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

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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.

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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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