class7_GE-production_

# class7_GE-production_ - Econ 51, Winter 2011 Class #7...

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1 Econ 51, Winter 2011 Class #7 • Reminders: – PS #2 due Friday – PS #3 posted Thursday. T od ay

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2 General Equilibrium with Production Tuesday, January 25, 2011
3 Overview GE w/ prod. If a general equilibrium is supposed to describe the economy, we need to model firms and production. We will introduce firms which have technologies to transform one commodity into another, and which maximize profits. Most of the conclusions from the pure exchange case are still valid. However, with production the model provides some important new insights. Tuesday, January 25, 2011

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4 Outline GE w/ prod. How can we model firms? How can we characterize Pareto efficiency if there are firms? Define Walrasian equilibrium with firms. Application: comparative advantage and free trade. Tuesday, January 25, 2011
5 Firms GE w/ prod. We characterize a firm by the set of all things it could produce. We call this the production possibility set , denoted it by Y . A production possibility set is the set of all possible production plans the firm can undertake. A production plan y is a vector of commodities, inputs are negative numbers, outputs positive numbers. E.g., if there are apples and bananas, a firm might be able to transform two apples to one banana. So a production plan might be y=(y a ,y b )=(-2,1) , i.e. the firm needs 2 apples to produce 1 banana. Tuesday, January 25, 2011

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6 Constant return to scale GE w/ prod. If, for any x 0 the firm can transform x apples into x bananas, its production possibility set will be Y={(y a , y b )| y a 0, y b =-y a }. This technology exhibits constant returns to scale , that is, doubling all inputs results in twice the output. A different example of a production possibility set is Y={(y a ,y b ):y a 0,y b =(-y a ) 2 }. – This technology exhibits increasing returns to scale , that is, doubling all inputs results in more than twice the output. Throughout, we’ll always assume constant returns to scale. Tuesday, January 25, 2011
Linear production sets GE w/ prod. When there is only one input, constant returns to scale means linear production functions (i.e., y b =- α y a for a constant α >0 ) For two inputs and one output, production functions are often assumed to be of the Cobb-Douglass form: y=x 1 α x 2 1- α The associated production possibility set is Y={(-x 1 ,-x 2 ,y):-x 1 0,-x 2 0, y = x 1 α x 2 1- α } This is also a technology of constant returns to scale, but is obviously not linear. To

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## This note was uploaded on 03/19/2011 for the course ECON 51 taught by Professor Tendall,m during the Winter '07 term at Stanford.

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class7_GE-production_ - Econ 51, Winter 2011 Class #7...

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