Notably, while both the Romer and Solow Models help to analyze @@ -588,68 +606,98 @@
- Based on the the motivations for creating this model, it is more - useful to first analyze the growth rates of equilibrium long run - output \(Y_t^*\). -
-- According to the production function, \[g_Y=g_A+\alpha - g_K+(1-\alpha)g_{L_{y}}\] -
-- From previous analysis it was found that - \(g_A=\bar{z}\bar{l}\bar{L}\). -
-- Based on the Law of Capital Motion, \[g_K=\frac{\Delta - K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}\] -
-- Because growth rates are constant on the Balanced Growth Path, - \(g_K\) must be constant as well. Thus, so is - \(\bar{s}\frac{Y_t}{K_t}-\bar{d}\); it must be that - \(g_K^*=g_Y^*\). -
-- The model assumes population is constant, so - \(g_{\bar{L}}=0\rightarrow g_{\bar{L}_yt}=0\) as well. -
-- Combining these terms, we find: - \[g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\] - \[\rightarrow g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\] -
-- Solving for \(Y_t^*\) is trivial after discovering \(g_K=g_Y\) - must hold on a balanced growth path. -
-- Invoking the Law of Capital Motion with magic chants, - \[g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow - K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}\] -
-- Isolating \(Y_t^*\), \[Y_t^*=A_t^* - (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha - ({(1-\bar{l})\bar{L}})^{1-\alpha}\] \[\rightarrow - {Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}\] -
-- Plugging in the known expressions for \(A_t^*\) and \(g_Y^*\), a - final expression for the Balanced Growth Path output as a function - of the endogenous parameters and time is obtained: \[ - Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\] -
- -- Intuitively, this means that idea-driving factors, as well as an - increased allocation of labor to output, will increase the - Balanced Growth Path (the level of long-run growth), - combining both the Romer and Solow model. -
++ Based on the the motivations for creating this model, it is more + useful to first analyze the growth rates of equilibrium long run + output \(Y_t^*\). +
++ According to the production function, \[g_Y=g_A+\alpha + g_K+(1-\alpha)g_{L_y}\] +
++ From previous analysis it was found that + \(g_A=\bar{z}\bar{l}\bar{L}\). +
++ Based on the Law of Capital Motion, \[g_K=\frac{\Delta + K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}\] +
++ Because growth rates are constant on the Balanced Growth Path, + \(g_K\) must be constant as well. Thus, so is + \(\bar{s}\frac{Y_t}{K_t}-\bar{d}\); it must be that + \(g_K^*=g_Y^*\). +
++ The model assumes population is constant, so + \(g_{\bar{L}}=0\rightarrow g_{\bar{L}_yt}=0\) as well. +
++ Combining these terms, we find: + \[g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\] + \[\rightarrow g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}\] +
++ Solving for \(Y_t^*\) is trivial after discovering \(g_K=g_Y\) + must hold on a balanced growth path. +
++ Invoking the Law of Capital Motion with magic chants, + \[g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow + K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}\] +
++ Isolating \(Y_t^*\), \[Y_t^*=A_t^* + (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha + ({(1-\bar{l})\bar{L}})^{1-\alpha}\] \[\rightarrow + {Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}\] +
++ Plugging in the known expressions for \(A_t^*\) and \(g_Y^*\), a + final expression for the Balanced Growth Path output as a + function of the endogenous parameters and time is obtained: \[ + Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\] +
++ 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 \(\frac{d}{dg_A^*}Y_t^*>0\). +
++ Expectedly, output has a positive relationship with the savings + rate and a negative relationship with the depreciation rate. +
++ However, do economics grow faster/slower the + further below/above they are from their Balanced + Growth Path, as initially desired? While this can be + mathematically proven (of course), sometimes a visualization + helps. +
+