A suggestion for simplifying some macro math

She took a lot of heat for saying it, but I agree with Talking Barbie: math class is hard. It's especially hard in economics. And I find it much harder than most economists. So I have always been on the hunt for devious little tricks to avoid doing hard math. Here's one such trick:

Think back to the very simple Keynesian Cross model. There were two types of math problems the students are forced to do:

Type 1. Assume G=100. Solve for Y.

Type 2. Assume full-employment Y=200. Solve for G that gets us to full employment Y.

Most of the time, economists solve Type 1 problems. And when they have the solution, they use that solution to see whether an increase in G would increase or decrease Y.

But what policymakers really want is the solution to Type 2 problems. For example, a central bank targeting 2% inflation faces a Type 2 problem.

Sometimes there are two Y's for one G, and solving the Type 1 problem should let you see that. But other times there are two G's for one Y, and solving the Type 2 problem should let you see that. That problem cuts both ways.

In the simple Keynesian Cross model, Type 1 and Type 2 problems are equally hard to solve. But when we introduce expectations and intertemporal optimisation into the model, Type 1 problems can be much harder to solve than Type 2 problems.

So pose the question as a Type 2 problem:

"Assume inflation is always at 2% and output is always at potential. So agents always expect 2% inflation and always expect output to be at potential. Solve for the monetary and/or fiscal policies that would make that happen."

That's much easier than the Type 1 version.

That's the devious little trick I used in this post on fiscal policy in New Keynesian models to skive off doing the (for me) impossibly hard math of a Type 1 problem. The solution to the Type 2 problem popped out very easily. And it taught me something the much cleverer economists solving Type 1 problems seem to have missed.

Comments

[Kevin Donoghue]

You need a way to weed out solutions like: promise to implement severe austerity 50 years from now, with G(t) dropping to zero and staying there forever. The model will tell you that works fine as a way of eliminating output shortfalls. The immortal representative agent gratefully revise consumption plans in the light of the lower future tax burden and C(t) rises. There's a reason why those clever NK economists don't spend much time on policy rules of that sort.

[Nick Rowe]

Kevin: I think it's the other way around.

If we set it up as a Type 2 problem, we should be able to find *all* the policy rules (for Gdot(t) and r(t) in this case) that let us hit the target. We can then pick the one(s) we like best, and that don't lead to bad outcomes 50 years from now. In this case it would be something like: "cut Gdot when the ZLB is a binding constraint, and raise Gdot when it isn't".

If we set it up as a Type 1 problem, we are only looking at that narrow class of policy rules the modeller happened to try out and figured would be easy to find solutions for.

[Nick Rowe]

In fact, any halfway competent math economist (i.e. anyone but me) could solve for the policy rules for G(t) and r(t) that maximise the present value of the SWF subject to the constraints that r(t) = n(t) - alphaGdot(t) and r(t) > the ZLB.

They couldn't do that with the Type 1 method.

[Min]

"Type 2. Assume full-employment Y=200. Solve for G that gets us to full employment Y. . . .
"But what policymakers really want is the solution to Type 2 problems."

Would it were so!

[Nick Rowe]

Min: Touche! (But that's not our department.)

[Dave]

In macro, solving type 2 problems is known as calibration! Specify the equilibrium equations and outcomes, then find the parameters that make it happen.

[Nick Rowe]

Dave: Hmmm. I think calibration is something different again. Because you are solving for the parameters, not the policy variables. Related, but different.