D28: Minimizing Production Costs

A firm produces its product from inputs of labor (L) and capital (K) according to the production function `Q=K^0.5 \times L^0.5`. It is interested in producing 30 units of the product and would like to know its minimum cost mix of inputs, and its total cost, for three capital price scenarios: `P_k=$70`, `P_k=$60`, and `P_k=$50`, For all three scenarios it would like the price of labor to be set at $50.

Use the spreadsheet in Teams to find the cost minimizing mix of inputs, and the firm's total cost, in each of the scenarios. Here are some tips on how to set up the spreadsheet_:

1. Use algebra to derive an equation that shows the amount of labor, L, the firm would need in order to make Q units of output when it is using K units of capital.
2. Fill in the Q with the target quantity and the `P_k` and `P_l` cells with the data for the first scenario.
3. In the Calculations section of the spreadsheet, fill in the K column with numbers from 20 to 35.
4. Insert an Excel Table in the area you just created.
5. Add the labor demand equation you derived in step 1 to the L column.
6. Add the output (Q) equation to the Q column.
7. Add an appropriate cost calculation to the Cost column.
8. Use Color Scales conditional formatting on the "Cost" column with the scale set to Red-Yellow-Green (i.e., red is largest).
9. Sort the table by cost from lowest to highest and record the appropriate data in the Results section.
10. Change the price of capital to the value for the second scenario and repeat the process of finding the lowest cost bundle and recording the results.
11. Repeat the process for the third case.
12. Answer the question at the bottom.

Save and submit the results!