Chapter 5 introduced the national accounts and the business cycle. This chapter builds the workhorse models of intermediate macroeconomics: the IS-LM model for analyzing short-run fluctuations and the Solow growth model for understanding long-run economic performance.
These models occupy different time horizons. IS-LM asks: given existing productive capacity, how do shocks to demand or monetary policy affect output and interest rates in the short run? Solow asks: what determines a country's standard of living in the long run, and why are some countries rich and others poor?
The Keynesian cross is the simplest model of short-run output determination. It starts from a powerful idea attributed to Keynes: in the short run, aggregate demand determines output. If people want to spend more, firms produce more to meet that demand. If people want to spend less, firms cut production. Prices are held fixed — they adjust only in the longer run. (This sticky-price assumption will be formalized with microfoundations in Chapter 15.)
The model starts from the expenditure identity $Y = C + I + G + NX$ and makes planned expenditure a function of income.
If $c = 0.8$, then for every additional dollar of disposable income, the household spends 80 cents and saves 20 cents. The marginal propensity to save is \$1 - c = 0.2$.
Equilibrium condition: Actual output equals planned expenditure: $Y = PE$. Solving:
The term $\frac{1}{1-c}$ is the Keynesian multiplier. A \$1 increase in government spending raises equilibrium output by $\frac{1}{1-c}$.
Why does the multiplier exceed 1? Because of the feedback loop: Government spends an extra \$1 → GDP rises by \$1 → that becomes income, and $c$ of it is spent → GDP rises again by $c$ → and so on. Total: \$1 + c + c^2 + c^3 + \ldots = \frac{1}{1-c}$.
Tax multiplier. A tax cut of $\Delta T$ has a smaller multiplier: $-c/(1-c)$. With $c = 0.8$, the tax multiplier is $-4$ vs. the spending multiplier of \$1$. The balanced-budget multiplier is 1.
Figure 8.1. The Keynesian cross. Equilibrium occurs where the planned expenditure line crosses the 45-degree line. Drag sliders to see how the multiplier amplifies changes in $G$, $T$, and $c$.
The Keynesian cross holds investment fixed. Now let investment depend on the interest rate: $I = I_0 - dr$, where $d > 0$ captures the sensitivity of investment to the real interest rate $r$. Higher interest rates raise the cost of borrowing, reducing investment.
Substituting into the equilibrium condition:
This gives a negative relationship between $r$ and $Y$: higher interest rates reduce investment, which reduces output through the multiplier. This is the IS curve — named because, in equilibrium, investment equals saving.
What shifts IS: Increase in $G$ or decrease in $T$: IS shifts right (fiscal expansion). Increase in consumer confidence ($C_0$): IS shifts right. Decrease in investment confidence ($I_0$): IS shifts left.
The LM curve describes equilibrium in the money market. Money demand depends on income (transactions motive) and the interest rate (opportunity cost):
Money market equilibrium: real money supply equals real money demand:
Solving for $r$:
The LM curve slopes upward: higher income increases money demand, and with fixed money supply, the interest rate must rise to restore equilibrium.
What shifts LM: Increase in $M/P$ shifts LM right (lower $r$ at each $Y$). Decrease in $M/P$ shifts LM left.
Simultaneous equilibrium in both goods and money markets occurs where IS and LM intersect.
Given: $C = 200 + 0.75(Y-T)$, $T = 100$, $G = 100$, $I = 200 - 25r$, $M/P = 1000$, $L = Y - 100r$.
IS: $Y = 1700 - 100r$ | LM: $r = (Y - 1000)/100$
Solving: $Y^* = 1350$, $r^* = 3.5\%$
Figure 8.4. IS-LM equilibrium. The IS curve (goods market) slopes down; the LM curve (money market) slopes up. Drag sliders to shift the curves and see how equilibrium output and interest rates respond. The dashed curves show the baseline position for comparison.
An increase in $G$ shifts IS right. The new equilibrium has higher $Y$ and higher $r$.
$G$ rises from 100 to 200 ($\Delta G = 100$). New IS: $Y = 2100 - 100r$.
