The Phillips Curve in a Liquidity Trap

This note discusses the Phillips curve with special emphasis on understanding it not as a single equation standing alone but how it interacts with other equilibrium relationships. The modern Phillips curve has a specification that looks something like:

PC: Infl(t) = a*E(infl(t+1)) + b*OutputGap; a > 0, b < 0.

Notice that the first term on the right hand side has expected FUTURE inflation. This is the specification that results from a rigorous derivation based on an economy with forward looking, value maximizing firms that are constrained by nominal frictions, (for example an inability to continuously adjust wages to match fluctuations in the demand for their output). This makes the specification given here different from the empirically derived Phillips curves that failed so badly in the seventies. This version attempts to pre-empt the Lucas critique by explicitly specifying that expected inflation does not increase output, it only increases current inflation. (The specification of the underlying frictions is not fully articulated in the canonical version which leaves it open to a generalized version of the Lucas critique. There are however a variety of proposed improvements in the literature.)

Also notice that having b < 0 implies a definition of OutputGap such that a larger output gap means less output, the gap is how much below potential the economy is operating. However, the gap is allowed to be negative, that is, the economy is allowed to operate above “potential”. As such, “potential” is not a hard constraint but an elastic one. “Above potential” output is understood to mean that there are inflationary pressures.

One thing that is important to understand about the Phillips curve, in any of its theoretical incarnations, is that in no case does it say that high inflation causes higher output, the theory only says that high output and high inflation tend to coincide, generally due to both being caused by strong aggregate demand.

This and other similar considerations imply that to make use of the Phillips curve we need to relate it to the equilibrium determinants of aggregate demand. One of these is the consumption Euler equation, (see here as well), the other is the Fisher equation:

FE: 1+i = (1+r)*(1+E(infl)); i is the nominal interest rate, r is the real interest rate.

It is not correct to base an analysis on the Phillips curve alone, the time path of consumption, inflation and interest rates (both real and nominal) must satisfy all three equations simultaneously.

The Effect of an Increase in Expected Inflation

The first thing to notice is that the equation PC, on its own, implies that increasing expected inflation does not change output at all, it simply raises current inflation. The effect on output of an increase in expected inflation depends on the reaction of the nominal interest rate.

In particular, if the nominal interest rate does not change (perhaps because the central bank is targeting the nominal rate) then the Fisher equation implies that the real rate has fallen and this feeds into the consumption Euler equation to increase aggregate demand.

On the other hand, if the central bank increases the nominal rate one for one with expected inflation then the real rate and AD are unchanged. Finally, if the central bank increases nominal rates more than one for one then the real rate is actually raised and thus AD falls.

Two Important Conclusions for Policy in a Liquidity Trap

The Phillips curve and its relation to the other two equations makes clear two important points.

1) Higher inflation should be expected to precede a real recovery, it is unlikely to be the case that inflation only picks up after the output gap closes.

2) The real recovery will only happens if the central bank keeps nominal rates low for a time after inflation begins to rise. The central bank must accommodate some inflation for a period of time in order to increase output.

The three equations taken together imply that in a liquidity trap there is simply no way for monetary policy to generate a real recovery without accommodating a sufficient amount of inflation.

In a liquidity trap investment is a like a public good

This post picks up where the last two left off. The idea is that we’ve found ourselves in a situation where the natural real interest rate is negative but there isn’t enough expected inflation to make the real rate that agents believe they are faced with when making their consumption-savings decision as low as the natural rate. Thus, the consumption level that satisfies agent’s consumption Euler equations is lower than the consumption level at full employment or, put a different way, the savings level that satisfies the Euler equation is too high. At the same time, the extra savings does not get spent on real investment due to the marginal returns to investment being too low to compensate for the risk. In particular, with a risk premium the shortfall in investment demand can happen even if the marginal product of capital is still positive.

I’ve argued here http://canucksanonymous.blogspot.com/2009/05/what-do-liquidity-traps-have-to-do-with.html that simply increasing the money supply, without generating expectations of future inflation, accomplishes nothing no matter how much money is printed. The reason for this is that agents are caught in a sort of prisoners dilemma type equilibrium, even if everyone understands what’s happening there is no way to coordinate increased expenditures. Since each agent individually prefers to save rather than consume and each individually is unwilling to fund the available investments then, even if everyone understands it’s better for us all to spend more now, regardless of what you expect the rest of the economy to do you’re better not spending. If everyone else spends you get the benefit whether or not you take part and if nobody else spends you won’t see the benefit even if you do spend. Thus to not spend is the dominant strategy for everyone.

