L7 (3): Intertemporal Consumption Choice – Two-Period Model Study Notes

Recap & Context

• Lecture continues the series on inter-temporal consumption choice.
• Builds on the previously studied one-period model, now expanded to two periods: “today” (period $t$) and “tomorrow” (period $t'$).
• Labor–leisure decisions are abstracted away; focus is solely on how a consumer allocates consumption across time.
• Reminder: Students should be able to re-derive all equations/graphs from first principles—no rote memorisation.
• Copyright notice from ANU reiterated at the start (Section 113P, Copyright Act 1968).

Intertemporal Budget Constraint (IBC) – Review

• Lifetime budget must equate the present value of consumption to the present value of disposable income.
• Canonical form (two-period): $c + \frac{c'}{1+r} = (y - t) + \frac{(y' - t')}{1+r}$ where
  • $c, c'$ = consumption today / tomorrow
  • $y, y'$ = income today / tomorrow
  • $t, t'$ = taxes today / tomorrow
  • $r$ = real interest rate (assumed exogenous in the partial-equilibrium setup)
• Graphical representation:
  • Horizontal axis: $c$ (today).
  • Vertical axis: $c'$ (tomorrow).
  • Slope of IBC: $-(1+r)$ (opportunity cost of consuming 1 extra unit today).
  • Vertical intercept: $c' = (1+r)(y - t) + (y' - t')$ (consume nothing today).
  • Horizontal intercept: $c = (y - t) + \frac{(y' - t')}{1+r}$ (consume nothing tomorrow).
• Any bundle on the line is affordable; above is infeasible; below leaves unspent resources.

Preferences in a Two-Period Setting

• Assumed well-behaved: monotonic (more of either good raises utility) & strictly convex (diminishing MRS).
• Indifference curves:
  • Downward-sloping (trade-off between $c$ and $c'$).
  • Convex toward the origin (captures preference for consumption smoothing).
  • Higher indifference curves lie to the north-east, representing higher utility.
• Only difference from the static model: axes are now time-stamped consumption bundles rather than consumption vs. leisure.

Consumer Optimisation Problem

• Choose $(c^*, c'^*)$ such that:
  • Utility is maximised (highest attainable indifference curve).
  • IBC holds (choice is affordable).
• Formally: $\max<em>{c,c'}\ U(c,c')\ \text{s.t.}\ c + \frac{c'}{1+r}=PV</em>{income}$.
• The word “and” emphasised: both optimality & feasibility must hold simultaneously.

Optimal Consumption & Savings Scenarios

• Graphically illustrated with three coloured examples:
  1. Saver (blue)
  • Optimum lies left of endowment point (consume less today, more tomorrow).
  • s^* > 0 where $s = y - t - c$.
  1. Neither Borrower nor Saver
  • Optimum exactly at endowment point.
  • $s^* = 0$; consumes entire disposable income each period.
  1. Borrower (green)
  • Optimum lies right/below the endowment.
  • s^* < 0 (negative savings).
• Star (★) notation on the slide marks optimal saving level.

Comparative Statics: Shifts in Disposable Income

• Income today up ($y - t \uparrow$) or income tomorrow up ($y' - t' \uparrow$):
  • IBC shifts outward (parallel).
  • Both $c^*$ and $c'^*$ rise under normality of consumption.
• Permanent income change (both periods rise):
  • Effect is reinforced; larger increases in both $c$ and $c'$.
• Temporary income change (one period only):
  • Rise in income today → smaller rise in $c$ than the income increase; remainder saved.
  • Mechanism = consumption smoothing (spread gains across periods).

Consumption Smoothing Explained

• Definition: Adjusting consumption less than one-for-one with temporary income fluctuations so as to stabilise utility over time.
• Outcome: \Delta c < \Delta (y - t) if increase is temporary.
• Biblical analogy: Joseph advising Pharaoh to store grain in good years for the famine years.

Effect of a Change in the Real Interest Rate (Exercise 2 Preview)

• Increase in $r$ changes the intertemporal price of consumption, generating:
  1. Substitution effect
  • Consumption today becomes more expensive relative to tomorrow ⇒ tend to consume less today, more tomorrow.
  1. Income effect
  • For savers (lenders): Higher $r$ makes them richer ⇒ could raise both $c$ and $c'$.
  • For borrowers: Higher $r$ makes them poorer ⇒ could lower both.
• Net effect = substitution + income; sign depends on borrower vs. saver status.
• Full decomposition to be done in upcoming workshop.

Forthcoming Extensions

• Endogenise income streams: allow $y, y'$ to be choice variables or state-dependent.
• Endogenise $r$ within a general equilibrium (capital market clears via demand & supply of savings).
• Use extended model for business cycle analysis and policy evaluation.

Ethical / Practical Implications

• Highlights prudence in personal finance: smooth temporary shocks; avoid over-reacting to windfalls.
• Underpins policy debates on consumption taxes, interest-rate policy, and social insurance (helps households smooth consumption).

Key Terms & Definitions

• Disposable income: $y - t$ each period.
• Present value (PV): Current worth of future cash flows discounted by $1+r$.
• Savings: $s = y - t - c$ (positive ⇒ saving, negative ⇒ borrowing).
• Consumption smoothing: Choosing a stable consumption path despite volatile income.
• Substitution effect vs. Income effect: Standard decomposition when a relative price (here, $1+r$) changes.

Connections to Previous Material

• Builds directly on static utility maximisation (one-period) and introduces time as an additional dimension.
• Retains the same logic of highest indifference curve subject to a budget constraint.

Numerical / Symbolic References from Lecture

• Slope of IBC: $-(1+r)$.
• Optimality condition (first-order): $MRS_{c,c'} = 1+r$ (although not explicitly derived, implied by tangency).
• Savings sign convention: s^* > 0 (saving), $s^* = 0$ (neither), s^*<0 (borrowing).

Action Items for Students

• Practise drawing the IBC from scratch, labelling intercepts & slope.
• Work through textbook exercises on:
  1. Income shifts (temporary vs. permanent).
  2. Increase in the real interest rate—decompose effects for borrowers & savers.
• Prepare for the workshop: you may be asked to present your derivations.
• Stay up-to-date weekly; the course will shortly escalate to general-equilibrium & policy analysis.