In an economic model, gross investment /, is proportional to aggregate output where s is the constant of proportionality. Capital is replaced at the...

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1. In an economic model, gross investment /, is proportional to aggregate output where s is theconstant of proportionality. Capital is replaced at the depreciation rate 8, where 0 < 6, s < 1. Theaggregate production function is of the Cobb - Douglas type f (K, L) - aka,!-s, with a and abeing positive constants, a < 1. Labour / grows at a constant positive rate A. This means thatat/dt - A. Net investment In is the rate of change of the capital stock, that is, /,. - ". Furthermore7net investment is the difference between gross investment and the rate of replacement, that is,In = Ig - OK.(i)Show thatdk .!+ 8% - sa (4 )"14J(ii)The capital - labor ratio is k (t) = 10)=KID. By substituting into the differential equation youobtained in part (i), show that"+ (8 + 1)k = saka(You will need to use the quotient rule at some point. )[6]