Producers: main equations
\[\begin{equation}
Y = CH + I + G
\end{equation}\]
- Notional demand of factors
\[\begin{equation}
L^{n} = \left( \frac{Y}{PROG^{L}} \right) . \left( \left( \varphi^{L} \right) ^ {\rho^{KL}} \right) . \left( \frac{\left( \frac{w}{PROG^{L}} \right)}{c^{Y}} \right) ^ {\left( -\rho^{KL} \right)}
\end{equation}\]
\[\begin{equation}
K^{n} = \left( \frac{Y}{PROG^{K}} \right) . \left( \left( \varphi^{K} \right) ^ {\rho^{KL}} \right) . \left( \frac{\left( \frac{c^{K}}{PROG^{K}} \right)}{c^{Y}} \right) ^ {\left( -\rho^{KL} \right)}
\end{equation}\]
\[\begin{equation}
\varDelta \left(\operatorname{log} I\right) = \alpha^{I,Kn} . \varDelta \left(\operatorname{log} K^{n}\right) + \alpha^{I,I1} . \varDelta \left(\operatorname{log} I_{t-1}\right) + \alpha^{I,KnK1} . \operatorname{log} \frac{K^{n}_{t-1}}{K_{t-1}} - \alpha^{I,rK} . \varDelta \left(r - \frac{\varDelta \left(p\right)}{p_{t-1}}\right)
\end{equation}\]
Capital stock \[\begin{equation}
K = K_{t-1} \; \left( 1 - \delta \right) + I_{t-1}
\end{equation}\]
Notional production price \[\begin{equation}
p^{n} . Y = c^{Y} . Y . \left( 1 + m^{up} + dm^{up} \right)
\end{equation}\]
Notional mark-up \[\begin{equation}
\varDelta \left(\operatorname{log} \left(1 + m^{up,n}\right)\right) = \rho^{mupn,Ln} . \varDelta \left(\operatorname{log} \frac{L^{n}}{L}\right) + \rho^{mupn,Kn} . \varDelta \left(\operatorname{log} \frac{K^{n}}{K}\right)
\end{equation}\]
Notional unit cost production cost \[\begin{equation}
c^{Y} . Y = w . L^{n} + c^{K} . K^{n}
\end{equation}\]
Capital cost \[\begin{equation}
c^{K} . K = p^{K}_{t-1} \; K_{t-1} \; \left( \delta + r^{K}_{t-1} \right)
\end{equation}\]
Average price of the accumulated capital stock \[\begin{equation}
p^{K} . K = p^{K}_{t-1} \; K_{t-1} \; \left( 1 - \delta \right) + p_{t-1} \; I_{t-1}
\end{equation}\]
Average interest rate paid on the debt \[\begin{equation}
r^{K} . p^{K} . K = r^{K}_{t-1} \; p^{K}_{t-1} \; K_{t-1} \; \left( 1 - \delta \right) + p_{t-1} \; I_{t-1} \; r_{t-1}
\end{equation}\]
Profit of firms (before investment) \[\begin{equation}
PROF = p . Y - w . L - c^{K} . K
\end{equation}\]
Firms’ savings \[\begin{equation}
SAV^{F} = PROF - DIV - p . I
\end{equation}\]
Firms’ bank debt \[\begin{equation}
DEBT^{F} = DEBT^{F}_{t-1} \; \left( 1 - \varphi^{RD^{F}}_{t-1} \right) - SAV^{F}
\end{equation}\]
Adjustments: Minimizing an adjustment cost function
- To take into account that the changes in price are all the more costly that they are large propose to use quadratic adjustment cost models (Rotemberg, 1982).
- The firm defines the optimal price as a trade-off between the cost of adjusting and the cost of not been adjusted.
\[\begin{equation}
\Gamma_t\left(X_t\right)-\Gamma_t\left(X_t^n\right)=\Gamma_t^{\prime}\left(X_t^n\right)\left(X_t-X_t^n\right)+\Gamma_t^{\prime \prime}\left(X_t^n\right)\left(X_t-X_t^n\right)^2
\end{equation}\] The profit being maximum for \(X_t^n, \Gamma_t^{\prime}\left(X_t^n\right)=0\) and \(\Gamma_t^{\prime \prime}\left(X_t^n\right)<0\). As a first approximation, the adjustment cost, i.e. the loss of profit suffered by a company that is not in the optimum, is therefore: \[
C_D=\Gamma_t\left(X_t^n\right)-\Gamma_t\left(X_t\right)=C_D\left(X_t-X_t^n\right)^2
\] Where: \[
C_D=-\Gamma_t^{\prime \prime}\left(X_t^n\right)
\] Suppose that the adjustment cost is proportional to the square of the speed of adjustment: \[
C_A=c_A\left(X_t-X_{t-1}\right)^2
\] Where: \[
c_A>0
\] Minimizing the total cost function \(\left(C_t=C_D+C_A\right)\) is equivalent to solving: \[
C_t^{\prime}\left(X_t\right)=2 c_D\left(X_t-X_t^n\right)+2 c_A\left(X_t-X_{t-1}\right)=0
\] The condition of the second order \(\left(C_t^{\prime \prime}\left(X_t\right)>0\right)\) being always verified, the optimal adjustment which minimizes the total cost has the following dynamic process: \[
X_t=\alpha X_t^n+(1-\alpha) X_{t-1}
\] With: \[
\alpha=\frac{c_D}{\left(c_D+c_A\right)}
\]
With this simple model, the average adjustment time is: \[
\frac{\alpha}{(1-\alpha)}=\frac{c_D}{c_A}
\] The slower the adjustment, the higher the adjustment cost \(c_A\) compared to the cost of non being adjusted \(c_D\).
