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 difference in income. Endogenous growth theory models share the conclusion that technology is the key component for growth, though it is nonrival and thus cannot account for cross-country differences. 
There is much empirical work being done on cross-country income differences. A first group analyzes the marginal products (prices) of the factors of production to analyze their contributions to the income differences. It is strong because we can be confident in the conclusions, though it considers only immediate determinants of income. The second group looks at deep determinants, like political institutions, geography, and religion. Unfortunately there are too many variables to account for and too few countries, thus we cannot be confident in the empirical conclusions. 
The analysis in the following sections will cover the most obviously important inputs: human capital (a close proximate determinant of countries' incomes) and social infrastructure.

4.1  Extending the Solow Model to Include Human Capital

Assumptions

The model is set in continuous time, thus output is
\(Y = K(t) ^\alpha [A(t)H(t)]^{1 - \alpha}\)
Dynamics of capital and technology are also the same as in the Solow model:
\(\dot K(t) = sY(t) - \delta K(t)\)
\(\dot A(t) = gA(t)\)
The model revolves around the assumption that human capital accumulation depends both on the amount of human capital created by a given set of resources (a production function) and how many resources are given. The assumption for production is that the only determinant is a worker's years of education (more years, more human capital). With regard to the amount of resources devoted to human capital, the parallel is that of physical capital where investment is exogenously driven. This is given by:
\(H(t) = L(t) G(E)\)
Where L is the number of workers, and \(G(•)\) is a function giving human capital per worker as a function of years of education per worker. Per usual, the number of workers grows exogenously:
\(\dot L(t) = nL(t)\)
While we have every reason to assume that \(G'(•) > 0\) is positive because the number of years of education would improve human capital, but there is no reason to impose \(G''(•) > 0\). Microeconomic evidence suggests that each additional year of education increases an individual's wage by the same percentage amount. Therefore, we have:
\(G(E) = e^{\phi E} , \phi > 0\)

Analyzing the Model

Since the dynamics of capital are the same, the formula is identical to Solow's:
\(\dot k(t) = sk(t)^\alpha - (n + g + \delta)k(t)\)
Since the steady state is when the rate of capital converges to \(\dot k = 0\), we denote
\(k* = \frac{s}{(n + g + \delta)}^{\frac{1}{1 - \alpha}}\)
Output per worker is equal to output per unit of effective labor services times effective labor services per worker, thus:
\(AG(E) = \frac{Y}{L} = AG(E) y\)
This path is not affected by the change in the saving rate, since A rows at exogenous rate g and G(E) is constant. Looking at the long-run effects of a rise in the number of years of schooling per worker, we note that E does not enter the equation for the balanced-growth-path value of k. However, we can see from the above equation that it would increase output per worker by the same proportion that it increase G(E). 

Students and Workers

Since our analysis depends on output per worker, a change in the amount of time individuals spend in school changes the proportion of the population that is working, thus, output per person and output per worker would behave differently. 
Now we need more demographic assumptions. Naturally, we model that each individual has a fixed lifespan (T), and spends the first E years of life in school and the remaining (T - E) years working. Furthermore, for the the overall population to be growing at rate n, we would assume that the number of people born per unit of time grows at rate n. 
Using N(t) to denote the population at t and B(t) to denote the number of people born at t,
\(N(t) = \int_{\tau = 0} ^T B(t - \tau) d \tau\)
\(= \int_{\tau = 0} ^T B(t) e^{- n \tau} d \tau\)
\(= \frac{1 - e^{-n T}}{n} B(t)\)
Similarly, the number of workers at time t equals the number of individuals who are alive and no longer in school, or, the number of people born from t - T to t - E:
\(L(t) = \int_{\tau = E} ^T B(t - \tau) d \tau\)
\(= \int_{\tau = E} ^T B(t) e^{- n \tau} d \tau\)
\(\frac{L(t)}{N(t)} = \frac{e^{-nE} - e^{-n T}}{n} B(t)\)
Now we have the ratio of the number of workers to the total population:
\(\frac{L(t)}{N(t)} = \frac{e^{-nE} - e^{-n T}}{1 - e^{-nT}}\)
Now we can find the output per person (as opposed to per worker) on the balanced growth path. Output per person equals output per unit of effective labor services, y, times the amount of effective labor services supplied by the average person:
\((\frac{Y}{N})^* = y^* A(t) G(E) \frac{e^{-nE} - e^{-n T}}{1 - e^{-nT}}\)
Where y* equals f(k*), output per unit of effective labor services on the balanced growth path. Thus our analysis implies that a change in the amount of education each person receives alters output per person on the balanced growth path by the same proportion that it changes,
\(G(E)[\frac{(e^{-nE} - e^{-nT})}{(1 - e^{-nT})}]\)
A rise in education therefore has two effects on output per person: each worker has more human capital so G(E) rises, but a smaller fraction of the population is working. So in the long run, a rise in E can either raise or lower output per person. Since only the newest time period of population would receive the higher amount of education, the average level of education in the labor force does not reach its new balanced-growth-path value until date \(t_0 + T\).

