Output in an economy is given by the production function \(Y=A K^{0.3} N^{0.7}\), where \(Y\) is output, \(A\) measures productivity, the capital stock, \(K\), is fixed at 30 , and employment \(N\) is fixed at 100 . Output equals 100 in the year 2021 and equals 105 in 2022.

a. Find the Solow residual in the years 2021 and 2022, and its growth rate between those two years.

b. What is the relationship between the growth in the Solow residual between 2021 and 2022 and the growth in productivity (as measured by the parameter \(A\) ) in the same years? Assume that the rates of utilization of capital and labor remain unchanged.

c. Repeat part ( \(b\) ) under the assumption that utilization of labor increases by 3\% between 2021 and 2022. You will have to modify the production function along the lines of Eq. (10.2).

d. Repeat part \((b)\) under the assumption that the utilization rates of both labor and capital increase by 3\% between 2021 and 2022.