The role of credit constraints in a small open economy with frictional unemployment : a case for the Portuguese economy

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Master Thesis

The Role of Credit Constraints

In a Small Open Economy with Frictional

Unemployment

A Case for the Portuguese Economy

Duarte Marques Carapeto Carreiro

Student Number: 11375248

Master’s Programme: Economics

Supervisor: Sweder van Wijnbergen

Second Reader: Ward Romp

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Statement of Originality

This document is written by Student Duarte Marques Carapeto Carreiro who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Contents

3.3 Financial Sector – Banks ... 12

3.4 Production Sector – Operating firms ... 16

3.4.1 Financially constrained firms ... 17

3.4.2 Labour Market ... 20

3.4.3 Intermediate Firms ... 21

3.4.4 Value Function ... 22

3.4.5 Nash Wage Bargaining ... 22

3.4.6 Real Wage Rigidities ... 22

3.5 Default Probability ... 23

3.6 Domestic Retail Firms ... 24

3.7 Importers ... 25

3.8 Composite goods producer ... 25

3.9 Capital Producers ... 28 3.10 Exporters ... 29 3.11 Government ... 29 3.12 Monetary Policy ... 30 3.13 Market clearing ... 30 3.14 Current Account ... 30 4. Calibration ... 31 5. Results ... 32 5.1. Shocks ... 32 5.2. Recapitalization ... 36 6. Conclusion ... 37 7. References ... 39 8. Appendix ... 42

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8.1. Retail firms ... 42

8.2. Importers ... 44

8.3. Price Dispersion ... 45

8.4. Current Account ... 46

8.5. Banks with Financial support ... 47

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1. Introduction

By the beginning of 2008, the world already knew which was later considered the major crisis ever since the Big Depression of the 1930’s; the Sub-Prime crisis and the following Big Recession. This derived from various problems that ranged from the lack of liquidity, to the overleveraging of banks and other financial institutions.

After the Lehman Brothers collapse, a multitude of damaging effects echoed throughout the world’s economies, leading to the failure of innumerous other banks, namely the Portuguese Private Bank (BPP) and Portuguese Bank of Business (BPN); this last one, ending up being nationalized to protect consumers from losing their savings. Without a functioning interbank market, banks on these periphery countries (namely Portugal, Greece and Ireland) making their operations a daily struggle for survival. All these problems combined with the eventual financial crisis, led to skyrocketing levels of unemployment, deflation and Gross Domestic Product contraction; and so in 2011, Portugal requested international financial help, in order to keep the country afloat. Against these funds, national and international entities started to implement policies that envisioned the mitigation of contagion, control systemic risk in the financial sector and policies to incentivize banks to perform their daily operations in a more safely manner. An example of these policies was the utilization of newly imposed capital ratios by the Basel Committee; these made banks have a dire need for a recapitalization in order to meet these ratios.

With this set of policies, regulators hoped to contain the crisis and its adverse effects across the world and make consumers and markets begin to trust the financial sector once again; thus recapitalizing the banking sector and ensure lowers levels of leverage on financial institutions. This made the four biggest banks at the time – Portugal’s biggest bank would end up declaring bankruptcy years later, but mainly due to bad governance - to undertake measures in order to meet the national and international demands regards to their capitalization. With the help of the Portuguese government, close to 6.6 billion euros were injected into the banking sector.

But Portugal was brewing problems way before the crisis hit, and these just helped amplifying it and make it worse than what was thought initially. Low productivity and growth, enormous budget and current account deficits, and that this would most likely imply a period of competitive disinflation and high

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unemployment until the competitiveness of the country was reestablished (Blanchard, 2007). As showed by Carneiro et al., (2013), most of this rise in unemployment was due to lower numbers of vacancies posted, but also due to a higher level of job destruction (by defaulting firms or downsizing). One of the labour market characteristics that might have contributed to the bad shape of the Portuguese economy during the crisis was its high degree of wage rigidity (Blanchard 2007).

Regarding wages, most of the economy was on a halt: the public sector completely froze wages during the financial assistance program by the Troika and contained promotions. Having wages frozen over, makes impossible for this variable to absorb part of the shock, and impedes firms from hiring more workers at a lower price, thus rendering higher wage rigidities with lower levels of job creation and hiring.

This thesis tries to analyze the impact that of this credit constraints, as well as the part played by the firm’s profits volatility, in explaining the unprecedented levels of unemployment in Portugal, during and in the wake of the crisis. We do this using a New Keynesian dynamic stochastic general equilibrium model based of van Wijnbergen & Jakucionyte (2017), which incorporates an active banking sector with financial frictions. In order to further approximate this model to reality, we introduce unemployment into the model with a simple Search and Matching block along with real wage rigidities. In order to for our results to be as realistic as possible, the model is calibrated accordingly to the Portuguese economy. Recapitalization policy experiments are undertaken in order to see if a more swift response to the crisis would help improve the general paradigm of the economy, rather than a delayed recapitalization.

2. Literature Review

This section will build on past work on the subject relative to credit supply conditions to firms along with literature on labour markets with search frictions.

Ever since the 1970’s, dynamic stochastic general equilibrium models have been more and more used due to its micro foundations and the macroeconomics insights they can provide when compared against theory. Smets & Wouters (2003), introduce a number of features to the DSGE literature in order to explain some of the persistence seen until then on euro area data; still, they use a closed economy model which didn’t feature any type of financial frictions or banking

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sector. A step towards a more finely tuned model was made by Adolfson et al. (2007), which makes up a model on the lines of Christiano, Eichenbaum, and Evans (2005) ; they extended this model in order to accommodate the assumption of a small open economy for the Swedish economy, showcasing results in line with the microeconomic evidence. Still, the presented model lacks on explaining the importance of the financial sector effects on the overall economy due to its disregards on modeling of any financial frictions.

This quest was taken upon Bernanke, Gertler and Gilchrist (1999), who tried to shed some light on how credit market frictions could help justify some of aggregate variables movement of the economy. To do this, they developed what later became known has the Financial Accelerator: credit market frictions provide a path for shocks to propagate and magnify themselves and thus having substantial consequences on the macroeconomic level. It comes that the addition of the financial accelerator helps explain how small shock can have an amplified effect on the economy. Another way to model these financial frictions was proposed by Gertler & Karadi (2011) where they introduced a new layer to the baseline model: financial intermediaries in between the households and non-financial firms of the economy. These agents are subject to an agency problem and to a constrained balance sheet, thus making them a powerful player on the propagation of shock into the real economy. This improvement of the model was particularly useful in order to study the impact and mechanisms associated with the sub-prime crisis, since banks played a determinant role on the propagation of the shocks during this period.

Along with these financial sector frictions, another important feature missing from most DSGE models is a non-walrasian labor market; most models assume that the labor market clears in equilibrium, fact that is far from the truth in most economies, being that, when combined with financial frictions will end up outputting several changes on employment levels. Labour market rigidities were extensively studied in Mortensen and Pissarides (1999) where they provided a simple description on how to market fluctuation; ending up being one of the cornerstones of unemployment studies ever since. As shown by Chodorow-Reich (2013), financial frictions (namely on bank’s lending) are of importance when studying lenders financial health against the borrower’s ability to expand their hiring services. His results help explain how the aftermath of the financial crisis employment plummeted and how this affected households consumption preferences. Consumers and firms exposed to worse capitalized financial intermediaries will affect the the economics agents choice of inputs. This

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compounding effect worsened the initial shock and made it more and more persistent as years went by.

Torrachi (2017) took this fact and modeled an economy where labour market results depend on how healthy banks really are by taking financial intermediaries moral hazard problem into a closed economy model with a random search & match labour market. This establishes a nexus between the firms hiring decisions and the banks’ balance sheet and focuses on the supply side of the credit market. This linkage implies that an adverse shock to the financial intermediaries that restricts their lending opportunities will put downward pressures on the firm’s ability to hire. Banking sector’s health is an important factor to help explain economic downturns (and decrease in output) on the wake of a negative and strong shock to financial intermediaries.

Another important contribution was made by Antonella Trigari (2006); in this paper she models a standard NK DSGE model, and complements it a search and match labour market to study inflation dynamics. She goes one step further and includes wages that are determined through Nash Wage Bargaining to replicate the wage rigidities seen in the data. In Carneiro et al. (2013) that the main transmission channels of the financial crisis into the economy (in his paper, for the Portuguese case) are: credit constraints, the limited wage flexibility and labour market segmentation. All these contributed deeply to the recession period felt and may have impacted employers decisions, thus leading to the lower levels of employment, reduced consumption, declining tax revenues and halts in domestic production.

