Economics 370 Professor H.J. Schuetze Practice Problem Set 6 Solutions Read each question in its entirety before beginning, then answer the question as clearly and concisely as possible. Make sure to answer all of the questions. You may find it helpful to outline the important points first, and then fill in the details. 1. The following question refers to a model of imperfect monopsonistic wage discrimination against females like the one presented in class. a) Indicate how imperfect monopsonistic wage discrimination between males and females can explain the persistence of male-female wage differentials for equally productive workers while at the same time explain the existence of a mixed work force within the firm. The mechanics of an imperfect monopsonistic wage differentiation, which can give rise to sex discrimination, are explained in the class notes for section 4.1. There are two conditions which must be met in order for equally productive workers to be paid different wages yet still enter the same labour market. First, it is necessary to separate the two sexes, because all of the workers would want to join the ranks of the males to receive higher pay. This condition is easily met, as it is very uncommon for someone to change his or her sex. Second, there must be upward sloping supplies of labour (always true in instances of imperfect competition in the labour market) with differing elasticities. Without these differing elasticities, there is no basis to pay different wages, as the demand and supply conditions would be identical for men and women. Recall that all workers will receive less than their MRP N, as the employer is extracting surplus from all of them. The employer extracts less surplus from men, however. When differing supply elasticities are coupled with an identical value of the marginal product for men and women, the men, with the higher supply elasticity, are paid a wage that is closer to the common MRP. In the graph below, this type of situation is depicted. w MC m S m MC w MC* w m MC T S w w w MRP N N m N K N w N T N b) Would such a monopsonist ever hire an all female workforce? An all male workforce?
If the intersection of the MRP N and MC T curves occurs to the left of N K on the graph shown above, then it could be that the wage level at this intersection induces no men into the work force. This implies that the aforementioned intersection occurs at a wage level below the intersection of the S M curve and the vertical axis. On the other hand, given that the S M curve sits above the S W curve, there could not be a men-only work force. This is simply because any wage that induces men to enter the work force, will also induce women to enter. c) One of the conditions necessary for monopsonistic wage discrimination against women as a group is that their labour supply to the firm must be more inelastic than that of males, at their respective wages and employment. Would you expect this to be the case -- why or why not? Yes, one might expect this to be the case. First of all, to the extent that women suffer from labour market discrimination, their labour market opportunities are reduced. This is likely to make them less responsive to wage changes than would otherwise be the case. In other words, their elasticity of labour supply is lower. Second, in many cases the labour supply behavior of the wife is affected by the employment situation of the husband. If he is gainfully employed, her labour market mobility may be reduced, which would render her elasticity of labour supply lower. Third, in many families, particularly those with small children, the wife is the partner who is primarily responsible for the household production. This charge will normally greatly reduce her labour market opportunities, which, like the two other factors mentioned, is likely to make her less responsive to wage changes. d) Show that there are vacancies for both men and women in equilibrium under imperfect monopsonistic wage discrimination. (i.e. show that the firm would be willing to hire more of both men and women at the going wage). Would the monopsonist ever report such vacancies? Yes, there are vacancies of both male and female workers. At the going wages for both men and women, the marginal revenue product (demand for labour) of both sexes is greater than the supply of each type of worker. The firm will report such vacancies but will be unwilling to increase the wages in order to attract new workers. This is because the higher wage would have to be paid to not only the new workers, but also the existing ones. 2. You run a personnel office for a large manufacturing firm. Suppose that each workers marginal revenue product of labour (value to the firm) is constant over time and equal to $10 per hour. Your predecessors have told you that workers tend to shirk on the job, so you propose to the owner of the firm a deferred compensation program. In particular, the wage at every level of seniority is determined by the equation: Wage = 2/3 (# years in the firm) (i.e. after 10 years the person would earn W = 2/3(10) or $6.67 per hour) a) Draw the diagram of the deferred wage profile.
$ W=2/3 10 VMP N =$10 S B MR T(years) b) Could this profile persist? This wage profile can exist. The wage profile here [w=2/3(t)] where T is seniority or tenure is a bonding mechanism, and is economically feasible as long as it is accompanied by a mandatory retirement age. c) Solve for the breakeven point or length of time the person would have to stay with the firm so that their wage would just equal their productivity. Set the wage function so that it is equal to the value of their marginal product, and solve for T. 2 / 3T = 10 and T = 15 For the first 15 years of tenure, the employee will be given compensation less than his/her contribution to production (ie. a wage less than the value of his/her marginal product). At the 15 th year, they are paid exactly their value in production. d) If workers typically start working with the firm at age 35 and have a zero rate of discount (so that a dollar today is the same as a dollar in the future), what should the mandatory retirement age be at your firm? First, must calculate the mandatory retirement age in terms of T: What we want to find is the number of years of seniority at which the present value of wages is matched by the present value of MRPs. In this case, we can appeal to the constant slope of the wage profile equation to respond to this question. We know from part (c) that the worker will be paid under their value in production up until 15 years. Due to symmetry, the first 15 years of the employment relationship are the mirror image of the second 15 years of it. Thus, we know that at T=16 the person will be paid exactly above the value of their marginal product what they were being paid below at T=14. Therefore, if we draw a time line, we can see that: w( 17 ) MRPN = w(13) MRPN We are given no discount rate, therefore the mandatory retirement age, in terms of T, is T=30. Now we just add T=30 onto age 35, and the mandatory retirement age is 65 (point MR on the graph in (a))
