The macroeconomic effects of the ECB’s evolving QE programme: a model-based analysis

Abstract

This paper analyses the macroeconomic effects of the European Central Bank’s asset purchase programme in its initial version and subsequent modifications under the lens of a dynamic macroeconomic model, which includes assets of different types and maturity, and explicitly introduces asset purchases of long-term bonds (held by euro area and non-euro area residents) by the central bank. With imperfect substitutability between asset classes, portfolio rebalancing in the context of quantitative easing (QE) affects bond yields, stock prices, the exchange rate and the private sector’s saving decision. QE as announced in January 2015 generates 0.4% effective euro depreciation and raises real GDP in the euro area by 0.2% and prices by 0.3% by 2017 in the model. The subsequent extensions of the QE programme (extension in time and increase in volume) more than double the medium-term output and inflation effects according to the simulations.

Model Appendix

1.1 Model Overview

The analysis uses QUEST III (Ratto et al. 2009), a quarterly global macroeconomic model developed for macroeconomic policy analysis and research. QUEST III is a member of the class of New-Keynesian Dynamic Stochastic General Equilibrium (DSGE) models. It has rigorous microeconomic foundations derived from utility and profit optimisation and includes frictions in goods, labour and financial markets.

This paper uses a two-region setup of the model with the euro area (EA), and the rest of the world (RoW). Each region has one (tradable) production sector. The model distinguishes between Ricardian households that have full access to financial markets and liquidity-constrained households that do not access financial markets. In each region there are monetary and fiscal authorities, both following rule-based stabilisation policies.

Monetary policy in “normal times” follows a Taylor rule reacting to inflation and the output gap. In case of QE, the central bank purchases long-term (government) bonds, with the aim of reducing the interest spread between short and long maturities (i.e. flattening the yield curve). In our simulations, the central bank finances the bond purchases by providing additional liquidity to the private sector rather than by reducing central bank profits.

1.2 Firms

Each firm produces a variety of the sectoral good, which is an imperfect substitute for varieties produced by other firms. Given this imperfect substitutability, firms are monopolistically competitive in the goods market and face a demand function for goods. Firms sell consumption goods to domestic and foreign private households and governments, and they sell investment goods to domestic and foreign firms and governments. The firms are owned by intertemporally optimising households.

Output (Y _t ) is produced with a Cobb-Douglas technology using capital (K _t ), production workers (N _t  − LO _t ) and public infrastructure (KG _t ):

$$ {Y}_t={\left({ucap}_t{K}_t\right)}^{1-\alpha }{\left({N}_t-{LO}_t\right)}^{\alpha }{KG}_t^{\alpha_g} $$

(4)

The variables ucap _t and LO _t  respectively denote capacity utilisation and overhead labour.¹ Firm-level employment N _t is itself a CES aggregate of labour services supplied by individual households h:

$$ {N}_t\equiv {\left(\underset{0}{\overset{1}{\int }}{N_t^h}^{\frac{\theta -1}{\theta }} dh\right)}^{\frac{\theta }{\theta -1}} $$

(5)

The parameter θ > 1 determines the degree of substitutability among different types of labour.

The objective of the firm is to maximise real profits (Pr _t ):

$$ {\Pr}_t={p}_t{Y}_t-{w}_t{N}_t-{i}_t^k{p}_t^I{K}_t-\left({adj}^P\left({P}_t\right)+{adj}^N\left({N}_t\right)+{adj}^{ucap}\left({ucap}_t\right)\right) $$

(6)

with respect to the goods price, where ${i}_t^K$ is the rental rate of capital. Firms face technology and regulatory constraints that restrict their decisions about hiring/firing, pricing, and capacity utilisation. The constraints are modelled here as adjustment costs with the following convex functional forms:

$$ {adj}^N\left({N}_t\right)\equiv \frac{\gamma_N}{2}{w}_t{\left(\varDelta {N}_t\right)}^2 $$

(7a)

$$ {adj}^P\left({P}_t\right)\equiv \frac{\gamma_P}{2}{\left(\frac{P_t-{P}_{t-1}}{P_{t-1}}\right)}^2{Y}_t $$

(7b)

$$ {adj}^{ucap}\left({ucap}_t\right)\equiv {p}_t^I{K}_t\left({\gamma}_{ucap,1}\left({ucap}_t-1\right)+\frac{\gamma_{ucap,2}}{2}{\left({ucap}_t-1\right)}^2\right) $$

(7c)

The firms determine labour input, capital services, capacity utilisation and goods prices optimally in each period given the technology and administrative constraints as well as the demand conditions. The first-order conditions (FOC) are given by:

$$ \frac{\partial {\Pr}_t}{\partial {N}_t}=>\frac{\partial {Y}_t}{\partial {N}_t}{\eta}_t-{\gamma}_N{w}_t\varDelta {N}_t+{\gamma}_N{E}_t\left({\beta \lambda}_{t,t+1}^r{w}_{t+1}\varDelta {N}_{t+1}\right)={w}_t $$

(8a)

$$ \frac{\partial {\Pr}_t}{\partial {K}_t}=>\frac{\partial {Y}_t}{\partial {K}_t}{\eta}_t={i}_t^k{p}_t^I $$

(8b)

$$ \frac{\partial {\Pr}_t}{\partial {ucap}_t}=>\frac{\partial {Y}_t}{\partial {ucap}_t}{\eta}_t={p}_t^I{K}_t\left({\gamma}_{ucap,1}+{\gamma}_{ucap,2}\left({ucap}_t-1\right)\right) $$

(8c)

$$ \frac{\partial {\Pr}_t}{\partial {Y}_t}=>{\eta}_t=1-1/\sigma -{\gamma}_P{E}_t\left({\beta \lambda}_{t,t+1}^r{\pi}_{t+1}-{\pi}_t\right)\ \mathrm{with}\kern0.5em {\pi}_t\equiv {P}_t/{P}_{t-1}-1 $$

(8d)

where η _t is the Lagrange multiplier of the technology constraint, β is the rate of time preference, and ${E}_t\left({\beta \lambda}_{t,t+1}^r\right)$ is the stochastic discount factor of the intertemporally optimising households. Firms equate the marginal product of labour, net of marginal adjustment costs, to wage costs. As can be seen from the left-hand side of Eq. (8a), the convex part of the adjustment cost function penalises changes in employment. Eqs. (8b–c) jointly determine the optimal capital stock and optimal capacity utilisation by equating the marginal value product of capital to the rental price and the marginal product of capital services to the marginal cost of increasing capacity. Eq. (8d) defines the price mark-up factor as function of the elasticity of substitution and changes in inflation. The steady-state price mark-up equals the inverse of the price elasticity of demand. QUEST follows the empirical literature and allows for additional backward looking elements in price setting by assuming that a fraction 1-sfp of the firms indexes price increases to inflation in t-1. This leads to the following specification:

$$ {\eta}_t=1-1/\sigma -{\gamma}_P{E}_t\left({\beta \lambda}_{t,t+1}^r\left( sfp{\pi}_{t+1}+\left(1- sfp\right){\pi}_{t-1}\right)-{\pi}_t\right)\kern0.5em 0\le sfp\le 1 $$

(8d’)

for price mark-ups.

1.3 Long-Term Government Bonds

Following Woodford (2001), Chen et al. (2012), and Liu et al. (2017), long-term government debt is modelled through bonds for which the nominal coupon c, which is a fraction of the principal, depreciates over time at rate δ _b . The price in period t of a long-term bond issued in t (${P}_t^N$) equals the discounted value of future payments:

$$ {P}_t^N=\sum \limits_{n=0}^T\frac{\delta_b^n}{{\left(1+i\right)}^{1+n}}c $$

(9)

where T is the maturity period of the bond. Analogously, the price in period t of a long-term bond issued in t-1 (${P}_t^O$) equals the discounted sum of outstanding payments:

$$ {P}_t^O=\sum \limits_{n=0}^{T-1}\frac{\delta_b^{1+n}}{{\left(1+i\right)}^{1+n}}c $$

(10)

If δ _b /(1 + i) < 1 and T is large, the price in t of long-term bonds issued in t-1 corresponds (approximately) to the price of newly issued long-term bonds times the depreciation rate:

$$ {P}_t^O={\delta}_b{P}_t^N $$

(11)

Equation (11) shows that the price of the long-term bond that pays a declining coupon declines over time at the rate δ _b .

