What is the consumption function?
– The consumption function is a simple macroeconomic relationship that links total consumer spending to the level of income available for households. It is used at the aggregate (economy-wide) level to summarize how changes in income tend to translate into changes in consumption.
Key definitions
– Consumption function: an equation that predicts aggregate consumer spending as a function of disposable income and other parameters.
– Disposable income: income available to households after taxes and transfers; the money they can actually spend or save.
– Autonomous consumption: the portion of consumption that occurs even when disposable income is zero (basic subsistence spending, financed from past savings or borrowing).
– Marginal propensity to consume (MPC): the fraction of an additional dollar of disposable income that households will spend rather than save.
– Multiplier: a scalar that shows how a change in autonomous spending (or investment) is amplified into a larger change in overall income/output.
Basic formula
– The standard linear form is:
C = A + M × D
where
C = aggregate consumption,
A = autonomous consumption,
M = marginal propensity to consume (0 ≤ M ≤ 1),
D = real disposable income.
How it works (intuition)
– As disposable income rises, consumption typically rises too, but by less than the increase in income if M 10).
– Run endogeneity tests (Durbin–Wu–Hausman).
6) Robustness checks
– Try alternative controls (wealth, interest rates, unemployment, demographics).
– Estimate subgroups (age, wealth terciles).
– Test sample splits and structural breaks.
– Report confidence intervals, not only point estimates.
Worked numeric example: interpreting an estimated consumption function
– Suppose you estimate (real dollars, quarterly):
C_t = 200 + 0.75 Y_t
where C and Y are in billions.
– Interpretation:
– Estimated MPC = 0.75: an extra $1 of real income raises consumption by $0.75 on average, ceteris paribus.
– Intercept (autonomous consumption) = $200 billion: predicted consumption when measured income is zero (an extrapolation; interpret carefully).
– Average propensity to consume (APC) at Y = $2,000 billion: APC = C/Y = (200 + 0.75*
0.75*2000)/2000 = (200 + 1500)/2000 = 1700/2000 = 0.85.
– Meaning: At Y = $2,000 billion the model predicts consumption C = $1,700 billion, so the average propensity to consume (APC) = 85%. The marginal propensity to consume (MPC) is 0.75 (the slope) and the marginal propensity to save (MPS) = 1 − MPC = 0.25. Actual saving at that income is S = Y − C = $300 billion, so the average propensity to save (APS) = 300/2000 = 0.15. Note APS ≠ MPS in general; MPS is a marginal concept, APS is an average.
Illustrative policy simulation (simple Keynesian multiplier)
– Assumptions: closed economy, no taxes, no imports. Multiplier = 1/(1 − MPC).
– With MPC = 0.75: multiplier = 1/(1 − 0.75) = 4.
– If government spending increases by $100 billion, the (first-round) equilibrium change in output ΔY ≈ 4 × $100B = $400B.
– Change in consumption from that ΔY: ΔC = MPC × ΔY = 0.75 × $400B = $300B.
– Net change in saving: ΔS = ΔY − ΔC = $400B − $300B = $100B (equal to the initial spending in this simplified accounting).
Practical interpretation and caveats (what to watch for)
– Intercept (autonomous consumption) is an extrapolation. At Y = 0 the model predicts C = $200B, but real households rarely have zero measured income; this term often captures transfers, borrowing, or wealth effects and can be sensitive to omitted variables.
– Endogeneity: measured income is often endogenous (current consumption decisions and income can move together), biasing OLS estimates. Instrumental variables (IV) or lagged-income specifications may be needed.
– Dynamics: consumption responds over time. A contemporaneous static model omits habits, adjustment costs, and expectations. Consider distributed-lag, error-correction, or dynamic panel models.
– Aggregation and heterogeneity: the MPC can differ across household groups (by wealth, liquidity, age). Aggregate estimated MPCs are population-weighted averages and may mask large differences.
– Measurement and specification: choices about real vs nominal variables, smoothing, deflators, and sample period (business cycles, structural change) all affect estimates.
– Stationarity and cointegration: for long time series, test for unit roots and for cointegration between C and Y; otherwise spurious regression is possible.
Estimation checklist (step-by-step)
1. Inspect data plots and summary stats for C and Y (levels and growth rates).
2. Test stationarity (ADF, KPSS) and check for cointegration if nonstationary.
3. Decide frequency and deflation (real vs nominal). Justify choices.
4. Start with OLS on levels if cointegrated; otherwise use differences or an error-correction model.
5. Check residual diagnostics (autocorrelation, heteroskedasticity). Use HAC (Newey–West) or robust SEs as needed.
6. Address endogeneity: consider IVs (lagged variables, shocks to income like tax changes), or use instrumental approaches.
7. Try alternative specifications: add wealth, interest rate, unemployment, demographics, or interaction terms.
8. Run subgroup analyses (by age, wealth, liquidity) and robustness checks (sample splits, structural break tests).
9. Report MPC, intercept, APC at representative incomes, confidence intervals, and goodness-of-fit measures.
10. Interpret cautiously: emphasize economic significance, not just statistical significance.
Extensions and model variants (brief)
– Life-Cycle/Permanent-Income Hypothesis: consumption depends on expected lifetime resources, not just current income. Leads to smoother consumption profiles and forecasting implications.
– Habit formation models: consumption depends on past consumption (C_t−1), producing inertia.
– Credit/liquidity constraints: binding
binding: when households cannot borrow against future income, current income matters more for consumption. Liquidity constraints raise short‑run marginal propensities to consume (MPCs) out of transitory income because constrained people cannot smooth by borrowing. Empirically, this implies higher MPC estimates in low‑liquidity subsamples and stronger responses to temporary transfers.
– Buffer‑stock saving (precautionary saving): if consumers face income uncertainty and borrowing limits, they hold a buffer of wealth to self‑insure. That generates a consumption function that is less responsive to predictable permanent income changes and more responsive to unexpected or idiosyncratic shocks.
– Precautionary saving (definition): extra saving driven by uncertainty about future income or expenses. It steepens the consumption function at low wealth and flattens it at high wealth.
– Habit formation: consumption today depends on past consumption (C_t−1). This produces inertia: short‑run responses to income changes are damped, and dynamic specifications (e.g., C_t = α + βY_t + γC_{t−1} + ε_t) are common.
– Behavioral models: features like liquidity framing, mental accounting, or present‑bias discounting (hyperbolic discounting) modify the simple consumption function and can generate time‑inconsistent smoothing or large short‑run responses.
– Nonlinearities and subsistence: the C(Y) relationship can be nonlinear (e.g., Engel curves for necessities). Models sometimes include a subsistence level (C = c0 + c1·Y + … with c0 > 0) or allow MPC to vary with income or wealth.
– Micro vs macro evidence: micro (household) estimates of MPC from panel data or randomized transfers often show