Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 3

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**.

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

**Working** = trading time for money

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

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

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

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 \(c_2\).

c_1 = m_1 - s

c_2 = m_2 + s

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

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.

c_1 = m_1 - s

c_2 = m_2 + (1+r)s

c_2 = m_2 + (1+r)(m - c_1)

(1+r)c_1 + c_2 = (1+r)m_1 + m_2

c_1 = m_1 + b

c_2 = m_2 - (1+r)b

c_2 = m_2 - (1+r)(c_1 - m_1)

(1+r)c_1 + c_2 = (1+r)m_1 + m_2

(1+r)c_1 + c_2 = (1 + r)m_1 + m_2

"Future Value"

1.2c_1 + c_2 = 1.2 \times 30 + 24

1.2c_1 + c_2 = 60

c_1 + \frac{c_2}{1+r} = m_1 + \frac{m_2}{1 + r}

"Present Value"

c_1 + \frac{c_2}{1.2} = 30 + \frac{24}{1.2}

c_1 + \frac{c_2}{1.2} = 50

u(c_1,c_2) = v(c_1)+\beta v(c_2)

v(c) = \text{“within-period" utility}

\beta = \text{“between-period" discount factor}

v(c) = \ln c

v(c) = c

u(c_1,c_2) = \ln c_1 + \beta \ln c_2

Examples:

v(c) = \sqrt{c}

u(c_1,c_2) = c_1 + \beta c_2

u(c_1,c_2) = \sqrt{c_1} + \beta \sqrt{c_2}

u(c_1,c_2) = v(c_1)+\beta v(c_2)

MRS \text{ at endowment }= {v^\prime (m_1) \over \beta v^\prime (m_2)}

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)

u(c_1,c_2) = v(c_1)+\beta v(c_2)

MRS \text{ at endowment }= {v^\prime (m_1) \over \beta v^\prime (m_2)}

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

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

pollev.com/chrismakler

MRS(c_1,c_2) = {c_2 \over \beta c_1}

\text{Example: }v(c_t) = \ln c_t

u(c_1,c_2) = \ln c_1 + \beta \ln c_2

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

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

MRS(c_1,c_2) = {c_2 \over \beta c_1}

\text{MRS at endowment } =

\text{Price Ratio } =

\text{Example: }v(c_t) = \ln c_t \Rightarrow u(c_1,c_2) = \ln c_1 + \beta \ln c_2

m_1 = 30, m_2 = 24, \beta = 0.5

\text{Generic }m_1,m_2,r

\text{Borrow if } :

MRS(m_1,m_2) = {m_2 \over \beta m_1}

MRS(30,24) = {24 \over 0.5 \times 30}

1 + r

= 1.6

>

1 + r

1.6

>

1 + r

MRS

r < 0.6

1 + r

\text{Example: }v(c_t) = \ln c_t \Rightarrow u(c_1,c_2) = \ln c_1 + \beta \ln c_2

Tangency condition:

Budget line:

MRS(c_1,c_2) = {c_2 \over \beta c_1}

{c_2 \over \beta c_1} = 1 + r

(1 + r)c_1 + c_2 = (1 + r)m_1 + m_2

\Rightarrow c_2 = (1+r)\beta c_1

(1 + r)c_1 + (1+r)\beta c_1 = (1 + r)m_1 + m_2

c_1^* = {1 \over 1 + \beta}(m_1 + {m_2 \over 1 + r})

c_2^* = {\beta \over 1 + \beta}((1 + r)m_1 + m_2)

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

what is your optimal choice?

pollev.com/chrismakler

\text{Example: }v(c_t) = \ln c_t \Rightarrow u(c_1,c_2) = \ln c_1 + \beta \ln c_2

Tangency condition:

Budget line:

m_1 = 30, m_2 = 24, \beta = 0.25, r = 0.2

{c_2 \over \beta c_1} = 1 + r

{c_2 \over 0.25 c_1} = 1.2

(1 + r)c_1 + c_2 = (1 + r)m_1 + m_2

1.2c_1 + c_2 = 1.2 \times 30 + 24

1.2c_1 + c_2 = 60

\Rightarrow c_2 = 0.3 c_1

\Rightarrow c_2 = (1+r)\beta c_1

(1 + r)c_1 + (1+r)\beta c_1 = (1 + r)m_1 + m_2

1.2c_1 + 0.3 c_1 = 60

c_1^* = {1 \over 1 + \beta}(m_1 + {m_2 \over 1 + r})

c_2^* = {\beta \over 1 + \beta}((1 + r)m_1 + m_2)

c_1^* = 40

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

MRS(c_1,c_2) = {c_2 \over \beta c_1}

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.

MRS > 1 + r

MRS < 1 + r

BORROW

SAVE

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

c_2 = m_2 + (1 + r)(m_1 - c_1)

Suppose there is inflation,

so that each dollar saved can only buy

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

c_2 = m_2 + \left({1 + r\over 1 + \pi}\right)(m_1 - c_1)

Up to now, we've been just looking at

dollar amounts in both periods

\text{let }\rho = {1 + r \over 1 + \pi} - 1

c_2 = m_2 + (1 + \rho)(m_1 - c_1)

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

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

c_1 + \frac{c_2}{1+r} = m_1 + \frac{m_2}{1 + r}

"Present Value" for 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

s(1 + r) = x

s = {x \over 1 + r}

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

s(1 + r) = x

s = {x \over 1 + r}

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

s(1 + r)^2 = x_2

s = {x_2 \over (1 + r)^2}

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

s(1 + r)^2 = x_2

s = {x_2 \over (1 + r)^2}

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

s(1 + r)^t = x_t

s = {x_t \over (1 + r)^t}

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

s(1 + r)^t = x_t

s = {x_t \over (1 + r)^t}

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

PV(x_t) = {x_t \over (1 + r)^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