台湾清华大学 DSGE模型讲义 lecture1
基于DSGE模型的利率传导机制
实证分析方法
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模型选择
采用DSGE模型作为理论基础， 该模型能够较好地描述经济系统 的动态特征和传导机制。
参数估计
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03
பைடு நூலகம்
模拟分析
利用历史数据和计量经济学方法 ，对DSGE模型中的参数进行估 计。
利用估计得到的参数，对模型进 行模拟分析，探究利率变动对经 济变量的影响。
实证分析结果
利率传导机制的有效性
利率对通货膨胀的影响
研究显示，利率的调整对通货膨胀的影响具有滞后效应，且通货膨胀对利率变动的反应 存在非对称性。
研究展望
进一步完善模型
未来研究可以进一步改进和扩展DSGE 模型，以更好地描述和预测经济现象。
跨国家比较研究
比较不同国家利率传导机制的异同， 揭示不同经济环境下货币政策的有效
性。
政策效应评估
产业结构调整
基于DSGE模型分析，产业结构调整对经济有重要影响，政府应加大对新兴产业的支持 力度，推动产业升级和转型。
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结论与展望
研究结论
利率传导机制的有效性
研究结果表明，基于DSGE模型的利率传导机制能够有效地解释和预测货币政策对经济 的影响，为政策制定提供了有力的依据。
利率对产出的影响
研究发现，利率的调整对经济产出的影响具有非对称性，即紧缩性货币政策对产出的影 响大于扩张性货币政策。
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03
动态调整
DSGE模型中的利率动态特性具有动态 调整的特性，即利率水平会随着宏观 经济状况的变化而变化。
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基于DSGE模型的利率传导机制 实证分析
数据来源与处理
数据来源
主要来源于各大经济数据库、政府部门发布的数据以及公开的金融市场数据。
DSGE模型讨论之三——线形理性预期模型
Naturally we have its counterpart-backward-looking solution. We start again from yt = xt + bEt yt+1 which can be written as yt = xt + byt+1 + bεt+1 We seperate the stochastic component of Et yt+1 out as εt+1 . Move one period backward, yt−1 = xt−1 + byt + bεt Rearrange to seperate yt on the left-hand side, byt = yt−1 − xt−1 − bεt yt = b−1 yt−1 − b−1 xt−1 − εt (4) is our starting point, move one period backward, yt−1 = b−1 yt−2 − b−1 xt−2 − εt−1 And substitute back to (4), yt = b−1 [b−1 yt−2 − b−1 xt−2 − εt−1 ] − b−1 xt−1 − εt = b−2 yt−2 − b−2 xt−2 − b−1 εt−1 − b−1 xt−1 − εt Following this pattern, we get
m m
(4)
yt = −
k=0
b−k εt−k −
k=1
b−k xt−k + b−m yt−m
3
Again, transversality condition, lim b−m yt−m = 0
DSGE的学习
接下来，讨论DSGE的学习。
在教材上，DSGE模型的学习是三板斧。
第一个是DSGE的思想和技巧：Cooley，1995，Frontiers of Business Cycle Research。
推荐这本书，而不是其他，是有原因的。
在RBC中，竞争性均衡模型的构建语言是递归竞争性均衡，换言之，就是Sargent等所说的用“大K、小k技巧”来表述理性预期思想。
而在其他教材中，更多的是直接定义竞争性均衡：消费者最优、厂商最优、两者最优的结构构成所有市场的供给与需求、市场出清要求供求相等（价格随之决定）——如果竞争性均衡搞不明白，可以看看Williamson的宏观经济学这本教材，再加Varian的中微做补充。
这本书里面的重要知识是：RBC模型、社会计划者问题、竞争性均衡角度构建的非最优经济环境、校准。
不需要看的是模型的求解，因为选取的一些求解方法有些老旧了，现在不怎么常用。
第二个是动态规划：Stokey，Lucas with Prescott，1989，名字就不写了，大家都知道。
主要目的是三个：其一是用来做参考书，以便什么时候需要证明解的存在和唯一性时拿出来翻阅；其二是了解动态规划的术语；第三个是拿来提高数学（尤其是代数）功底。
第三个是动态规划的一系列应用：Ljungqvist and Sargent，2000，名字就不写了，大家都知道。
这本书不好啃，而且啃了未必有所裨益，因为里面的专题实在太多了，而且，一些详细介绍的研究方法存在严重的过时，而一些重要的研究方法又只有只言片语。
举例来说，线性二次型动态规划，现在用得很少，书上很多地方都是用它来求解；再者，Blanchard and Kahn 求解、King-Plosser-Rebelo求解（或Uhlig的tookit）、状态空间、极大似然、GMM 等等重要方法，都是只言片语。
以上三本书实际上有个良好的替代品：Dirk Kreuger的Macroeconomic Theory。
清华大学计量经济学课件
二、经济预测
• 计量经济学模型作为一类经济数学模型，是从 用于经济预测，特别是短期预测而发展起来的。
• 计量经济学模型是以模拟历史、从已经发生的 经济活动中找出变化规律为主要技术手段。
• 对于非稳定发展的经济过程，对于缺乏规范行 为理论的经济活动，计量经济学模型预测功能 失效。
• 模型理论方法的发展以适应预测的需要。
ln(人均食品需求量)=－2.0+0.5ln(人均收 入)－0.8ln(食品价格) +0.8ln(其它商品价格)
⑵ 统计检验 由数理统计理论决定 包括拟合优度检验 总体显著性检验 变量显著性检验
⑶ 计量经济学检验 由计量经济学理论决定 包括异方差性检验 序列相关性检验 共线性检验
⑷ 模型预测检验 由模型的应用要求决定 包括实际
例如：ln(人均食品需求量)=α+βln(人均收入) +γln(食品价格) +δln(其它商品价格)+ε
其中α 、β、γ、δ的符号、大小、 关系
二、样本数据的收集
⑴ 几类常用的样本数据 时间序列数据 截面数据 虚变量离散数据 联合应用
⑵ 数据质量 完整性 准确性 可比性 一致性
三、模型参数的估计
微观计量：
非
选择性样本模型
经
典
微观计量：
计
离散选择模型
量
经
济 学
时间序列：
协整理论—现代宏观计量
时间序列：
ARCH—现代金融计量
Heckman McFadden
Granger Engle
五、计量经济学在经济学科中的地位
△ 从现代西方经济学的特征看 △ 从西方经济学的发展历史看 △ 从世界一流大学经济学课程表看 △ 从国际经济学刊物论文看 △ 从经济学的“世界先进水平”看
动态随机一般均衡模型及其应用DSGEPPT课件
• 国际经济周期(Mendoza (1991), Backus et al. (1992), Stockman andTesar (1995))
• 能源价格与波动：Kim(1992)
2020/1/122013-12-02
DSGE
summary measure such as a neutral real
interest 2020/1/122013-12-02 rate would be DSGE unwise...
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模型调校
• Kydland and Prescott (1982) 建立了一种崭新 的数量化研究方法——模型调校（Calibration）
– 卢卡斯批判 – 外生结构冲击的结构性不足 – 模型的实证表现尚待改善
2020/1/122013-12-02
DSGE
2
一、动态随机一般均衡模型
2020/1/122013-12-02
DSGE
3
动态随机一般均衡模型的特征
• 所谓“动态随机一般均衡模型”是指模型 具有三大特征：
– 动态 – 随机 – 一般均衡
increases in output gaps and in inflation result from increases in the natural rate of interest that are not offset by a corresponding tightening of monetary policy...or alternatively from loosenings of monetary policy that are not justified by declines in the natural rate of interests.
