Columbia University Department of Economics Lecture 18 Economics - - PowerPoint PPT Presentation

Columbia University Department of Economics Lecture 18 Economics UN3213 Intermediate Macroeconomics Professor Mart ´ ın Uribe Spring 2019

Announcement Homework 5 due on Monday in class.

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Topics Today

• The End of the German Hyperinflation of 1923
• Cagan’s Nightmare

1

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

A Money-Based Inflation Stabilization Program

Consider an economy that starts in period 0 and lasts forever (t = 0, 1, 2, . . . ). We will study an experiment in which at some date T > 0, the central bank unexpectedly and permanently lowers the growth rate of the money supply from µ > 0 to 0. By unexpectedly, we mean that at any period t < T agents expect that money will grow forever at the rate µ > 0, or Mt = (1 + µ)tM0. But then in period T it is announced that the money supply will stop growing and stay constant forever Mt = MT for t ≥ T. We wish to characterize the behavior of prices and money demand induced by this stabilization program.

2

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Let’s recall the building blocks of the Cagan model: (1) Equilibrium in the money market: Mt/Pt = (1 + it)−ηY (2) The Fisher equation : 1 + it = (1 + r)(1 + Etπt+1) (3) Specification of how expectations are formed: Assume that expectations are rational. This means that Etπt+1 = πt+1 (perfect foresight), except in period T in which households are surprised by an unanticipated policy change, so we have that ET −1πT will not necessarily be equal to πT. (4) Monetary Policy: The money supply rule states that Mt is ex-

• genous and grows at the rate µ > 0 between periods 1 and T, with

the initial money supply, M0, given. In period T, the government announces, by surprise, that from that period on the money supply will stay constant forever.

3

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Breaking Down the Equilibrium Dynamics

Subperiod 1: The pre-reform period, t ≤ T − 1. Subperiod 2: The post-reform period, t ≥ T + 1. Subperiod 3: The reform period, t = T

4

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

(1) The Pre-Reform Period, t ≤ T − 1

Monetary Policy: Mt = (1 + µ)tM0 for t = 1, 2, . . . , T (and this is expected to last forever). Solution: use the guess and verify method. Guess that Etπt+1 = µ for t = 0, 1, 2, . . . , T − 1. This guess makes sense given the analysis of the previous lecture, because agents expect the money growth rate to be µ forever. This guess and the Fisher equation imply that the nominal interest rate satisfies 1 + it = (1 + r)(1 + µ); for t = 0, 1, 2, . . . , T − 1.

5

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Then, equilibrium in the money market implies that: Mt Pt = [(1 + r)(1 + µ)]−ηY, for t = 0, 1, 2, . . . , T − 1. (1) This means that Mt/Pt is constant for t = 0, 1, 2, . . . , T − 1. Since Mt grows at the rate µ so must Pt., that is, Pt/Pt−1 ≡ 1 + πt = 1 + µ, for t = 1, 2, . . . , T − 1. This expression says that inflation is equal to the money growth rate during the pre-stabilization period. This result also corroborates that the inflationary expectations we guessed are actually rational.

6

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Comments on the Pre-Reform Period

• The equilibrium dynamics of inflation, real balances, interest rate,

and prices are just like the constant money growth rate regime we analyzed in the last lecture. This makes sense, because agents expect the money growth rate to be equal to µ forever. In particular, they do not anticipate the stabilization program of period T.

• The interest rate in t = T −1 is still high at iT −1 = (1+r)(1+µ)−1

even though prices will be stabilized in T. Again, this is because agents expect money growth to continue to be µ indefinitely, so they expect inflation between T −1 and T to be µ, i.e., ET −1πT = µ, and hence the nominal interest rate is still 1 + iT −1 = (1 + r)(1 + ET −1πT) = (1 + r)(1 + µ). As we will see shortly, in period T actual inflation will be lower than expected in T − 1.

• Real balances are low throughout the pre-reform period.

Why? because interest rates are high, so the opportunity cost of holding money is high.

7

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

(2) The Post-Reform Period, t ≥ T + 1

Constant Money Supply: Mt = MT, for t = T + 1, T + 2, . . . Guess: Etπt+1 = 0, for t = T + 1, T + 2, . . . . That is, we guess that people expect prices to be constant over time. This guess makes sense given what we know about this model: since the money growth rate is expected to be nil forever, so is the expected rate of inflation. Then, the nominal interest rate is 1+it = (1+r)(1+Etπt+1) = 1+r,

• r it = r, for t = T + 1, T + 2, . . . , the nominal interest rate equals

the real interest rate. Real money balances are constant and equal to Mt Pt = (1 + r)−ηY, for t = T + 1, T + 2, . . . . (2) Because Mt/Pt and Mt are constant, Pt must also be constant for t = T +1, T +2 . . . . That is, inflation is nil, πt = 0 for t = T +2, T +3, . . . . Finally, expectations are rational Etπt+1 = πt+1 = 0 for t ≥ T + 1, validating our guess.

