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: guilherme.bandeira@treasury.nsw.gov.au†Universitat Autònoma de Barcelona and Barcelona GSE. e-mail: jordi.caballe@uab.es‡Corresponding author: MOVE, Universitat Autònoma de Barcelona and University of Sheffield. e-mail:

evgenia.vella@movebarcelona.eu

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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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