Advanced Macroeconomics 3. The Taylor Principle Karl Whelan School of Economics, UCD Spring 2020 Karl Whelan (UCD) The Taylor Principle Spring 2020 1 / 17
What is the Taylor Principle? We have assumed β π > 1. This which means the central bank reacts to a change in inflation by implementing a bigger change in interest rates. This means that real interest rates go up when inflation rises and go down when inflation falls. This is why the IS-MP curve slopes downwards: Along this curve, Higher inflation means lower output. Because Taylor’s original proposed rule had the feature that β π > 1, the idea that monetary policy rules should have this feature has become known as the Taylor Principle . We now discuss why policy rules should satisfy the Taylor principle. Karl Whelan (UCD) The Taylor Principle Spring 2020 2 / 17
Three Cases: 1. Taylor Principle Case ( β π > 1) Inflation in the IS-MP-PC model is given by t + (1 − θ ) π ∗ + θ ( γǫ y π t = θπ e t + ǫ π t ) where � 1 � θ = 1 + αγ ( β π − 1) Under adaptive expectations, π e t = π t − 1 and the model can be re-written as π t = θπ t − 1 + (1 − θ ) π ∗ + θ ( γǫ y t + ǫ π t ) Three different cases depending on different values of β π . β π > 1 1 ⋆ β π > 1 ⇒ αγ ( β π − 1) > 0 ⋆ β π > 1 ⇒ 1 + αγ ( β π − 1) > 1 ⋆ β π > 1 ⇒ 0 < θ < 1 Karl Whelan (UCD) The Taylor Principle Spring 2020 3 / 17
Three Cases: 2. β π falls below one Recall from our last set of notes that inflation in the IS-MP-PC model is given by t + (1 − θ ) π ∗ + θ ( γǫ y π t = θπ e t + ǫ π t ) where � 1 � θ = 1 + αγ ( β π − 1) Under adaptive expectations, π e t = π t − 1 and the model can be re-written as π t = θπ t − 1 + (1 − θ ) π ∗ + θ ( γǫ y t + ǫ π t ) Three different cases depending on different values of β π . � � 1 1 − < β π < 1: 2 αγ ⋆ β π < 1 ⇒ αγ ( β π − 1) < 0 ⇒ 1 + αγ ( β π − 1) < 1 � � ⋆ β π > 1 1 − ⇒ 1 + αγ ( β π − 1) > 0. αγ ⋆ θ > 1 Karl Whelan (UCD) The Taylor Principle Spring 2020 4 / 17
Three Cases: 3. Where β π is well below one Recall from our last set of notes that inflation in the IS-MP-PC model is given by t + (1 − θ ) π ∗ + θ ( γǫ y π t = θπ e t + ǫ π t ) where � 1 � θ = 1 + αγ ( β π − 1) Under adaptive expectations, π e t = π t − 1 and the model can be re-written as π t = θπ t − 1 + (1 − θ ) π ∗ + θ ( γǫ y t + ǫ π t ) Three different cases depending on different values of β π . � � 1 1 − : β π < 3 αγ � � 1 ⋆ β π < 1 − ⇒ 1 + αγ ( β π − 1) < 0. αγ � � 1 ⋆ β π < 1 − ⇒ θ < 0 αγ Karl Whelan (UCD) The Taylor Principle Spring 2020 5 / 17
Macro Dynamics and Difference Equations So the value of β π determines the value of θ – so what? To explain why this matters, we need to explain something about difference equations. A difference equation is a formula that generates a sequence of numbers. In economics, these sequences can be understood as a pattern over time for a variable of interest. After supplying some starting values, the difference equation provides a sequence explaining how the variable changes over time. For example, consider a case in which the first value for a series is z 1 = 1 and then z t follows a difference equation z t = z t − 1 + 2 This will give z 2 = 3, z 3 = 5, z 4 = 7 and so on. So the sequence of numbers generated is 1 , 3 , 5 , 7 , .... Karl Whelan (UCD) The Taylor Principle Spring 2020 6 / 17
A More Relevant Example More relevant to our case is the multiplicative model z t = bz t − 1 For a starting value of z 1 = x , this difference equation delivers a sequence of values x , xb , xb 2 , xb 3 , xb 4 .... . If b is between zero and one, the sequence converges to zero but if b > 1 it explodes to either plus or minus infinity depending on whether x is positive or negative. The same logic prevails if we add a constant term z t = a + bz t − 1 a If b is between zero and one, the sequence converges over time to 1 − b but if b > 1, the sequence explodes towards infinity. Add random shocks to the model z t = a + bz t − 1 + ǫ t where ǫ t is a series of zero-mean random shocks. This is called a first-order autoregressive or AR(1) model. Then if 0 < b < 1 the series tends to oscillate a above and below the average value of 1 − b while if b > 1 the series will tend to explode over time. Karl Whelan (UCD) The Taylor Principle Spring 2020 7 / 17
