Tight money as binding output quota, and upward-sloping IS curves

Like all metaphors (and like all models), it works up to a point. Let's see how far it takes us.

David Andolfatto has a lovely neat little model that he uses to engage my point that the IS curve may slope up -- so tight monetary policy may cause the real interest rate to fall as well as output and employment to fall. It's a simple Overlapping Generations model. The young have a perfectly inelastic labour supply, and save all their labour income. The old do not work, and consume all their past savings. There's a simple Neoclassical constant returns to scale production function Y=F(N,K)=C+I, with 100% depreciation on capital per generation so K(t)=I(t-1).

Money is not in the model explicitly, but David shows that if some entity he calls "the central bank" (somehow?) sets the real interest rate too low, the result is a fall in employment, output, and saving-and-investment. It looks like secular stagnation, but it is caused by bad "monetary policy".

I want to explore a slightly different approach. Suppose there is some entity (let us call it "the entity") that enforces a binding quota on output in David's model. What happens to the real rate of interest?

Start in full steady-state equilibrium in David's model. Let Y=100. What happens if the entity issues quotas for only 90 units of output?

Y falls by 10%, obviously, but what happens to employment of labour and employment of the existing stock of capital? One or both must fall, but which? There is a fixed existing stock of capital within the period, with zero opportunity cost, so the within-period supply of capital services is perfectly inelastic. And the supply of labour services in David's model is also perfectly inelastic. Taken literally, both real wages and real rentals would fall to zero, and quota owners would earn the whole of national income, with the capital/labour ratio actually used being indeterminate.

So let's change David's model just slightly. Assume there is an opportunity cost to labour (the young get utility from leisure). So capital stays fully-employed, and employment of labour drops when the quotas are imposed. So real wages fall as the economy slides down along the upward-sloping labour supply curve. But the ratio of real capital rentals to the real wage must equal the Marginal Rate of Technical Substitution for cost-minimising firms facing an output quota, and that ratio must fall as the labour/capital ratio falls. So real rentals must fall even more than real wages, percentage-wise.

If both real wages and real capital rentals fall when the entity imposes binding output quotas, what happens to the "adding up condition" of a Constant Returns to Scale production function? It gets violated. Because some of the output gets paid to whoever owns the quotas. You need extra quotas, as well as extra capital or labour, to produce extra output. Quotas are a joint input with either capital or labour.

The real rate of interest in David's model equals the real rental rate on capital (minus the 100% depreciation rate). So if we define "the IS curve" as that curve traced out in {real interest rate, real output} space by varying the number of output quotas issued by the entity, it is clear that the IS curve slopes up. A fall in Y is associated with a fall in r.

That tells us all we need to know about the immediate (within-period) effects of a binding output quota in David's (slightly modified) model. But if we want to know what happens next period we need to know who gets the rents from the output quotas. If it's the old they will consume those rents. If it's the young they will save those rents. If it's the old, then saving-and-investment will fall as a percentage of output, because real wages income falls as a percentage of income, so the real interest rate will rise over time as the capital stock falls. If it's the young, then saving-and-investment will rise as a percentage of income (because capital income falls) so the real interest rate will fall even more over time as the capital stock rises.

Now for the metaphor bit.

In a monetary exchange economy you need money to buy goods, and firms won't produce goods unless someone buys them. If the stock of money is too small, relative to the price level and the velocity of circulation, then money is very much like a binding quota on output. And the central bank that issues that money is like the entity that issues quotas. A tight monetary policy means issuing too little money, relative to the price level and velocity of circulation.

And it is clear that the effect of tight monetary policy in David's (slightly modified) model is to reduce the real rate of interest. The IS curve slopes up.

But who gets the quota rents? Are they saved-and-invested or are they consumed? What happens over time? Hmmm. If prices are sticky but wages and rents are perfectly flexible, then it is the "firms" that hire both capital and labour services that get those quota rents. And if it is the old in David's model who own the firms (though, strictly, there are no "firms" in David's model) then it is the old who get the quota rents, and those rents will be consumed. So the capital stock will fall over time, and so the real interest rate will rise over time. (Not to mention that the quotas themselves are a durable asset, and hence a savings vehicle that displaces investment a la Samuelson/Diamond/whoever.) But if wages are sticky and output prices and capital rents are perfectly flexible, it is the young workers who get those quota rents, so the capital stock will rise over time and the real interest rate will fall further over time. (Sticky capital rentals would lead to unemployment of capital, not labour, but this is implausible if firms own capital and rent it to themselves.)

(Which also, parenthetically, tells us something about the distribution of income.)

I think that's right.

Low real interest rates do not mean monetary policy is loose. Low real interest rates are a consequence of tight monetary policy. But this does not mean that to loosen monetary policy you should tell the Fed or Bank of Canada to "raise interest rates". Because they will totally misunderstand you, and do the exact opposite of what you want them to do. Given current operating procedures, they will think you are telling them to reduce the money supply, not increase it. They are funny like that.

