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Production Theory 1
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Short-Run v. Long-Run u Fixed input/factor of production: quantity of input is fixed regardless of required output level, e.g. capital or specialized labour u Variable input/factor of production: quantity of input used depends on the level of output u Short run: at least one input/factor is fixed u Long run: all inputs/factors are variable
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Production Function u A technology is a process by which inputs (e.g. labour and capital) are converted into output. u The output level is denoted by y. u The technology’s production function states the maximum amount of output possible from an input bundle.
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Production Function y = f(x) is the production function x’x Input Level Output Level y’ y’ = f(x’) is the maximum output level obtainable from x’ input units. One input
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Technology Set u The collection of all feasible production plans is the technology set.
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Technology Set y = f(x) is the production function. x’x Input Level Output Level y’ y” One input y” = f(x’) is an output level that is feasible from x’ input units.
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Technology Set x’x Input Level Output Level y’ One input y” The technology set
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Technology Set x’x Input Level Output Level y’ One input y” The technology set Technically inefficient plans Technically efficient plans
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Technology: Multiple Inputs u What does a technology look like when there is more than one input? u The two input case: Input levels are x 1 and x 2. Output level is y. u Example of production function is
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PREVIEW: ISOQUANT u An isoquant is the set of all combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output. u The slope of the isoquant = the marginal rate of technical substitution (MRTS) = the technical rate of substitution (TRS) u MRTS (TRS): The number of units of K that we can dispose of if one more unit of L becomes available while remaining on the original isoquant.
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Technologies with Multiple Inputs u The complete collection of isoquants is the isoquant map. u The isoquant map is equivalent to the production function. u Example
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Isoquants with Two Inputs Y=20 Y=40 L K
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Isoquants with Two Inputs u Properties Y/ K>0, Y/ L>0 2 Y/ K 2 <0, 2 Y/ L 2 <0 Diminishing marginal product (Diminishing marginal utility)
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x2x2 x1x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technology
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Marginal (Physical) Product u The marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed.
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Marginal (Physical) Product then the marginal product of input 1 is
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Marginal (Physical) Product then the marginal product of input 1 is
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Marginal (Physical) Product then the marginal product of input 1 is and the marginal product of input 2 is
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Marginal (Physical) Product then the marginal product of input 1 is and the marginal product of input 2 is
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Marginal (Physical) Product u The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if
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Technical Rate-of-Substitution x2x2 x1x1 y The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution.
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Technical Rate-of-Substitution u How is a technical rate-of-substitution computed?
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Technical Rate-of-Substitution u How is a technical rate-of-substitution computed? u The production function is u A small change (dx 1, dx 2 ) in the input bundle causes a change to the output level of
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Technical Rate-of-Substitution Along an individual isoquant, dy = 0, therefore the changes dx 1 and dx 2 must satisfy the following,
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Technical Rate-of-Substitution which rearranges to or
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Technical Rate-of-Substitution is the rate at which input 2 must be given up as input 1 increases so as to keep the output level constant. It is the slope of the isoquant = MRTS = TRS.
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