Chapter 8 built the workhorse models of introductory macroeconomics: the IS-LM model for short-run fluctuations, AD-AS for price-level determination, and the Solow model for long-run growth — all at the algebra level. This chapter rebuilds each of those pieces with calculus. The central move is micro-founding: deriving macroeconomic relationships from the optimizing behavior of households and firms.
The IS curve will emerge from an intertemporal Euler equation rather than an assumed consumption function. Investment will follow from Tobin’s $q$ theory with convex adjustment costs. The Phillips curve will gain an expectations mechanism, and eventually a preview of the New Keynesian derivation from monopolistic competition and sticky prices. The Solow growth model gets a full calculus treatment with differential equations and phase diagrams, preparing the ground for the Ramsey model in Chapter 13.
The mathematical level throughout is calculus: Lagrangians, first-order conditions, Euler equations, basic differential equations, and phase diagram analysis. We explicitly do not use Hamiltonians, Bellman equations, or dynamic programming — those are reserved for Chapters 13–14.
Prerequisites: Chapter 8 (IS-LM, AD-AS, Solow at algebra level), Chapter 6 (Lagrangians, constrained optimization). Mathematical prerequisites: single-variable calculus, constrained optimization, basic differential equations.
Named literature: 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.
In Chapter 8 we used the Keynesian consumption function $C = C_0 + c(Y - T)$, where the marginal propensity to consume $c$ was a behavioral parameter between zero and one. This function tells a simple story — households spend a fixed fraction of current income — but it has two deep problems. First, it treats $c$ as a constant, but empirical evidence shows that consumption responses depend on whether income changes are temporary or permanent, anticipated or surprising. Second, the parameter $c$ has no connection to deeper preferences: we cannot say how it changes when interest rates rise, when the population ages, or when uncertainty increases.
The micro-founded approach starts from first principles: a household with well-defined preferences maximizes lifetime utility subject to a budget constraint. The marginal propensity to consume is no longer assumed — it is derived from the optimization, and it depends on interest rates, income persistence, time preference, and risk aversion. This is the methodological essence of modern macro.
Consider a household that lives for two periods. It earns income $y_1$ in period 1 and $y_2$ in period 2. It can save or borrow at a real interest rate $r$. The household chooses consumption $c_1$ and $c_2$ to maximize lifetime utility:
where $u(\cdot)$ is a strictly concave, increasing utility function, and $\beta \in (0,1)$ is the discount factor. The household faces the intertemporal budget constraint:
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.Geometrically, Eq. 9.1 defines a straight line in $(c_1, c_2)$ space with slope $-(1+r)$. The endowment point $(y_1, y_2)$ always lies on this line. When $r$ increases, the budget constraint pivots clockwise around the endowment point: saving becomes more attractive.
The first-order conditions are: $u'(c_1) = \lambda$ and $\beta \, u'(c_2) = \lambda/(1+r)$. Dividing eliminates the multiplier $\lambda$:
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.This is one of the most important equations in macroeconomics. It says: at the optimum, the household is indifferent between consuming one more unit today and saving that unit, earning interest \$1+r$, and consuming \$1+r$ units tomorrow. If $\beta(1+r) > 1$, the household tilts consumption toward the future: $c_2 > c_1$. If $\beta(1+r) < 1$, the household front-loads: $c_1 > c_2$.
The most commonly used utility function in macro is the constant relative risk aversion (CRRA) family: $u(c) = \frac{c^{1-\sigma} - 1}{1-\sigma}$ for $\sigma > 0, \sigma \neq 1$, and $u(c) = \ln c$ when $\sigma = 1$. Here $\sigma$ is the coefficient of relative risk aversion, and \$1/\sigma$ is the intertemporal elasticity of substitution (IES). With CRRA utility the Euler equation becomes:
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.When $\sigma = 1$ (log utility), $c_2/c_1 = \beta(1+r)$. A higher interest rate raises the consumption growth rate, with the elasticity governed by \$1/\sigma$.
The two-period model delivers the PIH as a theorem. With log utility and $\beta(1+r) = 1$, so $c_1 = c_2 = c$, the budget constraint gives $c = \frac{1+r}{2+r}(y_1 + y_2/(1+r))$. A temporary income increase raises consumption by only about half the windfall; a permanent increase raises it nearly one-for-one.
