第8章构建了入门宏观经济学的主力模型：用于短期波动的IS-LM模型、用于价格水平决定的AD-AS模型，以及用于长期增长的索洛模型——全部在代数层面。本章用微积分重建这些模型。核心做法是微观基础化：从家庭和企业的优化行为中推导宏观经济关系。
IS曲线将从跨期欧拉方程而非假定的消费函数中推导出来。投资将遵循具有凸调整成本的托宾 q理论。菲利普斯曲线将获得预期机制，并最终预览从垄断竞争和价格粘性推导出的新凯恩斯版本。索洛增长模型获得完整的微积分处理，包括微分方程和相图分析，为第13章的拉姆齐模型做准备。
全章的数学水平是微积分：拉格朗日乘数法、一阶条件、欧拉方程、基本微分方程和相图分析。我们明确不使用Hamilton函数、Bellman方程或动态规划——那些留待第13-14章。
前置知识：第8章（IS-LM、AD-AS、代数层面的Solow模型），第6章（Lagrange乘数法、受约束优化）。数学前置知识：单变量微积分、受约束优化、基本微分方程。
相关文献：Fisher（1930）；Ramsey（1928）；Friedman（1957）；Hall（1978）；Modigliani & Brumberg（1954）；Tobin（1969）；Hayashi（1982）；Solow（1956）；Swan（1956）；Phelps（1966）；Friedman（1968）；Phelps（1967）；Lucas（1972）；Mundell（1963）；Fleming（1962）；Calvo（1983）；Galí（2015）。
This chapter's micro-foundations connect to four of the book's Big Questions. Each juncture appears after the section where the relevant model is developed.
在第8章中，我们使用了凯恩斯消费函数$C = C_0 + c(Y - T)$，其中边际消费倾向$c$是一个介于0和1之间的行为参数。这个函数讲述了一个简单的故事——家庭将当期收入的固定比例用于消费——但它存在两个深层问题。第一，它将$c$视为常数，但经验证据表明消费反应取决于收入变化是暂时性的还是永久性的、是预期中的还是意外的。第二，参数$c$与更深层的偏好没有联系：我们无法说明当利率上升、人口老龄化或不确定性增加时它如何变化。
微观基础方法从第一性原理出发：一个具有明确偏好的家庭在预算约束下最大化终身效用。边际消费倾向不再是假定的——它是从优化过程中推导出来的，取决于利率、收入持续性、时间偏好和风险厌恶。这就是现代宏观经济学的方法论精髓。
考虑一个存活两期的家庭。它在第1期获得收入$y_1$，在第2期获得收入$y_2$。它可以按实际利率$r$储蓄或借贷。家庭选择消费$c_1$和$c_2$以最大化终身效用：
其中$u(\cdot)$是严格凹的递增效用函数，$\beta \in (0,1)$是贴现因子。家庭面临跨期预算约束：
What this says: A household chooses how much to consume now vs. later to maximize lifetime happiness, subject to the constraint that total lifetime spending (in present value) cannot exceed total lifetime income.
Why it matters: This replaces the mechanical Keynesian assumption that people spend a fixed fraction of current income. Instead, consumption depends on lifetime wealth — a temporary bonus gets mostly saved, while a permanent raise gets spent. This is the foundation of the permanent income hypothesis.
See Full Mode for the derivation.从几何上看，公式9.1在$(c_1, c_2)$空间中定义了一条斜率为$-(1+r)$的直线。禀赋点$(y_1, y_2)$始终位于这条线上。当$r$增加时，预算约束绕禀赋点顺时针旋转：储蓄变得更有吸引力。
一阶条件为：$u'(c_1) = \lambda$和$\beta \, u'(c_2) = \lambda/(1+r)$。相除消去乘子$\lambda$：
What this says: At the optimum, a household is exactly indifferent between consuming one more dollar today and saving it. Saving earns interest (1+r) but the future is discounted by the impatience factor. The household balances these forces until the marginal benefit of consuming now equals the marginal benefit of waiting.
Why it matters: The Euler equation is the single most important equation in modern macro. It governs consumption timing: when interest rates rise, households shift spending to the future. When they become more patient (higher beta), they save more today. Every modern macro model — from DSGE to New Keynesian — builds on this condition.
See Full Mode for the derivation.这是宏观经济学中最重要的方程之一。它表明：在最优状态下，家庭对于多消费一单位和将其储蓄、赚取\$1+r$的利息、明天消费\$1+r$单位之间是无差异的。如果$\beta(1+r) > 1$，家庭将消费倾向于未来：$c_2 > c_1$。如果$\beta(1+r) < 1$，家庭提前消费：$c_1 > c_2$。
宏观经济学中最常用的效用函数是常相对风险厌恶（CRRA）族：当$\sigma > 0, \sigma \neq 1$时$u(c) = \frac{c^{1-\sigma} - 1}{1-\sigma}$，当$\sigma = 1$时$u(c) = \ln c$。这里$\sigma$是相对风险厌恶系数，\$1/\sigma$是跨期替代弹性（IES）。在CRRA效用下，欧拉方程变为：
What this says: With CRRA preferences, the ratio of future to current consumption depends on the interest rate and impatience. The parameter sigma controls how willing households are to shift consumption across time — high sigma means they strongly prefer smooth consumption and barely respond to interest rate changes.
