I need to define the cost function of a firm given the production function, rent and labor costs. Also about the equation that would minimize costs and the respective ratio.

Question 2
Since there are are 5 (fixed) assembly machines installed at the plant, the production function becomes:

q = 5*5*L
q = 25*L

In order to find the cost function, we must find out how many teams are needed to build a single engine. This is easily done by isolating L from the production function:

L = q/25

Now, the cost function will be:

Cost(q) = 2000*q + 5000*L + 50000

The 2000*q term is the cost of the raw materials. The 5000*L term is what the firm pays in wages ($5000 per team). Finally, the 50000 comes from the fact that there are 5 machines and each machine costs $10000.

Replacing L as a function of q in this equation, as we found earlier, gives:

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