清华大学中级微观经济学讲义
.
Short-Run Iso-Profit Lines
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Short-Run Profit-Maximization
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Economic Profit
The economic profit generated by (x1,…,xm,y1,…,yn) is
p 1 y 1 p n y n w 1 x 1 w m x m .
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For the time being, we restrict to the case of a competitive firm, which is a tiny relative to the market size and takes prices p1,…,pn w1,…,wm as given constants;
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Comparative Statics of Short-Run
Profit-Maximization
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Comparative Statics of Short-Run
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2024版清华大学运筹学课件(完整课件)
其他存储问题
多品种存储问题
研究多种物资同时存储时的优化策略，如联合订货、分类管理等。
有限期存储问题
考虑物资具有有限保质期的情况，研究如何在有限时间内合理安排进货和 销售计划。
带有约束条件的存储问题
研究在资金、仓库容量等约束条件下如何制定最优存储策略的问题。
THANKS
感谢观看
最大流可以增加。
Ford-Fulkerson算法
02
通过不断寻找增广路并增加流量来求解网络最大流问题。
Edmonds-Karp算法
03
在Ford-Fulkerson算法的基础上，使用BFS寻找增广路，
可以保证算法的时间复杂度为多项式级别。
最小费用最大流问题
01
最小费用流的定义
02
消圈定理
03
SPFA算法
运筹学的发展
战后，运筹学逐渐应用于经济、 管理、工程等领域，发展出线性 规划、整数规划、动态规划等分 支。
运筹学的研究对象与特点
研究对象
运筹学的研究对象主要是各种系统的优化问题，如生产、运输、库存、财务等系统的优 化。
特点
运筹学具有多学科交叉性，涉及数学、计算机科学、经济学等多个学科；强调系统性和 整体性，关注整体最优而非局部最优；注重定量分析和数学建模，通过数学模型描述和
整数规划问题的分类
根据整数变量的取值范围，可分为纯整数规划、混合 整数规划和0-1整数规划。
整数规划问题的数学模型
整数规划问题的数学模型与线性规划问题类似， 但需要加入整数约束条件。
分枝定界法
分枝定界法的基本思想
将原问题分解为若干个子问题，每个子问题对应原问题的 一个子集，通过求解子问题的最优解来逼近原问题的最优 解。
DSGE模型
Home bias,exchange rate disconnect,and optimalexchange rate policy qJian Wang *Research Department,Federal Reserve Bank of Dallas,2200N.Pearl Street,Dallas,TX 75201,USAJEL classification:E52F31F41Keywords:Home biasExchange rate volatilityExchange rate disconnectOptimal monetary policy a b s t r a c tThis paper examines how much the central bank should adjust the interest rate in response to real exchange rate fluctuations.The paper first demonstrates,in a two-country Dynamic Stochastic General Equilibrium (DSGE)model,that home bias in consumption is important to replicate the exchange rate volatility and exchange rate disconnect documented in the data.When home bias is high,the shock to Uncovered Interest rate Parity (UIP)can substantially drive up exchange rate volatility while leaving the volatility of real macroeconomic variables,such as GDP,almost untouched.The modelpredicts that the volatility of the real exchange rate relative to that ofGDP increases with the extent of home bias.This relation is supportedby the data.A second-order accurate solution method is employed tofind the optimal operational monetary policy rule.Our modelsuggests that the monetary authority should not seek to vigorouslystabilize exchange rate fluctuations.In particular,when the centralbank does not take a strong stance against the inflation rate,exchangerate stabilization may induce substantial welfare loss.The model doesnot detect welfare gain from international monetary cooperation,which extends Obstfeld and Rogoff’s [Obstfeld,M.,Rogoff,K.,2002.Global implications of self-oriented national monetary rules,Quar-terly Journal of Economics May,503–535]findings to a DSGE model.Ó2008Elsevier Ltd.All rights reserved.q I am grateful to Charles Engel for his priceless advice and encouragement.I would like to thank Menzie Chinn,Vasco Curdia ´,Kevin Grier,Bruce Hansen,Ben Keen,Robert Kollmann,Nelson Mark,Yongseok Shin,John Taylor,Ken West,and many seminar and conference participants for discussions and comments.I also want to thank the co-editor Michael Melvin and an anony-mous referee for close reading and constructive criticism.Nicole Ball provided excellent research assistance.All views are those of the author and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.*Tel.:þ12149226471;fax:þ12149225194.E-mail address:jian.wang@Contents lists available at ScienceDirectJournal of International Moneyand Financejournal homepage :www.else /locate/jimf0261-5606/$–see front matter Ó2008Elsevier Ltd.All rights reserved.doi:10.1016/j.jimonfin.2008.12.010Journal of International Money and Finance 29(2010)55–781.IntroductionMany countries adopted a monetary policy regime defined by John Taylor as a trinity:(1)a flexible exchange rate,(2)an inflation target,and (3)a monetary policy rule.John Taylor (2001)argues that the role of the exchange rate in the monetary policy rule is an important issue for this new policy regime.In this paper we first show that home bias in consumption can help to replicate two findings in the data:1.Exchange rates are much more volatile than other macroeconomic variables such as GDP (exchange rate volatility);2.The volatility of output does not respond to the volatility of exchange rates (exchange rate disconnect).Under this explanation of exchange rate volatility and disconnect,our model suggests that the central bank should not vigorously stabilize the real exchange rate in its monetary policy rule.There are two different strands of literature focusing upon exchange rate stabilization.The first one studies the tradeoff between exchange rate stabilization and the stability of the whole economy.Ball (1999)and Svensson (2000)find that the inclusion of the exchange rate into a monetary policy rule can stabilize output or inflation,or both.In contrast,Obstfeld and Rogoff (1995)warn policymakers that the required interest rate changes for exchange rate stabilization can aggravate instability elsewhere in the economy.In an empirical study on New Zealand,West (2004)finds that exchange rate stabilization would increase the volatility of output,inflation,and the interest rate.Another strand of literature uses welfare-based New Open-Economy Macroeconomic (NOEM)models to study the tradeoff between real exchange rate stabilization and the expenditure-switching effect.1Though elegant in allowing for analytical solutions,these NOEM models are usually static with no price persistence,and are therefore unable to address the tradeoff considered in the first strand of literature.Kollmann (2004)incorporates the tradeoffs in both strands of literature into a two-country Dynamic Stochastic General Equilibrium (DSGE)model.He compares the welfare effects of the exchange rate policy in a sticky-price dynamic model.Our paper is closely related,but we emphasize the connection between home bias in consumption and the exchange rate disconnect puzzle.Several authors have recently used pricing in the importer’s currency (Local Currency Pricing,or LCP)to model the low exchange rate pass-through documented in the data.2The short-run exchange rate pass-through into import prices is close to zero in those models.In industrial countries,although the pass-through into consumer prices is low,there is still a sizable short-run exchange rate pass-through into import prices.3In addition,the LCP in import prices is also