(2) Suppose a social security system will pay you 25% of your working income when you are retired. Now, with y =1,000,000, as in part (a), how much will you save and how much will you consume each period? (5 分)

3. The textbook Solow-Swan model gives you the following framework setup: \(\succ\) Aggregate production function: \(Y_{t} = AK_{t}^{\alpha}L_{t}^{1 - \alpha}\), where \(Y_{t}, K_{t}\), and \(L_{t}\) are aggregate output, aggregate physical capital, and total labor force at time \(t\), respectively. Total factor productivity \(A\) and the capital share \(\alpha\) are assumed constant. \(\succ\) Physical capital accumulation: \(K_{t + 1} = I_t + (1 - d)K_t\) , where \(I_{t}\) is the gross investment at time \(t\) and the capital depreciates at a constant rate of \(d\) > Labor force growth: \( L_{t+1} = (1 + n)L_t \), where \( n \) is a constant labor growth rate. > Aggregate output is distributed to aggregate consumption, \( C_t = (1 - s)Y_t \), and aggregate saving, \( S_t = sY_t \), where the saving rate \( s \) is constant. There is no government in this closed economy. Goods market is in equilibrium for all time. (1) Let \( k_{t} = K_{t} / L_{t} \) and \( y_{t} = Y_{t} / L_{t} \) be capital-labor ratio and output per worker, respectively, find the first-order difference equation for \( k_{t} \), that is \( k_{t + 1} = g(k_{t}) \). (5 分)

(2) Suppose a social security system will pay you 25% of your working income when you are retired. Now, with \( y = \)1,000,000, as in part (a), how much will you save and how much will you consume each period? (5 分)

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