Why it matters: This single equation determines whether a rate hike causes households to save more (substitution effect) or spend more (income effect). The answer depends on sigma, which is why it is one of the most debated parameters in macroeconomics.
See Full Mode for the derivation.当$\sigma = 1$（对数效用）时，$c_2/c_1 = \beta(1+r)$。更高的利率提高消费增长率，弹性由\$1/\sigma$决定。
两期模型将PIH作为定理推导出来。在对数效用且$\beta(1+r) = 1$（即$c_1 = c_2 = c$）的条件下，预算约束给出$c = \frac{1+r}{2+r}(y_1 + y_2/(1+r))$。暂时性收入增加只会使消费增加约一半；永久性增加则几乎一对一地提高消费。
欧拉方程假设可以按利率$r$自由借贷。当借贷限制约束生效（$c_1 \leq y_1$）时，消费跟随当期收入，暂时性收入的MPC趋近于1——这恰好就是凯恩斯消费函数。这解释了为什么凯恩斯模型对受流动性约束的家庭（约占人口的30-50%）是有效的。
图9.1. 两期消费模型。当利率变化时，预算约束绕禀赋点旋转。最优组合满足欧拉方程。
考虑一个具有对数效用$u(c) = \ln c$的家庭，收入$y_1 = 100$，$y_2 = 50$，实际利率$r = 0.10$，贴现因子$\beta = 0.95$。
第1步：拉格朗日函数。$\mathcal{L} = \ln c_1 + 0.95 \ln c_2 + \lambda[100 + 50/1.10 - c_1 - c_2/1.10]$。终身财富：$W = 100 + 45.45 = 145.45$。
第2步：欧拉方程。在对数效用下，$u'(c) = 1/c$，因此$c_2/c_1 = \beta(1+r) = 0.95 \times 1.10 = 1.045$。
第3步：求解。$c_2 = 1.045\,c_1$。预算约束：$c_1 + 1.045\,c_1/1.10 = 145.45 \implies 1.950\,c_1 = 145.45 \implies c_1^* = 74.59$，$c_2^* = 77.95$。
第4步：验证。预算：\$14.59 + 77.95/1.10 = 145.45$。✓ 欧拉：\$17.95/74.59 = 1.045 = \beta(1+r)$。✓
第5步：储蓄。$s = y_1 - c_1^* = 100 - 74.59 = 25.41$。家庭储蓄是因为当期收入超过消费平滑水平。
第6步：比较静态。如果$r$升至0.20，则$\beta(1+r) = 1.14$，所以$c_2/c_1 = 1.14$。更高的利率将消费倾向于未来。在对数效用（IES $= 1$）下，替代效应占主导，$c_1$下降。
第8章的IS曲线是$Y = A - br$：当期产出取决于自主支出$A$和利率$r$，对未来的预期不起作用。欧拉方程改变了这一点。我们将两期模型推广到多期并进行对数线性化。在CRRA效用和参数$\sigma$下，定义$\hat{c}_t = \ln c_t - \ln \bar{c}$和$\rho = 1/\beta - 1$：
What this says: Current consumption depends on expected future consumption and the gap between the interest rate and the household's impatience rate. When the interest rate exceeds impatience, households postpone consumption (consumption grows over time).
Why it matters: This log-linearized form is the building block of the New Keynesian IS curve. It makes expectations central: if households expect better times ahead, they spend more today. This forward-looking behavior is what distinguishes modern macro from the Keynesian cross.
See Full Mode for the derivation.在封闭经济中$Y_t = C_t$，定义产出缺口$x_t = \hat{y}_t - \hat{y}_t^n$和自然利率$r^n$：
What this says: Today's output gap depends on the expected future output gap and the real interest rate relative to its natural level. When the central bank sets interest rates above the natural rate, it depresses current demand; when it sets them below, it stimulates demand.
Why it matters: Unlike the Chapter 8 IS curve, this one is forward-looking. Expectations about the future directly affect today's spending. A credible promise of future stimulus raises output now, even before the stimulus arrives. This is why central bank communication and forward guidance matter.
