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Unformatted text preview: Chapter 7: Appendix CHAPTER 7 APPENDIX PRODUCTION AND COST THEORY— A MATHEMATICAL TREATMENT EXERCISES 1. Of the following production functions, which exhibit increasing, constant, or decreasing returns to scale? a. F(K, L) = K 2 L b. F(K, L) = 10K + 5L c. F(K, L) = (KL) 0.5 Returns to scale refer to the relationship between output and proportional increases in all inputs. This is represented in the following manner (let λ >0) : F( λ K , λ L ) > λ F( K, L ) implies increasing returns to scale; F( λ K , λ L ) = λ F( K, L ) implies constant returns to scale; and F( λ K , λ L ) < λ F( K, L ) implies decreasing returns to scale. a. Applying this to F( K, L ) = K 2 L , F( λ K , λ L ) = ( λ K ) 2 ( λ L ) = λ 3 K 2 L = λ 3 F( K, L ). This is greater than λ F( K, L ); therefore, this production function exhibits increasing returns to scale. b. Applying the same technique to F( K, L ) = 10 K + 5 L , F( λ K , λ L ) = 10 λ K + 5 λ L = λ F( K, L ). This production function exhibits constant returns to scale. c. Applying the same technique to F( K, L) = ( KL ) 0.5 , F( λ K , λ L ) = ( λ K λ L ) 0.5 = ( λ 2 ) 0.5 ( KL ) 0.5 = λ ( KL ) 0.5 = λ F( K, L ). This production function exhibits constant returns to scale. 2. The production function for a product is given by q = 100KL. If the price of capital is $120 per day and the price of labor $30 per day, what is the minimum cost of producing 1000 units of output? The cost-minimizing combination of capital and labor is the one where MRTS MP MP w r L K = = . The marginal product of labor is dq dL = 100 K . The marginal product of capital is dq dK = 100 L . Therefore, the marginal rate of technical substitution is 100 100 K L K L = ....
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- Spring '11
- Monotonic function, rental rate, $32, total cost function