Microeconomic Theory

Business

Quiz 9 :

Cost Functions

Quiz 9 :

Cost Functions

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Suppose that a firm produces two different outputs, the quantities of which are represented by q1 and q2. In general, the firm's total costs can be represented by C ( q 1 , q 2 ). This function exhibits economies of scope if img for all output levels of either good. a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two single-product firms producing each good separately. b. If the two outputs are actually the same good, we can define total output as q = q 1 + q 2. Suppose that in this case average cost (= C/q) decreases as q increases. Show that this firm also enjoys economies of scope under the definition provided here.
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a)
There are two goods - Good 1 and Good 2. The following function is given:
img 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
img The cost of a single product firm producing q 2 units of good 2 is
img 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
img +
img .
Moreover, the total cost of producing q 1 units of good 1 and q 2 units of good 2 in a multiproduct firm is
img .
Since the function states that
img +
img is greater than
img , 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.
img 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,
img Similarly,
img Now sum equations (1) and (2).
img Hence, it is shown that the multiproduct firm enjoys economies of scope.

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Professor Smith and Professor Jones are going to produce a new introductory textbook. As correct scientists, they have laid out the production function for the book as img 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. After having spent 900 hours preparing the first draft, time which he valued at $3 per working hour, Smith has to move on to other things and cannot contribute any more to the book. Jones, whose labor is valued at $12 per working hour, will revise Smith's draft to complete the book. a. How many hours will Jones have to spend to produce a finished book of 150 pages? Of 300 pages? Of 450 pages? b. What is the marginal cost of the 150th page of the finished book? Of the 300th page? Of the 450th page?
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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
img 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
img ……. (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.
img Thus, Jones has to spend 25 hours to produce a finished book of 150 pages.
Similarly, if q= 300, then from (1) J= 100
img Thus, Jones has to spend 100 hours to produce a finished book of 300 pages.
Similarly, if q= 450, then from (1) J= 225
img 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
img ……. (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
img Substituting J from (1) we have
img Therefore the marginal cost is given as
img ……….. (2)
If q= 150, then according to (2) MC =4
img Therefore, the marginal cost of the 150th page of the finished book is $4.
If q= 300, then according to (2) MC =8
img Therefore, the marginal cost of the 300th page of the finished book is $8.
If q= 450, then according to (2) MC =12
img Therefore, the marginal cost of the 450th page of the finished book is $12.

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Suppose that a firm's fixed proportion production function is given by q = min ( 5 k , 10 l ) : a. Calculate the firm's long-run total, average, and marginal cost functions. b. Suppose that k is fixed at 10 in the short run. Calculate the firm's short-run total, average, and marginal cost functions. c. Suppose v = 1 and w = 3. Calculate this firm's long-run and short-run average and marginal cost curves.
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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:
img The firms total cost function is given as follows:
img Where,
img Wages
img Rent
The quantity produced by using the fixed proportion is given below:
img Put the value of k and l in the total cost function as shown below:
img The firms average cost function is given below:
img The firms marginal cost function is given below:
img Therefore the firm's total cost function is given by
img . The average and marginal cost is given as
img .
b.
Suppose that k is fixed at 10 in the short run. Then the short run production function is given below:
img The short run cost function is given below:
img The quantity produced by using the fixed proportion is given below:
img Put the value of l in the short run total cost function as shown below:
img The firm's average cost function is given below:
img The firm's marginal cost function is given below:
img Therefore, in the short run the firm's total cost function is given by
img . The average and marginal cost is given as
img and
img .
c.
For
img and
img , the long run total cost function is given below:
img For
img and
img , the long run marginal and average cost function is given below:
img For
img and
img , the short run total cost function is given below:
img For
img and
img , the short run marginal and average cost function is given below:
img img

