(36 points) General Equilibrium: Consider the representative household, who chooses a path of consumption and leisure over an infinite horizon, {C+s, its),20, to maximize the following objective function: $=0 where u(c, 4) is a well-behaved utility function, and # is a discount factor. The household faces the following real budget constraint each period: where a, is real wealth, r, is the real interest rate, w, is the real wage rate, n, is labor supply, and T, is a lump-sum tax. The household also faces a unitary time endowment which holds each period: 1= 4+m Also consider the representative firm, who chooses a path of capital and labor input over an infinite horizon, {kilts, nts}20 to maximize the following real profit function: Prof = > (Itrets ) ( 1 (kits , mets ) - invits - Witemts ) 8=0 where f( ke, n) is a well-behaved production function, r, is the real interest rate, w is the real wage rate, and ko is given. For any period t, net investment is defined as: inv, = kitl - (1 - 6) k where o is the rate of capital depreciation. Finally, each period the government purchases an amount of real goods and services equal to real wage income tax revenue: 91 = T. so that government savings is always zero. (a) Derive the household's intertemporal and intratemporal optimality conditions in terms of the general utility function u(c, 4). (b) Derive the firm's intertemporal and intratemporal optimality conditions in terms of the general production function f(ke, n)