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notes5 - Chapter 5 Producer Theory Markets have two sides...

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Chapter 5 Producer Theory Markets have two sides: consumers and producers. Up until now we have been studying the consumer side of the market. We now begin our study of the producer side of the market. The basic unit of activity on the production side of the market is the fi rm. The task of the fi rm is take commodities and turn them into other commodities. The objective of the fi rm (in the neoclassical model) is to maximize pro fi ts. That is, the fi rm chooses the production plan from among all feasible plans that maximizes the pro fi t earned on that plan. In the neoclassical (competitive) production model, the fi rm is assumed to be one fi rm among many others. Because of this (as in the consumer model), prices are exogenous in the neoclassical production model. Firms are unable to a ff ect the prices of either their inputs or their outputs. Situations where the fi rm is able to a ff ect the price of its output will be studied later under the headings of monopoly and oligopoly. Our study of production will be divided into three parts: First, we will consider production from a purely technological point of view, characterizing the fi rm’s set of feasible production plans in terms of its production set Y . Second, we will assume that the fi rm produces a single output using multiple inputs, and we will study its pro fi t maximization and cost minimization problems using a production function to characterize its production possibilities. Finally, we will consider a special class of production models, where the fi rm’s production function exhibits constant returns to scale. 121
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Nolan Miller Notes on Microeconomic Theory: Chapter 5 ver: Aug. 2006 5.1 Production Sets Consider an economy with L commodities. The task of the fi rm is to change inputs into outputs. For example, if there are three commodities, and the fi rm uses 2 units of commodity one and 3 units of commodity two to produce 7 units of commodity three, we can write this production plan as y = ( 2 , 3 , 7) , where, by convention, negative components mean that that commodity is an input and positive components mean that that commodity is an output. If the prices of the three commodities are p = (1 , 2 , 2) , then a fi rm that chooses this production plan earns pro fi t of π = p · y = (1 , 2 , 2) · ( 2 , 3 , 7) = 6 . Usually, we will let y = ( y 1 , ..., y L ) stand for a single production plan, and Y R L stand for the set of all feasible production plans. The shape of Y is going to be driven by the way in which di ff erent inputs can be substituted for each other in the production process. A typical production set (for the case of two commodities) is shown in MWG Figure 5.B.1. The set of points below the curved line represents all feasible production plans. Notice that in this situation, either commodity 1 can be used to produce commodity 2 ( y 1 < 0 , y 2 > 0 ), commodity 2 can be used to produce commodity 1 ( y 1 > 0 , y 2 < 0 ), nothing can be done ( y 1 = y 2 = 0 ) or both commodities can be used without producing an output, ( y 1 < 0 , y 2 < 0 ). Of course, the last
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