New equilibrium: $Y^* = 1550$, $r^* = 5.5\%$. Output rises by 200, not 400.
Crowding out: Higher $Y$ → higher money demand → higher $r$ → investment falls by 50.
An increase in $M/P$ shifts LM right. New equilibrium: higher $Y$, lower $r$.
$M/P$ rises from 1000 to 1200. New equilibrium: $Y^* = 1450$, $r^* = 2.5\%$.
More money → buy bonds → interest rates fall → investment rises → output rises through multiplier.
| Policy | $\Delta Y$ | $\Delta r$ | Effect on investment |
|---|---|---|---|
| Fiscal ($\Delta G = 100$) | +200 | +2.0 pp | Crowded out (↓50) |
| Monetary ($\Delta M/P = 200$) | +100 | −1.0 pp | Stimulated (↑25) |
Figure 8.5. Side-by-side comparison. Fiscal expansion (left) shifts IS right — output and interest rates both rise, crowding out investment. Monetary expansion (right) shifts LM right — output rises while interest rates fall, stimulating investment.
IS-LM holds the price level $P$ fixed. The AD-AS model relaxes this.
The AD curve is derived from IS-LM by varying $P$ and tracing equilibrium output. Higher $P$ reduces real money balances $M/P$, shifting LM left, raising $r$, reducing investment, lowering output. AD slopes downward in $(Y, P)$ space.
Three reinforcing channels: (1) Interest rate effect (Keynes), (2) Wealth effect (Pigou), (3) Exchange rate effect (Mundell-Fleming).
SRAS slopes upward: firms expand output when actual prices exceed expectations. LRAS is vertical at potential output $Y_n$ — in the long run, expectations adjust so $P = P^e$.
Demand shock: AD shifts right → short-run: $Y$ and $P$ rise. Long run: SRAS shifts left, $Y$ returns to $Y_n$ at higher $P$.
Supply shock: SRAS shifts left → $Y$ falls and $P$ rises (stagflation). The central bank faces a dilemma: accommodate (restore $Y$ but raise $P$ further) or hold firm (lower $P$ but deepen recession).
An oil price shock shifts SRAS left. Initially the economy is at $Y = Y_n = 1000$, $P = 100$.
After the shock, the new short-run equilibrium: $Y = 900$, $P = 115$. Output falls below potential while prices rise — this is stagflation.
Policy dilemma:
Figure 8.6. AD-AS model. Drag sliders to apply demand shocks (shifts AD) and supply shocks (shifts SRAS). Watch the price level, output, and economic condition update. LRAS marks potential output.
We now shift from the short run to the long run. The Solow model explains why some countries are richer than others and what drives sustained economic growth.
Production: $Y = AK^\alpha L^{1-\alpha}$ (Cobb-Douglas, CRS). In per-effective-worker terms ($k = K/(AL)$, $y = Y/(AL)$):
Capital accumulation:
At steady state, $\dot{k} = 0$:
Key implications: (1) Higher saving rate raises steady-state $k^*$ and $y^*$ — but does NOT affect the long-run growth rate. (2) Long-run growth of output per worker is driven entirely by $g$ (technological progress). (3) Countries below their steady state grow faster (convergence).
For Cobb-Douglas: $s_g = \alpha$. If the economy saves more than $\alpha$, it is dynamically inefficient.
Parameters: $\alpha = 1/3$, $s = 0.24$, $n = 0.02$, $g = 0.02$, $\delta = 0.05$.
Break-even rate: $n + g + \delta = 0.09$.
$k^* = \left(\frac{s}{n+g+\delta}\right)^{1/(1-\alpha)} = \left(\frac{0.24}{0.09}\right)^{3/2} = (2.667)^{1.5} = 4.35$
$y^* = (k^*)^{1/3} = (4.35)^{1/3} = 1.633$
$c^* = (1-s)y^* = 0.76 \times 1.633 = 1.241$
Output per worker grows at rate $g = 2\%$ per year in steady state.
Using the parameters from Example 8.5, the golden rule saving rate is $s_g = \alpha = 1/3 \approx 0.333$.