8

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Comments on the Post-Reform Period

• With money supply expected to be constant from T onward, there

is zero inflation and the nominal interest rate is low, equal to the real rate of interest, r.

• Real balances are high in periods T + 1, T + 2, T + 3, . . . because

the opportunity cost of money, i.e., the nominal interest rate, is low.

9

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

(3) The Reform Period, t = T

Money Supply: MT = (1 + µ)MT −1 Guess: ETπT +1 = 0. Nominal Interest rate: (1 + iT) = (1 + r)(1 + ETπT +1) = (1 + r). Real Money Balances: MT PT = (1 + r)−ηY. (3) This is the same as in period T + 1. From (2) we have that MT+1

PT+1 =

(1+r)−ηY. Combining this expressin with (3) we have that MT/PT = MT +1/PT +1; and recalling that MT = MT +1, we have that PT = PT +1, or πT +1 = 0. This means that ETπT +1 = πT +1 = 0, which implies that expectations are rational, validating our guess.

10

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Now combine MT −1/PT −1 = [(1+r)(1+µ)]−ηY (equation (1)) with MT/Pt = (1 + r)−ηY (equation (3)) to get PT PT −1 = MT(1 + r)η/Y MT −1[(1 + r)(1 + µ)]η/Y = (1 + µ) (1 + µ)η = (1 + µ)1−η < 1 + µ. That is, πT < µ. This is remarkable, because it says that the inflation rate between periods T − 1 and T falls below µ even though the money supply grows at the rate µ between T − 1 and T. This is the most important prediction of this model for explaining what actually happens at the end of hyperinflations.

11

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Comments on the Reform Period

• Money growth in period T is still high at µ.

Nevertheless, real balances are already high, that is MT/PT = Y (1 + r)−η > Y [(1 + r)(1 + µ)]−η = MT −1/PT −1. In other words, real balances expand before money growth slows down!

• Inflation slows down before money growth slows down!
• It is even possible that inflation not only slows down but actually

becomes negative. Notice if the interest semi-elasticity of money demand is greater than one in absolute value, η > 1, which is the case of greatest empirical interest, then we would see deflation in the period of reform.

12

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

• The intuition for why inflation slows down even though the money

supply is still growing at the rate µ is that in period T everybody expects inflation in the future to be zero. As a result, the interest rate falls in period T, as banks don’t need to compensate depositors for future inflation. In turn, a low interest rate induces people to increase their money holdings. So, the increase in the money supply in period T doesn’t go to prices as before, because people don’t get rid of it, but keep it to rebuild their money holdings

• Actual inflation in period T is lower than was expected in period

T − 1. Recall that ET −1πT = µ and that πT < µ. But this is not a violation of Rational Expectations, since the stabilization program implemented in period T was unpredictable in period T − 1.

13

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Reliquefication

We noted that if the interest semi-elasticity of money demand is greater than unity in absolute value (η > 1), then prices can actually fall when the reform is implemented. This one-period deflation can be avoided by a one-time increase in the money supply in period T by a rate larger than µ to accommodate the increase in the demand for real moneay balances. Such an increase in nominal balances is called reliquefication. Let ˜ µT denote the growth rate of the money supply in period T that would prevent the price level from falling. Can you deduce ˜ µT in terms of µ and η?

14

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Comparison of Inflation and Money Demand Dynamics in the Cagan and QTM Models

Recall that in the QTM model: Mt¯ v = PtY It follows that in the QTM model, inflation in period T is given by 1 + πT ≡ PT PT −1 = MT¯ v/Y MT −1¯ v/Y = 1 + µ Compare this result with the prediction of Cagan’s model with ra- tional expectations: 1 + πT = (1 + µ)1−η. Under QTM price growth slows down only in T +1, once the money growth rate has slowed down. Under QTM no expansion in real money holdings in period T. In- stead, real money holdings are always constant and equal to Y/¯ v.

15

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Adaptive Expectations versus Rational Expectations

Contrary to the presentation given here, in his original work, Cagan assumed that expectations were adaptive: Etπt+1 = (1 − β)[πt + βπt−1 + β2πt−2 + β3πt−3 + . . . ] (∗)

16

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

What is the problem with this assumption? Go back to the inflation stabilization program we just studied. Starting in the period of reform, that is, in period T, under rational expectations everybody correctly expects future inflation to fall drastically, namely to 0. Under adaptive expectations, economic agents expect inflation to be some combination of current and past inflations. Since past inflations are high, expected inflation is also high.

