191T5 - ECON191(Spring 2009 9-10.3.2009(Tutorial 5 Chapter...

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1 ECON191 (Spring 2009) 9-10.3.2009 (Tutorial 5) Chapter 3 Production and Technology Chapter 4 Cost functions (Chapter 6 & 7 of textbook) Example : Assume that a firm uses two types of input in the production of a certain commodity. The production function is 2 2 2 1 2 1 5 . 0 5 . 0 100 X X X X Y - - = . What is the maximum output if the total amount that can be spent on inputs is $1000, the price of input 1 is $2, and the price of input 2 is $5? Answer At the optimal input combination with total cost $1000, two conditions must be satisfied. (1) 2 1 w w MRTS = (2) 1000 2 2 1 1 = + X w X w Example : Company A’s production function: 2 / 1 2 / 1 L K q = . Cost of capital is $2 and cost of labor is $5 Company A is currently producing output with 50 L and 375 K, is Company A producing efficiently? Answer Company A is not producing efficiently, as MRTS is not equal to the relative input price. 5 . 7 50 375 2 / 1 2 / 1 2 1 2 / 1 2 / 1 2 1 = = = = = - - L K L K L K MP MP MRTS K L while the price ratio is 2 5 = K L w w Company is currently producing 93 . 136 375 50 2 / 1 2 / 1 = = q at a cost of $1000. At the same cost of $1000, an efficient combination of inputs can be obtained, at the same time with a higher output level. L K w w L K MRTS K L 5 . 2 5 . 2 = = = = and 1000 5 2 = + L K K * = 250, L * = 100, output = 158.11
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2 Various types of production functions (1) The linear production technology (2) The Leontief production technology s Capital and labor must be used in a fixed proportion in order to produce output. b K and L must be increased in a same proportion if you want to increase your output. (K and L are not substitutable) s The Leontief production function : = b L a K Y , min where a and b are constants (3) The Cobb-Douglas production technology s Example: 1 K and 6 L need for producing 1 Y o = 6 , 1 min L K Y , if we have 2 K and 6 L , o 1 ) 1 , 2 min( 6 6 , 1 2 min = = = Y s Cost minimizing condition: Cost is minimized when b L a K = s Cobb-Douglas production function: β α K AL Y = where A
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This note was uploaded on 10/11/2009 for the course ECON 191 taught by Professor Chen during the Spring '08 term at HKUST.

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191T5 - ECON191(Spring 2009 9-10.3.2009(Tutorial 5 Chapter...

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