Terms of Trade Shocks and Monetary Policy in India

Abstract

Central banks in emerging market economies often grapple with understanding the monetary policy response to an inter-sectoral terms of trade shock. To address this, we develop a three sector closed economy NK-DSGE model calibrated to India. Our framework can be generalized to other emerging markets and developing economies. The model is characterized by a manufacturing sector and an agricultural sector. The agricultural sector is disaggregated into a grain and vegetable sector. The government procures grain from the grain market and stores it. We show that the procurement of grain leads to higher inflation, a change in the sectoral terms of trade, and a positive output gap because of a change in the sectoral allocation of labor. We compare the transmission of a single period positive procurement shock with a single period negative productivity shock and discuss the implications of such shocks for monetary policy setting. Our paper contributes to a growing literature on monetary policy in India and other emerging market economies.

Appendix

1.1 Derivation of the Demand Function of Each Variety of Good j: Eq. (11)

$$\begin{aligned}&\max _{C_{s,t}\left( j\right) }\left[ \int _{0}^{1}C_{s,t}\left( j\right) ^{ \frac{\theta -1}{\theta }}\;dj\right] ^{\frac{\theta }{\theta -1}} \text {subject to}\\&\qquad \int _{0}^{1}P_{s,t}\left( j\right) C_{s,t}\left( j\right) dj=Z_{s,t} \end{aligned}$$

for a given level of expenditure level, $Z_{s,t}.$ The above maximization problem can be written as the following Lagrangian,

$$\begin{aligned} \mathcal {L} =\left[ \int _{0}^{1}C_{s,t}\left( j\right) ^{\frac{\theta -1}{ \theta }}\;dj\right] ^{\frac{\theta }{\theta -1}}-\lambda _{t}\left( \int _{0}^{1}P_{s,t}\left( j\right) C_{s,t}\left( j\right) dj_{1}-Z_{s,t}\right) . \end{aligned}$$

The first-order condition is given by,

$$\begin{aligned} C_{s,t}^{\frac{1}{\theta }}C_{s,t}\left( j\right) ^{-\frac{1}{\theta } }=\lambda _{t}P_{s,t}\left( j\right) \end{aligned}$$

for all $\ j\in \left[ 0,1\right] .$ Using the above first order condition for any two varieties $j_{1},$ $j_{2}$ and eliminating $\lambda _{t}$ we get,

$$\begin{aligned} C_{s,t}\left( j_{1}\right) =C_{s,t}\left( j_{2}\right) \left( \frac{ P_{s,t}\left( j_{1}\right) }{P_{s,t}\left( j_{2}\right) }\right) ^{-\theta }. \end{aligned}$$

Now substituting $C_{s,t}\left( j_{1}\right)$ into $\int _{0}^{1}P_{s,t} \left( j_{1}\right) C_{s,t}\left( j_{1}\right)$ $dj_{1}=Z_{s,t}$ and putting $\left[ \int P_{s,t}\left( j_{1}\right) ^{1-\theta }\;dj_{1} \right] ^{\frac{1}{1-\theta }}=P_{s,t}$, the aggregate price index of sector s, we get

$$\begin{aligned} C_{s,t}\left( j_{2}\right) =\left( \frac{P_{s,t}\left( j_{2}\right) }{P_{s,t} }\right) ^{-\theta }\frac{Z_{s,t}}{P_{s,t}} \end{aligned}$$

for all $\ j_{2}\in \left[ 0,1\right] .$ Also, substituting the term, $C_{s,t}\left( j_{1}\right) ,$ in the expression, $\left[ \int _{0}^{1}C_{s,t} \left( j_{1}\right) ^{\frac{\theta -1}{\theta }}\;dj_{1}\right] ^{ \frac{\theta }{\theta -1}}=C_{s,t}$, we get

$$\begin{aligned} \int _{0}^{1}P_{s,t}\left( j_{2}\right) C_{s,t}\left( j_{2}\right) dj_{2}=P_{s,t}C_{s,t}=Z_{s,t}. \end{aligned}$$

Hence $C_{s,t}\left( j\right) =\left( \frac{P_{s,t}\left( j\right) }{P_{s,t}} \right) ^{-\theta }C_{s,t}$ for all $j\in \left[ 0,1\right]$ where $s=\textit{OG},$ V,  M.

1.2 Derivation of the Demand Function for Each Sector’s Good: Eqs. (7)–(10)

The optimization exercise is to,

$$\begin{aligned} \begin{array}{c} \displaystyle \max _{\{C_{A,t},C_{M,t}\}}\frac{\left( C_{A,t}\right) ^{\delta }\left( C_{M,t}\right) ^{1-\delta }}{\delta ^{\delta }(1-\delta )^{(1-\delta )}}\; \text {subject to }\\ \displaystyle P_{A,t}C_{A,t}+P_{M,t}C_{M,t}=Z_{t}, \end{array} \end{aligned}$$

for a given level of expenditure level, $Z_{t}.$ The above maximization problem can be written as the following Lagrangian,

$$\begin{aligned} \mathcal {L} =\frac{\left( C_{A,t}\right) ^{\delta }\left( C_{M,t}\right) ^{1-\delta }}{\delta ^{\delta }(1-\delta )^{(1-\delta )}}-\lambda _{t}\left( P_{A,t}C_{A,t}+P_{M,t}C_{M,t}-Z_{t}\right) . \end{aligned}$$

The first order conditions with respect to $C_{A,t}$ and $C_{M,t}$ are given by,

$$\begin{aligned} \frac{\delta \left( C_{A,t}\right) ^{\delta -1}\left( C_{M,t}\right) ^{1-\delta }}{\delta ^{\delta }(1-\delta )^{(1-\delta )}}= & {} \lambda _{t}P_{A,t} \\ \frac{(1-\delta )\left( C_{A,t}\right) ^{\delta }\left( C_{M,t}\right) ^{-\delta }}{\delta ^{\delta }(1-\delta )^{(1-\delta )}}= & {} \lambda _{t}P_{M,t} \end{aligned}$$

respectively. Eliminating $\lambda _{t},$ we get,

$$\begin{aligned} C_{M,t}=\frac{(1-\delta )}{\delta }C_{A,t}\left( \frac{P_{M,t}}{P_{A,t}} \right) ^{-1}. \end{aligned}$$

Now substituting the term, $C_{M,t},$ into the expression, $\frac{\left( C_{A,t}\right) ^{\delta }\left( C_{M,t}\right) ^{1-\delta }}{\delta ^{\delta }(1-\delta )^{(1-\delta )}},$ and setting $\left( P_{A,t}\right) ^{\delta }\left( P_{M,t}\right) ^{1-\delta }=P_{t}$, we obtain,

$$\begin{aligned} C_{A,t}=\delta \left( \frac{P_{A,t}}{P_{t}}\right) ^{-1}C_{t}. \end{aligned}$$

