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.
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:
Cost(q) = 2000*q + 5000*q/25 + 50000
Cost(q) = 2000*q + 200*q + 50000
Cost(q) = 2200*q + 50000
Notice that due to the form of the production function, there are no diminishing returns to labor; so the cost function is linear with respect to q.
Average cost is simply calculated as Cost/q. In this case we get:
Avg Cost = 2200 + 50000/q
Marginal cost is calculated as the first derivative of Cost with respect to q. This gives:
Marginal cost = 2200
As you can see, the marginal cost is constant ...
This job defines the cost function of a firm given the production function, rent and labor costs.