Good producing firms were the most hit by this credit crunch; being these highly dependent on the banking sector, when the banks are undercapitalized and generally “in bad shape”, loans to these firms get restricted; not being able to finance the inputs for their operating activities – buy capital and hire labour – leads to a higher probability of an eventual default and consequently, less jobs in the economy. There’s empirical evidence that these credit constraints had a measurable and significant impact on employment levels in Portugal. It’s then clear that, credit supply conditions and the levels of employment are determinant factors in order to understand the mechanics of an economy, especially in the wake of the recent financial and sovereign debt crisis.

In this thesis, we take the model by van Wijnbergen & Jakucionyte (2017) and complement it by introducing a non-walrasian labour market and wage

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rigidities. Our focus will be the Portuguese economy since this was one where unemployment reached unprecedented levels (close to 18%) during the period of financial assistance by the Troika and due to the traditionally high degree of wage rigidity (both in the private and public sector), generally greater than the rest of the small European open economies, thus rendering this model ideal to the study. We consider shock to the foreign funding premium of the financial intermediaries of the economy, shocks to volatility of firm’s future profits and how the financial support sector (recapitalization channel) might have help mitigate further adverse effects of these shocks.

3. Model

3.1 Description

This section has the purpose of describing the model used in this thesis. The model used as a baseline is van Wijnbergen & Egle (2017), which is a small open economy model comprised of households, financially constrained firms, financial intermediaries/banks, a production sector made of retail firms, importers, exporters, composite goods producers and capital producers. A public sector represented by the government and an independent central bank are also included. In order to enrich this model, a search and match labour and wage rigidities are introduced.

Firstly, a distinction between private and public sector should be established; public sector in the model presented is made up by the government structure and the mentioned central bank. The government function is to collect taxes and sell bonds; this revenue streams should be utilized to pay up its own expenditures or recapitalize the baking system in the country in question. On the other hand, the central bank focuses exclusively on setting the nominal domestic interest rate.

On the other end of this economy we can find the private sector; and this can be split into three different segments: banks, non-financial firms and households. These households face a decision on how to distribute their income through real consumption, deposits and the purchase of domestic bonds; this revenues are derived from labour income, return on both deposits and bonds along with profits from the firms owned by them; all this, netted from taxes. The non-financial sector is a more intricate with a lot of different types of firms; first regarding financially constrained firms: these only live for two periods and

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utilize capital on their operations. It also requires paying a fraction 𝜌 of the desired input level in advance which can be financed by two things: either by equity or by a loan provided by the bank; in order for the bank to provide it, the firm’s needs to pledge a fraction of its future revenues.

The rest of the non-financial firms are comprised of exporters, importers, retailers, intermediate firms, composite goods producers and capital producers. Regarding retailers, these firms use the output produced by the intermediate firms and transform it into differentiated products in order to sell it; they have market power to impose monopoly prices. Importers have a similar task to retailers, but these differentiate foreign composite goods in order to sell them in the domestic economy. The output from the importers and the domestic retail firms is bought by the domestic composite goods producer and transformed into a composite good to be sold to the other economic agents. The rest of the aggregate domestic goods that aren’t used by the composite goods producers are going to be bought by exporting firms in order to export this domestic goods and the price charged abroad will depend on the real exchange rate in place at the time. Finally, capital goods producers sell capital to the financially constrained firms, and buy it back (after depreciation) in the next period; they then transform this depreciated capital into regular one, by combining it with investment goods; the technology used in this process is costly.

3.2 Households

This economy is assumed to have a representative household, which comprises of employed and unemployed agents. Agents then face a choice of how much real consumption and domestic bonds they want for a certain period. The utility function is defined as in Greenwood-Hercowitz-Huffman et al. (1998), GHH onwards, and it’s used since it cancels out the wealth effect on the labour supply choice.

The household’s problem is then summarized by:

𝐸

∑ 𝛽

1

1 − 𝛾

[𝑐

−

𝜒

1 + 𝜑

(𝑒𝑚𝑝

)

_]

_(𝟏)

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This problem is subject to the household’s own budget constraint:

𝑐𝑡+ 𝑏𝑡+ 𝑑𝑡= 𝑤𝑡𝑒𝑚𝑝𝑡+ (1 − 𝑒𝑚𝑝𝑡)𝑏 + 𝑅_𝑡−1 𝜋𝑡 𝑏𝑡−1+ 𝑅_𝑡−1 𝜋𝑡 𝑑𝑡−1+ Π𝑡− 𝑡𝑡 (𝟐)

The 𝜋𝑡 represents the composite goods price inflation and 𝑐𝑡 refers to the

household’s consumption choice. Both 𝑑𝑡 and 𝑏𝑡 denote the asset decision the

household has to make between deposits and bonds, respectively, and being subject to the same rate, makes them perfect substitutes. It is further assumed that, these agents are indifferent between the purchase of domestic bonds and making bank deposits, and so, the same nominal gross interest rate, 𝑅𝑡, is applied

to both assets. Since in this model, the households own the financial institutions and the financially constrained firms, they derive lump-sum dividends from these entities, given by Π𝑡; part of this transfer comes from the firms and it’s

given by the firm’s profits, netted from the equity that the households provided in the beginning of the period. They’re taxed 𝑡𝑡 every period for a lump-sum

value. Finally, all parameters included on the household’s lifetime utility function – 𝜑, 𝜒 and 𝛾 – are strictly positive.

As mentioned before, households decide on the amount of real consumption and domestic bonds for each period. This yields only two first-order conditions for the household’s optimization problem; w.r.t real consumption and domestic bonds respectively:

𝜆

= ( 𝑐

−

𝜒 (

)

1 + 𝜑

)

_(𝟑)

𝐸

𝛽𝛬

𝑅

𝜋

= 1 (𝟒)

𝜆𝑡 , in order to simplify reading; 𝜆𝑡 is the Lagrangian

multiplier that arises from the household’s budget constraint and the marginal utility of consumption.

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The lump-sum dividends coming from the financially constrained firms consist of the profits of these firms, minus the equity transfer from the household’s to the firms at the beginning of the period; it takes the following form:

𝛱_𝑡𝑓𝑖𝑟𝑚𝑠 = 𝜔𝑓𝑖𝑟𝑚𝑠[ 𝑝_𝑡𝑅𝑦_𝑡𝑅 + 𝑞_𝑡(1 − 𝛿)𝑘_𝑡−1− (1 − 𝜌)𝑞𝑡−1𝑘𝑡−1 𝜋_𝑡 ] −𝜔𝑓𝑖𝑟𝑚𝑠 {(1 − 𝛷(𝑑1,𝑡−1)) 𝜅 [𝑝𝑡𝑅𝑦𝑡𝑅+ 𝑞𝑡(1 − 𝛿)𝑘𝑡−1] + 𝛷(𝑑2,𝑡−1)𝑅𝑡−1𝑅 𝑙_𝑡−1𝐷 𝜋_𝑡 } −𝑛_𝑡𝑓𝑖𝑟𝑚𝑠− 𝑧_𝑡

3.3 Financial Sector – Banks

As mentioned, domestic banks are owned by the domestic households, and their operational activity consists of providing loans to the domestic financially constrained firms. Each period there’s a probability 𝜔 that a fraction of these banks dies out, thus transferring its net worth to the rightful owners, the domestic households. If this probability doesn’t materialize, the banks will keep operating as usual.

The balance sheet of the bank is comprised of assets and liabilities; on the assets side there’s the loans provided to the financially constrained domestic firms. Against these assets, the bank holds a certain amount of equity, deposits and foreign debt. This debt is denominated in domestic currency, thus eliminating any arising currency mismatch on the bank’s balance sheet; it’s further assumed that domestic banks exclusively loan in their domestic currency. The balance sheet of bank j, in composite good’s units, is given by the following equation:

𝑛

+ 𝑑

+ 𝑟𝑒𝑟

𝑑

= 𝑙

The banks pays the domestic nominal interest rate

𝑅

and a foreign nominal interest rate 𝑅𝑡∗𝜉𝑡 on their foreign debt holdings; where 𝑅𝑡∗

follows an order 1 autorregressive process, AR(1). Also, the 𝜉𝑡 parameter gives

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in Schmitt-Grohé and Uribe (2003) in order to ensure model stationarity; and so, the premium depends on the banks level of foreign debt:

𝜉

= 𝑒𝑥 𝑝 [ 𝜅

𝑟𝑒𝑟

𝑑

+ 𝑟𝑒𝑟𝑑

𝑟𝑒𝑟𝑑

+

Ϛ

− Ϛ

Ϛ

] (𝟓)

The parameter Ϛ𝑡 is defined as an exogenous shock, following na AR(1) process.

