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Here is a composite goods production function:enter image description here

And here is the price ratio of Ys and Yg, derived from their marginal products: enter image description here

Then the author normalized the price of final goods Y to 1 and somehow got the following relationship:enter image description here

How is the last equation derived?

Thank you!

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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 $\frac{p_g}{p_Y} = p_g= \left(\frac{Y}{Y_g}\right)^{\frac{1}{\rho}}$.

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  • $\begingroup$ 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. $\endgroup$
    – user10158324
    Aug 1, 2018 at 19:36
  • $\begingroup$ 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 $1$. $\endgroup$
    – Green.H
    Aug 2, 2018 at 14:31

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