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

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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)

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, r = 0.2

\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 30\beta}

1 + r

1.2

= {4 \over 5\beta}

= {6 \over 5}

>

{6 \over 5}

{4 \over 5\beta}

>

1 + r

MRS

\beta < {2 \over 3}

\text{Borrow if }\beta < \frac{2}{3}?

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\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}

m_1 = 30, m_2 = 24, r = 0.2

\text{Generic }m_1,m_2,r

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

{c_2 \over \beta 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 = 1.2\beta c_1

\Rightarrow c_2 = 1.2\beta 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 + 1.2 \beta 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^* = {1 \over 1 + \beta} \times 50

c_2^* = {\beta \over 1 + \beta} \times 60

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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 > \frac{p_1}{p_2}

MRS < \frac{p_1}{p_2}

BUY MORE GOOD 1

BUY LESS GOOD 1

Sell good 1 and buy good 2 if your MRS at the endowment is less than the price ratio to sell good 1 / buy more good 2.

**Either way, it all comes down to the relationship between your MRS at the endowment and the relevant price ratios.**

Buy good 1 and sell good 2 if your MRS at the endowment is greater than the price ratio to buy more good 1 / sell good 2

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

\text{Let's call }p_2 = (1 + \pi)p_1\text{, where }\pi\text{ is the inflation rate}:

\text{Suppose the price of consumption in period $t$ is }p_t:

\text{Up to now, we've been just looking at dollar amounts in both periods.}

\text{Let's define }(1 + \rho) = \frac{1 + r}{1 + \pi}\text{. Then:}

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

"Present Value" for two periods

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" for three periods

Why stop at two periods?

\text{If you save }s \text{ for two periods at compound interest, it grows to }(1 + r)^2s...

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