Intertemporal Choice: Consumption, Saving, and Interest Rates

Intertemporal Choice

Introduction

Intertemporal choice analyzes how consumers allocate resources over time, specifically focusing on the trade-off between consumption and saving. The framework assumes a consumer with an endowment of income who must decide how to distribute it across different periods.

• Key Concept: Consumers face decisions about how much to consume now versus later.

• Application: This topic is central to understanding saving behavior, investment, and the impact of interest rates.

Intertemporal Budget Constraint

Two-Period Framework

To simplify analysis, we often use a two-period model where the consumer chooses consumption in period 1 (c1) and period 2 (c2). The consumer's income in each period is denoted m1 and m2.

• Composite Good: Assumes a single good representing all consumption.

• Endowment: The consumer's income stream over the two periods.

• Budget Constraint: Relates consumption in both periods to income and interest rates.

Budget Constraint with Zero Interest Rate

If there is no interest rate, the consumer can only transfer income between periods at a 1:1 ratio.

• Equation: $c_1 + c_2 = m_1 + m_2$

• Graph: The budget line has a slope of -1, passing through the endowment point.

• Interpretation: Each unit of consumption in one period reduces available consumption in the other period by one unit.

Budget Constraint with Interest Rate

When an interest rate (r) is present, the consumer can save or borrow, changing the trade-off between current and future consumption.

• Equation (Future Value Form): $c_2 = (1 + r)(m_1 - c_1) + m_2$

• Equation (Present Value Form): $c_1 + \frac{c_2}{1 + r} = m_1 + \frac{m_2}{1 + r}$

• Graph: The budget line has a slope of $-(1 + r)$, passing through the endowment point.

• Interpretation: The price of current consumption rises relative to future consumption as the interest rate increases.

Case: Borrowing and Lending at Different Rates

If the borrowing rate differs from the saving rate, the budget constraint may be kinked at the endowment point.

• Kinked Budget Line: Occurs when $r_B \neq r_S$ (borrowing rate not equal to saving rate).

• Implication: The consumer faces different trade-offs depending on whether they are borrowing or saving.

General Case: Perfect Capital Market

In a perfect capital market, the consumer can borrow and lend freely at a single interest rate.

• Budget Constraint: $c_1 + \frac{c_2}{1 + r} = m_1 + \frac{m_2}{1 + r}$

• Interpretation: The present value of consumption must equal the present value of income.

Relative Price Ratio

The slope of the budget constraint reflects the relative price of current versus future consumption.

• Price of Current Consumption: 1

• Price of Future Consumption: $\frac{1}{1 + r}$

• Opportunity Cost: As r increases, the opportunity cost of current consumption rises.

Optimal Intertemporal Choice

Indifference Curves and Preferences

The consumer's optimal choice is found at the intersection of the highest attainable indifference curve and the budget constraint.

• Typical Preferences: Consumers prefer to smooth consumption over time, avoiding extremes.

• Perfect Substitutes: Indifference curves are straight lines with slope -1.

• Perfect Complements: Indifference curves are L-shaped.

• Borrower: If $c_1 > m_1$, the consumer is a borrower.

• Lender/Saver: If $c_1 < m_1$, the consumer is a lender or saver.

Graphical Representation

The following diagram illustrates choices for a borrower and a lender:

• Borrower: Chooses $c_1 > m_1$, $c_2 < m_2$

• Lender: Chooses $c_1 < m_1$, $c_2 > m_2$

Effect of Changes in the Interest Rate

Budget Line Pivot

When the interest rate changes, the budget line pivots around the endowment point, altering the consumer's feasible set.

• Cases: Four scenarios depending on whether the consumer is a borrower or lender and whether the interest rate rises or falls.

	Borrower	Lender
Interest rate falls	(a)	(b)
Interest rate rises	(c)	(d)

Substitution and Income Effects

Interest rate changes affect consumption through both substitution and income effects.

• Substitution Effect: Higher interest rates make current consumption more expensive relative to future consumption, leading consumers to substitute away from current consumption.

• Income Effect: For borrowers, higher interest rates reduce overall consumption; for lenders, the effect is ambiguous and depends on preferences.

Generalization to More Than Two Periods

Multi-Period Budget Constraint

The present value form of the budget constraint can be extended to more than two periods.

• Three-Period Example: With constant interest rate r, the price of future consumption relative to present consumption is determined by the interest rate.

• Equation: $c_1 + \frac{c_2}{1 + r} + \frac{c_3}{(1 + r)^2} = m_1 + \frac{m_2}{1 + r} + \frac{m_3}{(1 + r)^2}$

Present Value and Investment Decisions

Comparing Income Streams

To compare alternative income streams, compute the present value (PV) of each stream.

• Present Value: The sum of future income discounted by the interest rate.

• Equation: $PV = \sum_{t=1}^{n} \frac{m_t}{(1 + r)^{t-1}}$

Net Present Value (NPV) and Investment

The principle of present value applies to investment decisions. An investment is worthwhile if the present value of the income stream exceeds the present value of the payments.

• Net Present Value (NPV): $NPV = PV_{income} - PV_{payments}$

• Decision Rule: Invest if $NPV > 0$

• Example: Buying an apartment block to rent out is profitable if the discounted value of rental income exceeds the purchase cost.

Additional info: These notes expand on the slides and text by providing full equations, definitions, and context for the two-period and multi-period models, as well as the effects of interest rate changes and investment decisions.