add adj costs to market clearing

docs/book/content/theory/market_clearing.md

@@ -111,11 +111,11 @@ We also characterize here the law of motion for total bequests $BQ_t$. Although
     Y_{m,t} = C_{m,t} \quad\forall t \quad\text{and}\quad m=1,2,...M-1
   ```
 
-  The output of the $M$th industry can be used for private investment, infrastructure investment, government spending, and government debt.[^M_ind] As such, the market clearing condition in the $M$th industry will look more like the traditional $Y=C+I+G+NX$ expression.[^RCrates_note]
+  The output of the $M$th industry can be used for private investment, infrastructure investment, government spending, and government debt.[^M_ind] As such, the market clearing condition in the $M$th industry will look more like the traditional $Y=C+I+G+NX$ expression.[^RCrates_note] Note also that adjustment costs are paid in units of capital, which is the same units as the output of the $M$th industry. Therefore we must include the adjustment costs in the market clearing condition for the $M$th industry.
 
   ```{math}
   :label: EqMarkClrGoods_M
-    Y_{M,t} = C_{M,t} + I_{M,t} + I_{g,t} + G_t + r_{p,t} K^f_t + r_{p,t}D^f_t - (K^f_{t+1} - K^f_t) - \bigl(D^f_{t+1} - D^f_t\bigr) \quad\forall t
+    Y_{M,t} = C_{M,t} + I_{M,t} + I_{g,t} + G_t + r_{p,t} K^f_t + r_{p,t}D^f_t - (K^f_{t+1} - K^f_t) - \bigl(D^f_{t+1} - D^f_t\bigr)  + \Psi_{M,t} \quad\forall t
   ```
   where
   ```{math}
@@ -124,6 +124,11 @@ We also characterize here the law of motion for total bequests $BQ_t$. Although
     &= K_{t+1} - (1 - \delta_{M,t})K_t \\
     &= (K^d_{t+1} + K^f_{t+1}) - (1 - \delta_{M,t})(K^d_t + K^f_t)
   ```
+and
+  ```{math}
+  :label: EqMarkClrGoods_IMt
+    \Psi_{M,t} &\equiv \sum_{m=1}^M \Psi(I_{m,t},K_{m,t}) \quad\forall t \\
+  ```
 
   In the partially open economy, we must add to the right-hand-side of {eq}`EqMarkClrGoods_M` the output paid to the foreign owners of capital $r_{p,t} K^f_t$ and to the foreign holders of government debt $r_{p,t}D^f_t$. And we must subtract off the foreign inflow component $K^f_{t+1} - K^f_t$ from private capital investment as shown in the first term in parentheses on the right-hand-side of {eq}`EqMarkClrGoods_M`. You can see in the definition of private investment {eq}`EqMarkClrGoods_IMt` where this amount of foreign capital is part of $I_{M,t}$.
 

docs/book/content/theory/stationarization.md

@@ -405,7 +405,7 @@ The stationarized version of the capital adjustment cost function and it's first
   ```{math}
   :label: EqStnrzMarkClrGoods_M
     \hat{Y}_{M,t} &= \hat{C}_{M,t} + \hat{I}_{M,t} + \hat{I}_{g,t} + \hat{G}_t + r_{p,t} \hat{K}^f_t + r_{p,t}\hat{D}^f_t ... \\
-    &\quad - \Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{K}^f_{t+1} - \hat{K}^f_t\Bigr) - \Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}^f_{t+1} - \hat{D}^f_t\Bigr) \quad\forall t
+    &\quad - \Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{K}^f_{t+1} - \hat{K}^f_t\Bigr) - \Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}^f_{t+1} - \hat{D}^f_t\Bigr) + \hat{\Psi}_{M,t} \quad\forall t
   ```
   where
   ```{math}
@@ -419,6 +419,11 @@ The stationarized version of the capital adjustment cost function and it's first
     &= e^{g_y}\bigl(1 + \tilde{g}_{n,t+1}\bigr)\hat{K}_{t+1} - (1 - \delta_{M,t})\hat{K}_t \\
     &= e^{g_y}\bigl(1 + \tilde{g}_{n,t+1}\bigr)(\hat{K}^d_{t+1} + \hat{K}^f_{t+1}) - (1 - \delta_{M,t})(\hat{K}^d_t + \hat{K}^f_t)
   ```
+  and
+  ```{math}
+  :label: EqStnrzMarkClrGoods_IMt
+    \hat{\Psi}_{M,t} &\equiv \sum_{m=1}^M \Psi(\hat{I}_{m,t},\hat{K}_{m,t}) \quad\forall t \\
+  ```
 
   We stationarize the law of motion for total bequests $BQ_t$ in {eq}`EqMarkClrBQ` by dividing both sides by $e^{g_y t}\tilde{N}_t$. Because the population levels in the summation are from period $t-1$, we must multiply and divide the summed term by $\tilde{N}_{t-1}$ leaving the term in the denominator of $1+\tilde{g}_{n,t}$.