Update: see update at very end.
Some people aren't as rational and future-oriented as agents in New Keynesian models. Some people aren't as present-oriented as agents in Old Keynesian models. A hybrid Old/New Keynesian model, with both types of agents, looks attractive.
But other people's hybrid models looked too complicated, so I decided to build my own. All you really need is one equation. (OK, plus a couple more little ones to help explain it.)
I discuss the first-best solution. Then the second-best solution, if the Zero Lower Bound on nominal interest rates is a binding constraint.
The results are not what you might expect. If the ZLB is a binding constraint, the government should reduce the growth rate of government spending, and/or increase the growth rate of taxes, to ensure the economy remains at potential output.
(Paul Krugman again misses this point. "If some of them are instead liquidity-constrained, the increase in income from the rise in G will lead to some increase in C as well, and we have a story that is even closer to the old Keynesian version." It is not the rise in G; it is the fall in the growth rate of G that does the work, which is far away from the Old Keynesian version.)
I explain this weird result, and reconcile it with the Old Keynesian intuition.
The Old Keynesian agents are straight out of the textbook. C(t)=A(t)+B(Y(t)-T(t)), where A is autonomous consumption, B is the marginal propensity to consume, and (Y-T) is current disposable income. I've written it A(t) to allow for shocks to autonomous consumption.
The New Keynesian agents are also straight out of the (different) textbook. If the real rate of interest r(t) exceeds their rate of time preference n(t), their current consumption will be less than their permanent income, and will be growing over time. Assume it's Cdot(t)=R(r(t)-n(t)), where R(.) is some function representing the Euler equation with the properties R(0)=0 and R' > 0, where Cdot(t) is consumption growth. I've written it n(t) to allow for shocks to the rate of time preference.
What we are looking for is a fiscal and monetary policy (I look at both together) that will keep output at potential at all times. Ignoring investment and net exports to keep it simple, that means we want C(t)+G(t)=Y*(t) for all t, where Y*(t) is potential output. Or, taking the derivative with respect to time, we want Cdot(t)+Gdot(t)=Y*dot(t) for all t. Consumption growth plus government expenditure growth needs to equal potential output growth.
Assuming some fraction k of agents are Old Keynesian, and (1-k) are New Keynesian, we want a fiscal/monetary policy that ensures:
k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] + Gdot(t) = Y*dot(t)
That's the model.
If k=1 and you integrate with respect to time (delete the "dot"s) you get the first-year Keynesian Cross model's solution for fiscal policy G(t) and T(t) to keep output at potential.
If k=0 you get the New Keynesian model, where Tdot(t) disappears because Ricardian Equivalence holds.
(I've cheated a little because strictly speaking k should represent expenditure shares rather than population shares, but I don't think this matters much.)
The government/central bank has three instruments: G(t), T(t) and r(t), and it only needs one instrument to keep the economy at potential. But if the government wants to maximise agents' utility, it will need to use all three instruments.
Philosophically this is a bit tricky, as Frances Woolley explains, because the government is acting in loco parentis for the non-rational Old Keynesian agents. If A(t) increases, for example, does this represent a change in their underlying preferences that should be accommodated, or is it just an irrational whim that should be suppressed?
The simplest assumption (though one that is certainly contestible) is that the true preferences of the Old Keynesian agents are exactly like the preferences of the New Keynesian agents; they just don't act on them. Given this assumption, we can solve for optimal fiscal and monetary policy.
The full solution is left as an exercise for the reader. Here are my answers:
T(t) should alone be used to offset shocks to A(t). If the Old Keynesian agents go on an irrational consumption binge, raise taxes enough to stop them, without changing G(t) or r(t).
Set G(t) so that the marginal utility of G(t) is equal to the marginal utility of C(t). (Taxes are assumed non-distorting.) This means that if Y*(t) increases, both C(t) and G(t) should increase.
Set r(t) to ensure that output is at potential, given G(t) and T(t). This means that r(t) must respond positively to n(t) and to Y*dot(t), and negatively to Gdot(t) and positively to Tdot(t). (Look at the equation!)
That is the first best solution for fiscal and monetary policy.
Now suppose that monetary policy is constrained by the Zero Lower Bound. The central bank cannot cut r(t) to implement the first-best solution. What are the second best solutions for G(t) and T(t)?
Inspect the equation above. To offset an r(t) that is too high (because of the ZLB), the government needs to reduce Gdot(t) and/or increase Tdot(t), if it wants to keep output growing at potential.
That result is very different from the Old Keynesian model, which says you should increase the level of G(t) and/or reduce the level of T(t), if you want to avoid a recession.
Here's the intuition:
Let's start in a stationary equilibrium, to keep it simple. All the dot variables are zero, and nothing else is changing. Then the rate of time preference n(t) falls. If G(t), T(t) and r(t) were unchanged, the drop in n(t) would cause the New Keynesian agents to want to cut C(t) and increase Cdot(t). (They want to save now, so they can grow their consumption over time.) But this would cause a recession. The first best optimal policy is to keep G(t) and T(t) constant, and reduce r(t), so that Cdot(t) stays at zero and C(t) stays constant. But if the ZLB prevents r(t) from being cut, the second best policy will be to reduce Gdot(t) and increase Tdot(t). Both these policies allow the New Keynesian agents to have a positive Cdot(t) without cutting C(t). By reducing the growth of government spending, and reducing the growth of consumption by the Old Keynesian agents (by growing taxes), the permanent income of the New Keynesian agents is now higher than their current income, and this offsets their reduced rate of time preference, and prevents them wanting to save part of their current income.
Why is this result counter-intuitive, if you approach this with an Old Keynesian intuition? It's because if an Old Keynesian economist hears that agents have an increased desire to save, he interprets that as a reduction in A(t), to which the first best optimal response is a cut in T(t), not a cut in r(t). A reduction in A(t) never causes the ZLB to be a binding constraint on the first-best solution in the first place, if we assume it's an irrational whim that should be suppressed.
Update: I have to confess: these results still creep me out. So I'm not surprised if you have the same reaction. Are they "right"? Well, if by "right" you mean "do they follow from the model?", then yes, I think they do. But would I implement them? No, I wouldn't. Despite my giving the intuition, there is still something weird going on here, and I don't know what it is. My hunch is that the indeterminacy problem of the New Keynesian model is at the root of all this. I'm beginning to wonder if the indeterminacy of equilibrium output (under interest rate control) mightn't be a feature, rather than a bug, of the New Keynesian/Neo-Wicksellian model. And policy recommendations that are based on solutions that assume away the indeterminacy problem (which is what NK macroeconomists do) are deeply flawed. The recession was just the indeterminacy problem telling us it refuses to be assumed away.
It's sort of weird. Macroeconomists like Roger Farmer are trying to build indeterminacy into their models to help them explain the world; while New Keynesians, who already have indeterminacy in their models to begin with, are trying to get rid of it by assuming it away.
Worthwhile Canadian Initiative 
Is it fair to consider an investment-free model here? Or, for that matter, one without money?
If there is no investment, then what precisely are consumers doing with their non-consumed surplus? For that matter, how are G(t) and T(t) not identical? Provided the central bank does not monetize the deficit (generally frowned upon), without the capacity for investment (or disinvestment) then the resulting bonds issued to finance government spending must exactly match the un-spent consumer surplus.
Permitting investment allows the accumulation or drawdown of some un-consumed but stored surplus, such as a granary full of wheat.
My intuition suggests that perhaps the differences are reconcilable if we allow for actual return on investment, much as in traditional “stimulus” programs.
Also, speaking as a mathie, I think you can get some very qualitatively different results if 0 < k < 1. Then instead of an algebraic, steady-state relationship (either after integrating once or directly), you end up with a bona-fide differential relationship that can admit a solution path.
MF: no worries. I was losing it a bit!
“I think I see what you are doing. If you could get all the agents in the model to expect that y=y* then what values would you have to set the various variables at t=0 [and for all times t] to be be consistent with meeting these expectations [and actions]?”
