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Artificial Intelligence For Games 2nd Edition Ian Millington, John Funge, Millington - Solutions
4. Determine the content by considering your readers’ frame of reference and your purpose in writing.
3. Understand your role in the organization as a writer and how your role should be reflected in what you write.
2. Know the goals you want your writing to achieve and the business context in which you need to communicate.
1. Determine as fully as possible who will read what you write.
(d) Compute the firm’s total wage bill.
(c) Compute the first-order condition from the union’s maximization problem. Use the demand curve and the first-order condition to solve for the wage and level of employment at the firm.
(b) Using this expression, write down the union’s maximization problem assuming the union maximizes the wage differential times the number of employed union members.
(a) Write down an algebraic expression for the wage differential times the number of employed union members.
5 Suppose a union has so much bargaining power over a firm that the union can set any wage it wants. In addition, suppose the union membership has collectively decided that their objective should be to maximize the wage differential times the number of employed union members at the firm.(The wage
(d) Compute the firm’s total wage bill.
(c) Compute the first-order condition from the union’s maximization problem. Use the demand curve and the first-order condition to solve for the wage and level of employment at the firm.
(b) Using this expression for the wage bill, write down the union’s maximization problem assuming the union maximizes the firm’s wage bill.
(a) Write down an algebraic expression for the firm’s total wage bill.
4 Suppose a union has so much bargaining power over a firm that the union can set any wage it wants. In addition, suppose the union membership has collectively decided that their objective should be to maximize the total wages received by employed union members from the firm. In other words,
(d) Compare the solutions from the two sets of bargaining costs. What feature of the Rubinstein model have you illustrated? Explain.
(c) Now suppose that the delay costs fall to 100 for the firm only, i.e. CF =100, while the delay costs for the union remain unchanged at 200.Sketch the bargaining diagram and compute the Rubinstein solution to the game.
(b) Assuming the union makes the first offer, compute the Rubinstein solution to this bargaining problem.
(a) Sketch the bargaining game, showing the bargaining frontier, the effects of delay, and the disagreement point.
3 Suppose the bargaining frontier in firm-union contract negotiations is given by:where U is the union’s utility and π is the firm’s profit. Let the delay cost to the union be 200, i.e. CU = 200 and the delay cost to the firm also be 200, i.e. CF = 200.
(d) Compare the solutions from the two sets of bargaining costs. What feature of the Rubinstein model have you illustrated? Explain.
(c) Now suppose that delay costs fall to 50 for the union and the firm, i.e.CU = CF = 50. Sketch the bargaining diagram and compute the Rubinstein solution to the game.
(b) Assuming the firm makes the first offer, compute the Rubinstein solution to this bargaining problem.
(a) Sketch the bargaining game, showing the bargaining frontier, the effects of delay, and the disagreement point.
2 Suppose the bargaining frontier in a firm-union contract negotiation is given by:where U is the union’s utility and π is the firm’s profit. Let the delay cost to the union be 100, i.e. CU = 100 and the delay cost to the firm also be 100, i.e. CF = 100.
(e) Is there a first-offer advantage in this game?
(c) If the union makes the first offer, what is the Rubinstein solution to the bargaining problem?(d) If the firm makes the first offer, what is the Rubinstein solution to the bargaining problem?
(b) On the same diagram, sketch the bargaining frontier after a one-period delay. Again, indicate the horizontal and vertical intercepts on the diagram.
(a) Sketch the bargaining frontier on a diagram with U on the vertical andπ on the horizontal axis. Indicate the horizontal and vertical intercepts of the frontier.
1 Suppose the bargaining frontier is given by the following equation:where U is the union’s utility and π is the firm’s profit. Let the delay cost to the union be 10, i.e. CU = 10 and the delay cost to the firm also be 10, i.e. CF = 10.
(d) What do economists call the wage you have determined?
(c) Based on your calculations, what wage do you think the firm should adopt? Explain.
(b) Compute the percentage wage changes and corresponding percentage output increases for the wage pairs $10.00 and $11.25, $11.25 and$12.00, and $12.00 and $12.50.
(a) Calculate the percentage change in the wage and the corresponding percentage change in output as the wage is increased from $8.00 to$10.00. Given that all other inputs at the firm are fixed, would you recommend the wage increase? Explain.
