feat: format

src/content/posts/algorithms/models-of-production.mdx

@@ -1,6 +1,6 @@
 ---
 title: "models of production"
-date: "2024-06-22"
+date: "22/06/2024"
 useKatex: true
 ---
 
@@ -22,9 +22,9 @@ With:
 In this simple model, the following statements describe the economy:
 
 1.  Output is either saved or consumed; in other words, savings equals investment
-2. Capital accumulates according to investment $I_t$ and depreciation $\bar{d}$, beginning with $K_0$ (often called the <u>Law of Capital Motion</u>)
-3. Labor $L_t$ is time-independent
-4. A savings rate $\bar{s}$ describes the invested portion of total output
+2.  Capital accumulates according to investment $I_t$ and depreciation $\bar{d}$, beginning with $K_0$ (often called the <u>Law of Capital Motion</u>)
+3.  Labor $L_t$ is time-independent
+4.  A savings rate $\bar{s}$ describes the invested portion of total output
 
 Including the production function, these four ideas encapsulate the Solow Model:
 
@@ -49,7 +49,13 @@ When investment is completely disincentivized by depreciation (in other words, $
 
 Using this equilibrium condition, it follows that:
 
-$$Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}$$ $$\rightarrow \bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}$$ $$\rightarrow K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}$$ $$\rightarrow Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}$$
+$$Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}$$
+
+$$\rightarrow \bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}$$
+
+$$\rightarrow K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}$$
+
+$$\rightarrow Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}$$
 
 Thus, the equilibrium intensive form (output per worker) of both capital and output are summarized as follows:
 
@@ -63,8 +69,8 @@ Using both mathematical intuition and manipulating the visualization above, we f
 - Capital is influenced by workforce size, TFP, and savings rate
 - Capital output share's $\alpha$ impact on output is twofold:
 
-    1. Directly through capital quantity
-    2. Indirectly through TFP
+  1. Directly through capital quantity
+  2. Indirectly through TFP
 
 - Large deviations in capital from steady-state $K^*$ induce net investments of larger magnitude, leading to an accelerated reversion to the steady-state
 - Economies stagnate at the steady-state $(K^*,Y^*)$—this model provides no avenues for long-run growth.
@@ -86,7 +92,9 @@ The Romer Model provides an answer by both modeling ideas $A_t$ (analagous to TF
 The Model divides the world into two parts:
 
 - <u>Objects</u>: finite resources, like capital and labor in the Solow Model
-- <u>Ideas</u>: infinite, [non-rivalrous](https://en.wikipedia.org/wiki/Rivalry_$economics$) items leveraged in production (note that ideas may be [excludable](blank), though)
+- <u>Ideas</u>: infinite,
+  [non-rivalrous](https://en.wikipedia.org/wiki/Rivalry_$economics$) items
+  leveraged in production (note that ideas may be [excludable](blank), though)
 
 The Romer Models' production function can be modelled as:
 
@@ -196,7 +204,7 @@ $$g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow K_t^*=\frac{\bar{s}Y
 
 Isolating $Y_t^*$,
 
-$$Y_t^*=A_t^* (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha ({(1-\bar{l})\bar{L}})^{1-\alpha}$$ 
+$$Y_t^*=A_t^* (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha ({(1-\bar{l})\bar{L}})^{1-\alpha}$$
 
 $$\rightarrow {Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}$$
 
@@ -206,7 +214,7 @@ $$Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\fr
 
 ### analysis
 
-First looking at the growth rate of output, $g*Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}$, idea-driving factors and an increased allocation of labor to output increase the equilibrium Balanced Growth Path—the _level* of long-run growth. Thus, this model captures the influences of both capital and ideas on economic growth.
+First looking at the growth rate of output, $g*Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}$, idea-driving factors and an increased allocation of labor to output increase the equilibrium Balanced Growth Path—the \_level\* of long-run growth. Thus, this model captures the influences of both capital and ideas on economic growth.
 
 Looking at $Y_t^*$, ideas have both a direct and indirect effect on output. Firstly, ideas raise output because they increase productivity (directly); second, with the introduction of capital stock, ideas also increase capital, in turn increasing output further (indirectly). Mathematically, this is evident in both instances of $g_A^*$ in the formula for output $Y_t^*$—note that $\frac{1}{1-\alpha},\frac{\alpha}{1-\alpha}>0$ for any $\alpha\in(0,1)$, so $\frac{d}{dg_A^*}Y_t^*>0$.