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A firm producing hockey sticks has a production function given by img In the short run, the firm's amount of capital equipment is fixed at k = 100. The rental rate for k is v = $1, and the wage rate for l is w = $4. a. Calculate the firm's short-run total cost curve. Calculate the short-run average cost curve. b. What is the firm's short-run marginal cost function? What are the SC, SAC, and SMC for the firm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks? c. Graph the SAC and the SMC curves for the firm. Indicate the points found in part (b). d. Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point. Suppose now that capital used for producing hockey sticks is fixed at img in the short run. e. Calculate the firm's total costs as a function of q , w , v , and k. f. Given q, w, and v , how should the capital stock be chosen to minimize total cost? g. Use your results from part (f) to calculate the long-run total cost of hockey stick production. h. For w = $4, v = $1, graph the long-run total cost curve for hockey stick production. Show that this is an envelope for the short-run curves computed in part (e) by examining values of k of 100, 200, and 400.
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An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by img The factories differ, however, in the amount of capital equipment each has. In particular, factory 1 has k 1 = 25, whereas factory 2 has k 2 = 100. Rental rates for k and l are given by w = v = $1. a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between the two factories? b. Given that output is optimally allocated between the two factories, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100th widget? The 125th widget? The 200th widget? c. How should the entrepreneur allocate widget production between the two factories in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both factories exhibited diminishing returns to scale?
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Suppose the total-cost function for a firm is given by img a. Use Shephard's lemma to compute the (constant output) demand functions for inputs l and k. b. Use your results from part (a) to calculate the underlying production function for q.
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Suppose the total-cost function for a firm is given by img a. Use Shephard's lemma to compute the (constant output) demand function for each input, k and l. b. Use the results from part (a) to compute the underlying production function for q. c. You can check the result by using results from Example 10.2 to show that the CES cost function with img generates this total-cost function
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In a famous article [J. Viner, ''Cost Curves and Supply Curves,'' Zeitschrift fur Nationalokonomie 3 (September 1931): 23-46], Viner criticized his draftsman who could not draw a family of SAC curves whose points of tangency with the U-shaped AC curve were also the minimum points on each SAC curve. The draftsman protested that such a drawing was impossible to construct. Whom would you support in this debate?
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Generalizing the CES cost function The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is img a. What is the total-cost function for a firm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is v/? and for a unit of labor input is w/?. b. If img it can be shown that this production function converges to the Cobb-Douglas form img as img r fi 0. What is the total cost function for this particular version of the CES function? c. The relative labor cost share for a two-input production function is given by wl / vk. Show that this share is constant for the Cobb-Douglas function in part (b). How is the relative labor share affected by the parameters ? and ?? d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in w/v? How is the direction of this effect determined by the elasticity of substitution,?? How is it affected by the sizes of the parameters img
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Input demand elasticities The own-price elasticities of contingent input demand for labor and capital are defined as img a. Calculate img for each of the cost functions shown in Example 10.2. b. Show that, in general, img c. Show that the cross-price derivatives of contingent demand functions are equal-that is, show that img . Use this fact to show that img are, respectively, the share of labor in total cost ( wl/C ) and of capital in total cost ( vk/C ). d. Use the results from parts (b) and (c) to show that img . e. Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input demand.
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The elasticity of substitution and input demand elasticities The definition of the (Morishima) elasticity of substitution sij in Equation 10.54 can be recast in terms of input demand elasticities. This illustrates the basic asymmetry in the definition. a. Show that if only img b. Show that if only img c. Show that if the production function takes the general CES img . This is the only case in which the Morishima definition is symmetric. Reference: img
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The Allen elasticity of substitution Many empirical studies of costs report an alternative definition of the elasticity of substitution between inputs. This alternative definition was first proposed by R. G. D. Allen in the 1930s and further clarified by H. Uzawa in the 1960s. This definition builds directly on the production function-based elasticity of substitution defined in footnote 6 of Chapter 9: img where the subscripts indicate partial differentiation with respect to various input prices. Clearly, the Allen definition is symmetric. a. Show img is the share of input j in total cost. b. Show that the elasticity of si with respect to the price of input j is related to the Allen elasticity by img c. Show that, with only two inputs, img for the Cobb-Douglas case and img for the CES case. d. Read Blackorby and Russell (1989: ''Will the Real Elasticity of Substitution Please Stand Up?'') to see why the Morishima definition is preferred for most purposes.
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