Dynamic firms across multiple industries

docs/book/content/theory/financial.md

@@ -3,7 +3,7 @@
 
 # Financial Intermediary
 
-Domestic household wealth $W^d_{t}=B_{t}$ and foreign ownership of domestic assets $W^f_{t}$, both in terms of the numeraire good, are invested in a financial intermediary. This intermediary purchases a portfolio of government bonds and private capital in accordance with the domestic and foreign investor demand for these assets and then returns a single portfolio rate of return to all investors.
+Domestic household wealth $B_{t}$ is invested in a financial intermediary. This intermediary purchases a portfolio of government bonds and private capital in accordance with the domestic demand for these assets and then returns a single portfolio rate of return to all investors.
 
 Foreign demand for government bonds is specified in section {ref}`SecMarkClrMktClr_G` of the {ref}`Chap_MarkClr` chapter:
 
@@ -19,40 +19,33 @@ This leaves domestic investors to buy up the residual amount of government debt:
     D^{d}_{t} = D_{t} - D^{f}_{t} \quad\forall t
   ```
 
-We assume that debt dominates the capital markets, such that domestic investor demand for capital, $K^{d}_{t}$ is given as:
+We assume that debt dominates the capital markets, such that domestic investor demand for equity, $V^{d}_{t}$ is given as:
 
   ```{math}
   :label: eq_domestic_cap_demand
-    K^{d}_{t} = B_{t} - D^{d}_{t} \quad\forall t
+    V^{d}_{t} = B_{t} - D^{d}_{t} \quad\forall t
   ```
 
-Foreign demand for capital is given in {ref}`SecMarkClrMktClr_K`, where $K^{f}_{t}$ is an exogenous fraction of excess capital demand at the world interest rate:
+Foreign demand for equity is given in {ref}`SecMarkClrMktClr_K`, where $V^{f}_{t}$ is an exogenous fraction of excess equity demand:
 
   ```{math}
   :label: eq_foreign_cap_demand
-    K^{f}_t = \zeta_{K, t}ED^{K,r^*}_t \quad\forall t
+    V^{f}_t = \zeta_{K, t}ED^{V,r^*}_t \quad\forall t
   ```
 
 The total amount invested in the financial intermediary is thus:
 
 ```{math}
-W_{t} & = W^d_{t} + W^f_{t} \\
-    & = D^d_t + K^d_t  + D^f_t + K^f_t \\
-    & = D_t + K_t
+    B_t & = D^d_t + V^d_t \\
 ```
 
-Interest rates on private capital through the financial intermediary and on government bonds differ. The return on the portfolio of assets held in the financial intermediary is the weighted average of these two rates of return. As derived in {eq}`EqFirms_rKt` of Section {ref}`EqFirmsPosProfits`, the presence of public infrastructure in the production function means that the returns to private factors of production ($r_t$ and $w_t$) exhibit decreasing returns to scale.[^MoorePecoraro] It is assumed that competition ensures a zero profit condition among firms and the returns to public infrastructure through the returns of firms are captured by the financial intermediary and returned to share holders. The return on capital is therefore the sum of the (after-tax) returns to private and public capital.
+The return on the portfolio of assets held in the financial intermediary is the weighted average of the return on equity and government debt. As derived in {eq}`EqFirms_rKt` of Section {ref}`EqFirmsPosProfits`, the presence of public infrastructure in the production function means that the returns to private factors of production ($r_t$ and $w_t$) exhibit decreasing returns to scale.[^MoorePecoraro]. These excess profits are returned to shareholders through the financial intermediary.
 
-```{math}
-:label: eq_rK
-  r_{K,t}  = r_{t} + \frac{\sum_{m=1}^M(1 - \tau^{corp}_{m,t})p_{m,t}MPK_{g,m,t}K_{g,m,t}}{\sum_{m=1}^M K_{m,t}} \quad\forall t
-```
-
-The return on the portfolio of assets held by the financial intermediary is thus a weighted average of the return to government debt $r_{gov,t}$ from {eq}`EqUnbalGBC_rate_wedge` and the adjusted return on private capital $r_{K,t}$ from {eq}`eq_rK`.
+The return on the portfolio of assets held by the financial intermediary is thus a weighted average of the return to government debt $r_{gov,t}$ from {eq}`EqUnbalGBC_rate_wedge` and the adjusted return on equity through the distribution of profits and capital gains:
 