The Taylor Principle and Macroeconomic Stability These considerations explain why the Taylor principle is so important. If β π > 1 then inflation dynamics π t = θπ t − 1 + (1 − θ ) π ∗ + θ ( γǫ y t + ǫ π t ) are described by an AR(1) model with 0 < θ < 1. Inflation and output will be stable around long-run average values. � � 1 If 1 − < β π < 1, then θ > 1 and inflation ends up exploding off to αγ either plus or minus infinity. Output either collapses or explodes. Why does β π matter so much for macroeconomic stability? Obeying the Taylor principle means that shocks that boost inflation raise real interest rates and thus reduce output, which contains the increase in inflation. In contrast, when the β π falls below 1, shocks that raise inflation result in lower real interest rates and higher output which further fuels the initial increase in inflation. Karl Whelan (UCD) The Taylor Principle Spring 2020 8 / 17
Graphical Representation We can use graphs to illustrate the properties of the IS-MP-PC model when the Taylor principle is not obeyed. The IS-MP curve is given by t − α ( β π − 1) ( π t − π ∗ ) + ǫ y y t = y ∗ t The slope of the curve depends on whether or not β π > 1. When β π > 1 the slope − α ( β π − 1) < 0. The IS-MP curve slopes down. When β π < 1 the slope − α ( β π − 1) > 0. The IS-MP curve slopes up. But when is the upward-sloping IS-MP curve steeper than the Phillips curve and when is it not? I won’t show it here but the condition for IS-MP curve to slope up and be � � 1 steeper than the Phillips curve is 1 − < β π < 1. In other words, this αγ graph corresponds to the second case considered above. This is the case we will show in graphs here. Karl Whelan (UCD) The Taylor Principle Spring 2020 9 / 17
� � 1 − 1 The IS-MP-PC Model when < β π < 1 αγ IS-MP ( Inflation PC ( ) Output Karl Whelan (UCD) The Taylor Principle Spring 2020 10 / 17
� � 1 − 1 An Increase in π e t when < β π < 1 αγ IS-MP ( PC 2 ( ) Inflation PC ( ) Output Karl Whelan (UCD) The Taylor Principle Spring 2020 11 / 17
� � 1 − 1 Explosive Dynamics when < β π < 1 αγ PC 3 ( ) IS-MP ( Inflation PC ( ) Output Karl Whelan (UCD) The Taylor Principle Spring 2020 12 / 17
� � An Increase in π ∗ when 1 − 1 < β π < 1 αγ IS-MP ( Inflation IS-MP ( PC ( ) Output Karl Whelan (UCD) The Taylor Principle Spring 2020 13 / 17
An Increase in π ∗ when β π < 1 This last result is consistent with our basic equation for inflation: t + (1 − θ ) π ∗ + θ ( γǫ y π t = θπ e t + ǫ π t ) because we are considering the case where θ > 1 so the coefficient on π ∗ is negative. Still, it seems odd. Shouldn’t a higher inflation target lead to higher inflation? The explanation is that this doesn’t happen with our monetary policy rule: i t = r ∗ + π ∗ + β π ( π t − π ∗ ) A higher inflation target lowers i t because it reduces the “inflation gap” π t − π ∗ but it increases i t because of the first term r ∗ + π ∗ which is required to maintain the natural real rate in the long term, a reduction in inflation must be matched by a reduction in nominal interest rates. When β π < 1 this second effect is bigger than the first effect. A higher inflation target leads to higher interest rates and lower inflation. Karl Whelan (UCD) The Taylor Principle Spring 2020 14 / 17
Does This All Depend On Adaptive Expectations? Do these results depend crucially on the assumption of adaptive expectations? No Without assuming adaptive expectations, we still have t + (1 − θ ) π ∗ + θ ( γǫ y π t = θπ e t + ǫ π t ) So, when β < 1 and thus θ > 1, the coefficient in the inflation equation on the central bank’s inflation target, π ∗ , is negative. If you introduced a more sophisticated model of expectations formation, the public would realise that the central bank’s inflation target doesn’t have its intended influence on inflation and so there would no reason to expect this value of inflation to come about. And if people know that changes in expected inflation are translated more than one-for-one into changes in actual inflation then this could produce self-fulfilling inflationary spirals, even if the public had a more sophisticated method of forming expectations than the adaptive one employed here. Karl Whelan (UCD) The Taylor Principle Spring 2020 15 / 17
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