This production set tells us that the firm can use

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Unformatted text preview: o the firm – is: {( ) } The “production set” in producer theory is the counterpart of the “consumption set” in consumer theory5. This production set tells us that the firm can use any non-negative amounts of the inputs. For inputs the “production {( }: ) set” is As in consumer theory where the consumer could choose to consume a bundle of goods in the interior or the on boundary of the consumption set, notice that the firm can choose to use a bundle of inputs in the interior or the on boundary of the production set (of course, to produce positive output the firm can use both or one inputs). We can express the firm’s production process by a mathematical equation defined over all “bundles” of inputs in the production set which can be represented by a mathematical equation called the “production function”: 5 Recall that we assumed the consumption set, the set of all bundles of goods physically ava ilable to the consumer for consumption, {( } ) was 2 ECO 204 Chapter 11: Producer Theory— the Basics (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ( ) Given the firm’s technology and management, the firm’s output from a bundle of inputs in the production set is: ⏟ ⏟ ⏟ ( ( ) ) Notice that when the firm is able to produce a higher output with the same amount of inputs as before, or alternatively, if variable inputs can be adjusted costlessly, means that the firm can use fewer inputs to produce the 6 same level of output as before . In other words, means that all inputs have become more “productive” (more on this below). From ECO 100, recall that the firm can be in the short run (where at least one input is fixed) or in the long run (where all inputs are variable)7. In ECO 204, we will denote fixed inputs by lower case letters and variable inputs by UPPER CASE letters. For example: Input Notation for fixed inputs Notation for variable inputs Capital Labor Materials If the firm is in the long run, then its output is: ( ⏟ ) For example, suppose a firm is in the long run and uses variable capital and labor as essential inputs (in the sense that production requires both labor and capital) and where the firm can produce the same level of output by substituting labor for capital at a variable rate of substitution. Using intuition from consumer theory, this production process can be modeled by the long run “Cobb-Douglas” production function: Here, labor and capital are essential inputs because to produce positive output, the firm needs some workers and capital, i.e. . We will demonstrate the “variable rate of substitution” below. 6 ( ) it implies that when In ECO 204, when we say that that all inputs have become more “productive”. In some models, especially macro, “productivity” impacts some inputs: for example, in macro, the output of an economy is often modeled by ( ) where makes labor more productive. 7 For example, the Prestige Data S...
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