criticized by Obstfeld and Rogoff (2000b)on the grounds that it generates counterfactual correlation between the exchange rate and the terms of trade.Therefore,in this paper,we follow Devereux and Engel (2007)by assuming that the imports and exports are priced in the producer’s currency (Producer Currency Pricing,or PCP),but final goods are priced in the consumer’s currency.We further assume that both import prices and consumer prices are sticky.When those prices are fixed in the short-run,the import prices in the importer’s currency vary with the exchange rate,but the consumer prices do not fluctuate with the exchange rate.In this way,our model allows a low exchange rate pass-through into consumer prices and a relatively high pass-through to import prices.Under this setup,we find that home bias in consumption is critical for our model to replicate the well-docu-mented disconnect between exchange rates and real macroeconomic variables.4We follow Devereux and Engel (2002)and Kollmann (2004)by using the Uncovered Interest rate Parity (UIP)shock to generate fluctuations in the nominal exchange rate.However,Devereux and Engel (2002)use the LCP for import prices to insulate the economy from exchange rate fluctuations.After we allow the exchange rate movements to pass through into the import prices,we find that only when the foreign market is a small portion of total output could the UIP shock in the financial market substantially increase the1For example,see Devereux and Engel (2003,2007),Obstfeld (2001,2002)and the references cited therein.An exception is Obstfeld (2004).He defends the flexible exchange rate regime in light of its function that allows the central bank to pursue an independent interest rate policy in a world of international capital mobility.2For instance,see Devereux and Engel (2002,2003),Chari,Kehoe,and McGrattan (2002),Duarte and Stockman (2005),and Kollmann (2004).3For instance,see Campa and Goldberg (2005),Mumtaz,Oomen and Wang (2006).4For empirical studies on the exchange rate disconnect,see Flood and Rose (1995)and Baxter and Stockman (1989).J.Wang /Journal of International Money and Finance 29(2010)55–7856J.Wang/Journal of International Money and Finance29(2010)55–7857 volatility of the exchange rate while keeping the volatility of real variables,such as GDP,almost unchanged.The multiple price-stickiness used in this paper also helps us in two additional ways.First,it helps to replicate that the cyclical behavior of CPI inflation generally differs from that of PPI inflation in the data. The latter is typically more volatile and less persistent than the former.5More importantly,the multiple price-stickiness incorporates into our model a new tradeoff in the exchange rate policy discussed by Devereux and Engel(2007):when prices of both imports andfinal consumption goods are sticky,the flexible exchange rate facilitates the expenditure-switching effect for imports and exports,but distorts the prices offinal consumption goods across countries.Based on the fact that the expenditure-switching effect is empirically weak,they argue,the exchange rate should be stabilized to eliminate price distortions forfinal consumption goods.We can investigate the quantitative importance of this tradeoff in a model with home bias in consumption.We limit our search for the optimal exchange rate policy to a class of simple operational policy rules. Our benchmark model suggests a very weak stance on exchange rate stabilization.Given that the central bank strongly stabilizes the inflation rate,the extra gain from exchange rate stabilization is negligible.Intuitively,the real exchange rate volatility in our model is primarily driven by home bias in consumption and the Uncovered Interest rate Parity(UIP)shock.The similarity between home and foreignfinal consumption bundles is low when consumption is highly biased toward domestic goods. Therefore,the CPI-based real exchange ratefluctuations do not necessarily imply significant price distortions across countries.In this case,the gain from exchange rate stabilization is small.However, the restriction on exchange rateflexibility obstructs the terms of trade adjustment for intermediate goods.What’s more,movements of the interest rate required for exchange rate stabilization induce prolonged deviations of the inflation rate from its steady-state level,which also lowers welfare.In addition,wefind that exchange rate stabilization may induce substantial welfare loss if the central bank takes a weak stance against the inflation rate.When the central bank takes a weak stance on inflation stabilization,aggressively stabilizing the real exchange rate will transmit exchange rate fluctuations into large and prolonged inflationfluctuations through the tradeoff between exchange rate and inflation stabilization.In our model,this policy could be very detrimental.We alsofind that international monetary cooperation does not generate significant welfare gain compared to the Nash equilibrium.This result extends the analysis of Obstfeld and Rogoff(2002)to a more complicated DSGE model.When both country-specific productivity shocks and price-stickiness exist in an open-economy macro model with imperfect international risk sharing,policymakers may have to strike a balance between mitigating international risk sharing and sticky-price distortions.Obstfeld and Rogoff(2002) discuss this tradeoff thoroughly by deriving an analytical solution to a static New Open-Economy Macroeconomic(NOEM)model.They show that the welfare gain from international monetary coop-eration is small.Wefind similar results in this paper.Thisfinding may reflect the fact that the gain from international risk sharing is usually found to be small in open-economy macroeconomic models.6 Compared to Obstfeld and Rogoff’s(2002)model with static price setting,deviating fromflexible price equilibrium is more costly in our model with Calvo-style price setting:it induces price dispersions.The Nash equilibrium and international cooperation coincide if the cost of price dispersions exceeds the benefit from international risk sharing.As the degree of home bias decreases,wefind it is more desirable to stabilize the real exchange rate.7Kollmann(2004)and Leith and Wren-Lewis(2006)find similar results.However,we want to be cautious in interpreting this result as offering support for exchange rate stabilization.In the case with5For example,see Clark(1999).This difference may be caused by the CPI smoothing policy of central banks rather than the generic difference in price-stickiness in these two sectors.However,the same pattern has also been found in the case of the mid-1930s(see Means(1935)),when the role of monetary policy was not as significant as it is today.Dong(2006)alsofinds this difference in estimating a small open-economy