In the full model version with cross-border bond holdings, total government debt consists of long-term bonds ${B}_t^L$ held by domestic (${B}_t^{L,H}$) and foreign private agents (${B}_t^{L,F}$), and by the central bank (${B}_t^{L, CB}$), and of short-term bonds ${B}_t^S$:

$$ {B}_t={B}_t^{L,H}+{B}_t^{L,F}+{B}_t^{L, CB}+{B}_t^S $$

(12)

Short-term and long-term bonds are imperfect substitutes in the model. In particular, households have a preference for holding a mix of short-term and long-term bonds, and deviations from the target value κ for the ratio of long-term over short-term debt induce quadratic adjustment costs (γ _b ). The same formulation of portfolio preferences or adjustment costs has been used previously by, e.g., Andrés et al. (2004), Falagiarda (2013), Harrison (2012), and Liu et al. (2017). As shown by Schmitt-Grohé and Uribe (2003), an analogous formulation for (net) foreign assets relative to the target ${\overline{B}}^{\ast }$ closes the international finance part of the model.

1.4 Households

The household sector consists of a continuum of households h ∈ [0, 1]. There are s ^l ≤ 1 households that are liquidity-constrained and indexed by l. These households do not trade on asset markets and simply consume their disposable income each period. A fraction 1 − s ^l of all households is Ricardian and indexed by r. The period utility function is identical for each household type. It is separable in consumption (${C}_t^h$) and leisure ($1-{N}_t^h$), allows for habit persistence in consumption and is given by:

$$ U\left({C}_t^h,{N}_t^h\right)=\ln \left({C}_t^h-{hC}_{t-1}\right)+\omega {\left(1-{N}_t^h\right)}^{1-\kappa } $$

(13)

where ω is the weight of the utility of leisure in total period utility and κ the inverse of the elasticity of labour supply.

Both types of households supply differentiated labour services to unions that maximise a joint utility function for each type of labour h. It is assumed that types of labour are distributed equally across both household types. Nominal wage rigidity is introduced through adjustment costs for changing wages. These adjustment costs are borne by the households.

1.4.1 Ricardian Households

Ricardian households receive labour income, returns on financial assets, income ${i}_t^k$ from lending capital to firms net of an (exogenous) risk/insurance premium given revenue uncertainty φ _t , and dividends D _t from firm ownership. K _t  = I _t  + (1 − δ _k )K _t − 1 is the capital stock as the sum of new effective investment I _t and the pre-period capital stock depreciated at rate δ _k . The government levies taxes ${t}_t^w$ on income from labour, ${t}_t^k$ on corporate income and ${t}_t^c$on consumption. The price in period t of a short-term (1-period) bond of nominal value ${B}_t^S$ is ${B}_t^S/\left(1+{i}_t\right)$, with i _t being the short-term nominal interest rate. Analogously, ${e}_t{B}_t^{\ast }/\left(1+{i}_t^{\ast}\right)$ is the price in domestic currency of a foreign bond ${B}_t^{\ast }$, where e _t is the nominal exchange rate as the value in foreign currency of one unit of domestic currency.

