20.4 Optimal Saving and Borrowing

Having established the budget line \(c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}\) and the utility function \(u(c_1,c_2) = v(c_1) + \beta v(c_2)\) in the last two sections, the procedure for finding the optimal bundle is the same as it always is: 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 \(c_2^\star = \beta(1+r)c_1^\star = {\beta \over 1 + \beta}[(1+r)m_1 + m_2]\) In other words, as the Cobb-Douglas “trick” would suggest, 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 endowment 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?