Working, Saving, and Borrowing

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 3

Working, Saving,
and Borrowing

Lecture 3

Two Applications of
Trading from an Endowment

Endowment of time and money.

Present vs. Future Consumption

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

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

Application 1:
Labor Supply

Leisure-Consumption Tradeoff

Leisure (R)

Consumption (C)

24 - L

M + \Delta C

You trade \(L\) hours of labor for some amount of consumption, \(\Delta C\).

You start with 24 hours of leisure and \(M\) dollars.

\Delta C

You end up consuming \(R = 24 - L\) hours of leisure,
and \(C = M + \Delta C\) dollars worth of consumption.

Selling Labor at a Constant Wage

Leisure (R)

Consumption (C)

24 - L

M + wL

You sell \(L\) hours of labor at wage rate \(w\).

You start with 24 hours of leisure and \(M\) dollars.

You earn \(\Delta C = wL\) dollars in addition to the \(M\) you had.

M + 24w

C = M + wL

C = M + w(24 - R)

...and you consume \(R = 24 - L\) hours of leisure.

wR + C = 24w + M

Budget Line Equation

Leisure (R)

Consumption (C)

M + 24w

wR + C = 24w + M

p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2

x_1: R

x_2: C

p_1: w

p_2: 1

e_1: 24

e_2: M

This is just an endowment budget line

Optimal Supply of Labor

Preferences are over the two "good" things: leisure and consumption

u(R,C)

wR + C = 24w + M

We've just derived the budget constraint in terms of leisure and consumption as well:

Maximize utility as usual, with one caveat:
you can only sell your leisure time, not buy it.

When will labor supply be zero?

Remember: you only want to sell good 1 (in this case, your time) if

MRS(e_1,e_2)<{p_1 \over p_2}

Application 2:
Intertemporal Choice

Present-Future Tradeoff

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

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.

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

Preferences over Time

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}

When to borrow and save?

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 paitient (\(\beta\) is high)

Most Important Takeaways

Both of these were applications of the model of trading from an endowment.

Key to modeling a new situation is understanding how the real-life elements (leisure time, income streams, wages, interest rates, patience, working, saving, borrowing) relates to the elements of the model (endowment, price ratio, preferences, net supply and demand).

In class: you'll do the math for the canonical examples from the textbook.

Working, Saving, and Borrowing