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Economics of monetary unionsChapter 5. Fragility of monetary union

Jamel Saadaoui
University of Strasbourg, BETA,CNRS
https://www.jamelsaadaoui.com/
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In this blog, I will show how to build a model of the fragility of an incomplete monetary union. The idea is to find the optimal inflation rule and to examine different configurations for fixed exchange rate and expectation of a devaluation.
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To this end, I will use Mathematica and the chapter 5 of this book (https://global.oup.com/ushe/product/economics-of-the-monetary-union-9780198849544), that I use in my lecture of Economics of Monetary Union. Allow me to explain the different steps in the notebook below. I would like to publicly thank Paul De Grauwe for providing his very clear lecture notes.

The model

​
In[]:=
ClearAll["Global`*"](*completelyclearglobalsymbolstostartfresh*)
◼
  • The Phillips curve:
    • U=
    U
    N
    +a(
    e
    π
    −π)+ε
    (
    1
    )
    In[]:=
    U=UN+a(pe-p)+ϵ
    Out[]=
    a(-p+pe)+UN+ϵ
    ◼
  • Alternatively, you can specify the supply curve
    • Y=
    Y
    n
    +θ(π−
    e
    π
    )+ε
    (
    2
    )
    ​
    Slide
    3
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    ​
    ◼
  • Loss function of authorities
    • L=
    2
    π
    +β
    2
    (U−
    -
    U
    )
    (
    3
    )
    In[]:=
    L=
    2
    p
    +β
    2
    (U-UB)
    ​​
    Out[]=
    2
    p
    +β
    2
    (a(-p+pe)-UB+UN+ϵ)
    with UB = target unemployment
    -
    U
    =λ
    U
    N
    (
    4
    )
    In[]:=
    UB=λUN
    Out[]=
    λUN
    where λ < 1
    ​
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    L
    Out[]=
    2
    p
    +β
    2
    (a(-p+pe)+UN+ϵ-λUN)
    In[]:=
    2
    p
    +β
    2
    (a(-p+pe)+UN+ϵ-λUN)
    Out[]=
    2
    p
    +β
    2
    (a(-p+pe)+UN+ϵ-λUN)
    L=
    2
    π
    +β
    2
    [(1−λ)
    U
    N
    +a(
    e
    π
    −π)+ε]
    (
    5
    )
    ◼
  • Workers set wages based on pe
    • ◼
  • Authorities minimize loss given these expectations
    • ◼
  • Note : authorities directly control inflation
    • ​
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    ◼
  • Authorities minimize loss given these expectations
    • dL
    dπ
    =2π+2β(−a)[(1−λ)
    U
    N
    +a(
    e
    π
    −π)+ε]=0
    (
    6
    )
    In[]:=
    CBLoss=D[L,p]
    Out[]=
    2p-2aβ(a(-p+pe)+UN+ϵ-λUN)
    In[]:=
    CBLoss1=Simplify[Collect[CBLoss,βa]/2]
    Out[]=
    p-aβ(a(-p+pe)+UN+ϵ-λUN)
    In[]:=
    Collect[CBLoss1,p]
    Out[]=
    -
    2
    a
    peβ-aUNβ+p(1+
    2
    a
    β)-aβϵ+aβλUN
    π(1+β
    2
    a
    )−βa(1−λ)
    U
    N
    −β
    2
    a
    e
    π
    −βaε=0
    (
    7
    )
    ​
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    π(1+β
    2
    a
    )−βa(1−λ)
    U
    N
    −β
    2
    a
    e
    π
    −βaε=0
    In[]:=
    p(1+βa^2)-βa(1-λ)UN-βa^2pe-βaϵ==0
    Out[]=
    -pe
    2
    βa
    +p(1+
    2
    βa
    )-βaϵ-UNβa(1-λ)0
    In[]:=
    Solve[-pe
    2
    βa
    +p(1+
    2
    βa
    )-UNβa(1-λ)-βaϵ0,p]
    Out[]=
    p
    UNβa+pe
    2
    βa
    +βaϵ-UNβaλ
    1+
    2
    βa
    
