L7 (2): Inter-Temporal Budget Constraint – Two-Period Consumption Model

Overview & Learning Goal

Lecture continues the introduction to inter-temporal choice – how a consumer allocates resources across time.
For pedagogical clarity the model is kept extremely simple:
- Two periods only (“today” vs “tomorrow”).
- No leisure / labour supply yet; only consumption is chosen.
- Endowments, taxes, and the interest rate are treated as exogenous parameters (will be endogenised later).
Aim: derive and interpret the Inter-temporal Budget Constraint (IBC), setting the stage for optimal choice analysis in the next video.

Discrete vs Continuous Time

Time is assumed discrete: only two observable dates exist.
- Could correspond to days, quarters, years, etc. depending on data frequency.
Continuous-time dynamic optimisation exists (Honours level), but discrete representation captures the essentials without heavy maths.

Model Variables

Endogenous choices (consumer can pick)

$c$ : consumption today.
$c'$ : consumption tomorrow (prime (') denotes next period).
$s$ : saving today (amount carried into tomorrow, can be negative).
- Key identity: choosing $s$ is equivalent to choosing $c'$ , exactly as choosing leisure equalled choosing labour supply in the static model.

Exogenous parameters (taken as given for now)

$y,\;y'$ : endowment income today & tomorrow.
$t,\;t'$ : lump-sum taxes today & tomorrow (subsidies if negative).
$r$ : real interest rate (market-determined return on saving).

Reminder: In more sophisticated models, these parameters may become endogenous (e.g., income depends on labour supply, r determined in general equilibrium). Do not memorise the exogenous status – it changes with the question being asked.

Savings – the Inter-temporal Link

If the consumer reduces today’s consumption by $1$ unit, she can invest it and obtain $1+r$ units tomorrow.
Conversely, if she borrows, a unit of additional consumption today requires repaying $1+r$ tomorrow.
Savings therefore tie the two single-period budget constraints together, creating genuine dynamics.

Building the Inter-temporal Budget Constraint

1. Today’s accounting

\begin{aligned}
c + s &= y - t\qquad (1) \
\text{“Expenditure today”} &= \text{“Disposable income today”}
\end{aligned}

2. Tomorrow’s accounting

\begin{aligned}
c' &\le (1+r)s + (y' - t')\qquad (2)\
\text{Consumption tomorrow} &\le \text{Savings income} + \text{Disposable income tomorrow}
\end{aligned}

Weak inequality ((\le)) allows for the (non-optimal) possibility of dying with unspent resources; optimum will exhaust it.

3. Eliminating $s$

Rearrange (1): $s = y - t - c$ .
Substitute into (2):
\begin{aligned}
c' &\le (1+r)(y - t - c) + (y' - t') \
\Rightarrow c' &= -(1+r)c + (1+r)(y - t) + (y' - t') \tag{3}
\end{aligned}

(3) is the two-period IBC expressed entirely in $(c,c')$ space.

4. Slope & Intercept

Slope: $-(1+r)$ (opportunity cost of one more unit of $c$ in terms of $c'$ ).
Vertical intercept (set $c=0$ ):
$c'_{\text{max}} = (1+r)(y - t) + (y' - t')$ .
Horizontal intercept (set $c'=0$ ):
$c_{\text{max}} = \dfrac{y - t}{1} + \dfrac{y' - t'}{1+r}$
(interpreted shortly via present value).

Present Value & Lifetime Disposable Income

Define present-value wealth (a.k.a. lifetime disposable income)
$\boxed{\; W_E \equiv (y - t) + \dfrac{y' - t'}{1+r} \;}$
Then the IBC can be written compactly in either of two equivalent forms:
1. Present-value form: $c + \dfrac{c'}{1+r} = W_E$ .
2. Future-value form: $(1+r)c + c' = (1+r) W_E$ .
Interpretation:
- Present value discounts tomorrow’s flows by $1/(1+r)$ .
- Future value compounds today’s values by $(1+r)$ .

Graphical Representation

Axes: horizontal $c$ , vertical $c'$ ; positive quadrant only.
Straight-line IBC with slope $-(1+r)$ .
Endowment point $E$:
- Coordinates $\bigl( c<em>E,\, c'</em>E \bigr) = (y - t,\; y' - t')$ .
- Occurs when $s=0$ (consume income exactly in each period).
Borrower vs Saver regions:
- If c > c_E (to the right of $E$ ): s<0 ⇒ consumer is a borrower.
- Must repay $|s|(1+r)$ tomorrow, so c' < c'_E.
- If $c < c_E$ (left of $E$ ): $s>0$ ⇒ consumer is a saver.
- Receives interest, so c' > c'_E.
Comparative-statics hints:
- Higher $r$ steepens the slope (rotates IBC around the intercepts).
- Changes in $y,t,y',t'$ shift the line parallel (alter intercepts but keep slope).

Conceptual & Methodological Remarks

Partial vs General equilibrium: treating $r,y,y',t,t'$ as given is like a partial-equilibrium, single-consumer analysis. In full equilibrium they interact with other agents.
Danger of rote learning: whether income or interest rate is exogenous is problem-specific; always start from first principles instead of memorising outcomes.
Analogy to static labour-leisure model:
- Old trade-off: leisure vs labour (wage as price).
- New trade-off: consumption today vs tomorrow ((1+r) as price).
Ethical / policy angles (briefly flagged):
- Taxes or subsidies shift the IBC, influencing saving behaviour.
- Governments use such models when calibrating macro forecasts or pension reforms.

Numerical Example (hypothetical)

Suppose $y=50,\;y'=55,\;t=t'=0,\;r=0.10$ .
Present-value wealth: $W_E = 50 + \dfrac{55}{1.1} = 50 + 50 = 100$ .
IBC: $c + \dfrac{c'}{1.1} = 100$ ⇒ $c' = 110 - 1.1c$ .
Endowment point: $(50,55)$ lies on the line.
If the individual increases $c$ to 60 (borrows 10):
- Savings $s = 50-60 = -10$ .
- Tomorrow’s consumption: $c' = 55 + (1.1)(-10) = 44$ (must repay 11).

Roadmap – What Comes Next

Add indifference curves (preferences over $(c,c')$ ) to locate the optimum.
Perform comparative-static experiments:
- How does a rise in $r$ alter optimal saving? (Substitution vs income effects.)
- How do tax changes across periods affect consumption smoothing?
Later lectures will re-introduce labour-leisure decisions and possibly move towards general equilibrium with production.

Quick Recap Checklist

[ ] Two-period framework understood?
[ ] Can write both single-period constraints and combined IBC?
[ ] Able to derive slope $-(1+r)$ and intercepts?
[ ] Know definition & interpretation of $W_E$ ?
[ ] Recognise saver vs borrower zones?
[ ] Appreciate when variables are exogenous vs endogenous?