See Full Mode for the derivation.这与第8章的IS曲线有根本性的不同：(1) 预期至关重要。$E_t x_{t+1}$意味着当期产出取决于家庭对未来的预期。(2) 实际利率是事前利率$i_t - E_t \pi_{t+1}$。(3) 斜率取决于$\sigma$。$\sigma$越大，IS曲线越陡。
图9.2. 微观基础IS曲线与教科书IS曲线的比较。教科书IS曲线不对预期未来产出做出反应；微观基础IS曲线随预期变动。
从前瞻性IS（公式9.6）出发，假设$\sigma = 1$，$E_t \pi_{t+1} = 2\%$，$r^n = 3\%$，$E_t x_{t+1} = 0$。则：$x_t = -(i_t - 0.05)$。
若$i_t = 0.07$：$x_t = -0.02$（产出低于潜在水平2%）。若$i_t = 0.03$：$x_t = 0.02$（产出高于潜在水平2%）。这看起来类似教科书IS曲线。
现在改变预期。假设$E_t x_{t+1} = 0.03$（可信的未来财政扩张）。则：$x_t = 0.03 - (i_t - 0.05)$。在$i_t = 0.07$时：$x_t = 0.01$（产出现在高于潜在水平）。对未来繁荣的预期刺激了当期支出。教科书IS曲线完全忽略了这一渠道。
You now have the Euler equation and the micro-founded IS curve. Forward-looking consumers change everything about the fiscal multiplier story.
When consumers optimize intertemporally via the Euler equation, a temporary tax cut doesn't change their permanent income — so they save it rather than spend it. The micro-founded IS curve has smaller fiscal multipliers than the ad hoc version because consumption responds to permanent income, not current income. A debt-financed increase in $G$ that will be repaid by future taxes leaves present-value wealth unchanged for a Ricardian consumer. In the pure theory, the fiscal multiplier on consumption is zero — only the direct $G$ component raises GDP.
The Ricardian result is internally consistent but empirically fragile. Most households are liquidity-constrained — they cannot borrow against future income even if they want to. Campbell and Mankiw (1989) estimate that roughly 50% of aggregate consumption tracks current income, not permanent income. The "rational, unconstrained consumer" is a theoretical benchmark, not a description of actual behavior. If half the population spends their tax cut immediately, the multiplier is far from zero.
The mainstream responded by modeling heterogeneous agents — some Ricardian optimizers, some hand-to-mouth consumers who spend all current income. The TANK (Two-Agent New Keynesian) framework splits the population into these two types. The more recent HANK (Heterogeneous Agent New Keynesian) models allow a full distribution of wealth and income, making the fraction of constrained households an endogenous outcome rather than an assumed parameter. The multiplier depends on the wealth distribution, not just the representative agent's Euler equation.
Pure Ricardian equivalence is a useful benchmark that almost certainly doesn't hold in full. The question shifts from "does fiscal policy work?" to "what fraction of households are constrained?" — and the empirical answer is roughly 30–50%. Fiscal policy works, but through the constrained households, not through the optimizing ones. The micro-foundations sharpen the debate rather than settling it.
Even with constrained consumers restoring a positive multiplier, monetary policy can offset fiscal effects by adjusting interest rates. Does fiscal policy matter at all when the central bank is actively targeting inflation? The answer flips at the zero lower bound. Come back in Chapter 15 (§15.7) — when interest rates hit zero, crowding out disappears and the fiscal multiplier may exceed the textbook value, possibly reaching 1.5–2.0.
With micro-founded consumption, printing money and handing it out works only if households are constrained. Ricardian agents save the transfer and wait for the inevitable tax.
入门第8章假设投资是利率的递减函数：$I = I_0 - br$。微观基础理论必须解释企业为什么投资、投资多少、以及多快调整资本存量。
What this says: Owning a machine for one period costs you the interest you forgo (you could have invested the money elsewhere) plus the depreciation (the machine wears out). A firm keeps investing until the machine's output just covers this rental cost.
Why it matters: This explains why high interest rates kill investment — they raise the hurdle rate that new projects must clear. Tax policies like accelerated depreciation or investment tax credits work by reducing the effective user cost.
See Full Mode for the derivation.企业投资直到资本的边际产出等于使用成本：$MPK = uc$。但这并没有说明调整的速度——在无摩擦的世界中，企业会瞬间跳到理想的资本存量，这与事实不符。
What this says: Tobin's q compares the stock market's valuation of a firm's capital to what it would cost to buy that capital new. If q exceeds 1, the market values existing capital more than replacement cost — it pays to build more. If q is below 1, it is cheaper to buy existing firms than to build new capacity.
Why it matters: This links Wall Street to Main Street. A stock market boom raises q and stimulates real investment. A crash lowers q and freezes capital spending. You can literally read investment signals from stock prices.
See Full Mode for the derivation.在凸调整成本下，一阶条件给出：
What this says: Investment is proportional to how far q exceeds 1, but adjustment costs slow the response. The higher the adjustment cost parameter phi, the more gradually firms respond to investment opportunities. This explains why investment responds sluggishly to news.
