Questions: The production function for a print shop is q=2000 * min(L, 3K), where q is the number of sheets printed per hour, L is the number of workers, and K is the number of printers. For example, if L=8 and K=2, then min(L, 3K)=6 and q=12,000. Draw the isoquants for this production function when q=2000, q=6000, and when q=12,000, you should draw three isoquants. Note what are K and L at each vertex.

The production function for a print shop is q=2000 * min(L, 3K), where q is the number of sheets printed per hour, L is the number of workers, and K is the number of printers. For example, if L=8 and K=2, then min(L, 3K)=6 and q=12,000. Draw the isoquants for this production function when q=2000, q=6000, and when q=12,000, you should draw three isoquants. Note what are K and L at each vertex.

Transcript text: 6. ( 15 points) The production function for a print shop is $q=2000 \cdot \min (L, 3 K)$, where $q$ is the number of sheets printed per hour, $L$ is the number of workers, and $K$ is the number of printers. For example, if $L=8$ and $K=2$, then $\min (L, 3 K)=6$ and $q=12,000$. Draw the isoquants for this production function when $q=2000, q=6000$, and when $q=12,000$, you should draw three isoquants. Note what are $K$ and L at each vertex.

Solution

Determine the isoquants for the production function $ q = 2000 \cdot \min (L, 3K) $ for different values of $ q $.

Identify the relationship between $ L $ and $ K $.

The production function is given by $ q = 2000 \cdot \min (L, 3K) $. This means that the output $ q $ is determined by the smaller of $ L $ and $ 3K $. Therefore, the isoquants are determined by the condition $ L = 3K $ or $ L = q/2000 $.

Calculate the isoquants for $ q = 2000 $.

For $ q = 2000 $, we have $ 2000 = 2000 \cdot \min (L, 3K) $. This implies $ \min (L, 3K) = 1 $. Therefore, the isoquant is defined by the line $ L = 1 $ or $ 3K = 1 $. The vertices are $ (L, K) = (1, 0) $ and $ (L, K) = (0, \frac{1}{3}) $.

Calculate the isoquants for $ q = 6000 $.

For $ q = 6000 $, we have $ 6000 = 2000 \cdot \min (L, 3K) $. This implies $ \min (L, 3K) = 3 $. Therefore, the isoquant is defined by the line $ L = 3 $ or $ 3K = 3 $. The vertices are $ (L, K) = (3, 0) $ and $ (L, K) = (0, 1) $.

Calculate the isoquants for $ q = 12000 $.

For $ q = 12000 $, we have $ 12000 = 2000 \cdot \min (L, 3K) $. This implies $ \min (L, 3K) = 6 $. Therefore, the isoquant is defined by the line $ L = 6 $ or $ 3K = 6 $. The vertices are $ (L, K) = (6, 0) $ and $ (L, K) = (0, 2) $.

$\boxed{\text{Isoquants are defined by the lines } L = 1, 3, 6 \text{ and } 3K = 1, 3, 6 \text{ for } q = 2000, 6000, 12000 \text{ respectively.}}$

{"axisType": 3, "coordSystem": {"xmin": 0, "xmax": 7, "ymin": 0, "ymax": 3}, "commands": ["L = 1", "L = 3", "L = 6", "3K = 1", "3K = 3", "3K = 6"], "latex_expressions": ["$L = 1$", "$L = 3$", "$L = 6$", "$3K = 1$", "$3K = 3$", "$3K = 6$"]}

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