You’ve got it! [with my little bit added in brackets, because they actually have to act so as to fulfill their expectation of full-employment, and they won’t do that if policy is wrong, even if they do expect full employment, and you have to do it not just when t=0 but for all t.]
But tell me, was it my gas pedal/speed limit analogy that helped you get it, and you found my other post didn’t explain it clearly? (As a teacher, I need to know things like that.)
Roger: “I think both equation 1 and 2 are correctly written.”
no. They make even less sense now.
“Our predictions must accommodate introductions of new capital (money) into our equations.”
Money and capital are not the same thing.
This is not getting us anywhere. Let’s stop.
Peter N:
“Increase N and you decrease A.”
Let’s say it the other way around: if A decreases the government needs to increase N.
Majromax: Is it fair to consider an investment-free model here?”
New Keynesians often do this, for simplicity, so if it’s fair for them it’s fair for me, since this post is me speaking to New Keynesians.
“Or, for that matter, one without money?”
Same answer. Money is not explicitly in the NK model, but it has to be there implicitly. (I think this is a big problem, but this is not something NK’s can complain about.)
The rest of your questions are intelligent questions but mostly first year questions. Sorry, but I’m not going to explain the answers, except to say “bonds and money”. This old post may help
“Also, speaking as a mathie, I think you can get some very qualitatively different results if 0 < k < 1. Then instead of an algebraic, steady-state relationship (either after integrating once or directly), you end up with a bona-fide differential relationship that can admit a solution path.”
I’m not sure I understand you. (Probably because I’m bad at math.)
I think it was the “AAAAAAAAAAAAAAAARRRRRRRRRRRRGGGGGGGHHHHHHHHHHHHHHHHH!!!!!!!!!!” that shocked me into reading your earlier post more carefully, after which it made sense.
(Also I had read some of your earlier posts on the hidden assumptions in the NK model about y=y*, and somehow had formed the opinion that this was a bad thing for a model to include)
MF: Aha! So I have now learned a new pedagogical technique! Saying “AAAAAAAAAAAAAAAARRRRRRRRRRRRGGGGGGGHHHHHHHHHHHHHHHHH!!!!!!!!!!”!
(Yep, my earlier stuff on the NK’s just assuming expected future Y=Y* would have confused it. Because I’m doing that too here.)
… but I do have one further question that I hope is slightly above 101-level. How do New Keynsian agents in this model work at the zero lower bound? In the derivation, you say:
Nick Rowe: “My equation is then a necessary condition for Y=Y* for all t, but it is not a sufficient condition. That’s because of the indeterminacy problem. There are multiple equilibria. And I worry about this.”
Don’t worry, be happy! 🙂
Multiple equilibria are a feature, not a bug. IMO, all human systems are chaotic (in the technical sense, so they can appear to be stable for a long time) or on the edge of chaos, typically with multiple (quasi-) equilibria. Dysfunctional systems are typically in a suboptimal equilibrium, which can last a long time. IMO the so-called New Normal is such a suboptimal equilibrium, and needs to be recognized as such, or it will become a self-fulfilling prophecy. One feature of a dysfunctional system is doubling down by policy makers. X isn’t working, so we need more X. See Europe and the US. {weak grin}
Min: “Multiple equilibria are a feature, not a bug.”
I think I said the same thing (or that they might be a feature not a bug) in a comment just recently, but now I can’t find it. (spam filter strikes again?) And recessions might just be the indeterminacy problem telling us we can’t just assume it away.
And I said how ironic it is that economists like Roger Farmer are trying to put indeterminacy into their models while New Keynesians are trying to just assume it out of their models!
After much thinking, I can better explain it to myself.
I’m going to rearrange to C=Y-G, so:
k[Adot(t)+B(Ydot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] = Ydot(t) – Gdot(t)
And now I can understand this, as at the ZLB by decreasing Gdot(t) we increase the r.h.s. allowing to increase consumption… but…
for the OK if we take C(t) = A(t) + B(Y*(t) – T(t)), we really should be substituting back
So we’d get:
Cdot(t) = k[Adot(t)+B(Cdot(t) + Gdot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))]
Cdot(t) * [1 – kB] = k[Adot(t)+B(Gdot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))]
Now I’ll put back the r.h.s.
