IDC Revision

Consumption and Saving

Consumption and Saving

Part of: Macro-Economics Key concepts: Euler Equation, Life-Cycle Hypothesis, Permanent Income Hypothesis, Intertemporal Choice, Consumption Smoothing, Borrowing Constraints

Why We Care — Stylised Facts

Consumption is ~67% of US GDP (FRED, 2025), and the single largest expenditure item in the national accounts. Understanding how households decide to consume vs. save is essential to understanding GDP, business cycles, and fiscal policy.

Components of private consumption

Category Lasts Example
Durable goods Long Cars, appliances
Non-durable goods Short Food, clothing
Services Intangible Haircuts, cleaning

Cyclical facts (US, 1947–2025)

  • Aggregate CC: correlation with GDP ≈ 0.8; ~0.85× as volatile as GDP.
  • Services CC: correlation 0.69; only 0.72× as volatile.
  • Non-durables CC: correlation 0.66; 0.77× as volatile.
  • Durables CC: correlation 0.58 but ~2.9× as volatile as GDP.

l2_c_cyclical Cyclical components of consumption and GDP, USA. Consumption tracks GDP closely (corr ≈ 0.8) but is smoother — the visual signature of consumption smoothing.

Key puzzle

Consumption is strongly correlated with income but not just a constant fraction of it. Households must be using savings to smooth consumption across periods. We need a theory of why and how they smooth.

The "Old" Keynesian Consumer

Definitions

  • Marginal propensity to consume (MPC): ∂C/∂Y\partial C / \partial Y — extra CC per extra YY.
  • Average propensity to consume (APC): C/YC / Y.

Keynesian consumption function

Ct=Cˉ+c YtC_t = \bar{C} + c\,Y_t

Assumption: MPC (cc) is constant.

Predictions:

  1. Consumption is determined by current income.
  2. Saving rises with current income.

Why it fails

Under this model, consumption differences must be perfectly correlated with income differences:

Ct+1−CtYt+1−Yt=c\frac{C_{t+1} - C_t}{Y_{t+1} - Y_t} = c

In the data, that correlation is only 0.4–0.65 across categories. The model also:

  • Ignores dependence on future income.
  • Ignores the interest rate.
  • Lacks micro-foundations.
What's missing

We need a model of inter-temporal choice and forward-looking behaviour — Fisher's two-period model.

Consumption, Saving, and Wealth

Household saving

s=y−cs = y - c

where ss = saving (can be negative), yy = disposable income, cc = consumption.

Aggregate saving

S=Y−C−GS = Y - C - G

Flow vs. stock

  • Saving is a flow (per period).
  • Wealth is a stock. Positive saving increases wealth.

Why Save or Borrow?

  1. Life-cycle — save for retirement.
  2. Life-cycle — borrow against future earnings (e.g. student loans).
  3. Precautionary — uncertain income → save against downside.
  4. Time discounting — preferences over today vs. tomorrow.

The Two-Period Model — Setup

Notation

Exogenous Endogenous
yy: current income cc: current consumption
yfy^f: future income cfc^f: future consumption
aa: initial assets afa^f: next-period assets
rr: real interest rate
β∈(0,1]\beta \in (0,1]: discount factor

Preferences — three assumptions

  1. More is better: u′(c)>0u'(c) > 0.
  2. Diminishing marginal utility: u′′(c)<0u''(c) < 0.
  3. Impatience: today's utility weighted more than tomorrow's via β\beta.

Plus the technical conditions lim⁡c→0u′(c)=∞\lim_{c \to 0} u'(c) = \infty and lim⁡c→∞u′(c)=0\lim_{c \to \infty} u'(c) = 0.

Objective — lifetime utility

U(c,cf)=u(c)+β u(cf)U(c, c^f) = u(c) + \beta\,u(c^f)

Budget constraint — two equivalent forms

Sequential (flow):

c+af=y+a,cf=yf+(1+r)afc + a^f = y + a, \qquad c^f = y^f + (1+r)a^f

Lifetime (PVLR):

c+cf1+r=a+y+yf1+r≡PVLR\boxed{c + \frac{c^f}{1+r} = a + y + \frac{y^f}{1+r} \equiv \text{PVLR}}

where PVLR = Present Value of Lifetime Resources.

Why these are equivalent

Solve the first-period constraint for afa^f, substitute into the second, divide by (1+r)(1+r). The sequential form shows cash flows each period; the lifetime form shows the single lifetime budget line.

The Euler Equation

Setting up the Lagrangian

L=u(c)+βu(cf)+λ[PVLR−c−cf1+r]\mathcal{L} = u(c) + \beta u(c^f) + \lambda\Bigl[\text{PVLR} - c - \tfrac{c^f}{1+r}\Bigr]

FOCs give u′(c)=λu'(c) = \lambda and βu′(cf)=λ/(1+r)\beta u'(c^f) = \lambda/(1+r). Combining:

u′(c)=β(1+r) u′(cf)\boxed{u'(c) = \beta(1+r)\,u'(c^f)}

This is the Euler equation — a necessary condition for an optimal consumption plan.

