<p>Within the paper's discussion of formal models, I'm struggling to understand how the authors arrive at a certain equation. First, a model of the growth rate of economy is presented, which I do understand and of which I'm not sure it's relevant to my actual problem, but I'll include for context:</p>
<p><a href="https://i.sstatic.net/V1d2U.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/V1d2U.png" alt="enter image description here" /></a></p>
<p>Now the part where I'm stuck:</p>
<p><a href="https://i.sstatic.net/6prCs.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6prCs.png" alt="enter image description here" /></a></p>
<p>I don't understand how the paper arrives at (2) frpm the Cobb-Douglas function and what's happening with the bolded alpha. It would seem to have something to do with the geometric average and setting the price of domestic goods to 1, but I don't see how and I'm hoping someone could explain. I often don't understand algebraic computations unless someone shows me every step of the way so apologies if it should be clear.</p>

CC BY-SA 4.0
2022-01-28T12:10:50.210Z
Looks like there is a mistake in the equation. Have you studied Cobb-Douglas utility or Cobb-Douglas production? If you write out the (implicit) cost of the final good you should be able to figure out the price. The answer will be similar, but it won't be the same.
LudwigNagasena
CC BY-SA 4.0
2022-01-28T13:12:02.793Z
(1) For a given level of $I$, find the cost-minimizing combination of $I^d$ and $I^m$. (2) Calculate these minimal costs. (3) Divide the result by $I$ to find the implicit price $P_I$ of the investment good. (4) You will arrive at $P_I=\alpha^{-1}(P^m)^{(1-a)}$, so there's indeed a mistake in the paper.
VARulle
CC BY-SA 4.0
2022-01-28T15:48:10.880Z
Cost minimization in the manner of $MPl/w=MPk/r$ with a $Q=LK$ Cobb-Douglas function then? In that case I don't understand your step (3) @VARulle . My cost-minimizing outcome is $a/(1-a).P_m.I_m=I_d$ (with $P_d=1$). I don't understand how dividing this by I gets me equation (2)
Nasan
CC BY-SA 4.0
2022-01-29T05:36:31.380Z
You have found the cost minimizing condition, but it is not the minimal cost. Here is the cost derivation of the in general case, at the start of page 3 there is the resulting equation: http://coin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/cobb_douglas.pdf
LudwigNagasena
<cobb-douglas>
Trying to understanding how an equation is arrived at in Sach's et al "Geography and Economic Development"
Nasan
# Trying to understanding how an equation is arrived at in Sach's et al "Geography and Economic Development"

(Asked by: username_0 on 2022-01-28)

Within the paper's discussion of formal models, I'm struggling to understand how the authors arrive at a certain equation. First, a model of the growth rate of economy is presented, which I do understand and of which I'm not sure it's relevant to my actual problem, but I'll include for context:

[![enter image description here](https://i.sstatic.net/V1d2U.png)](https://i.sstatic.net/V1d2U.png)

Now the part where I'm stuck:

[![enter image description here](https://i.sstatic.net/6prCs.png)](https://i.sstatic.net/6prCs.png)

I don't understand how the paper arrives at (2) frpm the Cobb-Douglas function and what's happening with the bolded alpha. It would seem to have something to do with the geometric average and setting the price of domestic goods to 1, but I don't see how and I'm hoping someone could explain. I often don't understand algebraic computations unless someone shows me every step of the way so apologies if it should be clear.

**Tags:** cobb-douglas
> **Comments:**
> **username_1**: Looks like there is a mistake in the equation. Have you studied Cobb-Douglas utility or Cobb-Douglas production? If you write out the (implicit) cost of the final good you should be able to figure out the price. The answer will be similar, but it won't be the same.  
> **username_2**: (1) For a given level of $I$, find the cost-minimizing combination of $I^d$ and $I^m$. (2) Calculate these minimal costs. (3) Divide the result by $I$ to find the implicit price $P_I$ of the investment good. (4) You will arrive at $P_I=\alpha^{-1}(P^m)^{(1-a)}$, so there's indeed a mistake in the paper.  
> **username_0**: Cost minimization in the manner of $MPl/w=MPk/r$ with a $Q=LK$ Cobb-Douglas function then? In that case I don't understand your step (3) @VARulle . My cost-minimizing outcome is $a/(1-a).P_m.I_m=I_d$ (with $P_d=1$). I don't understand how dividing this by I gets me equation (2)  
> **username_1**: You have found the cost minimizing condition, but it is not the minimal cost. Here is the cost derivation of the in general case, at the start of page 3 there is the resulting equation: http://coin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/cobb_douglas.pdf  

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## 0 answers

