Question

Does this production function have constant returns to scale? Explain. b) What is the per-worker production function, y = f(k) 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% of output each year and country B saves 30% 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 stead-state levels of income per worker and consumption per worker. d) Suppose that both countries start off with a capital stock

Answer 1

`k = ((0.1)/(0.2))^(3/2)=0.354`

In the steady state level of Country A, the capital per worker is 0.354

`k = ((0.3)/(0.2))^(3/2)=1.837`

In th steady state level of country B, the capital per worker is 1.387.

Now we can find the steady level income per worker like this:

Country A

`y=k^(1/3)= (0.354)^(1/3)=0.707`

Country B:

`y=k^(1/3)= (1.837)^(1/3)=1.225`

And if we want to find the consumption per worker we can apply this formula:

`c= (1-s) y`

Country A

`c=(1-0.1)0.707=0.636`

Country B

`c=(1-0.3)1.225=0.858`

Step-by-step explanation:

We assume that we have a common production function for country A and B given by:

`Y=F(K,L)= K^(1/3)L^(2/3)`

And we can find the per worker production function
`y=f(X)` like this:

`(Y)/(L)= (K^(1/3)L^(2/3))/(L)=K^(1/3)L^(-1/3)= ((K)/(L))^(1/3)`

And we can express the function just in terms of a constant like this:

`y=k^(1/3)`

From the info given by the problem we have:

Depressciation
`\delta =0.2`

Savings for A
`S_A = 0.1`

Savings for B
`S_B = 0.3`

And we assume that
`y=k^(1/3)`

Let's begin finding the steady state level of capital per worker, we need to satisfy the following condition:

`s f(x) = \delta k`

The reason is because. The growth of capital per worker
`\Delta k =` is given by investment per worker
`sf(k)` minus the depreciation per worker
`\deta k` , and we have then
`\delta k= sf(k)-\delta k` and if is steady then k=0.

`S_A f(x)= \delta k`

`0.1 k^(1/3)=0.2 k`

`0.1 =0.2 k^(2/3)`

`k = ((0.1)/(0.2))^(3/2)=0.354`

In the steady state level of Country A, the capital per worker is 0.354. For country B we can do a similar procedure like this:

`s f(x) = \delta k`

`S_B f(x)= \delta k`

`0.3 k^(1/3)=0.2 k`

`0.3 =0.2 k^(2/3)`

`k = ((0.3)/(0.2))^(3/2)=1.837`

In the steady state level of country B, the capital per worker is 1.387.

Now we can find the steady level income per worker like this:

Country A

`y=k^(1/3)= (0.354)^(1/3)=0.707`

Country B:

`y=k^(1/3)= (1.837)^(1/3)=1.225`

And if we want to find the consumption per worker we can apply this formula:

`c= (1-s) y`

Country A

`c=(1-0.1)0.707=0.636`

Country B

`c=(1-0.3)1.225=0.858`