The Lagrangian of the maximisation problem is:

$$ {\displaystyle \begin{array}{l}\max {L}^r={E}_0\sum \limits_{t=0}^{\infty }{\beta}^tU\left({C}_t^r,{N}_t^r\right)\\ {}-{E}_0\sum \limits_{t=0}^{\infty }{\lambda}_t{\beta}^t\left(\begin{array}{l}\frac{\left(1+{t}_t^c\right){P}_t^C}{P_t}{C}_t^r+\frac{P_t^C\left({K}_t-\left(1-{\delta}_k\right){K}_{t-1}\right)}{P_t}+\frac{B_t^S}{\left(1+{i}_t\right){P}_t}\\ {}+\frac{P_t^N{B}_t^{L,H}}{P_t}\left(1+\frac{\gamma_b}{2}{\left(\kappa \frac{B_t^S}{B_t^{L,H}}-1\right)}^2\right)+\frac{e_t{B}_t^{\ast }}{\left(1+{i}_t^{\ast}\right){P}_t}+\frac{\gamma_f}{2}{\left(\frac{e_t\left({B}_t^{\ast }-{\overline{B}}^{\ast}\right)}{P_t}\right)}^2\\ {}+\frac{e_t{P}_t^{N\ast }{B}_t^{L,H\ast }}{P_t}\left(1+\frac{\gamma_b^{\ast }}{2}{\left({\kappa}^{\ast}\frac{B_t^{L,H}}{B_t^{L,H\ast }}-1\right)}^2\right)-\frac{T{R}_t}{P_t}-\frac{c{B}_{t-1}^{L,H}}{P_t}-\frac{c^{\ast }{e}_t{B}_{t-1}^{L,H\ast }}{P_t}\\ {}-\frac{\delta_b{P}_t^N{B}_{t-1}^{L,H}}{P_t}-\frac{\delta_b^{\ast }{e}_t{P}_t^{N\ast }{B}_{t-1}^{L,H\ast }}{P_t}-\frac{B_{t-1}^S}{P_t}-\frac{e_t{B}_{t-1}^{\ast }}{P_t}-\frac{\left(1-{t}_t^w\right){W}_t{N}_t^r}{P_t}\\ {}-\left({i}_{t-1}^k-\left({i}_{t-1}^k-{\delta}_k\right){t}_{t-1}^k-{\varphi}_{t-1}\right)\frac{P_t^C}{P_t}{K}_{t-1}-\frac{D_t}{P_t}\end{array}\right)\end{array}} $$

(14)

Investment in physical capital is subject to convex adjustment costs, introducing a distinction between real investment expenditure (I _t ) and physical investment net of adjustment costs (J _t ):

$$ {I}_t={J}_t\left(1+\frac{\gamma_K}{2}\frac{J_t}{K_t}\right)+\frac{\gamma_I}{2}{\left(\varDelta {J}_t\right)}^2 $$

(15)

The maximisation problem (Eq. 14) provides the following first-order conditions (FOC):

$$ \frac{\partial {L}^r}{\partial {B}_t^S}\Rightarrow \beta {E}_t\left(\frac{\lambda_{t+1}}{\lambda_t}\right)={E}_t\left(\frac{P_{t+1}}{P_t}\right)\left(\frac{1}{1+{i}_t}+{\gamma}_b\kappa {P}_t^N\left(\kappa \frac{B_t^S}{B_t^{L,H}}-1\right)\right) $$

(16)

$$ \frac{\partial {L}^r}{\partial {B}_t^{L,H}}\Rightarrow \beta {E}_t\left(\frac{\lambda_{t+1}}{\lambda_t}\frac{P_t}{P_{t+1}}\right)={E}_t\left(\frac{P_t^N}{\delta_b{P}_{t+1}^N+c}\right)\left(\begin{array}{l}1+\frac{\gamma_b}{2}{\left(\kappa \frac{B_t^S}{B_t^{L,H}}-1\right)}^2\\ {}-{\gamma}_b\kappa \left(\kappa \frac{B_t^S}{B_t^{L,H}}-1\right)\frac{B_t^S}{B_t^{L,H}}\\ {}+{\gamma}_b^{\ast }{\kappa}^{\ast}\left({\kappa}^{\ast}\frac{B_t^{L,H}}{B_t^{L,H\ast }}-1\right)\frac{e_t{P}_t^{N\ast }}{P_t^N}\end{array}\right) $$

