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Emigration and Fiscal Austerity in a Depression
Online Appendix
(not intended for publication)
Guilherme Bandeira∗ Jordi Caballé† Eugenia Vella‡
June 24, 2020
∗New South Wales Treasury. e-mail:
[email protected]†Universitat Autònoma de
Barcelona and Barcelona GSE. e-mail:
[email protected]‡Corresponding author: MOVE, Universitat
Autònoma de Barcelona and University of Sheffield. e-mail:
[email protected]
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1 Graphical Illustration of the Model
Figure 1: Overview of the Model
(a) Household members: Residents and Migrants
HomeNationals
HomeResidents
Work, Buy foreign
consumption good,
Pay taxesAbroad
On-the-jobSearchAbroad
Migrants
EmployedUnemployed
HOUSEHOLD
Disutility
Searchers in Home
Searchers Abroad
Pecuni
ary co
st
Remittances
+
Foreign consumption good
(b) Firms
Non-tradable
Competitive
Use K, L(S&M
frictions)
IntermediateGoods
Non-tradable
Competitive
Use domestic +
foreign goods
Final Goods
FIRMS
Tradable
Monopolistic competition
(Sticky prices)
Retailers
S&M denotes search and matching; K,L denote capita and
labour, respectively.
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2 Model Equations
2.1 The Household’s Problem
The household’s Lagrangean can be written as
L = E0∑∞
t=0 βt
(Ct − ζC̃t
)1−η1− η
− χh1+ξt nt + h
1+ξe ne,t
1 + ξ− Ω(ne,t)
1+µ
1 + µ
−λc,t[
(1 + τ c) ct + it +bg,t+1rt− etbf,t+1
rf,t+ φ (zt)nt + ς (s̃tũt) stut − (1− τn)wthtnt − but
−[rkt − τ k
(rkt − δt
)]xtkt − bg,t + etbf,t − Πpt − T
+et ((1 + τc?) ce,t − (1− τn?)w?ne,t)
]−λu,t (nt + ne,t + ut − n̂)
−λk,t
[kt+1 −
[1− ω
2
(itit−1− 1)2]
it −(1− δ̄xιt
)kt
]
−λn,t[nt+1 − (1− σ − ψ?Hϕ (zt))nt − ψH,t (1− st)ut
]−λe,t [ne,t+1 − (1− σ?)ne,t − ψ?H (stut + ϕ (zt)nt)]
}.
We assume external habits in consumption, meaning that C̃t ≡
Ct−1 is taken as given inperiod t. The choice variables comprise
ct, kt+1, it, xt, bg,t+1, bf,t+1, nt+1, ne,t+1, ut, st, andzt. The
corresponding first order conditions are the following:ct :
λc,t (1 + τc) = (Ct − ζCt−1)−η . (1)
kt+1 :
λk,t/β = Etλc,t+1([rkt+1 − τ k
(rkt+1 − δt+1
)]xt+1
)+ Etλk,t+1 (1− δt+1) . (2)
it :
λc,t − λk,t
{1− ω
2
(itit−1− 1)2− ω
(itit−1− 1)
itit−1
}= βEtλk,t+1ω
(it+1it− 1)(
it+1it
)2(3)
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xt :
λk,tιδ̄ (xt)ι−1 = λc,t
{rkt − τ k
(rkt − (1 + ι) δt
)}. (4)
bg,t+1 :
1/β =Etλc,t+1λc,t
rt . (5)
bf,t+1 :
1/β =Etλc,t+1et+1
λc,tetrf,t . (6)
nt+1 :
λn,t/β = Etλc,t+1 [(1− τn)wt+1ht+1 − φ (zt+1)]− Etλu,t+1 −
χh1+ξt+11 + ξ
+Etλn,t+1 (1− σ − ψ?Hϕ (zt+1)) + Etλe,t+1ψ?Hϕ (zt+1) . (7)
ne,t+1 :
λe,t/β = Etλc,t+1 (1− τn?) et+1w?he − χh1+ξe1 + ξ
− Ω (ne,t+1)µ − Etλu,t+1
+Etλe,t+1 (1− σ?) . (8)
ut :
λu,t = λc,t (b− ς (s̃tũt) st) + λn,tψH,t (1− st) + λe,tψ?Hst .
