Figure

Production Process Rays in Linear Programming

Points along each process ray represent combinations of inputs L and K required for that production process to produce output.

Units of L employed per time period

Production process A

Increasing labor intensity

Production process B

Production process C

Units of L employed per time period

Production process A

Increasing labor intensity

Production process B

Production process C

Production process D

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Units of K employed per time period

Increasing capital intensity

Production process D

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Units of K employed per time period line segment 0Aj and thus represents twice as much output. Along production process ray A, the distance 0A1 = A1A2 = AA3 = A3A4 = A4A5. Each of these line segments indicates the addition of one unit of output using increased quantities of L and K in the fixed ratio of 15 to 1.

Output along the ray increases proportionately with increases in the input factors. If each input is doubled, output is doubled; if inputs increase by a factor of 10 percent, output increases by 10 percent. This follows from the linearity assumption noted previously: Each production process must exhibit constant returns to scale.

Output is measured in the same way along the three other production process rays in Figure 9.1. Point C1 indicates the combination of L and K required to produce 1 unit of Q using process C. The production of 2 units of Q using that process requires the combination of L and K indicated at point C2; the same is true for points C3, C4, and C5. Although production of additional units using process C is indicated by line segments of equal length, just as for process A, these line segments are of different lengths between the various production systems. Whereas each production process exhibits constant returns to scale, equal distances along different process rays do not ordinarily indicate equal output quantities.

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