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.

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 \( (L, K) = (0, 2) \).

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

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

{"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": ["$L = 1$", "$L = 3$", "$L = 6$", "$3K = 1$", "$3K = 3$", "$3K = 6$"]}