Spring Semester ’13-’14 Akila Weerapana
Lecture 19: The Zero Lower Bound (ZLB)
I. OVERVIEW
• Having discussed the theory and practice of monetary policy in conventional settings, we now turn to monetary policy in unconventional settings, namely, what happens when the nominal interest rate equals zero and the Fed is limited in its ability to pursue expansionary policy.
• This is obviously not a mere academic exercise: he Federal Reserve has moved interest rates to unprecedentedly low levels for unprecedented levels of time. The Fed funds rate is now in its FIFTH year at zero, and is likely to stay at zero at least till the end of this year.
• In economics, when the Fed is constrained in not being able to push rates lower to boost the economy we say that the Fed was hampered by the zero lower bound (ZLB). In today’s class, we explore this idea of a constraint on interest rates falling below zero in the context of the AD/IA model. As we have seen in the United States over the past few years, when monetary policy hits that zero lower bound, economic recovery becomes a real challenge. The purpose of this lecture is to develop the implications of this zero lower bound for our baseline model of the macroeconomy.
• In future lectures we tackle the policy implications of the ZLB and discuss academic papers that examine the effectiveness of the policy responses that the Fed has actually put into place.
II. IMPLICATIONS OF THE ZLB FOR THE MP CURVE
• In our MP curve we specified that the central bank chooses the real interest rate to satisfy a policy rule of the form
rt= ¯r∗+ γπ(πt− ¯π) + γyYˆ
• Keep in mind that the Central bank chooses this real interest rate by manipulating the nominal interest rate, i.e. since r = i − π, we can describe the central bank’s behavior using the following policy rule for the nominal interest rate of it− πt= ¯r∗+ γπ(πt− ¯π) + γyY , whichˆ simplifies to
it= ¯r∗+ (1 + γπ)πt− γππ + γ¯ yYˆ
• In reality, the Fed is choosing a nominal interest rate using a rule like the above. Since we need the real interest rate (which is what drives spending behavior in the IS curve in our model), we find it easier to describe the Fed behavior in terms of the real interest rate.
• But the Fed can’t lower nominal interest rates once the nominal interest rate reaches zero.
So in fact the setting of the nominal interest rate is given by it= max(0, πt+ ¯r∗+ γπ(πt− ¯π) + γyY )ˆ
• When the nominal interest rate is constrained at zero, the real interest rate becomes equal to rt= −πt, so the MP curve is given by the equation
rt= max(−πt, ¯r∗+ γπ(πt− ¯π) + γyY )ˆ
• A graphical description of this relationship is given below.
r
Yˆ
M P = M ax(rA, rB)
rtA= −πt
rBt = ¯r∗+ γπ(πt− ¯π) + γyYˆ
III. IMPLICATIONS OF THE ZLB FOR THE AD CURVE
• We can also think of the implications of the ZLB line for the AD curve’s shape. Mathemat- ically, we derived the AD curve by substituting in an expression for r from the MP curve into the IS curve. Now we have two different MP curves, and hence two different r values to substitute in.
• In the typical setting, rt = ¯r∗ + γπ(πt− ¯π) + γyY and plugging into the IS curve ˆˆ Yt =
−µbI(rt− rn) gives us the typical downward sloping AD curve Yˆt=
µbI 1 + µbIγy
(rn− ¯r∗) −
µbIγπ 1 + µbIγy
(πt− ¯π∗)
• But when the ZLB binds, r = −πt, so plugging into the IS curve we get the AD curve to be Yˆt= µbIπt+ µbIrn
• Note that in this relationship lower inflation implies a lower output gap, i.e. we would have a backward bending AD curve. If we are picking r to be the maximum of the two MP curves, then since ˆY and r are inversely related, we have that the AD curve will be
Yˆt= M in
µbI 1 + µbIγy
(rn− ¯r∗) −
µbIγπ 1 + µbIγy
(πt− ¯π∗), µbIπt+ µbIrn
π
Yˆ AD = M in( ˆYA, ˆYB)
YˆtA= µbIπt+ µbIrn
YˆtB =
µbI
1+µbIγy
(rn− ¯r∗) −
µbIγπ
1+µbIγy
(πt− ¯π∗)
IV. WHAT DETERMINES WHERE ZLB KICKS IN?
• As inflation and the output gap falls, the nominal interest rate chosen by the Fed also falls.
At some point when inflation and the output gap are both low, the nominal interest rate will hit the lower bound of zero.
• We can illustrate this with the following two diagrams. The first figure shows how the output gap and the inflation rate have evolved in the United States over the past two decades. As you notice, both values fell sharply during the most recent recession.
• The second figure uses values from the Taylor Rule (γπ = γy = 0.5 and ¯π = ¯r∗ = 2%) to illustrate what that Rule would have required nominal interest rates in the United States to be during this period. When you compare to the actual Federal Funds rate you can clearly see that the zero lower bound was a significant problem during this downturn and recovery.
