On the Estimation and Application of Structural Decompositions of the South African Business Cycle

Abstract

Alternative model specifications can affect the estimated structure and dynamics of the business cycle, which has important implications for policy analysis. This paper evaluates the consistency of model estimations in the extant literature and tests the sensitivity of alternative models tailored to the South African economy. We find that both parameter estimates and model dynamics are sensitive to model specification. Our findings suggest that a three-equation New-Keynesian model and a traditional open economy model provide qualitatively and quantitatively similar results to the benchmark medium-scale New-Keynesian model with sticky prices and wages, habit formation, and investment adjustment costs. However, significant differences from the benchmark New-Keynesian specification are revealed once financial frictions are included. In addition, the types of exogenous shocks included in the model are key determinants for the variation of results.

Appendices

Appendix 1: Ireland’s Three-Equation New-Keynesian Model

IS Curve

$$\displaystyle \begin{aligned} \lambda_t = i_t + \lambda_{t+1} - \pi_{t+1} \end{aligned} $$

(35)

New Keynesian Phillips Curve

$$\displaystyle \begin{aligned} \pi = \frac{\gamma_p}{(1 + \gamma_p\beta)}\pi_{t-1} + \frac{\beta}{(1 + \gamma_{p}\beta)}E_{t}[\pi_{t+1}] - \kappa_h(\lambda_t + \xi^b_t) + \xi^p_t ~, \end{aligned} $$

(36)

where $\kappa _h = \frac {(1-\theta _p)(1-\theta _p\beta )}{\theta _p(1 + \beta \gamma _p)}$.

Monetary Policy Rule

$$\displaystyle \begin{aligned}i_t = \rho_{i}i_{t-1} + \kappa_{\pi}\pi_t + \kappa_{y}\Delta{y}_t + \epsilon^i_t ~, \end{aligned} $$

(37)

where the growth rate of output is $\Delta {y}_t = y_t - y_{t-1} + \xi ^a_t$. The marginal utility of consumption with internal habit formation and log preferences (i.e., σ _c = 1):

$$\displaystyle \begin{aligned} (1 - \beta\phi)(1-\phi)\lambda_t = {\phi}y_{t-1} - (1 + \beta\phi^2)y_t + \beta{\phi}y_{t+1} + (1 - \beta\phi\rho_b)(1 - \phi)\xi^b_t - \phi\xi^a_t ~, \end{aligned} $$

(38)

where $\xi ^b_t$ is the preference shock. We do not include this shock in the other models. Instead, the dynamics and relative contribution are compared to two similar shocks in the SW-NK framework, namely the risk premium shock and the exogenous spending shock. All three of these shocks are variant aggregate demand shocks.⁴³ A quick comparison with the three-equation model of the foreign economy in HGW19 shows how similar these two linearized frameworks are (see Appendix 2 and Sect. 4). Another key difference is that $\xi ^a_t$ is a shock to the growth rate of output. A number of variables therefore need to be renormalized to remove the inherited unit root. Setting β = 0 in Eq. 38 gives the marginal utility of consumption with external habit:

$$\displaystyle \begin{aligned} \lambda_t = \frac{\phi}{(1-\phi)}y_{t-1} - \frac{1}{(1-\phi)}y_t + \xi^b_t - \phi\xi^a_t ~. \end{aligned} $$

(39)

The efficient level of output is

$$\displaystyle \begin{aligned}0 = {\phi}q_{t-1} - (1 + \beta\phi^2)q_{t} + \beta{\phi}q_{t+1} + \beta\phi(1 - \phi)(1- \rho^b)\xi^b_t - {\phi}\xi^a_t ~,\end{aligned}$$

where the output gap is: $y^{gap}_{t} = y_{t} - q_{t}$. The laws of motion for the preference shock, the (renormalized) cost-push shock, and the technology shock are: $\xi ^b_t = \rho ^b\xi ^b_{t-1} + \epsilon ^b_t$; $\xi ^p_t = \rho ^p\xi ^p_{t-1} + \epsilon ^p_t$; $\xi ^a_t = \epsilon ^a_t$.

Appendix 2: The Open Economy

The open economy system of equilibrium conditions shown here is a direct extension of the “benchmark” closed economy New-Keynesian model presented in Sect. 3. The corresponding SW-NK model has some slight differences particularly related to the investment-specific efficiency shock. Specifically, the shock in SW-NK directly affects the accumulation of investment and its efficiency (via an adjustment cost) in physical capital accumulation, whereas the shock in SOE-NK affects the adjustment cost only (i.e., the relative price of capital, or, as described in SW-NK, the arbitrage equation for the value of capital). Also, we shut off the capital utilization channel in SW-NK. The flexible price (efficient) equilibrium follows by setting all price and wage rigidities to zero.

1.1 Aggregate Demand

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{c}^{h}_{t} &\displaystyle =&\displaystyle \gamma_{c}\eta_c(\hat{rer}_{t} - \hat{\psi}^f_t) - (\gamma_{c}\eta_c)\hat{pr}^h_t + \hat{c}_t {} \end{array} \end{aligned} $$

(40)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{c}^{f}_{t} &\displaystyle =&\displaystyle (1-\gamma_c)(\eta_c)\hat{pr}^h_t + (\gamma_{c} - \eta_c)(\hat{rer}_{t} - \hat{\psi}^f_t) + \hat{c}_t {} \end{array} \end{aligned} $$

(41)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{c}_t &\displaystyle =&\displaystyle \frac{1}{(1+\phi)}\hat{c}_{t+1} + \frac{\phi}{(1+\phi)}\hat{c}_{t-1} - \frac{(1-\phi)}{\sigma_{c}(1+\phi)}(\hat{i}^b_t - \hat{\pi}_{t+1} + \hat{\mu}^{b}_t) {} \end{array} \end{aligned} $$

(42)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{rer}_{t} &\displaystyle =&\displaystyle \frac{\sigma_c}{1-\phi}(\hat{c}_t - {\phi}\hat{c}_{t-1}) - \frac{\sigma^{*}_c}{1-\phi^{*}}(\hat{c}^{*}_{t} - {\phi^{*}}\hat{c}^{*}_{t-1}) {} ~. \end{array} \end{aligned} $$

(43)

Equation 40 domestic consumption of home goods; Eq. 41 domestic consumption of foreign goods; Eq. 42 Euler equation; Eq. 43 is the international risk sharing condition (where $\hat {c}^{*}_{t} = \hat {y}^{*}_{t}$).⁴⁴

1.1.1 Investment Schedule

$$\displaystyle \begin{aligned} \hat{v}_{t} - \hat{k}_t = \beta{E_t}(\hat{v}_{t+1} - \hat{k}_{t+1}) + \frac{\beta{R^k}}{\kappa_{v}}E_{t}(\hat{r}^k_{t+1}) + \frac{\sigma_c}{\kappa_{v}}(\hat{c}_{t} - \hat{c}_{t+1}) + \frac{1}{\kappa_v}(\hat{\xi}^v_t - \beta{E_{t}}\hat{\xi}^v_{t+1}) ~, \end{aligned} $$

(44)

where R ^k = (1∕β − (1 − δ)) and $\hat {\xi }^v_t$ is the investment-specific shock, and $q^k_t = \kappa _v(v_t - k_t) - \tilde {\xi }^v_t$.

1.2 Aggregate Supply and Inflation

1.2.1 (Real) Wage-Setting Equation

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{w}_t &\displaystyle =&\displaystyle \Omega{\beta}E_{t}\hat{w}_{t+1} + {\Omega}\hat{w}_{t-1} + {\Omega}\Omega^{*}(\hat{mrs}_{t} - \hat{w}_{t}) \\ &\displaystyle &\displaystyle + \Omega{\beta}E_{t}\hat{\pi}_{t+1} - {\Omega}\hat{\pi}_{t} - {\Omega}\theta_{w}\beta\gamma_{w}\hat{\pi}_{t} + {\Omega}\gamma_{w}\hat{\pi}_{t-1} ~. \end{array} \end{aligned} $$

The real wage ($\hat {w}_t = w_t - p_t$) setting equation can be re-written in nominal wage inflation form as:

$$\displaystyle \begin{aligned} \hat{\pi}^{w}_{t} - \gamma_{w}\hat{\pi}_{t-1} = {\beta}E_{t}\hat{\pi}^{w}_{t+1} - {\theta_{w}\beta}\gamma_{w}\hat{\pi}_{t} + \Omega^{*}(\hat{mrs}_{t} - \hat{w}_{t}) , \end{aligned} $$

(45)

where $\Omega ^{*} = \frac {(1 - \theta _w)(1 - \theta _{w}\beta )}{\theta _{w}(1 + \xi ^{w}\sigma _{n})}$, $\Omega = \frac {1}{(1 + \beta )}$, and $\hat {mrs}_{t} = \frac {\sigma _{c}}{1 - \phi }(\hat {c}_t - {\phi }\hat {c}_{t-1}) + \sigma _{n}{\hat {n}_t}$.

1.2.2 Domestic Production and Inflation (for Consumption Goods)

$$\displaystyle \begin{aligned} \hat{\pi}^{h}_{t} = \frac{\gamma_{p}}{(1+\gamma_{p}{\beta})}\hat{\pi}^{h}_{t-1} + \frac{\beta}{(1+\gamma_{p}{\beta})}E_{t}\hat{\pi}^{h}_{t+1} + \kappa_{h}(\hat{mc}^{h}_{t} + \hat{\xi}^{p}_t) , \end{aligned} $$

(46)

where $\hat {mc}^{h}_{t} = \hat {\lambda }_{t}$ is the real marginal cost of production, and $\kappa _h = \frac {(1-\theta _p)(1-\theta _{p}{\beta })}{\theta _{p}(1+\gamma _{p}\beta )}$.

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hat{\lambda}_t &\displaystyle =&\displaystyle (\hat{w}_{t} - \hat{pr}^h_t) - (\hat{y}^{h}_t - \hat{n}_t) \end{array} \end{aligned} $$

(47)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{\lambda}_t &\displaystyle =&\displaystyle \hat{r}^{k}_t - (\hat{y}^{h}_t - \hat{k}_t) \end{array} \end{aligned} $$

(48)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{y}^{h}_{t} &\displaystyle =&\displaystyle \hat{\xi}^{a}_t + \alpha{\hat{k}_t} + (1-\alpha)\hat{n}_t ~. \end{array} \end{aligned} $$

(49)

1.2.3 Imported Inflation (for Foreign Consumption Goods)

$$\displaystyle \begin{aligned} \hat{\pi}^{f}_{t} = {\beta}E_{t}[\hat{\pi}^{f}_{t+1}] + \kappa_{f}\hat{\psi}^{f}_{t} , \end{aligned} $$

(50)

where $\kappa _f = \frac {(1-\theta _f)(1-\theta _{f}{\beta })}{\theta _{f}}$, and $\hat {\psi }^{f}_{t}$ measures the l.o.p gap⁴⁵:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{\psi}^{f}_{t} &\displaystyle =&\displaystyle \hat{\varepsilon}_{t} + \hat{p}^{f*}_t - \hat{p}^{f}_t , \\ {} &\displaystyle =&\displaystyle \hat{rer}_t - \hat{pr}^{f}_t ~. \end{array} \end{aligned} $$

(51)

1.2.4 Inflation Aggregation Equations

From the inflation aggregation equations we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{\pi}_{t} &\displaystyle =&\displaystyle (1-\gamma_{c})\hat{\pi}^{h}_t + \gamma_{c}\hat{\pi}^{f}_t ~. \end{array} \end{aligned} $$

Equation 52 can be re-written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} 0 &\displaystyle =&\displaystyle (1-\gamma_{c})\hat{pr}^{h}_t + \gamma_{c}(\hat{pr}^{f}_t) ~. {} \end{array} \end{aligned} $$

(52)

1.2.5 Evolution of Relative Prices

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hat{pr}^{h}_t &\displaystyle =&\displaystyle \hat{pr}^h_{t-1} + \hat{\pi}^h_t - \hat{\pi}_t \end{array} \end{aligned} $$

(53)

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hat{pr}^f_t &\displaystyle =&\displaystyle \hat{pr}^f_{t-1} + \hat{\pi}^f_t - \hat{\pi}_t \end{array} \end{aligned} $$

(54)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{rer}_{t} &\displaystyle =&\displaystyle \hat{rer}_{t-1} + \Delta{\hat{\varepsilon}_{t}} + \hat{\pi}^{f*}_{t} - \hat{\pi}_{t} \end{array} \end{aligned} $$

(55)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{s}_{t} &\displaystyle =&\displaystyle \hat{pr}^{f} - \hat{pr}^{h} \end{array} \end{aligned} $$

(56)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{w}_{t} &\displaystyle =&\displaystyle \hat{w}_{t-1} + \hat{\pi}^w_t - \hat{\pi}_t ~, \end{array} \end{aligned} $$

(57)

where Eq. 55 is the equation of motion for the relative purchasing power parity condition.⁴⁶ Here, we can think of nominal exchange rate changes $(\Delta {\hat {\varepsilon }_{t}})$ as the price adjustment mechanism that maintains equilibrium between foreign and domestic goods markets.

1.2.6 Evolution of Capital

$$\displaystyle \begin{aligned} \hat{k}_{t+1} = (1 - \delta)\hat{k}_t + \delta{\hat{v}_{t}}. \end{aligned} $$

(58)

1.3 Policy Rule

The central bank conducts monetary policy according to a Taylor (1993) rule:

$$\displaystyle \begin{aligned} \hat{i}^{b}_{t} = {\rho_{i}}\hat{i}^{b}_{t-1} + (1-\rho_{i})\kappa_{\pi}\hat{\pi}_t + (1-\rho_{i})\kappa_{y}(\hat{y}_t - \hat{y}_{t-1}) + \epsilon^{i}_t ~. \end{aligned} $$

(59)

The short-term nominal interest rate rises (falls) whenever inflation and output growth rise above (fall below) their average, or steady-state.

1.4 Foreign Economy

We assume a large open economy for the foreign market. This allows us to specify the foreign rate $\hat {i}^{b*}_t$, foreign inflation $\hat {\pi }^{*}_{t+1} = \hat {\pi }^{f*}_{t+1}$, and foreign consumption $\hat {y}^{*}_t = \hat {c}^{*}_t$ according to the standard three-equation New-Keynesian model, namely an IS curve, a Phillips curve, and a Taylor-type policy rate rule.

$$\displaystyle \begin{aligned} \hat{y}^{*}_t = \frac{1}{(1+\phi^{*})}\hat{y}^{*}_{t+1} + \frac{\phi^{*}}{(1+\phi^{*})}{\hat{y}}^{*}_{t-1} - \frac{(1-\phi^{*})}{\sigma^{*}_{c}(1+\phi^{*})}(\hat{i}^{b*}_t - E_{t}[\hat{\pi}^{*}_{t+1}] + \hat{\mu}^{b*}_t) \end{aligned} $$

(60)

$$\displaystyle \begin{aligned} \hat{\pi}^{*}_{t} = \frac{\gamma^{*}}{(1+\gamma^{*}{\beta})}\hat{\pi}^{*}_{t-1} + \frac{\beta}{(1+\gamma^{*}{\beta})}E_{t}[\hat{\pi}^{*}_{t+1}] + \kappa_{*}\hat{mc}^{*}_{t} , \end{aligned} $$

(61)

where $\hat {mc}^{*}_{t}$ is the real marginal cost of production, and $\kappa _{*} = \frac {(1-\theta _{*})(1-\theta _{*}{\beta })}{\theta _{*}(1+\gamma ^{*}\beta )}$.

$$\displaystyle \begin{aligned} \hat{mc}^{*}_{t} = \biggl( \frac{\sigma_{c}^{*}}{1-\phi^{*}} + {\sigma}^{*}_{n} \biggr)\hat{y}^{*}_{t} - \biggl( \frac{\sigma_{c}^{*}\phi^{*}}{1-\phi^{*}} \biggr)\hat{y}^{*}_{t-1} - (1+ \sigma^{*}_{n})\hat{a}^{*}_{t} , \end{aligned} $$

(62)

$$\displaystyle \begin{aligned} \hat{i}^{b*}_{t} = {\rho_{i*}}\hat{i}^{b*}_{t-1} + (1-\rho_{i*})\kappa^{*}_{\pi}\hat{\pi}^{*}_t + (1-\rho_{i*})\kappa^{*}_{y}(\hat{y}^{*}_t - \hat{y}^{*}_{t-1}) + \epsilon^{i*}_t. \end{aligned} $$

(63)

1.5 Aggregate Equilibrium

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{y}^{h}_{t} &\displaystyle =&\displaystyle \frac{C^h}{Y^h}\hat{c}^{h}_{t} + \frac{C^{h*}}{Y^h}\hat{c}^{h*}_{t} \\ &\displaystyle =&\displaystyle \frac{C^h}{Y^h}\hat{c}^{h}_{t} + \frac{(1-C^h)}{Y^h}(\hat{y}^{*}_{t} - {\xi}^{f*}(\hat{pr}^h_t - \hat{rer}_t)) ~,{} \end{array} \end{aligned} $$

(64)

where ξ ^f∗ is the foreign price elasticity of demand for domestic goods (i.e., the change in foreign demand for domestic goods given the foreign price of domestic goods relative to the foreign price of foreign goods).

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{y}_{t} &\displaystyle =&\displaystyle \frac{C}{Y}\hat{c}_{t} + \frac{V}{Y}\hat{v}_{t} + \frac{X}{Y}\hat{x}_{t} - \frac{M}{Y}\hat{m}_{t} \end{array} \end{aligned} $$

(65)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{x}_{t} = \hat{c}^{h*}_{t} &\displaystyle =&\displaystyle \hat{y}^{*}_{t} - {\xi}^{f*}(\hat{pr}^h_t - \hat{rer}_t) \end{array} \end{aligned} $$

(66)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \hat{m}_{t} &\displaystyle =&\displaystyle \frac{C^{f}}{M}\hat{c}^{f}_{t}. \end{array} \end{aligned} $$

(67)

1.6 Exogenous Shocks

We include 10 shocks in the model. For the domestic economy, the monetary policy shock $(\epsilon ^{i}_t)$, as given in Eq. 59, is i.i.d., whereas the domestic technology shock, the domestic price mark-up shock, the wage mark-up shock and the investment-specific shock follow AR(1) processes: $\hat {a}_t = \rho _{a}\hat {a}_{t-1} + \epsilon ^{a}_t$; $\hat {\xi }^{p}_t = \rho _{p}\hat {\xi }^{p}_{t-1} + \epsilon ^{p}_t$: $\hat {\xi }^{w}_t = \rho _{w}\hat {\xi }^{w}_{t-1} + \epsilon ^{w}_t$; $\hat {\xi }^{v}_t = \rho _{v}\hat {\xi }^{v}_{t-1} + \epsilon ^{v}_t$. Following Smets and Wouters (2007), we include the exogenous spending shock in the aggregate resource constraint: $\hat {\xi }^{g}_t = \rho _{g}\hat {\xi }^{g}_{t-1} + \epsilon ^{g}_t + \rho _{g,y}\epsilon ^{a}_t$.⁴⁷ Notably, ρ _g,y captures spillover effects from domestic technology shocks.⁴⁸ The foreign economy follows with an i.i.d. monetary policy shock $(\epsilon ^{i*}_t)$ and the following supply shock: $\hat {a}^{*}_t = \rho _{a*}\hat {a}^{*}_{t-1} + \epsilon ^{a*}_t$. In addition, the risk premium shocks on domestic-currency assets relative to the policy rate and for foreign-currency borrowing abroad (equivalent to negative demand shocks) are described as follows: $\hat {\mu }^{b*}_t = \rho _{b}\hat {\mu }^{b*}_{t-1} + \epsilon ^{b*}_{t}$ and $\hat {\mu }^{b}_t = \rho _{b}\hat {\mu }^{b}_{t-1} + \epsilon ^{b}_{t}$.

Appendix 3: The Financial Frictions Model

1.1 Households

Households’ Euler equation

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(\frac{1}{R^h} - \beta_h\right)\hat{\lambda}^h_{t} &\displaystyle =&\displaystyle \beta_h\left(\frac{\gamma}{1-\phi}(\hat{c}_{t+1} - {\phi}\hat{c}_{t}) + \hat{\pi}_{t+1}\right) \\ &\displaystyle &\displaystyle - \frac{1}{R^h}\left(\frac{\gamma}{1-\phi}(\hat{c}_{t} - {\phi}\hat{c}_{t-1})+ \hat{i}^h_t\right) . \end{array} \end{aligned} $$

(68)

Safe-assets demand

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{b}_{t} &\displaystyle =&\displaystyle \frac{\gamma}{(1-\phi)(1 - \beta_{h}R)}(\hat{c}_{t} - {\phi}\hat{c}_{t-1})\\ &\displaystyle &\displaystyle + \frac{\beta_{h}R}{1 - \beta_{h}R}\left(\hat{i}_t - \hat{\pi}_{t+1} - \frac{\gamma}{1-\phi}(\hat{c}_{t+1} - {\phi}\hat{c}_{t})\right) - \hat{\xi}_{b,t} , \end{array} \end{aligned} $$

(69)

where 1∕(1 − β _hR) is the asset-consumption ratio of households and is calibrated from the data as 0.856.

Equity price

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{q}^{\psi}_{t} &\displaystyle =&\displaystyle E_t\left[\hat{q}^{\psi}_{t+1} - \frac{\gamma}{1-\phi}(\hat{c}_{t+1} - {\phi}\hat{c}_{t})\right] + \frac{\gamma}{(1 - \Gamma_{\psi})(1-\phi)}(\hat{c}_{t} - {\phi}\hat{c}_{t-1}) \\ &\displaystyle &\displaystyle + ~ \frac{\Gamma_{\psi}}{1 - \Gamma_{\psi}}(\hat{\lambda}_{t} + \hat{\nu}_{h,t}) - \hat{\xi}_{\psi,t} , \end{array} \end{aligned} $$

(70)

where Γ_ψ = ((1∕R ^h) − β _h)ν _h(1 − ϕ _w).

Borrowing constraint

$$\displaystyle \begin{aligned} \hat{l}^h_t = \frac{\phi_w}{R^h}({\hat{w}}_t + \hat{n}_t) + \frac{(1 - \phi_w)}{R^h}\hat{q}^{\psi}_t - \hat{i}^h_t + \frac{1}{R^h}\hat{\nu}_{h,t} . \end{aligned} $$

(71)

The household flow of funds constraint

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{C^h}{Y}\hat{c}^{h}_{t} &\displaystyle =&\displaystyle \frac{(1-\alpha)}{X}(\hat{y}_t - \hat{x}_t) + \frac{B}{Y}R(\hat{i}_{t-1} + \hat{b}_{t-1} - \hat{\pi}_t) - \frac{B}{Y}b_t \\ &\displaystyle &\displaystyle + \frac{L^h}{Y}(\hat{l}^h_{t} - R^{h}\hat{i}^h_{t-1} - R^{h}\hat{l}^e_{t-1} + R^{h}\hat{\pi}_t) + \frac{Q^{\psi}\Psi}{Y}(\zeta_{\psi}\hat{q}_{t}^{\psi} -\hat{\xi}_{\psi,t}) \end{array} \end{aligned} $$

(72)

Note that the household’s optimality condition for labour supply enters in the union wage-setting equation (see Eq. 84). Specifically, in the flexible price (efficient) equilibrium the marginal rate of substitution will equal the real wage. To therefore accommodate the efficient equilibrium in our setup, we must use Eq. 72 to close the model.

1.2 Entrepreneurs in Firm Production

Labour demand

$$\displaystyle \begin{aligned} \hat{n}_t = \hat{y}_t - \hat{x}_t - {\hat{w}}_t . \end{aligned} $$

(73)

Entrepreneurs’ Euler equation

$$\displaystyle \begin{aligned} \left(\frac{1}{R^e} - \beta_e\right)\hat{\lambda}^e_{t} = \beta_e(\gamma^e(\hat{c}^e_{t+1}) + \hat{\pi}_{t+1}) - \frac{1}{R^e}({\gamma^e}(\hat{c}^e_{t})+ \hat{i}^e_t) . \end{aligned} $$

(74)

Investment schedule

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{v}_t - \hat{k}_{t} &\displaystyle =&\displaystyle \frac{\beta_{e}}{(1 - \Upsilon_k)}E_{t}[\hat{v}_{t+1} - \hat{k}_{t+1}] + \frac{(1 - \beta_{e}(1 - \delta_e) - \Upsilon_k)}{(1 - \Upsilon_k)\kappa_v}(\hat{y}_{t+1} - \hat{x}_{t+1} - \hat{k}_{t+1}) \\ &\displaystyle &\displaystyle + \frac{\Upsilon_k}{(1 - \Upsilon_k)\kappa_v}(\hat{\lambda}^{e}_{t} + \hat{\nu}_{e,t}) + \frac{\beta_{e}(1 - \delta_e)\gamma^e}{(1 - \Upsilon_k)\kappa_v}(\hat{c}^{e}_{t} - \hat{c}^{e}_{t+1}) + \hat{\xi}^{v}_{t} ~, \end{array} \end{aligned} $$

(75)

where Υ_k = ((1∕R ^e) − β _e)ν _eϕ _k and the shadow price of capital is

$$\displaystyle \begin{aligned} \hat{q}^k_t = {\kappa_v}(\hat{v}_{t} - \hat{k}_{t}) - \gamma^e{\hat{c}^e_t}, \end{aligned} $$

(76)

where Υ_k = 0 is the same as Iacoviello (2005, p. 760 (A3)).

Production function

$$\displaystyle \begin{aligned} \hat{y}_{t} = {\alpha}\hat{k}_{t} + {(1-\alpha)}\hat{h}_{t} + \hat{\xi}^{a}_{t} ~. \end{aligned} $$

(77)

Borrowing constraint

$$\displaystyle \begin{aligned} \hat{l}^e_t = \frac{\phi_k}{R^e}(\hat{q}^k_t + \hat{k}_{t}) + \frac{(1 - \phi_k)}{R^e}\hat{q}^{\psi}_t - \hat{i}^e_t + \frac{1}{R^e}\hat{\nu}_{e,t}. \end{aligned} $$

(78)

Capital accumulation

$$\displaystyle \begin{aligned} \hat{k}_{t+1} = (1-\delta_e)\hat{k}_{t} + \delta_{e}\hat{v}_t. \end{aligned} $$

(79)

The entrepreneur flow of funds constraint

$$\displaystyle \begin{aligned} \frac{C^e}{Y}\hat{c}^{e}_{t} = \frac{(1-\alpha)}{X}(\hat{y}_t - \hat{x}_t) + \frac{L^e}{Y}(\hat{l}^e_{t} - R^{e}\hat{i}^e_{t-1} - R^{e}\hat{l}^e_{t-1} + R^{e}\hat{\pi}_t) - \frac{\delta_{e}K}{Y}\hat{v}_t - \frac{Q^{\psi}\Psi^{e}}{Y}\zeta_{\psi}\hat{q}_{t}^{\psi}. \end{aligned} $$

(80)

1.3 Retailers and Labour Unions

The forward-looking Phillips curve with price indexation

$$\displaystyle \begin{aligned} \pi_t = \frac{\beta_R}{(1 + \beta_{R}\gamma_p)}E_{t}\pi_{t+1} + \frac{\gamma_p}{(1 + \beta_{R}\gamma_p)}\pi_{t-1} - \frac{(1-\theta_p)(1-\theta_p\beta_R)}{(1 + \beta_{R}\gamma_p)\theta_p}~{x}_t + \varepsilon^{p}_{t} . \end{aligned} $$

