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ECON 306

Macroeconomics

Winter 2022

Problem Set 6

Search and Unemployment

Please follow all instructions and answer all questions. Submit your work for credit via the dropbox feature on LEARN. If you are submitting scanned images of handwritten work, please (i) submit all images in one document,  (ii) check that the pages are in the correct order and properly oriented, and (iii) ensure that your work is legible.

Grading scheme:  Each (part of) each question will be graded out of 3.  Grades will be assigned according to the following rubric:

3/3 The answer/solution is complete and correct, except for very minor flaws (e.g.,

the omission of non-essential details).

2/3 The answer/solution is mostly correct but contains some aws (e.g., the omis-

sion of an essential detail).

1/3 The answer/solution contains serious flaws (e.g, the omission of a central ele-

ment, or an explanation that is unclear).

0/3 No answer/solution was submitted, or the submitted answer/solution fails to

make any progress towards answering the question.

Half marks may be used in cases where an answer appears to fit somewhere between two adjacent grading categories.

 

1    Short-Written-Answer Questions

1.  Suppose you are having a conversation with an out-of-work friend.  Maybe you are setting your sights too high,” you remark condescendingly before asking them, “Why did you quit your old job before you had a new one lined up?”  Explain how your friend may respond to you, assuming their behaviour is optimal given a labour market environment such as the one described in the notes.  [3 points]

2.  In the one-sided search model, suppose there are two kinds of works in the pop- ulation, type h and type l. These two types of workers are identical, but when type h workers are unemployed, they receive more utility from unemployment than type l workers do, even though both types of workers receive the same EI benefits. Which type has the higher unemployment rate, and which type tends to be employed in higher-wage jobs? Briefly explain your answer.  [3 points]

 

2    Problems

1. Consider a three-period version of the one-sided search model with a two-rung job ladder.  Let ut  denote the unemployment rate at time t = 0, 1, 2.  The remaining of labour force participants are employed, either at a low-wage job or a high-wage job at time t = 0, 1, 2. Let et(l)  and et(h)  denote the fraction of the labour forced employed in low-wage and high-wage jobs. Note that ut + et(l) + et(h)  = 1.

The unemployment benefit is denoted b, and the two wage levels are wl  and wh  satis- fying b < wl  < wh . Unemployed workers only receive low-wage offers. Specifically, φl is the probability that an unemployed worker receives a wage offer in a given period. For simplicity, assume there is no exogenous separation of workers from low-wage jobs (i.e., δl  = 0).  However, a low-wage worker may receive an offer to switch to a high-wage job. A high wage offer arrives with probability φh  each period, conditional on being employed at a low-wage job. A worker employed at wage wh  will experience an exogenous job separation with probability δh , in which case they transition to unemployment.

Workers make accept/reject decisions to maximize the present discounted expected value of lifetime income, where β = 1/(1 + r) is the discount factor and r > 0 is the discount rate. In terms of notation, let Vtu , Vtl , and Vth  denote the present discounted expected values of remaining lifetime income when unemployed, employed at a low- wage job, and employed at a high-wage job at time t = 0, 1, 2.

(i)  Consider the situation of an unemployed worker at time 1. Write down an equa- tion for their present discounted expected value of remaining lifetime income, V1u . Be sure to include the accept/reject decision problem in this equation. Im- pose the optimal accept/reject decision to write V1u  in terms of the parameters of the model (i.e., the discount factor, the probabilities, and income levels b, wl and wh ).  [3 points]

(ii)  Consider the situation of a worker in a low-wage job at time  1.   Using an approach similar to what you did in part (i), derive an expression for V1(l)  in terms of the parameters of the model.  [3 points]

(iii) Write down an expression for V1h in terms of the parameters of the model. [3 points]

(iv) Now consider the time 0 situation of a worker in a low-wage job.  Write down an equation for their present discounted expected value of lifetime income, V0(l) . Determine conditions under which the low-wage worker will accept an offer to work for wage wh  if it is offered at time 0, and when they will not.  Briefly explain these conditions.  [3 points]

(v)  Suppose b = 1/2, wl  = 1 and wh  = 2.  The exogenous probabilities are φl  = 3/10, φh  = 1/10, and δh  = 1/20. The discount factor is β = 0.99.  Finally, the initial distribution of workers across employment status is u0  = 1/10, e0(l)  = 3/10 and e0(h)  = 6/10. Solve for the unemployment rate in periods 1 and 2.  [3 points]