Consider a basic Solow–Swan model. Suppose the aggregate production function is 𝑌=𝐴𝐾12𝐿12 and that A=1 , the depreciation rate is 𝛿=0.06 and the saving rate is 𝑠=0.12. In steady state, which of the following is TRUE?
Question
Consider a basic Solow–Swan model. Suppose the aggregate production function is 𝑌=𝐴𝐾12𝐿12 and that A=1 , the depreciation rate is 𝛿=0.06 and the saving rate is 𝑠=0.12. In steady state, which of the following is TRUE?
Solution 1
In the Solow-Swan model, the steady state level of capital per worker (k*) is determined by the equation sA(k*)^1/2 = δk*. Given the values of s, A, and δ in the problem, we can substitute these into the equation to find the steady state level of capital per worker.
0.12 * (k*)^1/2 = 0.06 * k*
Solving this equation for k* gives us the steady state level of capital per worker.
First, divide both sides by 0.06:
2 * (k*)^1/2 = k*
Then, square both sides to get rid of the square root:
4 * k* = (k*)^2
Finally, divide both sides by k* to solve for k*:
k* = 4
So, in the steady state, the level of capital per worker is 4. This means that the statement "In the steady state, the level of capital per worker is 4" is TRUE.
Solution 2
In the Solow-Swan model, the steady state is the point at which the economy's output, capital stock, and labor force stop growing. In this case, the steady state capital per worker (k*) is determined by the equation:
sA(k*)^1/2 = δk*
where s is the saving rate, A is the technology parameter, k* is the steady state capital per worker, and δ is the depreciation rate.
Given that s=0.12, A=1, and δ=0.06, we can substitute these values into the equation to find k*:
0.12*(k*)^1/2 = 0.06k
Solving this equation for k* gives us the steady state capital per worker.
First, divide both sides by k* to get:
0.12/(k*)^1/2 = 0.06
Then, square both sides to get:
0.0144/k* = 0.0036
Finally, solve for k* to get:
k* = 0.0144/0.0036 = 4
So, in the steady state, the capital per worker is 4.
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