optimal combination of factors of production

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How does a firm choose the optimal combination of factors of production when it wants to minimize costs for a given output, and when it wants to maximize output for given cost?

As with the production function, the cost function or C = f (PKK + PLL) can be calculated for one or more than one variable factors of production. The more than one variable factor cost function corresponds to the long-run in which all factors are variable and a firm can choose at what level or scale to produce.
To maximize profit, a firm can either: (a) maximize output for a given cost; or, (b) minimize cost for a given level of output.
a) Maximize Output for Given Cost
Let us assume a firm wants to maximize output for a given cost = C. Assuming that PK and PL are fixed we can calculate: (i) the maximum amount of K; (ii) the maximum amount of L; and, (iii) all the various combinations ...