4. Vacancy costs and the labor market. Consider the search-theoretic model of the labor market of Diamond {1982), Mortensen (1982} and Pissarides (1985). a. The equilibrium tightness of the labor market, 6", solves the following equation -ny-w luWWW Plot the left and the right hand side of the above equation as functions of 9 and identify 6'. k=so b. Using the same yaph as before, identify the effect of an incraese in the vacancy cost it: on the equilibrium tightness of the labor market. Interpret your nding. c. The equilibrium unemployment, n", and the equilibrium vacancies, v", solve simultaneously the following system of equations Plot the solutions to these two equations in a graph that has it on the horizontal axis and v on the vertical axis (Le. plot the Beveridge curve and the market tightness curve]. Identify n\" and 11*. (1. Using the same graph as before, illustrate the eifect of an increase in the vacancy cost I: on the equilibrium unemployment and vacancies. Interpret your ndings. e. Should the government intervene to lower unemployment in response to an increase in k? In Exercises 9-16, a table is given telling the time needed for each of a number of tasks and which tasks must immediately precede them. Make a PERT diagram for each problem, and determine the project time and critical path. 9. Task Time Preceding Tasks 10. Task Time Preceding Tasks 11. Task Time Preceding Tasks None A None A Non B None B WN U 7 A. B 10 D 2 A, B D 6 A 6 C. D E B. D B. C E B. D C 7 C. D C. E. F 6 D. E. F E. F 12. 13. Task Time Preceding Tasks 14. Task Tine Preceding Tasks Tark Time Preceding Tasks 5 10 None A None None B A. D 12 None 2.1 None C 10 B. I C 15 None None 6 A. C D 7.2 None D None E B 3 A, B E 6.1 None 9 A.B 13 I C F 4.1 G 13 B. C 5 C. E. F H 2.0 F. G 9 None D. E 8.5 D. E. G D. H E, F 5.2 E, H 16. 15. Preceding Tasks Park Time Preceding Tasks Tark Time None A None 13 None A 12 None C .10 A, F B, C 14 None JOT A. C.D E .02 None -IOTMORE E 6 A. B. D 10 A. B. C -11 B. C. D .09 F. G D. H E, F, H 06 F. G. HThe Pristine River Case The Pristine River has two polluting firms on its banks. Acme Industrial and Creative Chemicals each dump 100 tons of glop into the river each year. The cost of reducing glop emissions per ton equals $10 for Acme and $100 for Creative. The local government wants to reduce overall pollution from 200 tons to 50 tons. a. If the government knew the cost of reduction for each firm, what reductions would it impose to reach its overall goal? What would be the cost to each firm and the total cost to the firms together? b. In a more typical situation, the government would not know the cost of pollution reduction at each firm. If the government decided to reach its overall goal by imposing uniform reductions on the firms, calculate the reduction made by each firm, the cost to each firm, and the total cost to the firms together. c. Compare the total cost of pollution reduction in parts (a) and (b). If the government does not know the cost of reduction for each firm, is there still some way for it to reduce pollution to 50 tons at the total cost you calculated in part (a)? Explain.2. Economic Growth. Suppose that Real GDP is given by the following ag- gregate production function: Y, = A, KO3 NOT a) Suppose that, in a given year, output grows by 3%, capital grows by 2.67% and labour grows by 1%. What is productivity growth? Bonus points to students who can derive the growth accounting formula. Hint: You need to make use of the following approxi- mation for a growth rate in variable X: gx = In X - In X-1 b) Suppose that the savings rate, s, is 0.5, the depreciation rate, d, is 0.09 and population growth, n, is 0.01. Productivity, A, is equal to 2. i. Derive the per capita aggregate production function and plot it on a diagram with k, = ~ on the horizontal axis and y, = # on the vertical axis. ii. Plot savings per capita, M, on the same graph. iii. Plot the amount of investment required to keep capital per capita constant. c) Solve for a steady state equilibrium. In particular, find capital, output, consumption, savings and investment in per capita terms. Show this equilibrium on your graph from part a). d) The golden rule savings rate is the one where consumption per capita is maximized in the steady state. Find the savings rate which is associated with the golden rule consumption level. Is it higher or lower than the savings rate used in part c)? Show the golden rule steady state on your graph. Hint: Consumption per capita in the steady state is given by: c = (1 -$)A You will need to use calculus to find the savings rate that maximizes consumption. Take the derivative with respect to the savings rate and set the first order condition equal to zero. Make use of the product and chain rules for the derivative with respect to s. You do not need to check the second order condition