7) A study by an economics consulting firm finds the following relationship between wages and output at a small firm hiring a fixed number of workers.Wage rate ($) Units of output 8.00 60 10.00 80 11.25 90 12.00 95 12.50 98
6 Suppose the value of marginal product for all workers in a firm is $20 per hour. Shirking is a major problem in the firm, so the manager of the firm proposes to institute a scheme of deferred wages. Assume that the wage of a worker is determined by the formula:Assuming that the discount rate is
(c) The worker is paid less in her first year at the job than her market value. Why do you think a worker would opt for such an arrangement?
(b) Using this condition, calculate the wage the worker will be paid in the second year of the job.
(a) What condition must be true in order for the firm not to overpay or underpay the worker during her tenure at the firm?
5 Suppose that a worker is expected to stay with a firm for two years. The worker’s value of marginal product is $20,000 in the first year, rising to$25,000 in the second year owing to experience gained in the job. Suppose that the interest rate is 10 percent and the worker is paid $18,000 in her
(e) What potentially important aspect of the manager’s pay does this compensation scheme ignore? Explain.
(d) Assuming the pay scheme takes the form of a fixed payment P to the owner with all residual revenues from the business going to the manager, calculate the value of P.
(c) Assuming for the moment that the owner can observe the amount of shirking, derive the profit-maximizing hours of shirking at the business.
(b) Derive the pay constraint for the small business. What does the pay constraint represent?
(a) Derive the revenue function of the small business.
4 Suppose the owner of a small business is considering hiring a manager and has decided to opt for a performance-based scheme for the manager’s pay. Suppose a manager’s utility function is given by U(m, s) = m +40s1/2, where m is the manager’s income and s the number of hours of shirking the
(c) Does the present value of the benefit from training exceed the present value of the cost?(d) Suppose the interest rate rises 15 percent. Will the training still be undertaken? Explain.
(b) Determine the worker’s wage in the second year.
(a) Determine the worker’s wage in the first year.
3 Suppose that the value of marginal product of an untrained worker is$20,000 per year and that one year of worker training raises productivity by 20 percent. Suppose training costs are $3,500 and that the worker will stay with the firm for two years. Assume that the worker agrees to pay
(c) Because the firm can exceed the present value of the cost?
(b) Determine the worker’s wage in the second year.
(a) Determine the worker’s wage in the first year.
2 Suppose that the value of marginal product of an untrained worker is$25,000 per year. One year of worker training costs $12,500 and raises the worker’s value of marginal product to $50,000. Suppose the worker will only remain with the firm two years and that the interest rate is 10 percent.
1 In each of the following cases, state whether the case is an example of general or specific training and give a brief explanation.(a) A Bachelor of Arts in Economics.(b) Flight training on a Boeing 767.(c) Training in the Registrar’s Office at the University of London.(d) Training to operate
(c) Determine the new equilibrium wage in the labor market. Why do you think the wage does not rise by the full $10 required to compensate workers to work in these jobs?
(b) Now suppose that a new study reveals that the work has previously unknown harmful health effects to the extent that each worker now requires a compensating differential of $10 in order to work in the industry. Find the new equation for the labor supply curve.
(a) Determine the equilibrium wage and level of employment in the labor market.
6 Suppose that the (inverse) demand for labor in a competitive market is given by:(and (inverse) labor supply is given by:
What is the intuition behind your answer? (Hint: compute the elasticities of demand and supply at the pre-tax equilibrium.)
(d) Explain why the group that bears the higher burden of the tax does so.
(c) Compare the workers wage before the payroll tax to the wage they receive after the tax is imposed. Likewise, compare the wage firms paid before the tax to the wage they pay after the tax is imposed. Who bears the higher burden of the tax proportionately?
(b) Suppose a payroll tax of 25 percent is introduced. Determine the equation of the labor demand curve incorporating the tax. Hence, determine the wage paid by the firms, the wage received by the workers, and the total tax revenue.
(a) Determine the equilibrium wage and level of employment in the labor market.
5 Suppose that the (inverse) demand for labor in a competitive market is given by:(and that the supply of labor is given by:
4 Suppose 100 firms in a labor market all have an identical short run production function given by f(L) = 4L1/2, and that the price of output is p= 100. Determine the market labor demand curve. If the supply of workers to the market is given by L = 1,000 w, determine the equilibrium wage in the
3 If a firm’s short-run production function is f(L) = 2L1/2, the price of output is p = 1,000, and the market wage of workers is w = 100, determine the profit-maximizing number of workers the firm will hire. Be sure to check the second-order condition.
(b) Assuming that the cost of the fixed inputs is $40 per day, calculate the firm’s total daily profit.
