**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(*d*X/*d*t)/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)))

I think you have an important goal. That said, the initial structure may be faulty.

I think there is a vital time-scale error in "The new structure is U(t)=U(C(t)) and C(t)=K(t)=(1-d)K(t-1)+Y(t).". I think that period consumption (C(t) CANNOT equal the enduring consumption (K(t). The two numbers should be different for every possible period.

My initial stab at an equation for C(t) (which is this year's measured consumption) would be

C(t) = d1*K(t) + d2*K(t) + d3*K(t)

where d1, d2, and d3 are three rates of depreciation. Add additional rates of depreciation to more accurately model consumption.

Then K(t) would equal this years output plus the remaining value of K(t-1)

K(t) = [(1-d1)*K(t-1) + (1-d2)*K(t-1) + (1-d3)*K(t-1)] + Y(t)

I think the terms d1, d2, d3, ..., would vary from year to year, depending upon interest rates and other factors.

I don't think I could translate these equations to continuous form. Perhaps others could.

Posted by: Roger Sparks | March 08, 2016 at 12:25 PM

Roger: I'm assuming that consumer durables depreciate at the same rate whether you use them or not. They rust, rather than wear out. Let K be the amount of stuff that I own (cars, houses, chairs, computers, canoes, etc.) and C the flow of services that I get from owning that stuff. With the right choice of units, C=K. Or just delete C from my setup entirely, and write the Utility function as U=U(K).

Posted by: Nick Rowe | March 08, 2016 at 12:41 PM

Nick: Yes, I understand that C is the consumption.

However, we are assuming that all output is durable, even if part is consumed in the year of output. We need to convert durable output (Y(t) to period consumption C(t). For durable output to be consumed in the period of output, we use a depreciation rate of 1. This accomplishment forces us to have other depreciation rates for other classes of stuff "(cars, houses, chairs, computers, canoes, etc.)".

That is my excuse for claiming C ≠ K.

Posted by: Roger Sparks | March 08, 2016 at 01:10 PM

In your notation, the relationship between Y, r and K is

K^ = r - (r+d)^ - b

Y/K = r + d - (r+d)^ - b

This gives Y and K as a function of r(t). As d goes to infinity, (r+d)^ goes to zero, and the first equation becomes the standard NK Euler equation. When d is finite, growth of the durables stock depends on the expected growth rate of r as well as the level. If r is expected to fall, people expect that it will become more profitable to hold their wealth in durables rather than bonds, and so they expect to increase their stock of durables.

*If* K is not allowed to jump, and if prices are fixed, so that i=r, an unexpected, temporary increase in i leads to a permanent increase in the capital stock and output. This is a strange result, but I don't think we should take it too seriously. In a fixed price, continuous time model, the least bad option is to let K be a jump variable, even though this means Y can equal plus or minus infinity. (Since there's no aggregate supply side, there's nothing in the model *as written* that prevents Y from being infinite.) Then the model works in the standard way: K falls after a contractionary shock, so that it can rise again in the future.

This is what you get in a discrete time model (e.g. with prices fixed one period in advance). The Euler equation becomes

K(t+1)/K(t) = B*(1+r(t))*(1-(1-d)/(1+r(t)))/(1-(1-d)/(1+r(t+1))).

With K(t)=(1-d)K(t-1)+Y(t), K(t) is a jump variable. A temporary increase in r(t), with prices fixed one period in advance, causes K(t) and Y(t) to fall. (Even after prices adjust at date t+1, r(t) remains temporarily high because the Wicksellian rate is high, as capital slowly returns to steady state and households are getting richer over time). But taking the continuous time limit, as the time period becomes very small, you would need a very low rate of output per unit time (in fact it could be negative) in order to reduce K(t).

If you want to stick with continuous time, but don't want infinite output, then I think you have to take a stand on the aggregate supply side of the model - since that is where the (sensible) constraint that Y must be finite comes from.

Posted by: Keshav | March 08, 2016 at 06:09 PM

Keshav: "As d goes to infinity, (r+d)^ goes to zero, and the first equation becomes the standard NK Euler equation."

That's good. That should be a good check that your answer is making sense.

"If you want to stick with continuous time, but don't want infinite output, then I think you have to take a stand on the aggregate supply side of the model - since that is where the (sensible) constraint that Y must be finite comes from."

The standard NK model just assumes that Y eventually converges to the same level Y*, after temporary shocks to r have passed. (I have done a couple of posts on this, like this one.) I want to stick with that assumption, if possible.

I'm going to have to think through your comment slowly. My brain is a bit tired right now. Thank you for doing this!

Posted by: Nick Rowe | March 08, 2016 at 07:19 PM

Keshav: my math is cr*p. I have a very strong hunch that your math is excellent.

I am therefore going to assume that your answer "K^ = r - (r+d)^ - b and Y/K = r + d - (r+d)^ - b" is correct. Especially since it feels right to me.

"*If* K is not allowed to jump, and if prices are fixed, so that i=r, an unexpected, temporary increase in i leads to a permanent increase in the capital stock and output. This is a strange result, but I don't think we should take it too seriously."

It *is* a strange result. But I think we *should* take it seriously. It ties in perfectly with my old posts where I argued that standard (simple consumption only setup) NK models *just assume* eventual convergence to some natural rate of output, following a temporary shock. The difference in my setup is we now have a state variable K. What this means is that, if r=b forever, then if K and Y are initially low they *must* remain low forever. Full-bore hysterisis, in other words.

This is rather exciting! I had a vague hunch that something like this might come out.

Posted by: Nick Rowe | March 08, 2016 at 07:58 PM

Nick:

OK, let's write out the full Lagrangian here where have U = U(C(t)) + BU(C(t+1)) (also, add in rational expectations operators where necessary):

L = U + l(t)((1-d)C(t-1) + Y(t) + (1+r(t-1))B(t-1) - C(t) - B(t) - T(t))

where B(t) is government bonds, T(t) is lump sum taxes, r(t) is the real interest rate on government bonds

This should yield

(1) U'(C(t)) - l(t) + l(t+1)(1-d) = 0

(2) -l(t) + l(t+1)(1 + r(t)) = 0

Bear with my algebra...

U'(C(t)) - l(t) + l(t)(1-d)/(1+r(t)) = 0

U'(C(t)) = (1 - (1-d)/(1+r(t)))l(t)

U'(C(t))/(1 - (1-d)/(1+r(t))) = l(t)

substitute into (2):

(3) U'(C(t))/(1 - (1 - d)/(1 + r(t))) = U'(C(t+1))/(1 - (1 - d)/(1 + r(t+1)))(1 + r(t))

There's your extremely complicated Euler equation. Alternatively, make it so there are also perishable consumer goods in the model, so you don't have to deal with really annoying Euler equations.

Posted by: John Handley | March 08, 2016 at 08:18 PM

I forgot to properly add in the discount factor. It goes in the normal places (basically just stick it on the RHS of (3)).

Posted by: John Handley | March 08, 2016 at 08:23 PM

John: Glad you have shown up!

Is your answer the same as Keshav's discrete time version (given that U'(C) = 1/C = 1/K )?

Posted by: Nick Rowe | March 08, 2016 at 08:33 PM

Nick:

The Euler equation is the same as normal at d = 1, not so sure about "As d goes to infinity, (r+d)^ goes to zero, and the first equation becomes the standard NK Euler equation."

Otherwise, if we rewrite the equation as

C(t+1)(1 - (1-d)/(1+r(t+1)))/B = C(t)(1 - (1-d)/(1+r(t)))(1+r(t)

and assume the model returns to the steady state in the next period (which allows us to make the entire RHS constant)

Constant = C(t)(r(t) + d)

then raising the real interest rate causes c(t) to fall, as is normal for NK Euler equations.

I'm pretty sure the only interesting thing here is that the future interest rate is involved and r(t) must be greater than d at all times, but I'll need to get some proper IRF's first...

I'll try to get some graphs and then post the links here.

Posted by: John Handley | March 08, 2016 at 08:55 PM

John: the appearance of r(t+1) is one difference between my setup and the standard setup. But here's the biggest difference:

Assume r=b forever. The standard setup has an infinite number of solutions for Y. So "animal spirits" can cause Y to suddenly jump up and stay at a new higher level. That can't happen in my setup, since K(t-1) is pinned down by history.

Posted by: Nick Rowe | March 08, 2016 at 09:12 PM

So I wrote down a whole model (no sticky prices because I'm too lazy):

Here are the IRF's

Here are the equations (once again, substitute in RE operators wherever necessary):

(1) C(t+1)*(1-(1-d)/(1+r(t+1))) = B*C(t)*(1-(1-d)/(1+r(t)))*(1+r(t));

(2) C(t) = (1-d)*C(t-1) + Y(t);

(3) W(t) = C(t)*N(t)^b*(1-(1-d)/(1+r));

(4) W(t) = A(t);

(5) Y(t) = A(t)*N(t);

(6) ln(A(t)) = p*ln(A(t-1)) - e(t);

where C(t) is the stock of durables, r(t) is the real interest rate, W(t) is the real wage, N(t) is employment, A(t) is TFP, Y(t) is output, and e(t) is the TFP shock.

I tried to make r(t) exogenous (leaving only the Euler equation and the equation governing the AR(1) real interest rate), but it apparently violated B-K conditions, so I had to do a full model. I figured a negative TFP shock would be somewhat comparable to an increase in r(t). I might try to add sticky prices and see what happens under a monetary policy shock (either an increase in the nominal interest rate or a reduction in the money supply, depending on how rigorous I want to be).

Posted by: John Handley | March 08, 2016 at 09:20 PM

"So "animal spirits" can cause Y to suddenly jump up and stay at a new higher level. That can't happen in my setup, since K(t-1) is pinned down by history."

Nick - I don't think that statement is true in the discrete time model that I wrote down, but the continuous time model *could* be tweaked to make it true, if you add a rationing rule. The Euler equation relates K(t) and K(t+1), not K(t-1) and K(t). If we write the durables accumulation equation as

K(t) = (1-d)K(t-1) + Y(t)

then K(t) and Y(t) are free variables, as in the standard setup. Animal spirits can cause households to go out and buy more durables at the beginning of time, and keep durable stocks at that high level forever.

(This is all in the fixed price model where we assume r(t) = b forever. In a flexible price model there would be another condition, Y(t) = Y*, and inflation would adjust to make real interest rates whatever they need to be to clear markets.)

In the continuous time model, K(0) is pinned down *if* we assume Y(t) has to be finite. If you take limits of the discrete time model as the time interval goes to zero, there aren't yet any conditions that ensure the flow rate of output per unit time must be finite. Writing down the aggregate supply side should yield such conditions. This could be a New Keynesian Phillips curve, or it could just be a rationing rule like

0 <= Y(t) <= Y^*

As things stands I didn't write any such rule, which means the yeoman farmers have to produce however much output the customers demand given the farmers' fixed prices and the customers' animal spirits. Even if that output is infinitely positive or negative. With a sensible rationing rule precluding infinite output, I think you would get your desired result (hysteresis, but no animal spirits).

One caveat - the standard NK solution to all these problems would be to write down a forward looking Phillips curve, restrict attention to bounded solutions, and argue that if Y(t) differs from Y^* in the long run, inflation must explode, so that can't happen. This argument might be good or bad, but in any case I think it applies equally to both the durables model and to the standard setup discussed in your previous posts (though I haven't checked the math for the durables model).

Posted by: Keshav | March 08, 2016 at 09:40 PM

I updated the model with (Rotemberg) sticky prices. All the IRF's are in the same place.