Put $C_{A,t}=\delta \left( \frac{P_{A,t}}{P_{t}}\right) ^{-1}C_{t}$ in the term, $C_{M,t}$, which gives

$$\begin{aligned} C_{M,t}=(1-\delta )\left( \frac{P_{M,t}}{P_{t}}\right) ^{-1}C_{t}. \end{aligned}$$

The above two equations can be re-written as

$$\begin{aligned} P_{A,t}C_{A,t}= & {} \delta P_{t}C_{t} \\ P_{M,t}C_{M,t}= & {} \left( 1-\delta \right) P_{t}C_{t} \end{aligned}$$

Adding the above two equations we get $P_{A,t}C_{A,t}+P_{M,t}C_{M,t}=P_{t}C_{t}$. Hence $Z_{t}=P_{t}C_{t}$. Similarly, maximizing $\frac{\left( C_{\textit{OG},t}\right) ^{(1-\mu )}\left( C_{V,t}\right) ^{\mu }}{\mu ^{\mu }(1-\mu )^{(1-\mu )}}$ subject to the constraint $P_{\textit{OG},t}C_{\textit{OG},t}+P_{V,t}C_{V,t}=Z_{A,t}$ we get Eqs. (9) and (10).

1.3 Derivation of the Euler and Labor Supply Equations ((13) and (14), respectively)

$$\begin{aligned} \max _{C_{t},N_{t},B_{t+1}}E_{0}\sum _{t=0}^{\infty }\left[ \frac{\left( \Gamma _{t}C_{t}\right) ^{1-\sigma }}{1-\sigma }-\frac{(N_{t})^{1+\psi }}{ 1+\psi }\right] \end{aligned}$$

subject to

$$\begin{aligned}&\int _{0}^{1}P_{\textit{OG},t}\left( j\right) C_{\textit{OG},t}\left( j\right) \; dj+\int _{0}^{1}P_{V,t}\left( j\right) C_{V,t}\left( j\right) \; dj+\int _{0}^{1}P_{M,t}\left( j\right) C_{M,t}\left( j\right) \; dj\\&\qquad +E_{t}\left\{ Q_{t+1}B_{t+1}\right\} = B_{t}+W_{t}N_{t}+T_{t}+Div_{t}. \end{aligned}$$

The Lagrangian for the above problem can be written as:

$$\begin{aligned} \mathcal {L}= & {} E_{0}\sum _{t=0}^{\infty }\beta ^{t}\left\{ \left[ \frac{\left( \Gamma _{t}C_{t}\right) ^{1-\sigma }}{1-\sigma }-\frac{(N_{t})^{1+\psi }}{ 1+\psi }\right] \right. \\&\left. -\,\lambda _{t}\left[ P_{t}C_{t}+E_{t}\left\{ Q_{t+1}B_{t+1}\right\} -B_{t}-W_{t}N_{t}-T_{t}-Div_{t}\right] \right\} . \end{aligned}$$

The first order conditions for $C_{t}$, $N_{t}$ and $B_{t+1}$ are given by:

$$\begin{aligned}&\displaystyle \frac{\partial \mathcal {L} }{\partial C_{t}} =\left( \Gamma _{t})^{1-\sigma }(C_{t}\right) ^{-\sigma }-\lambda _{t}P_{t}=0 \\&\displaystyle \frac{\partial \mathcal {L} }{\partial N_{t}} =-(N_{t})^{\psi }+\lambda _{t}W_{t}=0 \\&\displaystyle \frac{\partial \mathcal {L} }{\partial B_{t+1}} =-\beta ^{t}\lambda _{t}E_{t}\{Q_{t,t+1}\}+\beta ^{t+1}E_{t}\{\lambda _{t+1}\}=0, \end{aligned}$$

respectively. Using the first two conditions we get the labor supply Eq. (14), and using the first and the last condition we get the Euler equation (13). In the Euler equation, $R_{t}=\frac{1}{E_{t}\{Q_{t,t+1}\}}.$

1.4 Derivation of the Price Setting Equation: The Grain Sector Equation (27)

The optimization problem is given by,

$$\begin{aligned} \max _{P_{\textit{OG},t}\left( j\right) }\{P_{\textit{OG},t}\left( j\right) [Y_{\textit{OG},t}\left( j\right) +Y_{\textit{PG},t}]-\textit{MC}_{G,t}[Y_{\textit{OG},t}\left( j\right) +Y_{\textit{PG},t}]\}\; \end{aligned}$$

subject to the demand constraint

$$\begin{aligned} Y_{\textit{OG},t}\left( j\right) =\left( \frac{P_{\textit{OG},t}\left( j\right) }{P_{\textit{OG},t}} \right) ^{-\theta }Y_{\textit{OG},t}. \end{aligned}$$

The first order condition is given by:

$$\begin{aligned}&Y_{\textit{OG},t}\left( j\right) +Y_{\textit{PG},t}+P_{\textit{OG},t}\left( j\right) \frac{\partial Y_{\textit{OG},t}\left( j\right) }{\partial P_{\textit{OG},t}\left( j\right) }-\textit{MC}_{G,t}\frac{ \partial Y_{\textit{OG},t}\left( j\right) }{\partial P_{\textit{OG},t}\left( j\right) }=0.\\&\quad \text {Now }\frac{\partial Y_{\textit{OG},t}\left( j\right) }{\partial P_{\textit{OG},t}\left( j\right) } = -\,\theta \left( \frac{P_{\textit{OG},t}\left( j\right) }{P_{\textit{OG},t}} \right) ^{-\theta }\frac{1}{P_{\textit{OG},t}\left( j\right) }Y_{\textit{OG},t} =-\theta \frac{Y_{\textit{OG},t}\left( j\right) }{P_{\textit{OG},t}\left( j\right) } \end{aligned}$$

Simplifying we get,

$$\begin{aligned}&\displaystyle Y_{\textit{OG},t}\left( j\right) +Y_{\textit{PG},t}-\theta Y_{\textit{OG},t}\left( j\right) +\theta \textit{MC}_{G,t}\frac{Y_{\textit{OG},t}\left( j\right) }{P_{\textit{OG},t}\left( j\right) }=0,\\&\displaystyle \quad P_{\textit{OG},t}\left( j\right) \left( \left( 1-\theta \right) Y_{\textit{OG},t}\left( j\right) +Y_{\textit{PG},t}\right) =-\theta \textit{MC}_{G,t}Y_{\textit{OG},t}\left( j\right) ,\\&\displaystyle \quad P_{\textit{OG},t}\left( j\right) =\frac{\theta \textit{MC}_{G,t}}{\theta -1-\frac{Y_{\textit{PG},t}}{ Y_{\textit{OG},t}\left( j\right) }}. \end{aligned}$$

Similarly one can solve for the price setting equation in the vegetable sector as given in Eq. (28).

