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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
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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:

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:

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:

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:

Next: Nominal and Real Interest Rates
Copyright (c) Christopher Makler /