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

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

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.

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类似问题

Over the last several decades the observed relationship between capital per worker and real-GDP per worker for the United States was linear. What is the most plausible explanation for this observation?

Question5 Ref. Solow-Swan Model. The steady state is defined as the point where capital accumulation, is equal to the depreciation rate. the saving rate. the population growth rate. zero. the productivity growth rate. ResetMaximum marks: 1 Flag question undefined

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