Adjustments: main equations
Wage \[\begin{equation}
\varDelta \left(\operatorname{log} w\right) = \alpha^{W,Wn} . \varDelta \left(\operatorname{log} w^{n}\right) + \alpha^{W,W1} . \varDelta \left(\operatorname{log} w_{t-1}\right) - \alpha^{W,W1Wn1} . \operatorname{log} \frac{w_{t-1}}{w^{n}_{t-1}}
\end{equation}\]
Production price \[\begin{equation}
\operatorname{log} p = \alpha^{P,Pn} . \operatorname{log} p^{n} + \left( 1 - \alpha^{P,Pn} \right) . \left( \operatorname{log} p_{t-1} + \varDelta \left(\operatorname{log} p^{e}\right) \right)
\end{equation}\]
Expected production price inflation \[\begin{equation}
\varDelta \left(\operatorname{log} p^{e}\right) = \alpha^{Pe,Pe1} . \varDelta \left(\operatorname{log} p^{e}_{t-1}\right) + \alpha^{Pe,P1} . \varDelta \left(\operatorname{log} p_{t-1}\right) + \alpha^{Pe,Pn} . \varDelta \left(\operatorname{log} p^{n}\right)
\end{equation}\]
Households final consumption \[\begin{equation}
\operatorname{log} CH = \alpha^{CH,CHn} . \operatorname{log} CH^{n} + \left( 1 - \alpha^{CH,CHn} \right) . \left( \operatorname{log} CH_{t-1} + \varDelta \left(\operatorname{log} CH^{e}\right) \right)
\end{equation}\]
Expected Households final consumption growth
\[\begin{equation}
\varDelta \left(\operatorname{log} CH^{e}\right) = \alpha^{CHe,CHe1} . \varDelta \left(\operatorname{log} CH^{e}_{t-1}\right) + \alpha^{CHe,CH1} . \varDelta \left(\operatorname{log} CH_{t-1}\right) + \alpha^{CHe,CHn} . \varDelta \left(\operatorname{log} CH^{n}\right)
\end{equation}\]
Labor \[\begin{equation}
\operatorname{log} L = \alpha^{L,Ln} . \operatorname{log} L^{n} + \left( 1 - \alpha^{L,Ln} \right) . \left( \operatorname{log} L_{t-1} + \varDelta \left(\operatorname{log} L^{e}\right) \right)
\end{equation}\]
Expected labor growth \[\begin{equation}
\varDelta \left(\operatorname{log} L^{e}\right) = \alpha^{Le,Le1} . \varDelta \left(\operatorname{log} L^{e}_{t-1}\right) + \alpha^{Le,L1} . \varDelta \left(\operatorname{log} L_{t-1}\right) + \alpha^{Le,Ln} . \varDelta \left(\operatorname{log} L^{n}\right)
\end{equation}\]
Dividend for households \[\begin{equation}
\operatorname{log} DIV = \alpha^{DIV,DIVn} . \operatorname{log} DIV^{n} + \left( 1 - \alpha^{DIV,DIVn} \right) . \left( \operatorname{log} DIV_{t-1} + \varDelta \left(\operatorname{log} DIV^{e}\right) \right)
\end{equation}\]
Expected dividend for households \[\begin{equation}
\varDelta \left(\operatorname{log} DIV^{e}\right) = \alpha^{DIVe,DIVe1} . \varDelta \left(\operatorname{log} DIV^{e}_{t-1}\right) + \alpha^{DIVe,DIV1} . \varDelta \left(\operatorname{log} DIV_{t-1}\right) + \alpha^{DIVe,DIVn} . \varDelta \left(\operatorname{log} DIV^{n}\right)
\end{equation}\]
Interest rate of the Central Bank \[\begin{equation}
r = \alpha^{r} . r^{n} + \left( 1 - \alpha^{r} \right) . r_{t-1}
\end{equation}\]
Propensity to save \[\begin{equation}
\sigma = \alpha^{\sigma} . \sigma^{n} + \left( 1 - \alpha^{\sigma} \right) . \sigma_{t-1}
\end{equation}\]
Mark-up \[\begin{equation}
m^{up} = \alpha^{m,up} . m^{up,n} + \left( 1 - \alpha^{m,up} \right) . m^{up}_{t-1}
\end{equation}\]
Income taxe rate \[\begin{equation}
t^{inc} = \alpha^{t,inc} . t^{inc,n} + \left( 1 - \alpha^{t,inc} \right) . t^{inc}_{t-1}
\end{equation}\]
