Optimal Saving and Borrowing

Having established the intertemporal budget line \(c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}\) and the intertemporal utility function \(u(c_1,c_2) = v(c_1) + \beta v(c_2)\) the procedure for finding the optimal bundle is the same as in consumer theory: if the conditions for tangency are met, we find the point along the budget line where the MRS is equal to the slope, $1 + r$. If the conditions for tangency are not met, we use the gravitational pull argument to find the solution.

Let’s solve using the Cobb-Douglas utility function $u(c_1,c_2) = \ln c_1 + \beta \ln c_2$. In this case the tangency condition is \({c_2 \over \beta c_1} = 1 + r \Rightarrow c_2 = \beta(1+r)c_1\) Plugging this into the budget constraint gives us, \(\begin{aligned} c_1 + {\beta(1+r)c_1 \over 1 + r} &= m_1 + {m_2 \over 1 + r}\\ (1 + \beta)c_1 &= m_1 + {m_2 \over 1 + r}\\ c_1^\star &= {1 \over 1 + \beta}\left(m_1 + {m_2 \over 1 + r}\right) \end{aligned}\) and therefore \(c_2^\star = \beta(1+r)c_1^\star = {\beta \over 1 + \beta}[(1+r)m_1 + m_2]\) One way to interpret this is that the consumer will want to consume fraction $1/(1 + \beta)$ of the present value of their income in the first period, and fraction $\beta/(1 + \beta)$ of the future value of their income in the second period:

Note that the optimal bundle moves to the right along the budget line as $\beta$ decreases: that is, the less patient you are, the more you want to consume now (giving up future consumption).

Net demand for present consumption

If we subtract the first-period income $m_1$ from the optimal consumption in period 1, $c_1^\star$, we get the net demand for borrowing — that is, the amount by which the consumer would like to exceed their current income when facing current interest rates: \(\begin{aligned} c_1^\star - m_1 &= {1 \over 1 + \beta}\left(m_1 + {m_2 \over 1 + r}\right)-m_1\\ &= {1 \over 1 + \beta}\left({m_2 \over 1 + r}\right)-{\beta \over 1 + \beta}m_1 \end{aligned}\) We can see that this is:

• positive when ${m_2 \over \beta m_1} > 1 + r$, and negative when ${m_2 \over \beta m_1} < 1 + r$: you want to borrow when the MRS at your income stream is greater than the slope of the budget constraint $(1 + r)$, and save when it’s less.
• decreasing in $r$, $m_1$ and $\beta$: the higher the interest rate, more income you have now, or the more patient you are, the less you want to borrow/more you want to save
• increasing in $m_2$: the more income you have in the future, the more you want to borrow/less you want to save

You can use the graph below to play with any of these comparative statics. See how the net demand for borrowing, and the net supply of savings, is affected by changing each of the parameters:

Try using the above diagrams to answer the questions:

• If you’re completely patient ($\beta = 1$) and $m_1 > m_2$, is there any interest rate for which you would borrow money? Why or why not?
• If you’re very impatient ($\beta = 0.25$), for what sort of income stream $(m_1,m_2)$ would you save money?

Different Interest Rates for Borrowing and Saving

Interesting behavior can arise if there are different interest rates for borrowing and saving.

To find the optimal behavior, as always, we need to look at the relationship between the MRS at the initial point representing your income stream (your “endowment”) 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: