Let L be the number of person-hours of labor and K be the number of machine hours of capital needed to produce 20 units of output.
From the problem statement, we know that the production function is given by:
Q = f(L, K) = 1 unit of output = 5 person-hours of labor + 3 machine hours of capital
To produce 20 units of output, the firm needs:
20 units of output x (5 person-hours of labor + 3 machine hours of capital) per unit of output = 100 person-hours of labor + 60 machine hours of capital
The total cost of producing 20 units of output is:
Cost = (wage rate x person-hours of labor) + (price of capital x machine hours of capital)
Cost = ($10 x 100) + ($50 x 60)
Cost = $1,000 + $3,000
Cost = $4,000
To minimize the cost of production, the firm will use the combination of inputs that produces 20 units of output at the lowest cost. This means that the firm will use the input combination that satisfies the following condition:
Marginal Rate of Technical Substitution (MRTS) = (Marginal Product of Labor) / (Marginal Product of Capital) = (wage rate) / (price of capital)
The marginal product of labor (MPL) is the additional output produced by hiring an additional unit of labor, holding the amount of capital constant. In this case, MPL = 1/5 unit of output per person-hour of labor.
The marginal product of capital (MPK) is the additional output produced by using an additional unit of capital, holding the amount of labor constant. In this case, MPK = 1/3 unit of output per machine hour of capital.
Therefore, the MRTS is:
MRTS = (MPL) / (MPK) = (1/5) / (1/3) = 3/5
The firm will use the combination of inputs that satisfies the MRTS condition. Let x be the number of person-hours of labor used and y be the number of machine hours of capital used. Then, we have:
MRTS = (wage rate) / (price of capital) = (10) / (50) = 1/5 = (dL / dK) / (wage rate / price of capital) = (dL / dK) / MRTS