Chapter 7: The Multiplier Effect in Closed and Open Economies

Algebraic Derivation and Extensions

“The multiplier is not one number but a family of numbers, indexed by the model’s assumptions about money markets, expectations, and openness.”

Cross-reference: Principles Ch. 8 (Keynesian cross and all multiplier types); Ch. 26 (open economy, import leakages); Ch. 28 (fiscal policy in practice, ARRA) [P:Ch.8, P:Ch.26, P:Ch.28]

7.1 The Keynesian Cross as a Fixed-Point Problem¶

Principles Chapter 8 introduced the Keynesian cross by means of a diagram: draw the 45-degree line and the expenditure function; equilibrium is their intersection. Here we reformulate this as a fixed-point problem in $Y$, which yields both the existence and uniqueness of equilibrium and the precise multiplier formula from a single piece of mathematics.

The goods-market equilibrium condition is:

where $\mathcal{E}(Y)$ is aggregate planned expenditure as a function of income $Y$ (treating $G$, $T$, and $\bar{I}$ as parameters). Define the excess demand function:

where $\bar{A} = a - bT + \bar{I} + G$ is autonomous expenditure. Equilibrium requires $\phi(Y^*) = 0$, i.e., $\bar{A} = (1-b)Y^*$, giving $Y^* = \bar{A}/(1-b) = \kappa_G\bar{A}$.

The contraction mapping argument: Consider the iteration $Y_{n+1} = \mathcal{E}(Y_n)$. This defines a sequence $(Y_0, Y_1, Y_2, \ldots)$ starting from any $Y_0$.

Definition 7.1 (Contraction Mapping). A function $f: \mathbb{R} \to \mathbb{R}$ is a contraction on $\mathbb{R}$ if there exists $\lambda \in [0,1)$ such that $|f(x) - f(y)| \leq \lambda|x - y|$ for all $x, y$.

Theorem 7.1 (Banach Fixed-Point Theorem, Scalar Version). If $f$ is a contraction on a complete metric space, then $f$ has a unique fixed point $x^*$ and the iteration $x_{n+1} = f(x_n)$ converges to $x^*$ from any starting point $x_0$.

Application to the Keynesian cross. The expenditure function $\mathcal{E}(Y) = \bar{A} + bY$ is a contraction with Lipschitz constant $b = \mathcal{E}'(Y) \in (0,1)$ — precisely the MPC. Since $b < 1$, the iteration $Y_{n+1} = \mathcal{E}(Y_n) = \bar{A} + bY_n$ converges to the unique fixed point $Y^* = \bar{A}/(1-b)$ from any starting income $Y_0$.

The speed of convergence: $|Y_n - Y^*| = b^n|Y_0 - Y^*|$. Each “round” of the multiplier process — firms produce, workers earn income, households spend fraction $b$, firms produce again — shrinks the gap from equilibrium by factor $b$. After $n$ rounds, the remaining gap is $b^n$ of the original.

Definition 7.2 (The Keynesian Multiplier). The Keynesian (government spending) multiplier is:

It equals the sum of the geometric series $\sum_{n=0}^\infty b^n = 1/(1-b)$: the first-round effect (1 dollar of spending), plus the second-round effect ($b$ dollars of induced consumption), plus the third round ($b^2$), and so on.

7.2 All Six Multipliers: Complete Algebraic Derivation¶

The Keynesian cross with proportional income tax rate $t$ (so tax revenue is $T = tY + T_0$ with lump-sum component $T_0$) and transfer payments $TR$:

Collecting terms:

Definition 7.3 (Effective Multiplier Denominator). The expression $1 - b(1-t)$ is the effective leakage rate: the fraction of each dollar of income that “leaks out” of the spending stream through saving $(1-b)$ and taxes on marginal income $(bt)$.

Multiplier 1 — Government spending multiplier:

Multiplier 2 — Lump-sum tax multiplier:

Multiplier 3 — Proportional tax rate multiplier:

A higher tax rate reduces the effective multiplier and, at any given income level, reduces equilibrium income. The effect depends on $Y^*$ itself (because the tax base is income), making this a nonlinear comparative static.

