Question76 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 Output in steady state is 6(1/2) The level of Output in steady state is 3 The level of Output in steady state is 6(3/2) The level of Output in steady state is 12 The level of Output in steady state is 6 ResetMaximum marks: 2 Flag question undefined单项选择题
A
The level of Output in steady state is 6(1/2)
B
The level of Output in steady state is 3
C
The level of Output in steady state is 6(3/2)
D
The level of Output in steady state is 12
E
The level of Output in steady state is 6

Question

Question76 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 Output in steady state is 6(1/2) The level of Output in steady state is 3 The level of Output in steady state is 6(3/2) The level of Output in steady state is 12 The level of Output in steady state is 6 ResetMaximum marks: 2 Flag question undefined单项选择题

A

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

B

The level of Output in steady state is 3

C

The level of Output in steady state is 6(3/2)

D

The level of Output in steady state is 12

E

The level of Output in steady state is 6

BlackTom AI · Accepted Answer

We start by identifying the Solow-Swan setup and plug in the given parameters. The production function is Y = A K^α L^(1−α) with α = 0.5, A = 1, and L = 1. Therefore, Y = K^0.5 when L = 1. The dynamic condition in steady state with population growth n and depreciation d is sY = (n + d)K, but since Y = K^αL^(1−α) and L = 1, we can work with per-worker (or per effective worker) variables.

Step-by-step for the steady-state capital per worker k*: In per-worker terms, the condition is s f(k) = (n + d) k, where f(k) = k^α. Substituting α = 0.5 gives s k^0.5 = (n + d) k. Rearranging, s k^0.5 = (n + d) k ⇒ s = (n + d) k^0.5 ⇒ k^0.5 = s / (n + d) ⇒ k* = [s / (n + d)]^2.

Compute the numbers: n = 0.1, d = 0.10, so n + d = 0.20. Investment share s = 0.60. Then s / (n + d) = 0.60 / 0.20 = 3. Therefore k* = 3^2 = 9.

Now determine steady-state output Y* given Y = A K^α L^(1−α) with A = 1 and L = 1. This yields Y* = K*^α L^(1−α) = (K*)^0.5 × 1^0.5 = sqrt(K*). With K* = 9, Y* = sqrt(9) = 3.

Now evaluate each answer option:
- Option 1: The level of Output in steady state is 6(1/2). This seems to represent 6 raised to the power of 1/2, i.e., sqrt(6) ≈ 2.45, which does not match the computed Y* = 3. It misinterprets the relationship between parameters and the steady-state level.
- Option 2: The level of Output in steady state is 3. This exactly matches the calculation Y* = 3 from the preceding steps, given the specified s, n, d, α, A, and L. It is consistent with the model equations.
- Option 3: The level of Output in steady state is 6(3/2). This reads as 6 × (3/2) = 9. This would correspond to Y* = 9, which is not what the steady-state condition yields under the given parameters and simple Y = K^α L^(1−α) with L = 1.
- Option 4: The level of Output in steady state is 12. This value lacks a basis in the derived steady-state relation and does not arise from s/(n+d) or sqrt(k*). It is not supported by the equations.
- Option 5: The level of Output in steady state is 6. This would imply Y* = 6, which again does not align with the calculated Y* = 3 under the model, parameters, and L = 1.

In summary, only the second option matches the derived steady-state output given the Cobb-Douglas form, the specified parameter values, and L = 1. The other options either misinterpret the algebra (using a square root, a product with 6, or a flat constant) or produce values that are inconsistent with the steady-state condition s f(k) = (n + d) k for f(k) = k^0.5 under the supplied numbers.