The Model Economy

Following SW (2007) and Milani et al. (2012), the model reported below is in loglinearized form mainly to show the role that various shocks and expectations play. The building blocks of the model economy consist of a basic real business cycle (RBC) model, in which investment decisions, capital accumulation, households’ labour supply decisions on how many hours to work and shocks to total factor productivity play an important role. The stylised New Keynesian model allows for imperfect competition, nominal rigidities such as price, and wage stickiness and assumes an interest-rate rule for monetary policy. The UK MTM endogenous variables include:

{𝑦

𝑐

𝑖

𝑧

𝑟

𝑞

𝑘

𝜇

𝑙

𝑘

𝑤

𝜇

𝑚𝑝𝑘

𝜋

𝜋

𝑟

}

the exogenous variables are:

{𝜀

𝜀

𝜀

𝜀

𝜀

𝜀

𝜀

}

𝑦_𝑡= 𝑐_𝑦𝑐_𝑡+ 𝑖_𝑦𝑖_𝑡+ 𝑢_𝑦𝑢_𝑡+ 𝜀_𝑡^𝑔 (3.15) equation (3.15) represents the economy’s aggregate resource constraint. Output 𝑦_𝑡 is absorbed by consumption 𝑐_𝑡, by investment 𝑖_𝑡, and by the resources used to vary the capacity utilization rate 𝑢_𝑡. According to Milani et al. (2012), the government spending is assumed to be exogenous and captured by the disturbance 𝜀_𝑡^𝑔.

𝑐_𝑡 = 𝑐₁𝑐_𝑡−1+ (1 − 𝑐₁)𝐸_𝑡𝑐_𝑡+1+ 𝑐₂(𝑙_𝑡− 𝐸_𝑡𝑙_𝑡+1) − 𝑐₃(𝑟_𝑡− 𝐸_𝑡𝜋_𝑡+1+ 𝜀_𝑡^𝑏) (3.16)

53 For a description of the data, see Appendix 3A.

54 The estimation is done with Dynare 4.3.4. The scale factor for the jump distribution has been set in order to obtain acceptance rates around 30 percent.

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Equation (3.16) is the Euler equation for consumption, where the contemporaneous value for consumption depends on expectations about future consumption, on lagged consumption, on current and expected hours of work 𝑙_𝑡 and on the ex-ante real interest rate (𝑟_𝑡− 𝐸_𝑡𝜋_𝑡+1). The term 𝜀_𝑡^𝑏 indicates a risk-premium shock which is an exogenous shock that affects yields on bonds. It is, sometimes, substituted in the literature by a preference or discount factor shock, which enters in similar ways but with a converted sign in the Euler equation. The investment and capital stocks are formulated as:

𝑖_𝑡= 𝑖₁𝑖_𝑡−1+ (1 − 𝑖₁)𝐸_𝑡𝑖_𝑡+1+ 𝑖₂𝑞_𝑡+ 𝜀_𝑡^𝑖 (3.17) 𝑞_𝑡= 𝑞₁𝐸_𝑡𝑞_𝑡+1+ (1 − 𝑞₁)𝐸_𝑡𝑟_𝑡+1^𝑘 − (𝑟_𝑡− 𝐸_𝑡𝜋_𝑡+1+ 𝜀_𝑡^𝑏)

𝑞_𝑡 = 𝑅_∗^𝑘

𝑅_∗^𝑘+ 1 − 𝛿𝐸_𝑡{𝑚𝑝𝑘_𝑡+1} + 1 − 𝛿

𝑅_∗^𝑘+ 1 − 𝛿𝐸_𝑡{𝑞_𝑡+1} − 𝑟_𝑡+ 𝜀_𝑡^𝑏 (3.18) Equation (3.17) and (3.18) characterise the dynamics of investment. Current investment is influenced by expectations about future investment, by lagged investment and by the value of capital stock 𝑞_𝑡. The capital stock is driven by expectations about its future one-period-ahead value, by expectations about the rental rate on capital 𝐸_𝑡𝑟_𝑡+1^𝑘 , and by the ex-ante real interest rate. The disturbance terms 𝜀_𝑡^𝑖 and 𝜀_𝑡^𝑏 affect the behaviour of investment. 𝜀_𝑡^𝑖 denotes investment-specific technological charge, while 𝜀_𝑡^𝑏 is the same risk-premium shock that also enters the consumption Euler equation which helps in fitting the co-movement of the investment and consumption series.

𝑦_𝑡 = Φ_𝑝(𝛼𝑘_𝑡^𝑠+ (1 − 𝛼)𝑙_𝑡+ 𝜀_𝑡^𝑎 (3.19)

Equation (3.19) is a Cobb-Douglas production function: output is produced using capital services 𝐾_𝑡^𝑠 and labour hours 𝑙_𝑡. Neutral technological progress enters the expression as the exogenous shock 𝜀_𝑡^𝑎. The coefficient Φ_𝑝 captures fixed costs in production. The capital utilisations are expressed in the following equations:

𝑘_𝑡^𝑠 = 𝑘_𝑡−1+ 𝑢_𝑡 (3.20)

𝑢_𝑡 = 𝑢₁𝑟_𝑡^𝑘 𝑧_𝑡= 1 − 𝜓

𝜓 𝑚𝑝𝑘_𝑡 (3.21)

𝑘_𝑡= 𝛿𝑖_𝑡+ (1 + 𝛿)𝑘_𝑡−1+ 𝛿(1 + 𝛽)𝜑𝜀_𝑡^𝑖 (3.22) Equation (3.20) accounts for the possibility to vary the rate of capacity utilization. Capital services are a function of the capital utilization rate 𝑢_𝑡 and of the lagged capital stock 𝑘_𝑡−1. The degree of capital utilization itself varies as a function of the rental rate of capital, as evidenced by Equation (3.21). From Equation (3.25), the rental rate of capital is a function of the capital to labour ratio

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and of the real wage. Capital rate of depreciation is accumulated according to Equation (3.22). The equilibrium in the labour (HHC channel) and goods market takes the following form:

𝜇_𝑡^𝑝 = 𝛼(𝑘_𝑡^𝑠− 𝑙_𝑡) − 𝑤_𝑡+ 𝜀_𝑡^𝑎 (3.23)

𝜋_𝑡= 𝜄_𝑝

1 + 𝛽𝜋_𝑡−1+ 𝛽

1 + 𝛽𝐸_𝑡{𝜋_𝑡+1} − 1 1 + 𝛽

(1 − 𝜉_𝑝)

𝜉_𝑝 𝜇_𝑡^𝑝+ 𝜀_𝑡^𝑝 (3.24) 𝑟_𝑡^𝑘 = −(𝑘_𝑡− 𝑙_𝑡) + 𝑤_𝑡

Also as:

𝑚𝑝𝑘_𝑡 = −(𝑘_𝑡^𝑠− 𝑙_𝑡) + 𝑤_𝑡

(3.25)

𝜇_𝑡^𝑤 = 𝑤_𝑡− (𝜎_𝑙𝑙_𝑡− 1

1 − ℎ/𝛾(𝑐_𝑡−ℎ

𝛾𝑐_𝑡−1)) (3.26) 𝑤_𝑡 = 𝑤₁𝑤_𝑡−1+ (1 − 𝑤₁) 𝐸_𝑡{𝑤_𝑡+1+ 𝜋_𝑡+1} − 𝑤₂𝜋_𝑡+ 𝑤₃𝜋_𝑡−1− 𝑤₄𝜇_𝑡^𝑤 + 𝜀_𝑡^𝑤 (3.27) Equation (3.23) to Equation (3.27) summarise the equilibrium in the goods and labour markets.

Inflation 𝜋_𝑡 is determined as a function of lagged inflation, expected inflation, and the price mark-up 𝜇_𝑡^𝑝, which is equal to the difference between the marginal product of labour 𝛼(𝑘_𝑡^𝑠− 𝑙_𝑡) + 𝜀_𝑡^𝑎 and the real wage 𝑤_𝑡. The real wage depends on lagged and expected future real wages, on past, current, and expected inflation, and on the wage mark-up 𝜇_𝑡^𝑤, which equals the difference between the real wage and the marginal rate of substitution between consumption and leisure. Inflation and wage dynamics are also affected by the exogenous price and wage mark-up shocks, 𝜀_𝑡^𝑝 and 𝜀_𝑡^𝑤, which are obtained by assuming a time-varying elasticity of substitution among differentiated goods.

