Back Matter
  • 1, International Monetary Fund


In this Appendix we report a detailed description of the model, excluding the fiscal policy part, the description of the households optimization problem that are reported in the main text.22

There are two countries, the Home country and rest of the monetary union, having different sizes and sharing the currency and the central bank. In each region there are households and firms. Each household consumes a final composite good made of non-tradable, domestic tradable and imported intermediate goods from the rest of the area. Households have access to financial markets and smooth consumption by trading a risk-free one-period nominal bond. They also own domestic firms and capital stock, which is rent to domestic firms in a perfectly competitive market. Households supply differentiated labor services to domestic firms and act as wage setters in monopolistically competitive markets by charging a markup over their marginal rate of substitution. A fraction of households, as said in the text, does not optimize over time but simply consume the overall wage income available in each period.

On the production side, there are perfectly competitive firms that produce the final goods and monopolistic firms that produce the intermediate goods. The three final goods (a private consumption, a private investment and a public consumption good) are produced combining all available intermediate goods in a constant-elasticity-of-substitution matter. Tradable and non-tradable intermediate goods are produced combining capital and labor in the same way. Tradable intermediate goods are split in domestically-consumed and export goods. Because intermediate goods are differentiated, firms have market power and restrict output to create excess profits. We assume that Home and the rest of the monetary union are segmented markets and the law of one price for tradables does not hold. Hence, each firm producing a tradable good sets two prices, one for the domestic market and the other for the export market. Since the firm faces the same marginal costs regardless of the scale of production in each market, the different price-setting problems are independent of each other.

To capture the empirical persistence of the aggregate data and generate realistic dynamics, we include adjustment costs on real and nominal variables, ensuring that, in response to a shock, consumption and production do not immediately jump to a new long-term equilibrium. On the real side, quadratic costs prolong the adjustment of the capital stock. On the nominal side, quadratic cost make wage and prices sticky.

Imperfect competition in product and labor markets is reflected in markups over marginal costs. The elasticity of substitution between products of different firms determines the market power of each profit-maximizing firm. The setup in the labor market is similar. Each worker offers a differentiated kind of labor services that is an imperfect substitute for services offered by other workers. The lower the degree of substitutability, for example because of skill differences or anti-competitive regulation, the higher is the markup and the lower employment in terms of hours. Hence, markups are modeled by a single parameter. In what follows we illustrate the Home economy. The structure of the Foreign economy (the rest of the monetary union) is similar and to save on space we do not report it.

A Final consumption and investment goods

There is continuum of symmetric Home firms producing Home final non-tradable consumption under perfect competition. Each firm producing the consumption good is indexed by x ∈ (0, s], where the parameter 0 < s < 1 is a measure of country size. Foreign firms producing the Foreign final consumption goods are indexed by by x* ∈ (s, 1] (the size of the monetary union is normalized to 1). The CES production technology used by firm x is:


where QHA, QFA and QNA are bundles of respectively Home tradable, Foreign tradable and Home non-tradable intermediate goods, ρ > 0 is the elasticity of substitution between tradables and ϕ > 0 is the elasticity of substitution between tradable and non-tradable goods. The parameter aH (0 < aH < 1) is the weight of domestic tradable, aT (0 < aT < 1) the weight of tradable goods.

The production of investment good is similar. There are symmetric Home firms under perfect competition indexed by y ∈ (0, s], and symmetric Foreign firms by y* ∈ (s, 1]. Output of Home firm y is:


Finally, we assume that public expenditure Cg has the same composition as that of private consumption.

B Intermediate goods Demand

Bundles used to produce the final consumption goods are CES indexes of differentiated intermediate goods, each produced by a single firm under conditions of monopolistic competition:


where firms in the Home tradable and non-tradable intermediate sectors and in the Foreign intermediate tradable sector are respectively indexed by h ∈ (0, s), n ∈ (0, s), f ∈ (s, 1]. Parameters θT, θN > 1 are respectively the elasticity of substitution between brands in the tradable and non-tradable sector. The prices of the non-tradable intermediate goods are denoted p(n). Each firm x takes these prices as given when minimizing production costs of the final good. The resulting demand for non-tradable intermediate input n is:


where PN, t is the cost-minimizing price of one basket of local intermediates:


We can derive QA(h,x),QA(f,x),CAg(h,x),CAg(f,x),PH and PF in a similar way. Firms y producing the final investment goods have similar demand curves. Aggregating over x and y, it can be shown that total demand for intermediate non-tradable good n is:


where CNg is non-tradable component of the public sector consumption. Home demands for Home and Foreign tradable intermediate goods can be derived in a similar way.


