# Why are firms taken to be profit-maximizing? Shouldn't that make them risk-neutral?

Intro texts normally explain that insurance firms (casinos, etc.) "work" by diversifying risk from many clients. Unsaid, then, seems to be that risk is bad for both firm and client.

But why should a firm even need such diversification, if they merely seek to maximize profits? Likewise why would a firm ever itself be a purchaser of insurance, or spend resources trying to "manage" risk? This seems to be a rather deep oddity in the basic picture we're taught from square one.

Why would a [risk-neutral] firm need to diversify if all it wanted to do was maximise profit?

Suppose there is a risk-neutral firm that has two strategies it could follow: risky and safe. The safe strategy (e.g. holding a diversified portfolio) yields a profit of $1$ every year forever. The risky strategy (e.g. not diversifying) yields a payoff of $2$ with (independent) probability $x$ each year. With probability $1-x$ a risk-taking firm goes bankrupt and earns no more money ever again. The firm discounts the future at rate $\delta$.

The lifetime profit of the safe strategy is $$\sum_{t=0}^\infty\delta^t=\frac{1}{1-\delta}.$$ The lifetime profit from the risky strategy $$\sum_{t=0}^\infty 2x^t\delta^t=\frac{2}{1-x\delta}.$$ The safe strategy yields a higher lifetime profit if $$x<\frac{2\delta-1}{\delta}\iff \delta>\frac{1}{2-x}.$$

In words, even though

• the firm is risk-neutral and
• the one-shot profit from the risky option (e.g. not diversifying) is higher,

the long-run profit from the safe option (e.g. diversification) can be greater over the lifetime of the firm.

In this example, the safe option maximises lifetime profit when (a) the probability of risky option leading to bankruptcy (and hence the loss of all future profit) is high or (b) the firm values the future highly relative to the present.

• (+1). Just for completeness, perhaps you should mention that the above incorporates the assumption that each time period is independent in probability from all the others (and so we get the lifetime profit form the risky strategy as written). Dec 18, 2016 at 19:00
• @NowIGetToLearnWhatAHeadIs Yes, it's fixed. Dec 19, 2016 at 8:27

Firms maximize profit, not expected profit. If they want to take a lower guaranteed value than the expected value because they are risk averse, then they're still maximizing profit based on their preferences and/or constraints. You're right that they wouldn't be maximizing expected profit. Profit could also have diminishing marginal returns.

To add a little more to my answer, given the bar has been raised [:^)], I'd like to give a short example from Mas-Colell, Whintson, and Greene's book, that I think might be helpful.

Imagine a risk neutral firm that maximizes expected profit, no matter how risky the expectation is. Is this the same as maximizing profit? If the firm's "preference" is risk-neutral, surely it seems to be the case. Let us suppose there is uncertainty in prices, and production occurs after resolving that uncertainty.

Say the firm has two scenarios they can take:

• Prices are uncertain.
• Prices are equal to the expected price vector.

A firm maximizing expected profit will actually prefer the first scenario to the second.

Say $\pi(\cdot)$ is the profit function and $F(\cdot)$ is the distribution function for the prices. Since $\pi(\cdot)$ is convex in price, by Jensen's inequality:

$$\int \pi(p) \ \text{d}F(p) \geq \pi\left(\int p \ \text{d}F(p)\right)$$

But the left hand side is expected profit from uncertain prices, and the right hand side is the profit under expected prices.

So you can reflect on whether this is how firms really behave. If a firm was faced with the possibility of selling their portfolio for 1 billion dollars, or losing 1 billion dollars, based on a coin flip, would they go through with this bet? Or would they sell off the portfolio for 0 dollars?

• You say "returns," but just what are said returns? For consumers/households, we have a device, the utility function, whose purpose is to characterize just this sort of behavior--the choices made (preferences had) with respect to risk and other factors. But this device is not used to characterize the behavior of firms. If they do things like display a certain amount of risk aversion, then why not? Your talk of "returns" and "maximizing profit based on their preferences" seems to underscore its glaring absence! Dec 18, 2016 at 18:30
• Firms are run by people with preferences. Dec 18, 2016 at 22:17
• But to address the more literal models mathematicians use, instead of the specific phrasing I use, a risk-neutral firm may still act to hedge against risk (as Ubiquitous's answer so skillfully illustrates). Dec 18, 2016 at 22:20

There are really three reasons why a risk neutral firm would buy insurance.

One is that the insurance market is under estimating the riskiness of an insurance product. In other words a firm might believe they have a 10% chance of losing \$1m but there are insurance companies willing to wear that risk for less than$100k. This can happen if insurance products are broadly defined and there's some specific niche firm whose risk of loss is higher than the broadly defined product.

The second is when the loss associated with an event is more than the business can pay. Rather than equations for this let's look at a simple everyday example. An electrician will have liability insurance because if they manage to burn someone's house down, odds are they simply do not have the funds or financing to cover a new house. That is to say, burning a house down isn't -300000 it's bankruptcy.

There's a third reason which balances out the short comings of the second. Of course some firms might decide that maximizing profits until bankruptcy is the correct course of action because the person running the firm isn't harmed when the firm is bankrupted. This is an agency problem but to counteract these decisions many people when they hire a service will only do so on the condition that that firm is insured.

I have a different take on the question:

Suppose a representative shareholder owns $$n$$ firms. The shareholder owns equal shares of the $$n$$ firms, and the firms are identical and independent. For simplicity, assume the shareholder has starting wealth one, and their preferences are an increasing function only of ending wealth. Firm $$j$$ has rate of return $$r_j$$ (e.g. $$r_j=1.05$$ would be a 5% rate of return), which is a random variable for each $$j$$. This implies that ending wealth for the representative shareholder is

$$\sum r_j / n$$

The expected utility for the representative shareholder is therefore

$$E[u(\sum r_j / n)]$$

By the assumption that the firms are identical but operate independently, I mean that the rates of return $$r_j$$ are independently and identically distributed; for notational simplicity, suppose they are distributed the same as the random variable $$r$$. For $$n$$ large, by the law of large numbers, $$\sum r_j / n$$ is approximately equal to $$E[r]$$. Thus

$$E[u(\sum r_j / n)] \approx E[u(E[r])] = u(E[r])$$

the last equality results because $$E[r]$$ is non-random. Thus, to maximize the shareholder's utility, the firm maximizes expected returns, $$E[r]$$.

To relate returns to profits, note that

$$r_j = (\pi_j + (1-\delta) K_j) / K_j$$

where $$K_j$$ is starting capital, $$\delta$$ is depreciation, and $$\pi_j$$ is profit. If $$\pi_j$$ is the only thing that is random in this expression, then maximizing $$E[r_j]$$ is the same as maximizing $$E[\pi_j]$$, that is, expected profits.

In the model I proposed here, the firm maximizes expected profits, and behaves in a risk-neutral fashion. The answer by @Ubiquitous, though insightful, is not a violation of the model I proposed here. In their answer, the firm faces a risky lottery versus a risk-free lottery, and the risk-free lottery has a higher expected value. Maximizing expected profits of course does not mean always choosing the more risky option; if by buying insurance the firm increases expected profits then it would do that.