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HW8

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Econ 300
Assignment 8

  1. Leonardo is a mechanically minded person who always builds things to help him understand his courses. To help in his understanding of average and marginal cost curves, he draws a TC-q axis pair on a board and attaches a thing wood pointer by a single nail through the origin. He now claims that he can find the level of output for which average cost is a minimum for any cost curve by the following mechanical process: (1) Draw the total cost cuve on his graph; (2) Rotate his pointer until it is precisely tangent to the total cost curve he has drawn; and (3) find the quantity that corresponds to this tangency. Leonardo claims that this is the quantity where average cost is minimized. Is he right? For which of the total cost curves in Figure 6.3 (in book) would this procedure work? When would it not work?

  2. Late Bloomer is taking a course in microecoomics. Grading in the course is based on 10 weekly quizzes, each with a 100 point maximum. On the first quiz, Late Bloomer recieves a 10. In each succeeding week, he raises his score by 10 points, scoring a 100 on the final quiz of the year.

    1. Calculate Late Bloomer's quiz average for each week of the semester. Why, after the first week, is his average always lower than his current week's quiz?

    2. To help Late Bloomer, his professer had decided to add 40 points to the total his quiz scores before computing the average. (What a nice professor she must be!) Recompute Late Bloomer's weekly averages given this professional gift.

    3. Explain why Late Bloomer's weekly quize averages now have a U shape. What is his lowest average during the term?

    4. Explain the relevance of this problem to the construction of cost curves. Why does the presence of a ''fixed cost'' of 40 points result in a U-shaped curve? Are Late Bloomer's average and marginal test scores equal at his miminum average?

  3. Jana is a mathematical whiz. She has been reading this chapter and remarks, ''All this short-run/long-run stuff is a trivial result of the mathematical fact that the minimum value for any function must be as samll as or smaller than the minimum value for the same function when some additional constraints are attached.'' Use Jana's insight to explain the following:

    1. Why short-run total costs must be equal to or greater than long-run total costs for any given output level.

    2. Why short-run average cost must be equal to or greater than long-run avergae cost for any given output level.

    3. Whether you can make a definite statement about the relationship between short-run and long-run marginal costs.

  4. A stuffed-wombat manufacturer determined that the lowest average production costs were achieved when eight wombats were produced at an average cost of $1000 each. If the marginal cost curve is a straight line intersecting the origin, what is the marginal cost of producing the ninth wombat?

  5. Trapper Joe, the fur trader, has found that his production function in acquiring pelts is given by

    $\displaystyle q=2\sqrt{H}$    

    where $ q=$ the number of pelts acquired in a day, and $ H$ = the number of hours Joe's employees sprend hunting and trapping in one day. Joe pays his employees $8 an hour.

    1. Calculate Joe's total and average cost curves (as a function of $ q$ ).

    2. What is Joe's total costs for the day if he acquires 4 pelts? 6 pelts? 8 pelts? What is Joe's average cost per pelt for the day if he acquires 4 pelts? 6 pelts? 8 pelts?

    3. Graph the cost curves from $ a$ and indicate the points from $ b.$

  6. Professor Smith and Professor Joenes are going to produce a new introductory textbood. As true economists, they have laid out the production function for the book as

    $\displaystyle q=\sqrt{SJ}$    

    where
    $\displaystyle q$ $\displaystyle =$ the number of pages in the finished book  
    $\displaystyle S$ $\displaystyle =$ the number of working hours spent by Smith  
    $\displaystyle J$ $\displaystyle =$ the number of working hours spent by Jones  

    Smith values her labor at $20 per working hour. She has spent 900 hours preparing the first draft. Jones, whose labor is valued at $80 per working hour, will revise Smith's draft to complete the draft.

    1. How many hours will Jones have to spend to produce a finished book of 150 pages? Of 300 pages? Of 450 pages?

    2. What is the marginal cost of the 150th page of the finished book? Of the 300th page? Of the 450th page?

  7. Suppose a firm's constant-returns-to-sclae production function requires it to use capital and laobr in a fixed ration of two workers per machine to produce 10 units and the the rental rates for capital and labor are give by $ v=1,w=3.$

    1. Calculate the firm's long-run total and average cost curves. What is the marginal cost of the 10th unit? The 25th unit? The 50th unit? The 100th unit?