Why it matters: Without adjustment costs, firms would jump instantly to the optimal capital stock — unrealistic. Convex costs mean firms spread investment over time, which generates the smooth, hump-shaped investment responses we see in the data.
See Full Mode for the derivation.投资-资本比率是$q$的线性函数，斜率为\$1/\phi$。当$q = 1$时，投资恰好为零。股市繁荣提高$q$并触发更高的投资；股市崩盘降低$q$并抑制投资。
图9.3. 托宾 q与投资。投资率是q的线性函数；调整成本是凸的。
一家企业有$K = 100$，$p_K = 1$，市场价值$V = 130$，调整成本$\phi = 5$。
第1步：$q = V/(p_K \cdot K) = 130/100 = 1.30$。
第2步：$I/K = (q-1)/\phi = 0.30/5 = 0.06$。计划投资：$I = 6$。
第3步：调整成本：$C(I) = (5/2)(0.06)^2 \times 100 = 0.90$。总成本：\$1 + 0.90 = 6.90$。
第4步：股市繁荣。$V \to 160 \Rightarrow q = 1.60$，$I/K = 0.12$，$I = 12$。调整成本：\$1.60$——增加了四倍（凸性）。由于凸调整成本，投资对新信息的反应是渐进的。
第8章在代数层面介绍了索洛模型。这里我们给出完整的微积分处理：微分方程、相图和黄金律优化。
假设Cobb-Douglas生产函数$Y = K^\alpha (AL)^{1-\alpha}$，其中$A$以速率$g$增长，$L$以速率$n$增长。定义$k = K/(AL)$和$y = Y/(AL)$：
What this says: The economy saves a fraction s of output and uses it to build new capital. But capital per worker erodes over time as machines wear out (depreciation), the population grows (more workers to equip), and technology advances (raising the bar for capital per effective worker). The economy grows when saving exceeds erosion, and shrinks when it doesn't.
Why it matters: This differential equation is the engine of the Solow model. It tells you the economy always converges to a steady state where saving exactly offsets erosion. Countries below steady state grow fast; countries near it grow slowly. This is conditional convergence — the most testable prediction in growth economics.
See Full Mode for the derivation.令$\dot{k} = 0$：
What this says: The steady-state capital stock depends on how much the economy saves (s) relative to how fast capital erodes (n + g + delta). Countries that save more or have slower population growth end up richer in steady state.
Why it matters: This is the Solow model's answer to why some countries are rich and others poor. But the answer is incomplete — calibrated versions can only explain a factor of 2-3x in income differences through capital alone, while the actual gap between rich and poor countries is 50x or more. The rest must be technology and institutions.
See Full Mode for the derivation.图9.4. 索洛相图。稳态k*是全局稳定的：箭头从两侧指向它。
什么储蓄率使稳态消费最大化？$c^*(s) = (1-s)(s/(n+g+\delta))^{\alpha/(1-\alpha)}$。在黄金律处：
What this says: There is a "just right" saving rate that maximizes long-run consumption. Save too little and you don't build enough capital. Save too much and you are pouring resources into capital whose diminishing returns don't justify the sacrifice. The sweet spot equals the capital share in output (alpha).
Why it matters: If a country saves more than the golden rule, it is dynamically inefficient — everyone could consume more, in every period, by saving less. Most real economies appear to save below the golden rule, meaning higher saving would raise future consumption but at the cost of consuming less during the transition.
See Full Mode for the derivation.What this says: The economy closes the gap to its steady state at a rate of about 5-6% per year, implying a half-life of roughly 12 years. A country that starts at half its steady-state capital will be halfway to steady state in about 12 years.
Why it matters: This predicts conditional convergence — poor countries (relative to their own steady state) should grow faster than rich ones. The prediction matches cross-country data reasonably well once you control for saving rates, population growth, and education. But the pace is slow enough that convergence takes decades, not years.
See Full Mode for the derivation.半衰期为$t_{1/2} = \ln 2 / \lambda$。对于$\alpha = 1/3$，$n = 0.02$，$g = 0.015$，$\delta = 0.05$：$\lambda = 0.0567$，$t_{1/2} \approx 12.2$年。
图9.5. 索洛黄金律。稳态消费在$s = \alpha$时最大化。
参数：$\alpha = 1/3$，$n = 0.02$，$g = 0.015$，$\delta = 0.05$。收支平衡线：$n+g+\delta = 0.085$。
第1步：$k^*(s) = (s/0.085)^{3/2}$。
第2步：黄金律。$s_g = \alpha = 1/3$。则$k_g = (0.333/0.085)^{1.5} = 7.76$，$y_g = 1.98$，$c_g = 1.32$。
第3步：凯拉尼（$s = 0.15$）。$k^* = (0.15/0.085)^{1.5} = 2.35$，$y^* = 1.33$，$c^* = 1.13$。
第4步：由于$s = 0.15 < s_g = 0.333$，凯拉尼是动态有效的，但远低于黄金律。通过提高储蓄率，消费可以增加17%，但代价是转型期间消费降低。
You now have the Solow model with calculus — capital accumulation, steady states, convergence dynamics, and the golden rule. Here's what it explains and what it can't.
