Lecture 11: Intertemporal Choice

Financial Economics

For the next two lectures, we’ll use the tools of consumer theory to analyze two of the fundamental principles of financial economics: to move money across time and to try to minimize risk.

If you think about finance, the first thing that comes to mind is probably the phrase “stocks and bonds.” The second is probably that stocks are riskier assets, while bonds provide a generally safer rate of return. These thoughts highlight the two primary aspects of financial economics: time and risk.

Time

One of the primary purposes financial instruments serve is the ability to shift consumption across time. I get a paycheck from Stanford University every two weeks, but I don’t rush out that day and spend all the money: it goes into a bank account, which allows me to use the money at a time that’s optimal for me. Furthermore, I’m now fortunate enough that I earn enough money that I don’t spend all of my pay every two weeks; so with the extra, I invest it in an interest-bearing account that allows it to grow until I will use it later.

Earlier in life, I faced the opposite scenario: I was earning less each month than I was spending. This was particularly true in graduate school, when I had a small stipend from TA’ing while my wife and I were raising our first child. At that time I needed to borrow money, which I paid back later in life when I got a job and my earnings increased.

Both saving and borrowing are examples of shifting money across time. Saving may involve just not spending money (e.g. by keeping money in your wallet or purse), or it may involve putting money in a savings account or buying a bond. Likewise, borrowing may be as simple an act as borrowing a few bucks from a friend that you pay back later, or it could mean taking out student loans to pay for college that you pay back over a lifetime.

Risk

The other primary purpose of financial instruments is to address risk and uncertainty. The world is full of uncertainty: no one can know for certain what the future holds. Nonetheless, we have to make choices in the face of such uncertainty. Part of the financial industry allows us to pay money to reduce our risk – for example, through insurance policies that help pay your bills if you get sick, or if you are in a car accident, or if your house burns down. Another part pays us to take on risk: the more risky an investment is, the higher a return it needs to pay to attract capital.

Indirect utility and the value of consumption

To analyze each of these aspects, we’ll use the tools of consumer theory that we developed in the first three modules of this course. However, instead of goods, we’re going to think about “good 1” and “good 2” as money in different states of the world. Today, to analyze intertemporal choice, we’ll think about “good 1” as “money in the present,” and “good 2” as “money in the future.” On Wednesday, as we think about risk, these two goods will represent the amount of money you consume if some event happens (like you gen into a car accident) and the amount of money you consume if that event doesn’t happen.

To analyze our preferences over money in different states of the world, we’re going to use the notion of an indirect utility function that we developed last time. Recall that we defined an indirect utility function $V()$ as the utility of the utility-maximizing bundle \(V(p_1,p_2,m) = u(x_1^\star(p_1,p_2,m),x_2^\star(p_1,p_2,m))\) In other words, it told you the maximum amount of utility you could attain if you had some amount of money $m$, and faced prices $p_1$ and $p_2$.

For the next two lectures, we’re going introduce a related concept: a “value” function for consumption \(v(c_i)\) which represents how much utility you would get by consuming $c$ dollars’ worth of goods in state of the world $i$. One way of thinking about this is that your choice of $c_i$ in this model determines the $m$ in the consumer theory model; however, as you’ll see later on, we might consider $c$ to be measured in “real” rather than “nominal” terms when we want to look at inflation.

Importantly, we’re going to make the incredibly inaccurate assumption that this function is independent of the state of the world: in other words, that you have some stable relationship between consumption and utility that doesn’t depend on how old you are, or whether or not you got into a car accident. If you study behavioral economics, you’ll see empirical evidence for why this is a terrible assumption about human nature. But for now we’ll use it as a baseline case, because it will make the math much easier and we can still derive some pretty great results, even with this assumption.

So, with that out of the way, let’s dive into an analysis of intertemporal choice.

The intertemporal budget constraint

The analysis of consumer and producer theory thus far has assumed a single time period within which a decision was being made. The scene opens; you have €20 in your pocket to spend on apples and bananas; you choose how to divide your money; you eat the fruit, and the scene ends. This is called a “static” or “one-period” model.

