1. Suppose you divide your life into two periods: working age and retirement age. When you work, you earn labor income $y$; when retired, you earn no labor income, but must live off your savings and the interest it earns. While working, you save the amount $s$, earning interest at a constant rate $r$, so you have $(1 + r)s$ to live on when retired. Because you don't need to consume as much when retired, you want to set consumption when working (c^w) twice as high as consumption when retired (c^r). (1) Suppose you earn $1,000,000 over your working life, and the real interest rate is 50%. How much will you save and how much will you consume in each part of your life? (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)

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