Solow says steady-state income $y^*$ depends on the saving rate $s$, population growth $n$, and depreciation $\delta$. Countries that save more and have slower population growth are richer in steady state. Conditional convergence holds: countries with similar parameters should converge to similar income levels, with poorer countries growing faster along the transition path. The speed of convergence $\lambda = (1-\alpha)(n+g+\delta)$ implies a half-life of roughly 12–15 years — not fast, but finite.
Solow explains income levels but not sustained growth — that depends entirely on the exogenous technology parameter $A$. Worse, calibrated Solow models can explain at most a factor of 2–3 in cross-country income differences through capital alone, but the actual gap is a factor of 50+. The residual — total factor productivity — accounts for most of the difference. As Moses Abramovitz put it, TFP is "a measure of our ignorance." Attributing the wealth of nations to $A$ is not an explanation; it's a confession that the model doesn't know the answer.
Mankiw, Romer, and Weil (1992) augmented the Solow model with human capital, which explains a larger share of cross-country variation — raising the effective capital share narrows the residual. But the fundamental problem remains: what determines $A$? This dissatisfaction launched two research programs: endogenous growth theory (Chapter 13), which tries to make technological progress a choice variable, and institutional economics (Chapter 18), which argues that the deep cause lies in political and economic institutions.
Solow is essential scaffolding. Its most important result is negative: capital accumulation alone cannot explain the wealth gap. Diminishing returns to capital mean that even large differences in saving rates produce modest differences in steady-state income. The real action is in TFP — and figuring out what drives it is the central question of growth economics.
What determines TFP? Is it technology and ideas — the ability to invent and adopt new methods? Come back in Chapter 13 (§13.3–13.5), where endogenous growth theory makes innovation the engine of long-run growth. Or is it institutions — property rights, rule of law, and checks on political power? Chapter 18 (§18.3–18.4) makes that case. The Solow model tells you where to look; it doesn't tell you what you'll find.
Dambisa Moyo argued that decades of aid to Africa have been actively destructive — fostering dependency and corruption. If the problem is insufficient capital, aid should accelerate convergence. If the problem is TFP, pouring in capital hits diminishing returns. The Solow model sharpens this debate.
中级弗里德曼-费尔普斯的关键洞见：菲利普斯曲线必须包含预期通胀：
What this says: Inflation equals expected inflation plus a boost from the output gap plus supply shocks. When the economy runs hot (output above potential), inflation rises above expectations. When it runs cold, inflation falls below expectations.
Why it matters: The Friedman-Phelps revolution: there is no permanent tradeoff between inflation and unemployment. You can temporarily reduce unemployment by generating surprise inflation, but once expectations adjust, you're back at the natural rate with higher inflation. The only way to keep unemployment below the natural rate is accelerating inflation — an unsustainable path.
See Full Mode for the derivation.代入得：$\Delta \pi_t = \alpha (Y_t - Y^*)/Y^* + \varepsilon_t$。
What this says: Under adaptive expectations, the change in inflation (not its level) depends on the output gap. Holding output above potential doesn't just cause inflation — it causes accelerating inflation, with each period's inflation higher than the last.
Why it matters: This is the accelerationist hypothesis. It implies the long-run Phillips curve is vertical: the only output level consistent with stable inflation is potential output. Policymakers cannot buy permanently lower unemployment with permanently higher (but stable) inflation.
See Full Mode for the derivation.在长期，$\Delta \pi = 0$要求$Y = Y^*$：长期菲利普斯曲线在自然率处是垂直的。通胀与产出之间不存在长期权衡。
在理性预期和完全可信度下，反通胀可以是无成本的——牺牲率为零。在适应性预期下，牺牲率很大。沃尔克反通胀（1979-1983年）的牺牲率约为2.5，与部分前瞻性、主要后顾性的预期一致。
图9.8. 预期增广菲利普斯曲线。短期菲利普斯曲线随预期通胀移动；长期曲线是垂直的。
经济处于$\pi = 8\%$，目标$\pi = 2\%$。菲利普斯曲线斜率$\alpha = 0.5$。
适应性预期。$\pi^e_t = \pi_{t-1}$。要每年降低通胀1个百分点：$-0.01 = 0.5 \cdot x_t \Rightarrow x_t = -0.02$。六年内产出低于潜在水平2%。累计损失：GDP的12%。牺牲率：\$12/6 = 2.0$。
理性预期且具有可信度。$\pi^e$跳至2%。当$x_t = 0$时：$\pi_t = 2\%$。无成本反通胀。牺牲率：0。
现实（沃尔克，1979-83年）：约4年，牺牲率$\approx 2.5$。部分前瞻性（一定的可信度），主要后顾性（工资和合同的惯性）。
You now have the expectations-augmented Phillips curve and dynamic AD-AS. The model can distinguish demand shocks from supply shocks — and the policy implications are opposite.
