# Intertemporal Choice

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 3

## Today's Agenda

Part 1: Baseline Case

Part 2: Extensions and Applications

Modeling present-future tradeoffs

The intertemporal budget constraint

Preferences over time

Optimal saving and borrowing

Different interest rates for borrowing and saving

Credit constraints

Real and nominal interest rates

Beyond two time periods

Saving and borrowing is a huge part of the U.S. economy.

Endowment of **time** and **money**.

### Today: Present vs. Future Consumption

Endowment of **money in different time periods** (an "income stream")

### Last time: Leisure vs. Consumption

**Working** = trading time for money

**Saving** = trading present consumption for future consumption

**Borrowing** = trading future consumption for present consumption

# For Each Context:

## Determine the **budget line**

## Analyze **preferences**

## Solve for **optimal choice**

## Comparative statics: analyze net **supply** and **demand**

**Do you have to spend all the money you earn in the period when you earn it?**

# Present-Future Tradeoff

Your endowment is an **income stream** of \(m_1\) dollars now and \(m_2\) dollars in the future.

What happens if you don't consume all \(m_1\) of your present income?

Two "goods" are present consumption \(c_1\) and future consumption \(c_2\).

Let \(s = m_1 - c_1\) be the amount you save.

# Saving and Borrowing with Interest

If you **save** at interest rate \(r\),

for each dollar you save today,

you get \(1 + r\) dollars in the future.

You can either **save** some of your current income, or **borrow** against your future income.

If you **borrow** at interest rate \(r\),

for each dollar you borrow today,

you have to repay \(1 + r\) dollars in the future.

"Future Value"

"Present Value"

# Preferences over Time

Examples:

# When to borrow and save?

Save if MRS at endowment < \(1 + r\)

Borrow if MRS at endowment > \(1 + r\)

(high interest rates or low MRS)

(low interest rates or high MRS)

If we assume \(v(c)\) exhibits **diminishing** marginal utility:

MRS is higher if you have less money today (\(m_1\) is low)

and/or more money tomorrow (\(m_2\) is high)

MRS is lower if you are more patient (\(\beta\) is high)

Save if MRS at endowment < \(1 + r\)

Borrow if MRS at endowment > \(1 + r\)

pollev.com/chrismakler

If \(m_1 = 30\), \(m_2 = 24\), and \(\beta = 0.5\),

what is the highest interest rate at which you would borrow money?

## Borrow or Save?

## Optimal Bundle

Tangency condition:

Budget line:

If \(m_1 = 30\), \(m_2 = 24\), \(\beta = 0.25\), and \(r = 0.2\),

what is your optimal choice?

pollev.com/chrismakler

## Optimal Bundle

Tangency condition:

Budget line:

Since you start with \(m_1 = 30\), this means you **borrow 10**.

### Supply of Savings and

Demand for Borrowing

In general, **net demand** is \(x_1^* - e_1\)

In this context, net demand is the demand for borrowing.

If it's negative, then it's the supply of saving.

# Different Interest Rates

BORROW

SAVE

**What if the interest rate is different for borrowing and saving?**

# When a Budget Line Ends...

# Inflation and Real Interest Rates

Suppose there is inflation,

so that each dollar saved can only buy

\(1/(1 + \pi)\) of what it originally could:

Up to now, we've been just looking at

dollar amounts in both periods

We call \(r\) the "nominal interest rate" and \(\rho\) the "real interest rate"

For low values of \(r\) and \(\pi\), \(\rho \approx r - \pi\)

"Present Value" for two periods

# Beyond Two Periods

If you save \(s\) now, you get \(x = s(1 + r)\) next period.

The amount you have to save in order to get \(x\) one period in the future is

Remember how we got this...

If you save \(s\) now, you get \(x = s(1 + r)\) next period.

The amount you have to save in order to get \(x\) one period in the future is

If you save for **two** periods, it grows at interest rate \(r\) again, so \(x_2 = (1+r)(1+r)s = (1+r)^2s\)

Therefore, the amount you have to save in order to get \(x_2\) two periods in the future is

If you save for **two** periods, it grows at interest rate \(r\) again, so \(x_2 = (1+r)(1+r)s = (1+r)^2s\)

Therefore, the amount you have to save in order to get \(x_2\) two periods in the future is

If you save for \(t\) periods, it grows at interest rate \(r\) each period, so \(x_t = (1+r)^ts\)

Therefore, the amount you have to save in order to get \(x_t\), \(t\) periods in the future, is

Therefore, the amount you have to save in order to get \(x_t\), \(t\) periods in the future, is

We call this the **present value **of a payoff of \(x_t\)

Application: Social Cost of Carbon

Obama Admin: $45

Uses a 3% discount rate; includes global costs

Trump Admin: less than $6

Uses a 7% discount rate; only includes American costs

PV of $1 Trillion in 2100:

$86B for Obama, $4B for Trump

### Most of what we did was to extend the endowment framework to another context.

### The **price ratio** is the real interest rate: i.e., the cost of increasing consumption today, which is foregone consumption tomorrow - based on interest rate and inflation.

### The **utility function** is a combination of a **within-period utility** \(v(c_t)\) and **between-utility discounting** \(\beta\).

## Conclusion and Next Steps

### Other important new feature: **kinked budget constraints** representing different prices for buying and selling.

### Next time: similar utility function for consumption in **different states of the world**.

#### Econ 51 | 03 | Intertemporal Choice

By Chris Makler

# Econ 51 | 03 | Intertemporal Choice

We apply the framework of consumer choice theory to the choice of how to allocation money across time, investigating saving and borrowing.

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