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.

  • 1
    $\begingroup$ Eq (5) results from "[m]aximization of the utility function (4) subject to the budget constraints (1)-(3)". Why don't you do that, and then tell us where exactly you're getting stuck. As it is, your question is simply a "do-it-for-me" post, which will be closed. $\endgroup$
    – Herr K.
    Aug 11, 2018 at 23:56

2 Answers 2


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.

  • 1
    $\begingroup$ Number of times the word "socialism" appears in this 37 page paper: 0. Number of times the word "social" appears in the paper: 1. From this it seems just as likely to me that the paper is a defense of social as that it is in some way a critique of the Italian novel "The Decameron". $\endgroup$
    – Giskard
    Aug 11, 2018 at 21:14
  • $\begingroup$ @denesp Analyzing a model in Economics takes much more effort than doing a Ctrl-F[ind] for certain keywords and reporting back with the word counts. $\endgroup$ Aug 11, 2018 at 21:20
  • $\begingroup$ @IñakiViggers: Two questions (1) why socialism? (2) Why is the utility formulation problematic? The authors are describing warm-glow altruism. $\endgroup$
    – erik
    Aug 11, 2018 at 21:51
  • 1
    $\begingroup$ @erik (1) Because the Chong-Gradstein paper essentially equates institutional quality with uniform levels of income; and it proposes the notion that, in strong institutions, $r$ (and, thus, a household's appropriation of a larger share of the resource) should be zero. (2) As explained above, that utility formulation is problematic because (i) its double-counting of $c$ overstates the role of consumption as source of utility (if V's 2nd parameter is $y$), or (ii) it leads to the absurdity that an improvement of institutional quality makes everyone worse off (if V's 2nd parameter is $r$). $\endgroup$ Aug 11, 2018 at 23:28
  • $\begingroup$ There is no "double counting" of the same consumption for the utility of the same person. In period $t$ the utility function is of the parent. In period $t+1$ the utility function is of the person that was a child in period $t$ and it is a parent in period $t+1$. So the expected future consumption of the child creates utility now for the parent, and utility tomorrow for the child, when materialized. There is no methodological problem here, just a standard consequence of allowing for "altruism" in preferences. And realistic too. $\endgroup$ Aug 13, 2018 at 2:20

There is nothing methodologically/technically wrong with the definition of the utility function (eq. $(4)$ of the paper). It is a utility function where parents derive utility also from the future income of their children, and more over without discounting it, even though it is in the future and even though it won't be their personal consumption. We can debate the realism of this formulation, namely whether it describes the majority of real-world parents as regards how they view the relation with their children, but technically it has no flaws.

As regards the optimal decision rules in equation $(5)$, they are correct. We have to

$$\max_{r_{t+1}}[V=\ln c_t + \ln y_{t+1}]$$

Using $(3)$ forwarded once we have $y_{t+1} = \varepsilon_{t+1} a_{t+1}$, while from $(2)$ we have

$$a_{t+1} = Ar_{t+1}^{w_{t+1}} \left (\int_0^1 r_{i,t+1}^{w_{t+1}}di\right)^{-1}$$

Using also the budget constraint $(1)$, the objective function can be re-written

$$V=\ln (y_t-r_{t+1}) + \ln \varepsilon_{t+1} + \ln a_{t+1}$$

$$= \ln (y_t-r_{t+1}) + \ln \varepsilon_{t+1} + \ln A + w_{t+1} \ln r_{t+1} - \ln \left (\int_0^1 r_{i,t+1}^{w_{t+1}}di\right)$$

Now note that $\varepsilon_{t+1}$ is a random variable, not a decision variable. Also, $A$ is a constant, and the last component that contains the integral is not differentiated w.r.t individual decision variables, because every individual has "measure zero", which is the fancy way to say that it is negligible with respect to the whole, and does not consider the effects that his actions has on the whole.

So the f.o.c will be, simply

$$\frac {\partial V}{\partial r_{t+1}}=-\frac {1}{y_t-r_{t+1}} + \frac {w_{t+1}}{r_{t+1}} =0$$

and from that we obtain eq. $(5)$.

I didn't read the whole paper, because I don't like its opaqueness as regards the mathematical steps. In other words, what I wrote above, however simple, should be spelled out at least in a Technical Appendix. Moreover, there is a curious mention about "productive investment" three lines below eq. $(4)$, that appears nowhere in the first pages of the paper.Perhaps in an initial version they had also productive investment in the model but then simplified it. But that's sloppy.

  • $\begingroup$ Thank you a lot for your explanations. Why not use any optimization method? $\endgroup$
    – Carl
    Aug 13, 2018 at 3:39
  • $\begingroup$ @Carl I am not sure I understand your question. What do you mean by "use any optimization method"? $\endgroup$ Aug 13, 2018 at 9:45

Not the answer you're looking for? Browse other questions tagged or ask your own question.