Lecture 13: Time and Risk: Applications and Extensions

The quiz on this reading has not yet been published.

In this lecture I’ll be reinforcing the concepts we developed on Monday and Wednesday by looking at some of the topics we weren’t able to get to then.

Demand for borrowing and supply of saving

When dealing with an endowment problem, there are actually two types of demand functions: gross demand and net demand. Gross demand describes the quantity you want to end up with; net demand describes the transaction you want to engage in to get there.

For example, on Monday we saw a situation in which you started with the income stream $(30,24)$, and facing an interest rate of 20%, your optimal choice was to consume at the bundle $(40,12)$.

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, or the “net supply” of saving — that is, the amount by which the consumer would like to exceed their current income when facing current interest rates: $\begin{aligned} c_1^\star - m_1 &= {1 \over 1 + \beta}\left(m_1 + {m_2 \over 1 + r}\right)-m_1\\ &= {1 \over 1 + \beta}\left({m_2 \over 1 + r}\right)-{\beta \over 1 + \beta}m_1 \end{aligned}$ If this is negative, of course, it represents the amount they would like to save rather than borrow. In particular, 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 initial “endowment” 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:

See interactive graph online here.

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?

Taking inflation into account: Nominal and real interest rates

An important consideration when thinking about consumption across time is whether there is likely to be inflation. For example, suppose all you spend money on is apples, and that today apples cost $€1.00$ each. Therefore, if you have $€100$, then at today’s price of apples, that could buy 100 apples.

Now let’s think about what would happen if you saved some of your money. Suppose you can earn 10% interest on any savings. If you saved all you $€100$, therefore it would grow to $€110$ in a year. If the price of apples is unchanged, you could therefore afford 110 apples. So far, so good.

However, what would happen if the price of apples increased – say, to $€1.08$? Then with your $€110$, you could only afford $110/1.08 \approx 102$ apples. In terms of purchasing power, your savings only grew by about 2% (from 100 apples to about 102 apples).

How can we incorporate this into our model? The intertemporal budget constraint we’ve been dealing with has been $c_1 + {1 \over 1+r}c_2 = m_1 + {1 \over 1 + r}m_2$ where $r$ is the nominal interest rate (10% in the example above). Instead of thinking of the axes of an intertemporal choice model as “dollars in the present” and “dollars in the future,” let’s think of them as “consumption in the present” and “consumption in the future.” (So, in the simple example above, it would be “apples in the present” and “apples in the future”.) Let’s assume that consumption in the present has a price of $p_1$, and consumption in the future has a price of $p_2$. Let’s also assume that we can express the income stream $(m_1, m_2)$ in real terms: for example, that $m_1$ is the amount of consumption your period-1 income can buy in period 1, and $m_2$ is the amount of consumption your period-2 income can buy in period 2. (One interpretation of this is that your wages are indexed to inflation, which is a feature of some negotiated labor contracts.) Then the intertemporal budget constraint becomes $p_1c_1 + {p_2 \over 1+r}c_2 = p_1m_1 + {p_2 \over 1 + r}m_2$ (Note that all we’ve done here is to add in the prices $p_1$ and $p_2$.) Inflation is what relates $p_1$ and $p_2$: that is, if $\pi$ is the inflation rate, then $p_2 = (1 + \pi)p_1$ Substituting this in for $p_2$ gives us $p_1c_1 + {p_1(1+\pi) \over 1+r}c_2 = p_1m_1 + {p_1(1+\pi) \over 1 + r}m_2$ Now every term has a $p_1$, so we can cancel all those terms, leaving us with $c_1 + {1+\pi \over 1+r}c_2 = m_1 + {1+\pi \over 1 + r}m_2$ Lastly, let’s define the “real” interest rate $\rho$ as: $1 + \rho = {1 + r \over 1 + \pi}$ Substituting this into the budget constraint gives us $c_1 + {1 \over 1+\rho}c_2 = m_1 + {1 \over 1 + \rho}m_2$ This is just the exact same budget constraint as we started out with, only with the real interest rate $\rho$ in place of the nominal interest rate $r$. In short, inflation simply has the effect of reducing the slope of the budget line, making it less advantageous to save and also less onerous to borrow.

See interactive graph online here.

Note that we can write $\rho$ as $\rho = {1 + r \over 1 + \pi} - 1 = {1 + r \over 1 + \pi} - {1 + \pi \over 1 + \pi} = {r - \pi \over 1 + \pi}$ which is approximately equal to $r - \pi$ as long as $\pi$ is fairly low.

