Question

Suppose Noah and Naomi's short-run weekly production function for garden benches is

            F(L)=min{0,L−3},F(L)=min⁡{0,L−3},
    
where L represents the number of hours of labor employed. The wage rate is $12 an hour. For non-negative amounts of output, what is their short-run cost function?

	C = 36 + 12Q.
	C =12 +36Q.
	C =12Q.
	C = 36Q.
	C =36 - 12Q.

Answer 1

Production function: F(L)=min{0,L−3}

It means that a labour at work takes minimum 3 hours to prepare for production; and then it is able to give output. It means that first three hours of work does not convert into garden bench production. It further means that wages for these compulsory three hours will be $ 36 fixed (@ $12 per hour), that has to be paid anyway.

Now, the 'L-3' part in the production function represents that, if 4 labour hours are put, then 1 garden bench will be ready. If 5 labour hours are put, 2 garden benches will be ready. And so on...

From here, let us evaluate the various choices of cost function in the question:

C = 36 + 12Q. It keep $36 as compulsory labour cost and in addition to the same it also keeps labour for each bench produced be paid @ $12. This is the right cost function.
C =12 +36Q. It keeps 12 as compulsory labour cost, instead of $36, hence incorrect choice
C =12Q. It doesn't talk about compulsory labour cost or $36, hence incorrect choice
C = 36Q. It doesn't talk about compulsory labour cost, hence incorrect choice
C =36 - 12Q. It does the things right mostly, but errs on the minus sign. $36 have to paid in addition to the 12Q, This minus sign here should have been a + sign. This is also an incorrect choice.