Thus, ideas \(A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t\). Finally, output can be solved the production function: \[Y_t=A_t - L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})(1-\bar{l})\bar{L}\] + L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}\]
@@ -488,10 +488,10 @@Further, all economy continually and perpetually grow along a - "Balanced Growth Path" as previously defined by \(Y_t\) as a - function of the endogenous variables. This directly contrasts - the Solow model, in which an economy converges to a steady-state - with transition dynamics. + constant "Balanced Growth Path" as previously defined by \(Y_t\) + as a function of the endogenous variables. This directly + contrasts the Solow model, in which an economy converges to a + steady-state via transition dynamics.
Changes in the growth rate of ideas, then, alter the growth rate @@ -569,12 +569,12 @@ Intuitively, incorporating capital into output via the Solow Model's production function, as well as including the Law of Capital Motion seems like one way to legitimately - create the so-called 'Romer-Solow' model: + create this so-called "Romer-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: \[ + Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}\] +
+ ++ 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. +