Production and Cost Minimization <–

Click the link above to access an explanation of cost minimization.
Let y – firm’s output, TC(y) – total cost, ATC(y)= TC(y)/y – average total cost. Given TC = VC+FC – the sum of fixed and variable cost. And, it must be borne in mind – the short run differs from the long run by the presence of the fixed cost, because all inputs are variable in the long run.

MC(y) = dTC(y)/dy .

The reason that MC intersects ATC and AVC in their minimums is that whatever MC curve below ATC and AVC curves, the latter will decrease (because MC – T additional cost and if MC is less then prior average cost so AC decrease), and vice versa, as long as MC curve exceeds ATC and AVC, the latter will increase.

Formal derivation:
MC(y) = dTC(y)/dy, ATC(y) = TC(y)/y – let’s find a minimum point. We equate the derivative to zero.

(ATC(y))’ = d(ATC(y))/dy = d(TC(y)/y)/dy = (y* MC(y) – TC(y) )/y^2 = 0

y^2 > 0, hence, y* MC(y) – TC(y) = 0, therefore
MC(y) = ATC(y) in the ATC minimum point (minimum, because in this point (ATC)’ < 0 under the condition MC < ATC, and under MC > ATC (ATC)’ > 0).

The case with AVC is similar, because d(TC(y)-FC)/dy = MC(y) too.


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