Multiplier 4 — Transfer payment multiplier:

Transfers are less stimulative than direct spending: a dollar of transfers raises disposable income by one dollar, of which only fraction $b$ is spent. A dollar of government purchases creates a full dollar of first-round demand.

Multiplier 5 — The Balanced-Budget Multiplier (Haavelmo’s theorem):

Suppose $\Delta G = \Delta T_0 = \Delta B$ (spending and lump-sum taxes increase equally):

Theorem 7.2 (Haavelmo’s Theorem, General Form). When government spending and lump-sum taxes both increase by $\Delta B$, output rises by:

With $t = 0$ (lump-sum taxes only), this reduces to $\Delta Y^* = \frac{1-b}{1-b}\Delta B = \Delta B$: the balanced budget multiplier is exactly 1.

Proof (general case, $t = 0$):

$\Delta Y^* = \kappa_G\Delta B + \mu_{T_0}\Delta B = \frac{1}{1-b}\Delta B - \frac{b}{1-b}\Delta B = \frac{1-b}{1-b}\Delta B = \Delta B.$ $\square$

The economic intuition: government spends every dollar of tax revenue, while households would have saved fraction $(1-b)$. The government therefore contributes more to aggregate demand per dollar taxed than households would have. The net gain equals exactly the saved fraction: $\Delta Y^* = 1 - |\mu_{T_0}| = 1 - b/(1-b) \cdot (1-b) = 1$... but wait — with $t = 0$, $\mu_G = 1/(1-b)$ and $|\mu_{T_0}| = b/(1-b)$, so $\Delta Y^* = [1/(1-b) - b/(1-b)]\Delta B = \Delta B$. The balanced budget multiplier is 1, independent of $b$. A remarkable result.

Multiplier 6 — Investment multiplier:

An autonomous increase in investment — an animal spirits boom [P:Ch.15.4] — has the same multiplied effect on output as an equivalent increase in government spending.

7.3 The Open-Economy Multiplier and Import Leakages¶

In an open economy, some of each round of induced spending falls on imports rather than domestically produced goods [P:Ch.26.5]. This import leakage reduces the multiplier.

With import function $IM = m_0 + m_Y Y$ (where $m_Y > 0$ is the marginal propensity to import), the open-economy equilibrium:

where $X$ is autonomous exports. Collecting:

Multiplier 7 — Open-economy government spending multiplier:

The import leakage $m_Y$ in the denominator reduces the multiplier: income generated by fiscal expansion partly flows abroad as import demand, weakening the domestic income circuit.

Definition 7.4 (Propensity to Spend on Domestic Output). Define $\tilde{b} = b(1-t) - m_Y$ as the net marginal propensity to spend on domestic output — the fraction of each additional dollar of income that re-enters the domestic spending stream. The open-economy multiplier is simply:

This unifies all six multipliers: they all take the form $1/(1-\tilde{b})$ where $\tilde{b}$ accounts for whatever leakages are present (saving, taxes, imports, and in the IS–LM context, the interest rate feedback via crowding out).

Table of leakages and their effects on multipliers:

7.4 The Propensity to Spend: Sensitivity Analysis¶

The multiplier $\kappa = 1/(1-b)$ is extremely sensitive to the MPC $b$ near $b = 1$:

At $b = 0.75$: $\kappa = 4$. At $b = 0.80$: $\kappa = 5$. At $b = 0.90$: $\kappa = 10$. At $b = 0.95$: $\kappa = 20$.

This sensitivity has important implications for policy analysis. The MPC is not a structural constant — it varies across:

1. Households by income and wealth: liquidity-constrained households have $b \approx 1$ (they spend every dollar of income); wealthy households have $b \approx 0.3$–0.5 (Kaplan et al., 2018, HANK model [P:Ch.25.3]).

2. Type of fiscal transfer: lump-sum cash transfers have high MPCs for constrained recipients; corporate tax cuts have low MPCs if they flow to wealthy shareholders.

3. Expectations about permanence: permanent income changes have higher MPCs than transitory ones (PIH, [P:Ch.11.2]).

The elasticity of the multiplier with respect to the MPC:

At $b = 0.75$: $\varepsilon = 3$. A 1% increase in the MPC raises the multiplier by 3%. This high elasticity means that the policy effectiveness of fiscal stimulus depends critically on who receives it — and whether recipients are constrained.