𝑟_𝑡 = 𝜌𝑟_𝑡−1+ (1 − 𝜌𝑟 ) (𝜒_𝑛𝜋_𝑡+ 𝜒_𝑦(𝑦_𝑡− 𝑦_𝑡^∗) + 𝜒_Δy(Δ𝑦_𝑡− Δ𝑦_𝑡^∗) + 𝜀_𝑡^𝑟 (3.28)

𝑟_𝑡 = 𝑟_𝑡^𝑛+ 𝐸_𝑡{𝜋_𝑡+1} (3.29)

Equations (3.28 and 3.29) describe a type of Taylor rule. The monetary authority sets the interest rate 𝑟_𝑡 in response to changes in inflation and the output gap. The policy rate also responds to the growth in the output gap. The term 𝜀_𝑡^𝑟 captures random deviations from the systematic policy rule.

The coefficients in the model are composite functions of the “deep” preference and technology parameters, such as the degree of habits in consumption, the elasticities of intertemporal substitution and of labour supply and the Calvo price rigidity coefficients, among others (Milani, 2012; SW, 2007).

The model governs the dynamics of 17 endogenous variables, which play in the movement of shocks in the transmission mechanism. The sources of uncertainty are given by Equation (3.21) random shocks: to government spending, risk-premium, investment-specific and neutral technology, price and wage mark-up and monetary policy. All exogenous shocks, often with the

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exception of the monetary policy shock (assumed i.i.d.), are assumed to follow AR(1) or ARMA(1,1,) processes. Seven expectation (expectations channel) terms directly enter the model:

expectations about future consumption 𝐸_𝑡𝐶_𝑡+1, hours of work 𝐸_𝑡𝑙_𝑡+1, inflation 𝐸_𝑡𝜋_𝑡+1, investment 𝐸_𝑡𝑖_𝑡+1, value of capital 𝐸_𝑡𝑞_𝑡+1, rental rate of capital 𝐸_𝑡𝑟_𝑡+1^𝑘 , and wages 𝐸_𝑡𝑤_𝑡+1. The expectations are typically modelled as being formed according to the REH. The notation 𝐸_𝑡 in the model denotes model-consistent rational expectations, that is, the mathematical conditional expectation based on time 𝑡 information set and derived from the above equations.

The model features monopolistic competition in product and labour markets as well as nominal rigidities in prices and wages that allow for backward inflation indexation. In order to match the data various other features such as, habit formation, costs of adjustment in capital accumulation and capacity utilisation, are introduced to the DSGE model. The interest rate channel is assumed to be the main channel that influences the UK MTM with the presence of the credit channel to account for the financial stress. The inclusion of wage and price mark-ups highlights the price and wage rigidities that imply the changes in the nominal interest rate, affecting the real interest rate based on the decisions on the intertemporal allocation of consumption of the agents. As in SW (2007), consumption and leisure are treated separately and the standard Dixit-Stiglitz aggregator are used for prices and wages (as in CN, 2011) instead of the Kimball aggregator of SW (2007). The share of fixed cost in the production function is set to zero, and finally, the study assumes no steady state growth for the UK economy. The Taylor type interest rate rule is modified to fit the model and assumed to be implemented by the central bank:

𝑟_𝑡= 𝜌𝑟_𝑡−1+ (1 − 𝜌)[𝜌_𝜋𝜋_𝑡+ 𝜌_𝑦𝑦_𝑡^𝐺𝐷𝑃] + 𝜖_𝑡 (3.30) where 𝑦_𝑡^𝐺𝐷𝑃 is the weighted sum (with weights equal to the steady state shares) of real consumption, real investment and real government spending that represents real output. As there is sufficient size of the sample used for two periods, the model does not need to be simplified as in CN (2011) and reducing parameter space is not required because the time series is sufficient to respond to the prior parameter restrictions.