Final answer:
To find the minimum cost of production for 1,000 units given a fixed capital and varying labor, one would use the inverted production function to determine the required labor, then calculate costs using wages and rent for labor and capital respectively. However, without adequate data from the production function, the exact cost cannot be determined.
Step-by-step explanation:
The student has asked about the minimum cost of production for a firm that uses a production function with both labor (L) and capital (K) inputs. Given the equation q(L,K) = AL¹K¹ where A=20, a=1.0, and b=1.0, with fixed capital in the short-run (K=8), we need to determine how many workers (L) are required to produce 1,000 units (Q) and the associated cost.
In the short-run scenario, we can simplify the production function, considering capital (K) is fixed, hence the function becomes Q = f[L, K] or simply Q = f[L]. The provided table can help us 'invert' the production function to show L = f(Q). Unfortunately, the necessary part of the provided table is missing, so we must infer how many workers are needed to produce 1,000 units (widgets).
Assuming we have the correct part of the inverted production function data that correlates the number of widgets to the number of workers, we could identify the L needed for 1,000 units. The cost of labor is found by multiplying the number of workers (L) by the wage rate (5), and the cost of capital is fixed since K is given and must be multiplied by the rent (40). Therefore, Minimum Cost = (Wage * L) + (Rent * K).
Without the specific number from the table, we can't calculate the exact figure, but the process involves finding L for Q = 1,000 from the inverted function, then plugging L into the cost formula: Minimum Cost = (5 * L) + (40 * K), where K=8.