumption, N_t is labour, H_t is housing and j is the housing preference share. Then j would capture the utility weight of housing and the proportionality between housing services and its stock. Hence calibrating it to one would give us a one to one relationship. Have I correctly interpreted you above comment?