It is possible to extend the production function so that output is produced with labor input, \(N\); capital input, K; and carbon-intensive energy input, \(E\).

If we write a production function as \(Y=N^{(1 / 3)} K^{(1 / 3)} E^{(1 / 3)}\)

a. Using the input values below, does this production function exhibit constant returns to scale?

b. Solve for the necessary amount of energy to produce the specified level of output in each row below using the production function \(Y=N^{(1 / 3)} K^{(1 / 3)} E^{(1 / 3)}\)

c. Comment on the following statement using your results from partb.
"Because of constant returns to scale, in order for output to rise, we must increase our consumption of energy."

d. Use words to describe how output can increase without an increase in the use of energy. The text points out that there is widespread agreement among economists that a tax on carbon emissions energy is the best way to reduce carbon emissions. How does this proposition fit into the production function above?

Units of N Units of K Units of E Output 27 125 64 54 250 128 81 375 192