Interpretation

Consider saving ε\varepsilon more today:

  • Cost: lose ε⋅u′(c)\varepsilon \cdot u'(c) in utility now.
  • Benefit: gain β(1+r)ε⋅u′(cf)\beta(1+r)\varepsilon \cdot u'(c^f) in utility tomorrow.

At the optimum the two are equal. If they're not, reallocate until they are.

Consumption-saving trade-off

Condition Implication
β(1+r)=1\beta(1+r) = 1 c=cfc = c^f (perfect smoothing) — the market rate just offsets impatience
β(1+r)>1\beta(1+r) > 1 c<cfc < c^f — market over-compensates for impatience → save
β(1+r)<1\beta(1+r) < 1 c>cfc > c^f — impatience dominates → borrow / front-load

General (T-period) form

u′(ct)=β(1+r) u′(ct+1)u'(c_t) = \beta(1+r)\,u'(c_{t+1})

Graphical Picture

  • Budget line in (c,cf)(c, c^f) space: slope −(1+r)-(1+r), passes through the endowment (y,yf)(y, y^f).
  • Indifference curves: convex, slope −1βu′(c)u′(cf)-\tfrac{1}{\beta}\tfrac{u'(c)}{u'(c^f)} = MRS.
  • Optimum: tangency where MRS=−(1+r)\text{MRS} = -(1+r) — exactly the Euler equation.

type: consumption-choice
figure: two-period
The optimum is where the highest reachable indifference curve is tangent to the budget line — the slope condition MRS=1+rMRS = 1+r is precisely the Euler equation. The consumer can choose any point on the budget line through the endowment by borrowing or lending.

Jensen meets economics

A concave utility function means the utility of the average exceeds the average of the utilities — i.e. smoothed consumption is preferred to volatile consumption with the same mean. This is the mathematical core of the consumption-smoothing motive.

Comparative Statics

The discount factor β\beta

Higher β\beta ⇒ more patient ⇒ u′(c)/u′(cf)u'(c)/u'(c^f) must rise ⇒ lower cc, higher cfc^f (save more).

β\beta appears in the Euler equation but not the budget constraint — a pure preference shift.

The interest rate rr — the ambiguous case

rr affects both the budget constraint (via PVLR) and the Euler equation. Two effects:

Effect Direction
Substitution effect — current cc gets relatively more expensive c↓c \downarrow
Income effect (borrower) — PVLR falls c↓c \downarrow
Income effect (lender) — PVLR rises c↑c \uparrow
Course convention

For a borrower, substitution and income effects both push cc down — unambiguous. For a lender, they oppose — ambiguous. Unless told otherwise, assume r↑⇒c↓r \uparrow \Rightarrow c \downarrow.

Generalisation to TT Periods

PVLR:

PVLR0=a+y0+y11+r+y2(1+r)2+⋯+yT(1+r)T\text{PVLR}_0 = a + y_0 + \frac{y_1}{1+r} + \frac{y_2}{(1+r)^2} + \dots + \frac{y_T}{(1+r)^T}

Budget:

∑t=0Tct(1+r)t=PVLR0\sum_{t=0}^{T} \frac{c_t}{(1+r)^t} = \text{PVLR}_0

Utility:

U=∑t=0Tβtu(ct)U = \sum_{t=0}^{T} \beta^t u(c_t)

Euler:

u′(ct)=β(1+r) u′(ct+1)u'(c_t) = \beta(1+r)\,u'(c_{t+1})

The Life-Cycle Hypothesis (Modigliani)

Stylised setup

Assume r=0r = 0, β=1\beta = 1, a=0a = 0, constant income yy for NN working periods, zero income for T−NT - N retirement periods.

Because β(1+r)=1\beta(1+r) = 1, the Euler equation gives perfect smoothing: ct=cˉc_t = \bar{c} for all tt.

PVLR =N⋅y= N \cdot y, and with r=0r = 0:

cˉ=N yT\bar{c} = \frac{N\,y}{T}

Implications:

Phase Income Consumption Saving
Working (t≤Nt \le N) yy Ny/TNy/T y−Ny/T>0y - Ny/T > 0
Retirement (t>Nt > N) 00 Ny/TNy/T −Ny/T<0-Ny/T < 0 

type: consumption-choice
figure: lifecycle
The life-cycle picture: income is high during working years then drops at retirement, but consumption is held flat (smoothed) across the whole life. The gaps are saving (working years) and dissaving (retirement).

Wealth profile: rises during working years, peaks at retirement, then depletes toward zero at death. This hump-shape is seen clearly in Survey of Consumer Finances (US), StatCan, and IFS (UK) data.

l2_lch_data_us Hump-shaped wealth by age, US Survey of Consumer Finances — wealth accumulates through working life, peaks near retirement, then is drawn down, exactly as the LCH predicts.

