Competitive And Unfair Insurance Flashcards
What do we assume for competitive insurance
Perfect competition
Perfect competition assumptions (5)
Free entry/exit
Homogenous goods
Perfect info
Large amount of firms/buyers.
And therefore profit=0
How can this 0 profit be proved mathematically
Profit of insurance firm is the difference between premium collected yK (price of insurance x quantity) , and the payments made in case of accident πaK.
Then include no accident is shown by probability of (1-πa) and cost is 0.
So…
Profit= yK - πaK - (1-πa)0
= yK - πaK
Then we can set profit = 0
0 = (y-πa)K
y=πa which is fair insurance
When is insurance fair
y=πa
The price of one unit of wealth insured = probability of accident
Then sub the fair insurance equation into the optimal contingent consumption plan. What can we find?
Sub our fair insurance equation (y=πα) into Y
y/1-y = πa/1-πa = πα MU(Ca) / πna MU(Cna)
Rewrite πna as 1-πa
πa/1-πa = πα MU(Ca) / 1-πa MU(Cna)
Then we find MU(Ca) = MU(Cna)
Which shows marginal utility of income must be the same in both states.
So how much fair insurance does a risk-averse consumer buy in a competitive market?
As MU(Ca)=MU(Cna),
Ca=Cna
When insurance is fair, a risk averse consumer will buy full insurance.
What does profit expression look like if insurance is unfair, do insurers make profit?
(y-πa)K > 0
So insurers make positive profit. Not in a PCM.
This means y>πa……
This means y>πa… what does this mean
y/1-y > πa/1-πa
Rational choice requires
y/1-y = πα MU(Ca) / πna MU(Cna)
Meaning MU(Ca) > MU(Cna) , so for a risk-adverse consumer Ca<Cna , (convexity and DMU)
Meaning a risk-adverse consumer will buy less than full insurance if it is unfair (i.e economic profit>0)
So in fair insurance and unfair insruance, what does a risk-adverse consumer buy?
Fair = buys full insurance as y=πa and Ca=Cna
Unfair = buys less than full insurance as y>πa so Cna>Ca
