Question
Show that the following generating function is homogeneous and find the degree of nationality: Q = 2K1/2L3/2 Do operations to show whether this function exhibits diminishing returns to scale, constant returns to scale or increasing returns to scale? Explain the answer your. B) The number, N, of daily infections by a virus, n days after the start of one pandemic grows exponentially and can be exemplified by the relationship: N= 7 x 1.2 (raised to n) B.1) Find the number of infections when n=10. B.2) After how many days will the number of new infections exceed 1000?
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Answer to a math question Show that the following generating function is homogeneous and find the degree of nationality: Q = 2K1/2L3/2 Do operations to show whether this function exhibits diminishing returns to scale, constant returns to scale or increasing returns to scale? Explain the answer your. B) The number, N, of daily infections by a virus, n days after the start of one pandemic grows exponentially and can be exemplified by the relationship: N= 7 x 1.2 (raised to n) B.1) Find the number of infections when n=10. B.2) After how many days will the number of new infections exceed 1000?
Timmothy
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1. Given \( Q = 2K^{1/2}L^{3/2} \), check homogeneity:
- \( Q(kK, kL) = 2(kK)^{1/2}(kL)^{3/2} = 2k^{1/2}K^{1/2} \cdot k^{3/2}L^{3/2} = k^{1/2 + 3/2} 2K^{1/2}L^{3/2} = k^{2}Q(K, L) \)
- Homogeneous of degree 2.
2. Check returns to scale:
- Since \( Q(kK, kL) = k^{2}Q(K, L) \), the function displays increasing returns to scale for \( k > 1 \).
3. For the infection model \( N = 7 \times 1.2^n \):
- B.1) When \( n = 10 \):
N = 7 \times 1.2^{10} \approx 43
- B.2) To find \( n \) such that \( N > 1000 \):
7 \times 1.2^n > 1000
1.2^n > \frac{1000}{7} \approx 142.857
Taking logarithms:
n \log 1.2 > \log 142.857
n > \frac{\log 142.857}{\log 1.2} \approx 37
So, after 37 days.
- \( Q(kK, kL) = 2(kK)^{1/2}(kL)^{3/2} = 2k^{1/2}K^{1/2} \cdot k^{3/2}L^{3/2} = k^{1/2 + 3/2} 2K^{1/2}L^{3/2} = k^{2}Q(K, L) \)
- Homogeneous of degree 2.
2. Check returns to scale:
- Since \( Q(kK, kL) = k^{2}Q(K, L) \), the function displays increasing returns to scale for \( k > 1 \).
3. For the infection model \( N = 7 \times 1.2^n \):
- B.1) When \( n = 10 \):
- B.2) To find \( n \) such that \( N > 1000 \):
Taking logarithms:
So, after 37 days.
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