1.5 Derivation of the Price Setting Equation: Manufacturing Sector Equations (29) and (36)

The optimization problem is given by,

$$\begin{aligned} \max _{P_{M,t}^{*}\left( j\right) }E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}[P_{M,t}^{*}\left( j\right) Y_{M,t+k}\left( j\right) -\textit{MC}_{M,t+k}Y_{M,t+k}\left( j\right) ]\; \end{aligned}$$

subject to the demand constraint

$$\begin{aligned} Y_{M,t+k}\left( j\right) =\left( \frac{P_{M,t}^{*}\left( j\right) }{ P_{M,t+k}}\right) ^{-\theta }Y_{M,t+k}. \end{aligned}$$

The first order condition is given by:

$$\begin{aligned}&E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}\left[ Y_{M,t+k}\left( j\right) +P_{M,t}^{*}\left( j\right) \frac{\partial Y_{M,t+k}\left( j\right) }{\partial P_{M,t}^{*}\left( j\right) }-\textit{MC}_{M,t+k}\frac{ \partial Y_{M,t+k}\left( j\right) }{\partial P_{M,t}^{*}\left( j\right) } \right] =0\\&\quad \text {Now }\frac{\partial Y_{M,t+k}\left( j\right) }{\partial P_{M,t}^{*}\left( j\right) } =-\theta \left( \frac{P_{M,t}^{*}\left( j\right) }{ P_{M,t+k}}\right) ^{-\theta }\frac{1}{P_{M,t}^{*}\left( j\right) } Y_{M,t+k} =-\theta \frac{Y_{M,t+k}\left( j\right) }{P_{M,t}^{*}\left( j\right) }. \end{aligned}$$

Simplifying we get,

$$\begin{aligned}&\displaystyle E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}\left[ Y_{M,t+k}\left( j\right) -\theta Y_{M,t+k}\left( j\right) +\theta \textit{MC}_{M,t+k}\frac{ Y_{M,t+k}\left( j\right) }{P_{M,t}^{*}\left( j\right) }\right] =0,\\&\displaystyle P_{M,t}^{*}\left( j\right) E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}\left( 1-\theta \right) Y_{M,t+k}\left( j\right) =-E_{t}\sum _{k=0}^{\infty }\left( \beta \alpha _{M}\right) ^{t}\theta \textit{MC}_{M,t+k}Y_{M,t+k}\left( j\right) ,\\&\displaystyle P_{M,t}^{*}(j)=\frac{\theta }{\theta -1}\frac{E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}Y_{M,t+k}(j)\textit{MC}_{M,t+k}}{E_{t}\sum _{k=0}^{\infty }\alpha _{M}^{k}Q_{t,t+k}Y_{M,t+k}(j)}. \end{aligned}$$

We know that

$$\begin{aligned} P_{M,t}\equiv \left( \int _{0}^{1}P_{M,t}(j)^{1-\theta }dj\right) ^{\frac{1}{ 1-\theta }}, \end{aligned}$$

is the aggregate price index of this sector. Since demand for each variety of goods in this sector is symmetric and all firms revise their prices with a common maximization problem we can drop the $^{\prime }j^{\prime }$ so that $P_{M,t}^{*}\left( j\right) =P_{M,t}$ for all j. For all the firms who do not get to choose their prices $P_{M,t}\left( j\right) =P_{M,t-1}\left( j\right) .$ Hence, the aggregate price index can be written as

$$\begin{aligned} P_{M,t}^{1-\theta }=\int _{0}^{1}P_{M,t}\left( j\right) ^{1-\theta }dj=(1-\alpha _{M})(P_{M,t}^{*})^{1-\theta }+\alpha _{M}\int _{0}^{1}P_{M,t-1}\left( j\right) ^{1-\theta }dj. \end{aligned}$$

Note that the expression, $\alpha _{M}\int _{0}^{1}P_{M,t-1}\left( j\right) ^{1-\theta }dj,$ is simply a subset of prices in $t-1$, with each price appearing in the period t distribution of unchanged prices with the same relative frequency as in the period $t-1$ price distribution (Woodford 2003, Chap. 3). Therefore,

$$\begin{aligned} P_{M,t}=\left[ (1-\alpha _{M})(P_{M,t}^{*})^{1-\theta }+\alpha _{M}(P_{M,t-1})^{1-\theta }\right] ^{\frac{1}{\left( 1-\theta \right) }}. \end{aligned}$$

1.6 Market Clearing: Derivation for Eq. (35)

Equation (34) can be re-written as,

$$\begin{aligned} Y_{t}= & {} C_{t}+\frac{P_{\textit{OG},t}}{P_{t}}Y_{\textit{PG},t} \\= & {} C_{t}+\frac{P_{\textit{OG},t}}{P_{A,t}}\frac{P_{A,t}}{P_{t}}Y_{\textit{PG},t} \\= & {} C_{t}+\frac{P_{\textit{OG},t}}{\left( P_{\textit{OG},t}\right) ^{1-\mu }\left( P_{V,t}\right) ^{\mu }}\frac{P_{A,t}}{\left( P_{A,t}\right) ^{\delta }\left( P_{M,t}\right) ^{1-\delta }}Y_{\textit{PG},t} \\= & {} C_{t}+\left( T_{\textit{OGV},t}\right) ^{\mu }\left( T_{\textit{AM},t}\right) ^{(1-\delta )}Y_{\textit{PG},t}. \end{aligned}$$

1.7 Derivation of Steady States: Sect. 3.2

Using the fact that $Q_{t,t+k}=\beta ^{k}\left( \frac{\Gamma _{t+1}}{\Gamma _{t}}\right) ^{1-\sigma }\left( \frac{C_{t+1}}{C_{t}}\right) ^{-\sigma }\left( \frac{P_{t}}{P_{t+1}}\right)$, in the steady state $Q_{t,t+k}=\beta ^{k}$. Thus Eqs. (29) and (31) in the steady state can be written as,

$$\begin{aligned} P_{M}^{*} =\frac{\theta }{\theta -1}\frac{E_{t}\mathop {\displaystyle \sum }\limits _{t=0}^{\infty }(\beta \alpha _{M})^{t}Y_{M}{} \textit{MC}_{M}}{E_{t}\mathop {\displaystyle \sum }\limits _{t=0}^{\infty }(\beta \alpha _{M})^{t}Y_{M}}, \quad P_{M}^{*} =\frac{\theta }{\theta -1}{} \textit{MC}_{M}, \end{aligned}$$

and

$$\begin{aligned} (P_{M})^{1-\theta }=\alpha _{M}\left( P_{M}\right) ^{1-\theta }+(1-\alpha _{M})\left( P_{M}^{*}\right) ^{1-\theta }\;\text {respectively.} \end{aligned}$$

The above equation implies,

$$\begin{aligned} P_{M}^{*} = P_{M} = \frac{\theta }{\theta -1}{} \textit{MC}_{M}. \end{aligned}$$

Similarly considering the price setting equation in the grain sector,

$$\begin{aligned} P_{\textit{OG}}=\frac{\theta \left( 1-c_{p}\right) }{\left( \theta -1\right) \left( 1-c_{p}\right) -c_{p}}{} \textit{MC}_{G},\quad \text { where }c_{p}=\frac{Y_{\textit{PG}}}{Y_{G}}, \end{aligned}$$

and in the vegetable sector,

$$\begin{aligned} P_{V}=\frac{\theta }{\theta -1}{} \textit{MC}_{V}. \end{aligned}$$

The aggregate price index at the steady state is:

$$\begin{aligned} P=(P_{\textit{OG}})^{(1-\mu )\delta }\left( P_{V}\right) ^{\mu \delta }\left( P_{M}\right) ^{1-\delta }. \end{aligned}$$

Using Eq. (22), $\textit{MC}_{s}=W$ for $s=G,$ V,  M,  as $A_{s}=1$. Substituting these values in the above aggregate price index we get,

$$\begin{aligned} P= & {} \left( \frac{\left( \theta -1\right) \left( 1-c_{p}\right) }{\left( \theta -1\right) \left( 1-c_{p}\right) -c_{p}}\right) ^{(1-\mu )\delta } \frac{\theta }{\theta -1}W. \\ P= & {} \gamma ^{-(1-\mu )\delta }\frac{\theta }{\theta -1}W\quad \text { where}\; \gamma =\frac{\left( \theta -1\right) \left( 1-c_{p}\right) -c_{p}}{\left( \theta -1\right) \left( 1-c_{p}\right) }. \end{aligned}$$

Since, $P_{M}=P_{V}=\frac{\theta }{\theta -1}W$ and $P_{\textit{OG}}=\frac{\theta \left( 1-c_{p}\right) }{\left( \theta -1\right) \left( 1-c_{p}\right) -c_{p}} W,$

$$\begin{aligned} \frac{P_{V}}{P} = \frac{P_{M}}{P}=\gamma ^{(1-\mu )\delta }\;\text {and}\;\frac{P_{\textit{OG}}}{P} = \gamma ^{(1-\mu )\delta -1}. \end{aligned}$$

Now from the demand functions,

$$\begin{aligned}&\displaystyle \frac{C_{\textit{OG}}}{C} =\frac{(1-\mu )\delta P}{P_{\textit{OG}}} = (1-\mu )\delta \gamma ^{-\delta \left( 1-\mu \right) +1}\\&\displaystyle \frac{C_{V}}{C} =\frac{\mu \delta P}{P_{V}} = \mu \delta \gamma ^{-\delta \left( 1-\mu \right) },\;\text {and,} \\&\displaystyle \frac{C_{M}}{C} =\frac{(1-\delta )P}{P_{M}} = \left( 1-\delta \right) \gamma ^{-\delta \left( 1-\mu \right) }. \end{aligned}$$

We can re-write the steady state labor supply Eq. (36) in the steady state as,

$$\begin{aligned} N= & {} N_{\textit{OG}}+N_{\textit{PG}}+N_{V}+N_{M} \\= & {} \frac{Y_{\textit{OG}}}{A_{G}}+\frac{Y_{\textit{PG}}}{A_{G}}+\frac{Y_{V}}{A_{V}}+\frac{Y_{M} }{A_{M}} \\= & {} C_{\textit{OG}}+C_{V}+C_{M}+Y_{\textit{PG}}\;\;\text {(Goods Market Equilibrium).} \end{aligned}$$

Using the above values from the steady state consumption demands,

$$\begin{aligned} N=\gamma ^{-\delta (1-\mu )}\left[ 1+\left( \gamma -1\right) (1-\mu )\delta \right] C+Y_{\textit{PG}} \end{aligned}$$

1.8 Derivation of the Log-Linearized Model: Eqs. (39), (40), (41a), (36), (47) and (51) in Sect. 3.3

Equation (39): Using a first order Taylor approximation in Eq. (13) yields,

$$\begin{aligned} E_{t}\left\{ \begin{array}{l} \beta R+\beta R\left( \frac{R_{t+1}-R}{R}\right) +\left( 1-\sigma \right) \beta R\left( \frac{\Gamma _{t+1}-\Gamma }{\Gamma }\right) \\ -\,\left( 1-\sigma \right) \beta R\left( \frac{\Gamma _{t}-\Gamma }{\Gamma } \right) -\sigma \beta R\left( \frac{C_{t+1}-C}{C}\right) +\sigma \beta R\left( \frac{C_{t}-C}{C}\right) \\ +\,\beta R\left( \frac{C_{t}-C}{C}\right) +\beta R\left( \frac{P_{t}-P}{P} \right) -\beta R\left( \frac{P_{t+1}-P}{P}\right) \end{array} \right\} \approx 1. \end{aligned}$$

Now for variable $X_{t,}\frac{X_{t}-X}{X}\approx \ln \left( X_{t}\right) -\ln \left( X\right) \approx \widehat{X}_{t}$. Using the steady state value of Euler Equation, $\beta R=1,$ we get

$$\begin{aligned} E_{t}\left\{ \widehat{R}_{t}+\left( 1-\sigma \right) \widehat{\Gamma } _{t+1}-\left( 1-\sigma \right) \widehat{\Gamma }_{t}-\sigma \widehat{C} _{t+1}+\sigma \widehat{C}_{t}+\widehat{P}_{t}-\widehat{P}_{t+1}\right\} \approx 0. \end{aligned}$$

Re-arranging terms and using $\widehat{P}_{t+1}-\widehat{P}_{t}=\pi _{t+1},$ we get

$$\begin{aligned} \widehat{C}_{t}=E_{t}\{\widehat{C}_{t+1}\}-\frac{1}{\sigma }[(\widehat{R} _{t}-E_{t}\{\pi _{t+1}\})+(1-\sigma )E_{t}\{\Delta \widehat{\Gamma } _{t+1}\}]. \end{aligned}$$

Equation (40): Using a first order Taylor approximation in Eq. (14), we have

$$\begin{aligned}&\frac{N^{\psi }}{\Gamma ^{1-\sigma }C^{-\sigma }}+\psi \frac{N^{\psi }}{ \Gamma ^{1-\sigma }C^{-\sigma }}\left( \frac{N_{t+1}-N}{N}\right) \\&\qquad -\left( 1-\sigma \right) \frac{N^{\psi }}{\Gamma ^{1-\sigma }C^{-\sigma }}\left( \frac{\Gamma _{t}-\Gamma }{\Gamma }\right) +\sigma \frac{N^{\psi }}{\Gamma ^{1-\sigma }C^{-\sigma }}\left( \frac{C_{t}-C}{C}\right) \\&\quad \approx \frac{W}{P}+\frac{W}{P}\left( \frac{W_{t}-W}{W}\right) -\frac{W}{P} \left( \frac{P_{t}-P}{P}\right) . \end{aligned}$$

This implies that,

$$\begin{aligned} \widehat{W}_{t}-\widehat{P}_{t}=\psi \widehat{N}_{t}+\sigma \widehat{C} _{t}-(1-\sigma )\widehat{\Gamma }_{t} \end{aligned}$$