(a) Determine the profit-maximizing hours of labor input the firm hires per day and, hence, the firm’s total daily output.
2 A firm uses labor as its only variable input. Its daily production function is given by:where L is measured in hours of labor time per day. All markets are perfectly competitive. The price of output is 10 cents per unit and the wage rate is $4.80 per hour.
(d) Suppose the wage rises to $800 per week, how many workers will the firm hire? (Assume P = 300.) Again, illustrate the answer on a diagram. What have you just shown?
(c) Suppose the price of bicycles rises to $300. Determine the new VMPL and, hence, determine the number of workers the firm will hire after the price increase. Illustrate your answer on the diagram. What characteristic of labor demand have you just illustrated?
(b) If the wage of workers is set at $600 per week, how many workers will the firm hire? Draw a diagram to illustrate your answer.
(a) Calculate, and record in a table, the marginal product of labor (MPL)and the value of marginal product of labor (VMPL) at each level of employment.
1 A manufacturing firm produces bicycles using labor as the only input according to the following weekly production table. The bicycles are sold in a competitive market for a price P = $200 each.Number of workers Output of bicycles 0 0 1 5 2 12 3 18 4 21 5 23
(d) Show that profits are greater at the merged firm than at when the two firms were operating as separate entities. What problems is the merger likely to face?
(c) Suppose the two firms were to merge and maximize joint profits.Determine the price the upstream division will charge the downstream division, i.e. determine the optimal transfer price.
(b) Determine the price the upstream firm will charge the downstream firm for the output produced by the upstream firms
(a) Determine the demand for output at the upstream firm from the downstream firm as a function of the price charged by the upstream firm.
6 Suppose there are two separate firms. The upstream firm is the only producer of an input required by the downstream firm and the upstream firm has no other customers. One unit of input at the downstream firm is required to produce one unit of output at the downstream firm. The cost functions for
(d) Demonstrate that the firm’s profits are greater when the optimal transfer price, as opposed to the transfer price set by the manager of the production division, is set.
(c) Suppose the firm allows the manager of the production division to set the transfer price. Determine the transfer price the manager will charge the marketing division, assuming the manager maximizes profits at the production division.
(b) Calculate the output of the two divisions and the price the marketing division charges its customers if the production division sets the optimal transfer price.
(a) Determine the profit maximizing transfer price for the firm.
5 Suppose there is a firm with two divisions: an upstream division and a downstream division. One unit of output from the upstream or production division is used for each unit of output of the downstream division. The upstream division is the only producer of the input for the downstream division.
(d) Show that total profit is higher for the optimal prices than it is for the prices using the inverse elasticity rule.
(c) Now suppose that the firm takes into account the relationship between the two goods and sets the prices accordingly. Determine p1 and p2. Are the prices higher or lower than the prices using the inverse elasticity rule? Explain the reason for your answer.
(b) Suppose the firm ignores the relationship between the two goods and uses the inverse elasticity formula to set the prices of the two goods as a markup over marginal cost. Calculate p1 and p2.
(a) Are these two goods substitutes or complements? Explain.
4 Consider a multiproduct firm that has a monopoly in the production of each of the two goods. Suppose the demand functions are given by:The total cost functions for production of each of the two goods are given by:
(d) Show that total profit is higher for the optimal prices than it is for the prices using the inverse elasticity rule.
(c) Nowsuppose that thefirm takes into account the relationship between the two goods and sets the prices accordingly. Determine p1 and p2.Are the prices higher or lower than the prices using the inverse elasticity rule? Explain the reason for your answer.
(b) Suppose the firm ignores the relationship between the two goods and uses the inverse elasticity formula to set the prices of the two goods as a markup over marginal cost. Calculate p1 and p2.
(a) Are these two goods substitutes or complements? Explain.
3 Consider a multiproduct firm that has a monopoly in the production of each of the two goods. Suppose the demand functions are given by:The total cost functions for production of each of the two goods are given by:
(d) Determine the price charged by the firm. Compute costs at the two plants and, hence, determine profits at the firm.
(c) Show that the profit maximizing rule for output allocation between the plants is satisfied.
(b) Use the profit maximizing rule to determine the output produced at each of the two plants and, hence, the total output of the firm.
(a) State the profit maximizing rule that determines how much output the monopoly will produce at each plant.
2 Consider a multiplant monopoly facing demand given by:The firm operates two plants with total cost functions given by:
(d) Determine the price charged by the firm. Compute costs at the two plants and, hence, determine profits at the firm.
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