Quick Lube production function:

Q=9.8L-0.5L^2

The price of oil changes is, P= $10

Quick Lube uses only labour as input. The lost of one labour is C=$18.

Let L be the labours employed.

• Profit = Revenue- Cost
• =P*Q - C*L
• =10*(9.8L-0.5L^2)-18L
• =98L-5L^2-18L
• =80L-5L^2

Using First order condition, therefore, differentiating w.r.t. L

80-10L=0

L=8

The optimal number of labour to hires are 8.

Q=9.8*8-0.5*8^2

=78.4-32

=46.4

46.4 oil changes will be completed.

Profit=80L-5L^2

=80*8-5*8^2

=640-320

=320

The profit of Quick Lube is $320.

Now, one morning a group of concerned community members (numbering 15 people) show up at Quick Lube and offer to work all day for free. Therefore, the cost is zero for hiring new labour.

Profit= Revenue-Cost

Since cost=0,

Profit= Revenue

=P*Q

=10(9.8L-0.5L^2)

=98L-5L^2

Using FOC,

0=98-10L

L=9.8

We want only 9.8 workers to work. While using 9.8 labour, our profit is $480.2. We will send remaining workers to go back. Because, if we use all 15 workers, then our profit will decreases.

Profit=98*15-5*15^2

=1425-1125

=$300