These banks further face an agency problem as in Gertler and Karadi (2011); where in the end of every period, the bankers face a decision of diverting a fraction 𝜆𝐿 of the bank’s assets and go bankrupt, or, not divert and continue operating as usual. Creditors take this problem into account and so, they only lend up to the point where the continuation value is bigger or equal to the value that the bankers can divert, thus ensuring that there are no incentives to go bankrupt. This condition puts a cap on the (infinite) expansion of the bank’s balance sheet for a certain amount of equity.

The fact that the banks of this economy only hold one asset makes them much more susceptible to an eventual default of this holding. The more probable the default of one of these firms that take up these loans, the lower the expected returns of the bank get; this implies that the return on the bank’s assets is influenced by the profits of the financially constrained firms. This gives the banks two options: they either get a total repayment of the loans they provided, or they’ll cease an amount 𝜅 of the firms output and the remaining capital that is sold at the end of production.

𝐸_𝑡 {𝑅𝑗,𝑡 𝐿 𝜋𝑡+1 𝑙_𝑗,𝑡} ≡ 𝐸_𝑡min {𝑅_𝑗,𝑡𝑅 𝑙𝑗,𝑡 𝐷 𝜋𝑡+1 , 𝜅(𝑝_𝑡+1𝑅 _𝑦 𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝑘𝑗,𝑡)} ⇒ 𝐸_𝑡 {𝑅𝑗,𝑡 𝐿 𝜋𝑡+1 𝑙_𝑗,𝑡} ≡ 𝐸_𝑡 {(1 − 𝛷(𝑑_1,𝑡)) 𝜅(𝑝_𝑡+1𝑅 _𝑦 𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝑘𝑗,𝑡) + 𝛷(𝑑2,𝑡)𝑅𝑗,𝑡𝑅 𝑙_𝑗,𝑡𝐷 𝜋𝑡+1 }

Another two variables are also defined with the bank’s problem: the default spread and the banking friction spread. The former is denoted as the difference

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between the interest rate the bank charges for the loan, minus the expected return the bank will have on the loan:

𝐸𝑡(

𝑅_𝑗,𝑡𝑅 − 𝑅_𝑗,𝑡𝐿 𝜋𝑡+1 )

The bigger this difference gets the higher the default probability and accordingly, the charged rate on the loans increases to compensate the risk taken by the bank.

The latter is given by the difference between the expected return on the loans provided by the bank, netted from the expected funding costs of the bank:

𝐸_𝑡 (𝑅𝑗,𝑡 𝐿 𝜋_𝑡+1− 𝑅_𝑡∗_𝜉 𝑡 𝜋_𝑡+1∗ 𝑟𝑒𝑟_𝑡+1 𝑟𝑒𝑟_𝑡 )

By adding both spread equations to each other, a final equation is reached – overall spread - that translates into the following: the higher the credit spread faced by the banks, the tighter the borrowing conditions will get. To note that movements on the real exchange rate could affect the bank’s expected cost of funding 𝐸_𝑡( 𝑅𝑗,𝑡 𝑅 𝜋_𝑡+1− 𝑅_𝑡∗𝜉𝑡 𝜋_𝑡+1∗ 𝑟𝑒𝑟𝑡+1 𝑟𝑒𝑟_𝑡 )

So, the optimization problem for each j bank can be summed by the following: 𝑉𝑗,𝑡= max {𝑑𝑗,𝑡,𝑑𝑗,𝑡∗ ,𝑙𝑗,𝑡} 𝐸𝑡 [𝛽𝛬𝑡,𝑡+1{(1 − 𝜔)𝑛𝑗,𝑡+1+ 𝜔𝑉𝑗,𝑡+1}] subject to: 𝑉_𝑗,𝑡 ≥ 𝜆𝐿_𝑙 𝑗,𝑡 𝑛𝑗,𝑡+ 𝑑𝑗,𝑡+ 𝑟𝑒𝑟𝑡𝑑𝑗,𝑡∗ = 𝑙𝑗,𝑡

𝑛

=

𝑙

−

𝑑

−

𝑟𝑒𝑟

𝑑

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yielding the following Lagrange:

𝐿 = (1 + 𝑣1,𝑡) 𝐸𝑡 𝛽𝛬𝑡,𝑡+1{(1 − 𝜔) ( 𝑅_𝑗,𝑡𝐿 𝜋𝑡+1 𝑙𝑗,𝑡− 𝑅_𝑡 𝜋_𝑡+1 𝑑𝑗,𝑡− 𝑅_𝑡∗ _𝜉 𝑡 𝜋 _𝑡+1∗ 𝑑𝑗,𝑡 ∗ _𝑟𝑒𝑟 𝑡+1) + 𝜔𝑉𝑗,𝑡+1} −𝑣1,𝑡𝜆𝐿𝑙𝑗,𝑡 −𝑣2,𝑡( 𝑅_{𝑗,𝑡−1}𝐿 𝜋𝑡 𝑙𝑗,𝑡− 𝑅_𝑡−1 𝜋𝑡 𝑑𝑗,𝑡−1− 𝑅_𝑡−1∗ 𝜉_𝑡−1 𝜋 _𝑡∗ 𝑟𝑒𝑟𝑡𝑑𝑗,𝑡−1∗ − 𝑙𝑗,𝑡+ 𝑑𝑗,𝑡+ 𝑟𝑒𝑟𝑡𝑑𝑗,𝑡∗ )

The first-order conditions follow:

𝑑𝑗,𝑡: (1 + 𝑣1,𝑡)𝐸𝑡 𝛽𝛬𝑡,𝑡+1{(1 − 𝜔) + 𝜔𝑣2,𝑡+1} ( 𝑅𝑡 𝜋𝑡+1 ) =𝑣_2,𝑡 (1) 𝑑𝑗,𝑡∗ : (1 + 𝑣1,𝑡)𝐸𝑡 𝛽𝛬𝑡,𝑡+1{(1 − 𝜔) + 𝜔𝑣2,𝑡+1} ( 𝑅𝑡∗𝜉_𝑡 𝜋_𝑡+1∗ 𝑟𝑒𝑟𝑡+1 𝑟𝑒𝑟𝑡 ) = 𝑣_2,𝑡 (2) 𝑙𝑗,𝑡: (1 + 𝑣1,𝑡)𝐸𝑡 𝛽𝛬𝑡,𝑡+1{(1 − 𝜔) + 𝜔𝑣2,𝑡+1} ( 𝑅_𝑗,𝑡𝐿 𝜋𝑡+1 ) =𝑣1,𝑡𝜆𝐿+ 𝑣2,𝑡 (3)

Both 𝑣1,𝑡 𝑎nd 𝑣2,𝑡 are lagrangian multipliers of the maximization problem; 𝑣1,𝑡

relates to the incentive constraint - 𝑉𝑗,𝑡 ≥ 𝜆𝐿𝑙𝑗,𝑡 – while 𝑣2,𝑡 relates to the

combination of the LoM the net worth and the balance sheet constraint.

Equation (1) represents the marginal cost a bank faces for each deposit unit against the marginal benefit of seeing its equity increased by one extra unit. Equation (2) follows the same logic: marginal cost of one extra unit of foreign debt is equal to the marginal benefit of one more extra unit of equity - taking into account the real exchange rate at the moment. In equilibrium, these two conditions mean that banks should be indifferent in-between taking deposits or foreign debt, and that both define the bank’s debt portfolio choice.

With regards to loans provided, the marginal cost needs to equal the marginal benefit of issuing another unit of loans, and this additional unit will earn the discounted risk adjusted return on loans. This return increase has to increase along with the marginal cost of issuing more debt to increase the balance sheet 𝑣_2,𝑡; along with the share of divertable assets and the marginal loss to the bank creditor in case the banker in fact diverts 𝑣1,𝑡. These two concepts relate to the

incentive constraint of the bank, meaning that, the tighter the leverage constraint gets, bank spread will also increase, directly implying less credit outflows.

