2 Production One Variable Input Ch 6

2 Production One Variable Input Ch 6 - 4/2/2008 Chapter 6...

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4/2/2008 1 Chapter 6 Production and Cost: One Variable Input The Production Function We begin by assuming that a firm produces one good, Y, using 2 inputs called input 1 and input 2. Let y = quantity of good Y z 1 = quantity of input 1 z 2 = quantity of input 2 The input bundle is (z 1 , z 2 ). Example: if the firm uses 10 units of input 1 and 6 units of input 2, the input bundle is (10, 6).
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4/2/2008 2 How inputs are combined to produce quantities of output varies depending on the technology the firm uses for production. We assume that there is one function that is technically efficient = it maximizes the qty of output that can be produced from a given bundle of inputs. We use the technically efficient technology to define: The production function y=F(z 1 , z 2 ) It tells us the maximum qty of good Y that can be produced from any input bundle (z 1 , z 2 ). Fixed-Proportions Production Function = the inputs are always used in a fixed ratio Example: one nut + one bolt for one fastener Example: 1 oz vodka + 6 oz orange juice + 2 oz ice for one screwdriver Note that the fixed proportion doesn’t have to be 1:1.
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4/2/2008 3 Example To make the best cranapple drink, need to use 150 ml of apple juice and 100 ml of cranberry juice. Let y = number of cranapple drinks z 1 = ml of apple juice z 2 = ml of cranberry juice Say you had z 1 = 600 ml of apple juice and z 2 = 500 ml of cranberry juice. You have enough apple juice to make 600 ml = 4 drinks i.e., z 1 / 150 150 ml You have enough cranberry juice to make 500 ml = 5 drinks i.e., z 2 / 100 100 ml The most cranapple drinks you could make is 4. You don’t have enough apple juice to make any more. The fixed-proportions production fn is y = min ( z 1 / 150, z 2 / 100 )
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4/2/2008 4 Functions of this type are known as Leontief production functions. Let’s do another example using more inputs. Example Seafood spaghetti at a fancy restaurant requires the following amounts for 4 servings: .8 kg spaghetti 8 tomatoes 12 scallops 4 garlic cloves So, for 1 serving, we need .8/4 = .2 kg spaghetti (input 1) 8/4 = 2 tomatoes (input 2) 12/4 = 3 scallops (input 3) 4/4 = 1 garlic clove (input 4) The production fn is y = min (z 1 /.2, z 2 / 2, z 3 /3, z 4 )
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4/2/2008 5 Suppose the cook has 4.9 kg spaghetti, 60 tomatoes, 54 scallops and 30 garlic cloves. What is the maximum number of servings possible? y = min (4.9/.2, 60/2, 54/3, 30) = min (24.5, 30,18,30) Therefore, y = 18 servings maximum. Variable-Proportions Production Function = the input mix can vary You can substitute by increasing the amount used of one input and decreasing the amount used of another input. Example: A Courier Service
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2 Production One Variable Input Ch 6 - 4/2/2008 Chapter 6...

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