4.2  Empirical Application: Accounting for Cross-Country Income Differences

A central goal of accounting-style studies of income differences is to decompose those differences into the contributions of physical-capital accumulation, human-capital accumulation, and other factors. Hall and Jones (1999) and Klenow and Rodríguez-Clare (1997) measure differences in the accumulation of physical and human capital to find if we should focus on factors that affect one or the other. 

Procedure 

Assuming Cobb-Douglas output:
\(Y_i = K_i ^\alpha (A_i H_i)^{1 - \alpha}\)
Dividing both sides by the number of workers, L, and taking logs:
\(\ln \frac{Y_i}{L_i} = \alpha \ln \frac{K_i}{L_i} + ( 1 - \alpha) \ln \frac{H_i}{L_i} + (1 - \alpha) \ln A_i\)
Since A is the residual, the main idea of these papers is to directly measure all other ingredients. However, it would be more useful to have a decomposition that attributes all the increase to the residual, so subtracting \(\alpha \ln (\frac{Y_i}{L_i})\) from both sides:
\((1 - \alpha) \ln \frac{Y_i}{L_i} = (\alpha \ln \frac{K_i}{L_i} - \alpha \ln \frac{Y_i}{L_i}) + (1 - \alpha) \ln \frac{H_i}{L_i} + (1 - \alpha) \ln A_i\)
\(\ln \frac{Y_i}{L_i} = \frac{\alpha}{1 - \alpha} \ln \frac{K_i}{Y_i} + \ln \frac{H_i}{L_i} + \ln A_i\)
This is most insightful, since it assigns the long-run effects of changes in labor services per worker and the residual entirely to those variables. 

Data,  Basic Results, and More Detail on Capital

From data in the Penn World Tables, Hall and Jones argue that microeconomic evidence suggests that the percent increase in earnings from an additional year of schooling falls as the amount of schooling rises. Thus they assume that the slope for the function is 0.134 for E below 4 years, 0.101 for E between 4-8 years, and 0.068 for E above 8 years.
Summarizing their results by comparing the five richest countries in their sample with the five poorest, they find that output per worker in the rich group exceeds the average in the poor group by a massive factor of 31.7. They also find a substantial correlation between physical capital, schooling, and the residual (meaning some of the unexplained difference would be due to the interaction of these variables). 
Expanding the analysis of human capital, a straightforward way to incorporate nuances would be comparing the wages earned in the same labor market of different countries. With immigration so common to the US, this can be done although there is certainly selection bias. The finding is that immigrants from lower income countries tend to earn less (holding the number of schooling years constant). Another way to measure is by separating low-skill and high-skill workers. 
This has also led to more analysis of physical capital. By examining the investment-output ratio, Hsieh and Klenow find that as we move from rich to poor countries, there is only a quarter decline in the investment-output ratio when reducing the saving rate. Almost none comes from increases in the price of investment goods. Thus we are still puzzled by the fact that nontradable consumption goods are cheaper in poorer countries. 

4.3  Social Infrastructure 

Hall and Jones define social infrastructure as the institutions and policies that align private and social returns to activities. There are two main categories of these returns: activities which are various types of investment and activities  which increase an individual's current benefit (diversions and production). Another way to describe these types are investments in future output, working toward current output, and diversions would be reallocating output, which is also called rent-seeking. 
There are also three aspects of social infrastructure: government's fiscal policy (i.e. tax treatments impact private and social returns), factors that determine the environment of private decisions (i.e. crime level), and the extent of rent-seeking (i.e. bribes). An excellent example of poor social infrastructure would be the Stalinist kleptocracy (& modern day... see Hudson Institute). 

