Expectations, Consumption & Investment

Intertemporal Consumption & Ricardian Equivalence

coreExam · highTA · TA4-Q3

Rational households choose consumption to maximise U(Cₜ, Cₜ₊₁) subject to an intertemporal budget constraint: Cₜ + Cₜ₊₁/(1+r) = Yₜᵈ + Yₜ₊₁ᵈ/(1+r) + WᶠᴴH. Under full credit access, consumption depends on total lifetime wealth, not current income. Ricardian equivalence: a tax cut today financed by a tax rise tomorrow (same PDV) leaves consumption unchanged — households save the tax cut to pay future taxes.

Derivation

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The Two-Period Problem

Households maximise $U(C_t, C_{t+1})$ subject to a single intertemporal constraint:

C_t + \frac{C_{t+1}}{1 + r} = Y^d_t + \frac{Y^d_{t+1}}{1 + r} + W^{FH}

The budget line in $(C_t, C_{t+1})$ space has slope $-(1+r)$ . The tangency with the indifference curve gives the optimum.

Temporary vs Permanent Income

A temporary income change is spread across both periods by smoothing → $\Delta C_t \ll \Delta Y_t$ . A permanent change moves both $Y_t$ and $Y_{t+1}$ , so consumption adjusts by almost the full amount.

Ricardian Equivalence

If the government cuts taxes today and raises them by the same PDV tomorrow:

\Delta T_t + \frac{\Delta T_{t+1}}{1 + r} = 0 \implies \text{lifetime wealth unchanged} \implies \Delta C = 0

The household saves the tax cut to pay the future tax. Fiscal policy is impotent in this case — a foundational result of rational-expectations macro.

When RE Fails

Credit constraints: borrowing-constrained households consume the tax cut because they cannot smooth on their own.
Finite horizons / no bequest: if some future taxes fall on others, RE is weakened.
Distortionary taxes: real taxes (not lump-sum) distort labour/savings margins.
Myopia: behavioural / Keynesian households use rules of thumb like $C = c_0 + c_1 Y_t$ .

TA4 Problem Structure

Write the intertemporal budget constraint and draw it in $(C_t, C_{t+1})$ space.
Consider the no-borrowing case: when does the constraint bind?
Apply Ricardian: $\Delta T_t = -\Delta T^e_{t+1}/(1+r)$ . Show $C$ is unchanged for the unconstrained case.
Then show a credit-constrained household's $C$ does change when the no-borrow constraint binds.

Worked Example

Two-period model, r = 0. Income Y_t = 100, Y_{t+1} = 100. Initial wealth zero. Government cuts T_t by 20, raises T_{t+1} by 20 (Ricardian).

Pre-policy: PDV of wealth = 100 + 100 = 200. Smoothed: C_t = C_{t+1} = 100.
Post-policy: Y^d_t = 120, Y^d_{t+1} = 80. PDV = 120 + 80 = 200 (unchanged).
Optimal C_t = C_{t+1} = 100 (still). Household saves 20 today to pay 20 more tomorrow.

Consumption unchanged at 100 in each period. Tax cut saved exactly to cover the future tax rise — classic Ricardian equivalence.

Common Mistakes

—Assuming fiscal policy is always ineffective — RE requires forward-looking, unconstrained households.
—Confusing permanent vs temporary income shocks — permanent shocks move C one-for-one; temporary shocks move C much less.
—Ignoring credit constraints — empirically many households consume a large fraction of transitory income.
—Dropping the discount factor (1+r) in the budget constraint — future flows must be PDV'd.

Exam Cues

→Budget: C_t + C_{t+1}/(1+r) = Y^d_t + Y^d_{t+1}/(1+r) + W^FH.
→Ricardian equivalence: if ΔT_t + ΔT_{t+1}/(1+r) = 0, ΔC = 0. Requires forward-looking + no credit constraint + no bequest motive.
→Keynesian consumer: C depends on current Y alone → tax cut raises C immediately.
→Credit-constrained case: cannot smooth, so MPC is high for tax cuts — fiscal multiplier is larger.