Supposed that a firm's daily output is
Q = 1.5L^0.76 K^0.24
Q = Daily output
L = number of workers
K = number of machines used per day
Price per output = $10
If wages of a worker is $30 a day, how many workers(L) per unit of output should the firm hire?
Here's what I did, to optimize cost, we use the equation marginal price for workers = marginal price for machines
MPL/PL = MPK/PK.
MPL = dQ/dL = 1.14(K/L)^0.24
MPK = dQ/dK = 0.36(L/K)^0.76 = 0.36(K/L)^(-0.76)
MPL/MPK = PL/PK = 1.14K/0.36L = 30/PK
Now here's where I'm stuck. What can i do to $10 per unit of output to answer the question? Do I even need that?
To determine the number of workers (L) per unit of output that the firm should hire, you need to set up an equation using the given information. However, the price per unit of output is not needed to solve this problem.
Start by rearranging the equation MPL/MPK = PL/PK to solve for L:
MPL/MPK = 1.14K/0.36L
Now, cross-multiply to get:
0.36L = 1.14K
Divide both sides of the equation by 0.36 to isolate L:
L = 1.14K / 0.36
Simplify the right side:
L = 3.17K
This equation tells you that for every unit of output the firm produces, they should hire approximately 3.17 workers.
So, the firm should hire about 3.17 workers per unit of output to optimize cost, given the wage of $30 per worker per day.