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in general the Leontief cost function is represented as:

$$c(w,y)=y\sum_{i=1}^kw_ix_i$$

where: $$y=\min\left[{\frac{x_1}{\alpha_1},...,\frac{x_n}{\alpha_n}}\right]$$

How does one go about estimating this econometrically if one does not know the underlying production function a firm exhibits (i.e we know the amount of $y$ produced however we don't know if the production technology is $\min\left[{\frac{x_1}{\alpha_1},...,\frac{x_n}{\alpha_n}}\right]$)?

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