1. (55) Declining Search Frictions and the Beveridge Curve. Using the search-theoretic model of unemployment and vacancies of Pissarides (1985), we want to understand the effect of declining search frictions. To this aim, assume that the matching function M(u, v) has the form A. M(u, v), where A > 0 is a parameter that controls the efficiency of the matching process (i.e., a parameter that is inversely related to search frictions) and M(u, v) is a constant returns to scale function that is increasing and concave in both u and v. a. (5) For $ = 1, the equilibrium tightness of the labor market, 0", solves the following equation (y - b) k = Aq(0)(1 - ") 5 + nAp(@) where p(0) = M(1, 0) and q(0) = M(01, 1). The left-hand side is the firm's cost of opening a vacancy. The right-hand side is the firm's benefit of opening a vacancy. Plot the left and the right hand side of the above equation as functions of e and identify 0*. b. (5) Using the equilibrium condition for #" identify and explain the two opposing effects of an increase in the efficiency A of the matching process on the firm's benefit of opening a vacancy. c. (10) Differentiate the firm's benefit of opening a vacancy with respect to A, and identify the dominant effect. d. (5) Using the same graph as in (a) and the result in (c), identify the effect of an increase in A on the equilibrium tightness of the labor market. . e. (5) The equilibrium unemployment, u", and the equilibrium vacancies, v*, solve simultaneously the following system of equations 1 = 6 + Ap(v/u)' v = bu. Plot the solutions to these two equations in a graph that has u on the horizontal axis and v on the vertical axis (i.e. plot the Beveridge Curve and the vacancy creation curve). Identify u* and v*. f. (10) Using the same graph as in (e), illustrate the effect of an increase in A on the Beveridge Curve. Interpret your findings. g. (5) Using the same graph as in (f) and the result in (d), illustrate the effect of an increase in A on u* and v*. Interpret your findings. h. (5) Below you see the Beveridge curve for the US, and the time-series of unemployment and vacancies. Can you think of reasons why techological progress that has ostensibly improved the efficiency of the matching process has not resulted in a progressive shift of the Beveridge curve and a downward trend in unemployment? 1927-1936 1937-1946 1947-1956 1957-1966 1967-1976 1977-1986 1987-1996 1897-2006 Vacancy Rate 2007-2018 15 Unemployment Rate Vacancy Rate 15 25 1809 1909 2090 2010 Unemployment Rate Time i. (5) Relating back to Question 1, can you think of reasons why technological progress that has improved the efficiency of monitoring has not resulted in a downward trend in unemployment