17

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Empirical Test of Cagan’s Assumption of Adaptive Expectations Test preparation. Start with the money demand function Mt Pt = (1 + it)−ηY Add the Fisher Equation: 1 + it = (1 + r)(1 + Etπt+1) Combine to obtain: Mt Pt = [(1 + r)(1 + Etπt+1)]−ηY

18

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Taking natural logarithm, ln(Mt/Pt) = −η ln(1 + r) + ln Y − η ln(1 + Etπt+1) Use the fact that ln(1 + Etπt+1) ≈ Etπt+1 and rearrange to get: Etπt+1 = constant − 1 η ln(Mt/Pt) (∗∗) According to this expression, there is a linear relationship between the expected rate of inflation and the natural log of real money balances.

19

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

So here is the empirical test of the validity of adaptive expectations: Gather data on inflation and real balances ( πt and Mt/Pt). Using the formula given in (*), that is, using Etπt+1 = (1 − β)[πt + βπt−1 + β2πt−2 + β3πt−3 + . . . ] (∗) construct the expected rate of inflation, Etπt+1, according to the adaptive expectations hypothesis. Then plot Etπt+1 against Mt/Pt. According to expression (**), that is, according to Etπt+1 = constant − 1 η ln(Mt/Pt) (∗∗) all points should lie on a line. If what you get is close to a line, then the adaptive expectations hypothesis is right. Otherwise, it is wrong. Fortunately, Cagan himself ran this experiment for us, using data stemming from the German Hyperinflation of 1920-1923. Look at the next slide.

20

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Cagan’s Nightmare

Source: reproduced from Cagan (1956) 21

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

What is Cagan plotting for us? The horizontal axis measures actual

• bserved real balances, (Mt/Pt), on a log scale. On the vertical axis

measures expected inflation. Each dot is a pair of the observed Mt/Pt and of Etπt+1 constructed according to Cagan’s assumed adaptive expectations hypothesis, given in equation (*). Four of those obser- vations are labeled, namely August 1923, September 1923, October 1923, and November 1923. The straight line in the graph is constant − 1/η ln(Mt/Pt) with η = 5.46. The parameter η was estimated using all nonlabeled ob- servations. Real balances are normalized with the highest level equal to 1. The lowest level is 0.02. The slope of the line is −1/η = −1/5.46 = −0.1832, so that if real balances were to in- crease by 100 percent (say from 0.5 to 1, or from 0.05 to 0.1), this should be associated with a decline in the expected monthly inflation rate of 18.32 percentage points (note that the vertical axis Etπt+1 is in per one but in the graph it is expressed in percent).

22

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

The title of the graph is mine, not Cagan’s. Why do I call it Cagan’s nightmare? Cagan’s money demand with adaptive expectations fits the data re- markably well for the period Sept 1920 until July 1923, as the dots line up almost on the straight lone. But then the theory breaks

• down. Clearly, the graph shows that Cagan’s model with adaptive

expectations cannot explain the pairs for real balances and expected inflation, (Mt/Pt, Etπt+1), observed in August, September, October, and November of 1923. Specifically, expected inflation is predicted to be extremely high in August, September, October and November

• f 1923. If expected inflation had been truly that high, then accord-

ing to the line constant − 1/ηEtπt+1, real balances should have been much smaller than they actually were.

23

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Why does Cagan’s model with adaptive expectations fail at the end

• f 1923? These months correspond to the end of the German hy-
• perinflation. In general, at the end of a hyperinflation, expectations

change dramatically. In Germany, in August 1923 a new govern- ment is elected and people start expecting reform. When a drastic political and economic-policy regime change occurs, past inflation rates provide little information about future inflation rates. In this situation, expectations are forward- rather than backward-looking (or rational rather than adaptive).

24

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Can we understand why real balances increase so much between October 1923 and November 1923 using the Cagan model with rational expectations? The monetary reform was announced in Oc- tober and implemented in November. Under rational expectations, Etπt+1 = πt+1. That is, agents would have demanded much more money in November than in October of 1923 if inflation in Decem- ber 1923 was much lower than in November of 1923. The next graph shows that this was indeed the case, suggesting that a model with rational expectations can explain the observed increase in real balances.

25

Econ UN3213 Interm. Macro Monetary Economics Lecture 18

Weekly Inflation in Germany October 1923 to December 1923

10−1 10−2 10−3 10−4 11−1 11−2 11−3 11−4 12−1 12−2 12−3 12−4 50 100 150 200 250 Weekly Inflation −−− Germany 1923 Percent per Week

Data Source: Webb 1989. 26