Mathematical Economics

Everything for the questions is below

1) Question. Resolve the representative intermediate goods firm's profit maximisation problem: max IIt = yt - wth - rekt-1 S.t. yt = atki-14", where parameter a E [0, 1] is to limn the output capital share. NB. Do not construct the Lagrangian equation, but directly substitute the production function into the profit equation and compute the first order conditions with respect to labour and capital. 2) Question. Envision the representative household's utility maximisation decentralised problem: Ct 1-de 11+01 max Eot s.t. 1 = 0 1-oc 1+o) at the - (1 - 8) kt-1 = wthe + rekt-1 + Ile lim BtAtki = 0 1-+00 such that at = pat_letter, whereby p E (0, 1) and et ~ N (0, o2) ; furthermore notice that BE (0, 1) . Utilise the definition of the representative intermediate goods firm's profit equation to derive an equation for capital or aggregate resources: take the representative household's budget constraint (viz. the first constraint above) and plug the implicit profit equation (sc. the one encompassing not the production function) into the same; cancel out terms