<p>Rewrite the first equation as
$$Y^{\frac{\rho-1}{\rho}} = Y_s^{\frac{\rho-1}{\rho}} + Y_g^{\frac{\rho-1}{\rho}}. $$</p>

<p>Divide both sides by $Y_s^{\frac{\rho-1}{\rho}}$ to obtain</p>

<p>$$\left(\frac{Y}{Y_g}\right)^{\frac{\rho-1}{\rho}} = 1 + \left(\frac{Y_s}{Y_g}\right)^{\frac{\rho-1}{\rho}}.$$</p>

<p>Substitute the price-ratio and production-ratio relationships as in your second equation and make some algebraic changes, and you will get your last equation.  Note: I interpreted ``the price of $Y$ normalized to $1$'' as $\frac{p_g}{p_Y} = p_g= \left(\frac{Y}{Y_g}\right)^{\frac{1}{\rho}}$.</p>

CC BY-SA 4.0
2018-08-01T19:36:21.980Z
Thank you! It seems to make sense. Just wondering how you got your last equation, i.e. price ratio between Pg and Py. I tried Marginal Product of g over Y but it doesn’t work.
user10158324
CC BY-SA 4.0
2018-08-02T14:31:45.753Z
I just used the fact that price ratio is equal to the reserve products ratio of some power in your second equation as well as the statement that the price of $Y$ is normalized to $1$.
Green.H
CC BY-SA 4.0
2018-08-01T16:23:38.337Z
2018-08-01T16:23:38.337Z
Green.H
<p>Here is a composite goods production function:<a href="https://i.sstatic.net/uBbF1.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uBbF1.jpg" alt="enter image description here"></a></p>

<p>And here is the price ratio of Ys and Yg, derived from their marginal products: <a href="https://i.sstatic.net/PfVzK.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PfVzK.jpg" alt="enter image description here"></a></p>

<p>Then the author normalized the price of final goods Y to 1 and somehow got the following relationship:<a href="https://i.sstatic.net/WvYCL.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WvYCL.jpg" alt="enter image description here"></a></p>

<p>How is the last equation derived?</p>

<p>Thank you!</p>

<microeconomics><production-function>
Price Derivation in a Production Function
user10158324
# Price Derivation in a Production Function

(Asked by: username_0 on 2018-08-01)

Here is a composite goods production function:[![enter image description here](https://i.sstatic.net/uBbF1.jpg)](https://i.sstatic.net/uBbF1.jpg)

And here is the price ratio of Ys and Yg, derived from their marginal products: [![enter image description here](https://i.sstatic.net/PfVzK.jpg)](https://i.sstatic.net/PfVzK.jpg)

Then the author normalized the price of final goods Y to 1 and somehow got the following relationship:[![enter image description here](https://i.sstatic.net/WvYCL.jpg)](https://i.sstatic.net/WvYCL.jpg)

How is the last equation derived?

Thank you!

**Tags:** microeconomics, production-function

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## 1 answers

### Answer by username_1 on 2018-08-01. Score: 1

Rewrite the first equation as
$$Y^{\frac{\rho-1}{\rho}} = Y\_s^{\frac{\rho-1}{\rho}} + Y\_g^{\frac{\rho-1}{\rho}}. $$

Divide both sides by $Y\_s^{\frac{\rho-1}{\rho}}$ to obtain

$$\left(\frac{Y}{Y\_g}\right)^{\frac{\rho-1}{\rho}} = 1 + \left(\frac{Y\_s}{Y\_g}\right)^{\frac{\rho-1}{\rho}}.$$

Substitute the price-ratio and production-ratio relationships as in your second equation and make some algebraic changes, and you will get your last equation. Note: I interpreted ``the price of $Y$ normalized to $1$'' as $\frac{p\_g}{p\_Y} = p\_g= \left(\frac{Y}{Y\_g}\right)^{\frac{1}{\rho}}$.

> **Comments:**
> **username_0**: Thank you! It seems to make sense. Just wondering how you got your last equation, i.e. price ratio between Pg and Py. I tried Marginal Product of g over Y but it doesn’t work.  
> **username_1**: I just used the fact that price ratio is equal to the reserve products ratio of some power in your second equation as well as the statement that the price of $Y$ is normalized to $1$.  

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