**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.**

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