In this sense investment seems like a public good in a liquidity trap, just like a road is. Everyone understands more investment spending is desirable but everyone prefers that someone else actually do the spending. So perhaps the solution is for the government to do the investment spending directly. Well, to the extent that the government issues new debt to finance the expenditures then this can be inflationary and thus break the trap. However, suppose that the central bank remains committed to price level stability and will tighten policy to avoid the inflation. While the government spending would raise income while the economy remains in the trap, without expected inflation it won’t succeed in breaking the trap.

One final possibility is that the central bank buy up the real investment with the intention of selling it back to the private sector. In particular here is the suggestion of a commenter:

the central bank buys the marginal investment project - with a negative return - for money - with zero return - and then re-sells the project in the next period at the purchase price. Everyone who did the switch avoids making a loss, which the central bank ends up with.

The problem here is that if investment truly has negative real returns then the real return to the portfolio of all the money being held by the private sector can’t have a zero real return. Suppose the real return to investment is -3%, well then when tomorrow comes and agents are ready to consume it’s not physically possible for them to all have made zero real return on each dollar they held because productive capacity expanded at less than one for one with the extra money. If the central bank leaves the money supply unchanged then this must necessarily translate into roughly 3% inflation. If the central bank wants to avoid this inflation then they must contract the money supply to keep prices unchanged. Thus, the result is that while each dollar that remains in circulation realizes a zero real return the sum of all the dollars does not. Some of the dollars are simply reclaimed by the central bank, and since the central bank doesn’t exchange these for stored consumption the aggregate real return can’t exceed the -3% that the investment generated.

Thus, if available investment returns are negative the central bank can only accomplish two things by buying it. One is to allow all outstanding currency to share equally in the real loss, that is, allow inflation. The second possibility is to have most of the money in the economy have zero real loss and have some small portion of it bear the entire loss

Euler Equations and the Liquidity Trap

This post picks up where the last one left off. The natural real rate of interest is the real rate that would obtain at full employment. This rate is determined jointly by the consumption Euler equation and firms profit maximization condition for investment which is that the real interest rate should equal the marginal product of capital.

Suppose that, for whatever reason, the real rate that prevailed in the economy is higher than the natural rate. Well, the Euler equation is just a condition for maximizing utility and so agents individually will always want to choose their consumption to satisfy it. Note that from the point of view of the individual agent today’s consumption level is the choice variable, the real rate is taken as given. Now, if the real rate is too high, relative to the natural rate, then 1/(1+r) is too low, thus in order to satisfy the Euler equation the ratio of marginal utility tomorrow to marginal utility today needs to be lower. The way to lower this ratio is to raise the marginal utility of consumption today and this means choosing less consumption today (since declining marginal utility means marginal utility is lower for higher consumption levels).

This is important in the liquidity trap case because a liquidity trap is manifestly a situation where the real rate prevailing in the economy is too high. If the natural rate is negative then a zero nominal rate will only translate into a negative real rate if inflation is expected. Suppose, on the other hand, that expected inflation is zero and so the real rate that agents infer is zero. Individual agents can’t change the prevailing real rate, thus, since they all want to maximize their utility they will reduce their consumption today to set up a consumption path from today to tomorrow that corresponds to utility maximization. The result is that the economy finds an equilibrium in which the real rate of zero is rationalized with aggregate demand being too low to support full employment.

Consumption Euler Equations Explained

The consumption Euler equation is nothing more than the intertemporal analogue of a standard result in basic economics that at a point of utility maximization the ratio of the marginal utilities of two goods is equal to the ratio of their prices. The intuition for this result is simple enough, the price ratio tells you how much of one good you need to give up to get an extra bit of the other good. To be concrete let’s say the choice is between pizza and beer (clear complements). If the price of a pizza is 3 time the price of a beer you must give up 3 beers to get a pizza. On the other hand, your marginal utility of pizza measures the utility gained from a pizza or the utility lost from giving up a pizza (taken to be the same because strictly speaking marginal utility refers to a derivative which means it’s the gain or loss from an infinitesimal amount of pizza added or subtracted from your consumption basket).

So, if your terms of trade are 3 beers per pizza and your marginal utility of more pizza was 4 times as high as your marginal utility of a beer, clearly you should make a trade and give up 3 beers to get a pizza. You lose the utility of 3 beers but gain the utility of 4 beers. Since such a trade clearly raises your total utility it means that the original point was not a point of utility maximization. You can only have maximized your utility if the ratio of marginal utilities equals the price ratio. A consumption Euler equation makes the same argument about your choice between consumption now and consumption in the future.