• So if lower inflation rates and lower output gaps are what drives the nominal interest rate lower, then at some point one or both of these variables will fall to a point such that the nominal interest rate is zero. We can then identify the set of points where the ZLB kicks in, which we call the ZLB line, by solving for
it = r¯∗+ (1 + γπ)πt− γππ + γ¯ yYˆ 0 = r¯∗+ (1 + γπ)πt− γππ + γ¯ yYˆ (1 + γπ)πt = −¯r∗+ γπ¯π − γyYˆ
πt = γππ¯∗− ¯r∗ 1 + γπ
−
γy
1 + γπ
Yˆ
• We can show the ZLB line on the same space that we show AD/IA. The ZLB line has the following properties
1. It is downward sloping - you can see that the slope is negative in the equation above.
You can also see this intuitively, if the output gap is positive and large then inflation has to be very low for the Fed to hit the ZLB; conversely, if the inflation rate is high then the output gap has to be quite negative for the Fed to be affected by the ZLB.
2. The intercept (where ˆY = 0) is equal to πt =
γππ¯∗−¯r∗ 1+γπ
. This value is lower than the target rate of inflation (¯π∗). Furthermore, it can be, but need not be, negative.
3. The slope of this ZLB line is − γ
y
1+γπ
. This means that the ZLB line is flatter than the AD line. The AD curve was given by the equation
Yˆt=
µbI 1 + µbIγy
(rn− ¯r∗) −
µbIγπ 1 + µbIγy
(πt− ¯π∗)
so it has a slope of -1+µb
Iγy
µbIγπ
, which simplifies to -
1 µbI+γy
γπ
. The numerator is larger and the denominator is smaller than in the ZLB line’s slope, which means the AD curve
is steeper than the ZLB line.
π
Yˆ
ZLB (Slope=-
γy
1+γπ
) 0
γπ¯π∗−¯r∗ 1+γπ
• Keep in mind that we always have a ZLB line in an AD-IA diagram. But if IA and AD intersect well above the ZLB line then its existence does not change the analysis we have been doing thus far. But the ZLB remains hidden beneath the surface, almost like a hidden reef or a hidden iceberg that only threatens a ship when the ocean becomes shallow.
• Note also from the ZLB line that the inflation rate where the zero lower bound kicks in need NOT be a rate that is consistent with deflation. If the output gap is negative enough then the ZLB can be reached even at positive levels of inflation.
V. WHEN DOES THE ZLB LINE MATTER?
• Let’s begin from our default position where inflation is at the target level and the output gap is zero. We can show the AD curve looking normal until the ZLB line is reached at which point it bends backwards.
π
Yˆ
ZLB: Slope=-
1 µbI+γy
γπ
0
γπ¯π∗−¯r∗ 1+γπ
AD
¯
π∗ IA
Then consider an adverse shock that shifts the AD curve in. The short run output gap falls to Yˆ1 and over time inflation falls moving us back to potential output. In this case even though inflation has fallen to zero, the economy still is able to return to long run equilibrium
π
Yˆ
ZLB 0
γπ¯π∗−¯r∗ 1+γπ
AD AD0
¯
π∗ IA
πLR IA0
• The danger of the zero lower bound shows up when the economy gets hit by a more severe shock that moves AD to AD00 in the diagram below. In that case the zero lower bound is reached at the current rate of inflation and the economy gets trapped in a recession where inflation keeps falling, potentially moving into deflation, while making the economy worse
off rather than better off. This can be seen in the diagram below where as IA falls, the output gap becomes more negative which leads to a further fall in inflation till in the end the economy is trapped in a deflationary recession where the monetary policymaker is powerless to intervene, this is called a liquidity trap.
π
Yˆ
ZLB 0
γππ¯∗−¯r∗ 1+γπ
AD0 AD
¯
π∗ IA
πLR IA0
• Notice that you do not need the shock to be that severe. Any negative shock that forces us to hit the ZLB before we get to zero will leave the economy in a liquidity trap. Any AD shift greater than the one shown below will suffice to put the economy in a liquidity trap (I use ADC as short-hand for the critical value of AD.
π
Yˆ
ZLB 0
γππ¯∗−¯r∗ 1+γπ
ADC AD
¯
π∗ IA
πLR IA0
• In such a situation, conventional monetary policy cannot work so the policymaker has to resort to other means such as quantitative easing, for example, to get the economy moving back towards potential.
• In such a setting, because of the relative impotence of monetary policy, fiscal policymakers need to play a big role in getting the economy to move back towards the long-run equilibrium.
Keep in mind that they have to push the AD curve out by enough to avoid the ZLB or else the expansionary fiscal policy may have been in vain