  8. Explain whether each of the following actions would affect the firm's profit-maximizing decision. (Hint: how would each affect MR and MC?)

    1. An increase in the cost of a variable input such as labor

    2. A decline in the output price for a price-taking firm

    3. Institution of a small fixed fee to be paid to the goverment for the right of doing business.

    4. Institution of a 50% tax on the firm's profits

    5. Institution of a per-unit tax on each unit the firm produces

    6. Receipt of a no-strings-attached grant from the government

    7. Receipt of a subsidy per unit of output from the government

    8. Receipt of a subsidy per worker hired from the government

  9. Sally Greenhorn has just graduated from a noted business school but does not have the foggiest idea about her new job with a shrink-wrapped dog biscuit firm. She has been given responsibility for a new line of turkey-flavored biscuits and must decide how many to produce. She opts for the following strategy: (1) Begin by hiring one worer and one dog biscuit machine; (2) if the revenues from this pilot project exceed its costs, add a second worker and machine; (3) if the additional revenues generated from the second worker/machine combination exceed what these cost, add a third; (4) stop this process when adding a worker/machine combination brings less in revenues that it costs. Answer the follwoing questions about SG's approach:

    1. Is SG using a marginal approach to her hiring of inputs?

    2. Does the approach adopted by SG also imply that she is following a MR=MC rule for finding a profit-maximizing output?

    3. SG's distinguished professor of marketing examines her procedures and suggests she is mistaken in her approach. He insists that she should instead measure the profit on each new worker/machine combination employed and stop adding new output as soon as the last one added earns a lower profit than the previous one. How would you evaluate this distinguised advice?

  10. Two features of the demand facing a firm will ensure that the firm must act as a price taker. (1) That other firms be willing to provide all that is demanded at the current price; and (2) That consumers of the firm's output regard it as identical to that of its competitors. Explain why both of these conditions are required if the firm is to treat the price of its output as fixed. Describe what the demand facing the firm would be like if one of the conditions held but not the other.

  11. John's Lawn Mowing Service is a small business that acts as a price-taker (MR=P). The prevailing market price of lawn mowing is $20 per acre. Although John can use the family mower for free, he has other costs given by,
    $\displaystyle TC$ $\displaystyle =$ $\displaystyle 0.1q^{2}+10q+50$  
    $\displaystyle MC$ $\displaystyle =$ $\displaystyle 0.2q+10$  

    where $ q=$ the number of acres John chooses to mow in a week.

    1. How many acres should John choose to move in order to maximize profits?

    2. Calculate John's maximum weekly profit.

    3. Graph these results and label John's supply curve.

  12. Consider again the profit-maximizing decision of John's Lawn Mowing Service from the previous problem. Suppose John's greedy father decides to charge John for the use of the family lawn mower.

    1. If the lawn mower charge is set at $100 per week, how will this affect the acre of lawns John chooses to mow What will this profit be?

    2. Suppose instaed that John's father requires John to pay 50% of his weekly profits as a mower charge. How will this affect John's profit-maximizing decision.

    3. If John's greedy father imposes a charge of $2 per acre for use of the family mower, how will this affect John's marginal cost function? How will it affect his profit-maximizing decision? What will his profits be now? How much will John's greedy father get?

    4. Suppose finally that John's father collects his $2 per acre by collecting 10% of the revenues from each acre John mows. How will this affect John's profit-maximizing decision? Explain why you get the same results here as for part $ c.$

  13. Suppose a firm faces the following demand curve:

    $\displaystyle q=60-2P.$    

    1. Calculate the total revenue curve for the firm (that is, TR in terms of $ q).$

    2. Using a tabular proof, show that the firm's MR curve is given by $ % MR=30-q.$

    3. Assume also that the firm has an MC curve given by $ MC=0.2q$ . What output level should the firm produce to maximize profits?

    4. Graph the demand, MC, and MR curves and the point of profit maximization.


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Jenn Thacher 2008-08-25
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