这个作业是分析用户成本以及住房市场的联系
Ec 502 Problem Set 4
1.用户费用和住房市场：
用户成本模型经常应用于房地产市场。这个问题走
您通过思考用户的住房资本成本。如果您感到困惑，
请参阅说明中的“购买的资金和用户成本”模型。
考虑一下住房市场的两阶段模型：
•房屋归房东所有，他们以R1的租金将房屋出租。
期间1和期间2中的R2。
•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
时期1，在时期2上线。每个时期，房屋贬值δ。
住房因此根据
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
他们的红利
D1 +
D2
1 + r
等于他们从Rt租用的收入减去所产生的任何费用
购买住房（或他们通过出售住房赚取的利润）。
（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
和R。
用边际成本和边际收益来解释这种情况。
房东盖房。
（d）如果房东获得资金的利率上升，租金会上升还是下降？
说明。
（e）如果预计房价会上涨，租金相对于P2 = P1会上升还是下降？
说明。
2.托宾（Tobin）问：公司具有生产功能
f（Kt）=θtKt
资本价格固定为1，实际利率固定为r。买资本
企业必须支付调整成本φ
2
一世
2
Ť
。资本以每期δ的比率贬值，
所以
Kt + 1 =（1 −δ）Kt +它
。
1个
没有不确定性。一共有三个时期，t = 1、2、3，最后一个时期
三，公司出售其资本而不进行投资。
（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
输入I1和I2的优化问题，将公司的问题写为优化问题
对于K2和K3。
（c）写下关于K2和K3的公司的一阶条件
托宾的Q，q1 = 1 +φI1，δ，r，θ2，θ3和q2。
（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
一年之内θ不上升的情况？直观地解释和对比
没有调整成本的世界。
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.