Romer or R&D model:
Assumes a separate sector for R&D, research and development.
-asumptions for simplifying:
        -Cobb-Douglas functions
        -fraction of output saved and fraction of labor force and capital stock is exogenous and constant.
Two sectors, production sector where output is made and R&D sector where knowledge stock is made.
Model function: \(Y(t)=[(1-a_K)K(t)]^{\alpha} [A(t)(1-a_L)L(t)]^{(1-\alpha)} , 0<\alpha<1\)
\(A^\cdot (t)= B[a_K K(t)]^\beta [a_L L(t)]^\gamma A(t)^\theta\)   \(B>0, \beta >0, \gamma>0\)
\(K^\cdot (t)=sY(t)\)           implying \(L^\cdot (t)=nL(t), n>0\)
When assuming no capital, the model:
\(Y(t)=A(t)(1-a_L)L(t)\)
\(A^\cdot (t)=B[a_L L(t)]^\gamma A(t)^\theta\)   
growth rate of A: \(g_A \equiv \frac{A^\cdot (t)}{A(t)}\)   \(=Ba_L ^\gamma L(t)^\gamma A(t)^{\theta -1}\)
\(\frac{g_A ^\cdot}{g_A} =\gamma n +(\theta -1)g_A(t)\)      or \(g_A ^\cdot =\gamma n g_A (t)+(\theta -1)[g_A (t)]^2\)
When \(\theta <1\),   \(g_A ^* =\frac{\gamma}{1-\theta} n\)   \(g_A\ converges\ to\ \ g_A^*\) regardless of initial conditions.
This is an example of endogenous growth.
Factors affecting \(a_L\) and not \(g_A ^\cdot\) do not change the steady state growth rate. For example, an increase in fraction of labor force to R&D. This conclusion is similar to that of the Solow model and rise in the saving rate.
When \(\theta >1\)\(g_A^{\cdot}>1\ for\ all\ of\ g_A\), or that \(g_A ^\cdot\) is increasing in \(g_A\).
When \(\theta =1\),   \(g_A (t)=Ba_L ^\gamma L(t)^\gamma\)  and \(g_A ^\cdot (t)=\gamma n g_A (t)\)
in this case, when population growth is positive, dynamics similar to \(\theta >1\). If pop growth is zero, \(g_A \) is constant and the economy is already growing steadily. Changes in \(a_L\) affect long-run growth rate.
GENERAL CASE of the model
\(K^\cdot (t)=s(1-a_K)^\alpha (1-a_L)^{1-\alpha} K(t)^\alpha A(t)^{1-\alpha} L(t)^{1-\alpha}\)
\(g_K(t) \equiv \frac{K^\cdot (t)}{K(t)} = c_K [\frac{A(t) L(t)}{K(t)}]^{1-\alpha}\)
\(\frac{g_K ^\cdot (t)}{g_K (t)} = (1-\alpha)[g_A(t)+n-g_K (t)]\)
\(g_A (t)=c_A K(t)^\beta L(t)^\gamma A(t)^{\theta -1}\)
\(\frac{g_A ^\cdot (t)}{g_A (t)} = \beta g_K (t) +\gamma n +(\theta -1) g_A (t)\)
When \(\beta + \theta <1\)\(\frac{(1-\theta)}{\beta} >1\).  \(g_A\) and \(g_K\) converge to the point where they equal. Thus satisfying: \(g_A ^* +n-g_K ^* =0\) and \(\beta g_K ^* +\gamma n +(\theta -1)g_A ^* =0\)
thus, \(g_A ^* =\frac{\beta +\gamma}{1-(\theta +\beta)} n\) and \(g_K ^*=g_A ^* + n\)
When \(\beta +\theta =1\) and \(n=0\), \(g_K =g_A\). This is a fully endogenous model.
Knowledge is nonrival, commonality. But it is excludable.
Because knowledge has these qualities, many view knowledge should be free and subsidized.
Three externalities from R&D: consumer-surplus effect, business -stealing effect, R&D effect.
Learning by doing is especially important for research advances:
\(Y(t)=K(t)^\alpha [A(t)L(t)]^{1-\alpha}\) this is the production function where all inputs are used in goods production.
\(A(t)=BK(t)^\phi\)                  \(B>0, \phi >0\)
\(Y(t)=K(t)^\alpha B^{1-\alpha} K(t)^{\phi (1-\alpha)} L(t)^{1-\alpha}\)               and since \(K^\cdot(t)=sY(t)\)    
\(K^\cdot (t)=sB^{1-\alpha} K(t)^\alpha K(t)^{\phi (1-\alpha)} L(t)^{1-\alpha}\)
When \(\phi =1, n=0\), then the production function becomes \(Y(t)=bK(t), b\equiv B^{1-\alpha}L^{1-\alpha}\)
and capital accumulation is given by \(K^\cdot(t)=sbK(t)\)
Romer model:
There are no transition dynamics shown; the economy essentially jumps to its new balanced growth path. 
\(Y=[\int _{i=0} ^A L(i)^\phi di]^{\frac{1}{\phi}} ,0<\phi <1\)         \(L(i)=\frac{L_Y}{A}\) for all of i  
\(Y=[A (\frac{L_Y}{A})^\phi ]^\frac{1}{\phi} =A^{{1-\phi}/\phi} L_Y\)
\(L(i)=[\frac{p(i)}{\lambda}]^\frac{1}{\phi -1} =[\frac{\lambda}{p(i)}]^\frac{1}{1-\phi}\)
\(L_A (t) +L_Y (t) =\overline {L}\)                              \(A^\cdot (t)=BL_A (t) A(t), B>0\)