The new model is:

(1) C(t+1) (1-(1-d)/(1+r(t+1))) = B C(t) (1-(1-d)/(1+r(t))) (1+r(t));

(2) C(t) = (1-d) C(t-1) + Y(t);

(3) W(t) = C(t) N(t)^b (1-(1-d)/(1+r(t)));

(4) 0 = 1 - E + E MC(t) - Phi pi(t) (1+pi(t)) + Phi pi(t+1) (1+pi(t+1)) (Y(t+1)/Y(t))/(1+r(t));

(5) W(t) = MC(t);

(6) Y(t) = N(t);

(7) m(t) = (1-p)m* + p*m(t-1) - e(t);

(8) 1 + i(t) = (1 + r(t))*(1 + pi(t+1));

(9) m(t) - p(t) = ln(Y(t)) - a*i(t);

(10) p(t) = p(t-1) + pi(t);

or

(7) i(t) = 1/B - 1 + a(pi(t)) + v(t)

(9) v(t) = p(v(t-1)) + e(t)

for the interest rate version.

Posted by: John Handley | March 08, 2016 at 10:13 PM

Keshav: "One caveat - the standard NK solution to all these problems would be to write down a forward looking Phillips curve, restrict attention to bounded solutions, and argue that if Y(t) differs from Y^* in the long run, inflation must explode, so that can't happen."

That brings out the Old Keynesian side of me. Why can't that happen, in a world where the central bank sets a nominal rate of interest? Why can't the economy explode or implode? Where is the automatic self-equilibrating tendency towards 'full employment"?

But yes, this can be a problem even in the standard setup.

I must go to bed, since I teach early in the morning.

Posted by: Nick Rowe | March 08, 2016 at 10:22 PM

Nick:

"Why can't the economy explode or implode?"

Blanchard-Kahn conditons (basically RE models don't work without this assumption, unless we all decide to switch to a different solution technique, but then we might risk one of the good aspects of B-K -- determinacy with active Taylor Rules). Otherwise, I would have to say that we should just add something into the monetary policy rule that says that the central bank will prevent explosions. I remember reading a paper about this that was really interesting, but I don't remember what it's called...

Posted by: John Handley | March 08, 2016 at 10:26 PM

Keshav: "The Euler equation relates K(t) and K(t+1), not K(t-1) and K(t). If we write the durables accumulation equation as

K(t) = (1-d)K(t-1) + Y(t)

then K(t) and Y(t) are free variables, as in the standard setup."

I missed seeing that bit. My brain is too tired.

Posted by: Nick Rowe | March 08, 2016 at 10:27 PM

Found it! http://www.patrickminford.net/wp/E2009_21.pdf

Posted by: John Handley | March 08, 2016 at 10:35 PM

OK. I've had a sleep, and my head is clearer.

Add the standard NK supply side to the model. So Y is bounded from below by 0, and has a finite upper bound too. Let there be a stationary equilibrium Y* and K* when r=b forever. Assume (though it's a problematic assumption), that the economy always returns to Y* K* eventually, if r eventually returns to b.

Let's work backwards. Suppose the central bank wanted to create a recession, so that Y fell, and then started rising again back to Y*. The central bank would need to raise r above b (a jump), and then *immediately* start reducing r again so it slowly returns to b.

***If there were *any* delay in r starting to fall again, so that r stays constant above b for a finite length of time, however short, Y would drop to 0, until r starts falling again.***

In other words, the model says that Y is extremely sensitive to the level of r.

Posted by: Nick Rowe | March 09, 2016 at 08:19 AM

Like Nick, I just had a good night's sleep. Now I am trying to correlate the comments and original text.

I am having some trouble with the definitions of terms:

In the original text, term B is the discount rate. FMI, what are we discounting and why?

Still in the original text, b (small B) is used. Is b the change in B between periods?

Logic is like a hanging chain--you climb the chain link by link. If the links are undefined (for any reason, including student error) the chain (logic) fails.