(17)

$$ \frac{\partial {L}^r}{\partial {B}_t^{L,H\ast }}\Rightarrow \beta {E}_t\left(\frac{\lambda_{t+1}}{\lambda_t}\frac{P_t}{P_{t+1}}\frac{e_{t+1}}{e_t}\right)={E}_t\left(\frac{P_t^{N\ast }}{\delta_b^{\ast }{P}_{t+1}^{N\ast }+{c}^{\ast }}\right)\left(\begin{array}{l}1+\frac{\gamma_b^{\ast }}{2}{\left({\kappa}^{\ast}\frac{B_t^{L,H}}{B_t^{L,H\ast }}-1\right)}^2\\ {}-{\gamma}_b^{\ast }{\kappa}^{\ast}\left({\kappa}^{\ast}\frac{B_t^{L.H}}{B_t^{L,H\ast }}-1\right)\frac{B_t^{L,H}}{B_t^{L,H\ast }}\end{array}\right) $$

(18)

$$ \frac{\partial {L}^r}{\partial {B}_t^{\ast }}\Rightarrow \beta {E}_t\left(\frac{\lambda_{t+1}}{\lambda_t}\right)={E}_t\left(\frac{e_t}{e_{t+1}}\frac{P_{t+1}}{P_t}\right)\left(\frac{1}{1+{i}_t^{\ast }}+{\gamma}_f\frac{e_t\left({B}_t^{\ast }-{\overline{B}}^{\ast}\right)}{P_t}\right) $$

(19)

$$ \frac{\partial {L}^r}{\partial {K}_t}\Rightarrow \beta {E}_t\left(\frac{\lambda_{t+1}}{\lambda_t}\right)={E}_t\left(\frac{P_{t+1}}{P_t}\frac{P_t^C}{P_{t+1}^C}\right)\frac{1}{\left(1+{i}_t^k-{\varphi}_t-{\delta}_k\right)-{t}_t^k\left({i}_t^k-{\delta}_k\right)} $$

(20)

$$ \frac{\partial {L}^r}{\partial {C}_t^r}\Rightarrow {U}_t^C=\frac{\left(1+{t}_t^c\right){P}_t^c}{P_t}{\lambda}_t $$

(21)

$$ \frac{\partial {L}^r}{\partial {N}_t^r}\Rightarrow {U}_t^N=\frac{\left(1-{t}_t^w\right){W}_t}{P_t}{\lambda}_t $$

(22)

The arbitrage condition for investment provides an investment rule linking capital formation to the shadow price of capital:

$$ \left({\gamma}_K\frac{J_t^K}{K_{t-1}}+{\gamma}_I\varDelta {J}_t^K\right)-{E}_t\left(\frac{1}{1+{r}_t+{\pi}_{t+1}^{GDP}-{\pi}_{t+1}^I}\varDelta {J}_{t+1}^K\right)=\frac{\xi_t}{p_t^I}-1 $$

(23)

where the shadow price of capital corresponds to the present discounted value of the rental income from physical capital:

$$ \frac{\xi_t}{p_t^I}={E}_t\left(\frac{1}{1+{r}_t+{\pi}_{t+1}^{GDP}-{\pi}_{t+1}^I}\frac{\xi_{t+1}}{p_{t+1}^I}\left(1-{\delta}_k\right)\right)+\left(\left(1-{t}_t^k\right){i}_t^k+{t}_t^k{\delta}_k\right)=0 $$

(24)

1.4.2 Liquidity-Constrained Households

Liquidity-constrained households do not optimise, but simply consume their entire disposable income at each date. Real consumption of household l is thus determined by the net wage and transfer income minus a lump-sum tax:

$$ \left(1+{t}_t^c\right){P}_t^c{C}_t^l=\left(1-{t}_t^w\right){W}_t{N}_t^l+{TR}_t^l-{T}_t^{LS,l} $$

(25)

The liquidity-constrained households share the same utility function as Ricardian households.