    π=
    βa(1−λ)
    U
    N
    1+β
    2
    a
    +
    β
    2
    a
    e
    π
    1+β
    2
    a
    +
    βaε
    1+β
    2
    a
    (
    8
    )
    ​
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    ​
    This is optimal policy rule of authorities given expectations
    ​
    Agents know this rule
    ​
    Thus when forming expectations pe they use this rule
    ​
    π=
    βa(1−λ)
    U
    N
    1+β
    2
    a
    +
    β
    2
    a
    e
    π
    1+β
    2
    a
    +
    βaε
    1+β
    2
    a
    (
    9
    )
    In[]:=
    p=
    βa(1-λ)UN+pe
    2
    βa
    +βaϵ
    1+
    2
    βa
    Out[]=
    pe
    2
    βa
    +βaϵ+UNβa(1-λ)
    1+
    2
    βa
    ​
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    ​
    1
    .
    Set p = pe and assume first no shocks (ϵ = 0)
    π=
    βa(1−λ)
    U
    N
    1+β
    2
    a
    +
    β
    2
    a
    e
    π
    1+β
    2
    a
    +
    βaε
    1+β
    2
    a
    (
    10
    )
    e
    π
    =
    βa(1−λ)
    U
    N
    1+β
    2
    a
    +
    β
    2
    a
    e
    π
    1+β
    2
    a
    (
    11
    )
    In[]:=
    Solve
    βa(1-λ)UN+pe
    2
    βa
    1+
    2
    βa
    -pe==0,pe
    Out[]=
    {{pe-UNβa(-1+λ)}}
    π=
    e
    π
    =βa(1−λ)
    U
    N
    (
    12
    )
    (*thereisaninflationbias(non-zeroinflation),whichincreaseswithβ,a.*)​​
    ​
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    ​
    ​
    When shocks (e . g ., deterioration of trade balance) occur, solution is
    π=
    e
    π
    +
    βa
    1+β
    2
    a
    ε=βa(1−λ)
    U
    N
    +
    βa
    1+β
    2
    a
    ε
    (
    13
    )
    Authorities react to shocks in a stabilizing manner depending on β
    ◼
  • if β = 0: no stabilization (strict inflation targeting)
    • ◼
  • if β > 0: some stabilization
    • ​
    Slide
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    ​
    Slide
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    ​
    Indifference curves between inflation and excess unemployment:
    Phillips curve with forward looking expectations:
    ​
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    Credibility of the monetary authority
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    Italy will not follow its commitment to the announced FX
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    ◼
  • If no cost of devaluation, the peg by Italy is not credible and will be attacked
    • ◼
  • Two countries must have same preferences
    • ◼
  • Even if they have same preferences, asymmetric shocks may reduce credibility
    • ​
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    Italy must now accept too high unemployment to maintain the FX
    ​
    ​
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    ◼
  • Italy must now accept too high unemployment
    • ◼
  • Italian authorities would like Germany to accommodate
    • ◼
  • If Germany refuses
    • ◼
  • conflict
    • ◼
  • loss of credibility
    • ◼
  • Attack on the system
    • ◼
  • Case study : EMS during 1992 - 1993
    • ◼
  • If there is a cost of devaluation, the fixed exchange rate can be made credible
    • ​
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    The model with multiple equilibria

    ​
    We will compute the loss of the authorities under different expectations of private agents
    ​
    1
    .
    Loss under discretion
    ​
    This is the loss when CB follows discretionary policy which is fully expected
    ​
    This is solution derived earlier
    ​
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    ​
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    2
    .
    Loss of fixed exchange rate (zero inflation) when agents expect fixed exchange rate
    ​
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    3
    .
    Loss when the CB devalues (cheats) while agents expect fixed exchange rate
    Take optimal inflation from first order condition and set pe = 0 (also ϵ = 0)
    ​
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    ​
    Substitute into loss function
    ​
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    ​
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    ​
    The difference between LFXZERO and LCHEAT and measures the temptation to devalue
    ​
    ​
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    This does not yet mean, however, that it will last
    ​
    ​
    ​
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    4
    .
    Loss when authorities maintain fixed exchange rate while agents expect devaluation (discretion)
    ​
    Call this loss under stabilisation
    ​
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    ​
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    ​
    ​
    Comparing LDIS with LSTAB allows us to find the cost of defending the peg
    ​
    ​
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    Three possible cases

    ​
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    ​
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    ​
    There are two possible equilibria
    ​
    Which on will prevail solely depends on expectations which become self - fulfilling
    ​
    ​
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    Importance of weight attached to stabilization (β)

    ​
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    ​
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    ◼
  • If β = 0 (no desire to stabilize) then C > 0 ensures there is never an attack
    • ◼
  • With β sufficiently ⇒ high always attack
    ​
    • ​
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    ​
    Two conclusions