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Profit Maximization, Cost Minimization

Prove that profit maximization implies cost minimization but not vice versa.

I'm looking for a mathematical proof (I think its involving convexity/concavity, I'm not quite sure?)

The types of proofs we learned in class are:

the proofs i learned in class are

Direct Proof. Assume that A is true, deduce various consequences and use them to show that B must also hold.

Contrapositive proof. Assume that B does not hold, then by deducing various consequences show that A cannot hold.

Proof by contradiction. Assume that A is true and B is not true, then show that these assumptions imply a logical contradiction.

I think the answer should satisfy one of the above. I'm not quite sure though.

Solution Preview

See the attached file where formatting is conserved.
Cost minimization means producing a given output quantity at minimal cost.

For Profit maximization output is no longer given. We assume that a company chooses inputs and output in order to maximize profits.

Profit maximization implies cost minimization but cost minimization does not imply profit maximization.

This is because:

Profit maximizing firms choose the optimal level of inputs to maximize profits and also choose the profit maximizing level of output (supply). In order to maximize profits firms have to be minimizing costs at the optimal level of output so profit maximization implies cost minimization.

Whereas

Cost minimizing firms choose the optimal level of input use to minimize costs for a given level of output but do ...

Solution Summary

A mathematical proof is given for proving that profit maximization implies cost minimization but not vice versa.

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