1.4.3 Wage Setting

A trade union is maximising a joint utility function for each type of labour h. It is assumed that types of labour are distributed equally over Ricardian and liquidity-constrained households with their respective population weights. The trade union sets wages by maximising a weighted average of the utility functions of these households. The wage rule is obtained by equating a weighted average of the marginal utility of leisure to a weighted average of the marginal utility of consumption times the net real consumption wage of both household types, adjusted for a wage mark-up (${\eta}_t^W$):

$$ \frac{\left(1-{s}^l\right){U}_{1-N,t}^r+{s}^l{U}_{1-N,t}^l}{\left(1-{s}^l\right){U}_{c,t}^r+{s}^l{U}_{c,t}^l}=\frac{1-{t}_t^w}{1+{t}_t^c}\frac{W_t}{P_t^C}{\eta}_t^W $$

(26)

Wage mark-ups fluctuate around 1/θ, which is the inverse of the elasticity of substitution between different varieties of labour services. The ratio of the marginal utility of leisure to the marginal utility of consumption is a natural measure of the reservation wage. If the ratio is equal to the consumption wage, the household is indifferent between supplying an additional unit of labour and spending the additional income on consumption, or not increasing labour supply. Fluctuations in the wage mark-up arise from wage adjustment costs and the fact that a fraction 1-sfw of workers indexes wage growth ${\pi}_t^W$ to inflation in the previous period:

$$ {\eta}_t^W=1-1/\theta -{\gamma}_W/\theta {E}_t\left({\beta \lambda}_{t,t+1}^r\left({\pi}_{t+1}^W-\left(1- sfw\right){\pi}_t\right)-\left({\pi}_t^W-\left(1- sfw\right){\pi}_{t-1}\right)\right)\kern0.5em 0\le sfw\le 1 $$

(27)

The (semi-)elasticity of wage inflation with respect to employment is given by κ/γ _W , i.e. it is positively related to the inverse of the elasticity of labour supply and inversely related to wage adjustment costs.

1.4.4 Aggregation

The aggregate value of any household specific variable ${X}_t^h$ in per-capita terms is given by ${X}_t\equiv {\int}_0^1{X}_t^h dh=\left(1-{s}^l\right){X}_t^r+{s}^l{X}_t^l$ since the households within each group are identical in theirs consumption and labour supply decisions. Hence, aggregate consumption is given by:

$$ {C}_t=\left(1-{s}^l\right){C}_t^r+{s}^l{C}_t^l $$

(28a)

and aggregate employment by:

$$ {N}_t=\left(1-{s}^l\right){N}_t^r+{s}^l{N}_t^l\ \mathrm{with}\ {N}_t^r={N}_t^l $$

(28b)

1.5 Policy

Fiscal policy and monetary policy are partly rules-based and partly discretionary. The fiscal rule stabilises government debt. The monetary policy rule stabilises inflation and output.

1.5.1 Fiscal Policy

Real government purchases (G _t ) and real government investment (IG _t ) are kept exogenous. The stock of public infrastructure, which enters the production function of firms (Eq. 4), develops according to:

$$ {KG}_t={IG}_t+\left(1-{\delta}^g\right){KG}_{t-1} $$

(29)

Nominal transfers (TR _t ) correspond to a CPI-adjusted exogenous transfer-to-GDP share (try):

$$ {TR}_t={tryP}_t^C $$

(30)

The government collects tax revenue from consumption, labour, corporate income and lump-sum taxes. The lump-sum taxes are a fixed share of GDP. Nominal government debt which is composed of short-term bonds and long-term bonds follows:

$$ {\displaystyle \begin{array}{l}\frac{B_t^S}{\left(1+{i}_t\right)}+{P}_t^N{B}_t^L={B}_{t-1}^S+\left({\delta}_b{P}_t^N+c\right){B}_{t-1}^L+{P}_t^C\left({G}_t+{IG}_t\right)+{TR}_t-{t}_t^c{P}_t^c{C}_t-{t}_t^w{W}_t{N}_t\\ {}-\left({t}_t^k\left({P}_t{Y}_t-{W}_t{N}_t-{\delta}_k{P}_t^I{K}_{t-1}\right)\right)-{T}_t^{LS}\end{array}} $$