Posted by: Roger Sparks | March 09, 2016 at 09:54 AM

Roger: B (or b) is the subjective discount rate. If B = 0.9 (roughly b=0.1) that means the agent is impatient. He prefers utility now than waiting a year. Next year's utility is only worth 90% of what this years utility is worth. He maximises V where V = U(Ct) + B U(Ct+1) + BBU(Ct+2) + etc.

Posted by: Nick Rowe | March 09, 2016 at 10:14 AM

John: lovely find, with that Patrick Minford paper! Good to see old monetarists are still firing away.

Posted by: Nick Rowe | March 09, 2016 at 10:16 AM

"Suppose the central bank wanted to create a recession, so that Y fell, and then started rising again back to Y*. The central bank would need to raise r above b (a jump), and then *immediately* start reducing r again so it slowly returns to b."

I agree, except for the jump in r at t=0. If there is a temporary recession (Y jumps down and then gradually returns to Y*) then K slowly falls for a while, and then gradually returns to K*. Solving the Euler equation forward, to see what r(t) is consistent with this, we get

K(t)/K(0) = [(r(0) + d)/(r(t) + d)]*exp{ \int_0^t (r(s) - b) ds }

(apologies for notation. \int_0^t is the integral from 0 to t. Again, note that as d -> infinity, this is the same as in the standard setup.)

If r(t) and K(t) both converge to *something*, then r(t) must converge to b. If K(t) converges to K(0), then r(0) must be equal to b. The interest rate *shouldn't* jump at t=0. Instead, it slowly increases for a while, and then slowly returns to b. The durable stock slowly falls for a while, and then gradually returns to K(0) = K*. Output falls sharply at t=0, and at some point returns to Y*.

So you're right, Y is extremely sensitive to the level of r - so sensitive that any jump in r, even a temporary one, would cause a permanent change in Y (and K).

Posted by: Keshav | March 09, 2016 at 11:35 AM

Thanks Nick.

I have the BEA pamphlet "A Guide to the NIPAs" open. GDP includes a term for "consumption of fixed capital (CFC)".

By including an increment of income derived from CFC, I think GDP has been converted into a measure of consumption-of-durable-goods, identical to the goal of this post.

The BEA would use the equivalent of your term K(t) to calculate CFC. I have not yet located a line (in the NIPAs) that reports the value of CFC, nor the running value of K(t).

As I read (and reread) this post and comments, I continue to try to mesh all of these details.

Posted by: Roger Sparks | March 09, 2016 at 11:55 AM

Keshav: "So you're right, Y is extremely sensitive to the level of r - so sensitive that any jump in r, even a temporary one, would cause a permanent change in Y (and K)."

Suppose we start at Y=Y* and K=K* and r=b. Then r jumps up and immediately starts falling. Y jumps down, and K starts falling. If r returns to b and stays there, Y and K stay permanently below Y* and K* ? In order to get back to Y=Y* and K=K*, we would need a period of time with r < b ?

Is that right?

Roger: "Consumption of Fixed Capital" (aka "Capital Consumption Allowance" is what accountants call depreciation.

Put that pamphlet down carefully, and slowly back away. National Income Accounting is a mare's nest. It is of negative help for understanding this post.

Posted by: Nick Rowe | March 09, 2016 at 04:28 PM

Nick - "In order to get back to Y=Y* and K=K*, we would need a period of time with r < b ?"

Yes, I think that's right. Ignore everything I wrote in my comment above starting with "If K(t) converges to K(0), then r(0) must be equal to b.", I was incorrect.

"r jumps up and immediately starts falling. Y jumps down, and K starts falling."

Actually, Y jumps *up*, and K starts *increasing*. A higher level of r, and an expected decline in r, both make households want to hold more durables in the future, relative to today.

Posted by: Keshav | March 09, 2016 at 05:33 PM

Kashav: "Actually, Y jumps *up*, and K starts *increasing*. A higher level of r, and an expected decline in r, both make households want to hold more durables in the future, relative to today."

