Question 1:
(i) Demand function for air travel between the U.S. and Europe has been estimated to be
ln Q = 2.737 - 1.247 ln P +1.905 ln I
where Q denotes number of passengers (in thousands) per year, P the (average) ticket price and I the U.S. national income.
Determine the price elasticity and income elasticity of demand.
(ii) It has been estimated that the price elasticity of demand for U.S. manufactured automobiles is -1.2, while the income elasticity of demand is 2.0 and the cross price elasticity of demand with respect to the foreign automobiles is 1.5. The current volume of sales for U. S. manufactured automobiles is 10 million per year. It is expected that over the next year the average income of the consumers in the U.S. will increase by 2.5 percent.
It has been determined that the price of the foreign imports will increase by 6% over the next year. By how much should the U.S. automakers adjust the price of their automobiles if they wish to increase the volume of their sales by 9.2% next year?
(iii)Bright Future, Ltd (BF) is a nonprofit foundation providing medical treatment to emotionally distressed children. BF has hired you as a business consultant to design an employment policy that would be consistent with its goal of providing the maximum possible service given its limited financial resources. You have determined that the service (Z) provided by BF is a function of its medical staff input (M) and sound service input (S) which is given by:
Z = M + .5S + .5 MS - S2
BF's staff budget for the coming year is $1,200,000. Annual employment costs are $30,000 for each social service staff member (S) and $60,000 for each medical staff member (M).
(a) Using the Lagrangean multiplier approach calculate the optimal (i.e., service maximizing) combination of medical and social staff. Determine the optimal amount of service provided by BF.
(b) Calculate BF's marginal cost. Explain your answer.
(c) Using Excel-Solver verify your answer to (a). (Show your work. Show the spreadsheets in detail. Provide print outs with Solver window. To print the solver window, use print screen command on your key board and then create a MS Word document using paste.)
(iv)The own price elasticity of demand for a pack of cigarettes is estimated to be -.4. Current price and consumption are $4.00 and 2 million units per year.
Assuming a linear demand relationship determine the demand equation for cigarettes. Show all your calculations.
Question 2
(i)The production function for a firm is given by
Q = LK
where Q denotes output; Land K labor and capital inputs. Wage rate and rental rate are given by w and r respectively.
(a) Show whether or not the above production function exhibits diminishing marginal productivity of labor.
(b) Determine the nature of the Return to Scale as exhibited by the above production function
(c) Using the Lagrangean Multiplier method, calculate the least cost combinations of labor and capital and the resulting long run total cost function for the above production function. Explain the economic significance of the Lagrangean Multiplier and calculate its value.
(d) Using the above cost function, calculate the numerical value of long run total cost when Q =225, w = 16 and r = 144.
(e) Using Excel- Solver verify your answer to (d) above.
(ii) As the manager of an 80-unit motel you know that all units are occupied
when you charge $60 a day per unit. Each occupied room costs $4 for service andmaintenance a day. You have also observed that for every x dollars increase in the daily rate above $60, there are x units vacant. Determine the daily price that you should charge in order to maximize profit.
Question 3
(i) Sleak Teak builds yard furniture using hardwoods and (in a smaller shop) handcrafted knick-knacks from the same sort of wood. The hardwood usage in the two lines of product are
Yard Furniture: Y = 2 Ty - .001Ty2
Knick-knacks: Z = 20Tz - .01 Tz2
where Y and Z are units of furniture and knick-knacks respectively. Ty and Tz are the amounts of hardwood used in Y and Z production respectively. Yard furniture can be sold at a profit (i.e., net of costs) of $100 per unit and knick-knacks can be sold at a profit of $25 per unit. Sleak Teak has 1300 units of hardwood available that can be allocated between these two lines of production.
(a) Using the Lagrangean Multiplier method, determine how should the hardwood be allocated between the two lines of product so that total profit can be maximized. Also calculate the optimal amounts of Y and Z and total profit from each product line.
(b) Using Excel Solver verify your answers to (a) above. (Show your work.)
(ii) You manage Dirt Diggers, an excavating firm that excavates roadside ditches for laying drainpipes. Its output follows the production function:
Q = 10L -.1L2
where L denotes labor hours and Q the length of the ditch in meters. You can hire labor at the going wage rate of $12 per hour. As the manager of DD you want to earn as high a profit as possible.
(a) You have received an offer to excavate 250 meters for a lump sum payment of $500. Should you accept the offer? Explain with appropriate calculations.
(b) Suppose that instead of the previous offer, you are now offered as much or as little excavation work at a price of $2.00 per meter dug. Should you accept the offer? Why or why not? If you accept the offer calculate DD's resulting profit. Also, calculate the optimal level of output (meter dug) and the level of labor usage.
(iii) As a manager of a firm you find the marginal cost of the firm to be $10 and the fixed cost $100. For the range of prices that you are planning to charge, own price elasticity of demand is believed to be -1.5. Calculate the optimal (profit maximizing) price that you should charge. Show all calculations."