Present value over multiple periods

Often in economics we want to evaluate the present value of a stream of payoffs. Among many examples:

in finance, we may want to know how to value an investment with a stream of payoffs in the future
in game theory, we may want to know the value of payoffs that occur in a multi-period game
in industrial organization and antitrust, we may want to know the value of investing in a collusive relationship

Let’s look at how to evaluate all of these by first looking at one future payoff; then several; then an infinite number.

Present value of a single payoff in the future

To think of the present value of a payoff in the future, think first of a concrete example: suppose you invested €1000 at 10% interest. After one year, it would grow to be worth $1.1 \times €1000 = €1100$; after two years, it would be worth $1.1 \times €1100 = 1210$; after three, $1.1 \times 1210 = €1331$; and so forth.

More generally, if you invested some present amount $v$ at an interest rate of $r$ (where $r = 0.1$ corresponds to a 10% interest rate), it would grow exponentially: $\begin{aligned} \text{Initial value} &= v\\ \text{Value after one year } &= (1 + r)v\\ \text{Value after two years } &= (1+r)^2v\\ \text{Value after three years } &= (1+r)^3v\\ & \vdots\\ \text{Value after }t\text{ years } &= (1+r)^tv\\ \end{aligned}$ Now let’s flip the script, and think about how much you would need to invest to achieve some target amount in the future. Let’s define $PV$ as the amount of money we would need to invest in order to be worth some amount $x$ at some point $t$ years into the future. Then by the above equation we have $(1+r)^t \times PV = x$ solving for $PV$ by dividing both sides by $(1 + r)^t$ gives us $PV(x,r,t) = {x \over (1 + r)^t}$ This relationship is visualized in the graph below. Try adjusting $x$ and $t$ (by dragging the dot) and $r$ (by adjusting the slider) to see how the present value of the future payoff changes:

See interactive graph online here.

Present value of a two-period payoff stream (now and in the future)

Now let’s think about a two-period income stream, comprised of present and future income. Let’s denote time in terms of periods into the future; so we’ll think of the “present” as being “period 0”, and the “future” as being “period 1;” so we can write our income stream as a payment of $x_0$ now and $x_1$ in the future. The interest rate between these two periods is $r$.

To calculate the present value of this income stream, imagine that we took our $x_0$ and invested it at an interest rate of $r$; in period 1, it would have grown to be worth $(1+r)x_0$. We would then receive our payment of $x_1$, meaning that our “future value” of this income stream after one period would be $FV = (1 + r)x_0 + x_1$ The present value would be the amount that, if invested in period 0, would grow to that future value after one period: $\begin{aligned} (1 + r)PV &= FV\\ (1 + r)PV &= (1 + r)x_0 + x_1\\ PV &= x_0 + {x_1 \over 1 + r} \end{aligned}$ We can see this equivalence in the following graph: the $PV$ grows at interest rate $r$ to become the same amount as you would end up with if you had the income stream:

See interactive graph online here.

Note that this equation for the present value is the same one we derived when looking at the budget constraint of a two-period intertemporal consumption model.

Present value of a multi-period (or infinite-period) stream of constant payments

We’ve established that the value of a payment of $x$ received $t$ years in the future (written $x_t$) at interest rate $r$ is given by $PV(x_t) = {x \over (1+r)^t}$ and that the value of receiving $x_0$ now and $x_1$ one period in the future is $PV(x_0,x_1) = x_0 + {x_1 \over 1+r}$ We can extend this to an arbitrary series of payments $(x_0, x_1, x_2, …, x_t)$ by just adding the value of the payment received in each period: $PV(x_0,x_1,x_2,...,x_t) = x_0 + {x_1 \over 1 + r} + {x_2 \over (1 + r)^2} + \cdots + {x_t \over (1 + r)^t}$ Now let’s think about the present value of a stream $n$ payments of $x$ dollars each, starting one period in the future: that is, $x_0 = 0$ and $x_1 = x_2 = \cdots = x_n = x$. The present value of this stream is $PV = {x \over 1 + r} + {x \over (1 + r)^2} + \cdots + {x \over (1 + r)^t}$ You can see this for various values of $n$ in the graph below. Try dragging $n$ from 1 to 5 to see how the present value changes:

See interactive graph online here.