(81)

The forward-looking sticky (real) wage equation with price indexation (where $\hat {w}_t = w_t - p_t$)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{w}_t &\displaystyle =&\displaystyle \Phi{\beta}E_{t}\hat{w}_{t+1} + {\Phi}\hat{w}_{t-1} + {\Phi}\Phi^{*}({mrs}_{t} - \hat{w}_{t}) \\ &\displaystyle &\displaystyle + \Phi{\beta}E_{t}{\pi}_{t+1} - {\Phi}{\pi}_{t} - \Phi\theta_{w}\beta\gamma_{w}{\pi}_{t} + \Phi\gamma_{w}{\pi}_{t-1} ~, \end{array} \end{aligned} $$

(82)

where $\Phi ^{*} = \frac {(1 - \theta _w)(1 - \theta _{w}\beta )}{\theta _{w}(1 + \xi ^{w}\sigma _{n})}$, $\Phi = \frac {1}{(1 + \beta )}$. The marginal rate of substitution (mrs _t) follows from the households optimality condition for labour supply:

$$\displaystyle \begin{aligned} {\hat{mrs}}_t = {aa}\frac{\gamma}{1-\phi}(\hat{c}_{t} - {\phi}\hat{c}_{t-1}) + {\eta}\hat{n}_t - {aa}\left(\frac{1}{R^h} - \beta_h\right){\nu_h}{\phi_w}(\hat{\lambda}^h_{t} + \hat{\nu}_{h,t}) , \end{aligned} $$

(83)

where aa is ${1}/{(1 + \big (\frac {1}{R^h} - \beta _h\big ){\nu _h}{\phi _w})}$, and ${\hat {w}}_t$ is the real wage. Under no borrowing constraints, ${mrs}_{t} = \frac {\gamma }{1 - \phi }(c_t - \phi {c}_{t-1}) + \eta {h_t}$.

The real wage ($\hat {w}_t = w_t - p_t$) setting equation can be re-written in nominal wage inflation form as:

$$\displaystyle \begin{aligned} \hat{\pi}^{w}_{t} - \gamma_{w}\hat{\pi}_{t-1} = {\beta}E_{t}\hat{\pi}^{w}_{t+1} - {\theta_{w}\beta}\gamma_{w}\hat{\pi}_{t} + \Phi^{*}(\hat{mrs}_{t} - \hat{w}_{t}) ~. \end{aligned} $$

(84)

1.4 Banking Sector

Interbank rate

$$\displaystyle \begin{aligned} \hat{i}^{c}_t = \hat{i}_t - \frac{\kappa_k}{r}{\tau^3}(\hat{k}^B_t - \hat{l}_t - \xi_{\tau,t}). \end{aligned} $$

(85)

Bank capital accumulation

$$\displaystyle \begin{aligned} \hat{k}_t^B = (1-\delta_B)\hat{k}^B_{t-1} + \delta_B\hat{\omega}_{B,t-1} + \phi_{\psi}(\hat{q}^{\psi}_t - \hat{q}^{\psi}_{t-1}) - (1 - \phi_{\psi})\hat{\pi}_t. \end{aligned} $$

(86)

Profit function

$$\displaystyle \begin{aligned} \frac{\omega_B}{L}\hat{\omega}_{B,t} = r^h\frac{L^h}{L}(\hat{i}^h_{t} + \hat{l}^h_t) + r^e\frac{L^e}{L}(\hat{i}^e_{t} + \hat{l}^e_t) - r\frac{B}{L}(\hat{i}_{t} + \hat{b}_t) - \frac{Q^{\psi}\Psi_B}{L}{\zeta_\psi}(\hat{q}^{\psi}_{t}). \end{aligned} $$

(87)

Retail loan rate setting to households

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{i}^h_t &\displaystyle =&\displaystyle \frac{\kappa_h}{(1 - \nu_B)(\varepsilon^h - 1) + (1+\beta_B)\kappa_h}\hat{i}^h_{t-1} \\ &\displaystyle &\displaystyle + \frac{\beta_{B}\kappa_h}{(1 - \nu_B)(\varepsilon^h - 1) + (1+\beta_B)\kappa_h}E_t\hat{i}^h_{t+1} \\ &\displaystyle &\displaystyle + \frac{2(\varepsilon^h-1)}{(1 - \nu_B)(\varepsilon^h - 1) + (1+\beta_B)\kappa_h}\hat{i}^{c}_{t} \\ &\displaystyle &\displaystyle + \frac{(1 - \nu_B)(\varepsilon^h - 1)}{(1 - \nu_B)(\varepsilon^h - 1) + (1+\beta_B)\kappa_h}\mu_{h,t} , \qquad \end{array} \end{aligned} $$

(88)

where $\mu _{h,t} = {\varepsilon ^h_t}/({\varepsilon ^h_t-1})$ is the stochastic markup shock.