As with any tradeoff, any decision comes with an opportunity cost; and the “good 1 - good 2” model we have can easily be adapted to model the tradeoff between present and future consumption. For the purposes of this lecture, we’ll just think of a two-period model, where “good 1” is “consumption now” and “good 2” is “consumption in the future.” However, just as “good 1 - good 2” model could be extended to any number of goods, so too can we think of this as a simplification of a more general model in which we consider decisions made over a larger number of time periods.

To model the tradeoff between present and future consumption, let’s think of “good 1” as “present consumption,” denoted $c_1$; and “good 2” as “future consumption,” denoted $c_2$. For the purposes of interest rates, we might think of this as “consumption this year” and “consumption next year,” or “consumption in period 1” vs “consumption in period 2,” or just more generally “consumption today” and “consumption tomorrow,” in the metaphorical sense of “tomorrow.”

An income stream as an “endowment”

We’ll assume that an agent — let’s call her Rita — has an income stream of a certain amount of money now, and a certain amount she expects to receive in the future. We can think of this as, for example, a weekly paycheck; or that she has a certain amount of money in the bank now, and expects a future payment. Either way, we can represent her income stream of $m_1$ dollars today and $m_2$ dollars in the future as an endowment point in good 1 - good 2 space, shown by the point $M$ in the graph below. This is just a straightforward application of the endowment budget constraint we analyzed in the first part of this class!

If Rita has no access to banking or financial markets, she has a simple choice: she can either choose to spend all $m_1$ of her current income today, or save some of it as cash to increase her consumption tomorrow. That is, if she saves $s$ dollars today, she can consume $c_1 = m_1 - s$ dollars today and $c_2 = m_2 + s$ dollars tomorrow; that is, \(c_2 = m_2 + (m_1 - c_1)\) or more simply \(c_1 + c_2 = m_1 + m_2\) This is just an endowment budget line \(p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2\) if we interpret the variables as follows: \(\begin{aligned} x_1 &= c_1 & \text{ (present consumption)}\\ x_2 &= c_2 & \text{ (future consumption)}\\ e_1 &= m_1 & \text{ (present income)}\\ e_2 &= m_2 & \text{ (future income)} \end{aligned}\) and where the price ratio is 1, since a dollar saved today is a dollar consumed tomorrow.

Saving with interest

Now suppose that Rita has a bank account that will pay her an interest rate of $r$ on her money: that is, if she saves $s$ at interest rate $r$, in the future she will receive $(1 + r)s$. Now her future consumption will be \(c_2 = m_2 + (1 + r)s = m_2 + (1 + r)(m_1 - c_1)\) Collecting the $c$ terms on the left, this gives us \((1+r)c_1 + c_2 = (1+r)m_1 + m_2\) Here we have our endowment budget constraint again, though now the “price” of present consumption is $1 + r$ because spending one dollar today means giving up $1 + r$ dollars in the future.

Note that if $c_1 = 0$, we have $c_2 = (1+r)m_1 + m_2$. This is the vertical intercept of the budget line, and is called future value of the income stream.

Borrowing with interest

Alternatively, let’s suppose that instead of saving, Rita borrows some amount $b$ in the present. Like most loans, it comes with an interest rate $r$: that is, she needs to repay $(1 + r)b$ in the future. With this, her present consumption can be \(c_1 = m_1 + b\) and in the future she’ll have to pay back the loan, so \(c_2 = m_2 - (1+r)b\) Substituting $b = c_1 - m_1$ into the second equation gives us \(c_2 = m_2 - (1+r)(c_1 - m_1)\) or \((1 + r)c_1 + c_2 = (1 + r)m_1 + m_2\) which is exactly what we had before. Essentially, as long as Rita can borrow or save at the same interest rate $r$, her budget constraint will be a straight line passing through her endowment point with a slope of $1 + r$:

The present value of an income stream

Regardless of whether Rita saves or borrows, we found that the equation of her budget line was \((1 + r)c_1 + c_2 = (1 + r)m_1 + m_2\) If we divide through by $1 + r$, this becomes \(c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}\) This looks an awful lot like our budget constraint, \(p_1x_1 + p_2x_2 = m\) with $p_1 = 1$, $p_2 = {1 \over 1 + r}$, and \(m = m_1 + {m_2 \over 1 + r}\) But what does this expression for $m$ mean?

