20.7 Different Interest Rates for Borrowing and Saving

Up to now we’ve been dealing with a single interest rate at which an agent could both borrow and save. In reality, the interest rates people face for saving are often very different from the interest rates charged for borrowing. The reason, in general, is that individual borrowers are a much higher risk than, say, the U.S. Government. Therefore the government can borrow money at a much lower interest rate than an individual.

For example, the following chart from the St. Louis Fed shows the 30-year mortgage rate (a rate at which well-qualified homeowners can borrow money) and the 10-year treasury note rate (a rate which is generally considered a “risk-free return” on savings). As you can see, the mortgage (borrowing) rate is consistently higher than the treasury (saving) rate:

On a more day-to-day basis, it’s not unusual for a savings account in a bank to have an interest rate of nearly zero, while interest on credit card debt can be around 30% — and payday lenders like Cash Call can charge an interest rate that amounts to nearly 90% per year!

How can we picture this in our model of intertemporal consumption? It’s actually quite simple: starting from the endowment, we can can have one interest rate (slope of the budget line) that corresponds to savings, and another that corresponds to borrowing:

To find the optimal behavior, as always, we need to look at the relationship between the MRS at the endowment point and the slopes of the budget constraint in each direction:

• If the MRS at the endowment is less than the slope moving to the left, then the agent will want to save.
• If the MRS at the endowment is greater than the slope moving to the right, then the agent will want to borrow.
• If the MRS at the endowment is between the slopes, then the agent will want to just consume their income stream (and neither borrow nor save)

You can play with the graph below to see how this works in practice. Try adjusting the discount factor $\beta$ to see for which values you would optimally borrow, save, or neither: