(1)
Q = f(K, L) = (L/2) + (K)^{0.5}
(a)
When both inputs are doubled, new production function becomes
Q* = (2L/2) + (2K)^{0.5} = (2L/2) + (2)^{0.5} x (K)^{0.5}
The first component of production function (L/2) is exactly doubled, but since (2)^{0.5} < 2, the second component of production function is less than doubled. Therefore, output is less than doubled. Since doubling both inputs less than doubles the output, there are decreasing returns to scale.
(b)
When K = 4,
Q = (L/2) + (4)^{0.5} = (L/2) + 2
Marginal product (MPL) = dQ/dL = 1/2 = 0.5
Average product (APL) = Q/L = (1/2) + (2/L) = 0.5 + 2/L
Data table used for graph:
L | MPL | APL |
1 | 0.5 | 2.5 |
5 | 0.5 | 0.9 |
10 | 0.5 | 0.70 |
15 | 0.5 | 0.63 |
20 | 0.5 | 0.6 |
25 | 0.5 | 0.58 |
30 | 0.5 | 0.57 |
35 | 0.5 | 0.56 |
40 | 0.5 | 0.55 |
45 | 0.5 | 0.54 |
50 | 0.5 | 0.54 |
Graph:
NOTE: As per Answering Policy, 1st question is answered.
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