7.5 The Geometric Series Derivation: Round-by-Round Accounting¶

The multiplier formula $\kappa_G = 1/(1-b) = \sum_{n=0}^\infty b^n$ connects to a concrete narrative about how spending circulates through the economy.

Round 0: Government increases spending by $\Delta G = 1$. Output rises by 1.

Round 1: The dollar of output becomes income for workers and owners. They spend fraction $b$: induced consumption $\Delta C_1 = b$. Output rises by an additional $b$.

Round 2: The additional $b$ of output becomes income. Fraction $b$ is spent: $\Delta C_2 = b^2$. Output rises by $b^2$.

Round $n$: $\Delta C_n = b^n$.

Total: $\sum_{n=0}^\infty b^n = 1/(1-b) = \kappa_G$.

The sum converges because $b < 1$: each round the increment is smaller by factor $b$, and the infinite sum is finite. If $b = 1$ (no saving), the sum diverges — a $\$1$ spending becomes infinite output. If $b = 0$ (all saving), the sum is 1 — the multiplier equals one because no induced spending occurs.

In APL, the round-by-round accumulation is a natural scan operation:

7.6 Worked Example: The 2009 ARRA Multiplier¶

Cross-reference: Principles Ch. 28.2 (ARRA empirical evidence) [P:Ch.28.2]

The American Recovery and Reinvestment Act (2009) provided $\$787$ billion in stimulus over 10 years. A naive Keynesian cross calculation with $b = 0.75$ and no leakages gives $\kappa_G = 4$ — the stimulus should have raised GDP by over $\$3$ trillion. The actual impact was far smaller. Why?

Calibrating with leakages:

Let $b = 0.75$, $t = 0.28$ (effective marginal federal plus state income tax rate), $m_Y = 0.12$ (import propensity), $b_r = 1.5$ (investment-interest sensitivity), $h = 4$ (interest semi-elasticity), $k = 0.5$ (income elasticity of money demand).

IS–LM multiplier (closed):

Adding proportional tax:

Adding imports (IS–LM with open economy):

With the combined denominator $(1-b(1-t)+m_Y) = 0.46 + 0.12 = 0.58$ replacing $(1-b)$:

At the ELB (estimated): With $h\to\infty$ (zero crowding out), the ELB multiplier:

This is consistent with the Romer–Bernstein (2009) projection of 1.57 and the Nakamura–Steinsson (2014) cross-state estimate of approximately 1.5–2.0 [P:Ch.28.2].

The lesson: the naive multiplier of 4 reflects a closed economy at the liquidity trap with no taxes or imports. Each realistic modification reduces it. Fiscal policy remains stimulative, but the magnitude depends critically on the model calibration.

7.7 Programming Exercises¶

Exercise 7.1 (APL — Full Multiplier Family)¶

Write a dfn all_multipliers ← {b t m_Y ← ⍵ ⋄ ...} that returns a 6-element vector $(\mu_G, \mu_{T_0}, \mu_{TR}, \mu_{balanced}, \mu_{open}, \mu_{\bar{I}})$ for parameters $(b, t, m_Y)$. Verify: (a) $|\mu_{T_0}| < \mu_G$ always; (b) $\mu_{balanced} = (1-b)/(1-b(1-t)) \leq 1$; (c) $\mu_{TR} = b\cdot\mu_G$.

Exercise 7.2 (Python — MPC Sensitivity)¶

import numpy as np, matplotlib.pyplot as plt

b_vals = np.linspace(0.01, 0.99, 500)
# Three multipliers as functions of b (t=0.25, m_Y=0.10)
t, m_Y = 0.25, 0.10
mu_closed = 1 / (1-b_vals)
mu_tax    = 1 / (1-b_vals*(1-t))
mu_open   = 1 / (1-b_vals*(1-t)+m_Y)

fig, ax = plt.subplots()
ax.plot(b_vals, mu_closed, label='Closed, no tax')
ax.plot(b_vals, mu_tax,    label='Closed, with tax t=0.25')
ax.plot(b_vals, mu_open,   label='Open, t=0.25, m_Y=0.10')
ax.axvline(0.75, linestyle='--', color='gray', alpha=0.5, label='b=0.75')
ax.set_ylim(0, 15); ax.set_xlabel('MPC (b)'); ax.set_ylabel('Multiplier')
ax.legend(); plt.title('Keynesian Multiplier vs. MPC'); plt.show()