(31)

As seen previously (Eq. 12), total government debt consists of long-term bonds ${B}_t^L$ held by domestic (${B}_t^{L,H}$) and foreign private agents (${B}_t^{L,F}$), and by the central bank (${B}_t^{L, CB}$), and of short-term bonds ${B}_t^S$. The labour tax is used to control the debt-to-GDP ratio according to the following rule:

$$ \varDelta {t}_t^w={\tau}^B\left(\frac{B_{t-1}}{P_{t-1}{Y}_{t-1}}-{b}^{tar}\right)+{\tau}^{DEF}\varDelta \left(\frac{B_t}{Y_t{P}_t}\right) $$

(32)

where b ^tar is the government debt target. The consumption and corporate income tax rates are kept constant.

1.5.2 Monetary Policy

The operating profit of the central bank equals the sum of base money issuance and interest income minus the current expenditure on buying long-term bonds, where the latter equals the change of the value of long-term bonds on the central bank’s balance sheet:

$$ {PR}_t^{CB}=\varDelta {M}_t+{cB}_{t-1}^{L, CB}-\left({P}_t^N{B}_t^{L, CB}-{\delta}_b{P}_t^N{B}_{t-1}^{L, CB}\right) $$

(33)

Under the central bank’s budget constraint (Eq. 33), purchases of long-term government bonds can be financed either by increasing liquidity (money issuance), or by reducing the central bank’s operating profit.² Purchases of long-term bonds by the central bank are modelled as an exogenous path that replicates the announced ECB programme in timing and size.

Monetary policy in “normal times” follows a Taylor rule that allows for smoothing of the interest rate response to inflation and the output gap:

$$ {i}_t={\rho}_i{i}_{t-1}+\left(1-{\rho}_i\right)\left(\overline{r}+{\pi}^{tar}+{\tau}_{\pi}\left({\pi}_t^C-{\pi}^{tar}\right)+{\tau}_y{ygap}_t\right) $$

(34)

The central bank has an inflation target π ^tar, adjusts its policy rate when actual CPI inflation deviates from the target and also responds to the output gap (ygap).

As regards the implementation of the zero bound on nominal interest rates, we employ the simplifying assumption of an exogenous zero bound, i.e. keep the policy rate frozen for a defined period and (with gradual phasing-in) let it respond according to the Taylor rule in the model thereafter. The alternative of creating a recession baseline that generates a binding zero lower bound endogenously on which then to perform QE simulations yields similar results for a comparable length of the no-interest-change period.

The output gap is not calculated as the difference between actual and efficient output, but derived from a production function framework, which is the standard practice of output gap calculation for fiscal surveillance and monetary policy. Precisely, the output gap is calculated from the deviation of capital and labour utilisation from theirs long-run trends:

$$ {ygap}_t\equiv \alpha \ln \left({N}_t/{N}_t^{ss}\right)+\left(1-\alpha \right)\ln \left({ucap}_t/{ucap}_t^{ss}\right) $$

(35)

The variables ${N}_t^{ss}$ and ${ucap}_t^{ss}$ are employment and capacity utilisation trends:

$$ {N}_t^{ss}={\rho}^N{N}_{t-1}^{ss}+\left(1-{\rho}^N\right){N}_t $$

(36a)

$$ {ucap}_t^{ss}={\rho}^{ucap}{ucap}_{t-1}^{ss}+\left(1-{\rho}^{ucap}\right){ucap}_t $$

(36b)

that are restricted to move slowly in response to actual values.

1.6 Trade and the Current Account

So far, only aggregate consumption and investment demand have been determined, but not its allocation over domestic and foreign tradable goods.

