Endogenous Growth Models

Solow, RCK, and Diamond conclude that if capital's importance in production is measured by it's share in income, there is still no way to explain for the cross-country differences of income. Endogenous models include more proximate determinants of growth, such as human capital, social infrastructure, population, etc. 

R&D

Resources are divided between the goods/services sector and the R&D sector, where additions to the stock of knowledge are made. Output for the goods/services sector is thus
\(Y_t = [(1 - a_K) K_t]^\alpha [A_t (1 - a_L)L_t]^{1 - \alpha} \)
And the production of new ideas in the R&D sector is given by 
\(\dot A_t = B(a_K K_t) ^\beta (a_L L_t)^\gamma A_t ^\theta\), where the parameter theta reflects the effect of the existing stock of knowledge on the success of R&D.
Note that the growth rate of population is a major determinant of the growth of knowledge. Typically we expect theta to be less than 1, meaning decreasing returns to scale for ideas. This gives us a steady state value of the growth rate of knowledge:
 \(g_A ^* = \frac{\beta + \gamma}{1 - ( \theta + \beta)} n\), where n is population growth
When we add the growth rate of capital,
\(g_K ^* = g_A ^* + n\)
We can illustrate the dynamics using phase diagrams:

Kremer

The main prediction of endogenous growth models is that high population spurs technological change. Kremer tested this throughout history (rather than cross-country since knowledge is non-rival). 
\(Y_t = T^\alpha (A_t L_t)^{1 - \alpha}\), where T is a fixed stock of land
\(\dot A_t = B L_t A_t ^\theta\), meaning that additions to knowledge are proportional to population and dependent on previous knowledge
\(\frac{Y_t}{L_t} = \bar y\), meaning that population adjusts so that output per person always equals the subsistence level  
Thus, the important solutions to this model are:
\(\frac{T^\alpha (A_t L_t) ^{1 - \alpha}}{L_t} = \bar y\)
\(L_t = \frac{1}{\bar y} ^{\frac{1}{\alpha}} A_t ^{\frac{1 - \alpha}{\alpha}} T\)
\(\frac{\dot L}{L} = \frac{1 - \alpha}{\alpha} \frac{\dot A}{A}\), where the factor of population growth is \(\frac{1 - \alpha}{\alpha}\)
Therefore, the growth rate of population is proportional to the growth rate of technical progress; the growth rate of technical progress is roughly proportional to the level of population; and the population growth rate will be proportional to its level. This relatively simple model explains the spread of knowledge and growth around the world. 

Convergence

To get a better sense of what drives growth, researchers have moved to empirical methodologies like cross-country convergence regressions. They are implied from the dynamics of k:
\(\dot k = s k^\alpha - (n + g + \delta) k\)
\(\gamma _k \equiv \frac{\dot k}{k} = sk^{-(1 - \alpha)} - (n + g + \delta)\)
\(\frac{\partial \gamma _k}{\partial k} = -(1 - \alpha)sk^{-(2 - \alpha)} < 0\)
Meaning that the rate of the rate of growth of capital per effective worker is negative as you draw closer to the steady state level of capital, implying that poor countries (far from their steady state arguably) should "catch up."
This will be difficult to measure, since the steady-state level of capital should be country specific (given particular institutions, government, climate, resources, etc.) our output function will involve a country-specific parameter,
\(Y = CK^\alpha (AL)^{1 - \alpha}\)
Recall that once countries reach their steady-state, they continue to grow at the same rate (output per worker grows at rate g, of technology). Furthermore, each country's steady-state level of capital may be changing over time. 
We reach a point where there are more controls that countries: education, government, tax policy, trade liberalization, openness to FDI, privatization, deregulation, secure property rights, etc. 
To analyze this convergence, Diamond, Landes, Acemoglu, Johnson, and Robinson have taken a historical approach by looking at geography, The Industrial Revolution, and institutions respectively. 

IS-LM Model

IS curve

\(Y = C + I + G\)
\(C = C(Y - T, i - \pi ^e)\) where T is taxes, i is interest rate, and the pi term is expected inflation
\(I = I(i - \pi ^e, Y_{-1})\) where the last Y term is expected income
\(G = \bar G\)
Thus, the solution to this component is