It’s clear that as $n$ increases, the present value changes by a lot at first, then by less and less. The mathematical formula for the limit of this expression as $n$ approaches infinity is $PV = {x \over r}$ Indeed, if you check the “Show $PV$ for $n = \infty$” in the graph above, you can see this theoretical limit, which is pretty close to the $PV$ of $n = 5$ if $r$ is large!

There are two elegant ways of deriving this limit:

The economic explanation is pretty simple: if you put away some value $PV$ and never take it out, and it pays you a certain amount each period, the amount it pays each period is just the interest rate times the principal you’ve invested. So $x = r \times PV$or$PV = {x \over r}$There’s actually a special name of this kind of financial instrument, which is a perpetuity – one can think of it as a bond which never comes due. For example, an asset that pays €10 per year “in perpetuity” is worth €40 if the interest rate is 25%, but only €20 if the interest rate is 50%.
The mathematical proof is by recursion. The infinite sum may be written as $PV = {x \over 1 + r} + {x \over (1 + r)^2} + {x \over (1 + r)^3}\cdots$ Suppose we factor out $1/(1 + r)$ from every term in the right-hand side. This gives us $PV = {1 \over 1 + r}\left[ x + {x \over 1 + r} + {x \over (1 + r)^2} + \cdots\right]$ But notice that the bracketed term is just $x$ plus the infinite series which is equal to PV! So if we substitute $PV$ back in and solve, we get $\begin{aligned} PV &= {1 \over 1 + r}\left[ x + PV\right]\\ (1+r)PV &= x + PV\\ r \times PV &= x\\ PV &= {x \over r} \end{aligned}$

Insurance

Up to now, we’ve just talked about the preferences over lotteries. What about the market for risk? What does a “budget line” in this kind of “state space” mean? What does it mean to trade across lotteries?

Just as the interest rate measures how money may be shifted across time, there are various ways that money can be shifted across states of the world. One common such way is insurance.

Let’s think about a situation of someone with 100 dollars who has a smartphone. If they drop the phone and break the screen, they would have to pay 80 dollars to repair it, leaving them with 20 dollars. This can be represented by the point $(20, 100)$ in a diagram showing state 1 - state 2 space. In the language of exchange, we can think about this as their “endowment;” let’s call this point $(e_1,e_2)$.

Now suppose they can buy some insurance against this loss. In particular, suppose an insurer says: “If you pay me $P = 10$, I will pay you $K = 40$ if you drop your phone and break the glass. It won’t cover all the damage, but it will at least cushion the blow.” If they buy this insurance contract, they will face a different lottery:

If they don’t break their phone, they have $c_2 = e_2 - P = 100 - 10 = 90$
If they do break their phone, they have $c_1 = c_1 + K - P = 20 + 40 - 10 = 50$

Therefore, they are trading $P = 10$ in state of the world 2 for $K - P = 30$ in state of the world 1:

See interactive graph online here.

Notice that you have to pay the insurance company $P$ regardless of the state of the world. So in the good state, you pay $P$ and get no payout. In the bad state you pay $P$ and get $K$. The “market rate of exchange” between “good 2” and “good 1” is therefore ${P \over K - P} = {10 \over 40 - 10} = {1 \over 3}$

A more common scenario is one in which an insurance company offers a price for each dollar of insurance. (For example, if you ever ship something by FedEx, they ask you to estimate its value, and then charge you about $€1$ for every $€10$ of declared value. If they lose the package, they give you the declared value back.) If we let $\gamma$ be the price per dollar of insurance payout, then the price $P$ of buying $K$ dollars of insurance is $P = \gamma K$. In the above example, you bought $K = 40$ for $P = 10$, so $\gamma = {1 \over 4}$. Suppose you buy $K$ dollars of insurance. Then your payoff in the two states of the world would be $\begin{aligned} c_1 &= 20 + K - \gamma K\\ c_2 &= 100 - \gamma K \end{aligned}$ If we solve each of these for $K$, we get $K = {20 - c_1 \over 1 - \gamma} = {100 - c_2 \over \gamma}$ Cross multiplying and simplifying gives us the equation $\gamma c_1 + (1-\gamma)c_2 = \gamma \times 20 + (1 - \gamma) \times 100$ or more generally $\gamma c_1 + (1-\gamma)c_2 = \gamma e_1 + (1-\gamma)e_2$ This is just like any endowment budget constraint, with an implied price ratio is $\gamma/(1 - \gamma)$. In the example above, $\gamma = {1 \over 4}$, so $\gamma/(1-\gamma) = {1 \over 4}/{3 \over 4} = 1/3$.

