# Quiz 10: Cost Functions

a) There are two goods - Good 1 and Good 2. The following function is given: It is required to be explained why this function implies that the costs will be lower in a multiproduct firm than in two single-product firms. The cost of a single product firm producing q 1 units of good 1 is The cost of a single product firm producing q 2 units of good 2 is So, the total cost of producing q 1 units of good 1 and q 2 units of good 2 in two single-product firms producing each good separately is + . Moreover, the total cost of producing q 1 units of good 1 and q 2 units of good 2 in a multiproduct firm is . Since the function states that + is greater than , the function by definition states that the costs of producing q 1 units of good 1 and q 2 units of good 2 will be lower in a multiproduct firm than in two single-product firms. b) Given that the two outputs are actually of a single good, not two separate goods, and the average cost decreases as output increases, it is required to be shown that the multiproduct firm enjoys economies of scope. In other words, given that the two outputs are actually of a single good, not two separate goods, and the average cost decreases as output increases, it is required to be shown that the following functions holds true. If q 1 and q 2 are outputs of a single good, the total output (q) of this good is q 1 + q 2. Suppose q 1 and q 2 are positive amounts of output. Since average costs is lower for a higher level of output, Similarly, Now sum equations (1) and (2). Hence, it is shown that the multiproduct firm enjoys economies of scope.

Professor Smith and Professor Jones are going to produce a new introductory textbook. As true scientists, they have laid out the production function for the book as Where q = the number of pages in the finished book, S = the number of working hours spent by Smith, and J = the number of hours spent working by Jones. Smith values his labor as $3 per working hour. He has spent 900 hours preparing the first draft. Jones, whose labor is valued at $12 per working hour, will revise Smith's draft to complete the book. As Smith has worked 900 hours, then the production function for the book becomes ……. (1) a. If Jones have to finished book of 150 pages, then q=150; Smith has spent 900 hours preparing the first draft, hence, S=900. Jones, will revise Smith's draft to complete the book. Then to complete the book he has to spend 25 hours. Thus, Jones has to spend 25 hours to produce a finished book of 150 pages. Similarly, if q= 300, then from (1) J= 100 Thus, Jones has to spend 100 hours to produce a finished book of 300 pages. Similarly, if q= 450, then from (1) J= 225 Thus, Jones has to spend 225 hours to produce a finished book of 450 pages. b. As Smith has worked 900 hours, then the production function for the book becomes ……. (1) Smith values his labor as $3 per working hour. Then his total cost of work is $2700. Jones labor is valued at $12 per working hour, his total cost is 12J. Thus the total cost is given as Substituting J from (1) we have Therefore the marginal cost is given as ……….. (2) If q= 150, then according to (2) MC =4 Therefore, the marginal cost of the 150th page of the finished book is $4. If q= 300, then according to (2) MC =8 Therefore, the marginal cost of the 300th page of the finished book is $8. If q= 450, then according to (2) MC =12 Therefore, the marginal cost of the 450th page of the finished book is $12.

Total cost function shows the cost incurred in producing a particular amount of output. Average cost function shows the cost of producing per unit of output. Marginal cost is change in the total cost due to the change in output by one unit. Short run cost function is a function in which some inputs are fixed and can't be changed. Long run cost function is a function showing relation between all variable inputs and output. In long run all inputs are variable. a. Suppose that a firm's fixed proportion production function is given below: The firms total cost function is given as follows: Where, Wages Rent The quantity produced by using the fixed proportion is given below: Put the value of k and l in the total cost function as shown below: The firms average cost function is given below: The firms marginal cost function is given below: Therefore the firm's total cost function is given by . The average and marginal cost is given as . b. Suppose that k is fixed at 10 in the short run. Then the short run production function is given below: The short run cost function is given below: The quantity produced by using the fixed proportion is given below: Put the value of l in the short run total cost function as shown below: The firm's average cost function is given below: The firm's marginal cost function is given below: Therefore, in the short run the firm's total cost function is given by . The average and marginal cost is given as and . c. For and , the long run total cost function is given below: For and , the long run marginal and average cost function is given below: For and , the short run total cost function is given below: For and , the short run marginal and average cost function is given below:

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