Answer
See explanation
Work Step by Step
To verify whether the production will be doubled if both the amount of labor (\( L \)) and the amount of capital (\( K \)) are doubled for the Cobb-Douglas production function \( P(L, K) = 1.01L^{0.75}K^{0.25} \), let's analyze the behavior when both \( L \) and \( K \) are doubled:
1. If \( L \) is doubled: \( P(2L, K) = 1.01(2L)^{0.75}K^{0.25} \)
2. If \( K \) is doubled: \( P(L, 2K) = 1.01L^{0.75}(2K)^{0.25} \)
Let's evaluate these expressions:
1. \( P(2L, K) = 1.01(2L)^{0.75}K^{0.25} = 1.01 \times 2^{0.75}L^{0.75}K^{0.25} = 1.01 \times 2^{0.75}P(L, K) \)
2. \( P(L, 2K) = 1.01L^{0.75}(2K)^{0.25} = 1.01L^{0.75} \times 2^{0.25}K^{0.25} = 1.01 \times 2^{0.25}P(L, K) \)
So, doubling \( L \) results in a production of \( 2^{0.75} \) times the original production, and doubling \( K \) results in a production of \( 2^{0.25} \) times the original production.
Now, if both \( L \) and \( K \) are doubled:
\[ P(2L, 2K) = 1.01(2L)^{0.75}(2K)^{0.25} = 1.01 \times 2^{0.75} \times 2^{0.25}P(L, K) = 1.01 \times 2P(L, K) \]
So, doubling both \( L \) and \( K \) results in a production of \( 2 \times 1.01 = 2.02 \) times the original production.
In conclusion, for the Cobb-Douglas production function, doubling both labor and capital does not exactly double the production; instead, it increases it by a factor of \( 2.02 \), which is slightly more than double.
For the general production function \( P(L, K) = bL^\alpha K^{1-\alpha} \), you would follow a similar process to verify whether doubling both labor and capital doubles the production, where \( \alpha \) represents the share of capital in production.
