# Determine the optimal schooling choice (whether to complete high school or not) for both types.

Subject: Economics / General Economics

Econ 450 Problem Set #4

2017

Department of Economics

Problem 1 – High School Choice (25 points)

Consider the choice of taking a high school degree or dropping out. Suppose there are two types

of individuals in the population. Completing high school increases the present value of lifetime

earnings by different rates across types, so that type 1 individuals will experience a rate of increase

of r1 and type 2 individuals experience a rate of increase of r2 , where r1 > r2 . Thus, given a base

level of earnings B, the present value of lifetime earnings conditional on completing high school is

P V1HS = B(1 + r1 ) and P V2HS = B(1 + r2 ) for types 1 and 2, respectively. Present value lifetime

earnings in case the worker does not complete high school is P V N o = B no matter the type of the

worker. Completing high school has a cost C no matter the type of the worker.

Suppose B = 100, r1 = 0.35, r2 = 0.2 and C = 22.

1. Determine the optimal schooling choice (whether to complete high school or not) for both

types.

2. Suppose a social science scholar were to compare earnings for high school graduates against

workers with no high school degree to determine the rate of return of a high school degree.

What would the result of the analysis be?

3. In light of this analysis, suppose the government decides to give the types that drop out of

high school incentive to complete high school. The government decides to provide a cash

prize of P = 3 upon high school graduation. Is this sufficient incentive?

4. The labor market is perfectly competitive and earnings perfectly reflect productivity. Thus,

social returns to schooling are given by earnings minus education cost. Is the policy of

providing a cash prize upon graduation desirable from a social planner point of view? Problem 2 – A graduation prize (15 points)

High school graduates are observed to have lifetime earnings that are roughly 30-40% higher than

individuals with only some high school. Suppose it can be shown that by offering high school

students a cash prize for completing high school, one can significantly increase the high school

graduation rate. Discuss whether this is a socially desirable policy. Problem 3 – Principal-Agent Problem (30 points).

A restaurant is writing a contract with a waiter stating that he will be paid a weekly wage of $200

plus 20% of the sales (the tip). The sales are a function of his effort, e, and can be stated as,

?

S(e) = 100 e. Thus, the wage is given by, w = 200 + 0.2S(e). The waiter has a utility function

u(w, e) = w ? e. The restaurant cannot observe or verify the waiter’s effort level but the contract

states that it expects an effort level of at least ec = 150.

1 1. Determine how much effort the waiter will choose to provide, e? .

2. Given e? , what is the waiter’s weekly wage?

3. A contract is self-enforcing when agents have incentive to satisfy the demands in the contract

without legal enforcement. What is the minimum fraction of sales that the restaurant must

promise the waiter for the contract to be self-enforcing? Problem 4 – Schooling as a signal (30 points).

Consider an economy with two types of individuals, low ability and high ability. Each type is equally

represented in the population. Individuals themselves know their types but employers cannot

directly observe the skill level. To simplify assume a 2-period model. In the first period, individuals

choose a schooling level. In the second period they produce. A low ability individual produces a low

level of output YL = 1 whereas a high ability individual produces a high level of output, YH = 2.

The output price is P = 1. Furthermore, low ability individuals have higher schooling cost than high

ability individuals. Specifically the cost of schooling level e is CL (e) = ?e for a low ability individual

and CH (e) = ?e/2 for a high ability individual. Individuals can choose any schooling level e ? 0.

Think of e as education beyond high school. The cost function includes the opportunity costs of

time in the first period. An individual has payoff function ui (e, W ) = W ? Ci (e) for i = {L, H}.

That is, we are disregarding discounting and payoff is simply earnings minus schooling cost.

1. For a given ?, find the lowest level of schooling, e? , that supports a separating equilibrium

where high ability individuals choose eH = e? and low ability individuals choose eL = 0. Here

is a roadmap to follow:

(a) Suppose firms believe that any individual that chooses e ? e? is high productivity, Y = 2,

and that any individual with schooling e < e? is low productivity, Y = 1. The labor

market is perfectly competitive. Write up the wage in the labor market as a function of

schooling, W (e).

(b) Given the education conditional wage in the labor market, write up incentive compatibility constraints such that a high ability individual (weakly) prefers eH = e? to eH = 0,

and that the low ability individual (weakly) prefers eL = 0 to eL = e? . Find the lowest

e? so that both incentive compatibility constraints are satisfied.

2. What happens to e? if ? increases? Comment.

3. Assume ? = 1/2. There are 100 individuals. Calculate the total production minus education

cost for the two period economy.

4. Suppose the government were to ban education, meaning, everybody must choose e = 0.

(a) What is the wage in the labor market? (Hint: What is the expected productivity of a

worker when the firm cannot tell whether she is high or low ability?).

(b) Are firms’ profits any different in the two scenarios?

(c) Calculate total production minus education cost for the two period economy. Comment. 2

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