Hotelling Competition on Quality in the Health

Care MarketMarcello Montefiori

• “Health” is both horizontally and vertically differentiated.• Payment is financed by tax.• Analysis is done on a standard Hotelling line.• Hospitals can choose only quality• The author finds Cournot and Stackleberg equilibria and then

compares them• He analyzes a dynamic extension of the Stackleberg

equilibrium.• He also introduces uncertainty into the model.

The Stackleberg Equilibrium

The Consumer Utility Function

• The Hotelling line is of unit length• Depending on the hospital the consumer chooses, his utility is given by • For the indifferent consumer:

• If .• If demand for hospital A converges to 0.• This makes sense intuitively. • The model assumes a reservation utility .• We assume that at least one of the firms gives a utility greater than

(we assume ). • Taking into account the P.C., we write the demand equation as follows

The Consumer Utility Function

• Hospital’s cost function is given by • Since we have a specific functional form for demand, we can re-write

the cost function as

• Thus, the hospital’s optimization problem is • • s.t.

The Hospital’s Behavior

• We assume that the two hospitals compete on quality only.• The best response functions are• and the solution is • Now, taking into consideration and the profit function with dummy

variables (to account for reservation utility), the function to be optimized is

The Cournot Equilibrium

• The minimum symmetric equilibrium to cover the entire market is The Cournot Equilibrium

• The leader (a) optimizes .• s.t. and .• Follower’s (b’s) best response function is given by .• Solving the problem, we get and • Since , b gets a greater portion of the market.• Thus, in this scenario the leader is the loser.• This counter intuitive result is because the follower can always beat the

leader on quality just enough to get more of the market without increasing costs too much.

The Stackleberg Equilibrium

Comparison Between Cournot and Stackleberg

• Social Welfare in Cournot is more efficient than in Stackleberg• In both cases, quality sums up to

• The equilibrium outcome is not dynamically stable. • When the leader moves again, he will react to the follower’s quality.• They will keep reacting to each other and if the game goes on for long

enough, their qualities converge. In the long run it reaches the same equilibrium at Cournot.

Extending the Stackleberg Equilibrium

• Patients are generally unable to observe true quality. Hence, the model is now modified to work based on the consumer’s expectations• Perceived quality is given by .

• Firms maximize

• If average quality is equal to true quality and equilibrium is symmetric

Uncertainty

• We can see that even if both hospitals provide the same quality, we can they may attain different profits.• This is because people prefer less variance for the same mean.• In the case of Stackleberg, a miniscule incremental increase in quality s is

not sufficient from the firm that is reacting.• This leads to a faster convergence to Cournot than in the deterministic

case.• Thus, hospitals always have an incentive to reduce variance.

Uncertainty

• Relates the previous literature of horizontal and vertical differentiation to a real world example.• It eliminates price from the model, and allows us to focus only on quality.• Most of the results are intuitive.• The Stackleberg result is counter intuitive but the explanation is solid• The idea of a dynamic Stackleberg game makes a lot of sense.

Convergence makes the analysis all the more interesting.• The author talks about consumer uncertainty, which is something he had

not discussed yet.

Strengths

• Functional form is too specific. • The assumption that the hospitals are already maximally

differentiated in terms of location is very strong.• Demand is uniform (again).• Transaction costs do not depend on direction and are affect the

consumer linearly.

Weaknesses