DSGE model.6For instance,see Cole and Obstfeld(1991)and Tesar(1995).7Devereux and Engel(2007)have no home bias in their model andfind that the central bank should stabilize the exchange rate.When home and foreign consumption bundles are identical,the gain from exchange rate stabilization is higher than that in our benchmark model.Another reason why they get different results is because there is no tradeoff between exchange rate stabilization and inflation stabilization in their model.Therefore,the cost of exchange rate stabilization is lower in their model.little home bias,the volatility of the real exchange rate relative to that of GDP is much smaller than what is observed in the data.In addition,the disconnect between exchange rate and output volatilities exists only when the home bias is high.Intuitively,when the foreign market is only a very small portion of output (high home bias),the UIP shock in the financial market can drive up exchange rate volatility,but has very limited impact on output.In this sense,our model provides an interesting solution to the exchange rate disconnect puzzle:home bias in consumption.8Our results are consistent with Hau’s (2002)finding that the volatility of the real exchange rate is positively correlated with the level of home bias.We confirm this finding in our data.In addition,our model also predicts that there is an even stronger relation between the extent of home bias and the volatility of the real exchange rate relative to the volatility of GDP.We find empirical support for this prediction in our OECD-country data as well.Our benchmark model suggests that the central bank should not include the real exchange rate in its policy rule.This finding is robust under the preference of habit persistence,though the welfare cost of exchange rate variability is higher under habit persistence.9When household preferences are more risk sensitive,welfare loss is higher for given exchange rate variability.However,this does not guarantee that central banks should react to exchange rate variation.The cost of real exchange rate stabilization is also higher under habit persistence.Our model shows that the cost still exceeds the benefit for real exchange rate stabilization in this case.In this paper,we take home bias in consumption as exogenously given.However,Helpman (1999)argues that there is no clear evidence for such demand patterns.Our treatment of home bias can be taken as a shortcut for a model with no home bias in consumption but with high international trade costs,such as the model in Atkeson and Burstein (2005).Burstein,Neves,and Rebelo (2003)find a trade cost that is large enough to generate the same home bias level as in our model,even after controlling for nominal exchange rate fluctuations.The trade-depressing effect of exchange rate volatility is more plausible for low-frequency movements of the exchange rate for which hedging strategies are unavailable.10We doubt this effect would be strong enough to overturn our results at business cycle frequencies.We want to emphasize that our paper is not about the optimal choice of exchange rate regimes.We limit our discussion to a flexible exchange rate regime and study if the central bank should include the exchange rate in its monetary policy rule.Our model has abstracted away from several benefits of an exchange rate ernments may decide to adopt a peg because it may stimulate international trade,eliminate competitive devaluations,and impede exchange rate speculation.11Kollmann (2004)finds that a monetary union (an extreme case of a peg)can raise welfare because it eliminates UIP shocks.In this paper,we assume that the UIP shocks do not respond to monetary policy under a flexible exchange rate regime.We admit that our results may depend on the interaction between exchange rate stabilization and the volatility of UIP shocks.Incorporating this interaction requires a model that can endogenously generate the UIP puzzle and exchange rate volatility.We leave this for future research.We also abstract from tradability and trade frictions in our model.Leith and Wren-Lewis (2006)argue that exchange rate movements may cause a large misalignment between tradable and non-tradable sectors.They find,in a model with both tradable and nontradable sectors,that this misalignment provides an additional incentive for the central bank to stabilize the exchange rate,though as in our model,they find the gain of exchange rate stabilization is small.Kumhof,Laxton,and Naknoi (2008)show that the exchange rate enters the optimal policy rule in a model in which firms enter and exit foreign markets when the exchange rate fluctuates.Empirical studies on the impacts of business cycle frequency fluctuations of the exchange rate on firms’exporting decisions may be fruitful for future research.Our benchmark model suggests that the monetary authority should not seek to stabilize the real exchange rate.This result should depend on the way in which the exchange rate is modeled.There is no8Obstfeld and Rogoff (2000a)conjecture that the home bias in consumption caused by trade costs may be important in explaining the exchange rate disconnect puzzle.9Bergin et al.(2007)find similar results.10Empirical findings on this effect are mixed.For instance,see Asseery and Peel (1991)and Koray and Lastrapes (1989).11For instance,see Klein and Shambaugh (2006),Corsetti,et al.(2000),and Pastine (2002).J.Wang /Journal of International Money and Finance 29(2010)55–7858consensus among economists on exchange rate determination.For instance,Chari,Kehoe,and McGrattan(2002)and Devereux and Engel(2002)attribute exchange rate volatility to nominal stickiness.In contrast,this volatility is mainly driven by home bias infinal consumption bundles in Atkeson and Burstein(2005),Ghironi and Melitz(2005),and our model in this paper.Naknoi(2008) argues that exchange ratefluctuations are also affected by the endogenous tradability of goods.Further research is desirable to determine how robust our policy suggestion is in different models of exchange rate determination.The remainder of the paper is organized as follows:Section2lays out the theoretical model;Section 3provides details about calibration and compares the business cycle statistics of our model with those in the data;Section4discusses the solution method and presents ourfindings for policy evaluation; and Section5concludes and discusses potential future research directions.2.A two-country DSGE modelThe world economy consists of two symmetric countries:Home and Foreign.There are two sectors of production in each country:thefinal goods sector and the intermediate goods sector.Final goods are internationally nontradable,and are produced from the internationally traded Home and Foreign intermediate good composites.The intermediate