(a) To verify that the aggregate production function has constant returns to scale, we need to check if doubling both inputs (capital and labor) leads to a doubling of the output.

Let's denote the function as 𝐹(𝐾, 𝐿) = .2√(𝐾^2 + 𝐿^2).

If we double both inputs, we get 𝐹(2𝐾, 2𝐿) = .2√((2𝐾)^2 + (2𝐿)^2) = .2√(4𝐾^2 + 4𝐿^2) = .2*2√(𝐾^2 + 𝐿^2) = 2𝐹(𝐾, 𝐿).

So, the production function has constant returns to scale.

(b) To find how 𝑘𝑡+1 depends on 𝑘𝑡, we first express 𝐼𝑡 in terms of 𝑌𝑡. Since 20% of total output is invested in every period, 𝐼𝑡 = 0.2𝑌𝑡.

Substituting 𝐼𝑡 into the equation for 𝐾𝑡+1, we get 𝐾𝑡+1 = .95𝐾𝑡 + 0.2𝑌𝑡.

Dividing through by 𝐿𝑡+1, we get 𝑘𝑡+1 = .95(𝐾𝑡/𝐿𝑡) + 0.2(𝑌𝑡/𝐿𝑡+1).

Since 𝐿𝑡+1 = 𝐿𝑡 * 1.02 (due to the 2% annual population growth), we can substitute this into the equation to get 𝑘𝑡+1 = .95𝑘𝑡 + 0.2(𝑌𝑡/(1.02𝐿𝑡)) = .95𝑘𝑡 + 0.2(𝑌𝑡/𝐿𝑡)/1.02 = .95𝑘𝑡 + 0.2𝑘𝑡/1.02.

(c) Given 𝐾0 = 1 and 𝐿0 = 1, we have 𝑘0 = 𝐾0/𝐿0 = 1 and 𝑌0 = .2√(𝐾0^2 + 𝐿0^2) = .2.

We can use the equation from part (b) to compute 𝑘𝑡 and 𝑌𝑡 for 𝑡 = 10, 20, and 30.

For 𝑡 = 10, we would iterate the equation 10 times, each time using the computed 𝑘𝑡 and 𝑌𝑡 from the previous iteration as inputs for the next iteration.

We would do the same for 𝑡 = 20 and 𝑡 = 30.

Note: The actual computation would require a calculator or computer program.

Question

(a) To verify that the aggregate production function has constant returns to scale, we need to check if doubling both inputs (capital and labor) leads to a doubling of the output.

Let's denote the function as 𝐹(𝐾, 𝐿) = .2√(𝐾^2 + 𝐿^2).

If we double both inputs, we get 𝐹(2𝐾, 2𝐿) = .2√((2𝐾)^2 + (2𝐿)^2) = .2√(4𝐾^2 + 4𝐿^2) = .2*2√(𝐾^2 + 𝐿^2) = 2𝐹(𝐾, 𝐿).

So, the production function has constant returns to scale.

(b) To find how 𝑘𝑡+1 depends on 𝑘𝑡, we first express 𝐼𝑡 in terms of 𝑌𝑡. Since 20% of total output is invested in every period, 𝐼𝑡 = 0.2𝑌𝑡.

Substituting 𝐼𝑡 into the equation for 𝐾𝑡+1, we get 𝐾𝑡+1 = .95𝐾𝑡 + 0.2𝑌𝑡.

Dividing through by 𝐿𝑡+1, we get 𝑘𝑡+1 = .95(𝐾𝑡/𝐿𝑡) + 0.2(𝑌𝑡/𝐿𝑡+1).

Since 𝐿𝑡+1 = 𝐿𝑡 * 1.02 (due to the 2% annual population growth), we can substitute this into the equation to get 𝑘𝑡+1 = .95𝑘𝑡 + 0.2(𝑌𝑡/(1.02𝐿𝑡)) = .95𝑘𝑡 + 0.2(𝑌𝑡/𝐿𝑡)/1.02 = .95𝑘𝑡 + 0.2𝑘𝑡/1.02.

(c) Given 𝐾0 = 1 and 𝐿0 = 1, we have 𝑘0 = 𝐾0/𝐿0 = 1 and 𝑌0 = .2√(𝐾0^2 + 𝐿0^2) = .2.

We can use the equation from part (b) to compute 𝑘𝑡 and 𝑌𝑡 for 𝑡 = 10, 20, and 30.

For 𝑡 = 10, we would iterate the equation 10 times, each time using the computed 𝑘𝑡 and 𝑌𝑡 from the previous iteration as inputs for the next iteration.

We would do the same for 𝑡 = 20 and 𝑡 = 30.

Note: The actual computation would require a calculator or computer program.

Knowee AI · Accepted Answer