 ```{math}
 :label: eq_portfolio_return
-  r_{p,t} = \frac{r_{gov,t}D_{t} + r_{K,t}K_{t}}{D_{t} + K_{t}} \quad\forall t \quad\text{where}\quad K_t \equiv \sum_{m=1}^M K_{m,t}
+  r_{p,t} = \frac{r_{gov,t}D_{t} + \Pi^d_{m,t}  + (V^d_{t+1} - V^d_{t}) }{D_{t} + V^d_{t}} \quad\forall t \quad\text{where}\quad \Pi^d_t \equiv \sum_{m=1}^M \pi_{m,t} * \frac{V^d_{t}}{V_{t}}
 ```
 
 (SecFinfootnotes)=

docs/book/content/theory/market_clearing.md

@@ -62,32 +62,32 @@ We also characterize here the law of motion for total bequests $BQ_t$. Although
 
 
   (SecMarkClrMktClr_K)=
-  ### Private capital market clearing
+  ### Equity market clearing
 
-  Domestic firms in each industry $m$ rent private capital $K_t\equiv\sum_{m=1}^M K_{m,t}$ from domestic households $K^d_t$ and from foreign investors $K^f_t$.
+  Equity in domestic firms, $V_t\equiv\sum_{m=1}^M K_{m,t}$, is held by domestic households, $V^d_t$ and foreign investors $V^f_t$. The equity market clearing condition is thus:
 
   ```{math}
   :label: EqMarkClr_KtKdKf
-   K_t = K^d_t + K^f_t \quad\forall t \quad\text{where}\quad K_t \equiv  \sum_{m=1}^M K_{m,t}
+   V_t = V^d_t + V^f_t \quad\forall t \quad\text{where}\quad V_t \equiv  \sum_{m=1}^M V_{m,t}
   ```
 
-  Assume that there exists some exogenous world interest rate $r^*_t$. We assume that foreign capital supply $K^f_t$ is an exogenous percentage $\zeta_K\in[0,1]$ of the excess total domestic private capital demand $ED^{K,r^*}_t$ that would exist if domestic private capital demand were determined by the exogenous world interest rate $r^*_t$ and domestic private capital supply were determined by the model consistent return on household savings $r_{p,t}$. This percentage $\zeta_K$ is something we calibrate. Define excess total domestic capital demand at the exogenous world interest rate $r^*_t$ as $ED^{K,r^*}_t$, where $K^{r^*}_t\equiv\sum_{m=1}^M K^{r^*}_{m,t}$ is the capital demand by domestic firms at the world interest rate $r^*_t$, and $K^{d}_t$ is the domestic supply of private capital to firms, which is modeled as being a function of the actual rate faced by households $r_{p,t}$. Then our measure of excess demand at the world interest rate is the following.
+  Assume that there exists some exogenous world interest rate $r^*_t$. We assume that foreign equity holdings, $V^f_t$, is an exogenous percentage $\zeta_K\in[0,1]$ of the excess total supply of domestic equity, $ES^{K,r^*}_t$ that would exist if domestic firm values were determined by investment decisions made under the exogenous world interest rate $r^*_t$ and domestic equity demand were determined by the model consistent return on household savings $r_{p,t}$. This percentage $\zeta_K$ is something we calibrate. Define excess supply of equity at the exogenous world interest rate $r^*_t$ as $ES^{V,r^*}_t$, where $V^{r^*}_t\equiv\sum_{m=1}^M V^{r^*}_{m,t}$ is value of domestic firms at the world interest rate $r^*_t$, and $V^{d}_t$ is the domestic demand for equity, which is modeled as being a function of the actual rate of return faced by households $r_{p,t}$. Then our measure of excess supply at the world interest rate is the following.
 
   ```{math}
   :label: EqMarkClr_ExDemK
-    ED^{K,r^*}_t \equiv K^{r^*}_t - K^d_t \quad\forall t \quad\text{where}\quad K^{r^*}_t\equiv \sum_{m=1}^M K^{r^*}_{m,t}