goods are produced from capital and labor in each country.In the Homefinal goods sector,there is a continuum of differentiatedfinal goods Y t(i)indexed by i˛[0,1].The representative household of Home country uses them to form afinal good composite Y t according to equation(1)for consumption,investment,saving,and associated costs:Y t¼"Z1Y tðiÞqÀ1q d i#qqÀ1:(1)In equation(1),q is the elasticity of substitution between differentiatedfinal goods Y t(i).Each variety of final goods is produced from the Home and Foreign intermediate good composites Y Ht and Y Ft by a singlefinal goodsfirm.The Home(Foreign)intermediate good composite is composed of differen-tiated Home(Foreign)intermediate goods Y Ht(j)(Y Ft(j)).In the intermediate good sector,each variety of Home(Foreign)intermediate goods is produced by a singlefirm with capital and labor in the Home (Foreign)country.2.1.Final goods marketThefinal good market is monopolistically competitive.In the Home country,eachfinal goodsfirm produces a variety offinal goods from the Home and Foreign intermediate good composites according to equation(2):Y s tðiÞ¼ha1g Y dHtðiÞgÀ1gþð1ÀaÞ1g Y d FtðiÞgÀ1gi g gÀ1:(2)Y t s(i)is the supply offinal good i,and Y dHt ðiÞðY dFtðiÞÞis the Home(Foreign)intermediate good compositedemanded byfinal goodfirm i.a is the weight of the home intermediate good composite required for producingfinal consumption goods.Consumption is home biased when a>1/2.g is the elasticity of substitution between the home and foreign intermediate good composites.Symmetrically,the production function for Foreignfinal goods producer i isY sÃtðiÞ¼ha1g Y dÃFtðiÞgÀ1gþð1ÀaÞ1g Y dÃHtðiÞgÀ1gi g gÀ1:(3)The variables with asterisks in equation(3)are foreign counterparts of the variables in Home country. Due to the symmetry between the two countries,we will focus only on the Home country to describe our model.J.Wang/Journal of International Money and Finance29(2010)55–7859In each country,final goods prices are denominated in the consumer’s currency.In contrast,intermediate goods are priced in the producer’s currency.For given technology,the firms choose pricesto maximize the expected profit.We introduce staggered price setting a l a´Calvo (1983)and Yun (1996).In each period,an individual firm has a probability of 1Àl f to re-optimize its price.Otherwise,it will charge a price equal to last period’s price multiplied by the long-run inflation rate (p ).When a final good firm re-optimizes its price,it will choose a price P tt to maximize the expected life time real profit.12So the profit maximization problem can be written asmax P tt XN k ¼0E t 8<:l k f G t ;t þk P À1t þk 24 p k P tt Àmc t þk p k P tt P t þk !Àq Y d t þk 359=;;where G t ,t þk is the pricing kernel between period t and t þk .We assume all firms are owned by home households,and therefore G t ,t þk is the marginal rate of substitution between time t and time t þk consumption.From the first order condition,we find that the optimal price P tt satisfiesP tt ¼q P N k ¼0l f p Àq k E t h G t ;t þk mc t þk P q À1t þk Y t þk i ðq À1ÞP N k ¼0 l f p 1Àq E t h G t ;t þk P q À1t þk Y t þki :(4)When price is flexible (l f ¼0),the optimal price reduces to P tt ¼q mc t /(q À1).The monopolist charges a constant markup over its marginal cost.Under this staggered price setting environment,prices are not synchronized across firms.At any time t ,only a fraction of 1Àl f firms charge up-to-date optimal price P tt .A fraction of l f k (1Àl f )firms charge outdated price P t –k ,t for k ¼1,2,.The price of the final goods composite P t evolves according to equation (5):P t ¼h 1Àl f P 1Àq tt þl f ðp P t À1Þ1Àq i 11Àq :(5)2.2.Intermediate goods marketThe Home intermediate good composite used by final good producers is made from a continuum of differentiated intermediate goods indexed by j ˛[0,1]according to equation (6):Y Ht ¼"Z10Y Ht ðj Þf À1f d j #f f À1;(6)where f is the elasticity of substitution between differentiated intermediate goods.The Foreign intermediate good composite is made in the same way from Foreign differentiated intermediate goods.We suppose that intermediate goods firms set prices in the producer’s currency,and the Law of One Price (LOP)holds in this market.Let P Ht (j )be the price of Home intermediate good j in the Homemarket,and let P Ht *(j )be the price in the Foreign market.By LOP,we have P Ht (j )¼S t P Ht*(j ).The intermediate good producers rent capital and labor from households.The technology takes a standard Cobb-Douglas form:Y s Ht ðj Þ¼A t K j t ðj ÞL 1Àj t ðj Þ;(7)where Y Ht s (j )is the supply of intermediate good j .K t (j )and L t (j )are,respectively,capital and laborused by intermediate good company j .A t (A t*)is the technology shock in the Home (Foreign)country,12For the notation P tt ,the first time subscript denotes when the price is re-optimized,and the second time subscript gives the current time.For example,P tt þk means the price is re-optimized at time t and is still effective at time t þk .From our setup,P tt þk ¼p k P tt .J.Wang /Journal of International Money and Finance 29(2010)55–7860which is identical for all firms in that country.The logarithms of productivity shocks follow a VAR(1)process:log ðA t Þlog ÀA Ãt Á!¼ x 11x 12x 21x 22! log ðA t À1Þlog ÀA Ãt À1Á!þ 3t 3Ãt !;(8)where x 12¼x 21are technology spillovers.The vector containing 3t and 3t *is i.i.d.with means of zero and a variance-covariance matrix V .We follow the same way as in the final goods sector to introduce staggered prices.1Àl int is the probability for firm j to re-optimize its problem.When re-optimizing price,the intermediate good producer j chooses a price P Htt to maximize the discounted lifetime real profitmax P Htt X N k ¼0E t (l k int G t ;t þk P À1Ht þk " p k P Htt Àmc int t þk p k P Htt Ht þk!