4.4  Empirical Application: Social Infrastructure and Cross-Country Income Difference

A Regression Framework

Suppose income in country i is determined by social infrastructure and other forces:
\(\ln (\frac{Y_i}{L_i}) = a + bSI_i + e_i\)
Where output per worker equals the social infrastructure (SI) and other influences on income (e). Attempts to estimate this relationship confronts two major problems: how to measure SI and how to gain accurate estimates that measure SI without it being correlated with the residual variable (e). In short, the OLS estimate in the above equation will suffer from omitted variable bias (OVB). To counteract this, instrumental variables are chose that will absorb the interplay between the independent variables to identify the impact on the dependent variable. The equation becomes:
 \(\ln (\frac{Y_i}{L_i}) = a + b \hat {SI_i} + b(SI_i - \hat {SI_i}_i) + e_i\)
\(\equiv a + b \hat {SI_i} + u_i\)

Implementation and Results

Hall and Jones have assessed this question using data from consulting companies that measure the quality of countries' institutions for doing business. In arguing that the main channel through which incomes in the world are effected is Britain's social infrastructure, they propose four channels: fraction of native English speakers, fraction of major European speakers, distance from the equator, and measure of geographic influences on openness to trade. Unfortunately these instruments are less than convincing. Another approach is to use natural experiments, which are situations that bring about random tests by chance.   

4.5  Beyond Social Infrastructure

Although somewhat intuitive, the hypothesis that social infrastructure is crucial to income does not have clear implications for policy or measurability. Thus, current research is going beyond this in three main ways. 

Looking within Social Infrastructure

One way is to dig deeper into what specific features matter. Therefore, multiple authors ask whether policies, institutional constraints, property rights, lack of corruption, etc. are important to growth rates. However, it is still impossible to learn which subset of SI is most important since they are all so interrelated. 

The Determinants of Social Infrastructure

One set of speculations focuses on incentives, particularly in the stream of public choice thought. Similarly, a stream of study focuses on culture arising from religion, family structure, etc.  A third group of ideas looks at geography.

Other  Sources of Cross-Country Income Differences

These studies try to emphasize how culture and geography can impact income directly (instead of through social infrastructure). A big new idea is the externalities from capital -- that it might attract saving and thus spiraling into a virtuous circle. Finally, research has focused on colonization to study why there has been a "great reversal" where the most developed colonies are usually the least developed areas today. One theory has been the difference between extractive vs. settlers colonies. 

4.6  Differences in Growth Rates

Another way to go about this problem is by noticing that the incomes are not fixed, and thus asking why do they change when they do?

Convergence to Balanced Growth Paths

Assume the underlying determinants of long-run relative income per person across countries are constant over time. Therefore, output is determined by:
\(\frac{Y_i(t)}{L_i(t)} = A(t) f(k_i(t))\)
From before, we know that the rate of change of k is approximately proportional to its departure from its balanced-growth-path value:
\(\dot k_i = \lambda [k_i * - k_i(t)]\)
This implies that when a country is father below its balanced growth path, its capital per unit of effective labor rises more rapidly and so its growth in income per worker is greater. This is known as unconditional convergence. In the post-war period, differences in growth among industrialized countries are explained very well by this. Thus the deep factors must be the same within countries, and the variation was dependent on the original level. 

Changes in Fundamentals

Now we will assume that growth is not necessarily constant after shocks to growth. This will give us:
\(\Delta k_{it + 1} = \lambda (k^* _{it} - k_{it})\)
To consider the continuous time case, this becomes:
\(k_i(T) - k_i(0) = (1 - e^{-\lambda T}) [k_i ^* (0) - k_i (0)] + \int _{\tau = 0} ^T (1 - e^{-\lambda (T - \tau)} \dot k_i ^* (\tau) d \tau\)
The first term depends on the country's initial position relative to its balanced growth path (thus measures conditional convergence). The second term depends on changes in the balanced growth path during the time period. Again, it is tempting to infer that there are strong forces for worldwide convergence but evidence does not support this claim. 

Growth Miracles and Disasters

Very rapid or very slow growth can be caused by shocks that change the level on an economy's path or its fundamentals. The best example of the former may be West Germany after WWII. Such miracles and disasters also tend to be most common under dictators.