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A firm has a production function Q = F(K, L) with constant returns to scale. Input prices are r = 2 and w = 1. The output-expansion path for this production function at these input prices is a straight line through the origin. When it produces 5 units of output, it uses 2 units of K and 3 units of L. How much K and L will it use when its long-run total cost is equal to 70?

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Answer:

Total cost (for Q = 5) = 2 x 2 + 3 x 1 = $ 7

When LRTC= $ 70

Q = 70/7 x 5 = 50

K = 2 x 10 = 20

L = 3 x 10 = 30

LAC = LRTC/Q = 70/50 = 1.4

Explanation:

Based on the information given, it should be noted that the number of capital and labor will be 20 and 30 respectively.

Solving the production function.

The total cost at 5 units of output will be:

= (2 × 2) + (3 × 1)

= 4 + 3 = 7

The total cost is $7

Also, the long run total cost (LRTS) is given as $70. Therefore, Q will be:

= 70/7 × 5 = 50

Capital (K) will be:

= 2 × 10 = 20

Labor(L) will be:

= 3 × 10 = 30.

Lastly, LAC will be:

= 70/50 = 1.4

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