Cdot(t) * [1 – kB] = k[Adot(t)+B(Gdot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] = [Ydot(t) – Gdot(t)] * [1 – kB]
IF we want to preserve NK consumption, then I do think that you’re right about the indeterminacy. Since we’ve only specified a condition on the derivative, it is possible to integrate and to get a different constant (i.e., it is underspecified and shocks to absolute consumption are certainly possible). And I think I understand better now when you say:
Both these policies allow the New Keynesian agents to have a positive Cdot(t) without cutting C(t). By reducing the growth of government spending, and reducing the growth of consumption by the Old Keynesian agents (by growing taxes), the permanent income of the New Keynesian agents is now higher than their current income, and this offsets their reduced rate of time preference, and prevents them wanting to save part of their current income.
I think the tie-in to reduction of the current income (or at the least in making its rate decrease), isn’t well represented in the equations presented. It should be possible to capture that in some term.
Thanks for taking the time to respond to all our comments!
Seen this one?
An Analytical Critique of ‘New Keynesian’ Dynamic Model
Click to access asada.pdf
“The most notorious problem of the prototype NK dynamic model is the phenomenon that is called the sign reversal in New Keynesian Phillips curve, pointed out by Mankiw (2001) clearly.”
and
“Lemma 3.1(1) means that the prototype NK model is accompanied by equilibrium that is unstable in the mathematically orthodox sense.
The equilibrium of NK dynamic model will never be reached. This undermines the basis of the NK theory, however. Hence, the NK literature adopts the trick that makes the unstable system try to mimic a ‘stable’ system by using the so-called jump variable technique.”
They draw on this paper by Mankiw:
Click to access royalpap.pdf
Peter N: The Mankiw paper (I had seen before) is about an empirical problem with the Phillips Curve assumed in NK models. Mankiw is right, IMO, but that problem has nothing important to do with the rest of the NK model. Plus, Mankiw and others have been working on fixing it.
The Asada paper looks (I think) closer to the indeterminacy problem I am talking about. But he lost me when he said “Jacobian”, because I have forgotten what that means, and why it matters. Then it wandered off into Lemmas. It’s probably a good paper, but not good for me, unfortunately.
I think his Old Keynesian model is just a tarted up early 1970’s ISLM with adaptive expectations, but I didn’t look too closely.
“2. Formulation of the Prototype NK Dynamic Model
The simplest version of the prototype NK dynamic model may be formulated as follows.”
Philips curve derived from the optimizing behavior of the imperfectly competitive firms with costly price change
π(t) = E(t)[π(t+1)] + α(y(t) – y) + ε(t)
IS curve derived from the Euler equation of consumption
y(t) = E(t)[y(t+1)] -β(r(t) – E(t)[π(t+1)] – ρ0) + ξ(t)
Taylor rule
r(t) = ρ0 +π +γ1(π(t) -π) + γ2(y(t) – y)
Simplifying and assuming rational expectations, you have 2 equations in two variables Δπ and Δy. The jacobian is the matrix of the coefficients. From this you calculate the characteristic equation. Its roots determine the stability. If the absolute value of 1 root > 1 the equations are unstable giving a saddle point equilibrium. If 2 have absolute value > 1 the equations are totally unstable, then all paths diverge either monotonically or cyclically. In this case, at least 1 root is > 1.
…
“The equilibrium of NK dynamic model will never be reached. This undermines the basis of the NK theory, however. Hence, the NK literature adopts the trick that makes the unstable system try to mimic a ‘stable’ system by using the so-called jump variable technique. In the NK dynamic model, both of two endogenous variables π(t) and y(t) are considered to be jump variables or not-predetermined variables , the initial values of which are freely chosen by the economic agents. only the initial conditions of the endogenous variables are chosen so as to ensure the convergence to the equilibrium point.”
