Intermediate Microeconomics UA10
Practice Questions for Final Exam 2025
1. Consider a risk–neutral worker picking between two jobs A, B after a period of search. The jobs both have uncertain wages.
(a) For job A, high and low wages wH
A = 40, wL
A = 20 are equally likely.
(b) For job B, high and low wages wH
B = 50, wL
B = 25 are equally likely.
Wages are independent across jobs: learning about either job teaches nothing about the other. The worker maximizes expected earnings less total costs of search. Both jobs are available whether searched or not. The exact wage for either job can be identified for cost k > 0 and both jobs can be searched sequentially before taking a job if the worker pays search costs twice. Identify the optimal strategy of search and job choice as it depends on the level of k > 0.
2. Consider a binary-state world with µ1 = 0.5 the prior probability of state ω1 with two possible signals. In state ω1 the signal is
P(s1 | ω1) = 3/4
, P(s2 | ω1) = 1/4
,
while in state ω2 the signal is
P(s1 | ω2) = 1/2
, P(s2 | ω2) = 1/2
.
(a) Compute the unconditional probabilities of each signal, P(s1) and P(s2).
(b) Use Bayes’ rule to compute the posterior beliefs following each signal.
(c) Verify Bayes’ consistency by checking that the average posterior belief is the same as the prior belief.
3. Consider a two–state decision problem with two actions a, b with the following state–dependent payoffs:
u(a, ω1) = 3, u(a, ω2) = 1;
u(b, ω1) = 0, u(b, ω2) = 2.
(a) Draw the figure illustrating the expected utility of both actions as a function of the belief p1 that the state is ω1.
(b) Compute and draw the maximized expected utility Uˆ(p1) for all levels of p1, as defined by the upper envelope in the figure.
(c) Use the figure to illustrate the value of updating the probability of state ω1 from prior µ1 = 0.4 to the pair of posteriors γ1
L = 0.2, γ1
H = 0.8. (No computations needed.)
4. Consider a two–state decision problem with two actions a, b with the following state–dependent payoffs:
u(a, ω1) = 3, u(a, ω2) = 0;
u(b, ω1) = 0, u(b, ω2) = 1.
This is a different payoff structure from Question 3; the geometry is similar but the EU lines differ.
(a) Draw the figure illustrating the expected utility of both actions as a function of the belief p1 that the state is ω1.
(b) Compute and draw the maximized expected utility for all levels of p1, as defined by the upper envelope in the figure.
(c) Use the figure to illustrate the value of updating the probability of state ω1 from prior µ1 = 0.5 to the pair of posteriors γ1
L = 0.2, γ1
H = 0.9. (No computations needed.)
5. Consider the two–action, two–equiprobable–state tracking problem
u(a, ω1) = u(b, ω2) = 1;
u(a, ω2) = u(b, ω1) = 0.
(a) Show that the mistake rate is reduced from 50% to 20% by the pair of posteriors γ1
L = 0.2, γ1
H = 0.8, and illustrate this in the figure showing the expected utility of both actions as a function of the belief p1 that the state is ω1.
(b) Now consider a different form. of learning that results in posteriors γ˜1
L = 0.4, ˜γ1
H = 1. Compute the resulting mistake rate, and illustrate in the same figure.
6. Consider again the two–action, two–equiprobable–state tracking prob-lem
u(a, ω1) = u(b, ω2) = 1;
u(a, ω2) = u(b, ω1) = 0.
(a) Derive a formula for the mistake rate for any pair of posteriors
γL 1 = 0.5 − d, γ1
H = 0.5 + 2d, d ∈ (0, 0.25),
assuming each posterior occurs with probability 1/2. Illustrate for both d = 0.1 and d = 0.2 in the figure showing the expected utility of both actions as a function of the belief p1 that the state is ω1.
(b) Use the figure to illustrate that there are many different pairs of posteriors γ1
L < 0.5 < γ1
H that produce the same reduction in the error rate as do
γ¯1
L = 0.4,
γ¯1
H = 0.7.
7. Continue with the tracking problem and prior µ1 = 0.5. Let C(p1) be a symmetric, strictly convex, “Shannon-like” cost curve with C(0.5) = 0 and very steep near p1 = 0 and p1 = 1.
(a) On cost axes, sketch C(p1).
(b) Consider an experiment with posteriors γ1
L = 0.4 and γ1
H = 0.6, each with probability 1/2. Illustrate the expected cost
1/2C(γ1
L
) + 1/2C(γ1
H)
on your cost diagram, using the chord between C(γ1
L
) and C(γ1
H).
8. Continue again with the tracking problem and prior µ1 = 0.5. Take quadratic cost C(p1) = (p1−0.5)2
. Compute and illustrate the expected cost of the experiment that yields γ1
L = 0.3, γ1
H = 0.7, P
L = P
H = 1/2.
9. Consider the tracking problem with stakes
u(a, ω1) = u(b, ω2) = 1/4
, u(a, ω2) = u(b, ω1) = 0,
and quadratic cost
C(p1) = (p1 − 0.5)2
.
(a) Consider posteriors equidistant from the prior, γ1
L = 0.5−d, γ1
H = 0.5 + d and find the optimal distance d
ˆ that maximizes expected net utility.
(b) Draw a concavification diagram with net-utility curves, envelope, the two optimal posteriors, the chord between them, and the value of learning at p1 = 0.5. [No calculation is needed]