Monday, December 24, 2012

Rational Expectations and the Taylor Rule

The Taylor rule requires that the central bank lower its policy interest rate enough to close any output gap.   If output is currently below potential, then this requires real output to grow more quickly.   If  this more rapid output growth is anticipated, then  expectations real output growth will increase current real expenditure.    This is a force that tends to increase "the" natural interest rate.   This effect means that only a smaller decrease in the policy rate should be necessary to close the output gap.

An alternative framing is that expecations of more rapid growth in real output will tend to raise credit demands and put upward pressure on nominal  market interest rates.    While the money creation needed to expand nominal and real expenditures enough to close the output gap tends to put downward pressure on "the"  nominal market interest rate, the net effect on "the" nominal market interest rate is smaller than otherwise.

Market Monetarists sometimes argue that it is possible that a commitment to open-ended quantitative easing tied to an explicit target for nominal GDP could result in increases in all market interest rates,  a closing of the output gap, and no increase in inflation.

I have made this argument multiple times.    Be mindful of the could.   It is also possible, and perhaps more likely, that open-ended quantitative easing in service of a nominal GDP level target would result higher inflation and further some nominal interest rates might fall.   Market Monetarists simply insist that higher inflation or lower nominal interest rates are not essential to the recovery process.

Is this really possible?   The Fed is already committed to closing the output gap.  If the reason nominal GDP is below some particular growth path is solely because of an output gap, then closing the output gap will bring  nominal GDP up to that growth path.

If firms and households believe the Fed will lower the policy rate enough to close the output gap and this results in sufficient real spending now to require a higher policy rate to prevent real output from outstripping potential output, then there really shouldn't be an output gap at all, or at the very least, there should be a rapid recovery of output without any decrease in the policy rate.

The Market Monetarist argument appears to be that interest rates are low because real expenditures are expected to be low.   If real expenditures are expected to be higher, then interest rates would be higher.    How can we explore that possibility?

Consider an economy where market interest rates freely adjust with the natural interest rate.   Unfortunately, in this economy, households and firms sometimes get the vapours.   From time to time, the animal spirits are weak.

Households worried about depressed economic conditions in the future fear that if they lose their job, they will take a very long time to find another.   They reduce consumption and save more.   The supply of saving increases.  

Firms worried about depressed economic conditions in the future reduce investment.  They spend less on capital goods because they fear these capital goods would simply add to their excess capacity in the future.

The increase in the supply of saving and decrease in the demand for investment result in a lower natural interest rate.    By assumption, the market interest rate freely drops with the natural interest rate.  

The lower interest rate decreases the quantity of saving supplied and increases the quantity of investment demanded.   This is the same thing as an increase in spending on consumer goods by households and an increase in spending on capital goods by firms.  At a sufficiently low interest rate, saving and investment would again match. 

It is possible that the net effect would be more saving.  But any increase in saving would be matched by an increase in investment.   In other words, any decrease in consumption by households worried about difficulty in finding jobs would offset by an increase in spending on capital goods by firms.  

It is also possible that the net effect would be less investment. The firms are worried about excess capacity, but any decrease in investment would be matched by a decrease in saving.   Again, in other words, any decrease in investment by firms is offset by an increase in consumption by households.

Finally, with both saving increasing and investment decreasing, it is quite possible that the net effect is that the lower interest rate would leave saving and investment unchanged.   The composition of spending on output in these broad categories would not be affected at all.  

Now, if market interest rates do freely adjust with the natural interest rates, so that real expenditure is maintained, once households and firms notice that finding jobs is not especially difficult, and excess capacity is not a general problem, then their wrong expectations would likely dissipate.   The supply of saving would fall again, and the demand for investment would rise again.   Even though the expectations were wrong, market prices--market interest rates--would have adjusted to coordinate these expectations, leaving spending on output matching productive capacity.  These market prices would have adjusted to falsify the false expectations.

Now, suppose there is a central bank which believes that excessively low interest rates are undesirable.   Perhaps it worries about the interest income of  retirees.   Or, perhaps it worries that these low interest rates caused by incorrect expectations will motivate firms to undertake projects that would not be profitable at the higher "correct" interest rates.   The central bank might even worry that some might borrow at these low interest rates to purchase stocks or commodities.   The prices of these stocks or commodities would be too high--higher than they would be at the "right" interest rates.  And if that is not a speculative bubble, the appreciation of the assets might attract further purchases.    If some borrow to fund those purchases, that would clearly be a  credit-fueled bubble.  And, of course, since banks borrow short and lend long (or at least somewhat longer,) if banks project currently low funding costs into the future, then they might suffer losses or failure when their funding costs rise but they are locked into lending at lower long term interest rates.