In order to facilitate aggregation, private households and the government are assumed to have identical preferences across goods for consumption and investment. Let Z ∈ (C, G, I, IG) be the demand of an individual household or the government and their preferences over domestic versus imported goods given by the following utility function:

$$ {Z}_t={\left({\left(1-{s}_m\right)}^{\frac{1}{\sigma_m}}{Z_t^D}^{\frac{\sigma_m-1}{\sigma_m}}+{s_m}^{\frac{1}{\sigma_m}}{Z_t^F}^{\frac{\sigma_m-1}{\sigma_m}}\right)}^{\frac{\sigma_m}{\sigma_m-1}} $$

(37a)

where Z ^D and Z ^∗ are indexes of demand across the continuum of goods produced in the domestic economy and abroad respectively:

$$ {Z}_t^D\equiv {\left(\sum \limits_{d=1}^m{m}^{-\frac{1}{\sigma }}{Z_t^d}^{\frac{\sigma -1}{\sigma }}\right)}^{\frac{\sigma }{\sigma -1}} $$

(37b)

$$ {Z}_t^F\equiv {\left(\sum \limits_{f=1}^q{q}^{-\frac{1}{\sigma }}{Z_t^f}^{\frac{\sigma -1}{\sigma }}\right)}^{\frac{\sigma }{\sigma -1}} $$

(37c)

The elasticity of substitution between bundles of domestic and foreign goods is σ _m .

The aggregate consumer price index is given by:

$$ {P}_t^C={\left(\left(1-{s}_m\right){\left({P}_t\right)}^{1-{\sigma}_m}+{s}_m{\left({P_t}^F{e}_t\right)}^{1-{\sigma}_m}\right)}^{\frac{1}{1-{\sigma}_m}} $$

(38)

The steady-state import share (s _m ), and the elasticity of substitution between domestic and imported goods (σ _m ) are assumed to be constant across all expenditure components. The aggregate imports are given by:

$$ {M}_t={s}_m{\left(\frac{e_t{P}_t^{\ast }}{P_t^C}\right)}^{-{\sigma}_m}{Z}_t $$

(39)

Assuming equivalent demand functions in the rest of the world, exports can be treated symmetrically and are given by:

$$ {X}_t={s}_m^{\ast }{\left(\frac{P_t}{e_t{P}_t^{C^{\ast }}}\right)}^{-{\sigma}_x}\left({Z}_t^{\ast}\right) $$

(40)

The domestic economy’s trade balance is the net trade in value terms:

$$ {TB}_t\equiv {P}_t{X}_t-{e}_t{P_t}^{\ast }{M}_t $$

(41)

The law of motion for the NFA position is:

$$ {\displaystyle \begin{array}{l}{e}_t\left({B}_t^{\ast }+{P}_t^{N\ast }{B}_t^{L,H\ast}\right)-{P}_{\mathrm{t}}^N{B}_t^{L,F}=\left(1+{i}_{t-1}^{\ast}\right){e}_t{B}_{t-1}^{\ast }+\left({c}^{\ast }+{\delta}_b^{\ast }{P}_t^{N\ast}\right){e}_t{B}_{t-1}^{L,H\ast}\\ {}-\left(c+{\delta}_b{P}_t^N\right){e}_t{B}_{t-1}^{L,F}+{P}_t^X{X}_t-{P}_t^M{M}_t\end{array}} $$

(42)

The model’s external side is closed by a (small) country risk premium (risk), which depends on the NFA position (e.g. Schmitt-Grohé and Uribe 2003):

$$ {i}_t={i}_t^F+\frac{\varDelta {e}_{t+1}}{e_t}- risk\left(\frac{e_t{B}_t^{\ast }}{P_t{Y}_t}-{bwy}^T\right) $$

(43)

and rules out explosive NFA dynamics.

1.7 Parametrisation

The parameters are based on long-term averages (mainly national accounts data) for the EA economy as far as the steady state of the variables is concerned, and on model versions estimated with Bayesian methods (notably Ratto et al. 2009) for the parameters governing the adjustment dynamics (price and wage stickiness, employment and investment adjustment costs, habit persistence, and others). These parameter values are also compatible with stylised facts of the EA economy, such as average price and wage durations.

Table 3 Model parametrisation