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I need to show whether or not the following function is increasing, decreasing, or constant returns to scale, but I can't figure out how to extract the t from it when it is put in the form $y(tX_1, tX_2)$. $$ y = X_2\sin{\left(\frac{X_1}{X_2}-\pi \right)} $$ Is there any other way of showing IRS, DRS, and CRS? Any help would be appreciated!

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You have $$F(x_1,x_2)=x_2\sin\bigg(\frac{x_1}{x_2}-\pi\bigg).$$ If you multiply all factors by $t$ you get $$F(tx_1,tx_2)=t x_2\sin\bigg(\frac{tx_1}{tx_2}-\pi\bigg).$$ Simplify the fraction and you are almost done.

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in the case of for example a production function, returns to scale mean the effect when all factors of production are multiplied by a certain constant, with what constant is the production level multiplied? i) the same constant as the factors of production (constant returns to scale) ii) a larger constant (increasing returns to scale) iii) a smaller constant (decreasing returns to scale) So the right approach is to simply fill in a constant (let's call it c) in your function. cX2(sin((cX1/cX2)-pi) and try to rewrite it to d(X2(sin((X1/X2)-pi)) where d is the constant with which y is multiplied as a consequence of multiplying X1 and X2 with c

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