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Along with these first-order conditions, two more slackness conditions are defined: 𝑣_1,𝑡: 𝑣_1,𝑡(𝑉_𝑗,𝑡− 𝜆𝐿_𝑙 𝑗,𝑡) = 0 𝑣2,𝑡: 𝑣2,𝑡( 𝑅𝑗,𝑡−1𝐿 𝜋𝑡 𝑙𝑗,𝑡−1− 𝑅𝑡−1 𝜋𝑡 𝑑𝑗,𝑡−1− 𝑅𝑡−1∗ 𝜉𝑡−1 𝜋 _𝑡∗ 𝑟𝑒𝑟𝑡𝑑𝑗,𝑡−1∗ − 𝑙𝑗,𝑡+ 𝑑𝑗,𝑡+ 𝑟𝑒𝑟𝑡𝑑𝑗,𝑡∗ ) = 0

This only leaves undefined the bank’s aggregate net worth dynamics: the current aggregate net worth of banks is defined as the previous period non-defaulted bank’s net worth, plus the net worth of newly created banks – injected by the domestic households of size :

𝑛_𝑡 = 𝜔 (𝑅𝑗,𝑡−1 𝐿 𝜋_𝑡 𝑙𝑡−1− 𝑅_𝑡−1 𝜋_𝑡 𝑑𝑡−1− 𝑅_𝑡−1∗ 𝜉_𝑡−1 𝜋 _𝑡∗ 𝑟𝑒𝑟𝑡𝑑𝑡−1 ∗ _{) + 𝜄𝑛}

To finalize this sector’s description we’ll mention the financial support mechanism of the economy. It is modelled as in Kirchner and van Wijnbergen (2016), and it states that the government can come in during a crisis period and provide capital injections to the financial sector in order to mitigate its effects. The recapitalization mechanism for each individual bank j takes the form:

𝜏_𝑡𝐹𝐼 _{= 𝜅}

𝐹𝑆(𝑠ℎ𝑜𝑐𝑘𝑡−𝑙− 𝑠ℎ𝑜𝑐𝑘)𝑛𝑗,𝑡−1 , 𝜅𝐹𝑆 > 0, 𝑙 ≥ 0

The recapitalization can be done immediately or it can be provided with some lag. We specify that the shock that drives the crisis can be anything; a risk premium shock (𝜉𝑡 ≡ 𝑠ℎ𝑜𝑐𝑘𝑡) per example, and that banks accept this capital

injection without ever having to pay it back. This mechanism slightly changes the bank’s optimization problem: we go into further detail in the Appendix.

3.4 Production Sector – Operating firms

The production sector of this economy is characterized by a set of firms: financially constrained firms, intermediate firms, retail firms and composite goods producers. Firstly, the financially constrained firms purchase capital from which they produce homogeneous goods. These goods are then bought by the intermediate firms that combine it with labour to produce an intermediate good later purchased by the retail firm, which is then differentiates it without any type of cost and then sell it as a monopolistic market – following Dixit-Stiglitz (1977).

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Importing firms, buy differentiated foreign goods (from abroad). The composite goods producer then buys both home differentiated goods, and foreign differentiated from retailers and importers; it aggregates them to produce both a domestic aggregate good and an aggregate foreign good, with the corresponding price of 𝑝𝑡𝐻 and 𝑝𝑡𝐹.

3.4.1 Financially constrained firms

These firms live for two periods, and every period a new generation of firms is born into the economy; the total amount of firms operating is a continuum of mass one at all times. In their first period of life, these firms buy capital where a fraction 𝜌 needs to be payed upfront, thus creating a demand for working capital in the economy. After input level is chosen, the firms produce in the following period.

In order to be able to get these inputs firms have two sources of funding: it either receives an equity transfer from the households, 𝑛_𝑖,𝑡𝑓𝑖𝑟𝑚𝑠 – its owners – or it takes a loan 𝑙𝑖,𝑡 from the bank for the amount needed to pay for the inputs. In this

model, the loans can only be done in domestic currency 𝑙𝑖,𝑡 = 𝑙𝑖,𝑡𝑑 . To borrow

these funds, the firms have to pledge a fraction 0 < 𝜅 ≤ 1 of their future revenue as collateral for the loan. The initially decide how much to borrow before any shock manifests itself. The total amount borrowed is equal to the expected expenditure on working capital minus the equity transfer made by the households; it follows:

𝐸𝑡−1{𝑙𝑖,𝑡} + 𝐸𝑡−1{ 𝑛𝑖,𝑡 𝑓𝑖𝑟𝑚𝑠

} = 𝐸𝑡−1{𝜌(𝑞𝑡𝑘𝑖,𝑡)}

Where 𝑞𝑡, is the real price of capital, expressed in units of composite goods. The

loan is taken in period t, shocks occur, and the debt overhang generated from the loan taken will distort the firms incentives to buy production inputs. Both the equity transfer and the total loan value are denoted in units of composite goods.

Depending on the amount of equity in these companies, firms take out loans; in bad times firms have a low level of (corporate) equity, and so, to produce the same amount of goods they need to take on bigger loans – implying a leveraging of the firm and more risk of eventually defaulting. Vice-versa can also happen: firms with reasonable amounts of corporate equity need less external financing, thus taking smaller loans and having lower default risk. Since these firms die out,

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after their second period of life, future profits aren’t accounted for when making their input decision.

Due to the timing of these shocks, the demand for working capital will mostly be different than the value of the loans taken; owners might transfer lump-sum funds, 𝑧𝑖,𝑡 , to these firms to cover the difference between the loans and working

capital demand. The transfers from the households are only considered residual funding, and cannot be seen as the main source of funding for these firms. Since these transfers are lump-sum in nature, the household’s and the firm’s choice is not distorted in any way.

Since the interest rates on these loans are only defined in the aftermath of the shocks, the decided rate will be such that market clears. The loan value in units of composite goods is given by: 𝑅𝑖,𝑡𝑅

𝑙_𝑖,𝑡𝐷

𝜋𝑡+1where the 𝑅𝑖,𝑡

𝑅 _{is the nominal gross interest}

rate on the loan. In case of a firm default, a fraction 𝜅 of the continuum of firms operating will have to handout their revenue comprised of their goods sale and the selling of the depreciated capital in period t+1: 𝑝𝑡+1𝑅 𝑦𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝑘𝑖,𝑡. The

𝑝_𝑡+1𝑅 is simply the price of the homogeneous goods in units of composite good. The choice on either or not to default is rationally made by paying up the lower value of two options: the full payment of their loan contract or paying up the pledge collateral decided when the loan contract was made:

min {𝑅_𝑖,𝑡𝑅 𝑙𝑖,𝑡

𝐷

𝜋𝑡+1

, 𝜅 [𝑝_𝑡+1𝑅 𝑦_𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝜂𝑡+1𝑘 𝑘𝑖,𝑡]}

Being that the production function follows the technology: 𝑦_𝑡+1𝑅 = 𝐴_𝑡+1𝜂_𝑡+1𝑘 𝑘𝑡 .

After the shocks take place, the financially constrained firms maximize their profit function taking the loan as given and accounting for the eventual repayment or to lose the pledged fraction of production.

The firm’s problem comes down to choosing the amount of production inputs they desire; they make this decision in order to maximize their expected future revenues, minus the working capital expenditures and the debt repayment cost. Flows incoming to the firms at period t, must also be taken into account, and so,

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this is defined as the difference between the sum of loans taken and equity, netted from the working capital expenses.

max 𝑘𝑡 𝐸_𝑡 𝛽𝛬_𝑡,𝑡+1[𝑝_𝑡+1𝑅 𝑦_𝑡+1𝑅 + 𝑞_𝑡+1(1 − 𝛿)𝜂_𝑡+1𝑘 _𝑘 𝑖,𝑡 − (1 − 𝜌) 𝑞_𝑡𝑘_𝑖,𝑡 𝜋_𝑡+1] +𝐸𝑡 𝛽𝛬𝑡,𝑡+1min {𝑅𝑖,𝑡𝑅 𝑙_𝑖,𝑡𝐷 𝜋𝑡+1 , 𝜅 [𝑝_𝑡+1𝑅 𝑦_𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝜂𝑡+1𝑘 𝑘𝑖,𝑡]} +𝑙_𝑖,𝑡+ 𝑛_𝑖,𝑡𝑓𝑖𝑟𝑚𝑠+ 𝑧_𝑖,𝑡− 𝜌(𝑞_𝑡𝑘_𝑡) subject to: 𝐸𝑡−1{𝑙𝑖,𝑡} + 𝐸𝑡−1{ 𝑛𝑖,𝑡 𝑓𝑖𝑟𝑚𝑠 } = 𝐸_𝑡−1{𝜌(𝑞_𝑡𝑘_𝑖,𝑡)} This yields the following FOC w.r.t. 𝑘𝑖,𝑡 :

