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3 Production Many Variable Inputs Ch 7

3 Production Many Variable Inputs Ch 7 - Chapter 7...

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4/2/2008 1 Chapter 7 Production and Cost: Many Variable Inputs In most real situations, firms have the ability to vary more than one input during the relevant time period. They can substitute more of one input for less of another. We can illustrate all the different combinations of inputs that yield the same level of total output as a production isoquant. Example: Our Courier Recall: y = courier services measured in km z 1 = driver’s time measured in hours z 2 = gasoline measured in litres The production function is y = (1200z 1 z 2 ) 1/2 Suppose we want to cover 120 km. 120 = (1200z 1 z 2 ) 1/2 14400 = 1200z 1 z 2 12 = z 1 z 2
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4/2/2008 2 There are all kinds of input bundles that will satisfy this equation. For example, 2 hrs driving and 6 litres gas 4 hrs driving and 3 litres gas, etc. Suppose we want to cover 240 km. 240 = (1200z 1 z 2 ) 1/2 57600 = 1200z 1 z 2 48 = z 1 z 2 For 240 km, can drive 6 hrs with 8 litres gas drive 4 hrs with 12 litres gas… We can graph all the combinations of driving time and gas usage that will result in 120 and 240 km covered, respectively, on an isoquant map. The higher the isoquant, the greater the quantity of output.
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4/2/2008 3 Marginal Rate of Technical Substitution, MRTS The MRTS measures the rate at which one input can be substituted for the other, with output remaining constant. The MRTS is the absolute value of the slope of the isoquant. It tells us the rate at which we must increase the qty of input 2 per unit decrease in qty of input 1. Example: if MRTS = 2.5, if we decrease input 1 by 1, we must increase input 2 by 2.5. Note that MRTS uses a marginal reduction in qty of input 1. That is, since the slope is z 2 / z 1 , let z 1 b2right 0. Generally, it becomes progressively harder to substitute one input for another. You need more and more of input 2 to compensate for each unit decrease in input 1. MRTS gets smaller – diminishes – as we move from left to right along an isoquant. MRTS and Marginal Product When the qty of input 1 is decreased by z 1 , the change in total output y is approximately the MP of input 1 multiplied by z 1 y = MP(z 1 ) z 1 or z 1 = y / MP(z 1 ) The change in input 2, z 2 , must yield a corresponding change in output y y = MP(z 2 ) z 2 or z 2 = y / MP(z 2 )
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4/2/2008 4 Since MRTS, the |slope of the isoquant|, is z 2 / z 1 , Then, MRTS = z 2 / z 1 = [ y / MP(z 2 ) ] / [ y / MP(z 1 ) ] MRTS = MP(z 1 ) / MP(z 2 ) Numerical Example A production function is given by F(z 1, z 2 ) = 16z 1 z 2 MP(z 1 ) = δ F(z 1, z 2 ) / δ z 1 = 16z 2 MP(z 2 ) = δ F(z 1, z 2 ) / δ z 2 = 16z 1 MRTS ( z 1 , z 2 ) = MP(z 1 ) / MP(z 2 ) = 16z 2 16z 1 = z 2 / z 1 In general, for any Cobb-Douglas production function of the form F(z 1 ,z 2 ) = Az 1 α z 2 1 – α MRTS (z 1 ,z 2 ) = α z 2 (1 - α ) z 1
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4/2/2008 5 Inputs are perfect substitutes when one input can always be substituted for the other on fixed terms and the MRTS is constant .
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