In 1928, Charles Cobb and Paul Douglas published a study in which they modeled the growth of the US economy during the period 1599 to 1922. The function used to model production was of the form: [1} P[L, K} = bLaKIa where P is total production [monetary value of goods produced in the year}; L, the amount of work (total number of personhours worked in a year); and K, the amount of capital invested (monetary value of machinery, equipment and buildings). In these terms, the hypotheses made by Cobb and Douglas can be stated as follows: (i) tf either labor or capital cancel each other out, so does production. {ii} The marginal productivity of labor is proportional to the amount of output per unit of work. (iii) The marginal productivity of capital is proportional to the amount of production per unit of capital. 1. The output per unit of work is fa} Tick the alternative that, for some constant a, describes the hypothesis {ii} 6P H33 (13:13:? L'E \"331,, ELL (b) lfwe keep K constant, K = Kt], then the partial differential equation noted in part {a} becomes the ordinary differential equation: [JP P : gr E L\" Show that: In P{L, Kg} 2 0:111 L + C(KD}, C(KD} is a constant that can depend on Kt]. (c) Conclude that P[L.~Ko} = CltKnEs as in the previous item, constant C1 may depend on the value of KD