𝐸_𝑡 𝛽𝛬_𝑡,𝑡+1[𝑝_𝑡+1𝑅 𝜕𝑦𝑡+1 𝑅 𝜕𝑘_𝑖,𝑡 + 𝑞𝑡+1(1 − 𝛿)𝜂𝑡+1 𝑘 _{− (1 − 𝜌)} 𝑞𝑡 𝜋_𝑡+1] +𝐸𝑡 𝛽𝛬𝑡,𝑡+1{(1 − 𝛷(𝑑1,𝑡)) 𝜅 [𝑝𝑡+1𝑅 𝜕𝑦_𝑡+1𝑅 𝜕𝑘_𝑖,𝑡 + 𝑞𝑡+1(1 − 𝛿)𝜂𝑡+1 𝑘 _]} = 𝜕𝑐𝑜𝑣 (𝛽𝛬_𝑡,𝑡+1, min {𝑅_𝑖,𝑡𝑅 𝑙𝑖,𝑡 𝐷 𝜋_𝑡+1, 𝜅 [𝑝𝑡+1𝑅 𝑦𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝜂𝑡+1𝑘 𝑘𝑖,𝑡]}) 𝜕𝑘_𝑖,𝑡 + 𝜌𝑞𝑡 We define 𝑑1,𝑡= 𝑑2,𝑡+ 𝜎𝑦; where 𝑑2,𝑡 = 𝐸𝑡𝑙𝑛 𝜅 [𝑝𝑡+1𝑅 𝑦𝑡+1𝑅 +𝑞𝑡+1𝜂𝑡+1𝑘 (1−𝛿)𝑘𝑖,𝑡]− 𝐸𝑡𝑙𝑛(𝑅𝑖,𝑡𝑅 𝑙𝑖,𝑡 𝐷 𝜋𝑡+1) 𝜎𝑦

and the 𝜎𝑦is given by 𝑣𝑎𝑟(𝜋𝑡+1 𝜅 (𝑝𝑡+1𝑅 𝑦𝑡+1𝑅 + 𝑞𝑡+1(1 − 𝛿)𝑘𝑖,𝑡). In the next period,

after the shocks occurred, part of these firms end up defaulting, and so, the household gathers all the remaining net worth – of non-defaulted firms - and aggregates it according to the rule:

𝑛_𝑡𝑓𝑖𝑟𝑚𝑠 = 𝜔𝑓𝑖𝑟𝑚𝑠_{[ 𝑝} 𝑡𝑅𝑦𝑡𝑅 + 𝑞𝑡𝜂𝑡𝑘(1 − 𝛿)𝑘𝑡−1− (1 − 𝜌) 𝑞𝑡−1𝑘𝑡−1 𝜋_𝑡 ] −𝜔𝑓𝑖𝑟𝑚𝑠 _{{(1 − 𝛷(𝑑} 1,𝑡−1)) 𝜅 [𝑝𝑡𝑅𝑦𝑡𝑅 + 𝑞𝑡𝜂𝑡𝑘(1 − 𝛿)𝑘𝑡−1] + 𝛷(𝑑2,𝑡−1)𝑅𝑡−1𝑅 𝑙_𝑡−1𝐷 𝜋_𝑡 } +𝜄𝑓𝑖𝑟𝑚𝑠_{∗ 𝑛}𝑓𝑖𝑟𝑚𝑠

The first term defines the financially constrained firm’s revenues from both production and the sale of depreciated capital, netted from the working capital

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expenses; the second term is the firm’s expenditures when repaying the loans contracted and the third term being the equity injections. The domestic household cannot divert pooled equity funds anywhere else; the amount of these equity injections is fixed and it’s proportional to the aggregate net worth in the steady-state and does not depend on the household’s decision. The 𝜔𝑓𝑖𝑟𝑚𝑠, is a parameter close to unity and proxies the equity management costs of the household – it is later used to calibrate the steady-state corporate leverage.

3.4.2 Labour Market

Households supply labour to financially constrained firms in a frictional labour market. Every period a certain amount of matches occur in the economy, and this is given by the following matching function:

𝑚𝑡= 𝜎𝑒𝑢𝑡 𝜎𝑚_𝑣

𝑡 1−𝜎𝑚

The matching function is characterized by a scale parameter, 𝜎𝑒, a matching

efficiency parameter, 𝜎𝑚, unemployment level and vacancies.

We assume full labor force is normalized to one; this implies that the number of searching unemployed agents is given by:

𝑢_𝑡= 1 − (1 − 𝛺)𝑒𝑚𝑝_𝑡−1

At the beginning of each period a number of workers are matched with a job, 𝑚𝑡.

This implies that the level of household members that are employed will vary according to the following rule:

𝑒𝑚𝑝_𝑡 = (1 − 𝛺)𝑒𝑚𝑝_𝑡−1+ 𝑚_𝑡

These employment relationships might be severed each period at a rate 𝛺, which denotes the exogenous probability of separation between worker and firm. The equation above states that the number of matched workers at any given period is the amount of workers that survived from the previous period, plus the newly-formed matches. Note that in this model, unemployment is calculated at the beginning of the period, and so, unemployment after hiring process is denoted by:

𝑢_𝑡= 1 − 𝑒𝑚𝑝_𝑡

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𝑝𝑓𝑖𝑙𝑙(𝜃_𝑡) = 𝑚𝑡 𝑣_𝑡

Where 𝜃𝑡 denotes the labour market tightness at time t and is given by 𝜃𝑡 = 𝑣𝑡

𝑢𝑡.

The probability for a worker that is looking for a job, to be matched with an open vacancy is:

𝑝𝑓𝑖𝑛𝑑(𝜃𝑡) =

𝑚_𝑡 𝑢𝑡

3.4.3 Intermediate Firms

The homogeneous goods produced by the financially constrained firms are used bought and used as inputs by the intermediate firms; the firms combine it with labour in order to produce output 𝑦𝑡𝐼. The do with following the technology:

𝑦_𝑡𝐼 = (𝑦_𝑡𝑅)𝛼( 𝑒𝑚𝑝𝑡)1−𝛼

These goods are sold at price 𝑃𝑡𝐼, and they try to maximize their profit function:

max 𝑦𝑡𝑅, 𝑒𝑚𝑝𝑡, 𝑣𝑡 𝐸𝑡∑ 𝛽𝛬𝑡,𝑡+1 ∞ 𝑡=0 [𝑝_𝑡𝐼𝑦_𝑡𝐼− 𝑝_𝑡𝑅𝑦_𝑡𝑅− 𝑤_𝑡 𝑒𝑚𝑝𝑡− 𝜓𝑣𝑡]

subject to the law of motion of employment:

𝑒𝑚𝑝_𝑡 = (1 − 𝛺)𝑒𝑚𝑝_𝑡−1+ 𝑝𝑓𝑖𝑙𝑙_𝑡𝑣_𝑡

This maximization problem then yields the following conditions to 𝑒𝑚𝑝𝑡, 𝑣𝑡 and

𝑦_𝑡𝑅 _{respectively.} 𝐽𝑡 = 𝑝𝑡𝐼(𝑦𝑡𝑅)𝛼(𝑒𝑚𝑝𝑡)−𝛼(1 − 𝛼) − 𝑤𝑡+ 𝛽(1 − 𝛺)𝐸𝑡𝛬𝑡,𝑡+1 𝐽𝑡+1 𝜓 𝑝𝑓𝑖𝑙𝑙(𝜃𝑡) = 𝐽𝑡 𝑝_𝑡𝑅 _{= 𝑝} 𝑡𝐼 𝛼 (𝑦𝑡𝑅)𝛼−1( 𝑒𝑚𝑝𝑡)1−𝛼

We denote the Langrange multiplier associated with the constraint by 𝐽𝑡, the

shadow value of employment. By combining the first two conditions we get the job creation condition:

22 𝜓 𝑝𝑓𝑖𝑙𝑙(𝜃𝑡) =𝑝_𝑡𝐼_(𝑦 𝑡 𝑅₎𝛼_{(𝑒𝑚𝑝} 𝑡) −𝛼 (1 − 𝛼)− 𝑤_𝑡+ 𝛽(1 − 𝛺)𝐸_𝑡𝛬_𝑡,𝑡+1 𝜓 𝑝𝑓𝑖𝑙𝑙(𝜃_𝑡+1)

3.4.4 Value Function

Regarding the workers side, they can either be employed and collecting a wage or unemployed and collecting benefits from the government. Their surplus function is given by:

𝑆_𝑡= 𝑤_𝑡− 𝑏 + (1 − 𝛺)𝐸_𝑡𝛽𝛬_𝑡,𝑡+1[1 − 𝑝𝑓𝑖𝑛𝑑(𝜃𝑡+1)]𝑆𝑡+1

3.4.5 Nash Wage Bargaining

To make the model more realistic and close to the Portuguese economy, in this section we introduce efficient Nash wage bargaining. This means that workers and firms will bargain over the wage; this wage will come out of the joint maximization of both the firm’s and the worker’s surplus:

arg max

𝑤𝑡

(𝑆𝑡)𝜇( 𝐽𝑡− 𝑉𝑡)1−𝜇

where the 𝜇 denotes the bargaining power of the worker. In equilibrium, the value of a vacancy, 𝑉𝑡, is zero and so, the following equation needs to hold:

(1 − 𝜇)𝑆𝑡= 𝜇 𝐽𝑡

This will yield the following:

𝑤𝑡𝑛𝑎𝑠ℎ= 𝑏 + 𝜇 1 − 𝜇 [ 𝜓 𝑝𝑓𝑖𝑙𝑙(𝜃𝑡) − (1 − 𝛺)𝐸𝑡𝛽𝛬𝑡,𝑡+1[1 − 𝑝𝑓𝑖𝑛𝑑(𝜃𝑡+1)] 𝜓 𝑝𝑓𝑖𝑙𝑙(𝜃𝑡+1) ] ⇒ 𝑤𝑡𝑛𝑎𝑠ℎ= 𝑂𝑈𝑇𝑡+ 𝜇 1 − 𝜇 𝑉𝐴𝑅𝑡

The Nash wage can then be defined as the worker’s outside option – a fixed level of unemployment benefits. The variable side of this condition will vary with marginal cost of posting a vacancy, which is related directly with the labour market tightness. The labour disutility parameter is omitted from the wage equation to ensure modelling simplicity.