Equation (41a): Using a first order Taylor approximation of Eq. (23a), we get

$$\begin{aligned}&\textit{mc}_{G}+\textit{mc}_{G}\left( \frac{\textit{mc}_{G,t}-\textit{mc}_{G}}{\textit{mc}_{G}}\right) \approx \frac{1}{ A_{G}}\frac{W}{P}\left( T_{\textit{AM}}\right) ^{-\left( 1-\delta \right) }\left( T_{\textit{OGV}}\right) ^{-\mu }\\&\qquad -\frac{1}{A_{G}}\frac{W}{P}\left( T_{\textit{AM}}\right) ^{-\left( 1-\delta \right) }\left( T_{\textit{OGV}}\right) ^{-\mu }\\&\qquad \left[ \left( \frac{A_{G,t}-A_{G}}{A_{G}}\right) +\left( \frac{W_{t}-W}{W} \right) -\left( \frac{P_{t}-P}{P}\right) \right. \\&\qquad \quad \left. -\left( 1-\delta \right) \left( \frac{T_{\textit{AM},t}-T_{\textit{AM}}}{T_{AM}}\right) -\mu \left( \frac{T_{\textit{OG},V,t}-T_{\textit{OG},V}}{ T_{\textit{OG},V}}\right) \right] . \end{aligned}$$

Simplifying the above expression using the steady state expression, $\textit{mc}_{G}=\frac{1}{A_{G}}\frac{W}{P}\left( T_{\textit{AM}}\right) ^{-\left( 1-\delta \right) }\left( T_{\textit{OGV}}\right) ^{-\mu },$ we get

$$\begin{aligned} \widehat{mc}_{G,t}=\widehat{W}_{t}-\widehat{P}_{t}-\widehat{A} _{G,t}-(1-\delta )\widehat{T}_{\textit{AM},t}-\mu \widehat{T}_{\textit{OGV},t}. \end{aligned}$$

We can derive (41b) and (41c) in a similar way.

The log-linearized sectoral employment equations can be obtained by taking a first order Taylor approximation of Eq. (26) and noting that $N_{G,t}=\frac{1}{A_{G,t}}\left\{ Y_{\textit{PG},t}+ Y_{\textit{OG},t}Z_{\textit{OG},t}\right\} ,$ where a first order approximation to the dispersion term, $\widehat{Z}_{s,t}\approx 0$ (for details see Galí 2008, Chap. 3).

Note that:

$$\begin{aligned} \frac{P_{t}}{P_{A,t}}= & {} \frac{\left( P_{A,t}\right) ^{\delta }\left( P_{M,t}\right) ^{1-\delta }}{P_{A,t}}=\left( \frac{P_{A,t}}{P_{M,t}}\right) ^{-(1-\delta )}=\left( T_{\textit{AM},t}\right) ^{-(1-\delta )} \\ \frac{P_{t}}{P_{M,t}}= & {} \frac{\left( P_{A,t}\right) ^{\delta }\left( P_{M,t}\right) ^{1-\delta }}{P_{M,t}}=\left( \frac{P_{A,t}}{P_{M,t}}\right) ^{\delta }=\left( T_{\textit{AM},t}\right) ^{\delta } \\ \frac{P_{A,t}}{P_{\textit{OG},t}}= & {} \frac{\left( P_{\textit{OG},t}\right) ^{1-\mu }\left( P_{V,t}\right) ^{\mu }}{P_{\textit{OG},t}}=\left( \frac{P_{\textit{OG},t}}{P_{V,t}}\right) ^{-\mu }=\left( T_{\textit{OGV},t}\right) ^{-\mu } \\ \frac{P_{A,t}}{P_{V,t}}= & {} \frac{\left( P_{\textit{OG},t}\right) ^{1-\mu }\left( P_{V,t}\right) ^{\mu }}{P_{V,t}}=\left( \frac{P_{\textit{OG},t}}{P_{V,t}}\right) ^{1-\mu }=\left( T_{\textit{OGV},t}\right) ^{1-\mu }. \end{aligned}$$

We use the above four equations to re-write the demand functions $C_{\textit{OG},t},$ $C_{M,t},$ $C_{V,t}$ in terms of $C_{t}$ and the terms of trade terms $\left( T_{\textit{AM},t} \mathrm{and} T_{\textit{OGV},t}\right)$. Using the goods market equilibrium and the demand functions it is easy to derive Eqs. (43a)–(43c) using a first order Taylor’s approximation. Log linearization of the aggregate goods market clearing Eq. (35), gives us,

$$\begin{aligned} Y+Y\frac{\left( Y_{t}-Y\right) }{Y}\approx & {} C+\left( T_{\textit{OGV}}\right) ^{\mu }\left( T_{\textit{AM}}\right) ^{1-\delta }Y_{\textit{PG}}+\frac{\left( C_{t}-C\right) }{C}C \\&+\,\mu \left( T_{\textit{OGV}}\right) ^{\mu -1}\left( T_{\textit{AM}}\right) ^{1-\delta }Y_{\textit{PG}} \frac{\left( T_{\textit{OGV},t}-T_{\textit{OGV}}\right) }{T_{\textit{OGV}}}T_{OGV} \\&+\left( 1-\delta \right) \left( T_{\textit{OGV}}\right) ^{\mu }\left( T_{\textit{AM}}\right) ^{-\delta }Y_{\textit{PG}}\frac{\left( T_{\textit{AM},t}-T_{\textit{AM}}\right) }{T_{\textit{AM}}}T_{AM} \\&+\left( T_{\textit{OGV}}\right) ^{\mu }\left( T_{\textit{AM}}\right) ^{1-\delta }\frac{ \left( Y_{\textit{PG},t}-Y_{\textit{PG}}\right) }{Y_{\textit{PG}}}Y_{PG}\\ \widehat{Y}_{t}= & {} \frac{C}{Y}\widehat{C}_{t}+\frac{\left( T_{\textit{OGV}}\right) ^{\mu }\left( T_{\textit{AM}}\right) ^{1-\delta }Y_{\textit{PG}}}{Y}\left[ \mu \widehat{T} _{\textit{OGV},t}+\left( 1-\delta \right) \widehat{T}_{\textit{AM},t}+\widehat{Y}_{\textit{PG},t}\right] \end{aligned}$$

Note

$$\begin{aligned} \frac{\left( T_{\textit{OGV}}\right) ^{\mu }\left( T_{\textit{AM}}\right) ^{1-\delta }Y_{\textit{PG}}}{Y }=\frac{\gamma ^{-\mu }\gamma ^{-\left( 1-\mu \right) \left( 1-\delta \right) }}{Y}Y_{\textit{PG}}=\gamma ^{\left[ \delta (1-\mu )-1\right] }c_{p}s_{g}=\lambda _{c} \end{aligned}$$

and

$$\begin{aligned} \frac{C}{Y}=1-\lambda _{c}. \end{aligned}$$

Therefore,

$$\begin{aligned} \widehat{Y}_{t}=(1-\lambda _{c})\widehat{C}_{t}+\lambda _{c}\left[ \mu \widehat{T}_{\textit{OGV},t}+\left( 1-\delta \right) \widehat{T}_{\textit{AM},t}+\widehat{Y} _{\textit{PG},t}\right] . \end{aligned}$$

