L7 (1): Dynamic Consumption-Saving Choice in a Two-Period Model

Recap: One-Period (Static) Micro-Founded Macro Model

  • Began course with a prototype one-period general-equilibrium model.
    • Used to rationalise aggregate demand for labour, final goods, etc.
    • Labour market: firms’ labour demand vs. consumers’ labour supply.
    • Product market: demand = supply under competitive equilibrium (all markets clear simultaneously).
  • Experiments in that framework:
    • Policy shocks or external shocks analysed within the single period.
    • Limitation: consumers must spend all income immediately; no borrowing or lending.

Motivation for a Dynamic Framework

  • Real agents: borrow, lend, save, and postpone consumption.
  • Macroeconomic policy is inherently forward-looking; expectations about the future matter.
  • Need to track effects of shocks/policies on today and tomorrow.
  • Therefore extend the static model to a dynamic setting—the simplest analytic step is a two-period model.

Structure of the Two-Period Model

  • Periods: “today” (period 0) and “tomorrow” (period 1, denoted by the prime ′).
  • Initial simplification: production side suppressed—income flows YY (today) and Y′Y' (tomorrow) are exogenous (“fall from the sky”).
  • Link between periods: saving SS.
    • Positive SS ⇒ lend today, consume more tomorrow.
    • Negative SS ⇒ borrow today, repay tomorrow.
  • Associated intertemporal (relative) price: the real interest rate rr.
    • One unit of today’s consumption trades for (1+r)(1+r) units of tomorrow’s consumption (or vice-versa via discounting 1\/(1+r)).

Key Concepts & Notation

  • Goods are time-dated: CC (consumption today), C′C' (consumption tomorrow).
  • Intertemporal Budget Constraint (IBC): C+C′1+r=Y+Y′1+rC + \frac{C'}{1+r} = Y + \frac{Y'}{1+r}
    • LHS = present-value cost of chosen consumption bundle.
    • RHS = present-value wealth (income stream).
  • Indifference Curves now depict preferences over (C,C′)(C, C') bundles rather than over leisure/consumption.
  • Same micro toolkit applies: budget lines, indifference curves, tangency conditions, substitution & income effects.
  • Acronym caveat: IBC = Intertemporal Budget Constraint (not Irritable Bowel Condition!).

Optimal Consumption-Saving Choice

  • Two unknowns: CC and C′C'.
  • Need two conditions:
    1. First-order (tangency) condition – marginal rate of substitution equals the market rate:
      \text{MRS}_{C,C'} \equiv \frac{\partial U\/\partial C}{\partial U\/\partial C'} = (1+r)
      • Interpreted as “consumer’s subjective exchange rate = objective exchange rate.”
    2. Feasibility condition – the IBC above.
  • Solving the pair yields optimal choices (C<em>,C′</em>)(C^<em>, C'^</em>) and implied saving S∗=Y−C∗S^* = Y - C^*.

Carry-Over of Micro Techniques

  • Graphical depiction:
    • Horizontal axis: CC, vertical axis: C′C'.
    • IBC is a straight line with slope −(1+r)-(1+r) and intercepts Y + Y'\/(1+r) (horizontal) and (1+r)Y+Y′(1+r)Y + Y' (vertical).
    • Indifference curves convex toward origin; tangency gives optimum.
  • Comparative-statics experiments will mirror static-model analysis:
    • Income shock (ΔYY or ΔY′Y') → parallel shift of IBC.
    • Interest-rate change (Δrr) → pivot of IBC; decompose response into income effect vs. intertemporal substitution effect.

Real-World Illustrations of Dynamic Choice

  • Student life: “Do I sleep in today and do homework tomorrow?”
  • Researcher procrastination: delay vs. immediate effort.
  • National saving policies: sovereign wealth funds, superannuation (forced saving for retirement).
  • Environmental policy: mitigate now vs. pass costs to future generations—often modelled as a dynamic game between voters and politicians.

Roadmap of Upcoming Lectures

  • Step 1: Derive and plot the IBC.
  • Step 2: Overlay preferences (indifference curves) and locate optimal bundle.
  • Step 3: Conduct policy/shock experiments (income changes, interest-rate changes).
  • Step 4: Re-introduce production so that income becomes endogenous; then combine intertemporal consumer choice with firms’ intertemporal production decisions.
  • Throughout: utilise familiar tools—budget lines, indifference curves, substitution vs. income effects—now re-labelled as intertemporal substitution.

Take-Away Messages

  • Dynamic macro analysis = static micro toolkit + time dimension.
  • Saving is the mechanism that links periods; rr is the price of shifting resources across time.
  • Optimality condition: trade-off set by MRS = (1+r)(1+r) subject to the IBC.
  • Understanding intertemporal substitution is key for forecasting responses to policy and shocks.
  • Although terminology is “new,” the analytical structure remains the same—so prior knowledge transfers directly.