The Euler equation assumes free borrowing at rate $r$. When borrowing limits bind ($c_1 \leq y_1$), consumption tracks current income and the MPC out of temporary income approaches one — exactly the Keynesian consumption function. This explains why the Keynesian model works for liquidity-constrained households (roughly 30–50% of the population).
Figure 9.1. Two-period consumption model. The budget constraint pivots around the endowment point as the interest rate changes. The optimal bundle satisfies the Euler equation.
Consider a household with log utility $u(c) = \ln c$, income $y_1 = 100$, $y_2 = 50$, real interest rate $r = 0.10$, and discount factor $\beta = 0.95$.
Step 1: Lagrangian. $\mathcal{L} = \ln c_1 + 0.95 \ln c_2 + \lambda[100 + 50/1.10 - c_1 - c_2/1.10]$. Lifetime wealth: $W = 100 + 45.45 = 145.45$.
Step 2: Euler equation. With log utility, $u'(c) = 1/c$, so $c_2/c_1 = \beta(1+r) = 0.95 \times 1.10 = 1.045$.
Step 3: Solve. $c_2 = 1.045\,c_1$. Budget constraint: $c_1 + 1.045\,c_1/1.10 = 145.45 \implies 1.950\,c_1 = 145.45 \implies c_1^* = 74.59$, $c_2^* = 77.95$.
Step 4: Verify. Budget: \$14.59 + 77.95/1.10 = 145.45$. ✓ Euler: \$17.95/74.59 = 1.045 = \beta(1+r)$. ✓
Step 5: Saving. $s = y_1 - c_1^* = 100 - 74.59 = 25.41$. The household saves because current income exceeds the consumption-smoothing level.
Step 6: Comparative statics. If $r$ rises to 0.20, then $\beta(1+r) = 1.14$, so $c_2/c_1 = 1.14$. The higher interest rate tilts consumption toward the future. With log utility (IES $= 1$), the substitution effect dominates and $c_1$ falls.
The IS curve in Chapter 8 was $Y = A - br$: current output depends on autonomous spending $A$ and the interest rate $r$, with no role for expectations about the future. The Euler equation changes this. We generalize the two-period model to many periods and log-linearize. With CRRA utility and parameter $\sigma$, defining $\hat{c}_t = \ln c_t - \ln \bar{c}$ and $\rho = 1/\beta - 1$:
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.In a closed economy with $Y_t = C_t$, defining the output gap $x_t = \hat{y}_t - \hat{y}_t^n$ and the natural rate $r^n$:
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.This is profoundly different from the Chapter 8 IS curve: (1) Expectations matter. $E_t x_{t+1}$ means current output depends on what households expect about the future. (2) The real interest rate is the ex ante rate $i_t - E_t \pi_{t+1}$. (3) The slope depends on $\sigma$. A larger $\sigma$ makes the IS curve steeper.
Figure 9.2. Micro-founded vs textbook IS curve. The textbook IS does not respond to expected future output; the micro-founded IS shifts with expectations.
Starting from the forward-looking IS (Eq. 9.6), suppose $\sigma = 1$, $E_t \pi_{t+1} = 2\%$, $r^n = 3\%$, and $E_t x_{t+1} = 0$. Then: $x_t = -(i_t - 0.05)$.
If $i_t = 0.07$: $x_t = -0.02$ (output 2% below potential). If $i_t = 0.03$: $x_t = 0.02$ (output 2% above potential). This looks like the textbook IS.
Now change expectations. Suppose $E_t x_{t+1} = 0.03$ (credible future fiscal expansion). Then: $x_t = 0.03 - (i_t - 0.05)$. At $i_t = 0.07$: $x_t = 0.01$ (output now above potential). The expectation of future prosperity stimulates current spending. The textbook IS misses this channel entirely.
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.
IntroChapter 8 assumed that investment is a decreasing function of the interest rate: $I = I_0 - br$. A micro-founded theory must explain why firms invest, how much, and how fast they adjust their capital stock.
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.The firm invests until the marginal product of capital equals the user cost: $MPK = uc$. But this says nothing about the speed of adjustment — in the frictionless world, the firm jumps instantly to the desired stock, which is counterfactual.
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.With convex adjustment costs, the first-order condition yields:
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.The investment-to-capital ratio is linear in $q$, with slope \$1/\phi$. When $q = 1$, investment is exactly zero. A stock market boom raises $q$ and triggers higher investment; a crash lowers $q$ and depresses investment.