(9)
st :
λe,tψ?H − λc,tς (s̃tũt) = λn,tψH,t . (10)
zt :
λc,tφ′ (zt)
ϕ′ (zt)= ψ?H (λe,t − λn,t) . (11)
Combining equations (9) and (10) we get the following two
expressions
λu,t = λc,tb + λn,tψH,t , (12)
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λu,t = λc,t (b− ς (s̃tũt)) + λe,tψ?H . (13)
We then use (12) to replace λu,t+1 in (7),
λn,t/β = Etλc,t+1 [(1− τn)wt+1ht+1 − b− φ (zt+1)]− χh1+ξt+11 +
ξ
+Etλn,t+1 (1− σ − ψH,t+1 − ψ?Hϕ (zt+1)) + Etλe,t+1ψ?Hϕ (zt+1) .
(14)
and we use (13) to replace λu,t+1 in (8),
λe,t/β = Etλc,t+1 ((1− τn?) et+1w?he − b + ς (s̃t+1ũt+1))−
χh1+ξe1 + ξ
− Ω (ne,t+1)µ
+Etλe,t+1 (1− σ? − ψ?H) . (15)
The last two expressions correspond to equations (14) and (15)
in the paper.
2.2 The Wage-Hours Bargaining Problem
2.2.1 Household’s Surplus
The surplus for workers consists of the asset value of
employment net of the outside option(value of being unemployed):
SHt ≡ V
EHt − V UHt . The asset value of employment V EH is
given by,
V EHt ≡ (1− τn)wtht − φ (zt)−χ
λc,t
h1+ξt1 + ξ
+Etβt+1{
(1− σ − ψ?Hϕ (zt))VEHt+1 + σV
UHt+1 + ψ
?Hϕ (zt)V
EFt+1
},
where the value of being unemployed at Home V UH is given
by,
V UHt ≡ b + Etβt+1{ψH,tV
EHt+1 + (1− ψH,t)V
UHt+1
}.
Hence, the worker’s surplus SHt is given by,
SHt = (1− τn)wtht − φ (zt)− b−χ
λc,t
h1+ξt1 + ξ
+ Etβt+1 (1− σ − ψH,t)SHt+1
+Etβt+1ψ?Hϕ (zt)
{V EFt+1 − V
EHt+1
}.
In the previous expression, V EFt marks the value of being
employed abroad,
V EFt ≡ et (1− τn?)w?he −χ
λc,t
h1+ξe1 + ξ
− Ωλc,t
(ne,t)µ
+Etβt+1{
(1− σ?)V EFt+1 + σ?VUFt+1
},
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where we assume that when an emigrant looses her job she joins
the stock of job seekerssearching for a job abroad. The value of
job seeking abroad V UFt is given by,
V UFt ≡ b− ς (s̃tũt) + Etβt+1{ψ?HV
EFt+1 + (1− ψ?H)V
UFt+1
}.
Hence, migrant workers’ surplus SFh,t ≡ VEFt − V UFt is given
by,
SFh,t = et (1− τn?)w?he − b− ς (s̃tũt)−χ
λc,t
h1+ξe1 + ξ
− Ωλc,t
(ne,t)µ
+ (1− σ? − ψ?H) Etβt+1SFh,t+1 .
Optimality implies that the value of job seeking at home or
abroad must be equal (seeequation (9)). Hence, V UHt = V
UFt implies,
ψH,tEtβt+1SHt+1 = ψ
?HEtβt+1S
Fh,t+1 − ς (s̃tũt) .
2.2.2 Firm’s Surplus
For the firm, the surplus from a match is given by,
SFt = (1− α)py,tytnt− wtht + (1− σ − ψ?Hϕ (zt)) Etβt+1SFt+1
,
which, using equation (9) can be written as,
SFt = (1− α)py,tytnt− wtht + (1− σ − ψ?Hϕ (zt))
κ
ψF,t.
2.2.3 The Nash-Bargained Wage
Inserting the two surpluses into the splitting rule (1− ϑ) (1−
τn)SFt = ϑSHt and solving forthe wage yields,
wtht = (1− ϑ){
(1− α) py,tytnt
+ (1− ϕ (zt))ψH,tψF,t
κ
}+
ϑ
(1− τn)
{b +
χ
λc,t
h1+ξt1 + ξ
+ φ (zt)− ϕ (zt) ς (s̃tũt)
}. (16)
2.2.4 Hours Worked in Equilibrium
Hours are determined through negotiation over the joint surplus,
i.e. maxht
(SHt)1−ϑ (
SFt)ϑ
.
