A few questions in paper Chong 2007 I think it is hard to derive Eq(5) in paper Chong (2007).
 (http://www.bgu.ac.il/~grade/inst_ineq.pdf) Could anyone share any ideas on its derivations or intuitions behind it? Let's discuss it in more details to help us understand more about this paper. Thank you a lot.
Could you help me derive Eq(5) in paper Chong (2007)?


My initial derivations were wrong because I misread Eq(1) as $y_{i,t}=c_{i,t}+r_{i,t}$. It was until I read the answer posted by @AlecosPapadopoulos that I realized that the paper actually uses $y_{i,t}=c_{i,t}+r_{i,t+1}$.

However, apropos of the latest edit of OP's inquiry (now to discuss the intuitions behind [the paper]), I still posit that the paper is a subtle promotion of socialism premised on flawed or impractical notions.

The paper makes the point (and Alecos' answer presents a clear derivation of Eqs(5)) that the optimal value of rent seeking $r_{i,t+1}$ is $0$ for strong institutions, which are denoted by $w=0$. This means that no household should pursue rent seeking. Or equivalently (see Eq(2)), that no household should seek to appropriate a larger share of the resource (page 6). Regardless of the model's time horizon (generations instead of shorter periods), that differs very little (if at all) from socialism.

Additionally, since $y_{i,t}=c_{i,t}+r_{i,t+1}$, the optimal choice in strong institutions would be $y_{i,t}=c_{i,t}$. That is, the household should allocate its entire income to consumption. Obviously, that generalized policy renders a real-life economy highly vulnerable to contingencies and structural changes. That vulnerability inherently contradicts any notion of "strong" institution.

Furthermore, substituting $w=0$ and "optimal" $r_{i,t+1}=0$ in Eq(2) gives $a_{i,t+1}=A\frac{0^0}{\int^1_0{0^0di}}$. What are we supposed to do with that?

Another problem is the model's inter-generational framework. I appreciate Alecos' emphasis/clarification that different values of $t$ for a same household $i$ allude to different generations of that household. 

But that modeling horizon is hardly realistic because each household experiences too many or too profound transformations (oftentimes just from one generation to the next one) to be still considered a single, continued entity. Indeed, households go through marriages, divorces, remarriages, child bearing in each marriage, emigration, bankruptcy, and so forth.

It is not necessarily realistic to assume that $w$ is constant throughout generations either, as the paper subsequently assumes.