3.4.6 Real Wage Rigidities

Real wage rigidities are introduced in the line with the literature on the subject; we follow Torrachi (2017) and model the real wage as a weighted average

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between the steady-state value of the Nash bargained wage and the Nash wage bargained on the period in question. Being 𝜇𝑟𝑖𝑔 be the degree of wage rigidity attributed to the wage norm:

𝑤_𝑡= 𝜇𝑟𝑖𝑔𝑤𝑛𝑎𝑠ℎ_{+ (1 − 𝜇}𝑟𝑖𝑔_{) 𝑤} 𝑡 𝑛𝑎𝑠ℎ

The parameter 𝜇𝑟𝑖𝑔 ∊ (0,1); being that when it is equal to one, it’s totally rigid and when equal to zero, wages are completely flexible.

3.5 Default Probability

Since these financially constrained firms have limited, they receive a put option from creditors; this is priced before, but since investment is not contractible on the debt agreement, the moral hazard problem is still in place. This implies that the debt overhang friction will generate an additional term to these firms demand functions - 1 − 𝛷(𝑑1,𝑡−1) – which will work as a proxy for these firms

default probability and reduce the firm’s marginal productivity of capital. It also serves to acknowledge the differences from the private and social benefits from investing. As this probability of default increases, private benefits from investing decrease and so will the demand for capital, ending up on a sub-level of working capital when compared with the socially chosen one. This underinvestment translates into a depressed aggregate demand on impact, lower demand for investment and this lower investment level leads to smaller levels of capital stock in the future, slowing down economic recovery.

It also implies that the default probability will directly depend on the volatility of the 𝜎𝑦2 term – variance of future profits:

𝜎𝑦2 = 𝑣𝑎𝑟( 𝜋𝑡+1 {𝜅 [𝑝𝑡+1𝑅 𝑦𝑡+1𝑅 + 𝑞𝑡+1𝜂𝑡+1𝑘 (1 − 𝛿)𝑘𝑖,𝑡]})

This will makes it then depend on the exogenous productivity shocks, working capital and endogenous price volatility. Increasing uncertainty regarding the firm’s future collateral value implies lower chances of repayment. This factor is important since the uncertainty environment felt during this past crisis played a significant role on firm’s ability to borrow and willingness to borrow; and so, a shock to this volatility term is also modeled.

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3.6 Domestic Retail Firms

This layer of firms buys the output produced by the previous intermediate firms and, each firm j, differentiates the bought good. They incur in no costs to do this, and they sell this differentiated good, at a monopolistic price 𝑝𝑡𝐻(𝑗). It is

further assumed that each period a share (1 − 𝜔𝐻) of the domestic retail firms can update their prices in a Calvo (1983) manner to its optimal level; while the rest, can only adjust past prices by the rate 𝜋_𝑡𝑎𝑑𝑗. The part of the retail firms that can update their price to the optimum in period t is denoted by:

(𝑃𝑡#(𝑗) − 𝑃𝑡𝐼)𝑦𝑡𝐻(𝑗)

This will yield an aggregate price level for the domestic retail sector 𝑃𝑡𝐻, further

defined as: 𝑃_𝑡𝐻= ((1 − 𝜔𝐻)(𝑃_𝑡#₎1−𝜖𝐻_{+ 𝜔}𝐻_(𝑃 𝑡−1𝐻 𝜋𝑡 𝑎𝑑𝑗 )1−𝜖𝐻) 1 1−𝜖𝐻

By re-writing the whole expression in relative price, w.r.t. composite goods price level 𝑃𝑡, and by defining 𝑝̃𝑡𝐻 ≡

𝑃_𝑡# 𝑃_𝑡𝐻, we get: 1 = (1 − 𝜔𝐻)(𝑝̃_𝑡𝐻₎1−𝜖𝐻_{+ 𝜔}𝐻₍𝑝𝑡−1 𝐻 _𝜋 𝑡 𝑎𝑑𝑗 𝜋_𝑡𝑝_𝑡𝐻 ) 1−𝜖𝐻

The differentiated goods produced , 𝑦𝑡𝐻(𝑗), are then sold to the domestic

composite goods producers. The retail firm then ends up having to solve the optimization problem of how to set their optimal price choice, 𝑃𝑡# (𝑗), conditional

on keeping it in the future, in order to maximize their profits:

max 𝑃𝑡# (𝑗), 𝐸_𝑡∑(𝜔𝐻)𝑠 _𝛽𝑠_𝛬 𝑡,𝑡+𝑠 (𝑃_𝑡#(𝑗)(∏𝑗=𝑠_𝑗=1𝜋_𝑡+𝑗𝑎𝑑𝑗) − 𝑃_𝑡+𝑠𝐼 ) 𝑃_𝑡+𝑠 𝑦𝑡+𝑠 𝐻 _(𝑗) ∞ 𝑠=0

And subject to the demand for retail goods:

𝑦_𝑡𝐻(𝑗) = (𝑃𝑡 #_{(𝑗)(∏ 𝜋} 𝑡+𝑗 𝑎𝑑𝑗 𝑗=𝑠 𝑗=1 ) 𝑃_𝑡𝐻 ) −𝜖𝐻 𝑦_𝑡𝐻

25

The rest of the mathematical derivation is done in the Appendix.

3.7 Importers

As we do with domestic retailers, we assume there’s a continuum of monopolistically competitive importers that differentiate that take up foreign composite goods and differentiate it. A variety j of foreign goods, 𝑦𝑡𝐹(𝑗) is

purchased at price 𝑃𝑡∗, that is then sold to the composite goods producer at price

𝑃_𝑡𝐹_{(𝑗), in domestic currency. Another similarity between these firms is the fact}

that importers have monopolistic power over prices (on their country) and set prices as in Calvo (1983), thus allowing for incomplete exchange rate pass-through. A fraction (1 − 𝜔𝐹) of the importing firms is able to choose their optimal price, while 𝜔𝐹can only adjust by the 𝜋_𝑡𝑎𝑑𝑗rate. So, importer j solves its profits maximization problem by choosing its optimal price 𝑃𝑡#𝐹(𝑗), conditional

on not changing it in the future:

max 𝑃𝑡#𝐹(𝑗) 𝐸𝑡∑(𝜔𝐹)𝑠 𝛽𝑠𝛬𝑡,𝑡+𝑠[ 𝑃_𝑡#𝐹(𝑗)(∏𝑗=𝑠_𝑗=1𝜋_𝑡+𝑗𝑎𝑑𝑗) 𝑃𝑡+𝑠 − 𝑟𝑒𝑟𝑡+𝑠] 𝑦𝑡+𝑠𝐹 (𝑗) ∞ 𝑠=0

Subject to the following:

𝑦_𝑡𝐹(𝑗) = 𝜂 (𝑃𝑡 #𝐹_{(𝑗)(∏ 𝜋} 𝑡+𝑗 𝑎𝑑𝑗 𝑗=𝑠 𝑗=1 ) 𝑃_𝑡𝐹 ) −𝜖 𝑦_𝑡𝐹

The mathematical derivation is practically the same the one done with the domestic retail firms, and it’s further in detail in the Appendix.

3.8 Composite goods producer

The composite goods producer takes the differentiated domestic and imported goods and turns them into domestic and foreign aggregate goods, respectively. To do this, it is further assumed that this producer has some kind of aggregation

26

technology that facilitates this process at no cost. The aggregation technology for each type of good follows:

𝑦_𝑡𝐻= (∫ 𝑦_𝑡𝐻(𝑗)1− 1 𝜖𝐻𝑑𝑗 1 0 ) 𝜖𝐻 𝜖𝐻−1 𝑦_𝑡𝐹 = (∫ 𝑦_𝑡𝐹(𝑗)1− 1 𝜖𝐹𝑑𝑗 1 0 ) 𝜖𝐹 𝜖𝐹−1

Both these aggregated goods are then turned into composite goods, 𝑦𝑡𝐶, with an

aggregation technology that takes the taste parameter for foreign aggregate goods, η, as given: 𝑦_𝑡𝐶 ≡ [(1 − 𝜂)1𝜖_(𝑦𝑡𝐻_{− 𝑒𝑥𝑡)} 𝜖−1 𝜖 _{+ 𝜂} 1 𝜖 _𝑦𝑡𝐹 𝜖−1 𝜖 _] 𝜖 𝜖−1

This equation then implies that only part of the total domestic aggregate goods is effectively used to produce the composite goods; being that the remaining of it is fully exported – this happens since exports have no imported content in them. The 𝜖 parameter denotes the elasticity of substitution between domestic and foreign aggregate goods. The produced composite good is then sold to domestic households, the government and the capital goods producers at price 𝑃𝑡, given

that it operates in a perfectly competitive market.