Figure 9.3. Tobin’s q and investment. The investment rate is linear in q; adjustment costs are convex.
A firm has $K = 100$, $p_K = 1$, market value $V = 130$, adjustment cost $\phi = 5$.
Step 1: $q = V/(p_K \cdot K) = 130/100 = 1.30$.
Step 2: $I/K = (q-1)/\phi = 0.30/5 = 0.06$. Planned investment: $I = 6$.
Step 3: Adjustment cost: $C(I) = (5/2)(0.06)^2 \times 100 = 0.90$. Total cost: \$1 + 0.90 = 6.90$.
Step 4: Stock market boom. $V \to 160 \Rightarrow q = 1.60$, $I/K = 0.12$, $I = 12$. Adjustment cost: \$1.60$ — a fourfold increase (convexity). Investment responds gradually to news because of convex costs.
Chapter 8 introduced the Solow model at the algebraic level. Here we give the full calculus treatment: differential equations, phase diagrams, and golden rule optimization.
Assume Cobb-Douglas $Y = K^\alpha (AL)^{1-\alpha}$, with $A$ growing at rate $g$, $L$ at rate $n$. Define $k = K/(AL)$ and $y = Y/(AL)$:
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.Setting $\dot{k} = 0$:
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.Figure 9.4. Solow phase diagram. The steady state k* is globally stable: arrows point toward it from both sides.
What saving rate maximizes steady-state consumption? $c^*(s) = (1-s)(s/(n+g+\delta))^{\alpha/(1-\alpha)}$. At the golden rule:
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.The half-life is $t_{1/2} = \ln 2 / \lambda$. For $\alpha = 1/3$, $n = 0.02$, $g = 0.015$, $\delta = 0.05$: $\lambda = 0.0567$, $t_{1/2} \approx 12.2$ years.
Figure 9.5. Solow golden rule. Steady-state consumption is maximized at $s = \alpha$.
Parameters: $\alpha = 1/3$, $n = 0.02$, $g = 0.015$, $\delta = 0.05$. Break-even: $n+g+\delta = 0.085$.
Step 1: $k^*(s) = (s/0.085)^{3/2}$.
Step 2: Golden rule. $s_g = \alpha = 1/3$. Then $k_g = (0.333/0.085)^{1.5} = 7.76$, $y_g = 1.98$, $c_g = 1.32$.
Step 3: Kaelani with $s = 0.15$. $k^* = (0.15/0.085)^{1.5} = 2.35$, $y^* = 1.33$, $c^* = 1.13$.
Step 4: Since $s = 0.15 < s_g = 0.333$, Kaelani is dynamically efficient but far below the golden rule. Consumption could increase 17% by raising the saving rate, at the cost of lower consumption during the transition.
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.
IntermediateThe crucial Friedman-Phelps insight: the Phillips curve must include expected inflation:
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.Substituting: $\Delta \pi_t = \alpha (Y_t - Y^*)/Y^* + \varepsilon_t$.
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.In the long run, $\Delta \pi = 0$ requires $Y = Y^*$: the long-run Phillips curve is vertical at the natural rate. There is no long-run tradeoff between inflation and output.
Under rational expectations with full credibility, disinflation can be costless — the sacrifice ratio is zero. Under adaptive expectations, it is large. The Volcker disinflation (1979–1983) had a sacrifice ratio of about 2.5, consistent with partly forward-looking, mostly backward-looking expectations.
Figure 9.8. Expectations-augmented Phillips curve. The short-run Phillips curve shifts with expected inflation; the long-run curve is vertical.
Economy at $\pi = 8\%$, target $\pi = 2\%$. Phillips slope $\alpha = 0.5$.
Adaptive expectations. $\pi^e_t = \pi_{t-1}$. To reduce inflation by 1pp/year: $-0.01 = 0.5 \cdot x_t \Rightarrow x_t = -0.02$. Six years at 2% below potential. Cumulative loss: 12% of GDP. Sacrifice ratio: \$12/6 = 2.0$.
Rational expectations with credibility. $\pi^e$ jumps to 2%. With $x_t = 0$: $\pi_t = 2\%$. Costless disinflation. Sacrifice ratio: 0.
Reality (Volcker, 1979–83): ~4 years, sacrifice ratio $\approx 2.5$. Partly forward-looking (some credibility), mostly backward-looking (inertia in wages and contracts).
