Ec 502 Problem Set 4

1.用户费用和住房市场：

•房屋归房东所有，他们以R1的租金将房屋出租。

•住房在第1阶段的价格为P1，在第2阶段的价格为P2。
•房东最初拥有H1住房单元。他们在

H2 =（1 −δ）H1 + I1
•在第2期末，房东以P2的价格出售了所有房屋。
•房东以实际利率r折价并最大化

D1 +
D2
1 + r

（a）房东在第1和第2期有多少红利？
（b）写下房东的优化问题。
（c）根据r，δ，P1写下房东的一阶条件，
P2-P1
P1

（d）如果房东获得资金的利率上升，租金会上升还是下降？

（e）如果预计房价会上涨，租金相对于P2 = P1会上升还是下降？

2.托宾（Tobin）问：公司具有生产功能
f（Kt）=θtKt

2

2
Ť
。资本以每期δ的比率贬值，

Kt + 1 =（1 −δ）Kt +它

1个

（a）写下公司的优化问题。
（b）替换大写Kt + 1 =（1-δ）Kt + It的运动定律

（c）写下关于K2和K3的公司的一阶条件

（d）代入写入q1作为θ2，θ3，δ和r的函数。
（e）假设θ3增加。相对于第一阶段的投资今天上升还是下降

3. RBC Model With Only Capital:
The economy consists of a unit mass of households and a unit mass of firms and lasts
two periods. There is no uncertainty.
• Households have no initial wealth and receive income Y1 in period 1 and Y2 in
period 2, where Y1 and Y2 are output from firms that they own. They also own
shares of the firms and receive the dividends. They take the interest rate rt as
given, and the price of the consumption good is 1. They maximize
U (C1) + βU (C2)
where U
0 > 0 and U
00 < 0. • Firms produce the consumption good with a technology Yt = θtKα t They enter with capital K1 and invest I1 in period 1, which comes on line in period 2, at a cost of 1 per unit of capital. Capital that is used depreciates at rate δ. There are no adjustment costs, and the firm maximizes the PDV of dividends discounted at the interest rate r. • At the end of period 2, the firms sell the capital they own at a price of 1 per unit of capital after depreciation. Goods markets clear, so C1 + I1 = Y1 C2 = Y2 + (1 − δ) K2 (a) Write down the households’ optimization problem and optimality conditions. If you use results from the consumption section of the course, you may skip straight from the optimization problem to the optimality condition. 2 (b) Write down the firms’ optimization problem and optimality conditions. If you use results from the investment section of the course, you may skip straight from the optimization problem to the optimality condition. (c) Write down the equilibrium conditions. i. What pins down the net real interest rate r − δ? ii. Combine the equilibrium conditions into one equation in K2 and plot the two sides of the equation vs. K2 to show equilibrium diagrammatically. (d) What happens if the economy gets news at time t = 1 that productivity will rise at time t = 2? How does your answer compare to the empirical evidence? (e) Now assume there is a government that finances itself by lump-sum taxes. Assume government spending is valued at U (Ct) + V (Gt) and government spending is included in goods market clearing so that C1 + I1 + G1 = Y1 C2 + G2 = Y2 + (1 − δ) K2 Write down the government’s present-value budget constraint and the equilibrium conditions. How does the one equation in K2 change? (f) Use your diagram from part c to show what happens to K2 and r when government spending G1 increases. How does the change in government spending affect I1? Explain intuitively whether government spending crowds out or crowds in private savings and why. EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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