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Sagot :
Let's solve the problem in a detailed, step-by-step manner.
### Part (a) Compute Labor Productivity
Labor productivity is defined as the average number of carts produced per worker per hour.
#### Before the new equipment:
- Number of workers: 6
- Total carts produced per hour: 80
Labor productivity before = Total carts produced / Number of workers
[tex]\[ \text{Labor productivity before} = \frac{80\ \text{carts}}{6\ \text{workers}} = 13.333\ \text{carts per worker per hour} \][/tex]
#### After the new equipment:
- Number of workers: 5 (One worker was transferred to another department)
- Total carts produced per hour: 84
Labor productivity after = Total carts produced / Number of workers
[tex]\[ \text{Labor productivity after} = \frac{84\ \text{carts}}{5\ \text{workers}} = 16.800\ \text{carts per worker per hour} \][/tex]
So, the answers for part (a) are:
[tex]\[ \begin{tabular}{|c|c|} \hline Before & 13.333 \\ \hline After & 16.800 \\ \hline \end{tabular} \][/tex]
### Part (b) Compute Multifactor Productivity
Multifactor productivity is defined as the number of carts produced per dollar cost, where the cost includes both labor and equipment.
#### Before the new equipment:
- Number of workers: 6
- Worker wage per hour: [tex]$\$[/tex]16[tex]$ - Equipment cost per hour: $[/tex]\[tex]$30$[/tex]
- Total carts produced per hour: 80
Total labor cost per hour (before) = Number of workers * Worker wage per hour
[tex]\[ \text{Total labor cost (before)} = 6\ \text{workers} \times \$16\ \text{per hour} = \$96\ \text{per hour} \][/tex]
Total cost per hour (before) = Total labor cost per hour + Equipment cost per hour
[tex]\[ \text{Total cost (before)} = \$96\ \text{(labor cost)} + \$30\ \text{(equipment cost)} = \$126\ \text{per hour} \][/tex]
Multifactor productivity before = Total carts produced / Total cost
[tex]\[ \text{Multifactor productivity before} = \frac{80\ \text{carts}}{\$126} = 0.635\ \text{carts per dollar cost} \][/tex]
#### After the new equipment:
- Number of workers: 5
- Worker wage per hour: [tex]$\$[/tex]16[tex]$ - Equipment cost per hour: $[/tex]\[tex]$30 + \$[/tex]11 = \[tex]$41$[/tex]
- Total carts produced per hour: 84
Total labor cost per hour (after) = Number of workers * Worker wage per hour
[tex]\[ \text{Total labor cost (after)} = 5\ \text{workers} \times \$16\ \text{per hour} = \$80\ \text{per hour} \][/tex]
Total cost per hour (after) = Total labor cost per hour + Equipment cost per hour
[tex]\[ \text{Total cost (after)} = \$80\ \text{(labor cost)} + \$41\ \text{(equipment cost)} = \$121\ \text{per hour} \][/tex]
Multifactor productivity after = Total carts produced / Total cost
[tex]\[ \text{Multifactor productivity after} = \frac{84\ \text{carts}}{\$121} = 0.694\ \text{carts per dollar cost} \][/tex]
So, the answers for part (b) are:
[tex]\[ \begin{tabular}{|l|l|l|} \hline Before & & 0.635 \\ \hline After & & 0.694 \\ \hline \end{tabular} \][/tex]
### Part (a) Compute Labor Productivity
Labor productivity is defined as the average number of carts produced per worker per hour.
#### Before the new equipment:
- Number of workers: 6
- Total carts produced per hour: 80
Labor productivity before = Total carts produced / Number of workers
[tex]\[ \text{Labor productivity before} = \frac{80\ \text{carts}}{6\ \text{workers}} = 13.333\ \text{carts per worker per hour} \][/tex]
#### After the new equipment:
- Number of workers: 5 (One worker was transferred to another department)
- Total carts produced per hour: 84
Labor productivity after = Total carts produced / Number of workers
[tex]\[ \text{Labor productivity after} = \frac{84\ \text{carts}}{5\ \text{workers}} = 16.800\ \text{carts per worker per hour} \][/tex]
So, the answers for part (a) are:
[tex]\[ \begin{tabular}{|c|c|} \hline Before & 13.333 \\ \hline After & 16.800 \\ \hline \end{tabular} \][/tex]
### Part (b) Compute Multifactor Productivity
Multifactor productivity is defined as the number of carts produced per dollar cost, where the cost includes both labor and equipment.
#### Before the new equipment:
- Number of workers: 6
- Worker wage per hour: [tex]$\$[/tex]16[tex]$ - Equipment cost per hour: $[/tex]\[tex]$30$[/tex]
- Total carts produced per hour: 80
Total labor cost per hour (before) = Number of workers * Worker wage per hour
[tex]\[ \text{Total labor cost (before)} = 6\ \text{workers} \times \$16\ \text{per hour} = \$96\ \text{per hour} \][/tex]
Total cost per hour (before) = Total labor cost per hour + Equipment cost per hour
[tex]\[ \text{Total cost (before)} = \$96\ \text{(labor cost)} + \$30\ \text{(equipment cost)} = \$126\ \text{per hour} \][/tex]
Multifactor productivity before = Total carts produced / Total cost
[tex]\[ \text{Multifactor productivity before} = \frac{80\ \text{carts}}{\$126} = 0.635\ \text{carts per dollar cost} \][/tex]
#### After the new equipment:
- Number of workers: 5
- Worker wage per hour: [tex]$\$[/tex]16[tex]$ - Equipment cost per hour: $[/tex]\[tex]$30 + \$[/tex]11 = \[tex]$41$[/tex]
- Total carts produced per hour: 84
Total labor cost per hour (after) = Number of workers * Worker wage per hour
[tex]\[ \text{Total labor cost (after)} = 5\ \text{workers} \times \$16\ \text{per hour} = \$80\ \text{per hour} \][/tex]
Total cost per hour (after) = Total labor cost per hour + Equipment cost per hour
[tex]\[ \text{Total cost (after)} = \$80\ \text{(labor cost)} + \$41\ \text{(equipment cost)} = \$121\ \text{per hour} \][/tex]
Multifactor productivity after = Total carts produced / Total cost
[tex]\[ \text{Multifactor productivity after} = \frac{84\ \text{carts}}{\$121} = 0.694\ \text{carts per dollar cost} \][/tex]
So, the answers for part (b) are:
[tex]\[ \begin{tabular}{|l|l|l|} \hline Before & & 0.635 \\ \hline After & & 0.694 \\ \hline \end{tabular} \][/tex]
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