Not even sure where to start with part (b) for the problem below. For part (a) assuming the worker knows her own skill level and the prevailing wage, I got:
y1 > y 0, or
S1 - S0 > ln(w0/w1)
for part(b), I think we are looking for D(s0, s1) = 1 iff y1 > y0... where D is the joint density function? but not totally sure what that means or how to do it.
Problem:
Consider a population of workers, each with two possible skills: call them s0 and s1 .
Think of s0 as, say, programming skill and s as language skill, which is useful for becoming e.g. a lawyer.
Assume that earnings are determined as follows: there is a prevailing wage w0 > 0 in the tech sector and w 1 > 0 in the legal sector. Workers are paid by skill, so more skilled workers earn more, and in particular,
Y0 = w0*e^s0
Y1 = w1*e^s1
where Y0 and Y1 are potential earnings in each sector. Assume that the joint distribution of skills in the population is bivariate normal, with E[s0] = E[s1] = 0.
(Here skills are on a logarithmic scale, so the fact that they can be negative is not a problem.) Let the standard deviation of s0 be σ0 , the standard deviation of s1 be σ1 , and let cov(s0 , s1) = σ01 .
(a) Each worker has a choice about whether to become a programmer or a lawyer, given his or her endowment of skills (s 0 , s 1 ), but no one can work as both (and no one decides not to work at all). What is a reasonable decision rule for a worker facing this problem?
(b) Suppose workers behave according to the decision rule you wrote down in (a) above. Find an expression for the fraction of workers who become lawyers. Hint: use the fact that if (X, Y ) are jointly normally distributed with mean zero, then
Y = cov(X, Y)/ V[X] + U
where U is independent of X, U ∼ N (0, V U ), and V U = V [Y ] − cov(X, Y ) 2 /V [X].
Another hint: the sum of two normal random variables is also normally distributed.