Golden rule capital: $k_g = \left(\frac{0.333}{0.09}\right)^{1.5} = (3.704)^{1.5} = 7.13$
Golden rule output: $y_g = (7.13)^{1/3} = 1.925$
Golden rule consumption: $c_g = y_g - (n+g+\delta)k_g = 1.925 - 0.642 = 1.283$
Since the economy saves $s = 0.24 < s_g = 0.333$, it is below the golden rule. Raising the saving rate would increase long-run consumption but require a short-run sacrifice. The economy is not dynamically inefficient.
Figure 8.7. Solow diagram. The concave curve is investment $sf(k)$; the straight line is break-even investment $(n+g+\delta)k$. Steady state occurs at their intersection. The golden rule point (where consumption is maximized) is shown for comparison. Drag sliders to see how parameters affect the steady state.
Conditional convergence: Countries further below their steady state grow faster. The mechanism: when $k < k^*$, the marginal product of capital is high, so investment generates large output gains. As $k$ approaches $k^*$, the marginal product falls and growth slows.
Figure 8.8. Solow convergence. The trajectory shows capital per effective worker approaching steady state over time. Drag the initial $k_0$ slider to see how starting point affects convergence speed. Countries further from steady state grow faster initially.
The residual $\Delta A / A$ — total factor productivity (TFP) growth — is the Solow residual. It measures "what we don't know" but accounts for the bulk of growth in developed economies.
Figure 8.9. Growth accounting. The stacked bar shows how GDP growth decomposes into contributions from capital accumulation, labor growth, and TFP (the Solow residual). Drag sliders to explore different growth scenarios. Capital share $\alpha = 0.3$.
Kaelani faces a recession. Given: $C = 1 + 0.8(Y - T)$, $T = 2$, $G = 2.5$ (billions KD), $I = 1.5 - 10r$, $M/P = 4$, $L = 0.5Y - 20r$.
IS: $Y = 17 - 50r$ | LM: $r = 0.025Y - 0.2$
Equilibrium: $Y^* = 12$, $r^* = 10\%$.
A fiscal expansion of $\Delta G = 0.5$B shifts IS right: new $Y^* = 13.1$, $r^* = 12.8\%$. Output rises by 1.1B but crowding out is substantial.
Kaelani saves 15% of GDP ($s_K = 0.15$); neighbor Talani saves 25% ($s_T = 0.25$). Both: $\alpha = 1/3$, $n = 0.02$, $g = 0.01$, $\delta = 0.05$.
$y^*_K / y^*_T = (0.15/0.25)^{0.5} = 0.775$. The Solow model predicts Kaelani should be 77.5% of Talani's income — but the observed gap is 2×. The remaining gap must reflect differences in TFP ($A$), human capital, or institutions.
In 1936, Keynes published The General Theory during the Great Depression. The IS-LM model, formalized by Hicks in 1937, is the mathematical distillation of Keynes's argument that aggregate demand could be persistently deficient. It dominated macroeconomic policy analysis for decades and remains a useful first approximation.
| Label | Equation | Description |
|---|---|---|
| Eq. 8.1 | $C = C_0 + c(Y-T)$ | Consumption function |
| Eq. 8.2 | $PE = C_0 + c(Y-T) + I + G$ | Planned expenditure |
| Eq. 8.3 | $Y^* = \frac{1}{1-c}(C_0 - cT + I + G)$ | Keynesian cross equilibrium |
| Eq. 8.4–8.5 | IS curve | Goods market equilibrium |
| Eq. 8.6–8.8 | LM curve | Money market equilibrium |
| Eq. 8.9 | $Y = Y^* + \alpha(P - P^e)$ | Short-run AS |
| Eq. 8.10 | $y = k^\alpha$ | Per-effective-worker production |
| Eq. 8.11 | $\dot{k} = sk^\alpha - (n+g+\delta)k$ | Solow capital accumulation |
| Eq. 8.12–8.14 | Steady-state $k^*$ and $y^*$ | Solow steady state |
| Eq. 8.15 | $f'(k_g) = n+g+\delta$ | Golden rule |
| Eq. 8.16 | Growth accounting decomposition | TFP residual |