Suppose we normalize so the price of your consumption basket today is 1 and consider a zero coupon bond maturing at time T in the future. If r is the real yield of the bond then the price of a unit of your consumption basket at time T is 1/(1+r) and this is also the price ratio. Thus, by the reasoning above you are only maximizing your utility if the ratio of your marginal utility of consumption today to your marginal utility of consumption at T is equal to 1/(1+r). This remains true for any time T for which you have available a zero coupon bond of that maturity.

Of course, in practice most bonds are coupon bearing bonds. However, since the coupon bond can be seen as just a portfolio of zero coupon bonds their prices are still fundamentally related to the relative prices of consumption today versus consumption in the future.

I've ignored uncertainty here, this complicates matters but doesn't really negate the basic intuition. You just have to apply it even more widely.

Update: Let U(c(t+1), t) denote the utility at time t of c units of consumption consumed on date t+1. The Euler equation relating consumption at date t and consumption at time t+1 is

E[U'(c(t+1),t)]/U'(c(t),t) =1/(1+r);

where the prime denotes differentiation with respect to c(.), the E denotes expected value and the r is the real interest rate between t and t+1.

Usually it is assumed that U(.,t) is related to U(.,s) by a common utility function and discount factor: U(c(t+1),t) = b*U(c(t+1)); b<1.

Micro-foundations of Money

This post picks up where the IS-LM post left off to discuss two versions of the effects of an increase in the money supply. One version is my view and is based, loosely, on the idea of a cash-in-advance constraint. The other version is my attempt to interpret the view of Nick Rowe as I understand it, I think Nick's view is based, also perhaps loosely, on a money-in-the-utility-function type formulation.

Two versions of what happens when the money supply is increased:

My version: In my version, at the margin money is held to only to fund purchases of consumption goods. Notice I said at the margin, there could be a buffer stock held for unexpected expenditure needs. Thus, once enough real balances are held to cover real purchases (plus the buffer) any marginal extra currency is used to buy an interest bearing asset, that is a bond. Thus, in my story an increase in the money supply shifts the LM curve by the following causal chain:

1) Start at a point on the LM curve were supply and demand for real balances are equated and increase the money supply. Since real balances are high enough to fund desired purchases ALL of the extra money flows to the bond market and thus drives real interest rates down.

2) The lower real interest rates increase consumption demand via the consumption Euler equations as usual. With sticky prices the extra consumption demand increases output.

3) Higher consumption demand increases demand for real balances to fund the extra expenditures and LM equilibrium is re-established.

My interpretation of Nick’s version: Here money (that is, real balances) is held to satisfy a liquidity preference. It is a derived preference in the sense that it is still for the purpose of funding real consumption expenditures plus a buffer for unexpected expenditures, the difference is in the word “preference”. The liquidity preference acts like a standard preference with the usual, smoothly declining marginal utility. Thus, in this version an increase in the money supply shifts the LM curve by this causal chain:

1) Start at a point on the LM curve were supply and demand for real balances are equated and increase the money supply. Since we are at an optimum, at the margin we are indifferent to holding the extra money, buying a bond with it or spending it on consumption goods. Thus, potentially some of the marginal extra money gets spent on consumption goods. To the extent that some of the extra money is invested in bonds the logic is identical to my version. However, since some of the excess also gets spent on consumption goods the extra money can increase aggregate demand directly. (In particular, if interest rates are already zero like right now then the money is entirely spent on consumption goods.) So, assume the marginal dollars are spent on consumption, then:

2) With sticky prices the increased consumption demand increases output. (In the usual case, with nominal rates not stuck on zero, the extra output, relative to future output, lowers the real rate via the consumption Euler equation.)

3) The lower real interest rate (in the non-liquidity trap case) is a lower cost to holding real balances and thus the demand for real balances goes up via the usual maringal benefit equals marginal cost logic. The increased demand for real balances matches the extra supply and equilibrium is restored.

Note the fundamental difference in the two stories, in my version the transmission from increased money supply to higher aggregate demand is entirely due to the lower interest rate. Thus in my story, when nominal rates are at zero the only way to increase aggregate demand is to raise inflation expectations thus lowering the real rate. In Nick’s version, when nominal rates are zero, the increased money supply increases aggregate demand directly.

IS-LM explained

My purpose here is to explain the IS-LM model as I understand it and to set the context for a subsequent post that will explain a difference I have with Nick Rowe about the microfoundations of monetary theory. The IS-LM model shows how income and the real interest rate are jointly determined in the economy. The model is not really dynamic so I’ll only refer to two periods, today and tomorrow (where tomorrow is just some time in the future, perhaps next year). I’ll ignore government spending so there won’t be any discussion of fiscal stimulus.