Comments

[Hugo André]

Thank you for this. It was bothering me that both of your previous upward sloping IS line posts assume that the capital stock is kept constant. Your discussion here about what would happen under sticky wages vs prices is therefore very enlightening for me.

I may well be wrong but it looks like you may have left something out though. All capital depreciates at the end of the period and this makes it unclear what happens to the quota-induced interest rate reduction. Sure, the still-employed workers get a larger share of total income (and save it all) but some of them are now unemployed and can’t save anything. Keep in mind that the economy’s output is now only 90. Whether or not the interest reduction persists depends on what the quota does to the relative net capital/labour shares of income. If the relative shares remain constant, the Capital/Labour ratio (as well as the MRTS) reverts to its pre-quota level. If savings were 50 and consumption 50 when Y was 100, both have now changed to 45.

If I’m right, your paragraph about what happens in the next period is a bit wrong in places. Am I missing something?

[Hugo André]

That was sloppy of me. Of course I should have written that if relative income shares remain the same, the savings/consumption ratio returns to it's pre-quota level, BUT NOT 45 EACH since some of the income goes to the owners of the quota (whoever they are).

[Nick Rowe]

Thanks Hugo Andre. Yep, in the back of my mind I always tend to assume that recessions are short enough that both gross investment and depreciation over the duration of the recession are small relative to the existing stock of capital, so we can treat K as approximately fixed. But in David's model, with 100% depreciation per "period", we are forced to think what happens next period.

Yep. In my version of David's model income shares of both labour and capital fall in a recession, and capital's share falls more than labour's share, because quota owners' share increases.

(Watch the crazy conspiracy theorists come on here wildly speculating about who those mysterious quota owners are!)

[Lord]

Depends on capital depreciating through use or through time doesn't it?

[Nick Rowe]

Lord: yes. David assumes rust, not wear. Assuming wear would change the results a little, because the supply curve of capital services would not be perfectly inelastic.

[ThomasH]

I think this is right, but the political economy needs to be brought in. I suspect that the Fed (after a few months of distraction) knew in 2008-2015 that the economy needed more monetary stimulation. They were, however, politically (more than intellectually) constrained not to reduce interest rates below 0, not to buy "too many" LT assets (buying foreign exchange was just not in the cards), and not to allow the inflation rate to approach, much less exceed, 2%. These political constraints explains the Fed's tight money policy 2008-15 and looks like it (plus some independent yen for "normal" interest rates) will continue to explain it going forward. If you would work those constraints "They are funny like that" into your model, it would explain a lot more.

[Luc Hansen]

I think only (some) North American economists think what you express in the first two sentences of your final paragraph, and I'm damned if I know why.

Back to the books, I guess.

[Nick Rowe]

Luc: the books may not help you much on that one. A metaphor may help. You are balancing a broomstick upright on the palm of your hand. If you first move your hand North, so the broomstick starts to lean South, you must subsequently move your hand even further South than you started, to keep the broomstick upright. (Except broomsticks don't have expectations.)

[jonathan]

I must say, I don't think much of David's model:

(1) No money or price stickiness, so how is the central bank setting r? Not by changing the real money supply, that's for sure.

(2) He has the central bank fixing the *rental rate of capital* not the interest rate. That means that not only is the interest rate (going forward) set too low, but the contemporaneous rental rate can't rise when capital is destroyed. Nobody thinks the central bank can set the current rental rate. Essentially he's assuming a price control.

(3) He doesn't discuss how rationing occurs following this price control. He says that because r can't rise, w doesn't change either. But if K is rationed, it will no longer be the case that r = MPK, and thus it will no longer be the case that r fixed implies K/N fixed, and therefore w fixed.

I'm sorry, but I just can't take this model seriously. When I try to understand what's making this model tick, my head hurts.

[Nick Rowe]

jonathan: I was fine with David's model until he had the central bank fix r. Like you, I wondered how it would do that, and if it did it, how the rationing would play out. Which is why I wanted to start with an explicit quota on output. Assuming an explicit price control on capital rentals would be another way to go, but I can't see how it would cause unemployment of labour, simply an excess demand for capital services.

[jonathan]

Nick:

I was "fine" with the model up to that point, meaning I understood what the model was doing and thought it was reasonable, but I didn't like that he called it an "IS curve".

In his basic model, there's a negative relationship between (r,Y) because investment is a fixed fraction of income:

K' = I = a*Y

Thus higher Y => higher K' => lower MPK' => lower R'

This is already quite different from the Keynesian IS curve, where lower R' implies higher Y because lower R' raises demand, and output is demand-determined. Basically the causality is reversed: in his model, causality flows from Y to I to R', whereas in the Keynesian model it flows from R to I/C to Y.

[Nick Rowe]

Jonathan: agreed that it's different from the Keynes/Hicks IS curve. But I disagree a bit on causality flowing from R to I (and/or C) to Y. Causality could flow either or both ways (though Keynes, and many Keynesians, seem to talk about it flowing from R to Y). I see the big difference as being that the Keynes/Hicks IS curve tells us about the relationship between R and Y holding Y* constant, while David is talking about the relation between R and Y* assuming Y=Y*. (Y* = "potential").

