The Steady-State Levels Of Income Per Worker And Consumption Assignment Paper

Country A and country B both have the production function $Y=K_{1/3}L_{2/3}$ (a) Does this production function have constant returns to scale? Explain. (b) What is the per-worker production function, (c) Assume that neither country experiences population growth or technological progress and that 20 percent of capital depreciates each year. Assume further that country A saves 10 percent of output each year and country B saves 30 percent of output each year. Using your answer from part (b) and the steady-state condition that investment equals depreciation, find the steady-state level of capital per worker for each country. Then find the steady-state levels of income per worker and consumption per worker. (d) Suppose that both countries start off with a capital stock per worker of 1. What are the levels of income per worker and consumption per worker?

a) The production function has constant returns to scale if doubling all inputs results in a doubling of output. Let’s test this by looking at the production function with all inputs doubled:

[ Y=(2K)1/3 (2L)2/3=21/3 K1/3 22/3 L2/3=21/3 Y ]

Since the output is multiplied by a constant factor of $2^{1/3}$, the production function has increasing returns to scale.

(b) To find the per-worker production function, we divide the production function by $L$:

[ y=Y/L=K1/3 L−1/3 ]

(c) In the steady−state,investment equals depreciation,which means that 0.2�=��,where �=0.2 is the depreciation rate⋅Solving for � gives us �=5�⋅For country A,the saving rate is 10percent,which means that 0.1�=0.1�/�=0.1�1/3�−1/3⋅Using the steady−state condition,we know that investment is equal to 0.2�,so 0.1�1/3�−1/3=0.2�,or �/�=23⋅Therefore,�=8 and �=1,so the steady−state capital per worker for country A is 8⋅For country B,the saving rate is 30percent,so 0.3�=0.3�/�=0.3�1/3�−1/3⋅

Using the steady−state condition,we know that investment is equal to 0.2�,so 0.3�1/3�−1/3=0.2�,or �/�=(0.2/0.3)3⋅Therefore,�=2.37 and �=1,so the steady−state capital per worker for country B is 2.37⋅Using the per-worker production function, we can find the steady-state income per worker:

For country A: �=�1/3�−1/3=81/31−1/3=2

For country B: �=�1/3�−1/3=2.371/31−1/3=1.26

To find the steady-state consumption per worker, we use the fact that consumption equals output minus investment:

For country A: �=�−0.1�=0.9�=1.8

(d) If both countries start off with a capital stock per worker of 1, we can use the per-worker production function to find the levels of income per worker and consumption per worker.

The per-worker production function is:

[ y=k1/3 ℓ2/3 ]

where �is output per worker,�is capital per worker,and ℓ is labor per worker⋅If the initial capital per worker is 1,then �=1⋅To find the level of income per worker,we substitute �=1 into the production function:[ y=11/3 ℓ2/3=ℓ2/3 ]

To find the level of consumption per worker, we use the fact that all output is either consumed or saved. If we assume that all output is consumed, then consumption per worker is equal to income per worker:

Therefore, both countries have an initial level of income per worker and consumption per worker of The Steady-State Levels Of Income Per Worker And Consumption Assignment Paper