Provide solutions.to the following questions.

There are two types of individuals: high ability (type H) and low ability (type L). Regardless of their level of education, the lifetime productivity of an H type is VMPH, while the lifetime value of the marginal product of an L type is VMPL

1.

Suppose that if an L type is indifferent between getting SL years of education and being

viewed as an L type and getting SH years of education and being viewed as an H type, he/she

chooses SL. Derive the conditions that determine whether a separating equilibrium in which SH

years of schooling serves as a signal of high ability exists.

2.

Let S* denote the minimum SH such that a separating equilibrium exists. What happens

to S* if SL increases by one year? What happens to S* if CL decreases slightly?

Now suppose that after finishing their schooling, each type works for only two periods. Assume

that after a person works for one period, with probability p all the firms in the labour market learn

his/her true type (e.g., by observing his/her first-period output). With probability (1 - p) the firms

learn nothing new. Assume also that in each period, the L type produces VMPL = $40,000 while

the H type produces VMPH = $64,000. (Note that, unlike in part a, VMP is measured per period

here.) Finally, let CH = $4,000 and CL = $6,000, and assume that SL = 0.

3. Derive the conditions that determine whether a separating equilibrium in which SH years

of schooling serves as a signal of high ability exists.

4.

As before, let S* denote the minimum SH such that a separating equilibrium exists. What

happens to S* if it becomes more likely that firms learn workers' types from their output (i.e., if p

increases)?

3. For the function f (x) = e"(r - 2) (a) (2 points) Find the domain of f(I) (b) (2 points) Find lim f(x) (c) (2 points) Find lim f(x) (d) (4 points) Find any r-intercept(s) (e) (4 points) Find any y-intercept(s) (f) (3 points) Find the first derivative of f (I) (g) (6 points) Find any interval(s) where f(x) is increasing and where f(z) is decreasing (h) (4 points) Find any minimum(s) or maximum(s) (1) (3 points) Find the second derivative of f(I) () (3 points) Find any interval(s) where /(z) is concave up (k) (3 points) Find any interval(s) where f(x) is concave down (1) (4 points) Find any inflection point(s) (m) (4 points) Graph the function be sure to label on points (intercepts, extrema, and inflection points)20. Suppose /(x) is a function defined for all s whose derivative and second derivative are given by f'(x) = (r - 4)s and /"(x) = 3r] -4. (a) (4 points) Find the r-coordinates of all critical points of f(r). If there are none, write write "none' Show your work and use calculus to justify your answer(s). Critical point (s) at = (b) (6 points) Find the r-coordinates of all local extrema of f(2). If there are none of a particular type, write 'none'. Use calculus to justify your answer(s), and be sure to show enough evidence to demonstrate that you have found all local extrem Local min(s) at 3 = Local max(es) at a = (c) (5 points) Find the r-coordinates of all inflection points of f(r). If there are none, write none. Use calculus to justify your answer(s), and be sure to show enough evidence to demonstrate that you have found all inflection points. Point (s) of inflection z =