Questions on topic 4: Diamond-Dybvig model

The followings are the questions asked by one of your classmates. You can download my answers here. Or go to http://groups.google.com/group/economaru/files and find file Questions_DiamondDybvig.zip.

1. Under the Bank Run model, when we analyze the competitive equilibrium. Why the t=0 price of period 1 good is 1, period t=1 and t=0 price of period 2 good are 1/R? How can we figure out there numbers since the paper did not explain and show calculation on them? What method should I use (assume utility is equal with Autarky case??)?

2. For the social planner allocation, after we try to maximize the social utility function we will have U'(c1*) = RU'[......can't type here...]. So, what can we imply from this ugly equation? Since c1* is still being inside the utility function, so, how can we know the exact c1*? What are the conditions for c1* and c2*? (the paper use another equation so it does not really help)

3. On the Suspension of Convertibility model, why the fraction f^ for suspension must exceed 1/r1? (how could they get this number?) Also, to demonstrate the model, the paper say that let r1=c1* and let f^ is a set of [t, [(R-r1)/r1(R-1)]. Why the maximum boundary of f^ must be at [(R-r1)/r1(R-1)]? If this is a correct number, so can we put this in the V2 equation to show that V2 > V1?

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And here are some questions from Hedhood in the chatter...

First, I dont quite understand about the integral and the outcome about the relative risk aversion. What is the difference between the absolute risk and relative risk aversion. and Second, why C1 and C2 that are allocated by policy planner can't be equal?????

Let me recall briefly the steps involved when we establish the social planner's solutions. Equation (3) on page 7 of the lecture note gives us the social planner's optimal allocation, which is essentially u'(c1)=Ru'(c2), only that I have substituted in the budget constraint for c2.

The key question was, how are c1 and c2 under social planner's allocation different from those under Autarky? i.e. can c1=1, and c2=R? If yes, then we must have u'(1)=Ru'(R), as we must obey equation (3). If not, then perhaps the society wants people to be exposed to less risk, so their consumption is not too dependent on their type. In this case, we're interested in finding the conditions under which c1 is bigger than 1 and c2 is less than R, so that indeed people will be exposed to less risk.

If c1 is greater than 1 and c2 is less than R, then it must follow that Ru'(R) is less than u'(1) (using dimishing marginal utility). In other words, we have some risk sharing (or less exposure to risk), if Ru'(R)-u'(1)<0. Now Ru'(R)-u'(1) can be written as an integral as shown in the lectures...we just integrate 1 with respect to a variable ru'(r), and then evaluate the integral by the upper limit R and lower limit 1.

When you play with this integral, you find that it is negative whenever ru''(r)/u'(r)<-1 for all r. This expression ru''(r)/u'(r) turns out to be what microeconomists use as a measure of relative risk aversion. So as long as agents are risk averse according to the relative risk aversion definition, our social planner will find it optimal to do risk sharing.

Intuitively, greater relative risk aversion means agents are risk averse to percentage changes (rather than absolute changes) in their wealth. Namely agents prefer a certain percentage change in wealth to a lottery which gives the same expected change, but with uncertainty attached. Absolute risk aversion, as the name suggests, measures the degree of being risk averse to absolute changes in wealth. It is mathematically defined by u''(r)/u'(r), so it is not scaled by the consumption level r.

Lastly, we cannot have c1=c2 as the optimal social planner's outcome because this would imply from equation (3) that u'(c1)=Ru'(c1), which cannot be true because R>1.