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I'm struggling to derive the growth of consumption from a two-sector model with the traditional Cobb-Douglas function. The model I am speaking about incorporates the fractions used by physical and human capital inside the sectors with different elasticities and it is one mentioned in the Book: "Economic Growth" of Barro & Sala I. Martin, page 247.

Starting from the Hamiltonian: $$ J=u(C) {e^{ -\rho t}}+ \nu [A{(vK)^ \alpha}{(uH)^{1-\alpha}} - \delta K - C]+ \mu [B{[(1-v)K]^ \eta}{[(1-u)H]^{1- \eta }}- \delta H] $$

Where u(C) is the utility function, rho is the discount rate, $\nu$ and $\mu$ are shadow prices, A and B are the technology for each sector. K and H are the aggregates of physical and human capital respectively with $v$ and $u$ as fractions of each sector employed in the production. And the first-order conditions I calculated are given by:

$$\frac{\partial J}{\partial C}=0 = u'(C) {e^{ -\rho t}} -\nu \; \; (1) $$ $$\frac{\partial J}{\partial v}=0 = \nu A \alpha{(vK)^ {\alpha-1}(K)}{(uH)^{1-\alpha}} + \mu B \eta[(1-v)K]^{\eta-1}(1-v)[(1-u)H]^{1-\eta} \; \; (2) $$ $$\frac{\partial J}{\partial u}=0= \nu A {(vK)^ \alpha}(1-\alpha){(uH)^{-\alpha}(H)} + \mu B{[(1-v)K]^ \eta}(1-\eta){[(1-u)H]^{- \eta }(1-u)} \; \; (3) $$ $$ \frac{\partial J}{\partial K}=-\dot{\nu}=\nu A \alpha (vK)^{\alpha -1}(v){(uH)^{1-\alpha}} + \mu B\eta{[(1-v)K]^{\eta -1}(1-v)}{[(1-u)H]^{1- \eta }} \; \; (4) $$

$$ \frac{\partial J}{\partial H}=-\dot{\mu}= \nu A{(vK)^ \alpha}(1-\alpha){(uH)^{-\alpha}}(u) + \mu B{[(1-v)K]^ \eta}(1-\eta){[(1-u)H]^{- \eta}(1-u)} \; \; (5) $$

According to Barro & Sala I. Martin, wth Mino (1996) the consumption growth rate can be defined as:

$$\frac{\dot{C} }{ C}={\frac{1}{\sigma }}[A\alpha(vK)^{\alpha -1}{(uH)^{1-\alpha}}-\delta -\rho] $$ However, I cannot reach this answer, aside that net marginal product of capital is way different, they are missing fraction v in their answer. And manipulation of the differential equations relies on a non-specified result.

By putting as equal (2) and (3) I've found the relationship of elasticities with the fractions. Same answer up to that. However, regarding the growth of consumption, I used logs in (1) and then differentiate with respect to time. And overall consumption growth rate is:

$$\frac{\dot{C} }{ C}={\frac{1}{\sigma }}[-\frac{\dot{\nu}}{\nu}-\delta -\rho] $$

Using (4) I can just simply divide by $\nu$ and replace it in the general consumption growth rate, however... with (4) the answer implies that the growth rate of consumption also involves the production of the second sector (human capital sector) weighted by the shadow price $\nu$ or $\mu$. And $\mu$ can be cleared out from (2) or (3) or the equity in both.

Here's when I'm starting to get lost because I wasn't able to find a combination between the set of equations (1) to (5) to reach the expression of consumption they show in the book (which I also wrote up here). My closest form to their answer is clearing $\mu$ from (2) and replacing it (4) which leads to the result for me as:

$$\frac{\dot{C} }{ C}={\frac{1}{\sigma }}[Av\alpha(vK)^{\alpha -1}{(uH)^{1-\alpha}}[1+\frac{(1-v)}{v}] -\delta -\rho] $$

So... any ideas of how to get that consumption growth rate they explicitly define in the book? If anything else is needed to compute the answer, please ask away.

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    $\begingroup$ Sometimes it looks us in the face but we don't see it. $$[1+\frac{(1-v)}{v}] = \frac{v+1-v}{v} = \frac 1 v$$ which cancels out with the "$v$" in "$Av\alpha$" and you arrive at the same expression. $\endgroup$ – Alecos Papadopoulos Jan 7 at 9:07
  • $\begingroup$ That was the reason!!! thank you very much Dr. Alecos, I was struggling with it for over a month, and asking a lot of people but so clearly it was from the beginning !!. Lots of regards!! $\endgroup$ – John M. Riveros Jan 7 at 20:15

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