Àf Y w Ht þk #);where Y w Ht þk ¼Y d Ht þk þY d *Htþk is the world demand for the Home intermediate good composite.2.3.Household’s problemThe representative household maximizes expected lifetime utility,which is given in equation (9):U ¼E 0"XN t ¼0b t u t ðC t ;1ÀL t Þ#;(9)where E 0[$]is the conditional expectation operator,and b is the subjective discount factor.The period utility u t is a concave function of final goods composite C t and leisure 1ÀL t .The representative household sells labor and rents capital to domestic intermediate goods firms in a competitive market.The law of motion for capital takes the standard form of:K t þ1¼ð1Àd ÞK t þI t ;(10)where I t is the investment at time t ,and d is the capital depreciation rate.Capital and investment are in the form of the final good composite.Following the literature,we assume a quadratic capital adjustment cost:AC t ¼1F I t tÀd 2K t ;where F is a scale parameter of capital adjustment cost.Households can only trade non-state-contingent Home and Foreign nominal bonds.There is a quadratic real cost of holding bonds:F t ¼12f d B Ht þ1P t 2þ12f f S t B Ft þ1P t 2;where B Ht þ1(B Ft þ1)is the Home (Foreign)bond held by the household in the Home country between period t and period t þ1.All bonds are denominated in the issuing country’s currency.f d and f f are parameters of cost for holding domestic bonds and holding foreign bonds,respectively.13We introduce this cost to ensure stationarity of the model.By assigning very small values to f d and f f ,this cost has a negligible effect on model dynamics.1413Note that in Foreign country,f d is the cost of holding Foreign bonds,and f f is the cost of holding Home bonds.14See Schmitt-Grohe´and Uribe (2003)for more details.J.Wang /Journal of International Money and Finance 29(2010)55–7861Other incomes for households include profits from intermediate and final goods firms.The repre-sentative household uses these incomes to buy differentiated final goods and aggregate them into a final good composite (equation (1)).The final good composite can be used for consumption,investment or paying the costs of adjusting the capital stock and holding bonds.Based on the above setup,we obtain the budget constraint:C t þB Ht þ1P t þS t B Ft þ1P t þI t þ12F I t K t Àd 2K t þ12f d B Ht þ1P t 2þ12f f S t B Ft þ1P t 2 W t L t P t þR t K t P t þB Ht ð1þi t À1ÞP t þB Ft À1þi Ãt À1ÁS t P t þp int t P t þp ft P t :(11)For any given initial capital stock and asset position,the representative household chooses the paths of consumption C t ,labor supply L t ,capital investment I t ,and bond holdings B Ht þ1and B Ft þ1to maximize the expected life time utility subject to the above budget constraint.In this paper,we employ the period utility function as in equation (12):u t ðC t ;1ÀL t Þ¼C 1Às t 1Às Àr L t :(12)The linear form of disutility from labor is used to capture fluctuations in the labor market and can be justified by the indivisible labor assumption,as in Hansen (1985).The first order conditions of the optimal bond holdings approximately imply the uncovered interest rate parity (UIP)condition 15:E t G t ;t þ1ð1þi t ÞP t þ1=P t !z E t G t ;t þ1À1þi Ãt ÁðP t þ1=S t þ1Þ=ðP t =S t Þ!;(13)where G t ;t þ1¼b ðv u =v c t þ1Þ=ðv u =v c t Þis the marginal rate of substitution between time t and time t þ1consumption.As it is well-documented that the UIP condition is strongly rejected by data (see Engel,1996and Lewis,1995for surveys),we follow Kollmann (2004)in introducing a UIP shock into the first order condition:1þf f S t B Ft þ1t ¼4t E t G t ;t þ1À1þi Ãt Át þ1t þ1t t !;(14)where 4t is a UIP shock that can be interpreted as the bias of market expectation on time t þ1exchange rate.16It enters the bond holding condition symmetrically in Foreign country.2.4.Monetary policy rules and market clearing conditionsIn Home country,the monetary authority follows a modified Taylor rule:i t ¼i þX p log p t p þX s log Q t Q ;(15)where X p and X s are policy parameters determined by the monetary authority.The variables without a time script are steady-state values.Q t is the real exchange rate defined as S t P t */P t .In this modified Taylor rule,the monetary authority adjusts the interest rate to stabilize the inflation rate and the real exchange rate.17Unlike the standard Taylor rule,the interest rate here does not react to the output gap.In a closed economy,Schmitt-Grohe´and Uribe (2004b)find that the interest rate should not respond to 15To obtain this condition,we delete f d B Ht þ1=P t and f f S t B Ft þ1=P t from the first order conditions since these two terms are small by assumption.16Jeanne and Rose (2002)model the market bias with noise traders.17Our results hold if the monetary authority stabilizes the change of the nominal exchange rate.J.Wang /Journal of International Money and Finance 29(2010)55–7862。
研究生学习建议及DSGE模型基础
研究生学习建议及DSGE模型基础一、研究生阶段学习方法 (1)1、研究习惯的养成 (2)2、确定研究方向与目标 (2)3、专业基础学习与研究工具方法掌握 (2)4、论文写作要点 (2)5、论文撰写技巧 (2)二、动态随机一般均衡模型简介（DSGE） (3)1、动态随机一般均衡模型概念 (3)2、动态随机一般均衡模型的应用 (4)3、动态随机一般均衡的估计方法 (5)三、动态随机一般均衡模型学习建议 (5)1、数学基础 (5)2、动态随机一般均衡模型方面的书籍 (7)3、软件应用 (7)四、动态随机一般均衡模型学习资源链接 (7)1、DSGE模型的讨论及基础数学知识解释 (7)2、DSGE课程的入门级课件 (7)3、DSGE课程的较高级课件 (8)4、DSGE研究上的一些牛人主页或有用资源 (8)五、数量经济学专业学习书籍推荐 (9)1、数学书 (9)2、金融计量类书籍 (15)六、学习计划与任务安排 (19)研究生阶段学习方法1、研究习惯的养成（1）自主学习与主动学习，研究生阶段与本科阶段的本质区别之一，从被动的接受性学习，向主动的研究性学习，从由教师讲授为主的学习向自主学习的转变。
（2）申报课题与撰写论文，在国家大力提倡创新型教育的前提下，研究生阶段的课题支持力度和数量在大幅度提高，而课题的申报与完成，则是培育研究生提出问题、发现问题和解决问题的有效途径。
因此，要积极的参与各类创新课题或项目的申报。
申报课题时，需要关注两个方面：（1）近期国家经济、社会生活中的热点和亟需解决的难点；（2）本学科领域发展的前沿。
前者属于应用对策类的咨询课题，后者属于基础研究类的课题。
要很好的把握前者，则需要对国内外经济生活中的新进展、新情况进行及时的掌握；要掌握本学科的发展的前沿，则需要阅读大量的文献，尤其是近期成果。
论文的写作习惯，要从研究生一年级即进行积极的培养和专门的训练。
这是与本科学习的根本区别之一，同时，也是体现自身优势的根本标志。
Stata DSGE模型入门指南说明书
Title Intro—Introduction to DSGE manualDescription Remarks and examples Also seeDescriptionDSGE stands for dynamic stochastic general equilibrium.DSGE models are multivariate time-series models that are used in economics,in particular,macroeconomics,for policy analysis and forecasting.These models are systems of equations that are typically derived from economic theory.As such, the parameters are often directly interpretable based on theory.DSGE models are unique in that equations in the system allow current values of variables to depend not only on past values but also on expectations of future values.The dsgenl command estimates parameters of nonlinear DSGE models.The dsge command estimates parameters of linear DSGE models.Remarks and examples We recommend that you read this manual beginning with[DSGE]Intro1and then continue with the remaining introductions.In these introductions,we will introduce DSGE models,show you how to use the dsgenl and dsge commands,walk you through worked examples of classic models,and present solutions to common stumbling blocks.[DSGE]Intro1and[DSGE]Intro2are essential reading.Read themfirst.Here you willfind an overview of DSGE models,descriptions of concepts used throughout the manual,discussion of assumptions,afirst worked example,and an introduction to the syntax.[DSGE]Intro1Introduction to DSGEs[DSGE]Intro2Learning the syntax[DSGE]Intro3focuses on classical DSGE models.It includes a series of examples that illustrate model solution,model estimation,and postestimation procedures for simple variants of common models.[DSGE]Intro3Classic DSGE examples[DSGE]Intro3a New Keynesian model[DSGE]Intro3b New Classical model[DSGE]Intro3c Financial frictions model[DSGE]Intro3d Nonlinear New Keynesian model[DSGE]Intro3e Nonlinear New Classical model[DSGE]Intro3f Stochastic growth model[DSGE]Intro4discusses some features commonly found in DSGE models and how to specify models with those features to dsge and dsgenl.The structural equations of the DSGE model must have a specific structure so that the model can be solved.Often,DSGE models are written using intuitive forms that do not have this structure.These intuitive forms can be rewritten in a logically equivalent form that has the structure required for solution.[DSGE]Intro4provides an overview of this topic and examples demonstrating solutions.12Intro—Introduction to DSGE manual[DSGE]Intro4Writing a DSGE in a solvable form[DSGE]Intro4a Specifying a shock on a control variable[DSGE]Intro4b Including a lag of a control variable[DSGE]Intro4c Including a lag of a state variable[DSGE]Intro4d Including an expectation dated by more than one periodahead[DSGE]Intro4e Including a second-order lag of a control[DSGE]Intro4f Including an observed exogenous variable[DSGE]Intro4g Correlated state variables[DSGE]Intro5–[DSGE]Intro8discuss technical issues.These introductions are essential reading, even though they are last.[DSGE]Intro5Stability conditions[DSGE]Intro6Identification[DSGE]Intro7Convergence problems[DSGE]Intro8Wald tests vary with nonlinear transforms[DSGE]Intro9,[DSGE]Intro9a,and[DSGE]Intro9b discuss Bayesian estimation.[DSGE]Intro9Bayesian estimation[DSGE]Intro9a Bayesian estimation of a New Keynesian model[DSGE]Intro9b Bayesian estimation of stochastic growth modelThe main command entries are references for syntax and implementation details.All the examples are in the introductions discussed above.[DSGE]dsge Linear dynamic stochastic general equilibrium models [DSGE]dsge postestimation Postestimation tools for dsge[DSGE]dsgenl Nonlinear dynamic stochastic general equilibrium models [DSGE]dsgenl postestimation Postestimation tools for dsgenl[DSGE]estat covariance Display estimated covariances of model variables[DSGE]estat policy Display policy matrix[DSGE]estat stable Check stability of system[DSGE]estat steady Display steady state of nonlinear DSGE model[DSGE]estat transition Display state transition matrix[BAYES]bayes:dsge Bayesian linear dynamic stochastic general equilibriummodels[BAYES]bayes:dsgenl Bayesian nonlinear dynamic stochastic general equilibriummodels[BAYES]bayes:dsge postestimation Postestimation tools for bayes:dsge and bayes:dsgenl [BAYES]bayesirf Bayesian IRFs,dynamic-multiplier functions,and FEVDsIntro—Introduction to DSGE manual3Also see[DSGE]Intro1—Introduction to DSGEsStata,Stata Press,and Mata are registered trademarks of StataCorp LLC.Stata andStata Press are registered trademarks with the World Intellectual Property Organization®of the United Nations.Other brand and product names are registered trademarks ortrademarks of their respective companies.Copyright c 1985–2023StataCorp LLC,College Station,TX,USA.All rights reserved.。
系统解剖与构建dsge框架
系统解剖与构建DSGE框架1. 介绍DSGE（动态随机一般均衡）模型是一种经济学模型，用于研究宏观经济的行为和变化。
DSGE模型基于一组方程，描述了各个经济变量之间的关系，并通过模拟分析来预测宏观经济的发展趋势。
本文将详细介绍系统解剖与构建DSGE框架的过程，包括模型的构建、参数估计、模型验证等方面。
2. DSGE模型构建DSGE模型的构建是一个复杂而繁琐的过程，需要考虑多个方面的因素。
以下是构建DSGE模型的一般步骤：步骤1：选择合适的理论框架选择合适的理论框架是构建DSGE模型的第一步。
常用的理论框架包括新古典主义合成体系、新凯恩斯主义体系等。
根据具体问题和研究目标选择合适的理论框架。
步骤2：确定经济变量和决策者行为在构建DSGE模型时，需要确定所研究经济系统的关键经济变量和决策者行为。
这些变量可以包括经济增长率、通货膨胀率、利率等。
决策者行为可以是家庭、企业或政府的行为。
步骤3：建立动态一般均衡方程DSGE模型的核心是建立动态一般均衡方程组。
这些方程描述了各个经济变量之间的关系，并反映了经济体内部的均衡条件。
通常使用微观基础理论来推导这些方程。
步骤4：确定模型参数在DSGE模型中，需要确定一组模型参数，这些参数描述了经济体内部的特征和行为规律。
参数可以通过历史数据估计得到，也可以通过其他方法进行估计。
3. DSGE模型的参数估计DSGE模型的参数估计是一个重要而复杂的过程。
以下是常用的参数估计方法：方法1：最大似然法最大似然法是一种常用的参数估计方法，它通过最大化样本数据出现的可能性来确定模型参数。
在DSGE模型中，最大似然法可以用于估计结构参数和状态空间参数。
方法2：贝叶斯方法贝叶斯方法是一种基于贝叶斯统计理论的参数估计方法。
它通过先验分布和样本数据来更新参数的后验分布，并得到参数的估计值。
贝叶斯方法在DSGE模型的参数估计中具有重要作用。
方法3：卡尔曼滤波卡尔曼滤波是一种递归算法，用于从观测数据中估计状态变量。
清华大学交流电机暂态PPT第一章
（假设条件） （《电机学》）
——用等值电路来简化实际的复杂模型：变压器等
值电路、异步电机等值电路（正弦稳态电路）
——运算等值电路：含有算子ຫໍສະໝຸດ 阻尼绕组的简化电路d
q
d
笼形回路
假设
简化
1d和1q阻尼回路
等值电路
r1
x1
I 1
′ x2
I m
rm
r2′
′ I 2
磁导分析法
FΛ = Φ 对均匀磁路或磁路的微元：磁导
Φ Λ B= =F S S
→
EG = I
S Λ=μ l
G =σ
Fλ = B
•
μ0 隐极同步电机：气隙均匀，气隙长度δ恒定→ δ （常数） λδ =
1 λ = μ （导磁系数） l
S l
• 凸极同步电机：不同位置的气隙长度δ (x)不同
μ0 → λδ (x ) = δ (x )
交交变频同步电机系统的数学模型
•多回路模型
阻尼绕组：按原始的笼形回路列写方程
阻尼损耗
i1 i i3 i4 i5 i6
D 2 D D D D D
= = = = = =
i1 q + i 3 d i2 q + i2 d i 3 q + i1 i 3 q − i1 i2 q − i2 i1 q − i 3
K R
合K瞬间： iL (0+)= iL (0-)= i0 ——闭合回路磁链不能突变 合K后： diL + RiL = U sin(t + θ ) L
dt
iL , ∞ ( t ) =
iL ( t ) =
DSGE模型
DSGE 模型在房产税影响分析的应用1.模型综述动态随机一般均衡模型（dynamic stochastic general equilibrium model ,即DSGE ），是以微观和宏观经济理论为基础，采用动态优化方法考察个行为主体（家庭、厂商等）的决策，即在家庭最大化其一生效用、厂商最大化其利润的假设下得到个行为主体的行为方程.各行为主体在决策时必须考虑其行为的当期影响,以及未来的后续影响,同时,现实经济中存在诸多的不确定性，因此，DSGE 模型在引入各种外生随机冲击的情况下，研究各主体之间的相互作用和相互影响. （Dynamic stochastic general equilibrium modeling (abbreviated DSGE or sometimes SDGE or DGE ） is a branch of applied general equilibrium theory that is influential in contemporary macroeconomics. The DSGE methodology attempts to explain aggregate economic phenomena, such as economic growth, business cycles, and the effects of monetary and fiscal policy ， on the basis of macroeconomic models derived from microeconomic principles 。
）其主要特征有：（1）动态“动态"指经济个体考虑的是跨期最优选择（Inter-temporal Optimal Choice ）.因此，模型得以探讨经济体系中各变量如何随时间变化而变化的动态性质。
（2）随机“随机”则指经济体系受到各种不同的外生随机冲击所影响。
07经济学模型DSGE模型讨论之七——最简单的DSGE模型的Dynare模拟和MLEBayesian估计
这个notes纯粹是扫盲用的。
我用了一个最简单的线性DSGE，只有两个方程。
先是我用手算的方法找到saddle-path 的policy function，然后手算出impulse response function。
这些我都用Dynare做了计算，程序和结果都写在note里面。
上面是我note的截图，这个DSGE模型实际上就是一个linear rational expecation model (LREM)，但DSGE的线性化后的本质也就是个LRE。
虽然这个note提供的模型非常简单，但是思路在于如何用Dynare来深入学习这个动态系统。
有几个事情需要大家自己来做：1. beta和rho的大小，大家从换很多次calibration，看能对IRF带来什么影响?2. beta和rho都大于1的时候，你应该怎么修改模型，为了维持模型的稳定性？3. 看修改shock的stardard deviation能对模型带了什么影响？4. 如果你再加一个方程进去呢？什么样子的方程？以上内容我都试验过了。
这个东西没法帮大家试验，所以大家必须自己试着做。
这样你可以学到很多关于动态系统的感性认识。
之后，我用最大似然估计对参数估计，然后我故意制造under-identification的问题，让大家看一下结果是什么样子。
最后就是Bayesian estimation，我只估计了1个参数，用了2条平行马尔科夫链，做了超超短程模拟（只有500次，正常情况都是100000），为了省时间(我电脑只用50秒左右)，所以我并没有让电脑跑很长的马尔科夫链和多个平行链条。
所以结果非常差，但是这不是的目的。
目的还是在于让从来没见过整个估计过程的同学看到一个全貌。
所以我没有提及理论内容，或者是一带而过。
对于Bayesian estimation，有个地方要注意的就是shock的个数必须大于等observable的个数，这是启动Kalman filter模拟likelihood function的充分条件。