We call this the present value, and is the horizontal intercept of the budget line. It represents the value of the income stream if it were all spent today: that is, the largest amount Rita could afford to borrow would be $b = m_2/(1+r)$, since that would grow to $(1 + r)b = m_2$ in the future, which is the maximum amount she could pay back with her future income.

Note that we could also look at the other end of her budget line, which is $(1+r)m_1 + m_2$. This is called the future value of the income stream.

Effect of a change in the interest rate

How does a change in the interest rate affect the budget line?

Let’s first think about an increase in the interest rate. If Rita wants to save money, it helps her: for any given amount of savings, $s$, she gets more in the future. On the other hand, if she wants to borrow money, it hurts her: for any amount of loan $b$, she has to repay more interest in the future. Thus when the interest rate rises, the vertical intercept shifts up, and the horizontal intercept shifts in, as the budget line pivots clockwise around her endowment point.

On the other hand, a decrease in the interest rate would have the opposite effect: it would make borrowers better off, and savers worse off.

Preferences over time: how do we discount future utility?

Technically, one could have any preferences over consumption in different time periods, just as one could have any preferences over any goods. However one models these preferences, though, there’s a fundamental breakdown between how one feels about consumption within a time period, and how one thinks about comparing consumption across time periods.

As discussed above, we’re going to assume there is some “value function” $v(c)$ which relates consumption $c$ with happiness (in utils) within a given time period. For now, we’ll just measure $c$ in dollars.

But what about across periods? Generally speaking, your present self doesn’t value current consumption and future consumption equally: if you’re impatient, or present-biased, you might value current your utility over your own future utility. Thus, your overall utility, as viewed from the present, might have a form something like \(u(c_1,c_2) = v(c_1) + \beta v(c_2)\) where $\beta < 1$ represents the amount by which you “discount” future utility.

In fact, we’ve already seen a number of utility functions which have this general form. In particular, if $v(c) = \ln c$, then intertemporal preferences may be represented by the Cobb-Douglas utility function \(u(c_1,c_2) = \ln c_1 + \beta \ln c_2\)

which is just a Cobb-Douglas utility function. The MRS of this function would be \(MRS = {c_2 \over \beta c_1}\) As a first approximation of modeling preferences over time, this has a lot going for it:

• it is decreasing in $c_1$: the more money you are consuming in the present, the less you’re willing to give up future consumption to increase present consumption.
• it is increasing in $c_2$: the more money you are consuming in the future, the more you’re willing to give up future consumption to increase present consumption.
• it is decreasing in $\beta$: the more patient you are, the less you’re willing to give up future consumption to increasing present consumption (or, more naturally, the more you’re willing to give up present consumption to increase future consumption)

While this works for the metaphor of two periods, for a long time economists took this a step further, postulating that people would discount a future stream of payments exponentially: \(u(c_1,c_2,...,c_T) = v(c_1) + \beta v(c_2) + \beta^2 v(c_3) + \cdots \beta^{T-1}v(c_T)\) This goes beyond saying that you’re just present-biased: it hypothesizes that “rational” people have time-consistent preferences. That is, it says the way you compare consumption today and consumption tomorrow is exactly the same way as you compare consumption, say, 1000 days from now with consumption 1001 days from now. A large literature in behavioral economics has pretty well disproved this hypothesis, and in its placed offered a range of other ways to think about how people make intertemporal choices.

But as with all our utility functions, realism is not our goal: we’re trying to model the kind of tensions that exist in evaluating how someone might think about distributing their consumption across time, and for those purposes the metaphor of a two-period model with a “patience” parameter $\beta$ works just fine.

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).

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 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:

This is perhaps a more natural way to get a “kinked” budget constraint than the examples we saw earlier in the course!

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