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(Edited) Here is the working paper. Enjoy. Rothschild, M., and J. E. Stiglitz, "Equilibrium in Competitive Insurance Markets," Technical Report No. 170, IMSSS Stanford University, 1975. https://drive.google.com/file/d/1MXb3OcOQc_lxNYC4CzsNTTax9_3Q_4UN/view?usp=sharing


6

Hidden information concerns characteristics that are unobservable by one side of the market. For example, a consumer's willingness to pay, a worker's productivity, the quality of a used car all fall under this category. The characteristics in question are typically assumed to be fixed or very costly to modify. Moral hazard concerns actions that are ...


3

I recommend reading 'Informational Equilibrium' by John Riley (1979). It discusses this setting more generally. How the "insurance model" of Rotschild and Stiglitz is a special case of this is presented on pg. 335, and then Theorem 3 on pg. 343 is what you need. The proof is not trivial, but - in my opinion - well presented.


3

We are given that $u$ is increasing and concave, and $u(0) = 0$. This implies that $\dfrac{u(t)}{t}$ is decreasing in $t$, and also, $\dfrac{u(t)}{t} > u'(t)$ for all $t$. Nicole's maximization problem : \begin{eqnarray*} \max_{x} \ q(x)(w-t(x))\end{eqnarray*} FOC : $q'(x)(w-t(x)) = q(x)t'(x)$ Suppose $x_N$ solves Nicole's problem. Therefore, it ...


2

A critical point here is to note that the total number of tickets is not set a priori. This is good, because it makes the expected utility function non-linear in $t_i$, and so permits us to proceed (half-way). Writing $S$ for the total number of tickets and $S_{-i}$ for the total number minus the purchases of bunny $i$, and simplifying, the expected ...


2

Suppose that $r'(\theta)>0$. The following figure should make clear that saying $r(\theta)\leq w$ is equivalent to saying that $\theta\leq r^{-1}(w)$ (where $r^{-1}(\cdot)$ is the inverse of $r$): So we can rephrase your question as 'why should $E(\theta|\theta<r^{-1}(w))$ be non-linear?' Let's calculate this expectation (assuming that $\theta$ is ...


2

I think its not a complex issue: a) You need to keep the high type agent from pretending to be the ow type agent, so you give him an extra compensation from showing his high type. But this nice deal pays him more than necessary to keep him in the game so his IR is more than satisfied, i.e. not binding. b) Similarly, you will make it painful to show that ...


2

So this seems to be a known issue. Quoting from the Wilson article of 1980, The Nature of Equilibrium in Markets with Adverse Selection: Using a variant of Akerlof's model of the used car market, we examine the equlibrium of the model under three distinct conventions: (1) an auctioneer sets the price; (2) buyers set the price; (3) sellers set the price. ...


2

In direct mechanism agents directly report their preferences (preferences are observable). In indirect mechanism agents don’t report their preferences directly. Preferences can be observed only indirectly through signals or behavior. By Revelation Principle if some outcomes can be implemented in indirect mechanism they must be also implementable in the ...


1

Think about an auction, where the designer is selling good and trying to sell it to the person that values it the most while collecting as much revenue as possible. A direct mechanism means that the seller asks buyers for how much they value the good and based on that decides who gets the good and how much they pay. Suppose that the designer uses a second-...


1

It is important to remember that figures are not proofs. Minor mistakes in the figure are hard to detect but can change the outcome. Algebra is much more reliable. Right now in your figure the low risk type consumers have indifference curves that are upward sloping somewhere. It seems though that is not critical here. The visualisation of subsidies seems ...


1

First: Pareto efficiency, optimality and dominance don't all describe equilibria. Loosely, they mean the following: A Pareto-efficient outcome is an outcome that satisfies the condition that no agent is worse off, and at least one agent is better off, as a result of the change. A Pareto-dominant outcome is an outcome that is both Pareto-efficient and ...


1

First, it's important to be clear about what Rochet proves. In theorem $1$ of Rochet (1987), he shows that cyclical monotonicity is equivalent to the implementability of a mechanism for general environments (i.e. arbitrary allocation/type spaces). (See the statement of theorem $1$, or a standard text such as Borgers (2015), for the definition of cyclical ...


1

In order to talk about whether the competitive equilibrium is also a Nash equilibrium, you first have to properly define a game. For instance, is the buyers setting the prices or the sellers or is there bargaining. How do they meet, are there search costs etc. And importantly, what is the order of events! Your reasoning implicitly assumed that buyers would ...


1

In the second case (information asymmetry) there is only a single market. This may be the market for the product with unknown quality or in case of adverse selection it may be a market for the bad quality product alone. Assuming there is no adverse selection the reservation price of a consumer w.r.t. the unknown good is the expected value of his reservation ...


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