Dynamic AD-AS with the expectations-augmented Phillips curve reveals that not all recessions are alike. A negative demand shock (falling investment confidence, fiscal contraction) reduces output below potential and pushes inflation below expectations — both output and inflation fall together. A negative supply shock (oil price spike, productivity collapse) reduces output but raises inflation — the dreaded stagflation. The policy prescription is diametrically opposed: demand shocks call for expansionary policy; supply shocks present a painful tradeoff between inflation and output stabilization.
If the economy self-corrects — expectations adjust, SRAS shifts, output returns to potential — why intervene at all? Because the self-correction mechanism (falling wages and prices) is itself contractionary. Irving Fisher's debt-deflation theory shows that falling prices increase the real burden of debt, triggering defaults, bank failures, and further demand contraction. The cure can be worse than the disease. More fundamentally, "the long run" in which self-correction occurs can mean years of elevated unemployment and permanent scarring of workers' human capital.
The speed-of-adjustment debate became central: monetarists argued adjustment is fast enough that activist policy is unnecessary (and often counterproductive given policy lags). Keynesians argued adjustment is slow enough that the output losses during self-correction are unacceptable. The truth likely varies by episode — some recessions are brief and self-correcting, while others (the Great Depression, the Great Recession) persist for years without intervention.
Dynamic AD-AS correctly captures the short-run/long-run distinction: recessions are departures from potential that eventually self-correct. But "eventually" can mean years of lost output and elevated unemployment. The expectations-augmented Phillips curve adds a crucial insight: inflation expectations anchor the short-run tradeoff. A central bank with credibility can disinflate at lower cost; one without credibility faces a steeper sacrifice ratio.
This framework describes the dynamics after a shock but doesn't explain why recessions happen. What generates the shocks? The RBC school (Chapter 14, §14.2) gives a radical answer: technology shocks, and recessions are efficient. The New Keynesian synthesis (Chapter 15, §15.8) merges demand and supply stories into a unified framework. Neither fully explains financial crises — the amplification through leverage, panic, and credit contraction that turned 2008 from a housing correction into a global catastrophe.
What this says: In an open economy, IS-LM gains two new channels: the exchange rate affects net exports (trade channel), and interest rate differentials drive capital flows (financial channel). The balance of payments requires that trade deficits are financed by capital inflows, and vice versa.
Why it matters: This is the Mundell-Fleming model — the workhorse for open-economy policy analysis. It reveals that whether fiscal or monetary policy is effective depends entirely on the exchange rate regime. Under fixed rates, fiscal policy works but monetary policy is powerless. Under floating rates, the reverse holds.
See Full Mode for the derivation.财政政策有效：IS右移 → $r$趋于高于$r^*$ → 资本流入 → 央行卖出本币 → LM内生右移 → $Y$上升。
货币政策无效：LM右移 → $r$降至$r^*$以下 → 资本流出 → 央行买入本币 → LM移回原位。$Y$不变。
财政政策无效：IS右移 → $r$趋于高于$r^*$ → 资本流入 → 本币升值 → NX下降 → IS移回原位。$Y$不变。
货币政策有效：LM右移 → $r$降至$r^*$以下 → 资本流出 → 本币贬值 → NX上升 → IS右移 → $Y$上升。
图9.6. 蒙代尔-弗莱明模型。财政政策在固定汇率下有效；货币政策在浮动汇率下有效。
图9.7. 不可能三角。一个国家必须在三者中选择两个：自由资本流动、固定汇率、独立的货币政策。
A部分——固定汇率。凯拉尼与TAD挂钩，$r_K = r^* = 5\%$。财政扩张$\Delta G = 0.5$B KD。
机制：IS右移 → $r$趋于高于$r^*$ → 资本流入 → 央行卖出KD/买入TAD → 货币供给扩大（LM右移）→ $Y$升至约12.5B KD。财政政策有效。
B部分——浮动汇率。相同的财政扩张。
机制：IS右移 → $r$上升压力 → 资本流入 → KD升值 → NX下降 → IS移回原位。$Y$几乎不变。财政政策无效——通过汇率渠道被挤出。
启示：在钉住汇率制下，凯拉尼拥有财政政策但没有货币政策。不可能三角：自由资本流动+固定汇率=没有独立的货币政策。
You now have the Mundell-Fleming model and the impossible trinity. The open economy complicates everything — monetary policy's power depends on the exchange rate regime.
The expectations-augmented Phillips curve delivers a sharp result: only unanticipated monetary policy moves real output. Once expectations adjust, the economy returns to the natural rate regardless of monetary policy. Mundell-Fleming adds the open-economy constraint: under a fixed exchange rate with free capital flows, monetary policy is completely impotent — the central bank must defend the peg, making the money supply endogenous. Under floating rates, monetary policy works, but partly through the exchange rate channel — a rate cut depreciates the currency, boosting net exports, which has international repercussions.
If only surprises matter, then systematic monetary policy is useless — the central bank can only affect the economy by doing things people don't expect, which is self-defeating as a long-run strategy. The rational expectations revolution (Lucas, Sargent) pushed this to its logical conclusion: the policy irrelevance proposition. Under rational expectations, any systematic monetary policy rule is fully anticipated and has no real effects. The central bank is a paper tiger.
Policy irrelevance was too strong. The New Keynesian response (Chapter 15) showed that sticky prices restore real effects of monetary policy even when expectations are rational — because not all firms can adjust prices simultaneously, monetary policy changes real demand. But the Lucas critique itself survived as a permanent methodological lesson: any model that ignores how behavior changes with the policy regime will give unreliable policy advice. Central bank models must be structural, not reduced-form.
Central banks face genuine constraints: the long-run neutrality of money, the impossible trinity, and the Lucas critique. But these constraints don't make monetary policy impotent — they make it more subtle. The question shifts from "can central banks control output?" to "can central banks control inflation and smooth business cycles within the constraints of expectations and exchange rate regimes?" The answer is a qualified yes — but only for countries with floating exchange rates and credible institutions.
How should central banks actually set policy in practice? The Taylor rule (Chapter 15, §15.5) provides the modern answer — but it breaks down at the zero lower bound, where the nominal interest rate can't go below zero and conventional monetary policy loses its bite. And the fiscal theory of the price level (Chapter 16, §16.5) raises a deeper challenge: perhaps it's fiscal policy, not monetary policy, that ultimately determines the price level. The debate about who's really in charge — the central bank or the treasury — is far from settled.
Mundell-Fleming says it depends on the exchange rate regime. Rational expectations say only surprises matter. The impossible trinity constrains everyone. The Fed has more power than most central banks — but less than most people think.
中级预期增广菲利普斯曲线假设产出缺口与通胀之间存在直接关系，但没有解释为什么。要使通胀反应具有粘性，我们需要两个要素：设定价格的企业（市场力量）和它们不连续调整的原因（粘性）。
每家企业面临向下倾斜的需求曲线，并将价格定为边际成本的加成$\mu = \varepsilon/(\varepsilon - 1)$，其中$\varepsilon$是Dixit-Stiglitz替代弹性。
每期有$(1 - \theta)$比例的企业重新设定价格，而$\theta$比例的企业保持不变。当$\theta = 0.75$时，平均价格持续时间为4个季度。最优重置价格：
What this says: Inflation today depends on expected future inflation and the current output gap. Firms that get to reset prices look forward — they set prices based on where they expect costs to go, not where costs have been. The slope kappa measures how sensitive inflation is to demand pressure.
Why it matters: This is the micro-founded replacement for the backward-looking Phillips curve. Because it is forward-looking, a credible commitment to low future inflation reduces inflation today — immediately. This is why central bank credibility matters: a trusted inflation target anchors expectations and flattens the short-run tradeoff. The full NK model (Chapter 15) builds on this equation.
