b. Suppose that the production function is given by Y = K1/2 N1
1/2. Assume that the size of the
population, the participation rate, and the unemployment rate are all constant.(Should writen mathetically)
i)
Is this production function characterized by constant returns to scale? Explain.
ii) Transform the production function into a relationship between output per worker and capital per
worker.
iii) Derive the steady state level of capital per worker in terms of the saving rate (s) and the
depreciation rate ( δ ).
iv) Derive the equations for steady-state output per worker and steady-state consumption per worker
in terms of s and δ .
v) Let δ= 08.0 and s = 0.16. Calculate the steady-state output per worker, capital per worker, and
consumption per worker.
vi) Let δ= 08.0 and s = 0.32. Calculate the steady-state output per worker, capital per worker, and
consumption per worker.
i) To determine if the production function is characterized by constant returns to scale, we examine what happens to output when we increase both capital (K) and labor (N) by a constant factor. In this case, we increase both K and N by λ.
Y' = (λK)^(1/2) (λN)^(1/2)
= λ^(1/2) λ^(1/2) K^(1/2) N^(1/2)
= λK^(1/2) N^(1/2)
= λY
Since Y' = λY, the production function exhibits constant returns to scale.
ii) To transform the production function into a relationship between output per worker (Y/N) and capital per worker (K/N), divide both sides of the production function by N:
(Y/N) = (K/N)^(1/2) N^(1/2)
= (K/N)^(1/2) √N
= (K/N)^(1/2) N^(1/2)
iii) To derive the steady state level of capital per worker (K/N), we set the saving rate (s) equal to the depreciation rate (δ):
s = δ
iv) To derive the equations for steady-state output per worker (Y/N) and steady-state consumption per worker (C/N) in terms of s and δ, we substitute the steady-state capital per worker (K/N) into the production function:
(Y/N) = (K/N)^(1/2) N^(1/2)
= (s/δ)^(1/2) N^(1/2)
Since consumption per worker (C/N) is equal to output per worker minus investment per worker, we have:
(C/N) = (1 - s) (Y/N)
= (1 - s) [(s/δ)^(1/2) N^(1/2)]
v) Given δ = 0.8 and s = 0.16, we can calculate the steady-state output per worker, capital per worker, and consumption per worker:
(Y/N) = (s/δ)^(1/2) N^(1/2)
= (0.16/0.8)^(1/2) N^(1/2)
= (0.2)^(1/2) N^(1/2)
= 0.447N^(1/2)
(K/N) = s/δ
= 0.16/0.8
= 0.2
(C/N) = (1 - s) [(s/δ)^(1/2) N^(1/2)]
= (1 - 0.16) [(0.16/0.8)^(1/2) N^(1/2)]
= 0.84 * 0.2^(1/2) N^(1/2)
= 0.168N^(1/2)
vi) Given δ = 0.8 and s = 0.32, we can calculate the steady-state output per worker, capital per worker, and consumption per worker using the same equations as in part v:
(Y/N) = (s/δ)^(1/2) N^(1/2)
= (0.32/0.8)^(1/2) N^(1/2)
= (0.4)^(1/2) N^(1/2)
= 0.632N^(1/2)
(K/N) = s/δ
= 0.32/0.8
= 0.4
(C/N) = (1 - s) [(s/δ)^(1/2) N^(1/2)]
= (1 - 0.32) [(0.32/0.8)^(1/2) N^(1/2)]
= 0.68 * 0.4^(1/2) N^(1/2)
= 0.272N^(1/2)
i) To determine whether the production function exhibits constant returns to scale, we need to analyze how changing the inputs affects the output.
Let's consider scaling both inputs (capital and labor) by a factor of λ.
If we increase capital and labor by λ, the new level of output would be:
Y' = (λK)^(1/2) (λN)^(1/2)
= λ^(1/2)K^(1/2) λ^(1/2)N^(1/2)
= λK^(1/2) N^(1/2)
= λY
Since the output is proportional to the scaling factor λ, the production function exhibits constant returns to scale.
ii) To transform the production function into a relationship between output per worker (Y/N) and capital per worker (K/N), we divide both sides of the production function by N:
Y/N = (K/N)^(1/2) N^(1/2-1)
= (K/N)^(1/2) N^(-1/2)
= (K/N)^(1/2) / N^(1/2)
iii) In the steady state, capital per worker (K/N) remains constant over time. To derive the steady state level of capital per worker, we set the investment (sY) equal to the depreciation (δK/N):
sY = δ(K/N)
Substituting the production function (Y = K^(1/2)N^(1/2)), we have:
s(K^(1/2)N^(1/2)) = δ(K/N)
Simplifying:
sK^(1/2)N^(1/2) = δK/N
Dividing both sides by K and N^(1/2):
sK^(-1/2) = δ/N^(3/2)
Rearranging, we obtain:
K^(-1/2) = (δ/s)N^(3/2)
Taking the inverse:
K^(1/2) = (s/δ)N^(-3/2)
Raise both sides to the power of 2:
K = (s/δ)^2 N^(-3)
iv) To derive the steady-state equations for output per worker (Y/N) and consumption per worker (C/N) in terms of s and δ, we substitute the steady-state equation for capital per worker (from iii) into the production function:
Y = K^(1/2) N^(1/2)
= ((s/δ)^2 N^(-3))^(1/2) N^(1/2)
= s/δ N^(-3/2) N^(1/2)
= s/δ N^(-3/2 + 1/2)
= s/δ N^(-1)
Hence, the steady-state output per worker is Y/N = s/δ N^(-1).
Consumption per worker is given by the equation C/N = (1-s)Y/N:
C/N = (1-s) (s/δ N^-1)
= (s/δ - s^2/δ) N^-1
= s/δ (1 - s/δ) N^-1
v) To calculate the steady-state output per worker, capital per worker, and consumption per worker with δ = 0.8 and s = 0.16, substitute these values into the equations derived in iv).
Steady-state output per worker: Y/N = (0.16/0.8) N^-1 = 0.2 N^-1
Steady-state capital per worker: K/N = (0.16/0.8)^2 N^(-3) = 0.04 N^-3
Steady-state consumption per worker: C/N = (0.16/0.8) (1 - 0.16/0.8) N^-1 = 0.16 (1 - 0.2) N^-1 = 0.128 N^-1
vi) To calculate the steady-state output per worker, capital per worker, and consumption per worker with δ = 0.8 and s = 0.32, substitute these values into the equations derived in iv).
Steady-state output per worker: Y/N = (0.32/0.8) N^-1 = 0.4 N^-1
Steady-state capital per worker: K/N = (0.32/0.8)^2 N^(-3) = 0.16 N^-3
Steady-state consumption per worker: C/N = (0.32/0.8) (1 - 0.32/0.8) N^-1 = 0.32 (1 - 0.4) N^-1 = 0.192 N^-1