So, to prevent these consequences, the central bank takes action to keep market interest rates from falling.   By contracting the quantity of money or even keeping it from rising to meet the demand to hold money, the central bank creates a liquidity effect so that market interest rates fail to decrease enough for saving to equal investment at a level of real income equal to productive capacity.   Real expenditure is permitted to fall below productive capacity.

Rather than interest rates falling enough so that the households and firms fears are shown to be an illusion, the central bank's policy causes unemployment and househods have great difficulty finding new jobs.   Firms, face substantial excess capacity, choose to invest less.   Their expectations have been confirmed.

We can imagine that the central bank keeps the market interest rate at the "correct" level.    If the market rates were consistent with the natural interest rate initially, then the increase in saving and decrease investment lead to reduced spending on output, and confirm the pessimistic expectations of households and firms.  

But suppose instead that market interest rates fall some, towards the new natural interest rate, and the central bank only blocks them from falling "too far."    If the central bank's policy changes, and it allows market interest rates to fall enough for real expenditures to recover,  then expectations will turn, savings will fall, investment will rise, and the market and natural interest rates will both rise above the central bank's previous "floor."   The market interest rate first falls towards the depressed natural interest rate, and then both the natural and market interest rates rise with more optimistic expectations.

If no one knows that an initially lower market interest rate will result in an economic recovery and so less saving and more investment, and then a higher market and natural interest rate, then this time path of interest rates appears necessary.    The market interest rates must at first fall towards the lower natural interest rate and then later rise once the recovery in real expenditures result in a higher natural interest rate again.

On the other hand, if everyone knows  that market interest rates will soon rise again and that any decease is only temporary, and further, everyone knows that everyone knows this, it would seem unnecessary for the central bank to end its block on lower interest rates.   There should be no need for market interest rates to fall to demonstrate that real expenditures will be maintained.   The current market interest rate is below the natural interest rate consistent with everyone expecting real expenditures to match productive capacity.  

Yet suppose that everyone knows this, but no one believes that anyone else knows.   Everyone believes that market interest rates must fall, so that everyone else will spend more, and that only once that happens, will everyone else see the light, and saving will fall and investment rise.   As far as each person is concerned, the central bank's block on keeping interest rates from falling too low is keeping spending too low.  Interestingly, if the block disappeared, everyone would immediately reduce saving and increase investment.   

In this scenario, it would seem possible that a change in policy by the central bank to allow market interest rates to fall enough to generate a recovery in real expenditure, would paradoxically result in an immediate increase in the natural interest rate and the market interest rate--both rising back to their initial level, above the central bank's floor.

Finally, suppose that there is a mixture of expectations.   Some know that saving and investment will readjust once the central bank ends its policy of keeping interest rates from falling too low.    But they correctly believe that others don't know.   However, once those others observe growing real expenditures, production, and employment, they will adjust their expectations and reduce saving and increase investment.

The central bank changes its policy.   Those who realize that now that interest rates will be allowed to fall as much as necessary for real expenditure to recover, immediately reduce their saving and expand their investment.  While their adjustment would not be sufficient to bring the natural interest rate and market interest rates back to the initial value, they could bring the natural rate back up past the central bank's artificial floor, pulling market interest rates up from that floor as well.

As real expenditures recover, those who had no idea that the central bank's floor was causing a problem respond to the improved economic conditions.   How fast do they respond?   Suppose that they don't simply expect the depressed past to continue but rather believe that while the depressed conditions could continue indefinitely, it is certainly possible for there to be a return to prosperity.   From their perspective, recession and recovery are a bit of a mystery.   With such  a scenario, the adjustment in saving, investment, the natural and market interest rates could be rapid indeed.

Perhaps these scenarios are incorrect.   However, rational expectations seems to limit us to considering situations where everyone knows and knows that everyone else knows.   If we imagine solving for savings supply and investment demand functions, that depend on expectations of aggregate real expenditure, and the expectations of aggregate real expenditure is the level of real expenditure generated by the solution, then the "low" market interest rates needed to coordinate saving and investment with pessimistic expectations cannot exist.  

With a more realistic scenario, where some people know, and they also know that many other people don't know, an expansionary monetary policy might well result in a rapid increase in all market interest rates, both nominal and real, without there being any increase in expected inflation. 


  1. A very difficult post Bill, but a very good one too (on the second reading).

    I like your assumption about a mixture of informed and uninformed expectations.

    I'm going to let this one mull over in my mind. Trying to draw an analogy to a conductor of an orchestra, or herding cattle, or something.

    Yep. Very important post. Related to learning models, I think.

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