Sovereign Risk, Monetary Policy and Fiscal Multipliers: A Structural Model-Based Assessment

Abstract

This paper briefly reviews the literature on fiscal multipliers and then presents results for the Italian economy obtained by simulating a dynamic general equilibrium model that allows for the possibility (a) that the zero lower bound may be binding and (b) that the initial public debt-to-GDP ratio may affect the financing conditions of the public and private sectors (sovereign risk channel). The results are the following. First, the public consumption multiplier is in general less than 1. Second, it goes above 1 only under extremely strong assumptions, namely the constancy of the monetary policy rate for an exceptionally long period (at least 5 years) and full time-coincidence between the fiscal and the monetary stimuli. Third, when the sovereign risk channel is active the government consumption multiplier is much lower. Fourth, in all cases tax multipliers are lower than government consumption multipliers. Finally, we make a tentative assessment of the fiscal consolidation measures enacted in Italy in 2011–2012: the evidence is that the impact on GDP was much weaker than the IMF had expected.

Appendix

In this appendix we report a detailed description of the model, excluding the fiscal and monetary policy part and the description of the households optimization problem that are reported in the main text.⁴³

There are three countries, Italy, the rest of the euro area (REA) and the rest of the world (RW). They have different sizes. Italy and the REA share the currency and the monetary authority. 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. Households have access to financial markets and smooth consumption by trading a risk-free one-period nominal bond, denominated in euro. 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.

On the production side, there are perfectly competitive firms that produce the final goods and monopolistic firms that produce the intermediate goods. Two final goods (private consumption and private investment) are produced combining all available intermediate goods according to constant-elasticity-of-substitution bundle. The public consumption good is a bundle of intermediate non-tradable goods.

Tradable and non-tradable intermediate goods are produced combining capital and labor in the same way. Tradable intermediate goods can be sold domestically or abroad. Because intermediate goods are differentiated, firms have market power and restrict output to create excess profits. We assume that goods markets are internationally segmented and the law of one price for tradables does not hold. Hence, each firm producing a tradable good sets three prices, one for the domestic market and the other two for the export market (one for each region). 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 react in a gradual way. On the real side, quadratic costs and habit prolong the adjustment of the investment and consumption. On the nominal side, quadratic costs make wage and prices sticky.

In what follows we illustrate the Italian economy. The structure of each of the other two regions (REA and the RW) is similar and to save on space we do not report it.

1.1 Final Consumption and Investment Goods

There is a continuum of symmetric Italian firms producing final non-tradable consumption under perfect competition. Each firm producing the consumption good is indexed by $x \in \left (0,s\right ]$, where the parameter 0 < s < 1 measures the size of Italy. Firms in the REA and in the RW are indexed by $x^{{\ast}}\in \left (s,S\right ]$ and $x^{{\ast}{\ast}}\in \left (S,1\right ]$, respectively (the size of the world economy is normalized to 1). The CES production technology used by the generic firm x is:

$$\displaystyle\begin{array}{rcl} & & A_{t}\left (x\right ) {}\\ & & \quad \equiv \left (\begin{array}{c} a_{T}^{ \frac{1} {\phi _{A}} }\left (a_{H}^{ \frac{1} {\rho _{A}} }Q_{\mathit{HA},t}\left (x\right )^{\frac{\rho _{A}-1} {\rho _{A}} } + a_{G}^{ \frac{1} {\rho _{A}} }Q_{\mathit{GA},t}\left (x\right )^{\frac{\rho _{A}-1} {\rho _{A}} }\left (1 - a_{H} - a_{G}\right )^{ \frac{1} {\rho _{A}} }Q_{\mathit{FA},t}\left (x\right )^{\frac{\rho _{A}-1} {\rho _{A}} }\right )^{ \frac{\rho _{A}} {\rho _{A}-1} \frac{\phi _{A}-1} {\phi _{A}} } \\ + \left (1 - a_{T}\right )^{ \frac{1} {\phi _{A}} }Q_{\mathit{NA},t}\left (x\right )^{\frac{\phi _{A}-1} {\phi _{A}} } \end{array} \right )^{ \frac{\phi _{A}} {\phi _{A}-1} }{}\\ \end{array}$$

where Q _HA , Q _GA , Q _FA and Q _NA are bundles of respectively intermediate tradables produced in Italy, intermediate tradables produced in the REA, intermediate tradables produced in the RW and intermediate non-tradables produced in Italy. The parameter ρ _A  > 0 is the elasticity of substitution between tradables and ϕ _A  > 0 is the elasticity of substitution between tradable and non-tradable goods. The parameter a _H (0 < a _H  < 1) is the weight of the Italian tradable, the parameter a _G (0 < a _G  < 1) the weight of tradables imported from the REA, a _T (0 < a _T  < 1) the weight of tradable goods.

