Question75 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 and adopting the assumptions made in lecture.Assume: s = 0.60, d = 0.10, α = 0.5, L= 1and A= 1. The level of the interest rate in steady state is 8.33% The level of the interest rate in steady state is 40.82% The level of the interest rate in steady state is 20.41% The level of the interest rate in steady state is 18% More information is required to compute the interest rate in equilibrium ResetMaximum marks: 2 Flag question undefined单项选择题
A
The level of the interest rate in steady state is 8.33%
B
The level of the interest rate in steady state is 40.82%
C
The level of the interest rate in steady state is 20.41%
D
The level of the interest rate in steady state is 18%
E
More information is required to compute the interest rate in equilibrium

Question

Question75 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 and adopting the assumptions made in lecture.Assume: s = 0.60, d = 0.10, α = 0.5, L= 1and A= 1. The level of the interest rate in steady state is 8.33% The level of the interest rate in steady state is 40.82% The level of the interest rate in steady state is 20.41% The level of the interest rate in steady state is 18% More information is required to compute the interest rate in equilibrium ResetMaximum marks: 2 Flag question undefined单项选择题

A

The level of the interest rate in steady state is 8.33%

B

The level of the interest rate in steady state is 40.82%

C

The level of the interest rate in steady state is 20.41%

D

The level of the interest rate in steady state is 18%

E

More information is required to compute the interest rate in equilibrium

BlackTom AI · Accepted Answer

We start by restating the setup and then examine each option in light of the Solow-Swan framework provided.

Question restatement: With Y = A K^α L^(1−α), L growing at rate n = 0, s = 0.60, d = 0.10, α = 0.5, L = 1, A = 1, determine the steady-state interest rate r. The investment function is I = sY, and the capital dynamics are dK/dt = sY − (d + n)K. The options given for the steady-state interest rate r are 8.33%, 40.82%, 20.41%, 18%, or that more information is required.

Step-by-step analysis of each option:
Option 1: 8.33%.
- In steady state, set dK/dt = 0, so sY = (d + n)K. With n = 0, this is sY = dK.
- Output is Y = A K^α L^(1−α). With A = 1, L = 1, α = 0.5, we have Y = K^0.5.
- Therefore the steady-state condition is 0.60 * K^0.5 = 0.10 * K. Dividing both sides by K^0.5 (assuming K > 0) gives 0.60 = 0.10 * K^0.5, hence K^0.5 = 6 and K = 36.
- The interest rate in this model is the (marginal) product of capital, r = ∂Y/∂K = α A K^(α−1) L^(1−α). With α = 0.5, A = 1, L = 1, we get r = 0.5 * K^(−0.5) = 0.5 / sqrt(K) = 0.5 / 6 = 0.08333 = 8.33%. This matches Option 1 as the steady-state interest rate.

Option 2: 40.82%.
- This would require a much smaller denominator in the MPK calculation or a different steady-state K value. Since r = 0.5 / sqrt(K), achieving r ≈ 0.4082 would require sqrt(K) ≈ 0.5 / 0.4082 ≈ 1.224, i.e., K ≈ 1.499, which contradicts the steady-state condition sY = dK that yields K = 36 for the given parameters. Therefore this value is inconsistent with the model’s steady-state equations here.

Option 3: 20.41%.
- Similar to Option 2, r = 0.5 / sqrt(K) implies sqrt(K) ≈ 0.5 / 0.2041 ≈ 2.449, so K ≈ 6.0. If K were 6, then Y would be sqrt(6) ≈ 2.449, and sY would be 0.60 * 2.449 ≈ 1.469, while (d + n)K would be 0.10 * 6 = 0.6, not equal. The steady-state condition would not hold, so this r is not consistent with the given steady-state setup.

Option 4: 18%.
- This would require sqrt(K) = 0.5 / 0.18 ≈ 2.777..., hence K ≈ 7.716. Again, this does not satisfy the steady-state condition sY = dK with the specified parameters, so it is not the correct steady-state r.

Option 5: More information is required to compute the interest rate in equilibrium.
- Given the Cobb-Douglas form, A, α, s, d, L, and n are specified, and a standard Solow setup with r defined as the marginal product of capital yields a unique r in steady state. Therefore, no additional information is required beyond what’s given to compute r in equilibrium for these parameter values.

Summary of the reasoning: By applying dK/dt = sY − (d + n)K with Y = K^α (since A = L = 1) and α = 0.5, we solved for the steady-state K from 0.60 K^0.5 = 0.10 K, obtaining K = 36. Then the steady-state interest rate is r = ∂Y/∂K = α K^(α−1) = 0.5 * K^(-0.5) = 0.5 / 6 = 0.0833 = 8.33%. Hence Option 1 is consistent with the model under the given assumptions, while the other numbers do not satisfy the steady-state conditions, and the information provided is sufficient to determine r.