Equation (36) can be written as,

$$\begin{aligned} N_{t}= & {} N_{\textit{OG},t}+N_{\textit{PG},t}+N_{V,t}+N_{M,t}, \\ N_{t}= & {} \frac{Y_{\textit{OG},t}}{A_{G}}+\frac{Y_{\textit{PG},t}}{A_{G}}+\frac{Y_{V,t}}{A_{V,t} }+\frac{Y_{M,t}Z_{M,t}}{A_{M,t}}. \end{aligned}$$

Log linearizing equation (36), we get

$$\begin{aligned}&N+N\left( \frac{N_{t}-N}{N}\right) \\&\quad \approx \frac{Y_{\textit{OG}}}{A_{G}}+\frac{ Y_{\textit{PG}}}{A_{G}}+\frac{Y_{V}}{A_{V}}+\frac{Y_{M}}{A_{M}}+\frac{Y_{\textit{OG}}}{A_{G,t}} \left[ \left( \frac{Y_{\textit{OG},t}-Y_{\textit{OG}}}{Y_{OG}}\right) -\left( \frac{ A_{G,t}-A_{G}}{A_{G}}\right) \right] \\&\qquad +\,\frac{Y_{\textit{PG}}}{A_{G,t}}\left[ \left( \frac{Y_{\textit{PG},t}-Y_{\textit{PG}}}{Y_{PG}}\right) -\left( \frac{A_{G,t}-A_{G}}{A_{G}}\right) \right] \\&\qquad +\,\frac{Y_{V}}{A_{V}}\left[ \left( \frac{Y_{V,t}-Y_{V}}{Y_{V}}\right) -\left( \frac{A_{V,t}-A_{V}}{A_{V}}\right) \right] \\&\qquad +\,\frac{Y_{M}Z_{M}}{A_{M,t}}\left[ \left( \frac{Y_{M,t}-Y_{M}}{Y_{M}} \right) +\left( \frac{Z_{M,t}-Z_{M}}{Z_{M}}\right) -\left( \frac{ A_{M,t}-A_{M}}{A_{M}}\right) \right] . \end{aligned}$$

Using $Z_{M}=1$ and $\widehat{Z}_{M,t}\approx 0$ (as shown in Galí 2008), we get

$$\begin{aligned} N\widehat{N}_{t}= & {} Y_{\textit{OG}}\left( \widehat{Y}_{\textit{OG},t}-\widehat{A}_{G,t}\right) +Y_{\textit{PG}}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) +Y_{V}\left( \widehat{Y}_{V,t}-\widehat{A}_{V,t}\right) \\&+\,Y_{M}\left( \widehat{Y}_{M,t}- \widehat{A}_{M,t}\right) \\ N\widehat{N}_{t}= & {} C_{\textit{OG}}\left( \widehat{C}_{\textit{OG},t}-\widehat{A}_{G,t}\right) +Y_{\textit{PG}}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) \\&+\,C_{V}\left( \widehat{C}_{V,t}-\widehat{A}_{V,t}\right) +C_{M}\left( \widehat{C}_{M,t}- \widehat{A}_{M,t}\right) . \end{aligned}$$

Using steady state Eqs. (37a)–(37b) in Sect. 3.2, we get

$$\begin{aligned} N\widehat{N}_{t}= & {} \gamma ^{-\delta \left( 1-\mu \right) }\left[ (1-\mu )(\gamma -1)\delta \left( \widehat{C}_{\textit{OG},t}-\widehat{A}_{G,t}\right) +\mu \delta \left( \widehat{C}_{V,t}-\widehat{A}_{V,t}\right) \right. \\&\left. +\,(1-\delta )\left( \widehat{C}_{M,t}-\widehat{A}_{M,t}\right) \right] C +Y_{\textit{PG}}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) \\ N\widehat{N}_{t}= & {} \gamma ^{-\delta \left( 1-\mu \right) }\left[ \widehat{C} _{t}-\widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}- \widehat{A}_{G,t}\right) \right] C\\&+Y_{\textit{PG}}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) \\ \text {where}\;\widehat{C}_{t}= & {} (1-\mu )\delta \widehat{C}_{\textit{OG},t}+\mu \delta \widehat{C}_{V,t}+(1-\delta )\widehat{C}_{M,t} \\ \widehat{A}_{t}= & {} (1-\mu )\delta \widehat{A}_{G,t}+\mu \delta \widehat{A} _{V,t}+(1-\delta )\widehat{A}_{M,t}. \end{aligned}$$

Using Eq. (38),

$$\begin{aligned} \widehat{N}_{t}=\frac{\gamma ^{-\delta \left( 1-\mu \right) }\left[ \widehat{C}_{t}-\widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}- \widehat{A}_{G,t}\right) \right] C+Y_{\textit{PG}}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) }{\gamma ^{-\delta \left( 1-\mu \right) }\left[ 1+(1-\mu )(\gamma -1)\delta \right] C+Y_{\textit{PG}}}. \end{aligned}$$

Using (35) at the steady state, $Y=C+\frac{P_{\textit{OG}}}{P} Y_{\textit{PG}},$

$$\begin{aligned} \frac{Y_{\textit{PG}}}{C}= & {} \frac{Y_{\textit{PG}}}{Y-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }Y_{\textit{PG}}}=\frac{\frac{Y_{\textit{PG}}}{Y_{G}}}{\frac{Y-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }Y_{\textit{PG}}}{Y_{G}}} \\= & {} \frac{c_{p}s_{g}}{1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }c_{p}s_{g}}\;\;\text {where }s_{g}=\frac{Y_{G}}{Y},c_{p}=\frac{Y_{\textit{PG}}}{Y_{G}}.\\ \widehat{N}_{t}= & {} \frac{\left( 1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }c_{p}s_{g}\right) \gamma ^{-\delta \left( 1-\mu \right) } \left[ \widehat{C}_{t}-\widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}-\widehat{A}_{G,t}\right) \right] }{\gamma ^{-\delta \left( 1-\mu \right) }\left[ 1+(1-\mu )\delta \gamma \right] \left( 1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }c_{p}s_{g}\right) +c_{p}s_{g}} \\&+\frac{c_{p}s_{g}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) }{\gamma ^{-\delta \left( 1-\mu \right) }\left[ 1+(1-\mu )\delta \gamma \right] \left( 1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }c_{p}s_{g}\right) +c_{p}s_{g}}.\\ \widehat{N}_{t}= & {} \Theta _{1}\left[ \widehat{C}_{t}-\widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}-\widehat{A}_{G,t}\right) \right] +\Theta _{2}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) , \\ \text {where}\;\Theta _{1}= & {} \frac{\left( 1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }c_{p}s_{g}\right) \gamma ^{-\delta \left( 1-\mu \right) }}{\gamma ^{-\delta \left( 1-\mu \right) }\left[ 1+(1-\mu )(\gamma -1)\delta \right] \left( 1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1 \right] }c_{p}s_{g}\right) +c_{p}s_{g}} \\ \Theta _{2}= & {} \frac{c_{p}s_{g}}{\gamma ^{-\delta \left( 1-\mu \right) } \left[ 1+(1-\mu )(\gamma -1)\delta \right] \left( 1-\gamma ^{\left[ \delta \left( 1-\mu \right) -1\right] }c_{p}s_{g}\right) +c_{p}s_{g}}. \end{aligned}$$

