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Math  /  Data & Statistics

Question\begin{tabular}{|c|c|c|c|c|} \hline & Unit resource & & & \\ \hline & prices & Method \#1 & Method \#2 & Method \#3 \\ \hline Land & \$1 & 6 & 3 & 4 \\ \hline Labor & 2 & 4 & 3 & 3 \\ \hline Capital & 3 & 3 & 4 & 6 \\ \hline Entrepreneurship & 4 & 2 & 4 & 1 \\ \hline \end{tabular} (a) Which method is most efficient? Why? (b) Given the above prices, will the firm adopt a new method which involves 10 units of land, 3 of labor, 2 of capital, and 2 of entrepreneurial ability? (c) Suppose the price of capital falls to \$1 without any other prices changing. Which of the methods will the firm now choose? Why?

Studdy Solution

STEP 1

1. Efficiency is determined by the total cost of resources used in each method.
2. The firm will choose the method with the lowest total cost.
3. The cost of each resource is calculated by multiplying the unit price by the number of units used.

STEP 2

1. Calculate the total cost for each method.
2. Determine the most efficient method.
3. Evaluate the cost of the new method and decide if the firm will adopt it.
4. Recalculate costs with the new capital price and determine the most efficient method.

STEP 3

Calculate the total cost for Method \#1:
Total Cost for Method #1=(1×6)+(2×4)+(3×3)+(4×2)\text{Total Cost for Method \#1} = (1 \times 6) + (2 \times 4) + (3 \times 3) + (4 \times 2)
=6+8+9+8=31= 6 + 8 + 9 + 8 = 31

STEP 4

Calculate the total cost for Method \#2:
Total Cost for Method #2=(1×3)+(2×3)+(3×4)+(4×4)\text{Total Cost for Method \#2} = (1 \times 3) + (2 \times 3) + (3 \times 4) + (4 \times 4)
=3+6+12+16=37= 3 + 6 + 12 + 16 = 37

STEP 5

Calculate the total cost for Method \#3:
Total Cost for Method #3=(1×4)+(2×3)+(3×6)+(4×1)\text{Total Cost for Method \#3} = (1 \times 4) + (2 \times 3) + (3 \times 6) + (4 \times 1)
=4+6+18+4=32= 4 + 6 + 18 + 4 = 32

STEP 6

Determine the most efficient method by comparing total costs:
- Method \#1: 31 - Method \#2: 37 - Method \#3: 32
The most efficient method is Method \#1 because it has the lowest total cost.

STEP 7

Calculate the total cost of the new method:
Total Cost for New Method=(1×10)+(2×3)+(3×2)+(4×2)\text{Total Cost for New Method} = (1 \times 10) + (2 \times 3) + (3 \times 2) + (4 \times 2)
=10+6+6+8=30= 10 + 6 + 6 + 8 = 30
Since the cost of the new method (30) is less than the cost of the current most efficient method (31), the firm will adopt the new method.

STEP 8

Recalculate the costs with the new capital price (\$1):
- Method \#1: $ (1 \times 6) + (2 \times 4) + (1 \times 3) + (4 \times 2) = 6 + 8 + 3 + 8 = 25 \]
- Method \#2: $ (1 \times 3) + (2 \times 3) + (1 \times 4) + (4 \times 4) = 3 + 6 + 4 + 16 = 29 \]
- Method \#3: $ (1 \times 4) + (2 \times 3) + (1 \times 6) + (4 \times 1) = 4 + 6 + 6 + 4 = 20 \]
The firm will choose Method \#3 because it now has the lowest total cost (20).
The solutions are: (a) Method \#1 is the most efficient initially. (b) The firm will adopt the new method with a total cost of 30. (c) With the new capital price, the firm will choose Method \#3.

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