Microeconomics Theory Exam Help

Microeconomic Theory Exam Help
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Microeconomic Theory
Problem
1. Show that the monotone likelihood ratio condition (f(x)/g(x) increasing in x, where x is a real
number) implies first-order stochastic dominance (F(x) G(x) for all x).
2. Let = 1 or 2 denote the state of nature in a two state world. Let p0 = Pr( = 1). Let f(p)
be an arbitrary distribution on the unit interval such that pf(p)dp = p0. Show that f(p) can be
construed as a distribution over posterior probabilities of p, stemming from an experiment
with outcomes y, a prior probability p = p0 and two likelihood functions p(y| 1) and p(y| 2).
[It may be easier to solve the problem if you assume that f(p) consists of two mass points so
that the experiment will have only two relevant outcomes, say, yL and yH.]
3. An agent has to decide between two actions a1 and a2, uncertain of the prevailing state of
nature s, which can be either s1 or s2. The agent’s payoff as a function of the action and the
state of nature is as follows:
The prior probability of state s1 is p = Pr(s1).
a. Give a graphical representation of the agent’s decision problem as a function of the
probability of state s1. If p = (.4), what is the agent’s optimal decision?
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b. Let the agent have access to a signal y before taking an action. Assume y has two possible
outcomes with the following likelihoods
c. Show that, irrespectively of the prior, the information system 1 = ½ and 2 = 0 is preferred
to the information system 1 = ½ + ½ and 2 = , where and are any numbers between 0
and 1. (Hint: show that latter system is garbling of former).
4. *An agent can work hard (e = eH) or be lazy (e = eL), where eH > eL > 0. There are two profit
levels, x1 and x2; x1 < x2. Hard work makes the high profit level more likely. Specifically, Prob(x =
x2) is fH if the agent works hard and fL < fH, if the agent is lazy. The principal can only observe
realized profits x. The principal is risk neutral and the agent is risk averse with preferences u(w) -
e. The utility function u is strictly concave and increasing.
a. Assume that the Principal’s reservation utility is 0; that is, the Principal has to be assured a
non-negative profit. Show how to solve for the Pareto Optimal action and incentive scheme s(xi)
= si, i = 1,2, where si represents the payment to the agent if the outcome is xi.
b. Suppose that the agent is given the opportunity to choose a third action eM with the features
that Prob(x2|eM) = ½fH + ½fL and eM > ½eH + ½eL. Can the Principal implement eM?
c. Suppose that the second-best solution in Part a is such that it is optimal to implement eH. Let
the agent have access to the action eM described above, but assume now that eM < ½eH + ½eL.
Will the solution to Part a stand once the agent is given access to eM?

5. Consider the risk-sharing problem we saw in class. Show that the Pareto frontier is concave
(in expected utility space), or, equivalently, that the feasible set of utility pairs is convex.

Solution
Definition 1 MLRP: g(x) is increasing in x.
Definition 2 FOSD: F (x) ≤ G(x)∀x
Solution 2. You want to design an experiment, that is a random variable Y (that takes value in
[0;1], for simplicity, and is characterized by the joint distribution p(θ,y) ) such that the
distribution of posteriors generated by this experiment is given by f(p). In the “experiment” the
posterior will be given by Pr(θ1|y) and the probability that this posterior arises is simply Pr(y).
In the statement, the probability that posterior p arises would be given by f(p). Therefore, we’d
like to take Pr(θ1|y)=p and Pr(y)=f(p) (for y=p) which directly defines p(θ1,y)=pf(p) ;Pr(θ2|y)=1-
p ; p(θ2,y)=(1-p)f(p). Does such a random variable exists ?

Therefore such an experiment exists (we can construct a random variable Y such that the joint
distribution of (Y,θ) is p(θ,y) since p is non negative and sums up to 1) and the prior is given by
Pr(θ1)=∫ p(θ1,y)dy =p0 as wanted We can then define the likelihood functions using Bayes rule
and we have Pr(y|θ1)= Pr(θ1| y)Pr(y)/Pr(θ1)=pf(p)/p0 and similarly for Pr(y|θ2) . You can then
directly verify that the experiment defined by the outcome y, these likelihood functions and the
prior p0 generate posteriors distributed according to f.
Solution 3
1. We are given u (a, s) therefore we can derive v (a, p) :
The upper envelope of v (a, p) will be the V (p) (it indicates the maximum expected utility that
the agent can reach if faced with a probability of s1 equal top):

At p = 0.4 the optimal decision is a2 since:
2. We are given the likelihood matrix:

The probability of observing the two signals is:
Let’s calculate the posterior probabilities:
• If λ1 = λ2 = the information system has no value because the same signal y1 is as likely to 2
appear in the two states of nature. Whenever λ1 = λ2 the information system has no value. 1
• If λ1 = and λ2 = 0 the the likelihood matrix is the following:
Posteriors in this case are:
And probabilities of the two signals are

When observe y1:
therefore the optimal choice as a function of the signal is:
Hence:
When observe y2:
therefore the optimal choice as a function of the signal is:

hence:
We can know calculate VY as:
We can now calculate the value of information as:
3. We know that the Blackwell theorem gives general conditions under which one information
system ¡ ¢ 1 1 is preferred to another. So we just have to prove that the information system λ1
= α + β, λ2 = β 2 2 ¡ ¢ 1 is a garbling of the information system λ1 = , λ2 = 0 . 2 We have to find
a Markov matrix M:

such that the following relationship between the likelihood matrices holds:
You can verify that such matrix M exists and is equal to the following:

Microeconomics Theory Exam Help

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