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Chapter 18 Technology First understand the technology constraint of a firm. Later we will talk about constraints imposed by consumers and firm’s competitors (the demand curve faced by the firm, the market structure) Inputs: labor and capital Inputs and outputs measured in flow units, i.e., how many units of labor per week, how many units of output per week, etc.

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Consider the case of one input (x) and one output (y). To describe the tech constraint of a firm, list all the technologically feasible ways to produce a given amount of outputs. The set of all combinations of inputs and outputs that comprise a technologically feasible way to produce is called a production set. A production function measures the maximum possible output that you can get from a given amount of input.

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Fig. 18.1

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Isoquant is another way to express the production function. It is a set of all possible combinations of inputs that are just sufficient to produce a given amount of output. Isoquant looks very much like indifference curve, but you cannot label it arbitrarily, neither can you do any monotonic transformation of the label. Some useful examples of production function. Two inputs, x 1 and x 2.

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Fixed proportion (perfect complement): f(x 1,x 2 )=min{x 1,x 2 } Perfect substitutes: f(x 1,x 2 )=x 1 +x 2 Cobb-Douglas f(x 1,x 2 )=A(x 1 ) a (x 2 ) b, cannot normalize to a+b=1 arbitrarily Some often-used assumptions on the production function Monotonicity: if you increase the amount of at least one input, you produce at least as much output as before

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Fig. 18.2

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Fig. 18.3

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Monotonicity holds because of free disposal, that is, can free dispose of any extra inputs Convexity: if y=f(x 1,x 2 )=f(z 1,z 2 ), then f(tx 1 +(1-t)z 1,tx 2 +(1-t)z 2 ) y for any t [0,1] Some terms often used to describe the production function Marginal product: operate at (x 1,x 2 ), increase a bit of x 1 and hold x 2, how much more y can we get per additional unit of x 1 ?

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Fig. 18.4

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Marginal product of factor 1: MP 1 (x 1,x 2 )=∆y/∆x 1 =(f(x 1 + ∆x 1,x 2 )- f(x 1,x 2 ))/∆x 1 (it is a rate, just like MU) Marginal rate of technical substitution factor 1 for factor 2: operate at (x 1,x 2 ), increase a bit of x 1 and hold y, how much less x 2 can you use? Measures the trade- off between two inputs in production MRTS 1,2 (x 1,x 2 )=∆x 2 /∆x 1 =? ∆y=MP 1 (x 1,x 2 )∆x 1 +MP 2 (x 1,x 2 )∆x 2 =0

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MRTS 1,2 (x 1,x 2 )=∆x 2 /∆x 1 =- MP 1 (x 1,x 2 )/MP 2 (x 1,x 2 ) (it is a slope, just like MRS) Law of diminishing marginal product: holding all other inputs fixed, if we increase one input, the marginal product of that input becomes smaller and smaller (diminishing MU) Diminishing MRTS: the slope of an isoquant decreases in absolute value as we increase x 1 (diminishing MRS)

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Short run: at least one factor of production is fixed Long run: all factors of production can be varied Can also plot the short run production function y=f(x 1,k) Returns to scale: if we use twice as much of each input, how much output will we get?

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Fig. 18.5

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constant returns to scale (CRS): for all t>0, f(tx 1,tx 2 )=tf(x 1,x 2 ) Idea is if we double the inputs, we can just set two plants and so we can double the outputs Increasing returns to scale (IRS): for all t>1, f(tx 1,tx 2 )>tf(x 1,x 2 ) Decreasing returns to scale (DRS): for all t>1, f(tx 1,tx 2 )<tf(x 1,x 2 ) MP, MRTS, returns to scale

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