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To say $Y$ differs by a factor of $X$ means $dY = X$, or equivalently $Y_2 - Y_1 = X$ I believe. But this does not make sense to me, as a factor is what multiplies another number, e.g. $2$ is a factor of $6$. In context:

Suppose we want to account for a difference of a factor of $X$ in output per worker between two economies on the basis of difference in capital per worker. If output per worker differs by a factor of $X$, the difference in log output per worker between the two economies is $lnX$. Since the elasticity of output per worker with respect to capital per worker is $\alpha_K$, log capital per worker must differ by $(lnX)/\alpha_K$. That is, capital per worker differs by a factor of $e^{(lnX)/\alpha_K} = X^{1/\alpha_K} $

I follow the derivation, it is simply the name "factor of $X$" that does not make sense, reading literally I think it means:

$$ X * \Delta Y = something $$

Where $\Delta Y = | Y_2 - Y_1 | $ Is the difference in output per worker

$E.g.$ If $Y_1 = 50, \; Y_2 = 100$ then intuitively $X$ is a factor of $2$ as $Y_2$ is twice as large as $Y_1$

$$ 2 *\Delta Y = 2 * 50 = 100 = max \{ Y_1, Y_2 \} $$

Otherwise, if they wrote ratio instead of difference, this makes much more sense

$$ \frac{Y_2}{Y_1} = X $$

Sorry if my question is confusing, I would like a clear mathematical expression for "$Y$ is a difference of a factor of $X$" please

Sunhwa
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    To differ by a factor means that one is a multiple of another. So $6$ and $2$ differ by a factor of $3$. In the example, one worker generated $X$-times the value of another worker. – Fly by Night Apr 21 '16 at 18:24
  • thank you, so $X * Y_1 = Y_2 $ which gives the ratio. thank you much! – Sunhwa Apr 21 '16 at 18:27

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