1. Consider a closed economy described by the Solow-Swan growth model with government in discrete time \( t \). Aggregate real output (\( Y_t \)) is allocated among aggregate consumption (\( C_t \)), gross investment (\( I_t \)), and government purchases (\( G_t \)). On the supply side, aggregate output is produced according to the Cobb-Douglas production function \( Y_t = AK_t^{\alpha}L_t^{1-\alpha} \), where \( A > 0 \), \( 0 < \alpha < 1 \), and \( K_t \) and \( L_t \) denote aggregate physical capital and labor, respectively. The capital depreciation rate is \( d \), and the population growth rate is \( n \). The government runs a balanced budget, so that \( G_t = \tau Y_t \), where \( \tau \in (0,1) \) is the proportional income tax rate. National saving is given by \( s(1 - \tau)Y_t \), where \( s \) is the saving rate and \( (1 - \tau)Y_t \) denotes aggregate disposable income. When the goods market is in equilibrium, national saving is equal to gross investment. Gross investment is the flow variable that governs the accumulation of physical capital over time.