Equation (47) is the New-Keynesian Phillips Curve for the manufacturing sector derived by log-linearizing (29) and (31) (for details see Galí 2008, Chap. 3).

Equation (51): Log-linearizing real marginal cost, $\textit{mc}_{G,t},$ as in (27), and using a first order Taylor approximation we get

$$\begin{aligned} \textit{mc}_{G,t}= & {} \frac{\theta -1}{\theta }-\frac{Y_{\textit{PG},t}}{\theta Y_{\textit{OG},t}}\\ \textit{mc}_{G}+\textit{mc}_{G}\left( \frac{\textit{mc}_{G,t}-\textit{mc}_{G}}{\textit{mc}_{G}}\right)\approx & {} \frac{ \theta -1}{\theta }-\frac{Y_{\textit{PG}}}{\theta Y_{\textit{OG}}}+\frac{Y_{\textit{PG}}}{\theta Y_{\textit{OG}}} \left( \frac{Y_{\textit{OG},t}-Y_{\textit{OG}}}{Y_{OG}}\right) \\&-\frac{Y_{\textit{PG}}}{\theta Y_{\textit{OG}}}\left( \frac{Y_{\textit{PG},t}-Y_{\textit{PG}}}{Y_{PG}}\right) \\ \textit{mc}_{G}\widehat{mc}_{G,t}= & {} \frac{Y_{\textit{PG}}}{\theta Y_{\textit{OG}}}\widehat{Y}_{\textit{OG},t}- \frac{Y_{\textit{PG}}}{\theta Y_{\textit{OG}}}\widehat{Y}_{\textit{PG},t} \\ \widehat{mc}_{G,t}= & {} \Phi \left( \widehat{Y}_{\textit{OG},t}-\widehat{Y} _{\textit{PG},t}\right) \;\text {where}\;\Phi =\frac{c_{p}}{(\theta -1)(1-c_{p})-c_{p}}. \end{aligned}$$

From (28) the real marginal cost $\left( V\right)$ is a constant and hence $\widehat{mc}_{V,t}=0.$

1.9 Derivation of the Flexible Price Equilibrium

The natural level of a variable is the flexible price equilibrium level. The natural level of the terms of trade in Eqs. (52) and (53) can be derived as (for Eq. 52)

$$\begin{aligned} T_{\textit{OGV},t}^{n} =\frac{P_{\textit{OG},t}}{P_{V,t}}=\frac{\frac{\textit{MC}_{G,t}}{\textit{mc}_{G,t}}}{ \frac{\textit{MC}_{V,t}}{\textit{mc}_{V,t}}}=\frac{\frac{W_{t}}{\textit{mc}_{G,t}A_{G,t}}}{\frac{W_{t}}{ \textit{mc}_{G,t}A_{V,t}}}=\frac{\textit{mc}_{V,t}}{\textit{mc}_{G,t}}\frac{A_{V,t}}{A_{G,t}}, \end{aligned}$$

where $\textit{MC}$ is nominal marginal cost and mc is real marginal cost.

$$\begin{aligned} \widehat{T}_{\textit{OGV},t}^{n}= & {} \widehat{mc}_{V,t}-\widehat{mc}_{G,t}+\widehat{A} _{V,t}-\widehat{A}_{G,t} \\= & {} -\Phi (\widehat{Y}_{\textit{OG},t}^{n}-\widehat{Y}_{\textit{PG},t})+\widehat{A}_{V,t}- \widehat{A}_{G,t}. \end{aligned}$$

Similarly $\widehat{T}_{\textit{AM},t}^{n}$ can be derived. For $\widehat{w}_{t}^{n}$ consider first the aggregate price index, $P_{t}^{n},$

$$\begin{aligned} P_{t}^{n}= & {} \left( P_{A,t}^{n}\right) ^{\delta }\left( P_{M,t}^{n}\right) ^{1-\delta }=\left( P_{\textit{OG},t}^{n}\right) ^{\left( 1-\mu \right) \delta }\left( P_{V,t}^{n}\right) ^{\mu \delta }\left( P_{M,t}^{n}\right) ^{1-\delta } \\= & {} \left( \frac{\textit{MC}_{G,t}^{n}}{\textit{mc}_{G,t}^{n}}\right) ^{\left( 1-\mu \right) \delta }\left( \frac{\textit{MC}_{V,t}^{n}}{\textit{mc}_{V,t}^{n}}\right) ^{\mu \delta }\left( \frac{\textit{MC}_{M,t}^{n}}{\textit{mc}_{M,t}^{n}}\right) ^{1-\delta } \\= & {} \left( \frac{W_{t}^{n}}{A_{G,t}{} \textit{mc}_{G,t}^{n}}\right) ^{\left( 1-\mu \right) \delta }\left( \frac{W_{t}^{n}}{A_{V,t}{} \textit{mc}_{V,t}^{n}}\right) ^{\mu \delta }\left( \frac{W_{t}^{n}}{A_{M,t}{} \textit{mc}_{M,t}^{n}}\right) ^{1-\delta } \\= & {} \frac{W_{t}^{n}}{\left( A_{G,t}{} \textit{mc}_{G,t}^{n}\right) ^{\left( 1-\mu \right) \delta }\left( A_{V,t}{} \textit{mc}_{V,t}^{n}\right) ^{\mu \delta }\left( A_{M,t}{} \textit{mc}_{M,t}^{n}\right) ^{1-\delta }}\\= & {} \frac{W_{t}^{n}}{A_{t}\left( \textit{mc}_{G,t}^{n}\right) ^{\left( 1-\mu \right) \delta }\left( \textit{mc}_{V,t}^{n}\right) ^{\mu \delta }\left( \textit{mc}_{M,t}^{n}\right) ^{1-\delta }}.\\ w_{t}^{n}= & {} \frac{W_{t}^{n}}{P_{t}^{n}}=A_{t}\left( \textit{mc}_{G,t}^{n}\right) ^{\left( 1-\mu \right) \delta }\left( \textit{mc}_{V,t}^{n}\right) ^{\mu \delta }\left( \textit{mc}_{M,t}^{n}\right) ^{1-\delta }. \end{aligned}$$