This price ratio of $\gamma/(1-\gamma)$ can be a little confusing. It’s clear why the “price of good 1” (i.e. one dollar of insurance payout) is $\gamma$. But why is the “price of good 2” $1 - \gamma$? The key is that the premium $P = \gamma K$ is paid in both states of the world. That is, you pay the insurance premium regardless of whether you break your phone or not. If you don’t break it, you’re out $\gamma K$. If you do break it, the insurance company pays you $K$, but you still paid the $\gamma K$ for insurance. So for every dollar of insurance you buy, you give up $\gamma$ in the good state but only get $1 - \gamma$ in the bad state; hence the price ratio is $\gamma/(1-\gamma)$. For example, if $\gamma = {1 \over 4}$, each dollar of insurance you buy costs you $€0.25$; so in the good state of the world you pay $€0.25$, and in the bad state of the world you get the dollar of insurance payoff but you’ve still paid the $€0.25$, so you’re only $€0.75$ better off than you were before; so the “price ratio” is $0.25/0.75 = 1/3$.

Utility maximization and the demand for insurance

Given this budget line, we can maximize utility as usual. Let’s assume your utility from money is $v(c) = \ln c$, and let the probability that your phone breaks be $\pi$. Then your expected utility from a lottery $(c_1,c_2)$ is $\mathbb{E}[v(c)] = \pi \ln c_1 + (1-\pi) \ln c_2$ Therefore your MRS at any lottery is ${\pi \over 1 - \pi} \times {c_2 \over c_1}$ If you can pay $\gamma$ for every dollar of insurance, and face the budget constraint we derived above, the slope of the budget line will be ${\gamma \over 1 - \gamma}$. Therefore your tangency condition will be ${\pi \over 1 - \pi} \times {c_2 \over c_1} = {\gamma \over 1 - \gamma}$ The usual Cobb-Douglas math gets us to $c_1^\star = \pi\left(e_1 + {1 - \gamma \over \gamma} \times e_2\right)$ What makes us buy more insurance (have a higher $c_1$), for a given lottery $(e_1,e_2)$?

A high value of $\pi$, meaning the probability of the bad thing happening
A low value of $\gamma$, meaning insurance is cheaper

All makes sense!

Pricing insurance: an insurance firm’s perspective

Let’s now think about insurance from the perspective of a firm offering an insurance policy. Let’s assume that the probability of the states of the world is known to both the insurance firm and the agent facing the risky lottery, and (importantly!) that there are no issues of moral hazard or adverse selection. Furthermore, let’s assume that the insurance company itself is risk neutral. Then the insurance company’s profits will be: $\text{Profit}=\gamma K - \pi K = (\gamma - \pi)K$ That is, for each dollar of insurance the firm sells, they get $\gamma$ (the price of insurance) for sure, and have to pay a dollar with probability $\pi$.

It’s clear that the firm will always set $\gamma \ge \pi$, because otherwise they would run a loss.

If the firm has market power, it will set a price $\gamma > \pi$ that maximizes this profit given the demand from the consumer. This is a complicated problem and beyond the scope of this course, but we can look at what it means visually: the higher the firm sets the price, the less insurance people will buy.

If the market for insurance is perfectly competitive, then profits will be driven down to zero: that is, we’ll end up with $\gamma = \pi$. This is called the actuarially fair price, and it has an interesting component: if risk-averse consumers face an actuarially fair price for insurance, they will fully insure: that is, consume at the point where $c_1 = c_2$. Why is this the case? Well, for the generic utility function $v(c)$, the consumer’s MRS will be $MRS(c_1,c_2) = {\pi \over 1 - \pi}{v^\prime(c_1) \over v^\prime(c_2)}$ Along the “line of certainty” where $c_1 = c_2$, therefore, the second fraction is just 1, not matter what the utility function is; so whenever $c_1 = c_2$, the MRS is $\pi / (1 - \pi)$. And if $\pi = \gamma$, it follows that the MRS is also $\gamma / (1 - \gamma)$, which is the price ratio, so the tangency condition will always occur along the line of certainty, meaning that a risk-averse consumer will always fully insure.