The production of investment good is similar. There are symmetric Italian firms under perfect competition indexed by $y \in \left (0,s\right ]$. Firms in the REA and in the RW are indexed by $y^{{\ast}}\in \left (s,S\right ]$ and $y^{{\ast}{\ast}}\in \left (S,1\right ]$. Output of the generic Italian firm y is:

$$\displaystyle\begin{array}{rcl} & & E_{t}\left (y\right ) {}\\ & & \quad \equiv \left (\begin{array}{@{}c@{}} v_{T}^{ \frac{1} {\phi _{E}} }\left (v_{H}^{ \frac{1} {\rho _{E}} }Q_{\mathit{HE},t}\left (y\right )^{\frac{\rho _{E}-1} {\rho _{E}} }\,+\,v_{G}^{ \frac{1} {\rho _{E}} }Q_{\mathit{GE},t}\left (y\right )^{\frac{\rho _{E}-1} {\rho _{E}} }\,+\,\left (1-v_{H}-v_{G}\right )^{ \frac{1} {\rho _{E}} }Q_{\mathit{FE},t}\left (y\right )^{\frac{\rho _{E}-1} {\rho _{E}} }\right )^{ \frac{\rho _{E}} {\rho _{E}-1} \frac{\phi _{E}-1} {\phi _{E}} } \\ + \left (1 - v_{T}\right )^{ \frac{1} {\phi _{E}} }Q_{\mathit{NE},t}\left (y\right )^{\frac{\phi _{E}-1} {\phi _{E}} } \end{array} \right )^{ \frac{\phi _{E}} {\phi _{E}-1} }{}\\ \end{array}$$

Finally, we assume that public consumption C ^g is composed by intermediate non-tradable goods only.

1.2 Intermediate Goods

1.2.1 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:

$$\displaystyle\begin{array}{rcl} Q_{\mathit{HA}}\left (x\right ) \equiv \left [\left (\frac{1} {s}\right )^{\theta _{T}}\int _{0}^{s}Q\left (h,x\right )^{\frac{\theta _{T}-1} {\theta _{T}} }\mathit{dh}\right ]^{ \frac{\theta _{T}} {\theta _{T}-1} }& &{}\end{array}$$

(16)

$$\displaystyle\begin{array}{rcl} Q_{\mathit{GA}}\left (x\right ) \equiv \left [\left ( \frac{1} {S - s}\right )^{\theta _{T}}\int _{s}^{S}Q\left (g,x\right )^{\frac{\theta _{T}-1} {\theta _{T}} }\mathit{dg}\right ]^{ \frac{\theta _{T}} {\theta _{T}-1} }& &{}\end{array}$$

(17)

$$\displaystyle\begin{array}{rcl} Q_{\mathit{FA}}\left (x\right ) \equiv \left [\left ( \frac{1} {1 - S}\right )^{\theta _{T}}\int _{S}^{1}Q\left (f,x\right )^{\frac{\theta _{T}-1} {\theta _{T}} }\mathit{df }\right ]^{ \frac{\theta _{T}} {\theta _{T}-1} }& &{}\end{array}$$

(18)

$$\displaystyle\begin{array}{rcl} Q_{\mathit{NA}}\left (x\right ) \equiv \left [\left (\frac{1} {s}\right )^{\theta _{N}}\int _{0}^{s}Q\left (n,x\right )^{\frac{\theta _{N}-1} {\theta _{N}} }\mathit{dn}\right ]^{ \frac{\theta _{N}} {\theta _{T}-1} }& &{}\end{array}$$

(19)

where firms in the Italian intermediate tradable and non-tradable sectors are respectively indexed by h ∈ (0, s) and n ∈ (0, s), firms in the REA by g ∈ (s, S] and firms in the RW by f ∈ (S, 1]. Parameters θ _T , θ _N  > 1 are respectively the elasticity of substitution across brands in the tradable and non-tradable sector. The prices of the intermediate non-tradable 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 intermediate non-tradable input n is:

$$\displaystyle{ Q_{A,t}\left (n,x\right ) = \left (\frac{1} {s}\right )\left (\frac{P_{t}\left (n\right )} {P_{N,t}} \right )^{-\theta _{N} }Q_{\mathit{NA},t}\left (x\right ) }$$

(20)

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

$$\displaystyle{ P_{N,t} = \left [\int _{0}^{s}P_{ t}\left (n\right )^{1-\theta _{N} }\mathit{dn}\right ]^{ \frac{1} {1-\theta _{N}} } }$$

(21)

We can derive $Q_{A}\left (h,x\right )$, $Q_{A}\left (f,x\right )$, $C_{A}^{g}\left (h,x\right )$, $C_{A}^{g}\left (f,x\right )$, P _H and P _F 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:

$$\displaystyle\begin{array}{rcl} & & \int _{0}^{s}Q_{ A,t}\left (n,x\right )\mathit{dx} +\int _{ 0}^{s}Q_{ E,t}\left (n,y\right )\mathit{dy} +\int _{ 0}^{s}C_{ t}^{g}\left (n,x\right )\mathit{dx} {}\\ & =& \left (\frac{P_{t}\left (n\right )} {P_{N,t}} \right )^{-\theta _{N} }\left (Q_{\mathit{NA},t} + Q_{\mathit{NE},t} + C_{N,t}^{g}\right ) {}\\ \end{array}$$

where $C_{N}^{g}$ is public sector consumption. Italy demands for (intermediate) domestic and imported tradable goods can be derived in a similar way.

1.2.2 Supply

The supply of each Italian intermediate non-tradable good n is denoted by N ^S(n):

$$\displaystyle{ N_{t}^{S}\left (n\right ) = \left (\left (1 -\alpha _{ N}\right )^{ \frac{1} {\xi _{N}} }L_{N,t}\left (n\right )^{\frac{\xi _{N}-1} {\xi _{N}} } +\alpha ^{ \frac{1} {\xi _{N}} }K_{N,t}\left (n\right )^{\frac{\xi _{N}-1} {\xi _{N}} }\right )^{ \frac{\xi _{N}} {\xi _{N}-1} } }$$

(22)

Firm n uses labor $L_{N,t}^{p}\left (n\right )$ and capital $K_{N,t}\left (n\right )$ with constant elasticity of input substitution ξ _N  > 0 and capital weight 0 < α _N  < 1. Firms producing intermediate goods take the prices of labor inputs and capital as given. Denoting W _t the nominal wage index and R _t ^K the nominal rental price of capital, cost minimization implies:

$$\displaystyle\begin{array}{rcl} & & L_{N,t}\left (n\right ) = \left (1 -\alpha _{N}\right )\left ( \frac{W_{t}} {\mathit{MC}_{N,t}\left (n\right )}\right )^{-\xi _{N} }N_{t}^{S}\left (n\right ) \\ & & \qquad K_{N,t}\left (n\right ) =\alpha \left ( \frac{R_{t}^{K}} {\mathit{MC}_{N,t}\left (n\right )}\right )^{-\xi _{N} }N_{t}^{S}\left (n\right ) {}\end{array}$$

(23)

where $\mathit{MC}_{N,t}\left (n\right )$ is the nominal marginal cost:

$$\displaystyle{ \mathit{MC}_{N,t}\left (n\right ) = \left (\left (1-\alpha \right )W_{t}^{1-\xi _{N} } +\alpha \left (R_{t}^{K}\right )^{1-\xi _{N} }\right )^{ \frac{1} {1-\xi _{N}} } }$$

(24)

The productions of each Italian tradable good, $T^{S}\left (h\right )$, is similarly characterized.

1.2.3 Price Setting in the Intermediate Sector

Consider now profit maximization in the Italian intermediate non-tradable sector. Each firm n sets the price p _t (n) by maximizing the present discounted value of profits subject to the demand constraint and the quadratic adjustment costs:

$$\displaystyle{ \mathit{AC}_{N,t}^{p}\left (n\right ) \equiv \frac{\kappa _{N}^{p}} {2} \left ( \frac{P_{t}\left (n\right )} {P_{t-1}\left (n\right )} - 1\right )^{2}Q_{ N,t}\text{ }\kappa _{N}^{p} \geq 0 }$$

paid in unit of sectorial product Q _N, t and where $\kappa _{N}^{p}$ measures the degree of price stickiness. The resulting first-order condition, expressed in terms of domestic consumption, is:

$$\displaystyle{ p_{t}\left (n\right ) = \frac{\theta _{N}} {\theta _{N} - 1}\mathit{mc}_{t}\left (n\right ) -\frac{A_{t}\left (n\right )} {\theta _{N} - 1} }$$

