AMSE Master Employment Policies: Exercises for

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1 AMSE Master Employment Policies: Exercises for Bruno Decreuse September 2015 There are five exercises, each containing five questions. The diffi culty of each question is evaluated by a number of *, one, two, or three. Feel free to contact me by if you struggle too much on this assignment. Exercise 1. Unemployment and stigma The purpose of this exercise is to model the effect of unemployment stigma on job-seekers behavior. We start from the standard job-search model in continuous time, and consider a risk-neutral and infinitelived individual who discounts time at instantaneous rate r. The individual receives job offers at rate m > 0. A wage offer is a random draw w from the cumulative distribution function F. While employed, the individual receives the wage w forever (no job loss risk). While unemployed the individual suffers stigma from the communauty. The utility flow attached to this stigma is s. *1.1 Let U and W denote, respectively, the value of unemployment and employment. Write these two value functions in recursive form. *1.2 Find the (implicit) reservation wage. What is the impact of s on the mean post-unemployment wage and on the job-finding rate? Explain your result. *1.3 As time passes, the stigma s may disappear, i.e. s goes to zero. This event occurs at rate p. Interpret this assumption. **1.4 Let U 1 denote the value of unemployment when the individual is stigmatized, and U 2 be the value of unemployment when he is not. Write U 1, U 2, and W. Remember that the stigma disappear at constant rate p when unemployed. **1.5 Let x 1 and x 2 be the reservation wages when the individual is stigmatized and when he is not. Show that x 2 > x 1. Explain this result. Exercise 2. On-the-job search The case of on-the-job search is treated in the course. However, we make a strong assumption whereby the search intensity is the same for unemployed and employees. The purpose of this exercise is to relax this assumption. We use the same notations as in the course. Unemployment income is b. Employees 1

2 lose their jobs at rate q. The discount rate is r. Unemployed job-seekers receive job offers at rate m 0 whereas employed workers receive offers at rate m 1. Each offer is a random draw according to the cdf Φ and associated pdf φ. *2.1 Write the Belmann equations defining the value of unemployment U and the value of having a job W (w). *2.2 Show that employed workers paid w accept any wage w > w. Let µ(w) be the job-to-job transition rate. Compute it and show that it decreases with w. Comment this result. **2.3 Show that x = b + (m 0 m 1 ) and comment the role played by m 1. Remember that dφ(w) = φ(w)dw. **2.4 Show that x [W (w ) W (x)]dφ(w ) (1) 1 Φ(w) x = b + (m 0 m 1 ) dw. (2) x r + q + m 1 [1 Φ(w)] Hint: remark that W (w) = 1/{r + q + m 1 [1 Φ(w)]} and integrate by part. **2.5 From question 2.4, show that unemployment income b reduces the overall magnitude of job-tojob transitions. Exercise 3. Explaining marriage duration The purpose of this exercise is to model the pattern of divorce rates by age. Time is continuous and goes from 0 onward. A population of men match and marry with a population of women. Marriage quality is a random variable. It is bad with probability p and good with complementary probability 1 p. Marriage quality determines the risk of divorce. We start by French facts and then consider two different ways to model divorce. *3.1 Comment Figure 1. Figure 1: Divorce rate by marriage duration for several divorce years, France,

3 *3.2 We model divorce as follows. In the small time interval dt, the probability that a good marriage ends is λ g dt, whereas the probability that a bad marriage ends is λ b dt, with λ b > λ g 0. Compute λ(t) the average divorce rate by marriage duration t. Is this pattern compatible with Figure 1? **3.3 Suppose that marriage quality is initially unobserved by men and women, but they can learn it. In other words, marriage is an experience good. The learning process is as follows. In all marriages, bad events can occur, like cheating by one of the partners. However, good marriages are more solid in that they resist to such events: partners can credibly commit that such events won t happen again. Thus the occurrence of a bad event is not suffi cient to end a marriage (it may happen to all marriages); however, the occurrence of a second one reveals that the marriage is bad and, therefore, leads to a divorce. We now focus on bad marriages. Assume that the marriage duration at which such a bad event happens is a random variable that follows the exponential law of parameter λ. Show that the marriage duration T at which a divorce occurs is such that Pr[T t] = H(t) = (1 e λt ) 2. (3) Some help: Let T 1 be the duration at which the first bad event occurs, and T 2 be the duration at which the second bad event occurs. Then, Pr(T t) = Pr(T 1 + T 2 t) = t t t1 f(t 0 0 1, t 2 )dt 1 dt 2, where f is the joint density of the two independent variables T 1 and T 2. *3.4 The divorce rate by marriage duration the so-called hazard rate conditional on being in a bad marriage is λ(t) = H (t) 1 H(t), (4) i.e., the ratio of the density H (t) of the divorce date divided by the survival function 1 H(t). Compute this divorce rate. How does it change with marriage duration? Assuming that all marriages are bad, can this model explain the pattern displayed by Figure 1? **3.5 Now we combine the two stories. Suppose that good marriages never end, whereas the divorce rate by marriage duration is λ(t) for bad marriages. Compute the proportion p(t) of bad marriages among marriages of duration t. Show that the mean divorce rate is λ(t) = Can this model explain the pattern displayed by Figure 1? 2pλe λt (1 e λt ) 1 p + p[2e λt e 2λt ]. (5) Exercise 4. False alarms In the US, between 94 and 99 percent of burglar-alarm calls turn out to be false alarms, and false alarms make up between 10 and 20 percent of all calls to police. In 2000 police responded to 36 million false calls at an estimated cost of $1.8 billion. Stephen Dubner, Freakonomics blog, 2012 The purpose of this exercise is to model the behavior of a risk-neutral policeman confronted to true and false alarm signals. Time is continuous and goes from 0 onward. During the small time interval dt, the policeman receives a crime-related alarm signal with probability λdt. He also receives a false alarm signal with probability f dt. The policeman cannot discriminate between the two signals. If available, 3