Note that $A_{t}=\left( A_{G,t}\right) ^{\left( 1-\mu \right) \delta }\left( A_{V,t}\right) ^{\mu \delta }\left( A_{M,t}\right) ^{1-\delta }$. Log-linearizing this we get,

$$\begin{aligned} \widehat{w}_{t}^{n}=\widehat{A}_{t}+\Phi \left( 1-\mu \right) \delta ( \widehat{Y}_{\textit{OG},t}^{n}-\widehat{Y}_{\textit{PG},t}). \end{aligned}$$

From the labor supply equation,

$$\begin{aligned} \widehat{w}_{t}^{n}=\psi \widehat{N}_{t}^{n}-\left( 1-\sigma \right) \widehat{\Gamma }_{t}+\sigma \widehat{C}_{t}^{n}. \end{aligned}$$

Substituting the value of $\widehat{N}_{t}^{n}=\Theta _{1}\left[ \widehat{C} _{t}^{n}-\widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y} _{\textit{OG},t}^{n}-\widehat{A}_{G,t}\right) \right] +\Theta _{2}\left( \widehat{Y} _{\textit{PG},t}-\widehat{A}_{G,t}\right)$ above we get,

$$\begin{aligned} \widehat{w}_{t}^{n}= & {} \psi \left[ \Theta _{1}\left[ \widehat{C}_{t}^{n}- \widehat{A}_{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}^{n}- \widehat{A}_{G,t}\right) \right] \right. \\&\left. +\,\Theta _{2}\left( \widehat{Y}_{\textit{PG},t}- \widehat{A}_{G,t}\right) \right] -\left( 1-\sigma \right) \widehat{\Gamma } _{t}+\sigma \widehat{C}_{t}^{n}. \end{aligned}$$

Replacing $\widehat{w}_{t}^{n}$ with $\widehat{A}_{t}+\Phi \left( 1-\mu \right) \delta (\widehat{Y}_{\textit{OG},t}^{n}-\widehat{Y}_{\textit{PG},t})$ yields

$$\begin{aligned} \widehat{A}_{t}+\Phi \left( 1-\mu \right) \delta (\widehat{Y}_{\textit{OG},t}^{n}- \widehat{Y}_{\textit{PG},t})= & {} \psi \Theta _{1}\left[ \widehat{C}_{t}^{n}-\widehat{A} _{t}+(1-\mu )(\gamma -1)\delta \left( \widehat{Y}_{\textit{OG},t}^{n}-\widehat{A} _{G,t}\right) \right] \\&+\,\psi \Theta _{2}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) -\left( 1-\sigma \right) \widehat{\Gamma }_{t}+\sigma \widehat{C}_{t}^{n}. \end{aligned}$$

Rearranging this to get $\widehat{C}_{t}^{n},$ we get Eq. (56)

$$\begin{aligned} \widehat{C}_{t}^{n}= & {} \frac{(\psi \Theta _{1}+1)}{\left( \psi \Theta _{1}+\sigma \right) }\widehat{A}_{t}-\frac{(\Phi \left( 1-\mu \right) \delta +\psi \Theta _{2})}{(\psi \Theta _{1}+\sigma )}\widehat{Y}_{\textit{PG},t}+\frac{ (1-\sigma )}{(\psi \Theta _{1}+\sigma )}\widehat{\Gamma }_{t} \\&+\frac{(\Phi \left( 1-\mu \right) \delta -\psi \Theta _{1}\left( \gamma -1\right) \left( 1-\mu \right) \delta )}{(\psi \Theta _{1}+\sigma )}\widehat{ Y}_{\textit{OG},t}^{n}\\&+\frac{(\psi \Theta _{1}\left( \gamma -1\right) \left( 1-\mu \right) \delta +\psi \Theta _{2})}{(\psi \Theta _{1}+\sigma )}\widehat{A} _{G,t}. \end{aligned}$$

1.10 Derivation of the Sticky Price Equilibrium: Eq. (59)

Using (41c) and (40) we get,

$$\begin{aligned} \widehat{mc}_{M,t}=\psi \widehat{N}_{t}+\sigma \widehat{C}_{t}-(1-\sigma ) \widehat{\Gamma }_{t}-\widehat{A}_{M,t}+\delta \widehat{T}_{\textit{AM},t}. \end{aligned}$$

Putting the value of$\ \widehat{N}_{t}$ from (45), we get

$$\begin{aligned} \widehat{mc}_{M,t}= & {} \left( \psi \Theta _{1}+\sigma \right) \widehat{C} _{t}-\psi \Theta _{1}\left[ \widehat{A}_{t}-(1-\mu )\left( \gamma -1\right) \delta \left( \widehat{Y}_{\textit{OG},t}-\widehat{A}_{G,t}\right) \right] \\&+\,\psi \Theta _{2}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) -(1-\sigma )\widehat{\Gamma }_{t}-\widehat{A}_{M,t}+\delta \widehat{T} _{\textit{AM},t}. \end{aligned}$$

At the natural level, $\widehat{mc}_{M,t}^{n}=0,$ which can also be written as,

$$\begin{aligned} 0= & {} \left( \psi \Theta _{1}+\sigma \right) \widehat{C}_{t}^{n}-\psi \Theta _{1}\left[ \widehat{A}_{t}-(1-\mu )\left( \gamma -1\right) \delta \left( \widehat{Y}_{\textit{OG},t}^{n}-\widehat{A}_{G,t}\right) \right] \\&+\,\psi \Theta _{2}\left( \widehat{Y}_{\textit{PG},t}-\widehat{A}_{G,t}\right) -(1-\sigma )\widehat{\Gamma }_{t}-\widehat{A}_{M,t}+\delta \widehat{T} _{\textit{AM},t}^{n} \\ \widetilde{mc}_{M,t}= & {} \widehat{mc}_{M,t}-\widehat{mc}_{M,t}^{n}=\left( \psi \Theta _{1}+\sigma \right) \left( \widehat{C}_{t}-\widehat{C} _{t}^{n}\right) +\delta \left( \widehat{T}_{\textit{AM},t}-\widehat{T} _{\textit{AM},t}^{n}\right) \\ \widehat{mc}_{M,t}= & {} \left( \psi \Theta _{1}+\sigma \right) \widetilde{C} _{t}+\delta \widetilde{T}_{\textit{AM},t} \end{aligned}$$

Using demand functions, $\widetilde{C}_{t}=\widetilde{Y}_{M,t}-\delta \widetilde{T}_{\textit{AM},t},$ the above equation can be written as,

$$\begin{aligned} \widehat{mc}_{M,t}=\left( \psi \Theta _{1}+\sigma \right) \widetilde{Y} _{M,t}-\delta \left( \psi \Theta _{1}+\sigma -1\right) \widetilde{T}_{\textit{AM},t}. \end{aligned}$$