Damn. Yes, that is what the math is saying. Trying to get my head around it, to get the intuition.

Posted by: Nick Rowe | March 09, 2016 at 06:55 PM

Nick and Kashav:

"Actually, Y jumps *up*, and K starts *increasing*. A higher level of r, and an expected decline in r, both make households want to hold more durables in the future, relative to today."

The stochastic simulation seems to agree; monetary policy shock results in lower r(t), then higher r(t+n) (because of everything returning to steady state), so C(t) falls.

Posted by: John Handley | March 09, 2016 at 09:14 PM

John: "The stochastic simulation seems to agree..."

Good to know.

Damn this is weird. But I can sorta see it.

In the standard setup, r > b means C must be growing, but that can happen by C jumping down when r jumps up, and then C is growing thereafter.

But it my setup, C *can't* jump down (at least with continuous time), and growing C means growing K which means higher Y.

Posted by: Nick Rowe | March 09, 2016 at 09:35 PM

In the standard setup, C can be growing in infinitely many ways. We can always pick C(0) so that C(t) returns to steady state, but there are infinitely many other solutions to the Euler equation. We need to use the aggregate supply side of the model, plus the standard (and perhaps arbitrary) NK assumption in favor of bounded solutions, to rule these others out.

In the model with durables, if markets clear and Y is finite, there's only one solution to the Euler equation - and it is exactly the kind of solution that standard NK practice would rule out, since it doesn't return to steady state. What happened to all the others? My *guess* is that the other solutions (in the standard setup) correspond to equilibria (in the durables model) in which one of the rationing constraints (the floor or ceiling on Y) binds. If this is the case, the Euler equations we derived aren't valid. Wherever u'(K) appears, it should be replaced with u'(K) + L, where L is a shadow price which is positive when Y=0 and negative when Y=Y*.

Suppose r(t) jumps up at t=0. Even though K can't jump, the shadow price can. Instead of K(0) falling, L(0) increases, which means Y(0)=0. As the interest rate jumps up, people would like to get rid of their durables, but the rate of decumulation is constrained by the nonnegativity constraint on investment. So they sit and let their stocks depreciate for a while.

But this is a guess and I haven't checked it. There might be interesting dynamics a la Neary-Stiglitz, since expected future rationing constraints affect rationing today.

Another conjecture is that if we had a Phillips curve instead of the rationing rule, then if i(t) jumps up at t=0 and slowly declines, the price level jumps down at t=0 and slowly rises. This creates just enough expected inflation that the real rate doesn't jump at date 0 (or it follows whatever path is consistent with the Euler equation we derived before).

Posted by: Keshav | March 09, 2016 at 10:36 PM

Keshav: "Another conjecture is that if we had a Phillips curve instead of the rationing rule, then if i(t) jumps up at t=0 and slowly declines, the price level jumps down at t=0 and slowly rises. This creates just enough expected inflation that the real rate doesn't jump at date 0 (or it follows whatever path is consistent with the Euler equation we derived before)."

That sounds like the "Neo-Fisherian" equilibrium. We can rule out a Phillips Curve that does that, in this sort of NK model where there is no stock of M. It doesn't make sense at the individual firm level. Let's not go there.

I think your guess may be correct. In a NK model with monopolistically competitive firms, the stationary equilibrium Y* will be less than the perfectly Competitive equilibrium Ypc, where all firms are at P=MC. And Ypc is the plausible upper bound on Y, because firms would ration customers if demand exceeded Ypc. So we know 0 <= Y <= Ypc.

But what this seems to suggest is that even an infinitesimally small mistake by the central bank will cause Y to hit the upper or lower bound.

Posted by: Nick Rowe | March 09, 2016 at 11:35 PM

And what that seems to suggest is that the standard NK model is very fragile. (Its results are extremely sensitive to the assumptions.) Because the standard setup is only a limiting case of my setup, and it's a limiting case that is very implausible. The average consumer good does not have zero durability.

