Question77 This question aims to explore some of the points discussed about the Solow-Swan model. Consider an economy with the general Cobb-Douglas production function:Y = A * Kα * L(1-α) The equation describing capital dynamics is:[math]Where d is a constant parameter that captures the depreciation rate. Investment follows a behavioural equation as discussed in class, ie, a constant 's' fraction of output is invested in every period.Answer the following questions assuming that labour grows at the rate n = 0.1 and adopting the assumptions made in lecture.Assume: s = 0.60, d = 0.10, α = 0.5, L= 1and A= 1. The level of capital in steady state is 6 The level of capital in steady state is 36 The level of capital in steady state is 6(1/2) The level of capital in steady state is 9 The level of capital in steady state is 12 ResetMaximum marks: 2 Flag question undefined单项选择题
A
The level of capital in steady state is 6
B
The level of capital in steady state is 36
C
The level of capital in steady state is 6(1/2)
D
The level of capital in steady state is 9
E
The level of capital in steady state is 12

Question

Question77 This question aims to explore some of the points discussed about the Solow-Swan model. Consider an economy with the general Cobb-Douglas production function:Y = A * Kα * L(1-α) The equation describing capital dynamics is:[math]Where d is a constant parameter that captures the depreciation rate. Investment follows a behavioural equation as discussed in class, ie, a constant 's' fraction of output is invested in every period.Answer the following questions assuming that labour grows at the rate n = 0.1 and adopting the assumptions made in lecture.Assume: s = 0.60, d = 0.10, α = 0.5, L= 1and A= 1. The level of capital in steady state is 6 The level of capital in steady state is 36 The level of capital in steady state is 6(1/2) The level of capital in steady state is 9 The level of capital in steady state is 12 ResetMaximum marks: 2 Flag question undefined单项选择题

A

The level of capital in steady state is 6

B

The level of capital in steady state is 36

C

The level of capital in steady state is 6(1/2)

D

The level of capital in steady state is 9

E

The level of capital in steady state is 12

BlackTom AI · Accepted Answer

To approach this steady-state question, we first identify the key Solow-Swan setup and plug in the given parameters.
- Production: Y = A K^α L^{1-α} with α = 0.5 and A = 1, so Y = K^{0.5} L^{0.5}.
- Capital accumulation with population growth: dK/dt = sY − (δ + n)K, where δ = d = 0.10 and n = 0.10. This leads to the per-worker form using k = K/L: dk/dt = s f(k) − (δ + n)k with f(k) = A k^α = k^{0.5}.
- Therefore the steady-state condition is s f(k*) = (δ + n) k*, i.e., 0.60 * sqrt(k*) = (0.10 + 0.10) k* = 0.20 k*.
- Dividing both sides by sqrt(k*) (assuming k* > 0) gives 0.60 = 0.20 sqrt(k*), so sqrt(k*) = 0.60 / 0.20 = 3, hence k* = 9.
- Finally, since L = 1 in the given setup, the steady-state level of capital is K* = k* × L = 9 × 1 = 9.

Now evaluate each answer option:

Option 1: The level of capital in steady state is 6
- This would require k* = 6 if L = 1. However, the steady-state condition derived above yields k* = 9, not 6. The mismatch arises from incorrect solving of 0.60 sqrt(k) = 0.20 k which does not simplify to sqrt(k) = 3.0 in this case if you mis-handle the algebra; the correct algebra gives sqrt(k) = 3, not sqrt(k) = 2.5 or some other value. Therefore this option is incorrect.

Option 2: The level of capital in steady state is 36
- If K* were 36 with L = 1, then k* = 36. Plugging k* = 36 into the steady-state condition: 0.60 sqrt(36) = 0.60 * 6 = 3.6, while 0.20 k* = 0.20 * 36 = 7.2. Since 3.6 ≠ 7.2, this does not satisfy the steady-state equation. Hence this option is incorrect.

Option 3: The level of capital in steady state is 6(1/2)
- Interpreting 6(1/2) as 6 × (1/2) = 3, this would imply k* = 3 if L = 1. Testing: 0.60 sqrt(3) ≈ 0.60 × 1.732 ≈ 1.039, while 0.20 k* = 0.20 × 3 = 0.6. Since 1.039 ≠ 0.6, this does not satisfy the steady-state condition. Therefore this option is incorrect.

Option 4: The level of capital in steady state is 9
- With L = 1, k* = 9 implies K* = 9. Check the steady-state equation: 0.60 sqrt(9) = 0.60 × 3 = 1.8, and 0.20 k* = 0.20 × 9 = 1.8. The two sides are equal, so the condition s f(k) = (δ + n) k holds, confirming that k* = 9 is indeed the steady-state capital per worker, and K* = 9 when L = 1. This option is consistent with the model equations and parameter values, making it correct.

Option 5: The level of capital in steady state is 12
- If K* = 12 with L = 1, then k* = 12. Evaluate: 0.60 sqrt(12) ≈ 0.60 × 3.464 ≈ 2.078, whereas 0.20 k* = 0.20 × 12 = 2.4. Since 2.078 ≠ 2.4, this does not satisfy the steady-state condition. Thus this option is incorrect.

In summary, only the option stating that the steady-state level of capital is 9 aligns with the derived condition s K^{α/2} = (δ + n) K and the given parameters. The other options fail the steady-state balance equation under the specified parameters.