Two simple NK pictures

I'm staying out of this argument. But I can't resist a challenge to show the New Keynesian model in pictures, with indifference curves, production functions, and budget lines.

I can't do it in one picture. I need two.

Here's the first picture:

There is no investment, government, or foreigners. Consumption and output are the same thing. The first picture is just like Steve's picture of the RBC model, except there's a wedge jammed in between the indifference curve and the production function. The thickness of that wedge represents the degree of firms' monopoly power, which causes the real wage to be below the marginal product of labour, so that at point A the indifference curve is flatter than the production function.

We only need the first picture to figure out where long run equilibrium Y* is, so we know where to draw the green lines in the second picture:

If the central bank sets the real interest rate exactly right, at r*, so the intertemporal budget line is tangent to the intertemporal indifference curve at Y* for both present and future, the equilibrium is at point A, and all is well with the world. (Or, as well as can be expected, given that firms are monopolistically competitive, so we get the blue wedge in the first diagram.)

Now suppose the central bank sets the real interest rate too high for one period, at r'. (But everyone expects the central bank will get it right next period, and so future consumption will return to Y*). That makes the budget line steeper than it should be. The equilibrium is at point B, where the brown budget line is tangent to the "worse indifference curve".

I have also drawn point B in the first diagram, just for completeness. But it doesn't really matter what happens to the real wage at point B in the first diagram. The same blue wedge would be a sloppy fit at point B, which tells us that firms will want to cut prices at point B, when the Calvo fairy gives them permission to do so.

If the central bank had made the opposite mistake, and had set the real interest rate too low, the economy would move horizontally right along the green line to the right of point A (not shown).

That's all folks.

Comments

[primedprimate]

Hmm... Are you sure we'd be to the RIGHT of point A along the green line?

If we're to the right of point A, presumably current Y is larger than Y*. Also, by assumption, future Y is set Y* (that's why we're moving along the green line). Wouldn't we invariably be on a better indifference curve?

Shouldn't the central bank set the interest rate to be as low as possible?

[Nick Rowe]

primed: Yes. I am sure. If the central bank sets r too low, we are to the right of point A, and we are on a higher indifference curve.

NK models assume monopolistic competition. Which means that Y* is too low. If we could permanently increase Y above Y* (to the point in the first picture where the production function is tangent to an undrawn indifference curve) we would want to do that. But we can't do that permanently, or even on average. Trying to do that permanently, or even on average, would simply result in explosive inflation. The best we can do is try to keep Y=Y* every period.

[primedprimate]

In the NK model, wouldn't wages and prices keep pace with each other on overage and over the long run?

If so, why is explosive inflation bad in the NK model if it allows for a higher permanent or average Y?

I am assuming agents only care about consumption and leisure (as is standard) and that prices by themselves don't enter into agents' utility functions.

Are we assuming (without explicitly modeling it) that explosive inflation has real consequences in terms of shoe-leather costs, undesirable redistribution from lenders to borrowers, and tax impacts due to nominal capital gains and tax brackets being in terms of nominal values?

[Nick Rowe]

primed: the NK Phillips curve says (roughly): actual inflation = expected inflation + (Y-Y*)

The only way to have Y > Y* permanently (or on average) is if actual inflation > expected inflation permanently (or on average). Can you fool people all the time? If not, and you try to do so, you end up in Zimbabwe, with no monetary policy at all.

This is the old "accelerationist" controversy. Trying to keep Y > Y* doesn't just lead to high inflation, it leads to ever-accelerating inflation. The central bank is always trying to create more inflation than people expect. What happens when people figure out that's what the central bank is trying to do? BANG goes the monetary system. There is no long run trade-off. Friedman and Phelps, and the vertical long run Phillips Curve.

But that debate is a whole other issue.

[Nick Rowe]

Plus, positive inflation is bad in the NK model, because firms don't change prices at the same time, so it causes relative price distortions. It causes the *mix* of consumption to be wrong, so utility is lower in a fully-anticipated inflation than at zero inflation.

[Frances Woolley]

Nick, have not read the post or done anything more than glance at the diagrams. But, Nick, it looks as if you've got a shaded area in a consumption/leisure picture. I've spent years yelling at students for shading in areas on budget constraint/indifference curve diagrams. The area is defined as 1/2*some consumption*some leisure. But multiplying consumption and leisure makes no sense (well, I guess it could, if one had an appropriately defined utility function, but...).

That wedge isn't really a wedge in the way that economists often draw wedges, it's basically being created by two lines, one showing the slope of the indifference curve (consumer price ratio) and one showing the slope of the production function (producer price ratio), and there's a wedge there because the consumer price ratio isn't the same as the producer price ratio.

So - like the picture, except for the hints of shading in the wedge.

How, by the way, would this be different from an analysis of a tax on leisure or a tax on consumption?

[Nick Rowe]

Frances: I was trying to make the wedge look solid, like a wedge you use (OK, I use) for splitting logs, so it couldn't get squeezed flat. But yes, the shaded area doesn't mean anything. What does mean something is the angle at the sharp end of the wedge. How do you micro people draw that?

It's exactly like a tax wedge, I think. A tax on consumption.

[primedprimate]

Thanks for the clear pictures and your explanations, Nick.