Posted by: Nick Rowe | March 09, 2016 at 11:40 PM

"And what that seems to suggest is that the standard NK model is very fragile."

If your problem is that NK models assume that the economy returns to equilibrium, then you should be criticizing DSGE in general, not just NK. Also, I'm pretty sure the main reason NK has the weird results that it does (e.g., explosive output gap and inflation) is that everyone uses linearized versions only, and no one seems to care much anymore about actually writing a gull model with LRAS (or, more accurately, stuff like a labor market that would reasonably prevent explosive solutions).

Posted by: John Handley | March 10, 2016 at 12:18 AM

John: that's a different problem to the fragility problem I'm worrying about here. (And the NK assumption of eventually returning to the natural rate is different again from the more general problem of stability in GE models.)

Posted by: Nick Rowe | March 10, 2016 at 12:34 AM

Nick:

I'm guessing then that you're talking about the slope of the IS curve, right?

Posted by: John Handley | March 10, 2016 at 01:08 AM

John: Yes. Or trying to. It *seems* to be sort of horizontal! Actually, even weirder than horizontal, if you add the lower and upper bounds. It's like a step, only a step going up, when it should be going down!

Posted by: Nick Rowe | March 10, 2016 at 01:16 AM

Nick: the discrete time version of the model still has a normal IS curve, though.

If we want to make this properly IS-LM (one period), that is:

Constant = C(t)(r(t) + d)

Higher r(t) -> lower C(t)

I think the problem only happens when you start dealing with continuous time, which is a strange way to think about IS curves anyway...

Posted by: John Handley | March 10, 2016 at 01:43 AM

Is "C(t)=K(t)=(1-d)K(t-1)+Y(t)" faulty?

I previously commented that C(t) could not equal K(t). Your reply was that it could with the correct choice of units, which is correct.

Now, after thinking about it, I think that the "K(t)=(1-d)K(t-1)+Y(t)" equality should be changed. Here is why:

You need to have

only onedepreciation rate. That rate would apply to a time period (in discrete time, one year). Current output Y(t) is an output of durable goods, only part of which will continue a lifespan into the next period.This creates a limit where the single depreciation rate needs to depreciate a large part of this year's output.We could accommodate this additional limit by writing

K(t)=(1-d)[K(t-1)+Y(t)]

The only change from your original is that term Y(t) is now modified by removing the short lived durable portion. In other words, term K(t) is the sum of the remaining longer term part of K(t-1) and any longer term durables from Y(t).

This change would have a large effect on the size of the single depreciation term, causing it to be bigger. This would change the relative sensitivities of changing term values. (Important when calibrating with existing measurements.)

We can check our work dimensions.

K is output greater than one year duration (a limit). Y has dimensions of "output-year" meaning the output of one entire year. Y needs to be converted to "output greater than one year duration". This can be done by multiplying Y with a term with dimensions containing a "output greater than one year duration per year" dimension. Depreciation has dimensions of ".......per year" and provides the needed conversion. The revised equality should be dimensionally correct. (Awkward but I think this is correct.)

Posted by: Roger Sparks | March 10, 2016 at 06:42 AM

John and Roger: measuring things once a year is only an accountant's convention. If you take the limit as the measuring period gets shorter, and you get different results at the limit than in the limit, something weird is going on. Strictly, K and Y have different units. K is a stock, and Y is a flow.

In the standard setup, C can be cut instantly down to zero. In my setup C can't be cut instantly at all. That's a genuine difference, that discrete time hides, by saying C can be cut by up to dK in "one period".

Posted by: Nick Rowe | March 10, 2016 at 07:12 AM

Here's another way to think about this economy: there are two production sectors. The first sector uses labour (with no capital) to produce capital goods. The second sector uses capital goods (with no labour) to produce consumption goods. The first has sticky prices and the second has perfectly flexible prices.

Posted by: Nick Rowe | March 10, 2016 at 08:01 AM

Good Morning!