Using the expresions for SHt and SFt derived above, together
with the wage’s splitting rule,
the solution to the negotiation problem over hours worked is
given by,
dSHtdht
= − (1− τn) dSFt
dht,
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which yields,
χh1+ξtλc,t
= (1− τn) (1− α)2 py,tytnt
. (17)
2.3 Production
2.3.1 Intermediate Goods Firms: Optimality Condition for
Capital
According to the first order condition with respect to effective
capital, the value of themarginal product equals the rental
rate,
rkt = αpy,tytxtkt
. (18)
2.3.2 Retailers
There is a continuum of monopolistically competitive retailers
indexed by i on the unitinterval. Retailers transform one unit of
intermediate goods into one unit of retail goods.The real marginal
cost is the relative price py,t of intermediate goods. Let yi,t be
the quantityof output produced by retailer i. These goods are
aggregated into a tradable good,
yr,t =
[∫ 10
(yi,t)�−1� di
] ��−1
,
where � > 1 is the constant elasticity of demand for each
variety. The aggregate tradable
good is sold at the nominal price Pr,t =(∫
(Pi,t)�−1 di
) 1�−1 , where Pi,t is the price of variety i.
The demand for yi,t depends on its relative price and on
aggregate demand,
yi,t =
(Pi,tPr,t
)−�yr,t .
Retailers reset prices with a probability 1−λp, choosing P ∗i,t
to maximize expected real profits,
Πt (i) = Et
∞∑s=0
(βλp)s λc,t+sλc,t
([Pi,tPt+s
− px,t+s]yi,t+s
),
subject to the demand schedule, where Pt is the final good
price. Since all firms are ex-anteidentical (except for the variety
they produce), P ∗i,t = P
∗r,t for all i. Taking into account
pr,t ≡ Pr,t/Pt , the resulting expression for the real reset
price p∗r,t ≡ P ∗r,t/Pt is,
p∗r,tpr,t
=�
(�− 1)NtDt, with
Nt = px,tyr,t + λpEtβt+1 (πr,t+1)�Nt+1 ,
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Dt = pr,tyr,t + λpEtβt+1 (πr,t+1)�−1Dt+1 ,
where πr,t≡ Pr,t/Pr,t−1 is the producer price inflation. Calvo
pricing implies,
(Pr,t)1−� = λp (Pr,t−1)
1−� + (1− λp)(P ∗r,t)1−�
.
The aggregate tradable good is sold domestically and abroad at
quantities yl,t and y?m,t ,
yr,t = yl,t + y?m,t , (19)
Note that y?m,t is the only variable with an asterisk ? that is
time dependent.
2.3.3 Final Goods Producers
Finally, perfectly competitive firms produce a non-tradable
final good yf,t by aggregatingdomestic yl,t and foreign ym,t
aggregate retail goods using a CES technology
yf,t =[($)
1γ (yl,t)
γ−1γ + (1−$)
1γ (ym,t)
γ−1γ
] γγ−1
, (20)
where $ denotes home bias and γ is the elasticity of
substitution. Final good producersmaximize profits yf,t− pr,tyl,t−
etp?rym,t, where pr,t ≡ Pr,t/Pt and p?r ≡ P ?r /P ? denote the
realprice of yl,t and ym,t, respectively, denominated in each
country’s numeraire. We assume thelaw of one price holds, i.e. pr,t
= etp
?r . Solving for the optimal demand functions gives
yl,t = $ (pr,t)−γ yf,t, (21)
ym,t = (1−$) (etp?r)−γ yf,t. (22)
We substitute out (21) and (22) into (20) to obtain
1 = $ (pr,t)1−γ + (1−$) (etp?r)
1−γ , (23)
Then we define implicitly the nominal consumer price index as
the value solving (23) for Pt.
2.4 Closing the Model
The final output must equal private and public demand (i.e., the
government uses final goodsto produce public goods and services).
Costs related to vacancy posting and search abroadreduce the amount
of resources available according to,
yf,t = ct + it + gt + κυt + ς (s̃tũt) stut . (24)
Aggregating the household budget constraint using the market
clearing conditions, the gov-ernment budget constraint, and
aggregate profits, we obtain the law of motion for net foreign
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assets,
et (rf,t−1bf,t−1 − bf,t) = nxt + etΞt , (25)
where net exports nxt are defined as
nxt ≡ pr,ty?m,t − etp?rym,t . (26)
Exports depend on the domestic price divided by the real
exchange rate,
y?m,t =
(pr,tet
)−γxy?m , (27)
where γx is the price elasticity and y?m is the steady-state
level of exports, pinned down bythe calibrated value of
steady-state net foreign assets. Real GDP is defined as,
gdpt = yf,t + nxt . (28)
The nominal exchange rate E is exogenously set. The nominal
interest rate on domesticgovernment bonds Rt is pinned down
endogenously through the Fisher equation,
rt =Rt
Etπt+1. (29)
where consumer price inflation πt is defined as πt = Pt /Pt−1
.