+    ES^{V,r^*}_t \equiv V^{r^*}_t - V^d_t \quad\forall t \quad\text{where}\quad V^{r^*}_t\equiv \sum_{m=1}^M V^{r^*}_{m,t}
   ```
 
-  Then we assume that total foreign private capital supply $K^f_t$ is a fixed fraction of this excess capital demand at the world interest rate $r^*$.
+  Then we assume that total foreign private capital supply $V^f_t$ is a fixed fraction of this equity supply at the world interest rate $r^*$.
 
   ```{math}
   :label: EqMarkClr_zetaK
-    K^{f}_t = \zeta_{K}ED^{K,r^*}_t \quad\forall t
+    V^{f}_t = \zeta_{K}ES^{K,r^*}_t \quad\forall t
   ```
 
-  This approach nicely nests the small open economy specification discussed in Section {ref}`SecSmallOpen` of Chapter {ref}`Chap_SmOpEcn` in which $\zeta_K=1$, foreigners flexibly supply all the excess demand for private capital, the marginal product of capital is fixed at the exogenous world interest rate $r^*$, and domestic households face the least amount of crowd out by government debt. The opposite extreme is the closed private capital market assumption of $\zeta_K=0$ in which $K^f_t=0$ and households must supply all the capital demanded in the domestic market. In this specification, the interest rate is the most flexible and adjusts to equilibrate domestic private capital supply $K^d_t$ with private capital demand $K_t$.
+  This approach nicely nests the small open economy specification discussed in Section {ref}`SecSmallOpen` of Chapter {ref}`Chap_SmOpEcn` in which $\zeta_K=1$, foreigners flexibly demand the excess supply of domestic equity, the domestic interest rate is fixed at the exogenous world interest rate $r^*$, and domestic households face the least amount of crowd out by government debt. The opposite extreme is the closed private capital market assumption of $\zeta_K=0$ in which $V^f_t=0$ and households must hold all the domestic firms' equity. In this specification, the interest rate is the most flexible and adjusts to equilibrate domestic equity demand, $V^d_t$, with the supply of equity, $V_t$.
 
-  For the intermediate specifications of $\zeta_K\in(0,1)$, foreigners provide a fraction of the excess demand defined in {eq}`EqMarkClr_ExDemK`. This allows for partial inflows of foreign private capital, partial crowd-out of government spending on private investment, and partial adjustment of the domestic interest rate $r_t$. This latter set of model specifications could be characterized as large-open economy or partial capital mobility.
+  For the intermediate specifications of $\zeta_K\in(0,1)$, foreigners provide a fraction of the excess supply of equity in {eq}`EqMarkClr_ExDemK`. This allows for partial inflows of foreign equity holdings, partial crowd-out of government spending on private investment, and partial adjustment of the domestic interest rate $r_t$. This latter set of model specifications could be characterized as large-open economy or partial capital mobility.
 
 
   (SecMarkClrMktClr_goods)=
@@ -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_{g,t}D^f_t - \Pi^f_{t} + (V^f_{t+1} - V^f_t) - \bigl(D^f_{t+1} - D^f_t\bigr)  + \Psi_{M,t} \quad\forall t
   ```
   where
   ```{math}
@@ -124,6 +124,16 @@ 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 \\
+  ```
+  and
+  ```{math}
+  :label: EqMarkClrGoods_Pi
+    \Pi^f_{t} &\equiv \sum_{m=1}^M \pi_{m,t} * \frac{V^f}{V} \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 $I_{M,t}$.