Question

Suppose there are high- and low-ability workers. The percentage of the population that is high-ability is ? and the percentage that is low-ability is 1 − ?. If the employer can distinguish the two types, it pays equal to the output of the two types: ?h and ??, respectively. If the employer cannot distinguish the two types it pays ?̅, which is equal to the expected (i.e., average) output of all workers it employs.

a. Suppose education is a continuous variable, where ?h is the years of schooling of a high- ability worker and ?? is the years of schooling of a lower-ability worker. (Only integer values of e are allowed: 1, 2, 3, etc.) The cost per period of education for these types of workers is ?h and ?? respectively, where ?? > ?h. Under what conditions is a separating equilibrium possible? How much education will each type of worker get?

Hint: You will need to explain the constraint on both high- and low-ability workers at equilibrium. These two constraints are the “conditions” the question asks for. Use these constraints to solve for the levels of education.

b. Under what conditions is a pooling equilibrium possible?

Hint: You will need to explain the constraint on high-ability workers and on cost generally. These two constraints are the “conditions” the question asks for. Explain (a) what levels of cost always give you a pooling equilibrium and (b) what combinations of cost and percentage of high-ability give you a pooling equilibrium.

Answer 1

The percentage of population i.e is high skilled workers is ?

The percentage of population i.e low skilled workers is 1- ?

a) Education is a signal i.e being sent by the workers to show that there ability. Let eh be the education level undertaken by high skilled worker and el be the education level undertaken by the skilled worker.The cost of education incurred by low skilled worker is more than that of high skilled workers.

?? > ?h

The conditions under which separating equilibrium can exist if

The level of education undertaken by the low skilled worker would be determined by

U(wl,0 /l) i.e the Utility of the low skilled worker when he undertakes zero level of education and the wage incurred by the worker is wl

U(wh,el/l) i.e is the Utility of the low skilled worker when he undertakes a positive level of education to pretend that he is a high skilled worker as he has an incentive of getting wh.

U(wh,el/l)= wh-c(el)

also ,U(wh,eh/h)>=U(wl,0/h)

U(wh,eh/h) i.e is the Utility of the high skilled worker when he undertakes a positive level of education.

U(wl,0/h) i.e is the Utility of the high skilled worker when he undertakes zero level of education.

In separating equilibrium

wl>=wh-c(el)..........................(1)

wl<=wh-c(eh).........................(2)

If (1) and (2) would be satisfied then separating equilibrium would exist i.e el* =0 and eh* lies between (el,eh)

where u is the probability that the person is a high skilled worker at a given level of education and in equilibrium it is equal to 1 if e>= eh*

wage function = wh s/t e=eh*

wage function = wl is e <eh*

b) Pooling Equilibrium would exist if

U(wl,0 /l) <= U( ?wh+(1- ?),el/l)

i.e wl <= ?wh+(1- ?)-c(el^ )                                  ........................(3)

and

U(?wh+(1- ?),el/h) >=U(wl,0/h)

wl<=?wh+(1- ?)-c(el^)                                    ........................(4)

Now,in case of pooling equilibrium el* belongs to (0,el^) and el* is equal for both high ability and low ability worker.

where u is the probability that the person is a high skilled worker at a given level of education el* and in equilibrium it is equal to ?.

wage function = ?wh+(1- ?) wls/t e=el*

wage function = wl is e <el*

One would also get a pooling equilibrium of ?? = ?h as the low skilled worker would undertake the same level of education as a high skilled worker to get a wage higher than wl.