The supply of each Home non-tradable intermediate good n is denoted by Ns(n):


Firm n uses labor LN,tp(n) and capital KN, t(n) with constant elasticity of input substitution ξN > 0 and capital weight 0 < αN < 1. Firms producing intermediate goods take the prices of labor inputs and capital as given. Denoting Wt the nominal wage index and RtK the nominal rental price of capital, cost minimization implies:


where MCN, t(n) is the nominal marginal cost:


The productions of each Home tradable good, Ts (h), is similarly characterized.

Price setting in the intermediate sector

Consider now profit maximization in the Home country’s nontradable intermediate sector. Each firm n sets the price pt(n) by maximizing the present discounted value of profits subject to demand constraint (18) and the quadratic adjustment costs:


paid in unit of sectorial product QN, t and where κNp measures the degree of price stickiness. The resulting first-order condition, expressed in terms of domestic consumption, is:


where mct (n) is the real marginal cost and A (n) contains terms related to the presence of price adjustment costs:


The above equations clarify the link between imperfect competition and nominal rigidities. As emphasized by Bayoumi et al.(2004), when the elasticity of substitution θN is very large and hence the competition in the sector is high, prices closely follow marginal costs, even though adjustment costs are large. To the contrary, it may be optimal to maintain stable prices and accommodate changes in demand through supply adjustments when the average markup over marginal costs is relatively high. If prices were flexible, optimal pricing would collapse to the standard pricing rule of constant markup over marginal costs (expressed in units of domestic consumption):


Firms operating in the intermediate tradable sector solve a similar problem. We assume that there is market segmentation. Hence the firm producing the brand h chooses pt (h) in the Home market and pt*(h) in the Foreign market as to maximize the expected flow of profits (in terms of domestic consumption units):


subject to quadratic price adjustment costs similar to those considered for nontradables and standard demand constraints. The term Et denotes the expectation operator conditional on the information set at time t, Λt,τ is the appropriate discount rate and mcH, t (h) is the real marginal cost. The first order conditions with respect to pt (h) and pt*(h) are:


where θT* is the elasticity of substitution of tradable intermediate goods in the Foreign country, while A (h) and A* (h) involve terms related to the presence of price adjustment costs:


where κHp>0(κHp*>0) measure the degree of nominal rigidity in the Home (Foreign) country. If nominal rigidities in the (domestic) export market are highly relevant (that is, if is relatively large), the degree of inertia of Home goods prices in the Foreign market will be high. If prices were flexible (κHp=κHp*) and θT=θT*, then optimal price setting would be consistent with the cross-border law of one price:


C Labor Market

In the case of firms in the nontradable intermediate sector, the labor input LN (n) is a CES combination of differentiated labor inputs supplied by domestic agents and defined over a continuum of mass equal to the country size (j ∈ [0, s]):


where L (n, j) is the demand of the labor input of type j by the producer of good n and ψ > 1 is the elasticity of substitution among labor inputs. Cost minimization implies:


where W (j) is the nominal wage of labor input j and the wage index W is:


Similar equations hold for firms producing intermediate tradable goods. Each household is the monopolistic supplier of a labor input j and sets the nominal wage facing a downward-sloping demand, obtained by aggregating demand across Home firms. The wage adjustment is sluggish because of quadratic costs paid in terms of the total wage bill:


where the parameter κW > 0 measures the degree of nominal wage rigidity and L is the total amount of labor in the Home economy.

D Monetary Policy

The monetary authority controls the short-term rate according to a Taylor rule of the form:


The parameter ρi (0 < ρi < 1) captures inertia in interest rate setting, while parameters ρπ and ρGDP are respectively the weights of currency union’s CPI inflation rate ΠMU, t and GDP GDPMU, t. The CPI inflation rate is a geometric average of CPI inflation rates in the Home and Foreign country (respectively Πt and Πt*) with weights equal to the correspondent country size:


The union-wide GDP is the sum of the Home and Foreign GDPs (respectively GDPt and GDPt*), both evaluated at the steady state prices:


where rer is the Home real exchange rate, defined as the ratio of rest of the monetary union to Home consumer prices.


