This is an individual submission. You may discuss the assignment with others, but never in front of a piece of paper, board, or screen. An immaculate submission will yield 50 points. Each problem splits its points evenly over the Roman letters

1. (15 points) A taxi driver is considering the pro?tability of her/his job. A customer ride takes 15 minutes and pays $10, on average. Fuel, taxes, license and the car depreciation average a total of $100 per shift. Assume these are all ?xed costs with no variable component.
(i) What is the pro?t function for the taxi driver on an 8hr-shift? Draw it accurately on the Cartesian plane and show the coordinates of all the relevant points.
(ii) How may rides are needed on average to make a $180 pro?t per shift?
(iii) If there is a demand of three rides every hour, what is the average pro?t?
(iv) If the ?xed costs ranged between $50 and $150 per shift, how would the maximum pro?t range?
(v) The taxi driver is willing to work overtime. There is no additional ?xed or variable cost, but there is an additional mixed cost that increases by $12 for each additional hour worked. In fact, the taxi driver values the ?rst extra hour worked $12 worth of stress accumulated, the second hour $24, and so on. Assuming the rate of utilization of the taxi in (iii) and including the additional pro?ts and disutility in your analysis, how many extra hours should the taxi driver work? Please give an integer answer.

2. (15 points) You have two data plan options for your cellular-enabled device: A) a 2Gb data ?at rate plan for $50/mo, or B) a 500Mb data plan for $30/mo plus ¢1.5 per each additional Mb until you reach 2Gb.
(i) Assuming both plans o?er 4G speed and you expect to consume something between 0 and 2Gb each month, draw the two cost functions on the same graph and ?nd the point of indi?erence.
(ii) Using Excel’s “value of” option in Solver or the “Goal Seek” tool (“Data” ? “What if analysis”) to compute again the Point of Indi?erence. As an answer, you may write down the cell references, along with their content and formulas, and the steps you used to ?nd the solution, or print your spreadsheet and the Solver or Goal Seek windows. (To do this, press ctrl + the ˜ key to show all the formulas, then press alt + prt sc to copy the active worksheet or window, and paste it on your document.)
(iii) Assume now you may consume up to 5Gb a month. In that case, after you used 2Gb, plan A allows you unlimited data free of charge but throttled at 2G speed, while plan B keeps charging the ¢1.5/Mb until 3Gb and then stops the cellular service until the end of the month. In order to compare the di?erent services, you have created a utility function that assigns a value of ¢4 for each Mb consumed at full speed, ¢1 for each Mb consumed at 2G speed, and ?¢2 for each Mb that you were not able to use.
(The latter applies only to your expected data usage in excess of 3Gb when opting for plan B.) Draw the two pro?t (utility ? cost) functions, or use Excel (the “if” function might come handy) to compute the break-even points, the points of indi?erence and state when it is optimal to use plan A, plan B, or no plan, according to your expected monthly data usage.

3. (20 points) Refer to the “Wrong Diet” spreadsheet available on BlackBoard.
(i) From the Case 1 worksheet, change the maximum daily fat intake from 100 to 80. Provide a graphical solution either using the corner point method or the gradient method.
(ii) How would the optimal value change if the maximum daily calorie intake increased by 1? Decreased by 1? Increased by 2? Decreased by 2? How do your answers compare to the ?gures in the Sensitivity report?
(iii) How would you explain the shadow price, allowable increase and allowable decrease values of the ?rst and third constraint?
(iv) From the Case 2 worksheet, what is the optimal solution if the cost of one ice cream cup is changed from $1 to $2? How would you change that cost such that the two corner points (8/5,14/5) and (3,0) are both optimal solutions?

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