(25)

where $\mathit{mc}_{t}\left (n\right )$ is the real marginal cost and $A\left (n\right )$ contains terms related to the presence of price adjustment costs:

$$\displaystyle\begin{array}{rcl} A_{t}\left (n\right )& \approx & \kappa _{N}^{p} \frac{P_{t}\left (n\right )} {P_{t-1}\left (n\right )}\left ( \frac{P_{t}\left (n\right )} {P_{t-1}\left (n\right )} - 1\right ) {}\\ & & -\beta \kappa _{N}^{p}\frac{P_{t+1}\left (n\right )} {P_{t}\left (n\right )} \left (\frac{P_{t+1}\left (n\right )} {P_{t}\left (n\right )} - 1\right )\frac{Q_{N,t+1}} {Q_{N,t}} {}\\ \end{array}$$

The above equations clarify the link between imperfect competition and nominal rigidities. As emphasized by Bayoumi et al. (2004), when the elasticity of substitution θ _N 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):

$$\displaystyle{ p_{t}\left (n\right ) = \frac{\theta _{N}} {\theta _{N} - 1}\mathit{mc}_{N,t}\left (n\right ) }$$

(26)

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 $p_{t}\left (h\right )$ in the Italian market,a price $\ p_{t}^{{\ast}}\left (h\right )$ in the REA and a price $p_{t}^{{\ast}{\ast}}\left (h\right )$ in the RW to maximize the expected flow of profits (in terms of domestic consumption units):

$$\displaystyle{ E_{t}\sum _{\tau =t}^{\infty }\Lambda _{ t,\tau }\left [\begin{array}{c} p_{\tau }\left (h\right )y_{\tau }\left (h\right ) + p_{\tau }^{{\ast}}\left (h\right )y_{\tau }^{{\ast}}\left (h\right ) + p_{\tau }^{{\ast}{\ast}}\left (h\right )y_{\tau }^{{\ast}{\ast}}\left (h\right ) \\ -\mathit{mc}_{H,\tau }\left (h\right )\left (y_{\tau }\left (h\right ) + y_{\tau }^{{\ast}}\left (h\right ) + y_{\tau }^{{\ast}{\ast}}\left (h\right )\right ) \end{array} \right ] }$$

subject to quadratic price adjustment costs similar to those considered for non-tradables and standard demand constraints. The term E _t denotes the expectation operator conditional on the information set at time $t$, $\Lambda _{t,\tau }$ is the appropriate discount rate and $\mathit{mc}_{H,t}\left (h\right )$ is the real marginal cost. The first order conditions with respect to $p_{t}\left (h\right )$, $p_{t}^{{\ast}}\left (h\right )$ and $p_{t}^{{\ast}{\ast}}\left (h\right )$ are:

$$\displaystyle\begin{array}{rcl} p_{t}\left (h\right ) = \frac{\theta _{T}} {\theta _{T} - 1}\mathit{mc}_{t}\left (h\right ) -\frac{A_{t}\left (h\right )} {\theta _{T} - 1} & &{}\end{array}$$

(27)

$$\displaystyle\begin{array}{rcl} p_{t}^{{\ast}}\left (h\right ) = \frac{\theta _{T}} {\theta _{T} - 1}\mathit{mc}_{t}\left (h\right ) -\frac{A_{t}^{{\ast}}\left (h\right )} {\theta _{T} - 1} & &{}\end{array}$$

(28)

$$\displaystyle\begin{array}{rcl} p_{t}^{{\ast}{\ast}}\left (h\right ) = \frac{\theta _{T}} {\theta _{T} - 1}\mathit{mc}_{t}\left (h\right ) -\frac{A_{t}^{{\ast}{\ast}}\left (h\right )} {\theta _{T} - 1} & &{}\end{array}$$

(29)

where θ _T is the elasticity of substitution of intermediate tradable goods, while $A\left (h\right )$ and $A^{{\ast}}\left (h\right )$ involve terms related to the presence of price adjustment costs:

$$\displaystyle\begin{array}{rcl} A_{t}\left (h\right )& \approx & \kappa _{H}^{p} \frac{P_{t}\left (h\right )} {P_{t-1}\left (h\right )}\left ( \frac{P_{t}\left (h\right )} {P_{t-1}\left (h\right )} - 1\right ) {}\\ & & -\beta \kappa _{H}^{p}\frac{P_{t+1}\left (h\right )} {P_{t}\left (h\right )} \left (\frac{P_{t+1}\left (h\right )} {P_{t}\left (h\right )} - 1\right )\frac{Q_{H,t+1}} {Q_{H,t}} {}\\ A_{t}^{{\ast}}\left (h\right )& \approx & \theta _{ T} - 1 + \kappa _{H}^{p} \frac{P_{t}^{{\ast}}\left (h\right )} {P_{t-1}^{{\ast}}\left (h\right )}\left ( \frac{P_{t}^{{\ast}}\left (h\right )} {P_{t-1}^{{\ast}}\left (h\right )} - 1\right ) {}\\ & & -\beta \kappa _{H}^{p}\frac{P_{t+1}^{{\ast}}\left (h\right )} {P_{t}^{{\ast}}\left (h\right )} \left (\frac{P_{t+1}^{{\ast}}\left (h\right )} {P_{t}^{{\ast}}\left (h\right )} - 1\right )\frac{Q_{H,t+1}^{{\ast}}} {Q_{H,t}^{{\ast}}} {}\\ A_{t}^{{\ast}{\ast}}\left (h\right )& \approx & \theta _{ T} - 1 + \kappa _{H}^{p} \frac{P_{t}^{{\ast}{\ast}}\left (h\right )} {P_{t-1}^{{\ast}{\ast}}\left (h\right )}\left ( \frac{P_{t}^{{\ast}{\ast}}\left (h\right )} {P_{t-1}^{{\ast}{\ast}}\left (h\right )} - 1\right ) {}\\ & & -\beta \kappa _{H}^{p}\frac{P_{t+1}^{{\ast}{\ast}}\left (h\right )} {P_{t}^{{\ast}{\ast}}\left (h\right )} \left (\frac{P_{t+1}^{{\ast}{\ast}}\left (h\right )} {P_{t}^{{\ast}{\ast}}\left (h\right )} - 1\right )\frac{Q_{H,t+1}^{{\ast}{\ast}}} {Q_{H,t}^{{\ast}{\ast}}} {}\\ \end{array}$$

where $\kappa _{H}^{p}$,$\kappa _{H}^{{p}^{{\ast}}}$,$\kappa _{H}^{{p}^{{\ast}{\ast}}}$ > 0 respectively measure the degree of nominal rigidity in Italy, in the REA and in the RW. If nominal rigidities in the (domestic) export market are highly relevant (that is, if is relatively large), the degree of inertia of Italian goods prices in the foreign markets will be high. If prices were flexible ($\kappa _{H}^{p} = \kappa _{H}^{p{\ast}} =\kappa _{ H}^{p{\ast}{\ast}} = 0$) then optimal price setting would be consistent with the cross-border law of one price (prices of the same tradable goods would be equal when denominated in the same currency).

1.3 Labor Market

In the case of firms in the intermediate non-tradable sector, the labor input $L_{N}\left (n\right )$ 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 $\in$ $\left [0,s\right ]$):

$$\displaystyle{ L_{N,t}\left (n\right ) \equiv \left (\frac{1} {s}\right )^{\frac{1} {\psi } }\left [\int _{0}^{s}L_{t}\left (n,j\right )^{\frac{\psi -1} {\psi } }\mathit{dj}\right ]^{ \frac{\psi }{\psi -1} } }$$

(30)

where $L\left (n,j\right )$ 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:

$$\displaystyle{ L_{t}\left (n,j\right ) = \left (\frac{1} {s}\right )\left (\frac{W_{t}\left (j\right )} {W_{t}} \right )^{-\psi }L_{ N,t}\left (j\right ) }$$

(31)

where $W\left (j\right )$ is the nominal wage of labor input j and the wage index W is:

$$\displaystyle{ W_{t} = \left [\left (\frac{1} {s}\right )\int _{0}^{s}W_{ t}\left (h\right )^{1-\psi }\mathit{dj}\right ]^{ \frac{1} {1-\psi }} }$$

(32)

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 Italian firms. The wage adjustment is sluggish because of quadratic costs paid in terms of the total wage bill:

$$\displaystyle{ \mathit{AC}_{t}^{W} = \frac{\kappa _{W}} {2} \left ( \frac{W_{t}} {W_{t-1}} - 1\right )^{2}W_{ t}L_{t} }$$

(33)

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

1.4 The Equilibrium

We find a symmetric equilibrium of the model. In each country there is a representative agent and four representative sectorial firms (in the intermediate tradable sector, intermediate non-tradable sector, consumption production sector and investment production sector). The equilibrium is a sequence of allocations and prices such that, given initial conditions and the sequence of exogenous shocks, each private agent and firm satisfy the correspondent first order conditions, the private and public sector budget constraints and market clearing conditions for goods, labor, capital and bond holdings.