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.Fiscal policy is effective: IS shifts right → $r$ tends above $r^*$ → capital inflows → central bank sells domestic currency → LM shifts right endogenously → $Y$ rises.
Monetary policy is ineffective: LM shifts right → $r$ falls below $r^*$ → capital outflows → central bank buys domestic currency → LM shifts back. No change in $Y$.
Fiscal policy is ineffective: IS shifts right → $r$ tends above $r^*$ → capital inflows → currency appreciates → NX falls → IS shifts back. No change in $Y$.
Monetary policy is effective: LM shifts right → $r$ falls below $r^*$ → capital outflows → currency depreciates → NX rises → IS shifts right → $Y$ rises.
Figure 9.6. Mundell-Fleming model. Fiscal policy is effective under fixed exchange rates; monetary policy is effective under floating rates.
Figure 9.7. The impossible trinity. A country must choose two of three: free capital flows, fixed exchange rate, independent monetary policy.
Part A — Fixed exchange rate. Kaelani pegs to TAD, $r_K = r^* = 5\%$. Fiscal expansion $\Delta G = 0.5$B KD.
Mechanism: IS shifts right → $r$ tends above $r^*$ → capital inflows → central bank sells KD/buys TAD → money supply expands (LM shifts right) → $Y$ rises to ~12.5B KD. Fiscal policy effective.
Part B — Floating exchange rate. Same fiscal expansion.
Mechanism: IS shifts right → $r$ pressure → capital inflows → KD appreciates → NX falls → IS shifts back. $Y$ barely changes. Fiscal policy ineffective — crowded out through the exchange rate.
Lesson: Under the peg, Kaelani has fiscal policy but no monetary policy. The impossible trinity: free capital + fixed rate means no independent monetary policy.
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.
IntermediateThe expectations-augmented Phillips curve assumes a direct relationship between the output gap and inflation without explaining why. For inflation to respond sluggishly, we need two ingredients: firms that set prices (market power) and a reason they don’t adjust continuously (stickiness).
Each firm faces a downward-sloping demand curve and sets its price as a markup $\mu = \varepsilon/(\varepsilon - 1)$ over marginal cost, where $\varepsilon$ is the Dixit-Stiglitz elasticity of substitution.
Each period, a fraction $(1 - \theta)$ of firms reset their prices, while fraction $\theta$ remain stuck. With $\theta = 0.75$, average price duration is 4 quarters. The optimal reset price:
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.The parameter $\kappa = \frac{(1-\theta)(1-\beta\theta)}{\theta} \cdot \gamma$ depends on price stickiness $\theta$, the discount factor $\beta$, and the sensitivity of marginal cost to the output gap $\gamma$. When $\theta$ is large, $\kappa$ is small — inflation responds weakly to the output gap.
The NKPC differs fundamentally from the backward-looking Phillips curve: inflation depends on expected future inflation, not past inflation. A credible commitment to low future inflation reduces $\pi_t$ immediately. The full three-equation NK model is covered in Chapter 15.
The Kaelani Republic (population 5 million, GDP ≈ 10 billion KD from Chapter 5, IS-LM baseline from Chapter 8) faces two intertwined challenges: choosing an exchange rate regime and raising long-run growth to close the gap with neighbor Talani.
Exchange Rate Regime (Mundell-Fleming). Kaelani maintains a fixed peg to the Talani dollar (TAD) with free capital mobility ($r_K = r_T = 5\%$). The government plans a fiscal expansion of $\Delta G = 0.5$B KD. Under the fixed rate, Mundell-Fleming predicts the expansion is effective: IS shifts right, capital inflows cause LM to shift right endogenously, $Y$ rises to ~12.5B KD. Under a floating rate, the same expansion would be neutralized by currency appreciation.
The central bank governor observes: “Under the peg, we have fiscal policy but no monetary policy. If we wanted to cut rates independently — say, during a recession that doesn’t hit Talani — we couldn’t.” This is the impossible trinity: free capital + fixed rate = no independent monetary policy.
Long-Run Growth (Solow with Calculus). Both economies: $\alpha = 1/3$, $n = 0.02$, $g = 0.015$, $\delta = 0.05$. Kaelani ($s = 0.15$): $k^* = 2.35$, $y^* = 1.33$. Talani ($s = 0.25$): $k^* = 5.04$, $y^* = 1.71$. Predicted income ratio: \$1.78$. Observed: \$1.50$. The gap is larger than Solow predicts — TFP differences (institutions, human capital) matter, foreshadowing Chapters 13 and 18.