The IS Curve.
The IS curve is made up of all combinations of real income Y and the real interest rate r such that real investment demand equals the demand for real savings. The IS curve traces out a downward sloping curve with r on the vertical axis and Y on the horizontal axis. Notice that everything is real, the curve relates the real interest rate to real income.

To begin we need to find one point on the curve. To do this consider the situation an full employment, the real income at full employment is entirely determined by the production technology and the real interest rate is equal to the natural real rate which is determined by expected consumption growth assuming full employment is maintained. Note that the difference between income and consumption is investment since we are ignoring government.

Next I’ll argue that the curve slopes downward:

1) Consumption: Start out in a case where we are in equilibrium so at the current interest rate you are happy with your consumption levels for today and tomorrow (ignore the uncertainty in tomorrow's consumption). Now I give you more income today but hold tomorrow's consumption unchanged (and prices/interest rates have not yet changed). Since you where at a maximum before you were indifferent between an extra bit of consumption today or tomorrow. Now you have extra consumption today and so by the usual declining marginal utility idea your marginal utility of today's consumption has fallen below your marginal utility of tomorrow's consumption. Thus, to re-establish your first order condition you need to reduce today's consumption and increase tomorrow's. Thus, the fact that you want to save some is not an assumption but comes from the fact that you're maximizing utility.

2) Investment: Still assuming prices/rates have not yet changed. A firm’s maximization problem says they want to employ capital until the marginal product of capital (MPK) equals the real interest rate. Since rates haven't yet changed there is no change in investment demand. (Firms also started in an equilibrium where they were happy with current investment plans).

3) We now have a situation where desired savings exceeds desired investment. How are they re-equated? Well, the excess saving starts to drive down the interest rate. This has two effects, it reduces saving demand by making tomorrow's consumption more expensive (in terms of today's) and it increases investment because reducing the real rate means firms need to increase the capital stock to re-establish MPK = r (declining MPK). The real rate falls until savings demand and investment demand are equal.

The LM Curve.

The LM curve gives the set of (Y,r) pairs such that money supply equals money demand. This curve is upward sloping when drawn with Y on the horizontal axis and r on the vertical axis. To derive the curve start at a point where supply and demand for real balances are equated and raise income.

1) Increasing income while holding the real rate constant increases consumption demand by the same utility maximization mechanism as above.

2) The increased demand for consumption purchases increases the demand for real balances to fund the extra expenditures. Recall that the money supply is unchanged.

3) The higher demand for cash prompts sales of bonds and thus raises interest rates.

Why do liquidity traps tend to follow investment booms?

My last post described my view on how a negative natural real interest rate gets turned into a liquidity trap, in particular, it is not the case that the negative real rate necessarily results in a recession. After all, it could just mean a huge investment boom. I said there that we'd fail to get the required level of real investment if the marginal product of capital was too low to generate a high enough risk premium due to a recent investment boom

My first post described a simple example of how a negative natural real rate can occur. That post was motivated in large part by discussions I've been having over on the WCI blog (http://worthwhile.typepad.com/worthwhile_canadian_initi/2009/05/could-the-natural-rate-of-interest-really-be-negative.html) about whether or not the natural real rate could really be negative. In that discusson Sthephen Gordon pointed out that one way to get a negative real interest rate is if the marginal product of capital was less than the depreciation rate and this could be the result of an overhang of essentially useless capital.

My point here is just that these arguments appear to explain why liquidity traps seem to follow gigantic investment booms. In the great depression, Japan's trap and the current recession, the trap followed a gigantic run up in asset prices.

There are, I think, two ways to interpret the asset price rallies that precede the springing of the traps. One interpretation is that future productivity growth was anticipated to be high and this growth was not regarded as being very uncertain. In this case, despite high asset prices, future returns were still believed to be high enough to compensate for the perceived risk. The trap would then be sprung by some shock, say a large run-up in commodities prices or an otherwise innocuous seeming monetary tightening that might make future output growth seem a bit more uncertain and thus, just slightly, raised risk premia. We then find ourselves in a situation where real investment and asset prices start to decline.

Of course another interpretation is that high prices are just another name for low returns. Thus, it may be that in each case a very low, possibly already negative, natural real rate preceded the liquidity trap by several years. Moreover, the sheer scale of the equity price rallies and valuations that preceded the crashes in all case imply that if future productivity growth was not expected to be enormous then both real rates and risk premia were very low. Again, some shock to the perceived distribution of future growth would raise risk premia and start a decline in real investment.

In either case, from there things would only get worse. The initial decline in asset prices in all cases destroyed the balance sheets of an over leveraged financial system. The sudden freezing of credit interemediation activity further interrupted investment and by reducing future output growth (by reducing the future capital stock) would cause further increases in both desired saving and risk premia.