ECON6003/6703
Assigment 2
Submission Deadline: 5:00pm, 20 May
1.Let A be a square matrix with columns a1, . . . , aN. Show the following
(i) If a′iaj = 1{i=j}, then A
′ is the inverse of A.
(ii) If A is positive definite, then trace(A) > 0.
2. Consider a bank that lends to two types of customers, type A and type B. Types are
not observable to the bank. Type A customers have a default probability of 8%. Type
B customers have a default probability of 10%. 90% of potential customers are of type
B, and 10% are of type A. Given a randomly selected customer, calculate on behalf of
the bank the default probability associated with this customer.
3. Consider the utility maximization problem
max
x1,x2
{0.5 log(x1) + log(x2)}
s.t. p1x1 + p2x2 = m, x1 > 0, x2 > 0,
where p1, p2 and m are strictly positive parameters. This problem has a unique maxi-
mizer (x∗1 , x
∗
2). Obtain it as a function of the parameters. Show your reasoning.