Warning: math-challenged economist at play. I want to see if I can sucker any readers into checking my math and doing the rest of the math for me. Do not read this post unless you think that might be fun.
Update: Keshav has solved the math. Now we are trying to understand what it means.
The simplest New Keynesian macro model assumes all output is consumer goods. The structure is U(t)=U(C(t)) and C(t)=Y(t), where C(t) is consumption and Y(t) is output at time t.
I want to assume instead that all output is consumer durables. The new structure is U(t)=U(C(t)) and C(t)=K(t)=(1-d)K(t-1)+Y(t). Where K(t) is the stock of consumer durables owned by the representative agent and d is the depreciation rate. Each unit of consumer durables provides one unit of consumption services per period.
I think New Keynesian macroeconomists will see this as a friendly amendment. Because theirs is a special case (where d=1) of my more general case. It is true that some consumption goods are more durable than others, but assuming the average consumer good has at least some durability is more plausible than assuming it has none. And since it is purchases of new durable goods (both consumer and producer durables) that usually gets hit hardest in a recession, I think it is important to do something like this. Plus, I'm just curious to see what difference it makes.