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.
We can think of the representative agent as renting consumer durables from himself, at a nominal rental R(t). R(t) is a perfectly flexible price, because the prices we charge ourselves are not sticky. We don't refuse to make utility-increasing trades with ourselves. R(t) is strictly a shadow price.
But the representative agent buys new consumer durables from the producers at a price P(t), and P(t) is sticky in exactly the same way as in the standard New Keynesian model.
What happens to the standard Consumption-Euler IS equation? Here we have to be careful.
There are two different real interest rates in my model, because there are two different inflation rates. There is the inflation rate of R(t), and there is the inflation rate of P(t).
If i(t) is the nominal interest rate, then define:
r(t) = i(t) - [EP(t+1)-P(t)]/P(t)
s(t) = i(t) - [ER(t+1)-R(t)]/R(t)
The consumption services Euler equation is just like in the standard model, only with the real interest rate s replacing r:
1. C(t) = EC(t+1)/B(1+s(t)) where B is the discount factor, assuming log utility function U=log(C) for simplicity.
Arbitrage ensures that the nominal interest rate equals the rate of return on owning consumer durables and collecting rents from oneself plus capital gains net of depreciation:
2. i(t) = R(t)/P(t) + [EP(t+1)-P(t)]/P(t) - d
We can rearrange 2 to get:
3. R(t) = [r(t)+d]P(t)
Leading 3 forward one period we get:
4. R(t+1) = [r(t+1)+d]P(t+1)
OK, that's the setup. But I'm now going to switch to continuous time, because the math is too hard and ugly in discrete time. But understand that all rates of change are forward-looking expectations (not backward-looking realisations), and that Y(t) is a jump-variable that will jump up or down discontinuously if new information arrives.
But since K cannot jump, neither can C jump. That's what makes my setup different from the standard New Keynesian setup.
This is probably wrong, because I'm bad at math.
X^ means expected rate of change of X. Or X^=E(dX/dt)/X
5. r = i - P^ (definition of r)
6. s = i - R^ (definition of s)
7. C^ = s - b where B=1/(1+b) (Consumption services Euler equation)
8. C^ = K^ = Y/K - d (Technology of accumulation of consumer durables)
9. R = [r+d]P (arbitrage condition)
10. R^ = [r+d]^ + P^ (follows directly from 9)
What I want to do is eliminate C, R, s, and P from this system, to find the relationship between Y and r given K. That will give us the modified New Keynesian IS equation. In particular, we need to answer this question:
Assume initially we are in steady state with r = b and K and Y constant. Then r suddenly jumps up, but is expected to slowly return to b. What happens to Y? Does there exist an equilibrium path where Y jumps down then slowly returns to its original level, like in the standard setup? How does the depreciation rate d affect that path? (Remember that as d gets bigger my setup approaches the standard setup.)
But that's not my comparative advantage. So now it's your turn, if you have read this far.
Update: Keshav in comments gets the following solutions:
For continuous time:
K^ = r - (r+d)^ - b
Y/K = r + d - (r+d)^ - b
For discrete time:
K(t+1)/K(t) = B*(1+r(t))*(1-(1-d)/(1+r(t)))/(1-(1-d)/(1+r(t+1)))