21.4 Insurance

In each of these applications of exchange theory, we’ve analyzed preferences and the interpretation of the slope of the budget line.

• In the chapter on labor supply, the two goods were time and money. Our preferences were over leisure and consumption (and could be any arbitrary preferences; we didn’t assert a specific structure). The slope of the budget line represented the wage rate: that is, the rate at which the market allowed us to sell leisure to buy consumption.

• In the chapter on intertemporal choice, the two goods were present consumption and future consumption. The preferences we analyzed were of the form \(u(c_1,c_2) = v(c_1) + \beta v(c_2)\) where the function $v(c_1)$ represented the utility from money within a time period. The slope of the budget line was derived from the interest rate: that is, the rate at which the market allowed us to buy future consumption with our present income via savings, or present consumption with our future income via borrowing.

In this chapter until now we’ve been concerned with preferences over risk. In an analogous way to intertemporal choice, we asserted that we were trying to maximize our expected utility, \(\mathbb{E}[u(c)] = \pi u(c_1) + (1-\pi) u(c_2)\)As with the intertemporal consumption, there is a tension between the utility within a state of the world, given by $u(c)$, and between states of the world, given by the probability $\pi$.

Let’s now think about how these preferences play out in a market scenario, by developing a “budget line.”

The insurance budget line

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:

(diagram forthcoming)

Note that the rate of exchange here is \({P \over K - P} = {10 \over 40 - 10} = {1 \over 3}\) 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 $K$.

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 $u(c) = \ln c$, and let the probability that your phone breaks be $\pi$. Then your expected utility from a lottery $(c_1,c_2)$ is \(\mathbb{E}[u(c)] = \pi \ln c_1 + (1-\pi) \ln c_2\) 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 $u(c)$, the consumer’s MRS will be \(MRS(c_1,c_2) = {\pi \over 1 - \pi}{u^\prime(c_1) \over u^\prime(c_2)}\) 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.