3 Quantitative Analysis
3.1 Simulated Shocks
The following table presents information about the shocks used
to match the path of con-sumption and investment in the Eurostat
data.
Table 1: Magnitude of the simulated shocks
t=1 t=2 t=3 t=4 t=5 t=6
risk premium shock 0.15 0.18 0.07 -0.03 -0.025 0.015
investment efficiency shock 0.06 0.07 0.09 0.04 -0.025 0.25
3.2 Migration Flows for Greece
The figure below depicts annual inflows and outflows of migrants
in Greece from 1991.
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Figure 2: Flows of Immigrants and Emigrants for Greece (vertical
axis: number of persons)
Source: Hellenic Statistic Authority
3.3 More Counterfactuals
Figure 3: Quantitative Analysis: Counterfactual Exercises
(a) The role of labour income tax hikes
Notes: See Figure 4 of the paper.
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Figure 4: Quantitative Analysis: Counterfactual Exercises
(continued)
(a) The role of cuts in total spending
(b) The role of cuts in productive spending
Notes: See Figure 4 of the paper.
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Figure 5: Quantitative Analysis: Counterfactual Exercises
(continued)
(a) The role of cuts in utility-enhancing spending
(b) The role of cuts in wasteful spending
Notes: See Figure 4 of the paper.
3.4 Intensive and Extensive Margin
To explore the role of the intensive versus the extensive
margin, we modify the utility functionas follows,
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U (Ct, gct , ht, ne,t) =
Φ1−η
1− η− χ
(h1+ξt nt + h
1+ξe ne,t
)1 + ξ
− Ω(ne,t)1+µ
1 + µ+X
l1−ϕlt1− ϕl
,
where X > 0 is the relative preference for leisure, which is
pinned down in steady stateby the first-order condition with
respect to unemployment (equation (9)), setting in steadystate l =
1/3, and ϕl is the inverse of the Frisch elasticity of labour
supply, which takes thestandard value 4 in our calibration. The
Figure below reports our simulations for the fullmodel (with
migration of the unemployed and the employed).
Figure 6: Results of Quantitative Analysis: Intensive and
Extensive Margins (Full Model)
Notes: See Figure 4 of the paper.
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4 Investment Efficiency Shocks
Figure 7: A Negative Shock to the Marginal Efficiency of
Investment
(a) Migration and Labour Market Variables
(b) Output and Monetary Variables
Notes: See Figure 9 of the paper.
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5 AR(1) Fiscal Shocks
Figure 8: A Tax Shock Inducing a 1% of GDP Rise in Labour Income
Tax Revenue
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: See Figure 6 of the paper.
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Figure 9: A 1% Cut in Wasteful Public Spending
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: See Figure 6 of the paper.
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6 The Role of Price Stickiness
Figure 10: Labour Tax Hikes: The Role of Price Stickiness
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: We investigate the impact or raising the degree of price
rigidities from 0.25 to 0.75. Theblack line in the Debt/GDP panel
reports the path for the debt-to-GDP target. See also Figure 6of
the paper.
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Figure 11: (Wasteful) Spending Cuts: The Role of Price
Stickiness
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: We investigate the impact or raising the degree of price
rigidities from 0.25 to 0.75. Theblack line in the Debt/GDP panel
reports the path for the debt-to-GDP target. Notes: See alsoFigure
6 of the paper.
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7 Fiscal Consolidation Shocks: Additional Results
Figure 12: Comparison in the Model without Migration
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: The black line in the Debt/GDP panel reports the path for
the debt-to-GDP target. Seealso Figure 6 of the paper.
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Figure 13: Comparison in the Model with Migration of the
Unemployed
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: The black line in the Debt/GDP panel reports the path for
the debt-to-GDP target. Seealso Figure 6 of the paper.
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Figure 14: Comparison in the Model with Migration of the
Unemployed and the Employed
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: The black line in the Debt/GDP panel reports the path for
the debt-to-GDP target. Seealso Figure 6 of the paper.
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Figure 15: Spending Cuts: The Case of Utility-Enhancing
Expenditure
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: The black line in the Debt/GDP panel reports the path for
the debt-to-GDP target. Seealso Figure 6 of the paper.
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Figure 16: Spending Cuts: The Case of Productive Expenditure
(a) Migration and Labour Market Variables
(b) Output and Fiscal Variables
Notes: The black line in the Debt/GDP panel reports the path for
the debt-to-GDP target. Seealso Figure 6 of the paper.
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