[notsneaky]

Nick and jonathan, I think part of what you're getting at is that there's no reason in David's model why the disequilibrium must occur in the labor market, with excess labor supply. It could just as easily occur in the capital-rental market, where firms simply wish to rent more capital then exists, and tough noogies to them. In the latter case, the "IS curve" (if I'm understanding what is meant by that correctly) would be vertical.

(I take the "IS curve" to be a relationship between the departure of the central bank fixed rental rate from it's equilibrium level and output)

I think I'm right.

[notsneaky]

In particular there could be a continuum of valid disequilibria, depending on the rationing and quota rents. You could have a little bit of a disequilibrium in one market and a whole bunch of disequilibrium in another. What you wind up with depends on the way the quotas are rationed and how "sticky" the real wage is (I know David says it's flexible, but he actually assumes it's fixed).

[Nick Rowe]

notsneaky @9.57: I *think* that's right. If there's a binding price ceiling on capital rental rates, I think that would simply cause an excess demand to rent capital, with owners of firms collecting profits that should have gone to owners of capital. Real wages and employment and output would stay the same. So the "IS curve" is vertical at Y* in that thought experiment.

But I don't agree (or understand) with your @12.24. I *think* the only way David gets employment to fall is that he insists on R=MPK despite binding rent controls, so a fall in N is the only way to get that, and a rise in W is the only way to keep W=MPL.

[jonathan]

Thinking about it a little more, maybe David is thinking about firms as they are in the New Keynesian model.

In the New Keynesian model, firms that can't adjust their prices are forced to produce exactly enough to meet the demand they face. Suppose all firms are in this position, with fixed prices p=1.

Now this doesn't mean the firm problem is trivial -- firms still optimally choose between capital and labor, subject to the constraint that they produce enough output to meet demand (cost minimization problem). What changes is that firms' real marginal cost of production need not equal 1.

In this case, we're one condition short of specifying equilibrium. Then we just follow the New Keynesian practice of letting the Fed pick the equilibrium, which corresponds to a particular R.

Now in the NK model, there's an obvious "story" behind this policy: in the background, there's money demand, and the Fed adjusts the money supply to change interest rates, which then affect demand through the incentives for consumption and investment.

But that doesn't happen in David's model, because C and I (the two components of demand) are entirely determined by income in his model. The young save/invest all their income, and the old consume all the income. So the Fed can't change demand by adjusting interest rates.

So what is the Fed doing in David's model? Effectively, it's picking the level of output, and then R (and w) adjust so that firms produce the desired amount. In this case, a low R(t) corresponds to *low* demand in period t. (By contrast, in the NK model the Fed picks R(t+1), not R(t)).

How does the Fed set the level of demand in this model? It's not really clear -- it just picks the level of output (setting NGDP expectations?). What matters is that there are many paths of output possible, and he assumes that the Fed is able to pick one. A low level of R corresponds to the Fed picking a low path of output.

If you want to call that "monetary policy", it sounds more like contractionary monetary policy to me! But then, this is essentially my whole complaint with Neo-Fisherians in the first place: They observe that if the Fed magically sets high growth expectations the interest rate goes up, and then conclude that the Fed should raise interest rates to raise growth!

[Nick Rowe]

jonathan: I think that's right.

"(By contrast, in the NK model the Fed picks R(t+1), not R(t))."

Hmmm. I missed seeing that.

[jonathan]

Actually, the model I described above would cause w to fall under his thought experiment, so it doesn't exactly coincide to what he's describing.

He seems to be assuming that the Fed gets to pick labor supply.

[jonathan]

"(By contrast, in the NK model the Fed picks R(t+1), not R(t))."

Actually, I reread David's post (after writing my long comment above), and he might have used R(t) a little inconsistently. At one point he defines R(t) as the rental rate in period t+1, and thus the interest rate on savings in period t, but then later talks about R(t) as the rental rate in period t (I think).

In any event, he's assuming the Fed picks the path of output y(t), which then determines the rental rate in every period.

[notsneaky]

The way I was thinking about it, and it might not be right, is that the interest rate is given by the (too low, since the shock is a fall in capital) steady state and "the entity" sets the price of a quota (right to rent a unit of capital), q. Of course that sort of begs the question - why not just set a different interest rate to begin with? But if it is done that way (and assuming quota revenue is redistributed lump sum to somebody) then the usual r=MPK condition becomes q=MPK-r, or another words, the demand for quotas. K is given, so q is a function of L (employment). We still have w=MPL, which is also a function of L. So we have three unknowns; q, L and w and two equations so you have an indeterminacy - a continuum of disequilibria. If "the entity" also picked the quota price q (again, why not just set a different r), we can then solve for L and w.

The problem with that is that if there is excess labor supply any worker could go to a firm and propose working at a real wage and there's still nothing stopping the wage rate from falling. But letting that happen amounts to assuming there's always full employment and no change in output.