# Constrained optimization when lending money between two periods

The question is: A consumer has utility function $u= x_{11}x_{12}+\beta x_{21}x_{22}$ where $x_{ti}$ is amount of good $i$ consumed in period $t$, $\beta$ is a positive parameter.

We are asked to maximize the utility with the following constraints:

$p_1x_{11} + p_2x_{12} = m_1\\ p_1x_{21} + p_2x_{22} = m_2$

where $p_i$ is the price of $x_{ti}$ and is the same between periods, and $m_t$ is the budget for each time period $t$.

I have no problem with the original optimization question. But then a later part of the question says:

Assume now that it is possible to borrow or lend income between the two periods at a fixed interest rate r. Write down the optimization problem.

I understand how to do constrained optimization with Lagrangians (as was taught in class). What I don't have a clue about is how to actually write the constraints.

A bit of googling led me to something like this from a different example:

$p_1x_{11} + p_2x_{12} = m_1 + (1+r)m_2\\ p_1x_{21} + p_2x_{22} = m_2 + (1+r)m_1$

But I don't think it's quite right and I also don't understand the reasoning behind these constraints if they are correct.

Can someone help me figure out how to write the equation for the constraints?

(Also not sure if this really belongs on the Math website but I couldn't find a stackexchange for Econ HW questions).

You are lending $y$ dollars from period 2 to period 1. $y$ can be negative, but you can't save more than $m_1$ or lend more than you can pay back in $m_2$. If you lend $y$ you need to replay $y(1+r)$. Your constraints are now \begin{align}p_1x_{11}+p_2x_{12}&=m_1+y \\p_1x_{21}+p_2x_{22}&=m_2-y(1+r)\\ m_1+y&\ge 0 \\ m_2-y(1+r)&\ge 0\end{align} $y$ is an additional variable to be optimized over.
• A "loan" is when $y$ is positive, and "saving" is when $y$ is negative. Feb 3 '14 at 14:41