See Full Mode for the derivation.参数$\kappa = \frac{(1-\theta)(1-\beta\theta)}{\theta} \cdot \gamma$取决于价格粘性$\theta$、贴现因子$\beta$和边际成本对产出缺口的敏感度$\gamma$。当$\theta$较大时，$\kappa$较小——通胀对产出缺口的反应较弱。
NKPC与后顾性菲利普斯曲线有根本区别：通胀取决于预期未来通胀，而非过去的通胀。对低未来通胀的可信承诺会立即降低$\pi_t$。完整的三方程NK模型将在第15章介绍。
凯拉尼共和国（人口500万，GDP约100亿KD，来自第5章；IS-LM基准来自第8章）面临两个相互交织的挑战：选择汇率制度和提高长期增长以缩小与邻国塔拉尼的差距。
汇率制度（蒙代尔-弗莱明）。凯拉尼维持与塔拉尼元（TAD）的固定钉住，资本完全自由流动（$r_K = r_T = 5\%$）。政府计划进行$\Delta G = 0.5$B KD的财政扩张。在固定汇率下，蒙代尔-弗莱明模型预测扩张是有效的：IS右移，资本流入导致LM内生右移，$Y$升至约12.5B KD。在浮动汇率下，同样的扩张将被本币升值所抵消。
央行行长指出：“在钉住汇率制下，我们有财政政策但没有货币政策。如果我们想独立降息——比如在塔拉尼未受影响的衰退期间——我们做不到。”这就是不可能三角：自由资本流动+固定汇率=没有独立的货币政策。
长期增长（微积分版索洛模型）。两个经济体：$\alpha = 1/3$，$n = 0.02$，$g = 0.015$，$\delta = 0.05$。凯拉尼（$s = 0.15$）：$k^* = 2.35$，$y^* = 1.33$。塔拉尼（$s = 0.25$）：$k^* = 5.04$，$y^* = 1.71$。预测收入比率：\$1.78$。实际观测值：\$1.50$。差距大于索洛模型的预测——全要素生产率差异（制度、人力资本）很重要，这为第13章和第18章埋下伏笔。
凯拉尼是动态有效的（$s = 0.15 < s_g = 0.333$），但远低于黄金律。收敛速度：$\lambda = 0.0567$，半衰期$\approx 12.2$年。
微观基础消费。一个凯拉尼家庭获得$y_1 = 2{,}000$ KD的收入，预期$y_2 = 2{,}400$ KD，$r = 5\%$，$\beta = 0.95$。欧拉方程给出$c_2^*/c_1^* = 0.9975 \approx 1$：近乎完美的平滑。家庭在第1期借入约195 KD，因为它预期未来收入更高。200 KD的一次性刺激大部分被储蓄；每期200 KD的永久补贴则几乎全部被消费。
本章结束时的状态：凯拉尼的宏观框架现已具有微观基础（欧拉方程、微积分版索洛模型、蒙代尔-弗莱明）。固定汇率制约束了货币政策。储蓄率低于黄金律。索洛模型只能部分解释收入差距。线索延续至第13章（拉姆齐增长模型）、第15章（NK货币政策）和第18章（制度）。
| 标签 | 公式 | 描述 |
|---|---|---|
| 公式9.1 | $c_1 + \frac{c_2}{1+r} = y_1 + \frac{y_2}{1+r}$ | 跨期预算约束 |
| 公式9.2 | $\mathcal{L} = u(c_1) + \beta u(c_2) + \lambda[\cdots]$ | 拉格朗日函数（两期） |
| 公式9.3 | $u'(c_1) = \beta(1+r)\,u'(c_2)$ | 消费欧拉方程 |
| 公式9.4 | $(c_2/c_1)^\sigma = \beta(1+r)$ | CRRA欧拉方程 |
| 公式9.5 | $\hat{c}_t = E_t\hat{c}_{t+1} - (1/\sigma)(r_t - \rho)$ | 对数线性化欧拉方程 |
| 公式9.6 | $x_t = E_tx_{t+1} - (1/\sigma)(i_t - E_t\pi_{t+1} - r^n)$ | 前瞻性IS曲线 |
| 公式9.7 | $uc = (r + \delta)p_K$ | 资本使用成本 |
| 公式9.8 | $q = V / (p_K \cdot K)$ | 托宾q |
| 公式9.9 | $I/K = (q - 1)/\phi$ | 最优投资 |
| 公式9.10 | $y = k^\alpha$ | 每有效工人产出 |
| 公式9.11 | $\dot{k} = sk^\alpha - (n+g+\delta)k$ | 索洛资本积累ODE |
| 公式9.12 | $k^* = [s/(n+g+\delta)]^{1/(1-\alpha)}$ | 索洛稳态 |
| 公式9.13 | $f'(k_g) = n + g + \delta$ | 黄金律条件 |
| 公式9.14 | $s_g = \alpha$ | 黄金律储蓄率 |
| 公式9.15 | $\lambda = (1-\alpha)(n+g+\delta)$ | 收敛速度 |
| 公式9.16 | $\pi_t = \pi^e_t + \alpha(Y_t-Y^*)/Y^* + \varepsilon_t$ | 预期增广菲利普斯曲线 |
| 公式9.17 | $\pi^e_t = \pi_{t-1}$ | 适应性预期 |
| 公式9.18 | $\Delta\pi_t = \alpha(Y_t-Y^*)/Y^* + \varepsilon_t$ | 加速主义菲利普斯曲线 |
| 公式9.19 | $Y = C(Y-T) + I(r) + G + NX(e)$ | 开放经济IS |
| 公式9.20 | $NX(e) + KA(r - r^*) = 0$ | BP曲线 |
| 公式9.21 | $r = r^*$ | 完全资本流动 |
| 公式9.22 | 三元悖论约束 | 不可能三角 |
| 公式9.23 | $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t$ | 新凯恩斯菲利普斯曲线 |
| 公式9.24 | $p_t^* = \mu + (1-\beta\theta)\sum(\beta\theta)^j E_t[mc_{t+j}]$ | 卡尔沃最优重置价格 |
Coming in Part IV: econometrics gives you the tools to TEST the models. Advanced micro gives the foundations for everything in Part V.