Retail loan rate setting to entrepreneurs

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{i}^e_t &\displaystyle =&\displaystyle \frac{\kappa_e}{(1 - \nu_B)(\varepsilon^e - 1) + (1+\beta_B)\kappa_e}\hat{i}^e_{t-1} \\&\displaystyle &\displaystyle + \frac{\beta_{B}\kappa_e}{(1 - \nu_B)(\varepsilon^e - 1) + (1+\beta_B)\kappa_e}E_t\hat{i}^e_{t+1} \\ &\displaystyle &\displaystyle + \frac{2(\varepsilon^e-1)}{(1 - \nu_B)(\varepsilon^e - 1) + (1+\beta_B)\kappa_e}\hat{i}^{c}_{t} \\&\displaystyle &\displaystyle + \frac{(1 - \nu_B)(\varepsilon^e - 1)}{(1 - \nu_B)(\varepsilon^e - 1) + (1+\beta_B)\kappa_e}\mu_{e,t} , \qquad \end{array} \end{aligned} $$

(89)

where $\mu _{e,t} = {\varepsilon ^e_t}/({\varepsilon ^e_t-1})$ is the stochastic markup shock.

Interbank spread and retail spread definitions (indexed by z = h, e)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{s}_t &\displaystyle =&\displaystyle \hat{i}^c_{t} - \hat{i}_t , \end{array} \end{aligned} $$

(90)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{s}^z_t &\displaystyle =&\displaystyle \hat{i}^z_{t} - \hat{i}^c_t . \end{array} \end{aligned} $$

(91)

1.5 Monetary Policy and Market Clearing Conditions

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{i}_t = {\kappa_i}\hat{i}_{t-1} + \kappa_{\pi}(1-\kappa_{i})\hat{\pi}_{t} + \kappa_{y}(1-\kappa_{i})(\hat{y}_t - \hat{y}_{t-1}) + \xi_{i,t} , \end{array} \end{aligned} $$

(92)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{y}_t = \frac{C}{Y}\hat{c}_t + \frac{C^e}{Y}\hat{c}^e_t + \delta_e\frac{K}{Y}\hat{v}_t + \frac{K^B}{Y}{\delta_B}\hat{k}_{t-1}^B , \end{array} \end{aligned} $$

(93)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{l}_t = \frac{L^h}{L}\hat{l}^h_t + \frac{L^e}{L}\hat{l}^e_t . \end{array} \end{aligned} $$

(94)

Appendix 4: Tables and Figures

See Table 6 and Figs. 19, 20, 21, 22, 23, and 24.

Table 6 Calibrated parameters

Appendix 5: Data and Sources

Data sources retrieved from the Federal Reserve Bank of St. Louis (FRED), the South African Reserve Bank (SARB), Eurostat, and OECD.stat:

• 1. Consumer Price Index of All Items in the USA [CPIAUCSL], UK [GBRCPIALL], Euro area [EZCCM086NEST], Japan [JPNCPIALL], and South Africa [ZAFCPIALL] retrieved from FRED (Copyright, 2018, OECD)

• 2. Real Gross Domestic Product by Expenditure for the USA [GDPC1], UK [GBQ661S], Euro area [EURSCAB1GQEA19], Japan [JPQ661S], and South Africa [ZAQ661S] retrieved from FRED (Copyright, 2018, OECD)

• 3. Interest Rates, Government Securities, 3-Month Treasury Bills for the USA [Gs3M], UK [GBM193N], Euro area [EZQ193N], Japan [JPM193N], and South Africa [ZAM193N] retrieved from FRED (Copyright, 2018, IMF and Eurostat)

• 4. Population: USA (Civilian Noninstitutional Population) [CNP16OV], Japan (15 and over) [JPQ647S], UK (Total) [POPNC; GBQ647S], and Euro area (Total) [POPNC; EZQ647S] (Copyright, 2018, OECD)

• 5. SARB, Final consumption expenditure by households: Total (PCE) [KBP6007L]

• 6. SARB, Gross fixed capital formation (Investment) [KBP6009L]

• 7. SARB, Nominal unit labour costs in the non-agricultural sectors [KBP7015L]

• 8. SARB, Total employment in the non-agricultural sectors [KBP7009L]

• 9. SARB, Balance of payments statistics [KBP5000L - KBP5010L]

• 10. SARB, Banking statistics