Is leaving the Euro the road to end austerity? What are the implications with respect to inflation? We examine the potential cost.

1. On the Risk of Leaving the Euro
Manuel Macera, Albert Marcet, Juanpa Nicolini
March 29, 2017

2. • Proposals to leave the Euro for South-European countries.
— So you can devalue the currency.
— End austerity.
• Nothing to say about the first.
• Is leaving the Euro the road to end austerity?
• What are the implications with respect to inflation?

3. • We focus on a potential cost
• Initial conditions:
— Positive and persistent deficits (total or primary?).
— High debt levels.
— Unable to sell bonds after leaving the Euro. (Summer of 2012...)
• Budget deficits imply money creation and inflation: How much?

4. • Interaction of two features
1. Volatile and persistent deficits
2. Learning (uncertainty regarding new regime?).
• Can get a lot of inflation.
• Learning model nests the rational expectations solution. Robustness.

5. Plan
1. Heterogenous agents monetary model with incomplete markets.
2. The rational expectations case.
3. The model with learning.
4. Calibration and simulation.

6. • Constant cohort size overlapping generations model.
• Each household lives for two periods.
• The endowments for agent  ∈ [0 1] born at time  are 1 when young,
and 

+1 when old, where


+1 = +1 + 

+1
• 

+1 are zero-mean agent-specific shocks, independent among  and
independent of the aggregate shock +1
• Preferences are given by
ln 

 +  ln 

+1

7. • When young, agents observe 

+1.
• Only asset agents can hold is fiat money.
• When young, and given  agents choose consumption and money
balances given
 ≤ 

 + 

 
• When old they consume 

+1 given the realization of the price level,
according to


 + 

+1+1 ≤ +1

+1 for all +1

8. • Then, given a probability distribution over next period price, optimality
implies
1



= 





+1(+1)+1(+1)

• Using the budget constraints
1
 − 


= 





 + 

+1+1

• Because +1 is in the denominator, no closed form solution exists.

9. • A linearized version




= 
Ã
1 − 




+1

!

where
 =

(1 + )
 and 

 =


+1

• Equilibrium in the money market is given by

 =
Z 1
0

³
 − 



 +1
´


10. • If expectations are homogeneous, so 

 +1 = +1 for all  then
 =
Z 1
0
 −
Z 1
0






 +1
= 

1 + 
− +1
Z 1
0
+1 + 

+1
1 + 

• The solution is given by
 =

1 + 
 − +1
+1
1 + 
=  − +1
• Solving that difference equation one obtains the equilibrium distribu-
tion of +1 as functions of the shocks +1 +1 and +1

11. • This is the probability distribution of +1 used by agents in their
optimization problem under rational expectations.
• Agents need to know this aggregate demand in order to know the
mapping from the exogenous variables to the endogenous one, 
• Need to assume agents observe the aggregate variable, +1.
• If agents cannot separate the individual from the aggregate component
in


+1 = +1 + 

+1
a non-trivial inference problem arises.

12. • An agent that knows (  

+1) cannot infer +1
• Each agent will have a different view on the probability distribution of

• 

 =

+1
 correlated with 

 +1.
• Solving the market clearing condition
 =
Z 1
0


 −
Z 1
0






 +1
requires each agent knowing the inference problem solved by all other
agents in the economy.

13. • Forecasting the forecast of others.
• A solution - Bayesian learning - is to assume that each agent has some
prior regarding the distributions of 

+1 for all  ∈ [0 1]
• Then, once data on the price level is generated, agents update their
priors.
• Entertains a continuum of priors updated using a single observation:

• We directly model expectations of inflation.

14. • Thus, agent’s beliefs are defined on the space of sequences
{  +1}∞
=0
• Agents do not have the information required to compute the mapping
from exogenous variables to the endogenous one, 
• Important during regime changes?
• Given this beliefs, they maximize expected utility.

15. The belief system
• Beliefs are consistent with the properties of inflation under RE.
• But agents are uncertain regarding a parameter.
• Agents rationally use the data generated by the model to obtain a
more precise estimate of that parameter.
• Beliefs are not an exact description of the economy they live in, but are
not inconsistent with the data generated by the model, given sample
sizes longer than a decade.

16. • Consider now the limiting case of vanishing heterogeneity and 

+1 =

• The model is then given by the money demand


= (1 − 

+1

)
• and a government budget constraint
 = −1 + 
• Seniorage {}, evolves according to
 = (1 − ) + −1 + (1 − )
where  is 

17. Rational expectations
• Combining the money demand equation and the government budget
constraint delivers
 =
 +  − 

−
1
−1
where  is the gross rate of inflation.
• This equation governs the dynamics of inflation under rational expec-
tations.

18. f(·, dt)
⇡t 1
⇡t
⇡l
(dt) ⇡h
(dt)
Inflation Dynamics with Rational Expectations

19. • A log linearization of the low-inflation equilibrium implies
b = b−1 + 
• In forecasting inflation, and as long as the economy lives close to the
steady state, one could use past inflation instead of the true state
variable.

20. The Model with Learning
• The main two equations are given by
−1 =
 − 
−1
 − 
 − 
and
 = −1 + (1 − ) + (1 − )
• Thus, we allow for the long run value of the deficit (and therefore of
inflation) to be time varying.
• We assume agents make forecast using the log-linearization.

21. • They know the persistency parameter  but not the value 
• ... which means they know that inflation follows
 = −1 + (1 − ) + 
for some unknown value of 
• We assume that  has permanent and transitory shocks, so we let
agent’s belief system for inflation to be
 − −1
(1 − )
=  + 
 = −1 + 
where   are i.i.d. and independent.

22. • The one period ahead forecast is

+1 =  + (1 − ) 
where  is the best forecast of .
• We let −1 denote the prior entering period .
• By the end of the period, after  is realized, this prior is optimally
updated as follows:
 = −1 +
1

Ã
 − −1
1 − 
− −1
!
• where
 =
 + 


23. • If
— the prior is centered at the RE solution
— the volatility of  is arbitrarily small, so 1 → 0.
• We obtain the RE solution as a special case.

24. • Using beliefs in the solution for inflation, we obtain
−1 =
 − (−2 + (1 − ) −2)
 − (2−1 + (1 − ) −1) − 
• If −2 ' −2 ' −1 ' −1 then
−1 '
 − −1
 − −1 − 
governs the map from perceived inflation to observed inflation.

25. Mapping from Perceived to Observed Inflation

26. • A hyperinflation occurs when a sequence of large shocks bring the
economy to the unstable region.
• Eventually, a currency reform that fixes an exchange rate brings infla-
tion down.
• During the fixed exchange rate regime, foreign reserves must be used
to finance the deficit. (Cosmetic Reforms, IMF)
• This cycle can repeat till a permanent change in the stochastic process
for the deficit makes the stable region large enough.

27. Testing the belief system
 − −1 = (1 − ) ( + )
 = −1 + 
implies
 ≡ ( − −1) − (−1 − −2) = (1 − )
h
 +  + −1
i
• Let  = (  − −1)

28. • Then
1. E[−] = 0 for all  ≥ 2.
2. E[(( − −1) + (−1 − −2))] = 0.
3. 0Σ + E[−1]  0.

29. Calibration
• We estimate a process for the quarterly deficit on GDP for Greece,
Portugal, Italy and Spain, to calibrate  and the standard deviation
of the shock to the deficit.
• We use parameters for the money demand that match a maximum
of the Laffer of 5% of GDP and a rate that maximizes seniorage of
around 60% per quarter. (Argentinean data).

30. Parameters for Baseline Economy
Parameter Symbol Value
Persistence of deficit ρ .9584
SD of shocks to deficit σ .0097
Long run deficit δ {.02,.03}
Money Demand Parameter φ .36
Money Demand Parameter γ .39

31. Results:
1. Time series simulations
2. Tables with probabilities of hyperinflations.
3. Tests

32. 0 20 40 60 80 100
0
1
2
3
4
5
6
7
8
9
10
quarterlyinflationrate(logscale) 1/α = 0.05
1/α = 0.00

33. 0 20 40 60 80 100
0
1
2
3
4
5
6
7
8
9
10
quarterlyinflationrate(logscale) 1/α = 0.10
1/α = 0.00

34. Probability of n hyperinflations (βU = 4.00,T = 100)
Deficit mean δ = 1.0%, Initial Deficit d0 = 1.0%
1/α 0 1 2 ≥ 3
0.05 53.00 20.90 13.50 12.60
0.04 53.30 22.60 13.20 10.90
0.03 51.60 25.30 12.90 10.20
0.02 54.70 23.10 12.40 9.80
0.01 89.10 8.30 1.80 0.80
0.00 100.00 0.00 0.00 0.00
Deficit mean δ = 2.0%, Initial Deficit d0 = 1.0%
1/α 0 1 2 ≥ 3
0.05 39.90 23.10 17.90 19.10
0.04 39.70 25.60 16.20 18.50
0.03 38.80 28.00 14.70 18.50
0.02 40.60 27.30 16.30 15.80
0.01 91.60 6.10 1.70 0.60
0.00 100.00 0.00 0.00 0.00

35. Table 1: Rejection frequencies at the 5% level obtained from testing restrictions of the form
E[yt ] = 0 for yt = et qt , using simulated data. The set of instruments qt includes a constant
(Restriction 1a) and up to three lags of xt (Restrictions 1b-1d).
T
40 60 100 200
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 1a 0.0 % 0.0 % 0.0 % 0.0 %
Restriction 1b 6.3 % 6.6 % 8.3 % 8.2 %
Restriction 1c 7.9 % 7.9 % 9.1 % 9.0 %
Restriction 1d 8.2 % 7.7 % 8.5 % 7.7 %
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 1a 0.0 % 0.0 % 0.0 % 0.0 %
Restriction 1b 4.5 % 5.1 % 5.8 % 5.8 %
Restriction 1c 5.6 % 6.0 % 6.7 % 6.5 %
Restriction 1d 6.7 % 6.4 % 6.7 % 5.9 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 1a 0.0 % 0.0 % 0.0 % 0.0 %
Restriction 1b 6.6 % 6.7 % 8.4 % 18.2 %
Restriction 1c 8.9 % 8.3 % 9.9 % 20.1 %
Restriction 1d 9.3 % 8.3 % 8.7 % 17.5 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 1a 0.0 % 0.0 % 0.0 % 0.0 %
Restriction 1b 5.2 % 4.6 % 5.2 % 10.5 %
Restriction 1c 7.3 % 6.4 % 6.6 % 11.1 %
Restriction 1d 8.5 % 6.6 % 7.0 % 11.1 %

36. Table 2: Rejection Frequencies at the 5% level obtained from testing the Restrictions 2-3 of
Proposition 2 using simulated data.
T
40 60 100 200
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 2 11.9 % 13.0 % 15.1 % 18.4 %
Restriction 3 8.1 % 4.7 % 2.8 % 1.3 %
Deficit mean δ = 0.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 2 8.7 % 9.4 % 10.7 % 10.3 %
Restriction 3 6.1 % 3.8 % 2.0 % 0.9 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.01
Restriction 2 12.0 % 12.3 % 17.3 % 34.4 %
Restriction 3 4.5 % 1.8 % 0.3 % 0.0 %
Deficit mean δ = 4.0%, Initial Deficit d0 = 4.0%, and 1/α = 0.05
Restriction 2 8.8 % 8.3 % 9.9 % 18.4 %
Restriction 3 3.7 % 1.5 % 0.2 % 0.0 %

37. Conclusions
• Do not leave the Euro if
— potential sovereign debt crisis
— cannot generate surpluses