Consumption vs. expenditure

Banks, Blundell & Tanner (1998) found a "retirement-savings puzzle": expenditure drops sharply at retirement. Aguiar & Hurst (2005) resolved this — what falls is work-related expenditure (food spending, clothing) while time spent on home production rises to substitute. True consumption stays smooth.

Expenditure ≠ consumption

When households retire they buy fewer prepared meals and more raw ingredients and cook themselves. Measured spending falls but welfare-relevant consumption is roughly flat — as the LCH predicts.

The Permanent Income Hypothesis (Friedman)

Permanent income ypy^p: the hypothetical constant income stream that has the same PVLR as the actual (variable) income stream.

Core claim

Consumption depends on permanent income (i.e., PVLR), not current income alone.

Predictions for income shocks

Shock type Expected? Δc\Delta c today Δs\Delta s today
Expected rise in yy Yes 0 ++ (whole rise saved until realised)
Unexpected temporary rise in yy No Small ++ (split over TT periods) ++ (most of shock saved)
Unexpected permanent rise in yy No +Δy+\Delta y (full MPC ≈ 1) 0
Unexpected future rise in yy No ++ (borrow against it) −-
Key subtlety

Consumption re-optimises when new information arrives, not when the income change actually happens. An anticipated raise has already been priced into today's consumption — no smoothing response when it arrives.

Worked example (two periods, β(1+r)=1\beta(1+r) = 1)

Optimal plan: c=1+r2+r[y0+y11+r]c = \tfrac{1+r}{2+r}\bigl[y_0 + \tfrac{y_1}{1+r}\bigr].

For a shock xx to income:

Shock Change in cc today
One-off in period 0 1+r2+rx\tfrac{1+r}{2+r}x
One-off in period 1 12+rx\tfrac{1}{2+r}x
Permanent xx (full pass-through)

Log-utility example, β(1+r)≠1\beta(1+r) \neq 1

With u(c)=ln⁡cu(c) = \ln c, Euler gives c1=β(1+r)c0c_1 = \beta(1+r) c_0, and substituting:

c0=11+β[y0+y11+r]c_0 = \frac{1}{1+\beta}\left[y_0 + \frac{y_1}{1+r}\right]

Excess Sensitivity and Borrowing Constraints

The puzzle

Micro-data studies find consumption is too sensitive to current-income changes relative to the PIH benchmark.

Candidate explanations

  1. "Rule of thumb" consumers — a fraction just spend current income.
  2. Myopia / short horizons.
  3. Consumption vs. expenditure measurement issues.
  4. Borrowing constraints — can't borrow to smooth.

Borrowing constraint mechanics

If the household wants to borrow but can't, the Euler equation becomes:

u′(c)>β(1+r) u′(cf)u'(c) > \beta(1+r)\,u'(c^f)

Current consumption is below the unconstrained optimum. Extra income today goes straight into consumption (high MPC) until the constraint no longer binds.

type: consumption-choice
figure: borrowing
A borrower who would like to consume past the endowment cannot, so the household is stuck at c=yc = y — at that point u′(c)>β(1+r)u′(cf)u'(c) > \beta(1+r)u'(c^f), and any extra current income is spent immediately (high MPC).

Policy relevance

Borrowing-constrained households respond strongly to temporary fiscal transfers — this is why stimulus checks work even though PIH says they shouldn't. Targeting transfers to the constrained maximises stimulus per dollar.

"Wealthy hand-to-mouth" (Kaplan & Violante, 2014)

Not just the poor. Roughly 20% of US households have illiquid assets (housing, retirement accounts) but little liquid wealth — they behave as hand-to-mouth despite being rich. In France, Germany, Italy and Spain, the wealthy HtM outnumber the poor HtM by 3-to-1.

Implications:

  • Borrowing frictions ≠ poverty.
  • The MPC out of transitory income shocks can be high even in non-poor households.
  • Fiscal multipliers depend on the share of HtM consumers in the economy.

Summary — What This Lecture Teaches

  1. Stylised facts: CC is large, correlated with YY, and smoother than YY (except durables).
  2. Old Keynesian C=Cˉ+cYC = \bar{C} + cY is too simple — no role for future income, interest rates, or micro-foundations.
  3. Inter-temporal optimisation delivers the Euler equation u′(c)=β(1+r)u′(cf)u'(c) = \beta(1+r)u'(c^f) — the central equation of consumption theory.
  4. Life-cycle hypothesis: save during working years, dissave in retirement → hump-shaped wealth profile (confirmed in data).
  5. Permanent income hypothesis: consumption tracks PVLR; anticipated income changes don't move cc; information arrival does.
  6. rr has ambiguous effects — substitution vs. income. Convention: r↑⇒c↓r \uparrow \Rightarrow c \downarrow.
  7. Borrowing constraints explain excess sensitivity and give fiscal policy a transmission channel — relevant for the wealthy-HtM population, not just the poor.