In the paper, we investigate the pricing behavior of a risk averse monopoly. Since the focus is on the risk averse attitude of the firm, we ignore cost in our model. Demand is considered to be stoch astic demand: as price decreases, the expected number of customers increases, but it has a varia tion. Although demand is uncertain, it relates to the aggregation method of individual demands and the individual demand has the usual form. In our framework a risk neutral (or profit maximizer) monopoly does not change the product’s price as the number of clients increases. On product markets the risk averse monopoly with DARA utility function always increases the price as the number of clients grows, but in insurance markets the implication can be the opposite: the price of insurance may decrease as the number of clients increases.
Stochastic demand is not unknown in the economic literature. In management science models it is well-studied (see e.g. [
It is usual in microeconomic theory that increased demand increases market price. We can think of Marshallian cross for instance. There are known exceptions, such as the Giffen goods. Hoy and Robson [
Pricing of a monopoly differs from pricing of a competitive firm. It can happen that a monopoly reduces its price due to the increased demand, but the decrease is related to the costs (natural monopoly for instance) or the increase of the demand is not uniform. In our model we would like to focus on risk aversion of the decision maker, thus we ignore costs. On the other side, the increase of demand is uniform.
We provide a simple framework for investigating the pricing of a risk averse monopoly. We distinguish between product market and insurance market. In the product market increased demand will cause increased monopoly price, but in the insurance market it can happen, that increased demand results in reduced price.
In the insurance economic literature there are usually two kinds of models: insurer has many (or many type of) contracts, but the insurer is risk neutral (e.g. [
Insurance markets are dominated by a few (or not many) companies, which remind us to the market form of oligopoly. It depends on the situation whether the competitive economy or the monopoly is a more realistic mar- ket form. The presence of a monopolistic insurance company is accepted in the literature (see [
In Section 2, we present our theoretical model for both cases of product market and insurance market. In Section 3, we conclude the results.
In the model we consider a monopoly. It sells its product to (potential) customers. Cost is considered as sunk cost which does not affect the optimal price. The monopoly can fulfill the demand of arbitrary number of cus- tomers. An illustrative example is the case of a software vendor: the firm can sell any copies of a program with- out cost. Also an insurance company can supply insurance to many clients.
Customers can decide to buy the product or not, but they cannot buy fraction of the product. e.g. it is im- possible to buy a half of a software. In the insurance market it could be possible to buy fraction of the coverage, but we exclude this case. The clients’ decision is to buy full coverage or not to buy insurance. In the mo- nopolistic market the insurance company sets the contract’s properties and it is not interested in fractional cove- rages.
Each client has a reservation price (which differs from client to client), but the monopoly does not know the exact price, it knows only the distribution of the reservation prices within a greater community. So let
where
The monopoly is risk averse and maximizes its expected utility. Its behavior can be described with a concave (risk averse) utility function
In case of a product market we can assume that
The monopoly’s expected utility at price
where
We assume that the expression
If the monopoly is risk neutral (utility function is linear) then the expected utility becomes simple:
In Theorem 1 we prove that a risk averse monopoly applies a lower price than a risk neutral monopoly.
Theorem 1. A risk averse monopoly sets a lower price than a risk neutral firm.
Proof. Let us see the expected utility’s (1) derivative with respect to price:
A risk neutral decision maker sets a price
which is negative due to concavity of
The interpretation of Theorem 1 is that a risk averse monopoly is satisfied with a lower price which ensures higher probability of selling.
What happens if the number of clients increases? The risk aversion solely is not enough to change the mo- nopoly’s price. Let us consider e.g. exponential utility function
The optimal price is the same for all
If we assume decreasing absolute risk aversion (DARA), which is a more realistic assumption, we can state that the optimum price increases as market size increases:
Theorem 2. Let us consider a risk averse monopoly with a DARA (decreasing absolute risk averse) utility function. The monopoly determines a higher price in case of
The proof can be found in the Appendix.
From Theorem 1 and Theorem 2 we can have the intuition that the optimal price converges to the risk neutral price as the number of clients tends to infinity. Unfortunately this intuition is false.
Example 1. Let
If we set
In case of the product market there is only one source of uncertainty: whether the client buys the product or not. In case of insurance market there is another source of uncertainty: whether a claim occurs or not. For simplicity we investigate a two state model: a claim
From classical risk theory we know that
The expected utility of an insurance company:
The expected utility becomes simple for a risk neutral monopoly again:
For an exponential utility function, there is a closed formula for the expected utility:
The optimal price for an exponential utility function
The risk averse insurance monopoly takes two impacts. It would set a lower price than a risk neutral company to sell more contracts as in the case of a product market. On the other hand, we also know from the classical risk theory that a risk averse firm applies higher price for insurance than a risk neutral. So there are two opposite effects, we could not decide on whether the price will be higher or lower than the risk neutral price (we can give ex- ample for higher and lower prices as well).
From risk theory we know that the average claim amount disperses less for a greater risk community, al- though greater extreme losses can also happen. A greater risk community can be advantageous for a risk averse decision maker, and as market size increases this advantage may exceed the profit loss from a lower price. In Example 2, we demonstrate the previously described situation: the insurance company can decrease its in- surance’s price as the demand (number of clients) increases. It is an interesting situation: the increase of market size is a common interest for insurance company and insureds. In product market the interest of the monopoly and the clients are always in conflict.
Example 2. Let
As we can see, the utility function is a mixture of a risk neutral part (
the price for exponential utility function (CARA price hereafter). If
then as n grows, the common price will be closer to CARA price, if
Let parameter q and K take values such that
For the second condition it is enough if
For numerical check: let
and
# of Client | |||
---|---|---|---|
1 | 5.670 | 6.608 | 6.171 |
2 | 5.625 | 6.557 | 6.261 |
3 | 5.573 | 6.500 | 6.364 |
4 | 5.514 | 6.438 | 6.460 |
5 | 5.452 | 6.374 | 6.544 |
10 | 5.162 | 6.085 | 6.809 |
20 | 4.866 | 5.794 | 7.028 |
50 | 4.636 | 5.569 | 7.209 |
100 | 4.558 | 5.486 | 7.300 |
200 | 4.545 | 5.444 | 7.346 |
4.545 | 5.050 | 6.060 | |
5.834 | 6.828 | 8.825 | |
4.407 | 5.400 | 7.395 |
We can make two important remarks: let parameter
Let parameter
In the paper, we presented a microeconomic framework, in which a risk averse monopoly’s behavior can be investigated. We proved that in a product market a risk averse monopoly applies a lower price than a risk neutral (profit maximizer). If the risk aversion decreases with wealth (DARA), the market price will increase with the market size.
In the insurance market the price can decrease or increase with the market size even for utility function with DARA property. We have given numeric example for both cases.
I would like to thank Aegon Hungary insurance company, Insurance Education and Research Group, Corvinus University of Budapest (BOKCS) and Association of Hungarian Insurance Companies (MABISZ) for supporting my research. I thank Miklós Pintér, Péter Bíró, Bianka Ágoston and Tamás Polereczky for their suggestions and remarks. Naturally, all errors are mine.
Lemma 1. For monopoly’s expected utility the following recursive relationship holds:
Proof. We start from the recursive formula for binomial numbers:
For the sake of simplicity we introduce the following notations:
And
It is easy to check that the recursive relationship (6) also holds for expressions
Lemma 2
Proof.
where
By the Cauchy’s mean value theorem there exist
Now one thing is missing:
of utility functions with DARA property is also a utility function with DARA property (see [
we define
is equivalent to the decrease of
Proof of Theorem 2. The first order condition of
Which can be rearranged as
We investigate the sign of
mine the sign of
Using the recursive relationship (6):
Applying Lemma 2 and using some algebra we can state, that
Which actually means, that