Market with friction problem set
Problem Set #2 Econ 366-Markets with Frictions Prof. Guido Menzio New York University Spring 2024 Instructions: There are four questions worth a total of 100 points. You are encouraged to work in groups of up to 4 members. One submission per group. The problem set is due by midnight of March 6. Please email your solutions to the TA Nicolo’Ceneri at [email protected] . Good Luck! 1. (25) The determinants of the reservation wage. Consider the labor market model of Burdett and Mortensen (1998). Suppose that the distribution of wages o¤ered by di¤erent …rms is given by F (w) = w ( =2) , for w 2 h 2 ; + where is a parameter that determines the average wage o¤er, and the dispersion of wage o¤ers. Also, suppose that e = 0 and = 1. 2 i ; is a parameter that determines a. Write down and explain the equilibrium condition for the reservation wage R b. In the same graph, plot the left-hand and the right-hand side of the equilibrium condition for the reservation wage R. Identify the equilibrium value of R. c. What is the e¤ect of an increase in Interpret your …nding. on R? Answer the question either graphically or analytically. d. What is the e¤ect of an increase in Interpret your …nding. on R? Answer the question either graphically or analytically. e. What is the e¤ect of an increase in b on R? Answer the question either graphically or analytically. Interpret your …nding. 2. (25) Minimum Wage in a Frictional Labor Market. We want to understand the e¤ect of introducing a minimum wage on unemployment, employment and wages in the context of the theory of wage inequality of Burdett and Mortensen (1998). 1 a. First, consider a situation without a minimum wage. The equilibrium wage o¤er distribution F0 (w) is such that r y w e+ F0 (w) = 1 , all w 2 [w0 ; w0 ], ((WOD)) y R0 e with 2 w 0 = R0 , w0 = y + (y R0 ). ((HW)) e Plot the F0 (w) function with respect to w. Make sure to mark b, y, w0 and w0 in your picture. b. The equilibrium wage distribution G0 (w) is given by G0 (w) = + F0 (w) : F0 (w)) e (1 Plot the G0 (w) function with respect to w. Make sure to mark b, y, w and w in your picture. Compute the di¤erence between G0 and F0 . Which one is bigger? What does that mean? c. What is the probability that an unemployed worker …nds a job? What is the probability that an employed worker becomes unemployed? What is the stationary unemployment rate? d. Suppose that Congress passes new legislation imposing a minimum wage m > R0 . We want to derive the new equilibrium wage o¤er distribution F1 . Clearly, the lowest wage o¤ered by a …rm in equilibrium must now be w1 = m. Use this observation to derive the equilibrium level of pro…ts 1 (assume = 1). e. In equilibrium, every wage w 2 [w1 ; w1 ] on the support of must deliver the pro…t observation and the fact that the stationary distribution is given by u= + , G1 (w) = u + 1 to the …rm. Use this F1 (w) F1 (w)) e (1 to derive the wage o¤er distribution F1 (w). f. Plot the wage o¤er distribution F1 (w) against w. Make sure to mark b, y, m; R0 ; w1 and w1 in your picture. g. Compute the di¤erence between F1 (w) and F0 (w). Which one is smaller and what does that mean? How has the introduction of a minimum wage changed the wage o¤er distribution? What about the wage distribution? Explain. 2 h. How has the minimum wage a¤ected the unemployment rate? Explain. 3. (25) Search frictions and the gender gap. Consider the following version of the search model of the labor market by Burdett and Mortensen (1998). There are two labor markets, one for men (m) and one for women (f ). On one side of the m labor market, there is a continuum P1 of male workers with measure 1. Each worker has preferences described by the lifetime utility t=0 t ct , where denotes the worker’s discount factor and ct denotes the worker’s consumption in period t, which is equal to b if the worker is unemployed and w if the worker is employed. On the other side of the m labor market, there is a continuum of …rms. Each …rm maximizes the present value of pro…ts discounted at the factor . Assume = 1. Workers and …rms come together through a decentralized process m of search. In particular, a worker …nds a …rm with probability m u when unemployed and e when m m employed. Assume u = e = m . When a worker meets a …rm, it observes the wage of the …rm and decides whether to join the …rm or not. If he does, he produces y units of output per period until he either …nds a better job or he is forced into unemployment (an event which happens with probability per period). The labor market f is the same as the labor market m, except that fu = fe = f with f < m. a. Write down and compare the reservation wage for men, Rm , and for women, Rf . Discuss your …ndings. b. Write down the equilibrium wage o¤er distributions Fm (w) and Ff (w) in the labor markets m and f . Compute the di¤erence between Fm (w) and Ff (w). What does the sign of the di¤erence tell you about the wages o¤ered to men and women? c. Write down the equilibrium wage distributions Gm (w) and Gf (w) in the labor markets m and f . Which is greater: Gm (w) or Gf (w)? And what does this say about the wages earned by men and women? d. Imagine that a no-discrimination law is introduced by Congress, forcing …rms to o¤er the same wages to women as to men, i.e. Ff (w) = Fm (w). Compute the wage distributions Gm (w) and Gf (w) for men and women. Are they equal? Explain your …ndings. e. Did the no-discrimination policy succeed in reducing gender inequality? 4. (25) Drawing on the material covered in this or other classes, illustrate with an example the following quote by Lars Hansen (University of Chicago, Nobel Prize in Economics). Recently, I was reminded of the commonly used slogan “evidence-based policy.” Except for pure marketing purposes, I …nd this terminology to be a misnomer, a misleading portrayal of academic discourse and the advancement of understanding. While we want to embrace evidence, the evidence seldom speaks for itself; typically, it requires a modeling or conceptual framework for interpretation. Put another way, economists— and everyone else— need two things to draw a conclusion: data, and some way of making sense of the data. That’s where modeling comes in. Modeling is used not only to aid our basic understanding of 3 phenomena, but also to capture how we view any implied trade-o¤s for social well-being. The latter plays a pivotal role when our aim is to use evidence in policy design. This is intuitive if you think about the broad range of ideas and recommendations surrounding macroeconomic policy and the spirited, sometimes acrimonious way in which they’re debated. If everything were truly evidence based, to the extent we can agree on the accuracy of the evidence, why would there be such heterogeneity of opinion? The disagreement stems from the fact that people are using di¤erent models or conceptual frameworks, each with its own policy implications. Each of them might be guided by evidence, but policy conclusions can rarely be drawn directly from the evidence itself. 4
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