Let the aggregate price level of retail goods be 𝑃𝑡𝐻 ≡ [∫ 𝑃𝑡𝐻(𝑗)1−𝜖𝐻 𝑑𝑗 1

0 ]

1 1−𝜖𝐻_{, in}

domestic currency. We can now find the domestic demand for retail goods, by solving: max 𝑦_𝑡𝐻(𝑗) {𝑃𝑡 𝐻_𝑦 𝑡𝐻− ∫ 𝑃𝑡𝐻(𝑗)𝑦𝑡𝐻(𝑗)𝑑𝑗 1 0 } Subject to the technology:

𝑦_𝑡𝐻= (∫ 𝑦_𝑡𝐻(𝑗)1− 1 𝜖𝐻𝑑𝑗 1 0 ) 𝜖𝐻 𝜖𝐻−1

and the market clearing constraint:

𝑦_𝑡𝐻 _{= 𝑥}

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Being that 𝑥𝑡𝐻denotes the domestic good used by composite goods producer as

an inputs, and the amount that is exported. This will yield that the optimal demand for retail goods of variety j is given by:

𝑦_𝑡𝐻_{(𝑗) = (}𝑃𝑡 𝐻_(𝑗)

𝑃_𝑡𝐻 )

−𝜖𝐻

𝑦_𝑡𝐻

While this happens for the domestic aggregate goods, the foreign aggregate goods follow a similar problem. Defining the aggregate price level of importer’s goods by 𝑃𝑡𝐹 ≡ [∫ 𝑃𝑡𝐹(𝑗)1−𝜖𝐹 𝑑𝑗

1

0 ]

1

1−𝜖𝐹_{, in domestic currency values; the importers}

face a demand for their goods given by the solution to the following: max 𝑦𝑡𝐹(𝑗) {𝑃𝑡𝐹𝑦𝑡𝐹− ∫ 𝑃𝑡𝐹(𝑗)𝑦𝑡𝐹(𝑗)𝑑𝑗 1 0 } subject to: 𝑦_𝑡𝐹 _{= (∫ 𝑦} 𝑡𝐹(𝑗) 1−_𝜖1 𝐹𝑑𝑗 1 0 ) 𝜖𝐹 𝜖𝐹−1

and the market clearing condition stating that all foreign aggregate goods are to be utilized to satisfy the composite goods producer demand for it:

𝑦_𝑡𝐹 _{= 𝑥} 𝑡𝐹

Thus ending up with the optimal demand for importer’s production of variety j of: 𝑦_𝑡𝐹(𝑗) = (𝑃𝑡 𝐹_(𝑗) 𝑃_𝑡𝐹 ) −𝜖𝐹 𝑦_𝑡𝐹

Given all this, the composite goods transformation equation can now be written as: 𝑦_𝑡𝐶 ≡ [(1 − 𝜂)1𝜖(𝑥_𝑡𝐻) 𝜖−1 𝜖 _{+ 𝜂} 1 𝜖 _(𝑥𝑡𝐹₎ 𝜖−1 𝜖 _] 𝜖 𝜖−1

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The composite goods producer will then try to maximize its profits, 𝑃𝑡𝑦𝑡𝐶−

𝑃_𝑡𝐹_𝑥

𝑡𝐹− 𝑃𝑡𝐻𝑥𝑡𝐻, subject to the aggregation technology above. This will result in two

equations: 𝑥_𝑡𝐻= (1 − 𝜂)(𝑝_𝑡𝐻)−𝜖_𝑦 𝑡𝐶 𝑥𝑡𝐹 = 𝜂(𝑝𝑡𝐹)−𝜖𝑦𝑡𝐶 Being that 𝑝𝑡𝐻 ≡ 𝑃𝑡𝐻 𝑃𝑡 and 𝑝𝑡 𝐹 _≡ 𝑃𝑡𝐹 𝑃𝑡

3.9 Capital Producers

Capital producers buy, sell and restore capital; they sell it to financially constrained firms at a real competitive price 𝑞𝑡, buy the remaining depreciated

capital in the next period. In order to restore depreciated capital, this must be combined with composite goods (investment) 𝑖𝑡 following a technology subject

to investment adjustment costs, such that: 𝑘_𝑡 = (1 − 𝛿)𝜂_𝑡𝑘_𝑘 𝑡−1+ [1 − 𝛤 ( 𝑖𝑡 𝑖_𝑡−1)] 𝑖𝑡 where : 𝛤 (_𝑖𝑖𝑡 𝑡−1) = 𝛾 2 ( 𝑖𝑡 𝑖𝑡−1− 1) 2

Capital producers try to maximize their profits, in composite goods units, by choosing their optimal level of investment subject to the capital transformation technology, such that:

max 𝑖𝑡 𝛽𝐸_𝑡𝛬_𝑡,𝑡+1 {(1 − 𝜌) 𝑞𝑡 𝜋_𝑡+1𝑘𝑡} + 𝜌𝑞𝑡𝑘𝑡− 𝑞𝑡(1 − 𝛿)𝜂𝑡 𝑘_𝑘 𝑡−1− 𝑖𝑡 subject to: 𝑘_𝑡 = (1 − 𝛿)𝜂_𝑡𝑘𝑘_𝑡−1+ [1 − 𝛤 ( 𝑖𝑡 𝑖_𝑡−1)] 𝑖𝑡

During the optimization, capital producers will account for the fraction 𝜌 of capital purchases that is paid immediately, instead of the (1- 𝜌) share that is done the next period. The optimization solution yields the following:

29 1 𝑞𝑡 = 𝜌 [1 − 𝛤 ( 𝑖𝑡 𝑖𝑡−1 )] − 𝜌𝛾 ( 𝑖𝑡 𝑖𝑡−1 − 1) 𝑖𝑡 𝑖𝑡−1 + 𝜌𝛾 𝛽𝐸𝑡𝛬𝑡,𝑡+1 𝑞𝑡+1 𝑞𝑡 (𝑖𝑡+1 𝑖𝑡 − 1) (𝑖𝑡+1 𝑖𝑡 ) 2 + (1 − 𝜌)𝛽𝐸_𝑡𝛬_𝑡,𝑡+1𝑞𝑡+1 𝑞_𝑡 [ 1 − 𝛤 ( 𝑖𝑡 𝑖_𝑡−1) − 𝛾 ( 𝑖𝑡 𝑖_𝑡−1− 1) 𝑖𝑡 𝑖_𝑡−1] + (1 − 𝜌) 𝛾 𝛽2_𝐸 𝑡𝛬𝑡,𝑡+2 𝑞𝑡+1 𝑞𝑡 (𝑖𝑡+1 𝑖𝑡 − 1) (𝑖𝑡+1 𝑖𝑡 ) 2

3.10 Exporters

Exporters are assumed to operate in a perfectly competitive environment, and demand 𝑒𝑥𝑡 units of domestic aggregate good; these exported goods consist only

of domestic aggregate goods, they do not use a single unit of imported goods.

These exports are sold at price 𝑝𝑡𝐻

𝑟𝑒𝑟𝑡, which is basically, the domestic aggregate

goods price, adjusted by the real exchange rate, so that it is expressed in units of the foreign composite good. The foreign demand for domestic aggregate goods is then given by:

𝑒𝑥_𝑡 = 𝜂∗₍ 𝑝𝑡 𝐻

𝑟𝑒𝑟_𝑡)

−𝜖∗

𝑦_𝑡∗

In this equation we assume that all foreign economies are identical, so that their demand for the domestic aggregate good can be aggregated; we also can see that the demand for the domestic aggregate goods internationally is sensitive to variations on the real exchange rate, and consequently on the price at what exports are sold. Since our model assumes that home is a small open economy, meaning that 𝑃𝑡∗ and 𝑦𝑡∗ are taken as given. The 𝜖 parameter denotes the elasticity

of substitution between domestic aggregate goods and goods from other economies; while the 𝜂∗ defines the foreign households taste parameter for domestic aggregate goods.

