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Consider the following BBC model with investment adjustment costs. The process of accumulation of the capital stock (Kt) is dened as: Kn+1 : {17 5)K+[17W,It_l)]n where :5 is the depreciation rate of physical capital, It is gross investment, and 1H.) is the adjust- ment costs function. Following a popular assumption in the literature, we assume that the investment adjustment costs are quadratic (w is a positive parameter): The representative agent derives utility from a standard utility function. In particular, current eon- surnption (0;) and leisure (it) affect positively the utility. The utility function is specied as: En Z' [7105(Cc)+ (1 e 7) 108(k)] i=0 The representative agent has a time endowment of one unit per periodI which is split between leisure (It) and work (LE). The size of the labor force is normalized to 1 and it does not vary over time. We assume that the households own the capital stock, they undertake the investment decisions, and that their savings are equal to investment in every period (so that S: = I: is guaranteed). The budget constraint of the households is: Ct + II. : 11'ch + WK; In this model, there is no government. The production function is Cobb-Douglas: Y1 = Zth'LtbQ, where or > 0 is the capital share of income. We retain the stande source of business cycle uctuations, which is reected in a time-varying Total Factor Productivity (TFP), denoted by 2:. The natural log of Z; follows an ARU) stochastic process with normally distributed i.i.d. shocks: ln Z; = p2 l.ElZg_1 + 52, 62': ~ N(0,cr'z ).' 1. State the problem of the households, the problem of the rm, and the government's budget con- straint. Take the rst order conditions and obtain the equations representing the equilibrium dynamics. Comment on your results and argue whether the decentralized economy achieves the rst best