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Unit cost function is the minimum cost to produce one unit of output.

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The solution to the above problem is:

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where ai(w,r) is the amount of labour needed to produce 1 unit of good i at factor prices (w,r).

How do we arrive at this specific solution by finding the optimal values of K and L?

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Here $a_{iL}(w,r)$ and $a_{iK}(w,r)$ is just notation for the solutions to the cost minimization problem. There is no need for any derivation. It could only be possible to find a specific solution (i.e. solve the cost minimization problem to find the functions $a_{iL}$ and $a_{iK}$) if you were given a specific production function $f_i$.

[I suppose here $f_i$ must be constant returns to scale to make the idea of a unit cost function useful.]

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