[Frances Woolley]

Nick - a micro person wouldn't put something that looked like a supply-demand tax wedge on an indifference curve-production function diagram because they would know that people would calculate the size of the wedge and use it as a measure of dead weight loss and that would be wrong! Also with a wedge it's harder to see that these are two lines that are tangent to the indifference curve and production function respectively, and to shift the lines around.

The way I'd show the loss associated with less-than-full employment is similar to way that you've done it in the second diagram, and very much like the standard way of showing income and substitution effects. Can you show, on your first diagram, a full-employment equilibrium? Can you measure the difference in well-being between the actual and the full-employment equilibrium as a horizontal or vertical distance - i.e. the amount of consumption or the amount of leisure that would be equivalent to moving to the full-employment equilibrium? It's tricky because it's not simply the distance between the two indifference curves; it's the distance between two particular points.

[Nick Rowe]

Frances: If I wanted to show the size of the welfare loss in the first diagram, I would draw a second, "better indifference curve", that just kisses the production function at the social optimum, and say that the welfare loss is the difference in utility between those two indifference curves. Maybe draw a vertical line between those two indifference curves, starting at point A, and say that the length of that line is the CV or EV (I always get them muddled).

But I'm not using the wedge to show the size of the welfare loss. I'm using it to find point A. I know that the location of point A is determined by:

slope of I curve = (1-(1/E)).(slope of production function) where E is the elasticity of demand (assumed constant)

I want the wedge to represent (1-(1/E)), so that one side of the wedge is tangent to the indifference curve at point A, and the other side of the wedge is tangent to the production function at point A.

[RPLong]

Sorry if these are stupid comments. Maybe it's just because I learned macro from a bunch of Chicago Schoolers. But that first diagram sets off all kinds of alarm bells in my head. I am not sure it makes sense to talk about monopoly power at the macroeconomic level. You're representing the monopolists in both the green and red lines, so where does the wedge come from?

[Nick Rowe]

RPLong: not stupid.

Assume there are n firms. Each firm produces one variety of fruit, and has a monopoly on that variety. The representative agent likes to eat a variety of fruit (Dixit-Stiglitz preferences), so that each variety is an imperfect substitute for all the other varieties. So the individual firm faces a downward-sloping demand curve for its variety of fruit, as a function of its relative price. Labour produces fruit, and the labour market is perfectly competitive. Each firm sets the price of its fruit in terms of money, taking other firms' prices as given. Point A is the symmetric Bertrand-Nash equilibrium. Price is above marginal cost. Real wage is below the MPL.

[Nick Rowe]

Point A is also the symmetric Cournot-Nash equilibrium (with differentiated products), provided n is large.

[RPLong]

Makes sense, actually. I think I was reading too much into the picture. So long as there is product differentiation, it stands to reason that our collective indifference curve would fall some level "below" the PPF. But this also implies (to me) that the utility shortfall is a function of preference at least as much as it is a function of something on the production side. Can we really talk about collective preferences in such a way? I feel as though there is tension between that idea and the idea of a market equilibrium in general.

I'm just rambling at this point now, so I'll give it a rest. ;P

[Nick Rowe]

Think of each individual as a worker/firm. If all individuals worked more hours and produced more fruit then all would be better off. But if one individual worked more hours and produced more fruit he would be worse off, because his relative price would drop.

[Frances Woolley]

Nick, sorry if that sounded a bit snippy - you know that one of the reasons I agreed to be Associate Dean was that I just couldn't take students' messed up indifference curve/budget constraint diagrams any more, don't you?

With you on compensating and equivalent variation - CV and EV are one of the few things in intermediate micro I have to have a cheat sheet for - I simply cannot remember which is which.

The only thing that makes this different from a standard CV or EV calculation is that both curves are non-linear. So in a taxation world I would be able to have a concrete interpretation of the deadweight loss "this is the amount of extra revenue that could be raised by a lump sum tax that gave the consumer the *same* amount of well-being as the existing, distortionary tax". We want something like "this is the extra consumption that would be available to consumers - holding work effort constant - if we were able to move to this better equilibrium". But obviously it makes a difference at which point you hold work effort constant, because of the non-linearity of everything.

[Nick Rowe]

Frances: Yep! CV vs EV, and whether MRS of A for B means more A and less B or vice versa. I have to keep singing The Who song "substitute", but I can't remember if it's the same on both sides of the Atlantic.

[Frances Woolley]

"The simple things you see are all complicated?"

[Richard H. Serlin]

This is preliminary (a big useful thing commenting is for, which I consider intellectual dinner conversation without the physical limitations), but I'm about 3/4ths of the way through The Second Machine Age; the authors seem to imply that the digital computer revolution greatly increases monopoly power in the economy. And they flat out say that economics is very different with digital products, which tend to have little or no marginal cost -- This could really invalidate so much of what libertarian and freshwater economists claim, and it's a crucial point made as far back as by the great growth economist Paul Romer in the early 90s.

I definitely have to finish the book and digest it a while first before I engage in much more than dinner conversation/idea bouncing.

[Nick Rowe]

Richard: Here is my old post where I sketch a model of an economy where all goods have zero marginal cost. Just like those digital goods, where it costs nothing to make an extra copy. It's a very weird economy.