Kaelani is dynamically efficient ($s = 0.15 < s_g = 0.333$) but far below the golden rule. Speed of convergence: $\lambda = 0.0567$, half-life $\approx 12.2$ years.
Micro-Founded Consumption. A Kaelani household earns $y_1 = 2{,}000$ KD, expects $y_2 = 2{,}400$ KD, with $r = 5\%$, $\beta = 0.95$. The Euler equation gives $c_2^*/c_1^* = 0.9975 \approx 1$: near-perfect smoothing. The household borrows ~195 KD in period 1 because it expects higher future income. A one-time stimulus of 200 KD is mostly saved; a permanent 200 KD/period subsidy is consumed nearly one-for-one.
State at end of chapter: Kaelani’s macro framework is now micro-founded (Euler equation, Solow with calculus, Mundell-Fleming). The fixed rate constrains monetary policy. The saving rate is below the golden rule. The Solow model only partially explains the income gap. Threads continue into Chapter 13 (Ramsey growth), Chapter 15 (NK monetary policy), and Chapter 18 (institutions).
| Label | Equation | Description |
|---|---|---|
| Eq. 9.1 | $c_1 + \frac{c_2}{1+r} = y_1 + \frac{y_2}{1+r}$ | Intertemporal budget constraint |
| Eq. 9.2 | $\mathcal{L} = u(c_1) + \beta u(c_2) + \lambda[\cdots]$ | Lagrangian (two-period) |
| Eq. 9.3 | $u'(c_1) = \beta(1+r)\,u'(c_2)$ | Consumption Euler equation |
| Eq. 9.4 | $(c_2/c_1)^\sigma = \beta(1+r)$ | CRRA Euler equation |
| Eq. 9.5 | $\hat{c}_t = E_t\hat{c}_{t+1} - (1/\sigma)(r_t - \rho)$ | Log-linearized Euler equation |
| Eq. 9.6 | $x_t = E_tx_{t+1} - (1/\sigma)(i_t - E_t\pi_{t+1} - r^n)$ | Forward-looking IS curve |
| Eq. 9.7 | $uc = (r + \delta)p_K$ | User cost of capital |
| Eq. 9.8 | $q = V / (p_K \cdot K)$ | Tobin’s q |
| Eq. 9.9 | $I/K = (q - 1)/\phi$ | Optimal investment |
| Eq. 9.10 | $y = k^\alpha$ | Per-effective-worker production |
| Eq. 9.11 | $\dot{k} = sk^\alpha - (n+g+\delta)k$ | Solow capital accumulation ODE |
| Eq. 9.12 | $k^* = [s/(n+g+\delta)]^{1/(1-\alpha)}$ | Solow steady state |
| Eq. 9.13 | $f'(k_g) = n + g + \delta$ | Golden rule condition |
| Eq. 9.14 | $s_g = \alpha$ | Golden rule saving rate |
| Eq. 9.15 | $\lambda = (1-\alpha)(n+g+\delta)$ | Speed of convergence |
| Eq. 9.16 | $\pi_t = \pi^e_t + \alpha(Y_t-Y^*)/Y^* + \varepsilon_t$ | Expectations-augmented Phillips curve |
| Eq. 9.17 | $\pi^e_t = \pi_{t-1}$ | Adaptive expectations |
| Eq. 9.18 | $\Delta\pi_t = \alpha(Y_t-Y^*)/Y^* + \varepsilon_t$ | Accelerationist Phillips curve |
| Eq. 9.19 | $Y = C(Y-T) + I(r) + G + NX(e)$ | Open-economy IS |
| Eq. 9.20 | $NX(e) + KA(r - r^*) = 0$ | BP curve |
| Eq. 9.21 | $r = r^*$ | Perfect capital mobility |
| Eq. 9.22 | Trilemma constraint | Impossible trinity |
| Eq. 9.23 | $\pi_t = \beta E_t\pi_{t+1} + \kappa x_t$ | New Keynesian Phillips Curve |
| Eq. 9.24 | $p_t^* = \mu + (1-\beta\theta)\sum(\beta\theta)^j E_t[mc_{t+j}]$ | Calvo optimal reset price |
Coming in Part IV: econometrics gives you the tools to TEST the models. Advanced micro gives the foundations for everything in Part V.