Final answer:
To minimize cost, the firm needs to find the combination of labor and capital that produces 40 units of output at the lowest cost. The value of L∗ is 4000.
Step-by-step explanation:
To minimize cost, the firm needs to find the combination of labor and capital that produces 40 units of output at the lowest cost. The production function is Q(L,K)=L¹/³K²/³. We are given that the labor wage (w) is 2 and the capital rental rate (r) is 2∗w. To find the value of L∗, we need to determine the number of units of labor required to produce 40 units of output.
1. We can set the production function equal to 40 and solve for L∗:
40 = L∗¹/³K²/³
2. Since we are given a capital rental rate of 2∗w, we can substitute this into the production function:
40 = L∗¹/³(2∗w)²/³
3. Simplify the expression:
40 = L∗¹/³(4w²/³)
4. Raise both sides of the equation to the power of 3:
40³ = L∗(4w²)
5. Solve for L∗:
L∗ = (40³)/(4w²)
Substituting the given value of w=2, we can calculate L∗:
L∗ = (40³)/(4(2)²)
L∗ = (64000)/(16)
L∗ = 4000
Therefore, the value of L∗ is 4000.