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Chapter 8/Cost Functions v 37CHAPTER 8COST FUNCTIONSThe problems in this chapter focus mainly on the relationship between production and cost functions. Most of the examples developed are based on the Cobb-Douglas function (or its CES generalization) although a few of the easier ones employ a fixed proportions assumption. Two of the problems (8.9 and 8.10) make some use of Shephards Lemma since it is in describing the relationship between cost functions and (contingent) input demand that this envelope-type result is most often encountered.Comments on Problems8.1Famous example of Viners draftsman. This may be used for historical interest or as a way of stressing the tangencies inherent in envelope relationships .8.2An introduction to the concept of “economies of scope.” This problem illustrates the connection between that concept and the notion of increasing returns to scale.8.3A simplified numerical Cobb-Douglas example in which one of the inputs is held fixed.8.4A fixed proportion example. The very easy algebra in this problem may help to solidify basic concepts.8.5This problem derives cost concepts for the Cobb-Douglas production function with one fixed input. Most of the calculations are very simple. Later parts of the problem illustrate the envelope notion with cost curves. 8.6Another example based on the Cobb-Douglas with fixed capital. Shows that in order to minimize costs, marginal costs must be equal at each production facility. Might discuss how this principle is applied in practice by, say, electric companies with multiple generating facilities.8.7This problem focuses on the CES cost function. It illustrates how input shares behave in response to changes in input prices and thereby generalizes the fixed share result for the Cobb-Douglas.8.8This problem introduces elasticity concepts associated with contingent input demand. Many of these are quite similar to those introduced in demand theory.8.9Shows students that the process of deriving cost functions from production functions can be reversed. Might point out, therefore, that parameters of the production function (returns to scale, elasticity of substitution, factor shares) can be derived from cost functions as wellif that is more convenient.8.10Illustrates a cost function that arises from a very simple CES production function. Solutions8.1Support the draftsman. Its geometrically obvious that SAC cannot be at minimum because it is tangent to AC at a point with a negative slope. The only tangency occurs at minimum AC. 8.2a. By definition total costs are lower when both q1 and q2 are produced by the same firm than when the same output levels are produced by different firms C(q1,0) simply means that a firm produces only q1. b. Let q = q1+q2, where both q1 and q2 0. Because by assumption, . Similarly. Summing yields, which proves economies of scope.8.3a. J = 100 q = 300 J = 225 q = 450 b.Cost = 12 J = 12q2/900q = 150 MC = 4 q = 300 MC = 8 q = 450 MC = 12 8.4q = min(5k, 10l) v = 1 w = 3 C = vk + wl = k + 3l a.In the long run, keep 5k = 10, k = 2l b.k = 10 q = min(50, 10l)If l 5, q = 50 C = 10 + 3l MC is infinite for q 50. MC10 = MC50 = .3. MC100 is infinite. 8.5a. b. If q = 50, SC = 100 + If q = 100, SC = 100 + If q = 200, SC = 100 + c.d.As long as the marginal cost of producing one more unit is below the average-cost curve, average costs will be falling. Similarly, if the marginal cost of producing one more unit is higher than the average cost, then average costs will be rising. Therefore, the SMC curve must intersect the SAC curve at its lowest point.e.f. g. (a special case of Example 8.2) h.If w = 4 v = 1, C = 2q, SC = 200 = C for q = 100, SC = 400 = C for q = 200SC = 800 = C for q = 400 8.6a. To minimize cost, set up Lagrangian: . Therefore . b. SMC(125) = $2.00SMC(200) = $3.20 c.In the long run, can change k so, given constant returns to scale, location doesnt really matter. Could split evenly or produce all output in one location, etc. C = k + l = 2q AC = 2 = MC d.If there are decreasing returns to scale with identical production functions, then should let each firm have equal share of production. AC and MC not constant anymore, becoming increasing functions of q. 8.7a. .b. .c. .d. . Labors relative share is an increasing function of b/a. If 1 labors share moves in the same direction as v/w. If 1, labors relative share moves in the opposite direction to v/w. This accords with intuition on how substitutability should affect shares. 8.8a. The elasticities can be read directly from the contingent demand functions in Example 8.4. For the fixed proportions case, (because q is held constant). For the Cobb-Douglas, . Apparently the CES in this form has non-constant elasticities.b. Because cost functions are homogeneous of degree one in input prices, contingent demand functions are homogeneous of degree zero in those prices as intuition suggests. Using Eulers theorem gives . Dividing by gives the result. c.Use Youngs Theorem: Now multiply left by . d. Multiplying by shares in part b yields . Substituting from part c yields . e.All of these results give important checks to be used in empirical work. 8.9 From Shephards Lemma a. b.Eliminating the w/v from these equations: which is a Cobb-Douglas production function. 8.10As for many proofs involving duality, this one can be algebraically messy unless one sees the trick. Here the trick is to let B = (v.5 + w.5). With this notation, C = B2q. a. Using Shephards lemma, b.F