4 he starts an investigation, which duration follows a Poisson process of parameter q, i.e. investigation stops after the time interval dt with probability qdt. The investigation is successful only if there is a crime corresponding to the alarm signal. The utility gain for the policeman in case of success is S. The policeman discounts time at rate r. When the policeman does not investigate, he achieves regular tasks that give a constant utility flow y. *4.1 The probability of success is λ/(λ + f). How does it change with f and λ? Explain your result. *4.2 Let U be the expected utility of the policeman when available, and W when investigating. We have Explain these equations and solve them in U and W. ru = y + (λ + f) (W U), (6) rw = q(u + λ S W ). (7) λ + f *4.3 What is the effect of f on W U and U? Comment your result. **4.4 Now, the policeman decides if he investigates following an alarm signal. Show that he investigates if and only if q λ λ+f S > y. Explain this result. **4.5 An increase in the rate of false alarms diverts police resources away from non-alarm owners. Discuss this assertion in light of your answer to question 4.4. Exercise 5. Ageing and job search Senior workers struggle to find jobs. The purpose of this exercise is to study the impact of the retirement age on job search and hirings. Time is continuous. Bruno is a risk-neutral senior unemployed worker who discounts time at rate r, and searches for a job. When unemployed, he receives unemployment benefit b. The matching rate is m. When employed, he receives the net wage (1 τ) w, where τ is the payroll tax rate. Be careful, the wage offer distribution is degenerate: there is a unique possible wage. Bruno retires at age A, whether employed or unemployed, and then enjoys the pension p per unit of time. We suppose that (1 τ) w > b > p. **5.1 Let a denote Bruno s age. Write Bruno s expected utility when employed, W (a), and unemployed, U (a), in recursive form. Be careful, this is a nonstationary model. Let Z(a) = W (a) U(a) and show that for all a A, (1 τ) w b [ Z (a) = 1 e (r+m)(a a)], (8) r + m w (1 τ) [ W (a) = 1 e r(a a)] + p r r e r(a a), (9) U (a) = W (a) Z (a). (10) How do W, U, and W U vary with age? ***5.2 Bruno makes the search investment i. The effort cost is C (i) = c 0 i 2 /2. In exchange for such a cost he benefits from the matching rate mi. Let i (a) be the optimal search investment by age. Write 4

5 Bruno s maximization problem and show that rz (a) = (1 τ) w b [mz (a)] 2 / (2c 0 ) + Z (a) a A, (11) Z(A) = 0. (12) How does the hazard rate mi (a) vary with age? Remark: you have to solve a Riccati differential equation. Please visit **5.3 The retirement system must be modified to become sustainable. This can go through an increase in the tax rate τ, an increase in the retirement age A, or a decrease in the pension level p. Discuss the impacts of these parameters on the optimal search investment by age i (a). **5.4 We now study the labor demand for senior workers. A senior employed worker produces y and costs ω (net wage plus taxes) per unit of time. Let J (a) denote the value of a job held by a worker of age a. Write this value and show that, J (a) = y ω r ( 1 e r(a a)), a A. (13) How does J change with a? Suppose there is a training cost T that firms must pay when they hire a worker. Compute the threshold age a above which senior applicants are no longer considered. How does it change with A? **5.5 Forget the training cost. Senior workers may become sick and less productive as a result. This leads employers to discriminate against them. Suppose that people get sick at constant rate p. When sick, people lose cognitive skills and output falls from y g to y b, with y g > ω > y b. Assume that the process starts at age 0. Employers observe workers age, but do not observe their health status. Compute the proportion of workers in good health by age. Let a be the age above which employers refuse to hire senior applicants. Show that a = 1 [ ] ω p ln yb. (14) y g y b 5