Exercise 7.3 (Julia — Round-by-Round Convergence)¶

b = 0.80; n = 30
increments = b .^ (0:n-1)
cumulative = cumsum(increments)
limit = 1/(1-b)

println("Rounds to 90% of limit: ", findfirst(cumulative ./ limit .>= 0.90))
println("Rounds to 99% of limit: ", findfirst(cumulative ./ limit .>= 0.99))
println("Rounds to 99.9%:        ", findfirst(cumulative ./ limit .>= 0.999))
# Verify: at b=0.8, convergence is slower than b=0.75

Exercise 7.4 (R — ARRA Calibration Sweep)¶

# Sweep over (b, t, m_Y) to find range of plausible ARRA multipliers
b_vals <- seq(0.5, 0.9, 0.05)
t_vals <- seq(0.20, 0.35, 0.05)
m_Y    <- 0.12

# ELB multiplier: no crowding out
mu_grid <- outer(b_vals, t_vals, function(b, t) 1/(1-b*(1-t)+m_Y))
rownames(mu_grid) <- paste0("b=", b_vals)
colnames(mu_grid) <- paste0("t=", t_vals)
round(mu_grid, 2)

Exercise 7.5 — Haavelmo in the IS–LM Model ($\star$)¶

Prove that the balanced-budget multiplier in the IS–LM model (with $\Delta G = \Delta T_0 = \Delta B$) equals:

Show that this is strictly less than 1 (unlike the Keynesian cross balanced-budget multiplier of 1) because the crowding-out effect also applies to the spending expansion. What is the limit as $h\to\infty$ (liquidity trap)? As $h\to 0$ (classical case)?

Exercise 7.6 — Negative Multiplier? ($\star\star$)¶

The “expansionary austerity” hypothesis (Alesina and Ardagna, 2010) suggests that fiscal consolidation can raise output through confidence effects [P:Ch.28.5]. Formally, model confidence as $\bar{I}(G) = \bar{I}_0 - \gamma G$ where $\gamma > 0$ (higher government spending reduces private investment via an uncertainty channel). (a) Derive the multiplier $\partial Y^*/\partial G$ when confidence effects are present. (b) For what $\gamma$ does the multiplier turn negative (contractionary fiscal expansion)? (c) Calibrate with $b = 0.75$, $t = 0.25$, $\gamma = 0.5$: does the multiplier change sign? Interpret.

7.8 Chapter Summary¶

Key results:

• The Keynesian cross equilibrium is a fixed point of the expenditure function $\mathcal{E}(Y)$; the contraction mapping theorem guarantees existence and uniqueness when $b < 1$.

• The six multipliers for the closed economy with proportional taxes:

  • Spending: $\mu_G = 1/[1-b(1-t)]$

  • Lump-sum tax: $\mu_{T_0} = -b/[1-b(1-t)]$

  • Transfer: $\mu_{TR} = b/[1-b(1-t)]$

  • Balanced budget: $\mu_{BB} = (1-b)/[1-b(1-t)] \leq 1$

  • Investment: $\mu_{\bar{I}} = \mu_G$

  • Open economy: $\mu_G^{open} = 1/[1-b(1-t)+m_Y]$

• Haavelmo’s theorem: with lump-sum taxes and $t = 0$, the balanced-budget multiplier is exactly 1, independent of the MPC.

• Every multiplier takes the form $1/(1-\tilde{b})$ where $\tilde{b}$ is the net marginal propensity to spend on domestic output, unifying all cases.

• In APL: the round-by-round accumulation is +\ b * ⍳ n_rounds (scan of geometric sequence); the full multiplier table is generated via ∘.f outer product over parameter grids.

Connections forward: Chapter 9 derives the ELB multiplier — the analogous formula for the dynamic NK model at the zero lower bound, where crowding out is absent and forward-looking inflation expectations amplify the fiscal effect beyond $1/(1-\tilde{b})$.

Next: Chapter 8 — The AD–AS Model in Equation Form