Peter N: OK. I’m not worried about Y(t) being a jump variable. If new information arrives in a jump, you jump your spending up or down. And if you don’t, it means you have habit persistence, or adjustment costs, which need to be built into the model anyway. I’m more worried about inflation being a jump variable, but then I think the Calvo Phillips Curve is wrong, empirically.
Well, I have slept on this and done some more thinking about this. Back to basics.
Suppose a simple linear model that holds when y(t) = z(t). That is,
1) Ax(t) + B = y(t)
2) Ax(t) + B = z(t)
Now, starting when y = z, let the system be perturbed so that y > z, but the difference is small. Now we have
3) Ax(t) + B ≃ y(t)
4) Ax(t) + B ≃ z(t)
Ax(t) + B does not tell us how to return to the condition where y(t) = z(t).
But let’s see what we can get out of this. When y(t) = z(t) we have
A xdot(t) = ydot(t) = zdot(t).
If we hold y(t) constant, we can use A xdot(t) = zdot(t) to increase z(t) so that it equals y(t). With a small perturbation, that might work. Or we can hold z(t) constant and use A xdot(t) to decrease y(t) so that it equals z(t). Even assuming that one or the other works, we do not know which to try. And that is true even if one of y(t) or z(t) is exogenous, and we cannot raise or lower it. Exogeneity or endogeneity is not part of the equation. A kind of relativity holds, so that increasing z(t) to equal y(t) and decreasing y(t) to equal z(t) are equivalent operations. We simply cannot tell from the equation whether to increase x(t) or decrease x(t), even when doing one or the other will work.
OC, in real life, we can often try one and reverse course if it is not working. We can also use what we have learned from previous experience. 🙂
Min: I’m not 100% sure I understand you correctly, but assuming I do:
For the OK model, I can solve for Y as a function of G and shocks, or G as a function of Y and shocks, and I know what’s going on. I know if I increase G then Y increases. So if Y < Y*, I know I need to increase G.
For the NK model, it’s not that simple, but only because of the indeterminacy problem. There is only one path of r compatible with a given path of Y and shocks, but there are many paths of Y compatible with a given path of r and shocks. NK keep just assuming, that if Y < Y*, it means r > r*, so you need to cut r (which is true in an OK model like ISLM, but is not true in a NK model.
Nick Rowe: “For the OK model, I can solve for Y as a function of G and shocks, or G as a function of Y and shocks, and I know what’s going on. I know if I increase G then Y increases. So if Y < Y*, I know I need to increase G.”
Right. You have another equation that tells you about Y, whether Y = Y* or not. So you can use that information.
Nick Rowe: “For the NK model, it’s not that simple, but only because of the indeterminacy problem. There is only one path of r compatible with a given path of Y and shocks, but there are many paths of Y compatible with a given path of r and shocks. NK keep just assuming, that if Y < Y*, it means r > r*, so you need to cut r (which is true in an OK model like ISLM, but is not true in a NK model.”
Many thanks, Nick. 🙂
Then, if indeterminacy is a feature, the thing to do is to go empirical, to experiment, as Roosevelt did during the Great Depression. And as the US is not doing now.
Also, OC, if there are multiple equilibria, you can change the payoffs. But that means a level of gov’t intervention that is not now popular.
“OC” = ?
Or, recognise that restoring determinacy means putting M back into the model, and into the policy rule in the world, and not just r.
“You have another equation that tells you about Y, whether Y = Y* or not.”
It’s really the same equation, except you read it from left to right rather than right to left.
NK only has an r=F(Ydot) equation, not an r=F(Y) equation
Moi: “”You have another equation that tells you about Y, whether Y = Y* or not.”
Nick Rowe: “It’s really the same equation, except you read it from left to right rather than right to left.”
What I am trying to say is something like this.
1) Ax + B = y
is not the same as
2) Ax + B = y , given that y = y*.
The first equation gives you more information than the second, information that you can use.
Information about the relationship between x and y, I mean. 🙂
Moi: ” OC, if there are multiple equilibria, you can change the payoffs.”
Nick Rowe: “OC” = ?”
Changing the payoffs alters the game. I expect that you will still get chaos, but chaotic systems can remain relatively stable for a long time. Like the solar system. 🙂