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Discrete Dynamic Optimization:Six ExamplesDr.Tai-kuang HoAssociate Professor.Department of Quantitative Finance,National Tsing Hua University,No. 101,Section2,Kuang-Fu Road,Hsinchu,Taiwan30013,Tel:+886-3-571-5131,ext.62136,Fax: +886-3-562-1823,E-mail:tkho@.tw.Example1:the neoclassical growth model Uhlig(1999),section4Model principlesSpecify the environment explicitly:1.Preferences2.Technologies3.EndowmentsrmationState the object of study:1.The social planner’s problem2.The competition equilibrium3.The gameThe environment:Preferences:the representative agent experiences utility according to35U=E t241X t=0 t C1 t 110< <1>0Absolute risk aversion: u00(c)u0(c)Relative risk aversion: cu00(c)u0(c)is the coe¢cient of relative risk aversionTechnologies:we assume a Cobb-Douglas production functionY t=Z t K t N1tBudget constraint:K t+1=I t+(1 )K tC t+I t=Y tor equivalently+(1 )K tC t+K t+1=Z t K t N1t0< <10< <1Z t,the total factor productivity,is exogenously evolving according tolog Z t=(1 )log Z+log Z t 1+ tt i:i:d:N 0; 20<<1Endowment:N t=1K0Information:C t,N t,and K t+1need to be chosen based on all information I t up to time t.The social planner’s problemThe objective of the social planner is to maximize the utility of the representative agent subject to feasibility,max (C t;K t)1t=0E t241X t=0 t C1 t 1135s:t:K0;Z0C t+K t+1=Z t K t+(1 )K tlog Z t=(1 )log Z+log Z t 1+ tt i:i:d:N 0; 2To solve it,form the Lagrangian:L=max(C t;K t+1)1t=0E t1X t=024 t C1 t 11+ t t Z t K t+(1 )K t C t K t+135The…rst order conditions are:@L:0=C t+K t+1 Z t K t (1 )K t@ t@L:0=C t t@C t@L:0= t+ E t h t+1 Z t+1K 1t+1+(1 ) i @K t+1Notice that K t+1appears in period t and period t+1.The…rst order conditions are often also called Euler equations. One also obtains the transversality condition.0=limE0h T C T K T+1iT!1The transversality condition is a limiting Kuhn-Tucker condition.It means that in net present value terms,the agent should not have capital left over at in…nity.Rewrite the necessary conditions:C t=Z t K t+(1 )K t K t+1(1)+(1 )(2)R t= Z t K 1t1=E t" C t C t+1! R t+1#(3) log Z t=(1 )log Z+log Z t 1+ t; t i:i:d:N 0; 2 (4) Equation accounting.To solve for the steady state,dropping the time indices and yieldC= Z K +(1 ) K KR= Z K 1+(1 )1= RFrom the welfare theorems,the solution to the competitive equilibrium yields the same allocation as the solution to the social planner’s problem.The case for competitive equilibrium is provided in Uhlig(1999)section4.4.Example2:Hansen’s real business cycle model Uhlig(1999),section4Hansen’s real business cycle modelThe model is an extension of the stochastic neoclassical growth model. The main di¤erence is to endogenize the labour supply.The social planner solves the problem of the representative agent:max E t 1X t=1 t0@C1 t 11AN t1As:t:C t+I t=Y tK t+1=I t+(1 )K tY t=Z t K t N1tlog Z t=(1 )log Z+log Z t 1+ t; t i:i:d:N 0; 2 Hansen only consider the case =1,so that the objective function is:1X t=1 t(log C t AN t)E tProof:lim !1C1t 1 1=lim!1e(1 )ln C t 11=lim!1@h e(1 )ln C t 1i=@@[1 ]=@=lim!1e(1 )ln C t ( ln C t)1=e0 ( ln C t)1=ln C tTo solve it,form the Lagrangian:L=max(C t;N t;K t+1)1t=0E t1X t=02664 t C1 t 11 AN t!+ t t Z t K t N1 t+(1 )K t C t K t+13775The…rst order conditions are:@L@C t=C t t=0@L@N t= A+ t(1 )Z t K t N t=0@L@K t+1= t+ E t t+1 Z t+1K 1t+1N1 t+1+(1 ) =0After arrangementA=C t(1 )Y t Nt(1)R t= Y tK t+1 (2)1= E t" C t C t+1! R t+1#(3) The steady state for the real business cycle model above is obtained by droppingthe time subscripts and stochastic shocks in the equations above.A= C (1 ) Y N1= RR= YK+1Exercise:Equation accounting.Example 3:Brock-Mirman growth model IChow (1997),chapter 2Consider the Brock-Mirman growth model:max f c t gE t1Xt =0t ln c ts:t:k t+1=k t z t c tThe Lagrangian is:1X t=0h t ln c t+ t t(k t z t c t k t+1)i L=E tThe…rst order conditions are:@L @c t = t1c ttt!=0@L@k t+1= t t+ t+1E t t+1 k 1t+1z t+1 =0 The…rst order conditions can be rearranged as:1c t= tt= E t t+1k 1t+1z t+1 Solve the problem explicitly.k T+1=0c T=k T z Tc t=d k t z t(guess a solution)E t t +1k 1t +1z t +1=E t1c t +1k 1t +1z t +1!=E t1d k t +1z t +1k 1t +1z t +1!=E t1d 1k t +1!=1d 1k t z t c tt =E t t +1k 1t +1z t +1,1c t = 1d 1kt z t c tc t =d(k t z t c t )d k t z t=d(k t z t d k t z t)1=1(1 d)d=1)c t=d k t z t=(1 )k t z tExample 4:Brock-Mirman growth model II Chow (1997),chapter 2We shall use dynamic programming to solve the Brock-Mirman growth model. Consider the Brock-Mirman growth model:max f c t g E t 1X t =0 t ln c ts:t:k t+1=k t z t c t Control variable:c tState variables:k t;z tThe Bellman equation is:[ln c t+ E t V(k t+1;z t+1)]V(k t;z t)=maxc tWe conjecture value function takes the form:V(k t;z t)=a+b ln k t+c ln z ta,b,c are coe¢cients to be determined.E t V(k t+1;z t+1)V(k t+1;z t+1)=a+b ln k t+1+c ln z t+1=a+b ln(k t z t c t)+c ln z t+1*E t ln z t+1=0)E t V(k t+1;z t+1)=a+b ln(k t z t c t) It followsV (k t ;z t )=max c t [ln c t + E t V (k t +1;z t +1)]=max c t f ln c t + [a +b ln (k t z t c t )]g max c t fgFirst order condition.@fg @c t =1c t bk t z t c t =0)c t=k t z t1+ bSubstitute this optimal c t=k t z t1+ b into maxc t fg and equate the expression to thevalue function,we can solve the coe¢cients a,b,c,which after some algebraic manipulation are:a=(1 ) 1h ln(1 )+ (1 ) 1ln( )ib= (1 ) 1c=(1 ) 1Example5:the money-in-the-utility function(MIU) modelWalsh(2003),chapter2Utility function,labour supply,and production function.u c t;m t;l t =C1 t1 + l1 t1C t h ac1 b t+(1 a)m1 b t i11 bn t=1 l t=f(k t;n t;z t) y t=e z t k t n1tz t= z t 1+e tTwo state variables.a t t+(1+i t 1)b t 11+ t +m t 11+ tk t The objective function is:1X t=0 t u c t;m t;1 n tE ts:t:f(k t;n t;z t)+(1 )k t+a t=c t+k t+1+m t+b t The Bellman equation is:V(a t;k t)=max h u c t;m t;1 n t + E t V(a t+1;k t+1)i Use the de…nition of a t to substitute for a t+1.a t+1 t+1+(1+i t)b t1+ t+1+m t1+ t+1Use the budget constraint to eliminate k t+1.k t +1=f (k t ;n t ;z t )+(1 )k t +a t c t m t b tRewrite the Bellman equation as:V (a t ;k t )=max h u c t ;m t;1 n t+ E t V (a t +1;k t +1)i=max 26664u c t ;m t;1 n t+ E t V 0@ t +1+(1+i t )b t 1+ t +1+m t 1+ t +1;f (k t ;n t ;z t )+(1 )k t +a t c t m t b t1A 37775=maxc t ;b t ;m t ;n tfgThe…rst order conditions are:@fg@c t:u c c t;m t;1 n t E t V k(a t+1;k t+1)=0@fg @b t : E t1+i t1+ t+1!V a(a t+1;k t+1) E t V k(a t+1;k t+1)=0@fg@m t:u m c t;m t;1 n t + E t11+ t+1!V a(a t+1;k t+1) E t V k(a t+1;k t+1)=0@fg@n t: u n c t;m t;1 n t + E t V k(a t+1;k t+1)f n(k t;n t;z t)=0@V(a t;k t)@a t =@fg@a t:V a(a t;k t)= E t V k(a t+1;k t+1)@V(a t;k t)@k t =@fg@k t:V k(a t;k t)= E t V k(a t+1;k t+1)[f k(k t;n t;z t)+1 ]Resources constrains.y t+(1 )k t=c t+k t+1We can eliminate the partial di¤erentiations of value function in the F.O.C. and then simplify the equilibrium conditions into9equations with9endogenous variables to be determined.Equilibrium conditions include F.O.C.,de…nitions,and resources constraints. y t;c t;k t+1;m t;n t;R t; t;z t;u tu c c t;m t;1 n t =u m c t;m t;1 n t + E t 24u c c t+1;m t+1;1 n t+11+ t+135(1)u l c t;m t;1 n t =u c c t;m t;1 n t f n(k t;n t;z t)(2) u c c t;m t;1 n t = E t R t u c c t+1;m t+1;1 n t+1 (3) R t=1 +f k(k t;n t;z t)(4)y t+(1 )k t=c t+k t+1(5)y t=f(k t;n t;z t)(6)m t=1+ t1+ t!m t 1(7)z t= z t 1+e t(8) u t= u t 1+ z t 1+'t(9)Exercises:use Lagrangian to solve the MIU model.L =1Xt =0t uc t ;m t;1 n t+1Xt =0 t +1E t t +124f (k t ;n t ;z t )+(1 )k t + t +(1+i t 1)b t 11+ t +m t 11+ t c t k t +1 m t b t 35@L @ct ;@L @m t ;@L @n t ;@L @k t +1;@L @b tShow that combing the F.O.C.of the Lagrangian method gives us the same equations as by using the Bellman equation.Example6:the cash-in-advance(CIA)model Walsh(2003),chapter3,a certainty caseUtility function:1X t=0 t u(c t)CIA constraint:。