Here is another thought: Term d is a rate applied once per year. It collects all the Ks that have decayed during the year, no longer having a lifespan greater than one year. It turns those short-remaining-lifespan-Ks into Cs.

Term d is a collector term.

A thought for the morning!

Posted by: Roger Sparks | March 10, 2016 at 09:16 AM

"Strictly, K and Y have different units. K is a stock, and Y is a flow."

Is Y a flow? Or is Y the stock accumulation built during the year?

K should be stock extending over many years. But, stock K contains depreciating items that have lost their multi-year label. We annually apply the collecting term d to remove these items.

This should all reduce to continuous. We would need to reduce the value of the collecting term d to match the reduced value of the new stock term (which is measured over a shorter period of time). (Remember that even continuous is not truly

continuous. The time intervals are so small that many replications are used to expand the measuring period to real time. There is always an error in the calculation--it's just that the error is so small that it is below our ability (or desire) to measure.)The interest and discount terms would also need to be reduced to match the short time periods in continuous formulation.

Posted by: Roger Sparks | March 10, 2016 at 09:39 AM

It's OK to mix apples and oranges if ...........the result is labeled "box-apples-oranges".

We can apply this permission to the identity

C(t)=K(t)=(1-d)K(t-1)+Y(t)

by relabeling C(t)=C(t)(new durables,old durables). In other words, consumption-in-period-t (C(t) is a function containing the terms "new durables" and "old durables".

Utility and term K would be the same situation. The utility term is the combination of the utility of "new durables" and "old durables". Term K is the combination of "new durables" and "old durables".

I think Nick is right. The economy is indeed a mix of new output and old (and more durable) output. We should continue Nick's effort to put this concept into a tractable model.

Posted by: Roger Sparks | March 11, 2016 at 10:52 AM

Roger: the way GDP is defined is like this. If I buy a new car, that's included in GDP for the year when I buy it, whether I rent the car to you are use it myself (though in the first it counts as "Investment" and in the second it counts as "Consumption"). But if I own a car and rent it to you, the annual services of that car are included in GDP. But If I own a car and rent it to myself, those annual services are excluded.

Posted by: Nick Rowe | March 11, 2016 at 12:33 PM

Here's one thing I don't understand. Doesn't Keshav's solution:

Y/K = r + d - (r+d)^ - b

imply that r

cannotjump? Since if it did, Y/K would have to be infinite, since a derivative of r appears on the right-hand side?Posted by: nivedita | March 12, 2016 at 03:06 PM

Nick: I am impressed with your Durable Goods Model. Impressed enough to create my own post exploring debt from the Durable Goods perspective. I give you a lot of credit, maybe too much.

The post can be found at

http://mechanicalmoney.blogspot.com/2016/03/the-durable-goods-model-and-debt.html

I understand that the math for the model is not yet well described, as I suggested in comments to your post. I don't write about the math, only about the implications of considering all production to be durable and therefore having economic impact over several GDP measuring periods.

I will certainly welcome and consider any comments.

Posted by: Roger Sparks | March 12, 2016 at 08:33 PM

nive: Taken literally, it implies that r cannot *be expected* to jump. Or it it were expected to jump, Y could not be finite. But as Keshav says (later), if Y is bounded from above and below, (it can't go below zero, and it can't go above physical capacity), then those constraints needed to be added to the model, and we get some lambdas in the solution.

Posted by: Nick Rowe | March 13, 2016 at 02:57 PM

an economy where all the production is consumption goods would NOT imply that there is no investment

as investment, to make keynes multiplier get properly calculated needs to include any pre-existing wealth added to the system

so income could be enough for the income earners to pay for the goods and still have some income to save, make the economy viable

thats the income part

the fact that all the production is consumption goods that are durables means we have net production

thats the increase in wealth part

in all future periods discrete or continuous we will have to consider the increase or decrease in wealth by the wealth created (new production) minus the wealth depleted (depreciated, etc)

just general comments

Posted by: djb | March 17, 2016 at 08:26 AM