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

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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

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.

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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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