e) If a positive discount rate were assumed, what would that do to the mandatory retirement age? A positive discount rate means that the payments over the value of the marginal product (starting in T=16) are less important today than the payments under the value of the marginal product from T=1 14. Mathematically, we would express this as: T w( i) VMP 0 = N where δ 0 is the discount rate, and find the number of periods i 1 i= 1 (1 + δ ) (T) that solves this equation. In general, we can conclude that because the wages exceed the value of the marginal product in the latter periods, then the mandatory retirement age will be greater than T=30 in order to net out the losses incurred from T=1 14. f) If you had to downsize but wanted to maintain your reputation, you might want to offer an early retirement buyout package equal to the magnitude of the deferred compensation. What would that magnitude be for someone who has been at the firm for 15 years? (assume 2000 working hours in a year or 50 weeks at 40 hours per week) They would need to be compensated for the total cumulative amount that the worker was paid under his/her value in production. A table may be useful. We know that their value in production will equal $10/hr*2,000 hrs/year * 15 years and will be equal to $300,000. As for wage, we simply multiply yearly wage rate by the number of hours in a year, and sum up over years. T wage total wage bill 1 $0.67 $1,333.33 2 $1.33 $2,666.67 3 $2.00 $4,000.00 4 $2.67 $5,333.33 5 $3.33 $6,666.67 6 $4.00 $8,000.00 7 $4.67 $9,333.33 8 $5.33 $10,666.67 9 $6.00 $12,000.00 10 $6.67 $13,333.33 11 $7.33 $14,666.67 12 $8.00 $16,000.00 13 $8.67 $17,333.33 14 $9.33 $18,666.67 15 $10.00 $20,000.00 Total $80.00 $160,000.00 The total compensation required is $300,000 - $160,000 = $140,000
3. A researcher estimates the following wage equation for underwater construction workers: W = 10 + 5D Where W=the wage in dollars per hour and D=the depth underwater at which workers work, in meters. a) Based on this information, draw the offer curve and possible indifference curves for two workers, A and B: A works at a depth of 3 meters and B works at 5 meters. wage U A U B $10 (5, 12.50) (3,11.50) offer curve depth Referring to the graph above, A s wage at 3 metres is 10 +.5*3 = $11.50 per hour. At 5 metres, B s wage is 10 +.5*5 = $12.50 per hour. A s indifference curve must be tangent to the offer curve at 3 metres; B s must be tangent at 5 metres. b) At their current wages and depths, what is the rate at which each worker is willing to trade off one more meter of depth while holding utility constant? Because both indifference curves are tangent to a straight line, both must have the same slope at their points of tangency; therefore, both workers are willing to pay (or receive) 50 cents per hour for reduced (added) depth of one meter. c) At 3 meters of depth which worker has a greater willingness to pay for a reduction in depth? Worker A, who chooses to work at 3 metres, has a steeper indifference curve (a greater willingness to pay for reduced depth) at each level of depth; that is why worker A chooses to work at a shallower depth. 4. State whether you agree or disagree with each of the following statements. If you agree with the statement, explain why you agree, and if you disagree, explain why you disagree (include the correct statement in your answer). I encourage you to illustrate your answers using diagrams where appropriate. a) An increase in the price of employer provided health insurance is likely to lead to an increase in the number of hours worked by employees.
Disagree: This is very similar to an increase in the price of an input when discussing firm behavior. We can think of the firm as having a choice between two inputs number of workers (bodies) and number of hours per worker. An increase in the price of employer provided health will increase the price of bodies relative to hours. Thus, there will be both substitution and scale effects. (You should include a diagram to illustrate the two effects) Substitution effect: Since hours of work become relatively cheap the firm will substitute away from bodies towards greater hours. Scale effect: The costs of production will rise with the increase in price. This will cause firms to reduce the scale of operation. This implies that the firm will use less of both bodies and hours. This suggests that the overall effect on the number of hours worked by employees is indeterminate. b) All else equal, a firm that requires general training is more likely to enter into a deferred wage contract than a firm that requires firm-specific training. wage, MRP N wage MRP N 0 S T S B seniority Disagree. A firm that requires specific training will be willing to pay for the training while firms that require general training will not pay for the training. This gives firms that require specific training an added incentive to keep workers around. Because workers may look forward to collecting their deferred wage, such a payment plan may reduce turnover. This will reduce costs substantially for firms with high training and/or firing costs. An example of such a firm is one that requires firm specific training. In the above diagram, training takes place from time 0 to time S T. These costs are borne by the employer in the sense that the wage paid to the trainee is higher than the value of his or her product to the firm. These costs are reduced from time S T to time S B (known as the recouping period) when the worker is paid a wage below his or her marginal product. The employee is not likely to leave because their training is of no use at another firm. Although the wage paid by their present employer is below their marginal product,
it is still higher than the wage they would receive from any other competitive firm. If a firm trains an employee with general skills, they have no incentive to bear the training costs because, once trained, an employee can transfer to a firm that pays a wage equal to their marginal product. This means that there is no recouping period possible so there is no advantage to offering deferred wages. c) In a model of compensating wage differentials the indifference curves of risk lovers are steeper than those of risk-averse workers. Disagree. The slope of a worker s indifference curve in wage-safety space represents the marginal rate of substitution between wages and safety for the worker. In other words, moving up the indifference curve and to the left, it measures the wage increase required by the worker to give up one unit of safety holding total utility fixed. Workers with a higher preference for safety (risk-averse types) will require a bigger wage increase in exchange for a decrease in safety than would workers with a lower preference for safety (a risk lover). This means that at each level of safety the slope of the indifference curve of a risk-averse person will be steeper than it is for a risk lover. wage Risk lover Risk averse Slope = Marginal rate of substitution between wage and safety safety