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ajaz_eco204_2009_chapter_4.3

# ajaz_eco204_2009_chapter_4.3 - University of Toronto...

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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain ECO 204 2009 2010 S. Ajaz Hussain (Draft) Chapter 4.3: Complements & Perfect Substitutes Technologies CMP 1 Please help improve the course by sending me an e mail about typos or suggestions for improvements In chapter 4.2 we modeled the optimal choice of inputs for a firm that produces a single product (good or service), hires inputs as a price taker, is in the long run and deploys a Cobb Douglas technology. In this chapter we continue this discussion for complements or perfect substitutes technology. It is important to stress that we are modeling the optimal choice of inputs in the long run (all inputs are variable) and the firm is an inputs price taker (i.e. ܲ , ܲ are exogenously given). 1. Complements Production Function: CMP The complements production function is used to model technologies where inputs ‐‐ say, labor and capital ‐‐ are used as complements: ܳ ൌ minሺߙܮ, ߚܭሻ From chapter 4.1, recall that the complements technology combines inputs in fixed proportions. For example, for bus companies ( Greyhound , Fung Wah ): Suppose each bus has a driver. Then the number of passenger transported ܳ is a function of the number of buses ܤ and drivers ܦ : ܳ ൌ minሺܦ, ܤሻ. Suppose each bus has a driver and a conductor. Then the number of passenger transported ܳ is a function of the number of buses ܤ, drivers ܦ and conductors ܥ : ܳ ൌ minሺܦ, ܥ, ܤሻ. The CMP is: 1 Thanks: Matt Fantauzzi Note to self: for summer 2010 class use PTC case to illustrate complements production function and do more advanced sensitivity analysis of inputs and parameters using Cramer’s rule (check it’s done in MATH 133). 1 ECO 204 (Draft) Chapter 4.3

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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain min ௅,௄ ܥ ൌ ܲ ܮ ൅ ܲ ܭ s.t. ܣ minሺߙܮ, ߚܭሻ ൌ ݍ The CMP is depicted below. Since the complements production function isn’t differentiable everywhere, we cannot use the Langrangean method to solve the CMP. Instead, approach the problem by noting that the optimal solution must be at the corner of the target output ݍ isoq quant: L q C/P L C/P K Slope = P L /P K K K = ( α / β )L The corners of the L shaped iso quants are on the line: ܭ ൌ ߙ ߚ ܮ ߙܮ ൌ ߚܭ At the optimal solution ߙܮ ൌ ߚܭ , which substituted in the production function yields: ݍ ൌ ܣ minሺߙܮ, ߚܭሻ ൌ ܣ minሺߙܮ, ߙܮሻ ൌ ܣ ߙܮ ՜ ܮ ൌ ݍ ߙܣ ݍ ൌ ܣ minሺߙܮ, ߚܭሻ ൌ ܣ minሺߚܭ, ߚܭሻ ൌ ܣ ߚܭ ՜ ܭ ൌ ݍ ߚܣ Observe that inputs are used in fixed proportions : each unit of output requires ఈ஺ workers and ఉ஺ machines. 2 ECO 204 (Draft) Chapter 4.3
University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain Substituting the optimal labor and capital in ܲ yields the cost function (optimal cost of producing target output ݍ is) ܥ ൌ ܲ ܮ ൅ ܭ : ܥሺݍሻ ൌ ܲ ൅ ܲ ܭ ܮ ܥሺݍሻ ൌ ܲ ݍ ൅ ܲ ݍ

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ajaz_eco204_2009_chapter_4.3 - University of Toronto...

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