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We thank for useful comments C. Cottarelli, C. Kamps, A. Locarno, D. Muir, M. Guerguil, P. Manasse, A. Notarpietro and participants at the ECB Workshop” Challenges for sovereign debt management in the EU” (October 2011), at the Bank of Poland “Central Bank Macroeconomic Modeling Workshop” (September 2012) and in seminars at the IMF Fiscal Affair Department (November 2011) and at University of Bologna (December 2011).


Contagion and systemic crises are not the object of the analysis.


As long as there is a functioning secondary market for the debt, foreign holders have always the opportunity to sell their debt on the secondary market. Therefore if the fiscal authority when restructuring its debt tries to discriminate and imposes the haircut only on foreign holders, the latter would sell their bonds on the secondary market to the domestic residents (which should be willing to buy at a price close to the face value).


The empirical result of Cruces and Trebesch (2013) is consistent with the classical work of Eaton and Gersovitz (1981), where it is argued that sovereign borrowing can be supported as long as a restructuring is costly. Therefore, non-repayments have to be followed by punishments (in the form of high spreads or exclusion from international borrowing), and larger non-repayments by larger punishments.


We choose a fiscal rule defined in terms of lump-sum taxes for simplicity as it avoids the analysis of the distortions associated with other taxes.


For a description of the GEM and NAWM see Pesenti (2008) and Coenen et al. (2008), respectively. A detailed description of our model is reported in the Appendix.


For a model with similar fiscal features, see Forni et al. (2010).


As is standard in this class of models, bonds are one-period securities and each period is equal to one quarter. The actual average maturity of the debt is longer than one quarter. In this case the increase in spreads would bring about a gradual increase in interest costs. Assuming a longer average maturity of the debt in the model would produce similar effects on GDP and on the other variables, although these effects would materialize in a more gradual manner.

The GDP is defined as:

where PI, PEXP, pIMP are prices of respectively investment I, export EXP and import IMP while W represents nominal wage.

Given the presence of public employment, and consistently with common practice in the national accounts statistics, we include the public expenditure for wages in GDP.


There is only one (minor) difference between ϕgov and ϕb. It corresponds to the third term on the right-hand side of equation (4), that in the case of the government spread depends on the current and steady state values of the government bonds held by Home households. As said, the difference is quantitatively small, as we minimize the impact of that term on the dynamics. Finally, all revenues from the imposition of the spread are rebated in a lump-sum way to Foreign households (see Benigno 2009). For the latter, the spread does not enter neither in the government budget constraint nor in the Euler equations.


Foreign (rest of the monetary union) variables have a ‘*’.


For an analysis of the macroeconomic effects of different degree of markups in a model similar to the one used in this paper, see Forni et al. (2010).


The monetary union-wide consumer price inflation rate is weighted (by the country size) geometric average of the corresponding regional variables. The monetary union GDP is the sum of regional GDPs.


Since we assume that fiscal policy is managed by changing lump-sum transfers and that agents are Ricardian, results are not very different if we allow the debt level to increase after the restructuring.


The nominal interest rate set by the monetary authority does not greatly change, given the low weight of the Home country in monetary union and hence in the Taylor rule.


Real export and imports are evaluated at the initial steady-state prices.


As said, we assume that the Home country is relatively small compared to the rest of the monetary union and that there is no financial contagion.


In this case the spread on households’ financial position is set to a rather low value, to make the model stationary. See Benigno (2009).


As illustrated in the calibration section, in the baseline simulation it is assumed that Home households have an initial financial liability against the Foreign households equal to 100 percent of Home annualized GDP.


We have run similar simulations assuming a restructuring and a consolidation of public debt equal to 20 percent. Results, available upon request, suggest that also in this case the macroeconomic costs of restructuring are larger. GDP through would be equal to 10 percent (consumption and investment throughs would be equal to 10 and 30, percent respectively). Under consolidation, GDP through would be equal to 3 percent (consumption by 3 and 6 percent).


In some cases, the fiscal consolidation could have mild recessionary effects. For example, when it’s implemented through public spending cuts that would allow for reduction in both public debt and expected future taxes. See for example Forni et al. (2010).


For a detailed description of the main features of the model see also Pesenti (2008).

Macroeconomic Effects of Sovereign Restructuring in a Monetary Union: A Model-based Approach
Author: Lorenzo Forni and Massimiliano Pisani