3.11 Government

The government’s expenditures consist of an exogenous level of real government spending 𝑔𝑡 , the recapitalization mechanism for financial

intermediaries 𝜏𝑡𝐹𝑆 , the interest payment of previously issues bonds and the

30

taxes collected from the household’s and the issuing of new bonds. The government’s budget constraint, in units of composite goods, is as follows:

𝑔_𝑡+ 𝜏_𝑡𝐹𝑆 +𝑅𝑡−1

𝜋_𝑡 𝑏𝑡−1+ (1 − 𝑒𝑚𝑝𝑡)𝑏̅ = 𝑏𝑡+ 𝑡𝑡 It is further assumed that taxes follow the given rule:

𝑡_𝑡 = 𝑡 + 𝜅𝐵(𝑏_𝑡−1− 𝑏) + 𝜅𝐹𝑆𝜏_𝑡𝐹𝑆 + 𝑒_𝑡, 0 < 𝜅𝐵≤ 1; 0 < 𝜅𝐹𝑆 ≤ 1 Meaning that a share 𝜅𝐹𝑆of the recapitalization expenditures are financed by an increase on the lump-sum taxes, while the rest is done by issuing debt.

3.12 Monetary Policy

The central bank conducts monetary policy according to the Taylor rule: 𝑅𝑡 𝑅 = ( 𝑅𝑡−1 𝑅 ) 𝛾𝑅 (𝑦𝑡 𝐻 𝑦𝐻) (1− 𝛾𝑅)𝛾𝑌 (𝜋𝑡 𝐻 𝜋𝐻) (1− 𝛾𝑅)𝛾𝜋 exp(𝑚𝑝_𝑡)

The 𝑚𝑝𝑡 is modeled to be a monetary policy shock and the 𝜋𝐻 can be written as:

𝜋𝐻₌ 𝑝𝑡𝐻

𝑝_𝑡−1𝐻 𝜋_𝑡 .

3.13 Market clearing

The domestic household, the government and the capital producers are the agents on this economy that purchase the composite goods, and so, the supply of composite goods must equal the sum of the demands of these three agents:

𝑦_𝑡𝐶 = 𝑐_𝑡+ 𝑖_𝑡+ 𝑔_𝑡

3.14 Current Account

The equation that gives us the trade balance of the economy (in units of composite goods) is:

𝑡𝑏_𝑡= 𝑝_𝑡𝐻𝑒𝑥_𝑡− 𝑚_𝑡

where the 𝑚𝑡 expresses the value of imports and can also be represented by

𝑚_𝑡 ≡ 𝑟𝑒𝑟_𝑡𝐷_𝑡𝐹_𝑦

31

The current account is then given by the sum of the real trade balance and the real net income from abroad:

𝑐𝑎_𝑡 = 𝑡𝑏_𝑡+ 𝑛𝑖_𝑡

Since in this economy banks are the only agent that can borrow from abroad – and that nobody lends to foreign agents – the real net income from abroad is smaller than zero and equals to minus payments of bank’s foreign debt:

𝑐𝑎_𝑡 = 𝑡𝑏_𝑡− (𝑅_𝑡−1∗ 𝜉_𝑡−1− 1)𝑟𝑒𝑟_𝑡𝑑𝑡−1

∗

𝜋_𝑡∗

and since in equilibrium the current account needs to be exactly equal to the capital account, 𝑐𝑝𝑡, given by the change in bank’s foreign debt and in units of

composite goods is:

𝑐𝑝_𝑡= − (𝑟𝑒𝑟_𝑡𝑑_𝑡∗_{− 𝑟𝑒𝑟} 𝑡

𝑑_𝑡−1∗ 𝜋_𝑡∗ )

thus yielding the following condition:

𝑐𝑎_𝑡 = 𝑐𝑝_𝑡 ⇒ 𝑡𝑏_𝑡− (𝑅_𝑡−1∗ 𝜉_𝑡−1− 1)𝑟𝑒𝑟_𝑡𝑑𝑡−1 ∗ 𝜋_𝑡∗ = − (𝑟𝑒𝑟𝑡𝑑𝑡∗− 𝑟𝑒𝑟𝑡 𝑑_𝑡−1∗ 𝜋_𝑡∗ )

Further derivation and math developments will be in more detail in the Appendix.

4. Calibration

The model aforementioned is calibrated for the Portuguese economy as closely as possible. The calibration used departs from van Wijnbergen & Jakucionyte (2017) and it’s further complemented by Gertler and Karadi (2011) regarding the financial intermediaries agency problem. Regarding characteristic inherent to the Portuguese economy, a number of different articles are used.

The banking sector’s calibration follows closely Gertler and Karadi (2011) paper: the proportional transfer to entering banks is kept as in the original paper, while bank leverage is calibrated to the average value in the data between the years of 2001 and 2007, yielding a value of 10,25. This value is further adjusted by the fraction of loans to the private sector on bank’s balance

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sheet in the first quarter of 2007 of 0.26, thus getting a value of 2,665. The value of λL is then be calibrated to 0.3863 as in Van der Kwaak and Van Wijnbergen (2014).

Regarding the labour market we abstain for making inferences regarding wage rigidity and matching elasticity parameters, calibrating them to standard values of 0,5 as used in the search and match literature. The labor force is set to be equal to unity and the steady state value of unemployment is chosen to be 6,4% is the average in the data for the pre-crisis period. The unemployment benefits parameter is set to 0.58 as in OECD (2007); while the matching efficiency parameter 𝜎𝑒 is determined endogenously.

Concerning the parameters specific to the Portuguese economy we follow Almeida (2009) and are presented on the Table 1 in the Appendix along with the rest of the calibration.

5. Results

In this section we explain further into detail the dynamics of the financial frictions with the real sector and the non-walrasian labor market included into the model.

Firstly we take the baseline model that includes a flexible labor market to use it as a benchmark for our analysis; after we add the labor market frictions in order to see how the leverage constrained financial intermediaries influence the real economy’s dynamics.

5.1. Shocks

As expected in most models, when the economy in question faces shocks to its capital quality, suffers from loss in productivity, these perturbations will translate into episodes of economic downturn.

We should also make notice that in the presented model there’s an agency problem between depositors and the banks; this yields an endogenous credit spread that will control the financial intermediaries’ ability to lend money to the real economy according to their own leverage.

In the modelled economy, shocks arise from two different channels: technology shocks to financially constrained firms and from a decrease in the capital quality

33

available in the economy. The latter, an exogenous downward shock to credit quality produces the following effects on the economy: financially constrained firms will opt by choosing lower levels of capital for its production; this implies a then reduction of the future output of the firms (and lower collateral value against the loans contracted) – the loan is established based on the firm’s expectations; before any shock takes place and before capital is bought. This will drive down the financially constrained firm’s ability to repay their loans and also increasing the default risk for them. Another feature of this shock is that the amount of loans provided decreases significantly; this reduction on lending will cut down the price of capital and have a positive impact towards loan’s gross nominal rate – which adjusts after the shock in order to clear the loan market. The reduction seen on the amount of loans provided will force the banks to increase their net worth (from the higher rates charged on loans) in order to leverage down. As the shock starts to fade out, net worth goes down and the financial intermediaries start leveraging up again to steady state values. With the lower marginal productivity of capital, which came from the quality shock, intermediate firms will see their profits go down along with the real wage; employment will see a decrease on the wake of this shock. Household’s budget constraint will get squeezed out, implying lower levels of consumption for these agents.

As we can see, this type of crisis has long-lasting effects on the economy: variables like consumption, output, investment and capital (for example) haven’t returned to the levels seen before the shock after ten years. This shows how devastating this can be for an economy and for its agents – mostly consumers and producers.

Figure 1: Response to a Negative Capital Quality Shock ( ρ=1 )

34

Another factor that might help explain what happened in Portugal in the years that led to the crisis is its stagnated productivity growth; as mentioned by Reis (2013); during the last fifty years the Portuguese economy hasn’t seen a lot of productivity increases. We try to replicate theses effects with a negative and persistent shock to technology, which are amplified by the existence of the financial intermediaries.

With this contractionary shock, firms will have lower profits and will not be as incentivized to post more vacancies into the economy, thus increasing unemployment – the fixed separation rate creates jobless agents if not offsetted with the same amount of vacancies, meaning that the unemployment fluctuation are entirely due to job-creation slow down - and also reducing capital levels. With these lower levels of productivity, financially constrained firms will hold out on contracting loans with banks: their lower productivity makes more difficult to repay loans (even if these are low in value). As this lower demand for the bank’s assets appears, banks will try to rebalance their losses by increasing their interest rate on loans – further discouraging firms from contracting more loans. The fact that marginal productivity of labor goes down after this shock, makes the Nash wage chosen during the bargaining process decrease, but, since real wage rigidity plays a part, the real wage will be higher than desired by firms, which will further reinforce their desire to not post vacancies after this